# Patent

## Title
Cyclic error compensation in interferometry systems

## Abstract

An interference signal S(t) is provided from interference between two beams directed along different paths. The signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time. An error signal is provided to reduce errors in an estimate of {tilde over (L)}(t). The error signal is derived at least in part based on one or more collective properties of a distribution of multi-dimensional values. At least one of the multi-dimensional values in the distribution is generated from a plurality of samples of the signal S(t) (e.g., samples of the signal captured at different times).

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation-in-part of U.S. application Ser. No. 11/060,173 entitled “CYCLIC ERROR COMPENSATION IN INTERFEROMETRY SYSTEMS,” filed Feb. 17, 2005 now U.S. Pat. No. 7,428,685 which claims the benefit of U.S. Provisional Application No. 60/602,242 entitled “CYCLIC ERROR COMPENSATION IN INTERFEROMETRY SYSTEMS,” filed on Aug. 16, 2004 and is a continuation-in-part of U.S. application Ser. No. 10/616,504 entitled “CYCLIC ERROR COMPENSATION IN INTERFEROMETRY SYSTEMS,” filed Jul. 8, 2003 now U.S. Pat. No. 6,950,192 which claims the benefit of U.S. Provisional Application No. 60/394,418 entitled “ELECTRONIC CYCLIC ERROR COMPENSATION,” filed Jul. 8, 2002; each of which is incorporated herein by reference.


BACKGROUND
This invention relates to interferometers, e.g., displacement measuring and dispersion interferometers that measure displacements of a measurement object such as a mask stage or a wafer stage in a lithography scanner or stepper system, and also interferometers that monitor wavelength and determine intrinsic properties of gases.
Displacement measuring interferometers monitor changes in the position of a measurement object relative to a reference object based on an optical interference signal. The interferometer generates the optical interference signal by overlapping and interfering a measurement beam reflected from the measurement object with a reference beam reflected from the reference object.
In many applications, the measurement and reference beams have orthogonal polarizations and different frequencies. The different frequencies can be produced, for example, by laser Zeeman splitting, by acousto-optical modulation, or internal to the laser using birefringent elements or the like. The orthogonal polarizations allow a polarizing beam splitter to direct the measurement and reference beams to the measurement and reference objects, respectively, and combine the reflected measurement and reference beams to form overlapping exit measurement and reference beams. The overlapping exit beams form an output beam that subsequently passes through a polarizer. The polarizer mixes polarizations of the exit measurement and reference beams to form a mixed beam. Components of the exit measurement and reference beams in the mixed beam interfere with one another so that the intensity of the mixed beam varies with the relative phase of the exit measurement and reference beams. A detector measures the time-dependent intensity of the mixed beam and generates an electrical interference signal proportional to that intensity. Because the measurement and reference beams have different frequencies, the electrical interference signal includes a “heterodyne” signal portion having a beat frequency equal to the difference between the frequencies of the exit measurement and reference beams. If the lengths of the measurement and reference paths are changing relative to one another, e.g., by translating a stage that includes the measurement object, the measured beat frequency includes a Doppler shift equal to 2vnp/λ, where v is the relative speed of the measurement and reference objects, λ is the wavelength of the measurement and reference beams, n is the refractive index of the medium through which the light beams travel, e.g., air or vacuum, and p is the number of passes to the reference and measurement objects. Changes in the relative position of the measurement object correspond to changes in the phase of the measured interference signal, with a 2π phase change substantially equal to a distance change LRT of λ/(np), where LRT is a round-trip distance change, e.g., the change in distance to and from a stage that includes the measurement object.
Unfortunately, this equality is not always exact. Many interferometers include nonlinearities such as what are known as “cyclic errors.” The cyclic errors can be expressed as contributions to the phase and/or the intensity of the measured interference signal and have a sinusoidal dependence on the change in optical path length pnLRT. For example, a first order harmonic cyclic error in phase has a sinusoidal dependence on (2πpnLRT)/λ and a second order harmonic cyclic error in phase has a sinusoidal dependence on 2(2πpnLRT)/λ. Additional cyclic errors may include higher order harmonic cyclic errors, negative order harmonic cyclic errors, and sub-harmonic cyclic errors.
Cyclic errors can be produced by “beam mixing,” in which a portion of an input beam that nominally forms the reference beam propagates along the measurement path and/or a portion of an input beam that nominally forms the measurement beam propagates along the reference path. Such beam mixing can be caused by ellipticity in the polarizations of the input beams and imperfections in the interferometer components, e.g., imperfections in a polarizing beam splitter used to direct orthogonally polarized input beams along respective reference and measurement paths. Because of beam mixing and the resulting cyclic errors, there is not a strictly linear relation between changes in the phase of the measured interference signal and the relative optical path length pnL between the reference and measurement paths. If not compensated, cyclic errors caused by beam mixing can limit the accuracy of distance changes measured by an interferometer. Cyclic errors can also be produced by imperfections in transmissive surfaces that produce undesired multiple reflections within the interferometer and imperfections in components such as retroreflectors and/or phase retardation plates that produce undesired ellipticities in beams in the interferometer. For a general reference on the theoretical cause of cyclic error, see, for example, C. W. Wu and R. D. Deslattes, “Analytical modelling of the periodic nonlinearity in heterodyne interferometry,” Applied Optics, 37, 6696-6700, 1998.
In dispersion measuring applications, optical path length measurements are made at multiple wavelengths, e.g., 532 nm and 1064 nm, and are used to measure dispersion of a gas in the measurement path of the distance measuring interferometer. The dispersion measurement can be used to convert the optical path length measured by a distance measuring interferometer into a physical length. Such a conversion can be important since changes in the measured optical path length can be caused by gas turbulence and/or by a change in the average density of the gas in the measurement arm even though the physical distance to the measurement object is unchanged. In addition to the extrinsic dispersion measurement, the conversion of the optical path length to a physical length requires knowledge of an intrinsic value of the gas. The factor Γ is a suitable intrinsic value and is the reciprocal dispersive power of the gas for the wavelengths used in the dispersion interferometry. The factor Γ can be measured separately or based on literature values. Cyclic errors in the interferometer also contribute to dispersion measurements and measurements of the factor Γ. In addition, cyclic errors can degrade interferometric measurements used to measure and/or monitor the wavelength of a beam.
The interferometers described above are often crucial components of scanner systems and stepper systems used in lithography to produce integrated circuits on semiconductor wafers. Such lithography systems typically include a translatable stage to support and fix the wafer, focusing optics used to direct a radiation beam onto the wafer, a scanner or stepper system for translating the stage relative to the exposure beam, and one or more interferometers. Each interferometer directs a measurement beam to, and receives a reflected measurement beam from, a plane mirror attached to the stage. Each interferometer interferes its reflected measurement beams with a corresponding reference beam, and collectively the interferometers accurately measure changes in the position of the stage relative to the radiation beam. The interferometers enable the lithography system to precisely control which regions of the wafer are exposed to the radiation beam.
In practice, the interferometry systems are used to measure the position of the wafer stage along multiple measurement axes. For example, defining a Cartesian coordinate system in which the wafer stage lies in the x-y plane, measurements are typically made of the x and y positions of the stage as well as the angular orientation of the stage with respect to the z axis, as the wafer stage is translated along the x-y plane. Furthermore, it may be desirable to also monitor tilts of the wafer stage out of the x-y plane. For example, accurate characterization of such tilts may be necessary to calculate Abbe offset errors in the x and y positions. Thus, depending on the desired application, there may be up to five degrees of freedom to be measured. Moreover, in some applications, it is desirable to also monitor the position of the stage with respect to the z-axis, resulting in a sixth degree of freedom.
SUMMARY
Among other aspects, the invention features electronic processing methods that characterize and compensate cyclic errors in interferometric data. Because cyclic errors are compensated electronically, the interferometry system that produces the data has greater tolerance to optical, mechanical, and electronic imperfections that can cause cyclic errors, without sacrificing accuracy. The compensation techniques are especially useful for interferometric data used to position microlithographic stage systems.
In part, the invention is based on the realization that prior values of a main interferometric signal can be used to reduce the effect of cyclic errors on an estimate of a length being measured by the interferometry system (e.g., a length indicating the position of a stage). Any of a number of signal transformations such as a quadrature signal or a Fourier transform can be derived from the prior values of the main interferometric signal. These derived signal transformations can then be used to generate one or more error basis functions that represent one or more cyclic error terms in the main interferometric signal. Appropriate amounts of each error basis function (e.g., as determined by a coefficient for each error basis function) form an error signal that is subtracted from a signal from which the measured length is obtained. The corresponding reduction in cyclic errors increases the accuracy of the measured length.
For example, the prior values of the main interferometric signal can be used to calculate an estimate for a quadrature signal for the main interferometric signal. Algebraic combinations of such signals can yield error basis functions in the form of sinusoidal functions whose time-varying arguments correspond to particular cyclic error terms. In embodiments in which the interferometer beams have a heterodyne frequency splitting, one may also calculate the quadrature signal of the heterodyne reference signal, and the error basis functions may be derived from algebraic combinations the main signal, the reference signal, and the quadrature signals of the main and reference signals.
The error basis functions are used to isolate particular cyclic error terms in the main signal and characterize coefficients representative of each cyclic error term (e.g., its amplitude and phase). For example, algebraic combinations of the error basis functions and the main signal and its quadrature signal can move a selected cyclic error term to zero-frequency, where low-pass filtering techniques (e.g., averaging) can be used to determine its amplitude and phase. Such coefficients are stored. Thereafter, a superposition of the error basis functions weighted by the stored coefficients can be used to generate an error signal that can be subtracted from the main signal to reduce the cyclic errors therein and improve its accuracy.
The technique is particularly useful when the Doppler shift is small relative to the heterodyne frequency because the frequency of each cyclic error term is nearly equal to that of primary component of the main signal, in which case the estimate for the quadrature signal of the main signal is more accurate. This is an especially important property because it is precisely when the frequencies of the cyclic error terms are near that of the primary component of the main signal that the cyclic error terms are most problematic because they cannot be removed by frequency filtering techniques. Furthermore, at small Doppler shifts, one or more of the cyclic error frequencies may be within the bandwidth of a servo system used to position a stage based on the interferometric signal, in which the case the servo loop may actually amplify the cyclic error term when positioning the stage. Small Doppler shifts are actually quite common in microlithographic stage systems, such as when searching for an alignment mark, scanning in an orthogonal dimension to the one monitored by the interferometric signal, and changing stage direction. Moreover, at small Doppler shifts, selecting an integral relationship between the sampling rate of the detector and the heterodyne frequency (e.g., 6:1) yields an especially simple formula for the quadrature signal.
In addition, at small Doppler shifts, the main signal is nearly periodic with the heterodyne frequency, in which case prior data can be used to generate the error signal. As a result, correction of the main signal can be accomplished with only a single real-time subtraction of the error signal from the main signal, significantly reducing the computation time associated with the correction and thereby reducing data age errors in any servo system for position a microlithography stage.
Error basis functions can be derived from other combinations of prior values of the main interferometric signal besides a quadrature signal including a Fourier transform of the main interferometric signal. In the case of a Fourier transform, the resulting error signal is subtracted from a complex signal from which the measured length is obtained, as described in more detail below. In some cases (e.g., when the stage is moving slowly or is nearly stationary), the error basis functions can be derived from a distribution of values, where each of the values is generated from multiple values of the main interferometric signal, and the values in the distribution (e.g., multi-dimensional values distributed over a multi-dimensional space such as a complex space with real and imaginary dimensions) are not necessarily sequential in time.
We now summarize various aspects and features of the invention.
In one aspect, in general, a method includes providing an interference signal S(t) from interference between two beams directed along different paths. The signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time. The method includes providing an error signal to reduce errors in an estimate of {tilde over (L)}(t). The error signal is derived at least in part based on one or more collective properties of a distribution of multi-dimensional values. At least one of the multi-dimensional values in the distribution is generated from a plurality of samples of the signal S(t) (e.g., samples of the signal captured at different times).
Aspects include one or more of the following features.
One of the paths is associated with a position of a reference object and the other path is associated with a position of a moveable measurement object.
The position of the moveable object is controlled by a servo system having a bandwidth fS, and the samples of the signal S(t) from which the multi-dimensional values are calculated correspond to sample times at which the velocity v of the moveable measurement object relative to the reference object divided by a wavelength λ of the beams is less than ten times the bandwidth fS.
The servo system controls the position of the measurement object based on the signal S(t) and the error signal.
The two beams have different frequencies.
Providing the error signal comprises providing one or more coefficients representative of one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A1 cos(ωRt+φ(t)+ζ1), where A1 and ζ1 are constants, ωR is an angular frequency difference between the two beams before being directed along the different paths, and φ(t)=nk{tilde over (L)}(t), with k=2π/λ and λ equal to a wavelength for the beams.
The deviation can be expressed as




  
    
      ∑
      
        m
        ,
        p
      
    
    ⁢
    
      
    
    ⁢
    
      
        A
        
          m
          ,
          p
        
      
      ⁢
      
        cos
        ⁡
        
          (
          
            
              
                ω
                R
              
              ⁢
              t
            
            +
            
              
                m
                p
              
              ⁢
              
                φ
                ⁡
                
                  (
                  t
                  )
                
              
            
            +
            
              ζ
              
                m
                ,
                p
              
            
          
          )
        
      
    
  
  ,




where p=1, 2, 3 . . . , and m is any integer not equal to p, and where the provided coefficients comprise information corresponding to at least some of Am,p and Gm,p.

The coefficients are derived at least in part based on the one or more collective properties of the distribution of multi-dimensional values.
The error signal is generated from the coefficients and one or more error basis functions derived at least in part from a plurality of samples of the signal S(t).
Each of the error basis functions is derived at least in part from a linear combination of samples of the signal S(t).
Each of the error basis functions corresponds to a function that includes one or more leading sinusoidal terms having a time-varying argument that corresponds to a time-varying argument of an error term that represents a portion of the deviation of S(t) from the ideal expression.
The multi-dimensional values comprise two-dimensional values.
The one or more collective properties comprise one or more properties associated with a curve fit to at least some of the two-dimensional values in the distribution.
The one or more properties associated with the curve comprise values interpolated from the curve.
The curve comprises an ellipse.
The one or more properties associated with the curve comprise one or more of a center location, an ellipticity, and an angle of a major or minor axis.
Each of the two-dimensional values is represented as a complex number.
The complex number comprises a linear combination of samples of the signal S(t)
The linear combination comprises a transform signal Dq(t1) based on values of the signal S(t), for a frequency value determined by q, updated at a time t1.
The transform signal Dq(t1) comprises a sample of a discrete Fourier transform of the signal S(t).
The transform signal is represented as




  
    
      
        D
        q
      
      ⁡
      
        (
        
          t
          1
        
        )
      
    
    =
    
      
        ∑
        
          n
          =
          0
        
        
          N
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          W
          n
        
        ⁢
        
          S
          ⁡
          
            (
            
              
                n
                ⁢
                
                  
                
                ⁢
                τ
              
              +
              
                t
                1
              
            
            )
          
        
        ⁢
        exp
        ⁢
        
          {
          
            -
            
              j
              ⁡
              
                (
                
                  2
                  ⁢
                  
                    π
                    ⁡
                    
                      (
                      
                        n
                        -
                        
                          
                            N
                            -
                            1
                          
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    q
                    /
                    N
                  
                
                )
              
            
          
          }
        
      
    
  
  ,




where τ=1/fS, and Wn is a window function centered at





  n
  =
  
    
      
        N
        -
        1
      
      2
    
    .
  




The complex number is represented as a phase and a magnitude.
At least two phases associated with respective values in the distribution are separated from each other by at least 90 degrees.
The spacing between the phases of the values in the distribution varies by less than 20%.
The one or more collective properties of the distribution of multi-dimensional values comprise a statistical average derived from the multi-dimensional values in the distribution.
The distribution of multi-dimensional values includes at least 20 values.
Reducing errors in the estimate of {tilde over (L)}(t) comprises deriving the estimate of {tilde over (L)}(t) from a difference between the error signal and a discrete Fourier transform of samples of S(t).
The method further comprises directing the two beams along the different paths and measuring the interference signal S(t).
At least one of the beams is directed to reflect from a movable measurement object before producing the interference signal S(t).
The beam directed to reflect from the movable measurement object reflects from the measurement object multiple times before producing the interference signal S(t).
The beams are directed to reflect from different locations of the movable measurement object before producing the interference signal S(t).
The errors correspond to spurious beam paths.
A lithography method for use in fabricating integrated circuits on a wafer includes supporting the wafer on a moveable stage; imaging spatially patterned radiation onto the wafer; adjusting the position of the stage; and monitoring the position of the stage using an interferometry system, wherein monitoring the position of the stage comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the stage using the method described above.
A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method described above; and producing an integrated circuit from the wafer.
A lithography method for use in the fabrication of integrated circuits includes directing input radiation through a mask to produce spatially patterned radiation; positioning the mask relative to the input radiation; monitoring the position of the mask relative to the input radiation using an interferometry system, wherein monitoring the position of the mask comprises reducing errors in an estimate of a physical path difference associated with the position of the mask using the method described above; and imaging the spatially patterned radiation onto a wafer.
A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method described above; and producing an integrated circuit from the wafer.
A lithography method for fabricating integrated circuits on a wafer includes positioning a first component of a lithography system relative to a second component of a lithography system to expose the wafer to spatially patterned radiation; and monitoring the position of the first component relative to the second component using an interferometry system. Monitoring the position of the first component comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the first component using the method described above.
A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method described above; and producing an integrated circuit from the wafer.
A method for fabricating a lithography mask includes directing a write beam to a substrate to pattern the substrate; positioning the substrate relative to the write beam; and monitoring the position of the substrate relative to the write beam using an interferometry system. Monitoring the position of the substrate comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the substrate using the method described above.
An apparatus comprising a computer readable medium which during operation causes a processor to perform the method described above.
In another aspect, in general, an apparatus includes an interometry system. During operation the interometry system directs two beams along different paths and provides an interference signal S(t) from interference between the two beams. The signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time. The apparatus includes an electronic processor, which during operation receives the interference signal S(t) from the interferometry system and provides an error signal to reduce errors in an estimate of {tilde over (L)}(t). The error signal is derived at least in part based on one or more collective properties of a distribution of multi-dimensional values. At least one of the multi-dimensional values in the distribution is generated from a plurality of samples of the signal S(t).
Aspects can include one or more of the following features.
A lithography system for use in fabricating integrated circuits on a wafer includes a stage for supporting the wafer; an illumination system for imaging spatially patterned radiation onto the wafer; a positioning system for adjusting the position of the stage relative to the imaged radiation; and the apparatus described above for monitoring the position of the wafer relative to the imaged radiation.
A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system described above; and producing an integrated circuit from the wafer.
A lithography system for use in fabricating integrated circuits on a wafer includes a stage for supporting the wafer; and an illumination system including a radiation source, a mask, a positioning system, a lens assembly, and the apparatus described above. During operation the source directs radiation through the mask to produce spatially patterned radiation, the positioning system adjusts the position of the mask relative to the radiation from the source, the lens assembly images the spatially patterned radiation onto the wafer, and the apparatus monitors the position of the mask relative to the radiation from the source.
A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system described above; and producing an integrated circuit from the wafer.
A beam writing system for use in fabricating a lithography mask includes a source providing a write beam to pattern a substrate; a stage supporting the substrate; a beam directing assembly for delivering the write beam to the substrate; a positioning system for positioning the stage and beam directing assembly relative one another; and the apparatus described above for monitoring the position of the stage relative to the beam directing assembly.
A method for fabricating a lithography mask includes directing a beam to a substrate using the beam writing system described above; varying the intensity or the position of the beam at the substrate to form a pattern in the substrate; and forming the lithography mask from the patterned substrate.
As used herein “algebraic combinations” means combinations of operands (e.g., real or complex numbers including values of signals) according to one or more algebraic operations (e.g., addition, subtraction, multiplication, and division).
Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. In case of conflict with publications, patent applications, patents, and other references mentioned incorporated herein by reference, the present specification, including definitions, will control.
Other features and advantages of the invention will be apparent from the following detailed description.



DESCRIPTION OF DRAWINGS
FIG. 1a is a schematic diagram of a processing unit for generating cyclic error basis functions and characterizing cyclic error coefficients based on a main interference signal S(t) and a reference signal SR(t).
FIG. 1b is a schematic diagram of a processing unit for generating an error signal Sψ(t) from the cyclic error basis functions and characterized coefficients and using the error signal to reduce cyclic errors in the main interference signal S(t).
FIG. 1c is a schematic diagram of an exemplary measurement system for generating cyclic error basis functions and characterizing cyclic error coefficients based on a complex measurement signal.
FIG. 1d is a schematic diagram of an error estimator for the measurement system of FIG. 1c. 
FIG. 1e is a schematic diagram of a processing unit for the error estimator of FIG. 1d. 
FIG. 2 is a schematic diagram of an Mth order digital filter for use in low-pass filtering algebraic combinations of the main signal, the reference signal, their quadrature signals, and the error basis functions to yield the cyclic error coefficients.
FIG. 3 is a schematic diagram of an interferometry system including a high-stability plane mirror interferometer (HSPMI).
FIG. 4 is a schematic diagram of an embodiment of a lithography tool that includes an interferometer.
FIG. 5a and FIG. 5b are flow charts that describe steps for making integrated circuits.
FIG. 6 is a schematic of a beam writing system that includes an interferometry system.
FIGS. 7a and 7b are phasor diagrams showing effects of cyclic errors.
FIG. 8a-8c are plots of curves corresponding to a two-dimensional signal.



Like reference symbols in the various drawings indicate like elements.
DETAILED DESCRIPTION
Embodiments include an electronic cyclic error compensation (CEC) procedure for compensation of cyclic error effects in interferometry applications, such as heterodyne interferometry. In preferred embodiments, the compensation is achieved for low slew rates of a plane mirror measurement object attached to a stage or attached to a reference system with associated interferometers attached to a stage. When optical techniques are used to eliminate and/or reduce the amplitudes of certain cyclic errors such as sub-harmonic cyclic errors to ≲0.05 nm (3σ), the remaining cyclic errors of the harmonic type with amplitudes of 0.5 nm or less can be treated as having constant amplitudes with fixed offset phases and the required accuracy of the cyclic error compensation for a remaining cyclic error term is approximately 10% to meet a compensated cyclic error budget of 0.05 nm (3σ) or less. Further, the number of cyclic error terms that need to be compensated electronically are typically a small number, e.g., of the order of 3. In preferred embodiments, the processing operations of CEC at high digital processing rates can be limited to a single add operation, whereas the remaining processing operations, which require additions, subtractions, multiplications, and divisions, can be performed at lower rates using prior values of the interference signal.
Typically, cyclic error effects in heterodyne interferometry can be eliminated by filtering the heterodyne signal in frequency space, e.g., using Fourier spectral analysis, when the Doppler shift frequencies can be resolved by the frequency resolution of a phase meter used to determine the heterodyne phase. Unfortunately, such filtering techniques cannot be used to eliminate cyclic error effects at low slew rates of the stage (including, e.g., zero speed of the stage) when the corresponding Doppler shift frequencies cannot be distinguished from the frequency of the primary signal. Further complications arise with cyclic error frequencies are within the bandwidth of the servo system, in which case the cyclic errors can be coupled directly into the stage position through the servo control system, and even amplify the error in the stage position from a desired position.
Specific details of embodiments of the CEC are described further below. In one approach, the CEC procedure processes real time-sampled values of a digitized measurement signal (DMS) generated by an analog-to-digital-converter (ADC). Advantages of this “DMS approach” include a cyclic error correction signal that may be generated in a “feed forward mode,” where the feed forward mode can involve a simple discrete transform based on a translation in time and need not require a spectral analysis or the use of a discrete transform such as a discrete Fourier transform, such as a fast Fourier transform (FFT). Likewise, conjugated quadratures of the main interference signal and the reference signal can be generated by simple discrete transforms and need not require the use of a discrete transform such as a discrete Hilbert transform. Moreover, the feed forward mode can reduce the number of computer logic operations that are required at the very highest compute rates and can thereby reduce errors in data age that are introduced by incorporation of CEC.
In another approach, the CEC procedure processes complex values of a complex measurement signal (CMS) generated by a discrete Fourier transform (DFT) module after the ADC module. Advantages of this “CMS approach” include the ability to update the DFT (and the CEC computations) at a lower rate (e.g., 10 MHz) than the ADC sampling rate (e.g., 120 MHz). A reduction in the CEC update rate enables a simplified hardware architecture. For example, a reduction in CEC update rate by a factor of 12 can result in a hardware savings of greater than a factor of 12. The CMS approach also eliminates cyclic errors that are due to finite arithmetic precision of the samples generated by the ADC and of the DFT coefficients and calculations in the DFT module. The CMS approach is also less subject to noise than the DMS approach due to the number of samples and the window function used by the DFT module.
Another advantage of both the DMS approach and the CMS approach is that the cyclic error coefficients can be characterized at Doppler shift frequencies for which the phase meter cannot distinguish between the cyclic error frequencies from the frequency of the primary component of the interference signal. Furthermore, the cyclic error coefficients can be characterized and used for compensation over a range of Doppler shift frequencies that is small relative to heterodyne frequency, which is a range over which the cyclic error coefficients are typically frequency independent, thereby simplifying the cyclic error correction.
Prior to describing preferred embodiments of the CEC, it is useful to classify cyclic errors according to properties with respect to the cyclic error amplitudes. Three classifications are set out with reference to a standard high stability plane mirror interferometer (HSPMI). They are: Type 1—Constant amplitude cyclic errors; Type 2—Variable amplitude cyclic errors; and Type 3—Intermittent cyclic errors.
The amplitudes of Type 1 cyclic errors are independent of the orientation of a plane mirror measurement object of a plane mirror interferometer.
The amplitudes of Type 2 cyclic errors are dependent on the orientation of the plane mirror measurement object with a relative variability similar to that experienced by the amplitude of the primary component of the respective electrical interference signal, e.g., 20%.
The amplitudes of Type 3 cyclic errors are nominally zero except when the reflecting surface of the plane mirror measurement object is parallel to within 50 grad of a conjugate image of a reflecting or partially reflecting surface, e.g., the reference mirror, of the interferometer.
Examples of Type 1 cyclic errors are the harmonic cyclic errors generated by polarization mixing in the source of an input beam to the heterodyne interferometer, polarization mixing that is produced by a polarization beam-splitter having finite extinction ratios that splits the input beam into reference and measurement beams, and polarization mixing that is produced by a silver coated cube corner retroreflector such as commonly used in a high stability plane mirror interferometer (HSPMI). The amplitudes of Type 1 cyclic errors are typically ≲0.25 nm using techniques such as described in U.S. patent application Ser. No. 10/174,149 entitled “Interferometry System and Method Employing an Angular Difference in Propagation Between Orthogonally Polarized Input Beam Components” to Peter de Groot and Henry A. Hill and filed Jul. 17, 2002, in U.S. Pat. No. 6,201,609 B1 entitled “Interferometers Utilizing Polarization Preserving Optical Systems” to Henry A. Hill, and in U.S. Pat. No. 6,163,379 entitled “Interferometer with Tilted Waveplates for Reducing Ghost Reflections” to Peter de Groot, the contents of which is incorporated herein by reference.
Examples of Type 2 cyclic errors are the harmonic cyclic errors generated by spurious beams due to unwanted reflections by certain surfaces. The typical amplitude of a Type 2 cyclic error is 0.06 nm for a respective surface having a reflectivity of 0.0025. The amplitude of a particular Type 2 cyclic error will typically vary by ≲±20% as the stage mirror is scanned through a range of for example ±500 μrad in orientation with a physical separation of ≈0.7 m between the polarizing beam-splitter of an interferometer and plane mirror measurement object.
Examples of Type 3 intermittent cyclic errors are sub-harmonic cyclic errors that have relatively large amplitudes, e.g. 2 nm, when the conditions for Type 3 cyclic errors are met. The Type 3 cyclic errors can be eliminated or reduced to below 0.025 nm (3σ) by optical techniques where elements of the interferometer are rotated or tilted to eliminate or reduce certain cyclic non-linear errors such as described in U.S. Patent Application Publication No. 2003/0038947 entitled “Tilted Interferometer” to Henry A. Hill, the contents of which are herein incorporated in its entirety by reference. The elimination of Type 3 cyclic errors by the optical techniques considerably reduces the task left for CEC in achieving a compensated cyclic error residual of 0.025 nm (3σ) or less. Of course, in further embodiments of the invention, sub-harmonic cyclic errors (such as the Type 3 half-cycle error) may also be compensated, as described further below.
The variability of Type 2 cyclic errors expressed in terms of a displacement is typically ≲±0.010 nm in amplitude. As a consequence, Type 2 cyclic errors can be treated as cyclic errors having constant amplitudes with constant offset phases in implementation of CEC that has a compensated cyclic error budget of 0.01 nm (3σ) or less.
The number of cyclic error terms that need to be compensated when the Type 3 cyclic errors are eliminated or reduced by the optical techniques are low, e.g. of the order of 3, for a compensated cyclic error budget of 0.05 nm (3σ) or less. A particular cyclic error term after elimination of Type 3 cyclic errors may comprise one or more Type 1 and/or Type 2 cyclic errors. Nonetheless, in further embodiments, the CEC may also be used to compensate Type 3 cyclic errors.
Additional material describing quantifying and correcting cyclic errors are described in commonly owned U.S. Pat. No. 6,252,668, U.S. Pat. No. 6,246,481, U.S. Pat. No. 6,137,574, U.S. Pat. No. 6,747,744, and U.S. Pat. No. 6,806,961, the entire contents each of which are incorporated herein by reference.
We now describe the DMS approach for the CEC that operates in a feed forward mode of operation, in which a cyclic error correction signal Sψ(t) is subtracted from a corresponding electrical interference signal S(t) of an interferometer to produce a compensated electrical interference signal. The phase of the compensated electrical interference signal is then measured by a phase meter to extract relative path length information associated with the particular interferometer arrangement. Because cyclic error effects have been reduced, the relative path length information is more accurate. As a result, the compensated electrical interference phase can be used to measure and control through a servo control system the position of a stage, even at low slew rates including a zero slew rate where cyclic error effects can otherwise be especially problematic. The DMS approach is also described in published U.S. application Ser. No. 10/616,504 (publication number US 2004/0085545 A1).
Referring to FIGS. 1a and 1b, in a preferred embodiment, the CEC comprises two processing units. One processing unit 10 determines cyclic error basis functions and factors relating to the amplitudes and offset phases of cyclic errors that need be compensated. A second processing unit 60 of CEC generates cyclic error correction signal Sψ(t) using the cyclic error basis functions and the factors relating to the amplitudes and offset phases determined by first processing unit 10. The first processing unit 10 of CEC for the first embodiment is shown schematically in FIG. 1a and the second processing unit 60 of CEC of the first embodiment is shown schematically in FIG. 1b. 
Referring now to FIG. 1a, an optical signal 11 from an interferometer is detected by detector 12 to generate an electrical interference signal. The electrical interference signal is converted to digital format by an analog to digital converter (ADC) in converter/filter 52 as electrical interference signal S(t) and sent to the CEC processor. For example, the ADC conversion rate is a high rate, e.g., 120 MHz.
In the present embodiment, we focus on a particular set of four cyclic error terms that are compensated at low slew rates. Adaptation to compensate for a different set of cyclic errors will be evident to one skilled in the art based on the subsequent description. The electrical interference signal S(t) comprising the four cyclic error terms can be expressed in the form

S(t)=A1 cos(φR+φ+ζ1)+Sψ(t)  (1)

where

Sψ(t)=Sψ−1(t)+Sψ0+Sψ2(t)+Sψ3(t);  (2)

Sψ−1(t)=ε−1 cos(φR−φ+ζ−1),  (3)

Sψ0=ε0 cos(φR+ζ0),  (4)

Sψ2(t)=ε2 cos(φR+2φ+ζ2),  (5)

Sψ3(t)=ε3 cos(φR+3φ+ζ3);  (6)

φR is the phase of a reference signal SR(t) with dφR/dt=ωR corresponding to 2π times the frequency difference of the measurement beam and reference beam components of the input beam to the interferometer; A1 and ζ1 are the amplitude and offset phase, respectively, of the primary component of the electrical interference signal; ε−1, ε0, ε2, and ε3 are the amplitudes for the cyclic error terms; ζ−1, ζ0, ζ2, and ζ3 are the offset phases of the cyclic error terms;

φ=4kL  (7)

for a plane mirror interferometer such as a HSPMI (which involves two passes of the measurement beam to the measurement object); k is a wavenumber corresponding to wavelength λ of beam 11; and L is the difference between the one way physical length of the measurement path and the one way physical length of the reference path of the interferometer. Cyclic error amplitudes ε−1, ε0, ε2, and ε3 are much less than the A1, i.e. ≲( 1/50)A1. An example of the frequency difference ωR/2π is 20 MHz.

Note that there is generally a set of cyclic error terms whose phases are independent of φR. This set of cyclic error terms has been omitted from Equation (2) because they are eliminated by a high pass filter in converter/filter 52.
The factors relating to amplitudes εp and offset phases ζp of the cyclic error terms and the time dependent factors of the cyclic error terms are generated using measured values of both S(t) and reference signal SR(t). The factors relating to amplitudes εp and offset phases ζp are determined and the results transmitted to a table 40 for subsequent use in generation of the cyclic error correction signal Sψ(t). The time dependent factors of the cyclic error terms are obtained by application of simple discrete transforms based on trigonometric identities and properties of conjugated quadratures of signals.
Optical reference signal 13 is detected by detector 14 to produce an electrical reference signal. The optical reference signal can be derived from a portion of the input beam to the interferometer. Alternatively, the electrical reference signal can be derived directly from the source that introduces the heterodyne frequency splitting in the input beam components (e.g., from the drive signal to an acousto-optical modulator used to generate the heterodyne frequency splitting). The electrical reference signal is converted to a digital format and passed through a high pass filter in converter/filter 54 to produce reference signal SR(t). Reference signal SR(t) in digital format is written as

SR(t)=AR cos(φR+ζR)  (8)

where AR and ζR are the amplitude and offset phase, respectively, of the reference signal. The ADC conversion rate in 54 for SR(t) is the same as the ADC conversion rate in 52 for S(t). The quadrature signal {tilde over (S)}R(t) of SR(t) written as

{tilde over (S)}R(t)=AR sin(φR+ζR)  (9)

is generated by electronic processing using measured values of SR(t) according to the formula





  
    
      
        
          
            
              S
              ~
            
            R
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              (
              
                cot
                ⁢
                
                  
                
                ⁢
                
                  ω
                  R
                
                ⁢
                τ
              
              )
            
            ⁢
            
              
                S
                R
              
              ⁡
              
                (
                
                  t
                  -
                  
                    2
                    ⁢
                    τ
                  
                
                )
              
            
          
          -
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  ω
                  R
                
                ⁢
                τ
              
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  ω
                  R
                
                ⁢
                τ
              
            
            ⁢
            
              
                S
                R
              
              ⁡
              
                (
                
                  t
                  -
                  τ
                
                )
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where 1/τ is the ADC conversion rate of reference signal SR(t) in 54. For the example of a frequency difference for ωR/2π=20 MHz and an ADC conversion rate 1/τ of 120 MHz, Equation (10) reduces to a particularly simple form





  
    
      
        
          
            
              S
              ~
            
            R
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              1
              
                3
              
            
            ⁡
            
              [
              
                
                  
                    S
                    R
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      τ
                    
                    )
                  
                
                +
                
                  
                    S
                    R
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        2
                        ⁢
                        τ
                      
                    
                    )
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        11
        )
      
    
  




Reference signal SR(t) and quadrature signal {tilde over (S)}R(t) are conjugated quadratures of reference signal SR(t). The quadrature signal {tilde over (S)}R(t) is generated by processor 16 using Equation (11) or Equation (10) as appropriate.
The quadrature signal {tilde over (S)}(t) of S(t) is generated by processor 56 using the same processing procedure as that described for the generation of quadrature signal {tilde over (S)}R(t) Accordingly, for the example of a ratio of 1/τ and ωR/2π equal to 6,




  
    
      
        
          
            
              
                
                  
                    S
                    ~
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                =
                
                ⁢
                
                  
                    1
                    
                      3
                    
                  
                  ⁡
                  
                    [
                    
                      
                        S
                        ⁡
                        
                          (
                          
                            t
                            -
                            τ
                          
                          )
                        
                      
                      +
                      
                        S
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              2
                              ⁢
                              τ
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      A
                      0
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          φ
                          +
                          
                            ζ
                            1
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        S
                        ~
                      
                      ψ
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
      
    
    
      
        (
        12
        )
      
    
  
  
    
      
        
          
            
              S
              ~
            
            ψ
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              ɛ
              
                -
                1
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  
                    φ
                    R
                  
                  -
                  φ
                  +
                  
                    ζ
                    
                      -
                      1
                    
                  
                
                )
              
            
          
          +
          
            
              ɛ
              0
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  
                    φ
                    R
                  
                  +
                  
                    ζ
                    0
                  
                
                )
              
            
          
          +
          
            
              ɛ
              2
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  
                    φ
                    R
                  
                  +
                  
                    2
                    ⁢
                    φ
                  
                  +
                  
                    ζ
                    2
                  
                
                )
              
            
          
          +
          
            
              ɛ
              3
            
            ⁢
            
              
                sin
                ⁡
                
                  (
                  
                    
                      φ
                      R
                    
                    +
                    
                      3
                      ⁢
                      φ
                    
                    +
                    
                      ζ
                      3
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




Equation (12) is valid when the stage slew rate is low, e.g., when φ changes insignificantly over the time period 2τ. Signal S(t) and quadrature signal {tilde over (S)}(t) are conjugated quadratures of signal S(t).
Notably, the integral relationship between 1/τ and ωR/2π allows the generation of feed forward values S′(t) and {tilde over (S)}′(t) of S(t) and {tilde over (S)}(t), respectively, and S′R(t) and {tilde over (S)}′R(t) of SR(t) and {tilde over (S)}R(t), respectively, are according to the formulae

S′(t)=S(t−6mτ),  (14)

{tilde over (S)}′(t)={tilde over (S)}(t−6mτ),  (15)

S′R(t)=SR(t−6mτ),  (16)

{tilde over (S)}′R(t)={tilde over (S)}R(t−6mτ)  (17)

where m is an integer such that the error in the phases of feed forward signals with respect to corresponding phases of signals is less than predetermined values set by an end use application. In other words, prior values of the main interference signal and the reference signal can be used to generate the quadrature signals and subsequent error basis functions. In other embodiments in which the ratio between 1/τ and ωR/2π is an integer different from 6, Equations (14)-(17) are modified accordingly.

Using algebraic combinations of the signals S(t), {tilde over (S)}(t), SR(t), and {tilde over (S)}R(t), processing unit 10 generates cyclic error basis functions, which are sine and cosine functions that have the same time-varying arguments as the cyclic error terms, and then uses the cyclic error basis functions to project out respective cyclic error coefficients from S(t) and {tilde over (S)}(t) by low-pass filtering (e.g., averaging). The cyclic error basis functions for Sψ0(t), for example, are especially simple and correspond to the reference signal and its quadrature signal SR(t), and {tilde over (S)}R(t). In other words, to process the signals for information about the cyclic error term ε0 cos(φR+ζ0), signals SR(t) and {tilde over (S)}R(t) are used as time dependent factors in the representation of the cyclic error term ε0 cos(φR+ζ0).
To better understand the representation, it is beneficial to rewrite cyclic error term ε0 cos(φR+ζ0) in terms of the time dependent functions cos(φR+ζR) and sin(φR+ζR) with the results




  
    
      
        
          
            
              ɛ
              0
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    φ
                    R
                  
                  +
                  
                    ζ
                    0
                  
                
                )
              
            
          
          =
          
            
              ɛ
              0
            
            ⁡
            
              [
              
                
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                0
                              
                              -
                              
                                ζ
                                R
                              
                            
                            )
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                φ
                                R
                              
                              +
                              
                                ζ
                                R
                              
                            
                            )
                          
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              ζ
                              0
                            
                            -
                            
                              ζ
                              R
                            
                          
                          )
                        
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              φ
                              R
                            
                            +
                            
                              ζ
                              R
                            
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          
            ɛ
            0
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  φ
                  R
                
                +
                
                  ζ
                  0
                
              
              )
            
          
        
        =
        
          
            
              ɛ
              0
            
            ⁡
            
              [
              
                
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                0
                              
                              -
                              
                                ζ
                                R
                              
                            
                            )
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                φ
                                R
                              
                              +
                              
                                ζ
                                R
                              
                            
                            )
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              ζ
                              0
                            
                            -
                            
                              ζ
                              R
                            
                          
                          )
                        
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              φ
                              R
                            
                            +
                            
                              ζ
                              R
                            
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        19
        )
      
    
  




Equations (18) and (19) can be rewritten as

ε0 cos(φR+ζ0)=[A0 cos(φR+ζR)−B0 sin(φR+ζR)],  (20)

ε0 sin(φR+ζ0=[A0 sin(φR+ζR)+B0 cos(φR+ζR)]  (21)

where

A0=ε0 cos(ζ0−ζR),  (22)

B0=ε0 sin(ζ0−ζR).  (23)

Conjugated quadratures S(t) and {tilde over (S)}(t) and conjugated quadratures SR(t) and {tilde over (S)}R(t) are transmitted to processor 20 wherein signals Σ0(t) and {tilde over (Σ)}0(t) are generated. Signals Σ0(t) and {tilde over (Σ)}0(t) are given by the equations

Σ0(t)≡S(t)SR(t)+{tilde over (S)}(t){tilde over (S)}R(t),  (24)

{tilde over (Σ)}0(t)≡−S(t){tilde over (S)}R(t)+{tilde over (S)}(t)SR(t).  (25)

Using properties of conjugated quadratures and certain trigonometric identities, e.g., cos2 γ+sin2 γ=1, we have




  
    
      
        
          
            
              ∑
              0
            
            ⁢
            
              
            
            ⁢
            
              (
              t
              )
            
          
          =
          
            
              
                A
                R
              
              ⁢
              
                A
                0
              
            
            +
            
              
                A
                R
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            A
                            1
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                φ
                                +
                                
                                  ζ
                                  1
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            ɛ
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  -
                                  φ
                                
                                +
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ɛ
                            2
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  φ
                                
                                +
                                
                                  ζ
                                  2
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  φ
                                
                                +
                                
                                  ζ
                                  3
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        (
        26
        )
      
    
  
  
    
      
        
          
            
              ∑
              ~
            
            0
          
          ⁢
          
            
          
          ⁢
          
            (
            t
            )
          
        
        =
        
          
            
              A
              R
            
            ⁢
            
              B
              0
            
          
          +
          
            
              
                A
                R
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            A
                            1
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                φ
                                +
                                
                                  ζ
                                  1
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            ɛ
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  -
                                  φ
                                
                                +
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ɛ
                            2
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  φ
                                
                                +
                                
                                  ζ
                                  2
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  φ
                                
                                +
                                
                                  ζ
                                  3
                                
                                -
                                
                                  ζ
                                  R
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        27
        )
      
    
  




Notably, the generation of signals Σ0(t) and {tilde over (Σ)}0(t) project the coefficients associated with Sψ0(t) to zero frequency, where low-pass filtering techniques can be used to determine them. Thus, signals Σ0(t) and {tilde over (Σ)}0(t) are transmitted to low pass digital filters in processor 24, e.g., low pass Butterworth filters, where coefficients ARA0 and ARB0 are determined.

For a Butterworth filter Tn(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ0(t) and {tilde over (Σ)}0(t) are




  
    
      
        
          
            
              T
              n
            
            ⁡
            
              [
              
                
                  ∑
                  0
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  t
                  )
                
              
              ]
            
          
          =
          
            
              
                A
                R
              
              ⁢
              
                A
                0
              
            
            +
            
              
                A
                R
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            A
                            1
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    ω
                                    D
                                  
                                
                                )
                              
                            
                            n
                          
                        
                        +
                        
                          
                            ɛ
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    ω
                                    D
                                  
                                
                                )
                              
                            
                            n
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ɛ
                            2
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    2
                                    ⁢
                                    
                                      ω
                                      D
                                    
                                  
                                
                                )
                              
                            
                            n
                          
                        
                        +
                        
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    3
                                    ⁢
                                    
                                      ω
                                      D
                                    
                                  
                                
                                )
                              
                            
                            n
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        (
        28
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    ∑
                    2
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    t
                    )
                  
                
                ≡
                
                ⁢
                
                  
                    S
                    1
                  
                  ·
                  
                    
                      ∑
                      1
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        -
                        
                          
                            S
                            ~
                          
                          1
                        
                      
                      ·
                      
                        
                          ∑
                          ~
                        
                        1
                      
                    
                  
                
              
              ⁢
              
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      A
                      R
                    
                    ⁢
                    
                      
                        A
                        1
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    A
                                    1
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          φ
                                          R
                                        
                                        +
                                        
                                          2
                                          ⁢
                                          φ
                                        
                                        +
                                        
                                          2
                                          ⁢
                                          
                                            ζ
                                            1
                                          
                                        
                                        -
                                        
                                          ζ
                                          R
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  2
                                  ⁢
                                  
                                    ɛ
                                    
                                      -
                                      1
                                    
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          φ
                                          R
                                        
                                        +
                                        
                                          ζ
                                          1
                                        
                                        +
                                        
                                          ζ
                                          
                                            -
                                            1
                                          
                                        
                                        -
                                        
                                          ζ
                                          R
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  2
                                  ⁢
                                  
                                    ɛ
                                    2
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          φ
                                          R
                                        
                                        +
                                        
                                          3
                                          ⁢
                                          φ
                                        
                                        +
                                        
                                          ζ
                                          1
                                        
                                        +
                                        
                                          ζ
                                          2
                                        
                                        -
                                        
                                          ζ
                                          R
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  3
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        4
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        ζ
                                        3
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                  +
                  
                    
                      A
                      R
                    
                    ⁢
                    O
                    ⁢
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




where O(x) denotes a term of the order of x, ωc is the −3 dB angular cutoff frequency, and ωD=dφ/dt.

The terms on the right hand sides of Equations (28) and (29) with factors ARA1 are the sources of the largest errors and accordingly determine the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of processor 24 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ωD/ωc=7, the error terms on the right hand side of Equations (28) and (29) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the outputs ARA0 and ARB0 of the low pass filters in processor 24 are stored in table 40 and in processors 26 and 28 under the control of signal 72 (from processor 70).
Notably, in this preferred embodiment, ωD can vary by factors such as 2 or more during the period associated with output values of ARA0 and ARB0 that are stored in table 40.
Quadratures SR and {tilde over (S)}R are transmitted to processor 22 and the value for AR2 is determined in processor 22 by a procedure similar to that used in processor 20 and processor 24 using the formulae

Tn[SR(t)·SR(t)+{tilde over (S)}R(t)·{tilde over (S)}R(t)]=AR2.  (30)

The value of the order n need only be for example 2. Values of AR2 are transmitted to table 40 and stored under the control of signal 72.
Alternatively, detector 14, converter/filter 54, and processor 16 can be replaced by a lookup table synchronized to the reference signal. This can potentially reduce uncertainty caused by noise on the reference signal. If the value of AR2 is normalized to unity then some equations can be simplified and processor 22 can be removed.
The values for ARA0, ARB0, S(t), {tilde over (S)}R(t) SR(t), {tilde over (S)}R(t), and AR2 are transmitted to processor 26 and the values of ARA0, ARB0, and AR2 are stored in processor 26 under the control of signal 72 for the generation of conjugated quadratures S1(t) and {tilde over (S)}1(t) where




  
    
      
        
          
            
              
                
                  S
                  1
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  S
                  ⁡
                  
                    (
                    t
                    )
                  
                
                -
                
                  
                    
                      (
                      
                        
                          A
                          R
                        
                        ⁢
                        
                          A
                          0
                        
                      
                      )
                    
                    
                      A
                      R
                      2
                    
                  
                  ⁢
                  
                    S
                    R
                  
                
                +
                
                  
                    
                      (
                      
                        
                          A
                          R
                        
                        ⁢
                        
                          B
                          0
                        
                      
                      )
                    
                    
                      A
                      R
                      2
                    
                  
                  ⁢
                  
                    
                      S
                      ~
                    
                    R
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          R
                        
                        +
                        φ
                        +
                        
                          ζ
                          1
                        
                      
                      )
                    
                  
                
                +
                
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          R
                        
                        -
                        φ
                        +
                        
                          ζ
                          
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            2
                            ⁢
                            φ
                          
                          +
                          
                            ζ
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      ɛ
                      3
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            3
                            ⁢
                            φ
                          
                          +
                          
                            ζ
                            3
                          
                        
                        )
                      
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        31
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    S
                    ~
                  
                  1
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    S
                    ~
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                -
                
                  
                    
                      (
                      
                        
                          A
                          R
                        
                        ⁢
                        
                          A
                          0
                        
                      
                      )
                    
                    
                      A
                      R
                      2
                    
                  
                  ⁢
                  
                    
                      S
                      ~
                    
                    R
                  
                
                -
                
                  
                    
                      (
                      
                        
                          A
                          R
                        
                        ⁢
                        
                          B
                          0
                        
                      
                      )
                    
                    
                      A
                      R
                      2
                    
                  
                  ⁢
                  
                    S
                    R
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          φ
                          R
                        
                        +
                        φ
                        +
                        
                          ζ
                          1
                        
                      
                      )
                    
                  
                
                +
                
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          φ
                          R
                        
                        -
                        φ
                        +
                        
                          ζ
                          
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    2
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          φ
                          R
                        
                        +
                        
                          2
                          ⁢
                          φ
                        
                        +
                        
                          ζ
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    ɛ
                    3
                  
                  ⁢
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            3
                            ⁢
                            φ
                          
                          +
                          
                            ζ
                            3
                          
                        
                        )
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




The values for ARA0, ARB0, σ0(t), and {tilde over (Σ)}0(t) are transmitted to processor 28 and the values of ARA0 and ARB0 are stored in processor 28 under the control of signal 72 for the generation of conjugated quadratures Σ1(t) and {tilde over (Σ)}1(t) where




  
    
      
        
          
            
              
                
                  Σ
                  
                    
                      
                    
                    ⁢
                    1
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    Σ
                    0
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                -
                
                  
                    A
                    R
                  
                  ⁢
                  
                    A
                    0
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    A
                    R
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              
                                A
                                1
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    φ
                                    +
                                    
                                      ζ
                                      1
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              
                                ɛ
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      -
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      
                                        -
                                        1
                                      
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                ɛ
                                2
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      2
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              
                                ɛ
                                3
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      3
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        33
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    Σ
                    ~
                  
                  
                    
                      
                    
                    ⁢
                    1
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    
                      Σ
                      ~
                    
                    0
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                -
                
                  
                    A
                    R
                  
                  ⁢
                  
                    B
                    0
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    R
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              
                                A
                                1
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    φ
                                    +
                                    
                                      ζ
                                      1
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              
                                ɛ
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      -
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      
                                        -
                                        1
                                      
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                ɛ
                                2
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      2
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              
                                ɛ
                                3
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      3
                                      ⁢
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      3
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




Signals SR(t), {tilde over (S)}R(t), Σ1(t), and {tilde over (Σ)}1(t) are transmitted to processor 30 for the generation of conjugated quadratures Σ−1(t) and {tilde over (Σ)}−1(t) where




  
    
      
        
          
            
              
                
                  Σ
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    Σ
                    1
                  
                  ⁢
                  
                    S
                    R
                  
                
                +
                
                  
                    
                      Σ
                      ~
                    
                    1
                  
                  ⁢
                  
                    
                      S
                      ~
                    
                    R
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    A
                    R
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              
                                ɛ
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    +
                                    φ
                                    -
                                    
                                      ζ
                                      
                                        -
                                        1
                                      
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                A
                                1
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    -
                                    φ
                                    -
                                    
                                      ζ
                                      1
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                ɛ
                                2
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      φ
                                    
                                    -
                                    
                                      ζ
                                      2
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    φ
                                  
                                  -
                                  
                                    ζ
                                    3
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        35
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    Σ
                    ~
                  
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                
              
              ≡
               
              ⁢
              
                
                  
                    Σ
                    1
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      S
                      ~
                    
                    R
                  
                
                ⁢
                
                  
                
                +
                
                  
                
                ⁢
                
                  
                    
                      Σ
                      ~
                    
                    1
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    S
                    R
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    A
                    R
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              
                                ɛ
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    +
                                    φ
                                    -
                                    
                                      ζ
                                      
                                        -
                                        1
                                      
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                A
                                1
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    -
                                    φ
                                    -
                                    
                                      ζ
                                      1
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                ɛ
                                2
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      φ
                                    
                                    -
                                    
                                      ζ
                                      2
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    φ
                                  
                                  -
                                  
                                    ζ
                                    3
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




Signals Σ−1(t) and {tilde over (Σ)}−1(t) correspond to the cyclic errror basis functions for Sψ−1(t) in that the leading terms of Σ−1(t) and {tilde over (Σ)}−1(t) are sinusoids with the same time-varying argument as that of Sψ−1(t).
Coefficients AR2A1A−1 and −AR2A1B−1 are next determined through digital low pass filters, e.g., low pass Butterworth filters, in processor 32 where

A−1≡ε−1 cos(ζ−1+ζ1−2ζR),  (37)

B−1≡ε−1 sin(ζ−1+ζ1−2ζR).  (38)

The input signals for the digital filters are Σ4(t) and {tilde over (Σ)}S4(t). Input signals Σ4(t) and {tilde over (Σ)}4(t) are generated in processor 32 using signals S1, {tilde over (S)}1, Σ−1(t), and {tilde over (Σ)}−1(t) according to the formulae

Σ4(t)=[S1(t)Σ−1(t)+{tilde over (S)}1(t){tilde over (Σ)}−1(t)],  (39)

{tilde over (Σ)}4(t)=[S1(t){tilde over (E)}−1(t)−{tilde over (S)}1(t)Σ−1(t)].  (40)

Equations (39) and (40) are written in terms of A−1 and B−1 using Equations (37) and (38) as




  
    
      
        
          
            
              
                Σ
                4
              
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    
                      
                        
                      
                      ⁢
                      R
                    
                    
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    A
                    
                      -
                      1
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    
                      
                        
                      
                      ⁢
                      R
                    
                    
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    
                      A
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  A
                                  1
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          -
                                          2
                                        
                                        ⁢
                                        φ
                                      
                                      -
                                      
                                        ζ
                                        1
                                      
                                      -
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          R
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  2
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          -
                                          3
                                        
                                        ⁢
                                        φ
                                      
                                      -
                                      
                                        ζ
                                        1
                                      
                                      -
                                      
                                        ζ
                                        2
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          R
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              2
                              ⁢
                              
                                ɛ
                                2
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        -
                                        4
                                      
                                      ⁢
                                      φ
                                    
                                    -
                                    
                                      ζ
                                      1
                                    
                                    -
                                    
                                      ζ
                                      3
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  
  
    
      
        
          
            
              
                
                  Σ
                  ~
                
                4
              
              =
              
              ⁢
              
                
                  
                    -
                    2
                  
                  ⁢
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    B
                    
                      -
                      1
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    
                      
                        
                      
                      ⁢
                      R
                    
                    
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    
                      A
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  A
                                  1
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          -
                                          2
                                        
                                        ⁢
                                        φ
                                      
                                      -
                                      
                                        ζ
                                        1
                                      
                                      -
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          R
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  2
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          -
                                          3
                                        
                                        ⁢
                                        φ
                                      
                                      -
                                      
                                        ζ
                                        1
                                      
                                      -
                                      
                                        ζ
                                        2
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          R
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              2
                              ⁢
                              
                                ɛ
                                3
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        -
                                        4
                                      
                                      ⁢
                                      φ
                                    
                                    -
                                    
                                      ζ
                                      1
                                    
                                    -
                                    
                                      ζ
                                      3
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        R
                                      
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




Signals Σ4(t) and {tilde over (Σ)}4(t) are sent to low pass digital filters in processor 32, e.g., low pass Butterworth filters, where coefficients 2AR2A1A−1 and −2AR2A1B−1 are determined. For a Butterworth filter Tn(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ4(t) and {tilde over (Σ)}4(t) are




  
    
      
        
          
            
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    
                      Σ
                      4
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    A
                    
                      -
                      1
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        A
                        1
                      
                      ⁢
                      
                        
                          O
                          ⁡
                          
                            (
                            
                              
                                ω
                                c
                              
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  D
                                
                              
                            
                            )
                          
                        
                        n
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        ɛ
                        2
                      
                      ⁢
                      
                        
                          O
                          ⁡
                          
                            (
                            
                              
                                ω
                                c
                              
                              
                                3
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  D
                                
                              
                            
                            )
                          
                        
                        n
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        ɛ
                        3
                      
                      ⁢
                      
                        
                          O
                          ⁡
                          
                            (
                            
                              
                                ω
                                C
                              
                              
                                4
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  D
                                
                              
                            
                            )
                          
                        
                        n
                      
                    
                  
                  ]
                
                ,
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  
  
    
      
        
          
            
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    
                      
                        Σ
                        ~
                      
                      4
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    -
                    2
                  
                  ⁢
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    B
                    
                      -
                      1
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        A
                        1
                      
                      ⁢
                      
                        
                          O
                          ⁡
                          
                            (
                            
                              
                                ω
                                c
                              
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  D
                                
                              
                            
                            )
                          
                        
                        n
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        ɛ
                        2
                      
                      ⁢
                      
                        
                          O
                          ⁡
                          
                            (
                            
                              
                                ω
                                c
                              
                              
                                3
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  D
                                
                              
                            
                            )
                          
                        
                        n
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        ɛ
                        3
                      
                      ⁢
                      
                        
                          O
                          ⁡
                          
                            (
                            
                              
                                ω
                                C
                              
                              
                                4
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  D
                                
                              
                            
                            )
                          
                        
                        n
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




The terms on the right hand sides of Equations (43) and (44) with factors AR2A12 are the sources of the largest errors and accordingly determine the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of processor 32 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ωD/ωc=3.5, the error terms on the right hand side of Equations (43) and (44) will generate errors that correspond to ≲0.010 nm (3σ). The outputs 2AR2A1A−1 and −2AR2A1B−1 of low pass filters of processor 32 are divided by 2 to generate AR2A1A−1 and −AR2A1B−1 as the outputs of processor 32. When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the outputs AR2A1A−1 and −AR2A1B−1 of processor 32 are stored in table 40 and in processor 34 under the control of signal 72.
Signals S1(t) and {tilde over (S)}1(t) are transmitted to processor 30 for the generation of conjugated quadratures Σ2(t) and {tilde over (Σ)}2(t) where




  
    
      
        
          
            
              
                
                  Σ
                  2
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    S
                    1
                  
                  ·
                  
                    Σ
                    1
                  
                
                -
                
                  
                    
                      S
                      ~
                    
                    1
                  
                  ·
                  
                    
                      Σ
                      ~
                    
                    1
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    R
                  
                  ⁢
                  
                    
                      A
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  A
                                  1
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  
                                    -
                                    1
                                  
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        ζ
                                        
                                          -
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  2
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        3
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        ζ
                                        2
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              2
                              ⁢
                              
                                ɛ
                                3
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    +
                                    
                                      4
                                      ⁢
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      1
                                    
                                    +
                                    
                                      ζ
                                      3
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    R
                  
                  ⁢
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    Σ
                    ~
                  
                  2
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    S
                    1
                  
                  ·
                  
                    
                      Σ
                      ~
                    
                    1
                  
                
                -
                
                  
                    
                      S
                      ~
                    
                    1
                  
                  ·
                  
                    Σ
                    1
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    R
                  
                  ⁢
                  
                    
                      A
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  A
                                  1
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  
                                    -
                                    1
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        ζ
                                        
                                          -
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  ɛ
                                  2
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        R
                                      
                                      +
                                      
                                        3
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        ζ
                                        2
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              2
                              ⁢
                              
                                ɛ
                                3
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      R
                                    
                                    +
                                    
                                      4
                                      ⁢
                                      φ
                                    
                                    +
                                    
                                      ζ
                                      1
                                    
                                    +
                                    
                                      ζ
                                      3
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                
                ⁢
                
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




Signals Σ2(t) and {tilde over (Σ)}2(t) correspond to the cyclic error basis functions for Sψ2(t) in that the leading terms of Σ2(t) and {tilde over (Σ)}2(t) are sinusoids with the same time-varying argument as that of Sψ2(t).
Signals S1, {tilde over (S)}1, Σ1(t), {tilde over (Σ)}1(t), Σ2(t), and {tilde over (Σ)}2(t) and coefficients AR2A1A−1, and are −AR2A1B−1 transmitted to processor 34 and coefficients AR2A1A−1, and −AR2A1B−1 stored in processor 34 under the control of signal 72 for generation of conjugated quadratures Σ3(t) and {tilde over (Σ)}3(t) where




  
    
      
        
          
            
              
                
                  Σ
                  3
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    Σ
                    1
                  
                  ·
                  
                    Σ
                    2
                  
                
                -
                
                  
                    
                      Σ
                      ~
                    
                    1
                  
                  ·
                  
                    
                      Σ
                      ~
                    
                    2
                  
                
                +
                
                  3
                  ⁡
                  
                    [
                    
                      
                        
                          A
                          R
                          2
                        
                        ⁢
                        
                          A
                          1
                        
                        ⁢
                        
                          A
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          S
                          1
                        
                      
                      -
                      
                        
                          A
                          R
                          2
                        
                        ⁢
                        
                          A
                          1
                        
                        ⁢
                        
                          B
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            S
                            ~
                          
                          1
                        
                      
                    
                    ]
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            3
                            ⁢
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  φ
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  +
                                  
                                    ζ
                                    
                                      -
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  +
                                  
                                    ζ
                                    1
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            3
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              (
                              
                                
                                  φ
                                  R
                                
                                +
                                
                                  4
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  2
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          3
                          ⁢
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  φ
                                  R
                                
                                +
                                
                                  5
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  3
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    3
                    ⁡
                    
                      [
                      
                        
                          
                            A
                            R
                            2
                          
                          ⁢
                          
                            A
                            1
                          
                          ⁢
                          
                            A
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            S
                            1
                          
                        
                        -
                        
                          
                            A
                            R
                            2
                          
                          ⁢
                          
                            A
                            1
                          
                          ⁢
                          
                            B
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              S
                              ~
                            
                            1
                          
                        
                      
                      ]
                    
                  
                  +
                  
                    
                      A
                      R
                      2
                    
                    ⁢
                    
                      A
                      1
                    
                    ⁢
                    
                      O
                      ⁡
                      
                        (
                        
                          
                            ɛ
                            i
                          
                          ⁢
                          
                            ɛ
                            j
                          
                        
                        )
                      
                    
                  
                  +
                  ...
                
                ⁢
                
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  
  
    
      
        
          
            
              
                
                  Σ
                  3
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            3
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              (
                              
                                
                                  φ
                                  R
                                
                                +
                                
                                  4
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  2
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          3
                          ⁢
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  φ
                                  R
                                
                                +
                                
                                  5
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  3
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      A
                      R
                      2
                    
                    ⁢
                    
                      A
                      1
                    
                    ⁢
                    
                      O
                      ⁡
                      
                        (
                        
                          
                            ɛ
                            i
                          
                          ⁢
                          
                            ɛ
                            j
                          
                        
                        )
                      
                    
                  
                  +
                  ...
                
                ⁢
                
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        48
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    Σ
                    ~
                  
                  3
                
                ⁡
                
                  (
                  t
                  )
                
              
              ≡
              
              ⁢
              
                
                  
                    Σ
                    1
                  
                  ·
                  
                    
                      Σ
                      ~
                    
                    2
                  
                
                +
                
                  
                    
                      Σ
                      ~
                    
                    1
                  
                  ·
                  
                    Σ
                    2
                  
                
                -
                
                  3
                  ⁡
                  
                    [
                    
                      
                        
                          A
                          R
                          2
                        
                        ⁢
                        
                          A
                          1
                        
                        ⁢
                        
                          A
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            S
                            ~
                          
                          1
                        
                      
                      -
                      
                        
                          A
                          R
                          2
                        
                        ⁢
                        
                          A
                          1
                        
                        ⁢
                        
                          B
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          S
                          1
                        
                      
                    
                    ]
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            3
                            ⁢
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  φ
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  +
                                  
                                    ζ
                                    
                                      -
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  +
                                  
                                    ζ
                                    1
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            3
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  
                                    4
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  +
                                  
                                    ζ
                                    2
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          3
                          ⁢
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  φ
                                  R
                                
                                +
                                
                                  5
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  3
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    3
                    ⁡
                    
                      [
                      
                        
                          
                            A
                            R
                            2
                          
                          ⁢
                          
                            A
                            1
                          
                          ⁢
                          
                            A
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              S
                              ~
                            
                            1
                          
                        
                        +
                        
                          
                            A
                            R
                            2
                          
                          ⁢
                          
                            A
                            1
                          
                          ⁢
                          
                            B
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            S
                            1
                          
                        
                      
                      ]
                    
                  
                  +
                  
                    
                      A
                      R
                      2
                    
                    ⁢
                    
                      A
                      1
                    
                    ⁢
                    
                      O
                      ⁡
                      
                        (
                        
                          
                            ɛ
                            i
                          
                          ⁢
                          
                            ɛ
                            j
                          
                        
                        )
                      
                    
                  
                  +
                  ...
                
                ⁢
                
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    Σ
                    ~
                  
                  3
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            3
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  
                                    4
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  +
                                  
                                    ζ
                                    2
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      R
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          3
                          ⁢
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  φ
                                  R
                                
                                +
                                
                                  5
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  3
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                
                +
                
                  ...
                  ⁢
                  
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




Signals Σ3(t) and {tilde over (Σ)}3(t) correspond to the cyclic error basis functions for Sψ3(t) in that the leading terms of Σ3(t) and {tilde over (Σ)}3(t) are sinusoids with the same time-varying argument as that of Sψ3(t).
Coefficients ARA12A2 and −ARA12B2 (where A2 and B2 are cyclic error coefficients for Sψ2 and are given explicitly by Equations (67) and (68), respectively, further below) are next determined through digital low pass filters, e.g., low pass Butterworth filters, in processor 38. The input signals for the digital filters are Σ5(t) and {tilde over (Σ)}5(t), respectively. The input signals are generated in processor 38 using signals S1, {tilde over (S)}1, Σ2(t), and {tilde over (Σ)}2(t) according to the formulae

Σ5(t)≡[S1(t)Σ2(t)+{tilde over (S)}1(t){tilde over (Σ)}2(t)],  (51)

{tilde over (Σ)}5(t)≡[S1(t){tilde over (Σ)}2(t)−{tilde over (S)}1(t)Σ2(t)].  (52)

The expansions of Σ5(t) and {tilde over (Σ)}5(t), given by Equations (51) and (52), respectively, in terms of cyclic error and non-cyclic error terms are




  
    
      
        
          
            
              
                
                  Σ
                  5
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    A
                    R
                  
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    A
                    2
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  {
                  
                    
                      
                        
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  2
                                  ⁢
                                  
                                    sin
                                    ⁡
                                    
                                      (
                                      
                                        
                                          -
                                          φ
                                        
                                        +
                                        
                                          ζ
                                          
                                            -
                                            1
                                          
                                        
                                        -
                                        
                                          ζ
                                          R
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        
                                          -
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  φ
                                  +
                                  
                                    ζ
                                    1
                                  
                                  -
                                  
                                    ζ
                                    R
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            2
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    ζ
                                    2
                                  
                                  -
                                  
                                    ζ
                                    R
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            ɛ
                            3
                          
                          ⁡
                          
                            [
                            
                              
                                2
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        ζ
                                        3
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      -
                                      φ
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        1
                                      
                                    
                                    -
                                    
                                      ζ
                                      3
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                        
                      
                    
                  
                  }
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      A
                      R
                    
                    ⁢
                    
                      A
                      1
                    
                    ⁢
                    
                      O
                      ⁡
                      
                        (
                        
                          ɛ
                          i
                          2
                        
                        )
                      
                    
                  
                  +
                  ...
                
                ⁢
                
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        53
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    Σ
                    ~
                  
                  5
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    -
                    
                      A
                      R
                    
                  
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    B
                    2
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  {
                  
                    
                      
                        
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  2
                                  ⁢
                                  
                                    sin
                                    ⁡
                                    
                                      (
                                      
                                        
                                          -
                                          φ
                                        
                                        +
                                        
                                          ζ
                                          
                                            -
                                            1
                                          
                                        
                                        -
                                        
                                          ζ
                                          R
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        
                                          -
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  φ
                                  +
                                  
                                    ζ
                                    1
                                  
                                  -
                                  
                                    ζ
                                    R
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            2
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    φ
                                  
                                  +
                                  
                                    ζ
                                    2
                                  
                                  -
                                  
                                    ζ
                                    R
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            ɛ
                            3
                          
                          ⁡
                          
                            [
                            
                              
                                2
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        3
                                        ⁢
                                        φ
                                      
                                      +
                                      
                                        ζ
                                        3
                                      
                                      -
                                      
                                        ζ
                                        R
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      -
                                      φ
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ζ
                                        1
                                      
                                    
                                    -
                                    
                                      ζ
                                      3
                                    
                                    -
                                    
                                      ζ
                                      R
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                        
                      
                    
                  
                  }
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      A
                      R
                    
                    ⁢
                    
                      A
                      1
                    
                    ⁢
                    
                      O
                      ⁡
                      
                        (
                        
                          ɛ
                          i
                          2
                        
                        )
                      
                    
                  
                  +
                  ...
                
                ⁢
                
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        54
        )
      
    
  




where A2 and B2 are given by Equations (67) and (68), respectively, shown further below.

Signals Σ5(t) and {tilde over (Σ)}5(t) are sent to low pass digital filters in processor 38, e.g., low pass Butterworth filters, where coefficients ARA12A2 and −ARA12B2 are determined. For a Butterworth filter Tn(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ5(t) and {tilde over (Σ)}5(t) are




  
    
      
        
          
            
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    
                      Σ
                      5
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    A
                    R
                  
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    A
                    2
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                
                ⁢
                
                  A
                  1
                  2
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            2
                            ⁢
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            2
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            2
                            ⁢
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                        
                      
                    
                  
                  ]
                
                ,
              
            
          
        
      
    
    
      
        (
        55
        )
      
    
  
  
    
      
        
          A
          R
          2
        
        ⁢
        
          
            A
            1
            4
          
          .
        
      
    
    
      
        (
        56
        )
      
    
  




The terms on the right hand sides of Equations (55) and (56) with factors ARA13 are the sources of the largest errors and accordingly determine the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of processor 38 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ωD/ωc=7, the error terms on the right hand side of Equations (55) and (56) will generate errors that correspond to ≲0.010 nm (3σ). The outputs ARA12A2 and −ARA12B2 of low pass filters of processor 38 are the outputs of processor 38. When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the outputs ARA12A2 and −ARA12B2 of processor 38 are stored in table 40 under the control of signal 72.
Coefficients AR2A13A3 and −AR2A13B3 (where A3 and B3 are cyclic error coefficients for Sψ3 and are given explicitly by Equations (69) and (70), respectively, further below) are next determined through a digital low pass filter, e.g., a low pass Butterworth filter, in processor 36. The input signals for the digital filters are Σ6(t) and {tilde over (Σ)}6(t), respectively. The input signals are generated in processor 36 using signals S1, {tilde over (S)}1, Σ3(t), and {tilde over (Σ)}3(t) according to the formulae

Σ6(t)=[S1(t)Σ3(t)+{tilde over (S)}1(t){tilde over (Σ)}3(t)],  (57)

{tilde over (Σ)}6(t)=[S1(t){tilde over (Σ)}3(t)−{tilde over (S)}1(t)Σ3(t)].  (58)

The expansions of Σ6(t) and {tilde over (Σ)}6(t) given by Equations (57) and (58), respectively, in terms of cyclic error and non-cyclic error terms are




  
    
      
        
          
            
              
                
                  Σ
                  6
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    A
                    3
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  3
                
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    
                      
                        
                          
                            ɛ
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  φ
                                
                                -
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            A
                            1
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  φ
                                
                                -
                                
                                  ζ
                                  1
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ɛ
                            2
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                φ
                                -
                                
                                  ζ
                                  2
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                        +
                        
                          3
                          ⁢
                          
                            ɛ
                            2
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  3
                                  ⁢
                                  φ
                                
                                +
                                
                                  ζ
                                  1
                                
                                +
                                
                                  ζ
                                  2
                                
                                -
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    R
                                  
                                
                              
                              )
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        3
                        ⁢
                        
                          ɛ
                          3
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                3
                              
                              -
                              
                                2
                                ⁢
                                
                                  ζ
                                  R
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      A
                      R
                      2
                    
                    ⁢
                    
                      A
                      1
                      2
                    
                    ⁢
                    
                      O
                      ⁡
                      
                        (
                        
                          ɛ
                          i
                          2
                        
                        )
                      
                    
                  
                  +
                  ...
                
                ⁢
                
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        59
        )
      
    
  
  
    
      
        
          ɛ
          
            -
            1
          
        
        .
      
    
    
      
        (
        60
        )
      
    
  




where A3 and B3 are given by Equations (69) and (70), respectively.

Signals Σ6(t) and {tilde over (Σ)}6(t) are sent to low pass digital filters in processor 36, e.g., low pass Butterworth filters, where coefficients AR2A13A3 and −AR2A13B3 are determined. For a Butterworth filter Tn(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ6(t) and {tilde over (Σ)}6(t) are




  
    
      
        
          
            
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    
                      Σ
                      6
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    A
                    3
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  3
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      4
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            3
                            ⁢
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            3
                            ⁢
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      4
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                        
                      
                    
                  
                  ]
                
                ,
              
            
          
        
      
    
    
      
        (
        61
        )
      
    
  
  
    
      
        
          
            
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    
                      Σ
                      6
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    A
                    R
                    2
                  
                  ⁢
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    B
                    3
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  R
                  2
                
                ⁢
                
                  A
                  1
                  3
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      4
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            3
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            3
                            ⁢
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      4
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        62
        )
      
    
  




The terms on the right hand sides of Equations (61) and (62) with factors AR2A14 are the sources of the largest errors and accordingly determined the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of processor 36 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ωD/ωc=3.5, the error terms on the right hand side of Equations (61) and (62) will generate errors that correspond to ≲0.010 nm (3σ). The outputs AR2A13A3 and −AR2A1B3 of low pass filters of processor 36 are the outputs of processor 36. When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the outputs AR2A13A3 and −AR2A13B3 of processor 36 are stored in table 40 under the control of signal 72.
Finally, quadratures S and {tilde over (S)} are transmitted from processors 52 and 56 respectively to processor 18 for the purpose of determining a value for A12. First, a signal S(t)S(t)+{tilde over (S)}(t){tilde over (S)}(t) is generated where




  
    
      
        
          
            
              S
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              S
              ⁡
              
                (
                t
                )
              
            
          
          +
          
            
              
                S
                ~
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              
                S
                ~
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            A
            1
            2
          
          +
          
            [
            
              
                ɛ
                
                  -
                  1
                
                2
              
              +
              
                ɛ
                0
                2
              
              +
              
                ɛ
                2
                2
              
              +
              
                ɛ
                3
                2
              
            
            ]
          
          +
          
            2
            ⁢
            
              A
              1
            
            ⁢
            
              ɛ
              
                -
                1
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    φ
                  
                  +
                  
                    ζ
                    1
                  
                  -
                  
                    ζ
                    
                      -
                      1
                    
                  
                
                )
              
            
          
          +
          
            2
            ⁢
            
              A
              1
            
            ⁢
            
              ɛ
              0
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  φ
                  +
                  
                    ζ
                    1
                  
                  -
                  
                    ζ
                    0
                  
                
                )
              
            
          
          +
          
            2
            ⁢
            
              A
              1
            
            ⁢
            
              ɛ
              2
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    -
                    φ
                  
                  +
                  
                    ζ
                    1
                  
                  -
                  
                    ζ
                    2
                  
                
                )
              
            
          
          +
          
            2
            ⁢
            
              A
              1
            
            ⁢
            
              ɛ
              3
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      2
                    
                    ⁢
                    φ
                  
                  +
                  
                    ζ
                    1
                  
                  -
                  
                    ζ
                    3
                  
                
                )
              
            
          
          +
          
            
              O
              ⁡
              
                (
                
                  
                    ɛ
                    i
                  
                  ⁢
                  
                    ɛ
                    j
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        63
        )
      
    
  




The signal of Equation (63) is sent to a low pass digital filter in processor 18, e.g., a low pass Butterworth filter, where the coefficient A12 is determined. For a Butterworth filter Tn(x) of order n, the corresponding outputs of the low pass digital filter for the signal of Equation (63) is




  
    
      
        
          
            T
            n
          
          ⁡
          
            [
            
              
                
                  S
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ·
                
                  S
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              +
              
                
                  
                    S
                    ~
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ·
                
                  
                    S
                    ~
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            ]
          
        
        =
        
          
            A
            1
            2
          
          +
          
            [
            
              
                ɛ
                
                  -
                  1
                
                2
              
              +
              
                ɛ
                0
                2
              
              +
              
                ɛ
                2
                2
              
              +
              
                ɛ
                3
                2
              
            
            ]
          
          +
          
            
              
                A
                1
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          2
                          ⁢
                          
                            ɛ
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    2
                                    ⁢
                                    
                                      ω
                                      D
                                    
                                  
                                
                                )
                              
                            
                            n
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            ɛ
                            0
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    ω
                                    D
                                  
                                
                                )
                              
                            
                            n
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            +
                            2
                          
                          ⁢
                          
                            ɛ
                            2
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    ω
                                    D
                                  
                                
                                )
                              
                            
                            n
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            ɛ
                            3
                          
                          ⁢
                          
                            
                              O
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    c
                                  
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      ω
                                      D
                                    
                                  
                                
                                )
                              
                            
                            n
                          
                        
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        64
        )
      
    
  




The accuracy required for the determination of A12 is approximately 0.5% in order to limit errors generated in the computation of cyclic error signals Sψj to ≲0.010 nm (3σ). Therefore the error terms ε−12, ε02, ε22, and ε22 on the right hand side of Equation (64) are negligible. The terms on the right hand side of Equation (64) of the form




  
    O
    ⁡
    
      (
      
        
          ω
          c
        
        
          ω
          D
        
      
      )
    
  
  n




are the sources of the largest Doppler shift frequency dependent errors and accordingly determine the specifications of n and the minimum ratio for ωD/ωc that can be used when the output of processor 18 is stored in table 40. For a second order Butterworth filter, i.e., n=2, and a minimum ratio for ωD/ωc=3.5, the Doppler shift frequency dependent error terms on the right hand side of Equation (64) will generate errors that correspond to ≲0.010 nm (3σ). The output A12 of the low pass filter of processor 18 is the output of processor 18. When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the output A12 of processor 18 is stored in table 40 under the control of signal 72.

Referring now to FIG. 1b, processor 60 generates the compensating error signal Sψ. With respect to generating signal Sψ, it is beneficial to rewrite the ε−1, ε0, ε2, and ε3 cyclic error terms of Sψ in terms of the highest order time dependent terms of Σ−1(t), {tilde over (Σ)}−1(t), SR, {tilde over (S)}R, Σ2(t), {tilde over (Σ)}2(t), Σ3(t), and {tilde over (Σ)}3(t), i.e., cos(φR−φ−ζ1+2ζR), sin(φR−φ−ζ1+2ζR), cos(φR+ζR), sin(φR+ζR), cos(φR+2φ+2ζ1−ζR), sin(φR+2φ+2ζ1−ζR), cos(φR+3φ+3ζ1−2ζR), and sin(φR+3φ+3ζ1−2ζR) as




  
    
      
        
          
            S
            ψ
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            [
            
              
                
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            ζ
                            1
                          
                          +
                          
                            ζ
                            
                              -
                              1
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              ζ
                              R
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          -
                          φ
                          -
                          
                            ζ
                            1
                          
                          +
                          
                            2
                            ⁢
                            
                              ζ
                              R
                            
                          
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    
                      -
                      
                        ɛ
                        
                          -
                          1
                        
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            ζ
                            1
                          
                          +
                          
                            ζ
                            
                              -
                              1
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              ζ
                              R
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          -
                          φ
                          -
                          
                            ζ
                            1
                          
                          +
                          
                            2
                            ⁢
                            
                              ζ
                              R
                            
                          
                        
                        )
                      
                    
                  
                
              
            
            ]
          
          +
          
            

          
          ⁢
          
            [
            
              
                
                  
                    
                      ɛ
                      0
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            ζ
                            0
                          
                          -
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    
                      -
                      
                        ɛ
                        0
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            ζ
                            0
                          
                          -
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                  
                
              
            
            ]
          
          +
          
            

          
          ⁢
          
            [
            
              
                
                  
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              2
                            
                            ⁢
                            
                              ζ
                              1
                            
                          
                          +
                          
                            ζ
                            2
                          
                          +
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            2
                            ⁢
                            φ
                          
                          +
                          
                            2
                            ⁢
                            
                              ζ
                              1
                            
                          
                          -
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    
                      -
                      
                        ɛ
                        2
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              2
                            
                            ⁢
                            
                              ζ
                              1
                            
                          
                          +
                          
                            ζ
                            2
                          
                          +
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            φ
                            R
                          
                          +
                          
                            2
                            ⁢
                            φ
                          
                          +
                          
                            2
                            ⁢
                            
                              ζ
                              1
                            
                          
                          -
                          
                            ζ
                            R
                          
                        
                        )
                      
                    
                  
                
              
            
            ]
          
          +
          
            

          
          ⁢
          
            
              [
              
                
                  
                    
                      
                        ɛ
                        3
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                3
                              
                              ⁢
                              
                                ζ
                                1
                              
                            
                            +
                            
                              ζ
                              3
                            
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                R
                              
                            
                          
                          )
                        
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              φ
                              R
                            
                            +
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              3
                              ⁢
                              
                                ζ
                                1
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                ζ
                                R
                              
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        -
                        
                          ɛ
                          3
                        
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                3
                              
                              ⁢
                              
                                ζ
                                1
                              
                            
                            +
                            
                              ζ
                              3
                            
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                R
                              
                            
                          
                          )
                        
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              φ
                              R
                            
                            +
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              3
                              ⁢
                              
                                ζ
                                1
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                ζ
                                R
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        65
        )
      
    
  




Equation (65) for Sψ is next written in the form





  
    
      
        
          
            S
            ψ
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            [
            
              
                
                  A
                  
                    -
                    1
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      -
                      φ
                      -
                      
                        ζ
                        1
                      
                      +
                      
                        2
                        ⁢
                        
                          ζ
                          R
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  B
                  
                    -
                    1
                  
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      -
                      φ
                      -
                      
                        ζ
                        1
                      
                      +
                      
                        2
                        ⁢
                        
                          ζ
                          R
                        
                      
                    
                    )
                  
                
              
            
            ]
          
          +
          
            
          
          ⁢
          
            [
            
              
            
            ⁢
            
              
                
                  A
                  0
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      +
                      
                        ζ
                        R
                      
                    
                    )
                  
                
              
              -
              
                
                  B
                  0
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      +
                      
                        ζ
                        R
                      
                    
                    )
                  
                
              
            
            ]
          
          +
          
            
          
          ⁢
          
            [
            
              
            
            ⁢
            
              
                
                  A
                  2
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      +
                      
                        2
                        ⁢
                        φ
                      
                      +
                      
                        2
                        ⁢
                        
                          ζ
                          1
                        
                      
                      -
                      
                        ζ
                        R
                      
                    
                    )
                  
                
              
              -
              
                
                  B
                  2
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      +
                      
                        2
                        ⁢
                        φ
                      
                      +
                      
                        2
                        ⁢
                        
                          ζ
                          1
                        
                      
                      -
                      
                        ζ
                        R
                      
                    
                    )
                  
                
              
            
            ]
          
          +
          
            
          
          ⁢
          
            [
            
              
            
            ⁢
            
              
                
                  A
                  3
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      +
                      
                        3
                        ⁢
                        φ
                      
                      +
                      
                        3
                        ⁢
                        
                          ζ
                          1
                        
                      
                      -
                      
                        2
                        ⁢
                        
                          ζ
                          R
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  B
                  3
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        R
                      
                      +
                      
                        3
                        ⁢
                        φ
                      
                      +
                      
                        3
                        ⁢
                        
                          ζ
                          1
                        
                      
                      -
                      
                        2
                        ⁢
                        
                          ζ
                          R
                        
                      
                    
                    )
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        66
        )
      
    
  




where A−1, B−1, A0, and B0 are given by equations (37), (38), (22), and (23), respectively, and

A2=ε2 cos(−2ζ1+ζ2+ζR),  (67)

B2=ε2 sin(−2ζ1+ζ2+ζR),  (68)

A3=ε3 cos(−3ζ1+ζ3+2ζR),  (69)

B3=ε3 sin(−3ζ1+ζ3+2ζR).  (70)

Compensation error signal Sψ is generated in processor 44 using Equation (66), the coefficients transmitted from table 40 as signal 42, and the signals Σ−1(t), {tilde over (Σ)}−1(t), SR, {tilde over (S)}R, Σ2(t), {tilde over (S)}2(t), Σ3(t), and {tilde over (Σ)}3(t) (which comprise the cyclic error basis functions) under control of control of signal 74 (from processor 70). Explicitly,




  
    
      
        
          
            S
            ψ
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            [
            
              
                (
                
                  
                    
                      A
                      R
                      2
                    
                    ⁢
                    
                      A
                      1
                    
                    ⁢
                    
                      A
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        (
                        
                          A
                          R
                          2
                        
                        )
                      
                      2
                    
                    ⁢
                    
                      A
                      1
                      2
                    
                  
                
                )
              
              ⁢
              
                
                  ∑
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    +
                    
                      (
                      
                        
                          
                            -
                            
                              A
                              R
                              2
                            
                          
                          ⁢
                          
                            A
                            1
                          
                          ⁢
                          
                            B
                            
                              -
                              1
                            
                          
                        
                        
                          
                            
                              (
                              
                                A
                                R
                                2
                              
                              )
                            
                            2
                          
                          ⁢
                          
                            A
                            1
                            2
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∑
                      ~
                    
                    
                      -
                      1
                    
                  
                
              
            
            ]
          
          +
          
            

          
          ⁢
          
            [
            
              
                
                  (
                  
                    
                      
                        A
                        R
                      
                      ⁢
                      
                        A
                        0
                      
                    
                    
                      A
                      R
                      2
                    
                  
                  )
                
                ⁢
                
                  S
                  R
                
              
              -
              
                
                  (
                  
                    
                      
                        A
                        R
                      
                      ⁢
                      
                        B
                        0
                      
                    
                    
                      A
                      R
                      2
                    
                  
                  )
                
                ⁢
                
                  
                    S
                    ~
                  
                  R
                
              
            
            ]
          
          +
          
            

          
          ⁢
          
            [
            
              
                (
                
                  
                    
                      A
                      R
                    
                    ⁢
                    
                      A
                      1
                      2
                    
                    ⁢
                    
                      A
                      2
                    
                  
                  
                    
                      
                        A
                        R
                        2
                      
                      ⁡
                      
                        (
                        
                          A
                          1
                          2
                        
                        )
                      
                    
                    2
                  
                
                )
              
              ⁢
              
                
                  ∑
                  2
                
                ⁢
                
                  
                    +
                    
                      (
                      
                        
                          
                            -
                            
                              A
                              R
                            
                          
                          ⁢
                          
                            A
                            1
                            2
                          
                          ⁢
                          
                            B
                            2
                          
                        
                        
                          
                            
                              A
                              R
                              2
                            
                            ⁡
                            
                              (
                              
                                A
                                1
                                2
                              
                              )
                            
                          
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∑
                      ~
                    
                    2
                  
                
              
            
            ]
          
          +
          
            

          
          ⁢
          
            [
            
              
                (
                
                  
                    
                      A
                      R
                      2
                    
                    ⁢
                    
                      A
                      1
                      3
                    
                    ⁢
                    
                      A
                      3
                    
                  
                  
                    
                      
                        (
                        
                          A
                          R
                          2
                        
                        )
                      
                      2
                    
                    ⁢
                    
                      
                        (
                        
                          A
                          1
                          2
                        
                        )
                      
                      3
                    
                  
                
                )
              
              ⁢
              
                
                  ∑
                  3
                
                ⁢
                
                  
                    +
                    
                      (
                      
                        
                          
                            -
                            
                              A
                              R
                              2
                            
                          
                          ⁢
                          
                            A
                            1
                            3
                          
                          ⁢
                          
                            B
                            3
                          
                        
                        
                          
                            
                              (
                              
                                A
                                R
                                2
                              
                              )
                            
                            2
                          
                          ⁢
                          
                            
                              (
                              
                                A
                                1
                                2
                              
                              )
                            
                            3
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∑
                      ~
                    
                    3
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        71
        )
      
    
  




In other words, the compensation error signal is generated form a superposition of the error basis functions weighted by the cyclic error coefficients.

The compensating signal Sψ is subtracted from signal S in processor 46 under control of signal 76 (from processor 70) to generate compensated signal S−Sψ. Control signal 76 determines when signal S is to be compensated. The phase φ=4kL is then extracted from the compensated signal with a subsequent processor (not shown) to, for example, provide a more accurate measurement of the distance L.
In the presently preferred embodiment, the error compensation signal Sψ(t) is subtracted from prior values of the signals S(t). For example, feedforward signal S′(t), (as described in Equation (14)) may replace signal S(t). The delay, m, of the feedforward signal is chosen to be equal to the processing delay in calculating the error basis functions and subsequently Sψ(t) from signal S(t). In this manner, S′(t) and Sψ(t) represent the same time of input signal S(t).
In further embodiments, the cyclic error coefficients may be stored and updated at a lower data rate than that used to generate the cyclic error basis functions from the feedforward values. In such cases, the stored values for the cyclic error coefficients may used for the calculation of the cyclic error basis functions as necessary. Of course, in yet further embodiments, the coefficients and/or the error basis functions can be calculated in real time, without the use of the feedforward signals.
An important assumption in the preferred embodiment of the cyclic error compensation described above is that the Doppler shift frequency dφ(t)/dt is small enough relative to the heterodyne frequency ωR, that the quadrature signal {tilde over (S)}(t) can be approximated (in analogy to Equation (10) by the expression:




  
    
      
        
          
            S
            ~
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              (
              
                cot
                ⁢
                
                  
                
                ⁢
                
                  ω
                  R
                
                ⁢
                τ
              
              )
            
            ⁢
            
              S
              ⁡
              
                (
                
                  t
                  -
                  
                    2
                    ⁢
                    τ
                  
                
                )
              
            
          
          -
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  ω
                  R
                
                ⁢
                τ
              
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  ω
                  R
                
                ⁢
                τ
              
            
            ⁢
            
              S
              ⁡
              
                (
                
                  t
                  -
                  τ
                
                )
              
            
          
        
      
    
    
      
        (
        72
        )
      
    
  




or the simpler expression given by Equation (12). In further embodiments, the quadrature signal {tilde over (S)}(t) may be more accurately calculated according to:





  
    
      
        
          
            S
            ~
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              (
              
                cot
                ⁢
                
                  
                
                ⁢
                
                  ω
                  M
                
                ⁢
                τ
              
              )
            
            ⁢
            
              S
              ⁡
              
                (
                
                  t
                  -
                  
                    2
                    ⁢
                    τ
                  
                
                )
              
            
          
          -
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  ω
                  M
                
                ⁢
                τ
              
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  ω
                  M
                
                ⁢
                τ
              
            
            ⁢
            
              S
              ⁡
              
                (
                
                  t
                  -
                  τ
                
                )
              
            
          
        
      
    
    
      
        (
        73
        )
      
    
  




where ωM is the instantaneous rate of change of the phase of the main interference signal S(t), which can be determined with sufficient accuracy by assuming that the cyclic error contributions to S(t) are negligle.

Also, in further embodiments, the cyclic error compensation technique may be used for cyclic error terms different from those explicitly described in Equations (3)-(6). In particular, using algebraic combinations of the signals S(t), {tilde over (S)}(t), SR(t), and {tilde over (S)}R(t), a processing unit can generate cyclic error basis functions, which are sine and cosine functions that have the same time-varying arguments as the cyclic error terms that need to be compensated, and then use the cyclic error basis functions to project out respective cyclic error coefficients from S(t) and {tilde over (S)}(t) by low-pas filtering (e.g., averaging).
For example, to determine the coefficients for a half-cycle cyclic error of the form:




  
    
      
        
          S
          
            ψ
            ⁡
            
              (
              
                1
                /
                2
              
              )
            
          
        
        =
        
          
            ɛ
            
              1
              /
              2
            
          
          ⁢
          
            cos
            ⁡
            
              (
              
                
                  φ
                  R
                
                +
                
                  φ
                  2
                
                +
                
                  ζ
                  
                    1
                    /
                    2
                  
                
              
              )
            
          
        
      
    
    
      
        (
        74
        )
      
    
  




one can calculate cyclic error basis functions for the half-cycle cyclic error as follows. First calculate signals Σ7′(t) and {tilde over (Σ)}7′(t) as:





  
    
      
        
          
            
              ∑
              7
              ′
            
            ⁢
            
              (
              t
              )
            
          
          =
          
            
              
                
                  ∑
                  0
                
                ⁢
                
                  -
                  
                    (
                    
                      
                        
                          A
                          R
                        
                        ⁢
                        
                          A
                          0
                        
                      
                      -
                      
                        
                          A
                          R
                        
                        ⁢
                        
                          A
                          1
                        
                      
                    
                    )
                  
                
              
              
                2
                ⁢
                
                  A
                  R
                
                ⁢
                
                  A
                  1
                
              
            
          
        
        ,
      
    
    
      
        (
        75
        )
      
    
  
  
    
      
        
          
            
              ∑
              ~
            
            7
            ′
          
          ⁢
          
            (
            t
            )
          
        
        =
        
          
            
              
                
                  (
                  
                    
                      
                        A
                        R
                      
                      ⁢
                      
                        A
                        0
                      
                    
                    +
                    
                      
                        A
                        R
                      
                      ⁢
                      
                        A
                        1
                      
                    
                  
                  )
                
                -
                
                  ∑
                  0
                
              
              
                2
                ⁢
                
                  A
                  R
                
                ⁢
                
                  A
                  1
                
              
            
          
          .
        
      
    
    
      
        (
        76
        )
      
    
  




Notably, the leading term of Σ7′(t) is |cos(φ/2+ζ1/2−ζR/2)|, and the leading term of {tilde over (Σ)}7′(t) is |sin(φ/2+ζ1/2−ζR/2)|. Zero phase crossings in Σ7′(t) and {tilde over (Σ)}7′(t) are then measured to remove the absolute value operation and define Σ7(t) and {tilde over (Σ)}7(t), which have leading terms cos(φ/2+ζ1/2−ζR/2) and sin(φ/2+ζ1/2−ζR/2), respectively.

Half-cycle error basis functions Σ1/2(t) and {tilde over (Σ)}1/2(t) are then generated as:

Σ1/2(t)=Σ7(t)SR(t)−{tilde over (Σ)}7(t){tilde over (S)}R(t),  (77)

{tilde over (Σ)}1/2(t)={tilde over (Σ)}7(t)SR(t)+Σ7(t){tilde over (S)}R(t),  (78)

where Σ1/2(t) and {tilde over (Σ)}1/2(t) have as their leading terms sine and cosine functions, respectively, whose time-dependent argument is the same as that of Sψ(1/2). To determine the coefficients for Sψ(1/2), one projects half-cycle error basis functions onto S(t) and {tilde over (S)}(t) to move the half-cycle cyclic error component to zero frequency, for example:

Σ8(t)=S(t)Σ1/2(t)+{tilde over (S)}(t){tilde over (Σ)}1/2(t),  (79)

{tilde over (Σ)}8(t)={tilde over (S)}(t)Σ1/2(t)−S(t){tilde over (Σ)}1/2(t).  (80).

Low-pass filtering (e.g., with the Butterworth filter) of Σ8(t) and {tilde over (Σ)}8(t) then yield the half-cycle error coefficients in analogy to the extraction of the previously described cyclic error coefficients. In particular, the leading terms following the low-pass filtering are ARA1/2 cos(ζ1/2−ζ1/2−ζR/2) and ARA1/2 sin(ζ1/2−ζ1/2−ζR/2), respectively.

We now describe the CMS approach for the CEC, in which a cyclic error correction signal Dψ(t) is subtracted from a corresponding DFT signal D(t) of the electrical interference signal S(t) to produce a compensated DFT signal. FIG. 1c shows a simplified schematic diagram of a measurement using the CMS approach. The optical interference signal 111 is received and amplified by photoelectric receiver 112. The resulting electrical interference signal 113 is filtered by lowpass filter (LPF) 114 producing filtered signal 115. The LPF 114 is designed to prevent harmonics of the interference signal 111 from being aliased into the frequency range of interest. Filtered signal 115 is digitized by ADC 116, to produce digitized measurement signal 117. A typical ADC for a high performance displacement measuring interferometer may have 12 bits of resolution at sampling rates of 120 MHz. The digitized measurement signal 117 is processed by phase meter 120 (described below) to produce outputs magnitude 125 and phase 127 which represent the digitized measurement signal 117 as a transform. The magnitude output 125 is used for status and diagnostic purposes. The phase output 127 is used by position calculator 130 which is fully described in published U.S. application Ser. No. 10/211,435 (publication number US 2003/0025914 A1), incorporated herein by reference. Position calculator 130 calculates measured position 131 and estimated speed 133. The measured position 131 is filtered by digital filter 136, which is fully described in U.S. Pat. No. 5,767,972, incorporated herein by reference, to generate filtered position signal 137. Filtered position signal 137 represents the desired measurement of the distance L.
Phase meter 120 includes a Discrete Fourier Transform (DFT) processor 122, a cyclic error compensation (CEC) calculator 140, and a Coordinate Rotation by Digital Computer (CORDIC) converter 124. Signals 123, 143, 145, and 147 are complex values, which consist of both a real component and an imaginary component, as a+jb, where a is the real component, b is the imaginary component, and j is √{square root over (−1)}. (The symbol i is sometimes used in the literature instead of j.) Other representations of complex or quadrature values can be used, and may be expressed using other symbols such as, for example, I and Q, or X and Y, or A and Ã. Complex values may be converted from rectangular (real and imaginary) representation to polar (magnitude and phase angle) representation. The numeric representation of the digital signals may be integer, fractional, or floating point.
The DFT processor 122 converts a series of consecutive samples of digitized measurement signal 117 into a complex measurement signal 123 representing a transform of the digitized measurement signal 117 at a selected center frequency of DFT processor 122. The center frequency is determined by control circuitry (not shown) and the estimated speed 133 is determined by position calculator 130.
An exemplary DFT processor 122 is a 72-point windowed DFT performed at a 10 MHz rate. Since the DFT calculation is being updated in time, the complex measurement signal 123 is expressed as a function of time D(t). This 10 MHz update rate provides 83% overlap of samples of the digitized measurement signal 117 between one DFT calculation and the next for an ADC sampling rate of fS=120 MHz. A typical window function is the Blackman window, which reduces errors due to the discontinuities at the beginning and end of the series of digitized measurement signal samples used for the DFT.
The CEC calculator 140 calculates and compensates for certain of the cyclic errors. CEC error estimator 144 (described in more detail below with reference to FIG. 1d) calculates complex error compensation signal 145. Optional delay 142, and other delays (not shown) in CEC calculator 140 may be used to match the processing delay of the various calculations. Adder 146 combines delayed complex measurement signal 143 with complex error compensation signal 145 to produce compensated complex measurement signal 147, in which the certain cyclic error signals are substantially reduced.
CORDIC converter 124 converts the compensated complex measurement signal 147 to magnitude 125 and phase 127.
Referring to FIG. 1d, the CEC error estimator 144 includes two processing units. One processing unit 148 determines error basis functions and complex factors relating to the amplitudes and offset phases of the certain cyclic errors that need be compensated. A second processing unit 204 generates complex error compensation signal Dψ(t) 145 using the error basis functions and complex factors relating to the amplitudes and offset phases determined by first processing unit 148.
The first processing unit 148 for one embodiment is shown schematically in FIG. 1d and the second processing unit 204 of this embodiment is shown schematically in FIG. 1e. These processing units are incorporated into the architecture shown in FIG. 1c that may also include any of a variety of other techniques such as a glitch filter (as described in published U.S. application Ser. No. 10/211,435), dynamic data age adjustment (as described in U.S. Pat. No. 6,597,459, incorporated herein by reference), and digital filtering as described in U.S. Pat. No. 5,767,972.
Note that, as in the DMS approach, the set of cyclic error terms whose phases are independent of φR has been omitted from Equation (2) because, in this CMS approach, they are eliminated by LPF (Lowpass filter) 114 and the bandpass filtering inherent in DFT processor 122.
Note that there are conditions where the cyclic error terms ε2 or ε3 may be aliased and appear at or near the frequency of the measurement signal. Let fR be the reference frequency (fR=ωR/2π), fD be the Doppler shift frequency (fD=ωD/2π), and fM be the resulting measurement frequency. When fD=−(2⁄3)fR,




  
    
      
        
          
            f
            M
          
          =
          
            
              
                f
                R
              
              +
              
                f
                D
              
            
            =
            
              
                
                  f
                  R
                
                -
                
                  
                    2
                    3
                  
                  ⁢
                  
                    f
                    R
                  
                
              
              =
              
                
                  1
                  3
                
                ⁢
                
                  f
                  R
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            f
            
              ɛ
              2
            
          
          =
          
            
              
                f
                R
              
              +
              
                2
                ⁢
                
                  f
                  D
                
              
            
            =
            
              
                
                  f
                  R
                
                -
                
                  
                    4
                    3
                  
                  ⁢
                  
                    f
                    R
                  
                
              
              =
              
                
                  -
                  
                    1
                    3
                  
                
                ⁢
                
                  
                    f
                    R
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        81
        )
      
    
  




Also, when fD=−(1⁄2)fR,





  
    
      
        
          
            f
            M
          
          =
          
            
              
                f
                R
              
              +
              
                f
                D
              
            
            =
            
              
                
                  f
                  R
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    f
                    R
                  
                
              
              =
              
                
                  1
                  2
                
                ⁢
                
                  f
                  R
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            f
            
              ɛ
              3
            
          
          =
          
            
              
                f
                R
              
              +
              
                3
                ⁢
                
                  f
                  D
                
              
            
            =
            
              
                
                  f
                  R
                
                -
                
                  
                    3
                    2
                  
                  ⁢
                  
                    f
                    R
                  
                
              
              =
              
                
                  -
                  
                    1
                    2
                  
                
                ⁢
                
                  
                    f
                    R
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        82
        )
      
    
  




In both cases, the negative frequency is indistinguishable from the corresponding positive frequency, and measurement accuracy is affected.

Complex factors relating to amplitudes εp and offset phases ζp of the four cyclic error terms and time dependent factors of the cyclic error terms are generated using processed values D(t) 123 from DFT processor 122. The factors are stored in registers 162, 176, 186, and 192 for subsequent use in generation of the cyclic error correction signal Dψ(t) 145. The time dependent factors of the cyclic error terms are obtained by application of digital transforms based on trigonometric identities and properties of complex signals.
DFT processor 122 calculates the complex DFT of the digitized measurement signal 117 as:




  
    
      
        
          
            
              D
              q
            
            ⁡
            
              (
              
                t
                1
              
              )
            
          
          =
          
            
              ∑
              
                n
                =
                0
              
              
                N
                -
                1
              
            
            ⁢
            
              
                W
                n
              
              ⁢
              
                S
                ⁡
                
                  (
                  
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      τ
                    
                    +
                    
                      t
                      1
                    
                  
                  )
                
              
              ⁢
              exp
              ⁢
              
                {
                
                  -
                  
                    j
                    ⁡
                    
                      (
                      
                        2
                        ⁢
                        
                          π
                          ⁡
                          
                            (
                            
                              n
                              -
                              
                                
                                  N
                                  -
                                  1
                                
                                2
                              
                            
                            )
                          
                        
                        ⁢
                        
                          q
                          /
                          N
                        
                      
                      )
                    
                  
                
                }
              
            
          
        
        ,
      
    
    
      
        (
        83
        )
      
    
  




for 0<q<N−1, where τ=1/fS, Wn is a window function centered at





  
    n
    =
    
      
        N
        -
        1
      
      2
    
  
  ,




and t1 is the time at which the DFT calculation is updated. When q is selected by control circuitry (not shown) as an integer approximately equal to NfM/fS, to correspond to the center frequency of the primary component of the digitized measurement signal. A typical value for N is 72, and a typical window function Wn is the Blackman window function.

The equation for the DFT is arranged so that the center of the sequence of the N consecutive samples of the digitized measurement signal 117 (n=35.5 in this case) corresponds to zero phase and the center of the window function. Therefore, since the window function and DFT coefficients (the the complex exponential or equivalently the “cos+j sin” factors shown below) are symmetrical about the center, and window weighting is zero at the ends, a change in the value of q, has a reduced effect on the phase of the result.
The transform signal Dq(t1) is updated at a rate fU that is lower than the rate fS at which the signal S(t) is sampled. In this example, t1=lΔt1 (where l is an integer and Δt1≡1/fU is the update interval) and fU=10 MHz.
Alternatively, the DFT equation can be “folded” to reduce the number of multiplication operations that are performed and calculated as:




  
    
      
        
          
            D
            q
          
          ⁡
          
            (
            
              t
              1
            
            )
          
        
        =
        
          
            ∑
            
              n
              =
              0
            
            
              
                N
                2
              
              -
              1
            
          
          ⁢
          
            
              
                W
                n
              
              ⁡
              
                (
                
                  
                    
                      x
                      n
                    
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                  +
                  
                    
                      x
                      
                        N
                        -
                        1
                        -
                        n
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                
                )
              
            
            ⁢
            
              (
              
                
                  cos
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                          n
                          +
                          0.5
                        
                        N
                      
                      ⁢
                      
                        q
                        8
                      
                    
                    )
                  
                
                +
                
                  jsin
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                          n
                          +
                          0.5
                        
                        N
                      
                      ⁢
                      
                        q
                        8
                      
                    
                    )
                  
                
              
              )
            
          
        
      
    
    
      
        (
        84
        )
      
    
  




for 0<q<8N−1, where xn(t1) for n=0, . . . , N−1 are N consecutive samples (i.e., t=nτ) of S(t+t1), and Wn is centered at n=0. In this example, the larger range of q yields a more finely spaced resolution of 1⁄8 bin to reduce amplitude variations (or “picket fence” effect) as the frequency changes from one bin to the next.

The DFT function is equivalent to a mixing and a filtering operation. The mixing is a result of multiplying the input data by the complex exponential or equivalently the “cos+j sin” factor. The filtering is a result of the summation and the window function Wn.
Since the window function Wn makes terms zero outside the range of summation, the DFT expression can be written as a sum over all n. In a simplified expression for an “unfolded” DFT (as in equation (83)), the digitized measurement signal S(nτ+t1) 177 has been approximated by its primary term A1 cos(φR(nτ+t1)+φ(nτ+t1)+ζ1), where explicit sampling (t=nτ) has been included:




  
    
      
        
          D
          ⁡
          
            (
            
              t
              1
            
            )
          
        
        =
        
          
            A
            1
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
                W
                n
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        φ
                        R
                      
                      ⁡
                      
                        (
                        
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            τ
                          
                          +
                          
                            t
                            1
                          
                        
                        )
                      
                    
                    +
                    
                      φ
                      ⁡
                      
                        (
                        
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            τ
                          
                          +
                          
                            t
                            1
                          
                        
                        )
                      
                    
                    +
                    
                      ζ
                      1
                    
                  
                  )
                
              
              ⁢
              
                (
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          C
                        
                        ⁡
                        
                          (
                          
                            n
                            ,
                            q
                          
                          )
                        
                      
                      )
                    
                  
                  +
                  
                    jsin
                    ⁡
                    
                      (
                      
                        
                          φ
                          C
                        
                        ⁡
                        
                          (
                          
                            n
                            ,
                            q
                          
                          )
                        
                      
                      )
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        85
        )
      
    
  




where φR(nτ+t1)=ωRnτ+ωRt1 and φ(nτ+t1)=ωDnτ+ωDt1+φ0 is the sampled version of the Doppler phase shift φ(t)=ωDt+φ0, and





  
    
      φ
      C
    
    ⁡
    
      (
      
        n
        ,
        q
      
      )
    
  
  =
  
    2
    ⁢
    π
    ⁢
    
      nq
      N
    
  




is the phase of the DFT with q≈NfM/fS, yielding φC(n,q)≈2πfMnτ=ωMnτ. (The q in Dq(t1) has been suppressed so that D(t1)≡Dq(t1).) Using trigonometric identities, equation (85) can be expanded to (with time dependent arguments temporarily suppressed):





  
    
      
        
          D
          ⁡
          
            (
            
              t
              1
            
            )
          
        
        =
        
          
            A
            1
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
                W
                n
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  φ
                                  +
                                  
                                    ζ
                                    1
                                  
                                  +
                                  
                                    φ
                                    C
                                  
                                
                                )
                              
                            
                            +
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  φ
                                  +
                                  
                                    ζ
                                    1
                                  
                                  -
                                  
                                    φ
                                    C
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          +
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  φ
                                  +
                                  
                                    ζ
                                    1
                                  
                                  +
                                  
                                    φ
                                    C
                                  
                                
                                )
                              
                            
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    R
                                  
                                  +
                                  φ
                                  +
                                  
                                    ζ
                                    1
                                  
                                  -
                                  
                                    φ
                                    C
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        86
        )
      
    
  




The terms containing φR+φ+ζ1+φC are high frequency sinusoids varying with n that are filtered out in the summation including the window function Wn that covers many cycles. The constant 1⁄2 may be dropped for convenience. The terms containing φR+φ+ζ1−φC that are slowly varying in the summation over the window remain:




  
    
      
        
          D
          ⁡
          
            (
            
              t
              1
            
            )
          
        
        ≈
        
          
            A
            1
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
                W
                n
              
              ⁡
              
                [
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                ω
                                R
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              τ
                            
                            +
                            
                              
                                ω
                                R
                              
                              ⁢
                              
                                t
                                1
                              
                            
                            +
                            
                              
                                ω
                                D
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              τ
                            
                            +
                            
                              
                                ω
                                D
                              
                              ⁢
                              
                                t
                                1
                              
                            
                            +
                            
                              φ
                              0
                            
                            +
                            
                              ζ
                              1
                            
                            -
                            
                              
                                ω
                                M
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              τ
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  ω
                                  R
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                τ
                              
                              +
                              
                                
                                  ω
                                  R
                                
                                ⁢
                                
                                  t
                                  1
                                
                              
                              +
                              
                                
                                  ω
                                  D
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                τ
                              
                              +
                              
                                
                                  ω
                                  D
                                
                                ⁢
                                
                                  t
                                  1
                                
                              
                              +
                              
                                φ
                                0
                              
                              +
                              
                                ζ
                                1
                              
                              -
                              
                                
                                  ω
                                  M
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                τ
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        87
        )
      
    
  




Since ωM≈ωR+ωD the terms of the sinusoids including n drop out.

When the update interval Δt1≡1/fU is given by Δt1=m/fR (where m is an integer) the value of t1 is a multiple of 1/fR. Under this condition, for low stage velocities (i.e., ωD<<ωR)ωRt1≈m2π and DFT calculation becomes:




  
    
      
        
          
            
              
                D
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
              ≈
              
              ⁢
              
                
                  A
                  1
                
                ⁢
                
                  
                    ∑
                    n
                  
                  ⁢
                  
                    
                      W
                      n
                    
                    ⁡
                    
                      [
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  ω
                                  D
                                
                                ⁢
                                
                                  t
                                  1
                                
                              
                              +
                              
                                φ
                                0
                              
                              +
                              
                                ζ
                                1
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                
                                  ω
                                  D
                                
                                ⁢
                                
                                  t
                                  1
                                
                              
                              +
                              
                                φ
                                0
                              
                              +
                              
                                ζ
                                1
                              
                            
                            )
                          
                        
                      
                      ]
                    
                  
                
              
            
          
        
        
          
            
              ≈
              
              ⁢
              
                
                  A
                  1
                
                ⁡
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              D
                            
                            ⁢
                            
                              t
                              1
                            
                          
                          +
                          
                            φ
                            0
                          
                          +
                          
                            ζ
                            1
                          
                        
                        )
                      
                    
                    -
                    
                      jsin
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              D
                            
                            ⁢
                            
                              t
                              1
                            
                          
                          +
                          
                            φ
                            0
                          
                          +
                          
                            ζ
                            1
                          
                        
                        )
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        88
        )
      
    
  




The DFT result is a complex value which can represent either a positive or a negative frequency. Equation (88) could be written as a sum, rather than a difference, with appropriate changes in sign in other equations. This choice of sign is not important, provided that the choices result in a reduction in cyclic error.

Equation (88) can be combined with equations (1) to (6). This results in:

D(t1)=A1(cos(φ(t1)+ζ1)−j sin(φ(t1)+ζ1))+Dψ(t1)  (89)

where

Dψ(t1)=Dψ−1(t1)+Dψ0+Dψ2(t1)+Dψ3(t1);  (90)

Dψ−1(t1)=ε−1(cos(−φ(t1)+ζ−1)−j sin(−φ(t1)+ζ−1)),  (91)

Dψ0=ε0(cos(ζ0)−j sin(ζ0)),  (92)

Dψ2(t1)=ε2(cos(2φ(t1)+ζ2)−j sin(2φ(t1)+ζ2)),  (93)

Dψ3(t1)=ε3(cos(3φ(t1)+ζ3)−j sin(3φ(t1)+ζ3));  (94)

where φ(t1) is the value of the Doppler phase shift φ(t1)=ωDt1+φ0 that is updated with t1. For convenience, in the equations (98)-(116) below, t1 is written simply as t and φ(t1) is written simply as φ. Also, complex measurement signal D(t) 123 and complex error compensation signal Dψ(t) 145 are assumed to be updated at the rate fU such that D(t)≡D(t1) and Dψ(t)−Dψ(t1).

FIG. 1d shows a schematic diagram of CEC error estimator 144. In the following discussions the product of two complex signals frequently occurs. The following identities are used:

(cos(A)+j sin(A))·(cos(B)+j sin(B))=cos(A+B)+j sin(A+B),  (95)

(cos(A)+j sin(A))·(cos(B)−j sin(B))=cos(A−B)+j sin(A−B),  (96)

(cos(A)+j sin(A))*=cos(A)−j sin(A)=cos(−A)+j sin(−A)  (97)

The next step is the processing of signals for information about the cyclic error terms. In a first “lowpass filtering approach,” the signal D(t) is sent to LPF (Lowpass Filter) 160, for example an IIR (Infinite Impulse Response) Butterworth digital filter, an FIR (Finite Impulse Response), or CIC (Cascaded Integrator Comb) digital filter as described by Hogenauer (An Economical Class of Digital Filters for Decimation and Interpolation; E. B. Hogenauer; IEEE Transactions on Acoustics, Speech, and Signal Processing; Vol ASSP-29, No 2, April 1981, p 155-162, incorporated herein by reference). The CIC filter has the advantages in this implementation of simple design (using only integer addition) and decimation by large ratios. The implementation of an LPF for a complex signal uses two identical real LPF functions, one is used for the real component, and one is used for the imaginary component. The use of digital functions ensures precise matching of amplitude and phase response of the two filters.
For an LPF Tn(x) of order n, the complex output C0 is approximately:




  
    
      
        
          
            
              
                C
                0
              
              =
              
              ⁢
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    D
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ɛ
                    0
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            ζ
                            0
                          
                          )
                        
                      
                      -
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              ζ
                              0
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    
                      A
                      1
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              ω
                              D
                            
                          
                          )
                        
                      
                      n
                    
                  
                  +
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              ω
                              D
                            
                          
                          )
                        
                      
                      n
                    
                  
                  +
                  
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              2
                              ⁢
                              
                                ω
                                D
                              
                            
                          
                          )
                        
                      
                      n
                    
                  
                  +
                  
                    
                      ɛ
                      3
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              3
                              ⁢
                              
                                ω
                                D
                              
                            
                          
                          )
                        
                      
                      n
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        98
        )
      
    
  




where O(x) denotes a term of the order of x, ωc is the −3 dB angular cutoff frequency, and ωD=dφ/dt.

The term on the right hand sides of Equation (98) with factor A1 is the source of the largest error and accordingly determines the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of LPF 160 are stored in register 162. For a fourth order LPF, i.e., n=4, and a minimum ratio for ωD/ωc=7, the error terms on the right hand side of Equation (98) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the output C0 of the LPF 160 is stored in register 162 as C0R under the control of signal 161. This stage speed requirement reduces the possibility that sidebands of the primary Doppler signal caused by actual variations in the stage position or motion will be interpreted as cyclic errors.
An advantage of this approach is that ωD can vary by factors such as 2 or more during the period when output values of C0 are stored in register 162.
In a second approach to processing signals for information about the cyclic error terms, the CEC error estimator 144 stores values in the registers based on analysis of collective properties of a distribution of values of D(t). An advantage of this approach is that the stage can be nearly stationary, or moving at a speed such that the corresponding Doppler shift frequency ωD/2π is less than 10 times greater than the bandwidth of the stage servo control system. In this case, the measured motion typically has negligible sidebands that could be interpreted as cyclic errors. This “distribution analysis approach” is described in more detail below.
The values for C0R and D(t) are transmitted to processor 164 for the generation of complex signal Σ1(t) where




  
    
      
        
          
            
              
                
                  Σ
                  1
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  D
                  ⁡
                  
                    (
                    t
                    )
                  
                
                -
                
                  C
                  
                    0
                    ⁢
                    
                      
                    
                    ⁢
                    R
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            φ
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            φ
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              -
                              φ
                            
                            +
                            
                              ζ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              -
                              φ
                            
                            +
                            
                              ζ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              2
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              3
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              3
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        99
        )
      
    
  




Signal Σ1 is sent to processor 168, which calculates Σ−1 as the complex conjugate of Σ1.




  
    
      
        
          
            
              
                
                  Σ
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    Σ
                    1
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                *
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            φ
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            φ
                            -
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            φ
                            +
                            
                              ζ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            φ
                            -
                            
                              ζ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                2
                              
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                2
                              
                              ⁢
                              φ
                            
                            -
                            
                              ζ
                              2
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ɛ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                3
                              
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              3
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                3
                              
                              ⁢
                              φ
                            
                            -
                            
                              ζ
                              3
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        100
        )
      
    
  




Signal Σ1 is sent to processor 180, which calculates Σ2.




  
    
      
        
          
            
              
                Σ
                2
              
              =
              
              ⁢
              
                
                  (
                  
                    Σ
                    1
                  
                  )
                
                2
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            φ
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  O
                  ⁡
                  
                    (
                    
                      
                        ɛ
                        i
                      
                      ,
                      
                        ɛ
                        j
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        101
        )
      
    
  




Signal Σ2 is divided by two and sent to LPF 190, as described earlier for LPF 160.
For an LPF Tn(x) of order n, the complex output C4 is approximately:




  
    
      
        
          
            
              
                C
                4
              
              =
              
              ⁢
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      Σ
                      2
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  ζ
                                  1
                                
                                +
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      A
                      1
                      2
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              ω
                              D
                            
                          
                          )
                        
                      
                      n
                    
                  
                  +
                  
                    
                      A
                      1
                    
                    ⁢
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              3
                              ⁢
                              
                                ω
                                D
                              
                            
                          
                          )
                        
                      
                      n
                    
                  
                  +
                  
                    
                      A
                      1
                    
                    ⁢
                    
                      ɛ
                      3
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              2
                              ⁢
                              
                                ω
                                D
                              
                            
                          
                          )
                        
                      
                      n
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        102
        )
      
    
  




where O(x) denotes a term of the order of x, ωc is the −3 dB angular cutoff frequency, and ωD=dφ/dt.

The term on the right hand sides of Equation (102) with factors A12 is the source of the largest error and accordingly determines the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of processor 190 are stored in register 192. For a fourth order LPF, i.e., n=4, and a minimum ratio for ωD/ωc=3.5, the error terms on the right hand side of Equation (102) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the output C4 of LPF 190 is stored in register 192 as C4R under the control of signal 161.
Signals Σ1, Σ2, and C4R are sent to processor 200 which calculates Σ3. Signal Σ3A is calculated by combining signals C4R and Σ1 using multiplier 194:




  
    
      
        
          
            
              
                Σ
                
                  3
                  ⁢
                  A
                
              
              =
              
              ⁢
              
                
                  C
                  
                    4
                    ⁢
                    R
                  
                
                ⁢
                
                  Σ
                  1
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                              +
                              φ
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                              +
                              φ
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    
                      ɛ
                      
                        -
                        1
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              φ
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              φ
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                
                                  ζ
                                  
                                    
                                      -
                                      1
                                    
                                    +
                                  
                                
                                ⁢
                                2
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                              +
                              
                                2
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                
                                  ζ
                                  
                                    
                                      -
                                      1
                                    
                                    +
                                  
                                
                                ⁢
                                3
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                              +
                              
                                3
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  1
                
                ⁢
                
                  ɛ
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ,
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        103
        )
      
    
  




Signal Σ3B is calculated by combining signals Σ1 and Σ2 using multiplier 202:




  
    
      
        
          
            
              
                ∑
                
                  3
                  ⁢
                  B
                
              
              ⁢
              
                
              
              ⁢
              
                =
                
                ⁢
                
                  
                    ∑
                    1
                  
                  ⁢
                  
                    ∑
                    2
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              3
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                      -
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                φ
                              
                              +
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              φ
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                            
                            )
                          
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                φ
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  2
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                5
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  5
                                  ⁢
                                  φ
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                +
                                
                                  ζ
                                  3
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  O
                  ⁡
                  
                    (
                    
                      
                        ɛ
                        i
                      
                      ,
                      
                        ɛ
                        j
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        104
        )
      
    
  




Signal Σ3 is calculated by combining signals Σ3A and Σ3B using constant multiplier 196 and subtractor 198:




  
    
      
        
          
            
              
                Σ
                3
              
              =
              
              ⁢
              
                
                  Σ
                  
                    3
                    ⁢
                    B
                  
                
                -
                
                  3
                  ⁢
                  
                    Σ
                    
                      3
                      ⁢
                      A
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    
                      3
                      ⁢
                      
                        
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              3
                              ⁢
                              φ
                            
                            -
                            
                              3
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              3
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                5
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                5
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                3
                ⁢
                
                  A
                  1
                
                ⁢
                
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ,
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        105
        )
      
    
  




Signal Σ5 is calculated by combining signals Σ−1 and Σ2 using multiplier 182:




  
    
      
        
          
            
              
                Σ
                5
              
              =
              
              ⁢
              
                
                  Σ
                  
                    -
                    1
                  
                
                ⁢
                
                  Σ
                  2
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                -
                                
                                  ζ
                                  2
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              +
                              
                                
                                  ɛ
                                  
                                    -
                                    1
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            2
                                            ⁢
                                            
                                              (
                                              
                                                
                                                  cos
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      
                                                        -
                                                        φ
                                                      
                                                      +
                                                      
                                                        ζ
                                                        
                                                          -
                                                          1
                                                        
                                                      
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  jsin
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      
                                                        -
                                                        φ
                                                      
                                                      +
                                                      
                                                        ζ
                                                        
                                                          -
                                                          1
                                                        
                                                      
                                                    
                                                    )
                                                  
                                                
                                              
                                              )
                                            
                                          
                                          +
                                        
                                      
                                    
                                    
                                      
                                        
                                          (
                                          
                                            
                                              cos
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                (
                                                
                                                  
                                                    3
                                                    ⁢
                                                    φ
                                                  
                                                  +
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      ζ
                                                      1
                                                    
                                                  
                                                  -
                                                  
                                                    ζ
                                                    
                                                      -
                                                      1
                                                    
                                                  
                                                
                                                )
                                              
                                            
                                            -
                                            
                                              jsin
                                              ⁡
                                              
                                                (
                                                
                                                  
                                                    3
                                                    ⁢
                                                    φ
                                                  
                                                  +
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      ζ
                                                      1
                                                    
                                                  
                                                  -
                                                  
                                                    ζ
                                                    
                                                      -
                                                      1
                                                    
                                                  
                                                
                                                )
                                              
                                            
                                          
                                          )
                                        
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                A
                                1
                              
                              ⁡
                              
                                (
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        φ
                                        +
                                        
                                          ζ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    jsin
                                    ⁡
                                    
                                      (
                                      
                                        φ
                                        +
                                        
                                          ζ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              2
                              ⁢
                              
                                
                                  ɛ
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            φ
                                          
                                          +
                                          
                                            ζ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    -
                                    
                                      jsin
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            φ
                                          
                                          +
                                          
                                            ζ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              ɛ
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      
                                        2
                                        ⁢
                                        
                                          (
                                          
                                            
                                              cos
                                              ⁡
                                              
                                                (
                                                
                                                  
                                                    3
                                                    ⁢
                                                    φ
                                                  
                                                  +
                                                  
                                                    ζ
                                                    3
                                                  
                                                
                                                )
                                              
                                            
                                            -
                                            
                                              jsin
                                              ⁡
                                              
                                                (
                                                
                                                  
                                                    3
                                                    ⁢
                                                    φ
                                                  
                                                  +
                                                  
                                                    ζ
                                                    3
                                                  
                                                
                                                )
                                              
                                            
                                          
                                          )
                                        
                                      
                                      +
                                    
                                  
                                
                                
                                  
                                    
                                      (
                                      
                                        
                                          cos
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            (
                                            
                                              
                                                -
                                                φ
                                              
                                              +
                                              
                                                2
                                                ⁢
                                                
                                                  ζ
                                                  1
                                                
                                              
                                              -
                                              
                                                ζ
                                                3
                                              
                                            
                                            )
                                          
                                        
                                        -
                                        
                                          jsin
                                          ⁡
                                          
                                            (
                                            
                                              
                                                -
                                                φ
                                              
                                              +
                                              
                                                2
                                                ⁢
                                                
                                                  ζ
                                                  1
                                                
                                              
                                              -
                                              
                                                ζ
                                                3
                                              
                                            
                                            )
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    O
                    ⁡
                    
                      (
                      
                        ɛ
                        i
                      
                      )
                    
                  
                
                +
                
                  ⋯
                  ⁢
                  
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        106
        )
      
    
  




Signal Σ5 is sent to LPF 184, as described earlier for LPF 160.
For an LPF Tn(x) of order n, the complex output C5 is approximately:




  
    
      
        
          
            
              
                C
                5
              
              =
              
              ⁢
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    Σ
                    5
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    ζ
                                    1
                                  
                                
                                -
                                
                                  ζ
                                  2
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  1
                  2
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            2
                            ⁢
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            2
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            2
                            ⁢
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        107
        )
      
    
  




where O(x) denotes a term of the order of x, ωc is the −3 dB angular cutoff frequency, and ωD=dφ/dt.

The term on the right hand sides of Equation (107) with factor A13 is the source of the largest errors and accordingly determines the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of LPF 184 is stored in register 186. For a fourth order LPF, i.e., n=4, and a minimum ratio for ωD/ωc=7, the error terms on the right hand side of Equation (107) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the output C5 of LPF 184 is stored in register 186 as C5R under the control of signal 161.
Signal Σ6 is calculated by combining signals Σ−1 and Σ3 using multiplier 172:




  
    
      
        
          
            
              
                ∑
                6
              
              ⁢
              
                =
                
                ⁢
                
                  
                    ∑
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    ∑
                    3
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              -
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                              +
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              -
                              
                                ζ
                                
                                  -
                                  1
                                
                              
                              +
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                    4
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    
                      ɛ
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              
                                3
                                ⁢
                                
                                  (
                                  
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        
                                          
                                            3
                                            ⁢
                                            φ
                                          
                                          +
                                          
                                            ζ
                                            1
                                          
                                          +
                                          
                                            ζ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    -
                                    
                                      jsin
                                      ⁡
                                      
                                        (
                                        
                                          
                                            3
                                            ⁢
                                            φ
                                          
                                          +
                                          
                                            ζ
                                            1
                                          
                                          +
                                          
                                            ζ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              (
                              
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      φ
                                      +
                                      
                                        3
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        2
                                      
                                    
                                    )
                                  
                                
                                -
                                
                                  jsin
                                  ⁡
                                  
                                    (
                                    
                                      φ
                                      +
                                      
                                        3
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        2
                                      
                                    
                                    )
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                1
                              
                              +
                              ζ
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  3
                  ⁢
                  
                    A
                    1
                    2
                  
                  ⁢
                  
                    O
                    ⁡
                    
                      (
                      
                        
                          ɛ
                          i
                        
                        ⁢
                        
                          ɛ
                          j
                        
                      
                      )
                    
                  
                
                +
                
                  ⋯
                  .
                
              
            
          
        
      
    
    
      
        (
        108
        )
      
    
  




Signal Σ6 is sent to LPF 174, as described earlier for LPF 160.
For an LPF Tn(x) of order n, the complex output C6 is approximately:




  
    
      
        
          
            
              
                C
                6
              
              =
              
              ⁢
              
                
                  T
                  n
                
                ⁡
                
                  [
                  
                    ∑
                    6
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                    3
                  
                  ⁢
                  
                    
                      ɛ
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        -
                        
                          jsin
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  A
                  1
                  3
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            
                              ɛ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      4
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              A
                              1
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              +
                              3
                            
                            ⁢
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      3
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            
                              ɛ
                              2
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      ω
                                      D
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                          +
                          
                            3
                            ⁢
                            
                              ɛ
                              3
                            
                            ⁢
                            
                              
                                O
                                ⁡
                                
                                  (
                                  
                                    
                                      ω
                                      c
                                    
                                    
                                      4
                                      ⁢
                                      
                                        ω
                                        D
                                      
                                    
                                  
                                  )
                                
                              
                              n
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        109
        )
      
    
  




where O(x) denotes a term of the order of x, ωc is the −3 dB angular cutoff frequency, and ωD=dφ/dt.

The term on the right hand sides of Equation (107) with factor A14 is the source of the largest errors and accordingly determines the specifications of n and the minimum ratio for ωD/ωc that can be used when the outputs of LPF 174 is stored in register 176. For a fourth order LPF, i.e., n=4, and a minimum ratio for ωD/ωc=3.5, the error terms on the right hand side of Equation (109) will generate errors that correspond to ≲0.010 nm (3σ) When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the output C6 of LPF 174 is stored in register 176 as C6R under the control of signal 161.
The magnitude squared of signal D(t) is calculated by multiplier 152:

C1(t)=D(t)·D(t)*.  (110)

This may be simplified and implemented as:

C1(t)=Re(D(t))2+Im(D(t))2.  (111)

Alternate methods to calculate a value equivalent to the result of equation (111) are possible, for example the magnitude output 125 from CORDIC converter 124 could be squared, or signal Σ1 could be used instead of D(t).
When equation (111) is evaluated using the terms of D(t) from equations (89) to (94), the following result is obtained:




  
    
      
        
          
            
              
                
                  C
                  1
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  A
                  1
                  2
                
                +
                
                  [
                  
                    
                      ɛ
                      
                        -
                        1
                      
                      2
                    
                    +
                    
                      ɛ
                      0
                      2
                    
                    +
                    
                      ɛ
                      2
                      2
                    
                    +
                    
                      ɛ
                      3
                      2
                    
                  
                  ]
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          2
                          ⁢
                          φ
                        
                        +
                        
                          ζ
                          1
                        
                        -
                        
                          ζ
                          
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    ɛ
                    0
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        φ
                        +
                        
                          ζ
                          1
                        
                        -
                        
                          ζ
                          0
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    ɛ
                    2
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          -
                          φ
                        
                        +
                        
                          ζ
                          1
                        
                        -
                        
                          ζ
                          2
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    A
                    1
                  
                  ⁢
                  
                    ɛ
                    3
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            -
                            2
                          
                          ⁢
                          φ
                        
                        +
                        
                          ζ
                          1
                        
                        -
                        
                          ζ
                          3
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  O
                  ⁡
                  
                    (
                    
                      
                        ɛ
                        i
                      
                      ⁢
                      
                        ɛ
                        j
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        112
        )
      
    
  




The signal C1(t) is sent to LPF (Lowpass Filter) 154 as described earlier for LPF 160.
For an LPF Tn(x) of order n, the output C1 is approximately:




  
    
      
        
          C
          1
        
        =
        
        ⁢
        
          
            T
            n
          
          ⁡
          
            [
            
              
                C
                1
              
              ⁡
              
                (
                t
                )
              
            
            ]
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            A
            
              
                
              
              1
            
            2
          
          +
          
            [
            
              
                ɛ
                
                  -
                  1
                
                2
              
              +
              
                ɛ
                0
                2
              
              +
              
                ɛ
                2
                2
              
              +
              
                ɛ
                3
                2
              
            
            ]
          
          +
        
      
    
  
  
    
      
        
        ⁢
        
          2
          ⁢
          
            
              
                A
                1
              
              ⁡
              
                [
                
                  
                    
                      ɛ
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              2
                              ⁢
                              
                                ω
                                D
                              
                            
                          
                          )
                        
                      
                      2
                    
                  
                  +
                  
                    
                      ɛ
                      0
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              ω
                              D
                            
                          
                          )
                        
                      
                      2
                    
                  
                  +
                  
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              ω
                              D
                            
                          
                          )
                        
                      
                      n
                    
                  
                  +
                  
                    
                      ɛ
                      3
                    
                    ⁢
                    
                      
                        O
                        ⁡
                        
                          (
                          
                            
                              ω
                              c
                            
                            
                              
                                -
                                2
                              
                              ⁢
                              
                                ω
                                D
                              
                            
                          
                          )
                        
                      
                      n
                    
                  
                
                ]
              
            
            .
          
        
      
    
  




The accuracy required for the determination of C1 is approximately 0.5% in order to limit errors generated in the computation of cyclic error signals Sψj to ≲0.010 nm (3σ). Therefore the error terms ε−12, ε02, ε22, and ε22 on the right hand side of Equation (113) are negligible. The terms on the right hand side of Equation (113) of the form




  
    O
    ⁡
    
      (
      
        
          ω
          c
        
        
          ω
          D
        
      
      )
    
  
  n




are the sources of the largest Doppler shift frequency dependent errors and accordingly determine the specifications of n and the minimum ratio for ωD/ωc that can be used when the output of LPF 154 is held in register 156, providing signal C1R. For a second order LPF, i.e., n=2, and a minimum ratio for ωD/ωc=3.5, the Doppler shift frequency dependent error terms on the right hand side of Equation (113) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ωD/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ωD/ωc is satisfied, the output C1 of LPF 154 is stored in register 156 as C1R under the control of signal 161.

The low pass filtering approach to determining values from which error basis functions and their coefficients are derived is appropriate when the stage is moving at a speed such that the corresponding Doppler shift frequency satisfies constraints due to the low pass filter −3 dB cutoff ωc and the stage servo control system bandwidth. When the stage is being scanned slowly, or is nearly stationary (e.g., due to a change in scan direction), a distribution analysis approach can be used to calculate and store the values used to generate the compensating signal Dψ(t).
The distribution analysis approach includes performing error compensation calculations based on collective properties of a distribution of values derived at least in part from samples of the signal S(t). The values may represent, for example, samples of a multi-dimensional signal. As a function of time, the multi-dimensional signal defines a curve, and the values represent points on the curve. In the following example, the multi-dimensional signal is a two-dimensional signal, and the values in the distribution are two-dimensional values. A distribution of two-dimensional values that represent samples of the two-dimensional signal is compiled from the one-dimensional interference signal S(t). However, the two-dimensional values do not necessarily represent time sequential samples of the two-dimensional signal. Each two-dimensional value can be generated based on multiple samples of the one-dimensional interference signal S(t). By analyzing collective properties of the distribution of two-dimensional values, such as a distribution of their phases, or a curve fit to the values, the CEC error estimator 144 can calculate error basis function coefficients for a slow moving or nearly stationary stage. In this case, the two-dimensional signal used is the two-dimensional measurement signal D(t), which represents complex values of a DFT of the interference signal S(t) updated at time t.
The CEC error estimator 144 calculates coefficients to be stored in the registers based on analysis of collective properties of N updated values of D(t) generated within an analysis time T. If the stage is being scanned too slowly to use the low pass filtering approach described above, but fast enough to provide an adequate distribution of phases of the N values of D(t), then the CEC error estimator 144 uses the values to determine the coefficients and error basis functions for error compensation. An adequate distribution of the phases of D(t) corresponds to an approximately uniform sampling (e.g., in time) of a curve defined by D(t) over phase range corresponding to a full period (e.g., a 2π range), as described in more detail below. The CEC error estimator 144 calculates the coefficients and error basis functions as described above using a statistical averaging function mean(x) in place of low pass filtering function Tn(x). For example, the coefficients C0R and C1R to be stored in the registers are calculated as:

C0R=mean(D(t))  (114)

C1R=mean(Re(D(t))2+Im(D(t))2)  (115)

Similarly, values of other coefficients and the error basis functions can be calculated as described above, using statistical averaging in place of low pass filtering. The statistical averaging provides accurate coefficients and error basis functions as long as the distribution of the phases of the N values of D(t) are sufficiently uniform (e.g., spacing varies by less than 20% or less than 5%) and sufficiently dense (e.g., N>20 or N>100) to accurately represent the two dimensional curve defined by D(t).
If the stage is not being scanned fast enough to provide an adequate phase distribution of D(t), then CEC error estimator 144 performs analysis of collective properties of the values of D(t), such as curve fitting techniques, to generate interpolated values of D(t) that do have an adequate phase distribution. Even if the stage is not being scanned, there may be enough movement of the stage to generate values of D(t) with sufficient variation in phase to be used to fit a curve from which the interpolated values can be generated. For example, the stage may wander corresponding to a change in φ(t), even if the change is not linear in time.
The effects of the various cyclic error terms used for error correction can be analyzed to determine their effects on a ideal curve defined by an ideal measurement signal. Modified curves incorporating the effects of the cyclic error terms can then be fit to values of the two-dimensional measurement signal derived from the interference signal. For example, an ideal curve be in the form of a circle and added cyclic error terms may yield a modified curve that is elliptical and translated relative to the ideal curve. The center and ellipticity of the ellipse can be used as parameters to fit the modified curve to values of the two-dimensional measurement signal. The resulting fit curve can be used to derive cyclic error parameters, or to generate interpolated values (e.g., by “sampling” the fit curve) that can be averaged as described above.
If we consider equations (89) and (90), and examine the effect of just the cyclic error term Dψ0, the resulting complex measurement signal can be expressed as:




  
    
      
        
          
            
              
                D
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              φ
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              φ
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
                
                  
                    ɛ
                    0
                  
                  (
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          ζ
                          0
                        
                        )
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                jsin
                ⁡
                
                  (
                  
                    ζ
                    0
                  
                  )
                
              
              )
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        j
                        ⁡
                        
                          (
                          
                            
                              φ
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                    
                  
                
                +
                
                  
                    ɛ
                    0
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        jζ
                        O
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        116
        )
      
    
  




The effect of adding the cyclic error term Dψ0=ε0e−jζ0 to the ideal signal A1e−j(φ(t)+ζ1) is shown in the phasor diagram of FIG. 7a. As the phase φ(t)+ζ1 of the ideal signal shifts through 2π radians, the effect of the term Dψ0 is to shift the curve 700 defined by the ideal signal (a circle with radius A1) by a distance corresponding to the amplitude ε0 and in a direction corresponding to the phase ζ0, to a shifted curve 702 defined by the combined signal. The shifted curve 702 is also a circle with radius A1.
If we consider equations (89) and (91), and examine the effect of just the cyclic error term Dψ−1(t), the resulting complex measurement signal can be expressed as:




  
    
      
        
          
            
              
                D
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              φ
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                      -
                      
                        jsin
                        ⁡
                        
                          (
                          
                            
                              φ
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ɛ
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            -
                            
                              φ
                              ⁡
                              
                                (
                                
                                  t
                                  1
                                
                                )
                              
                            
                          
                          +
                          
                            ζ
                            
                              -
                              1
                            
                          
                        
                        )
                      
                    
                    -
                    
                      jsin
                      ⁡
                      
                        (
                        
                          
                            -
                            
                              φ
                              ⁡
                              
                                (
                                
                                  t
                                  1
                                
                                )
                              
                            
                          
                          +
                          
                            ζ
                            
                              -
                              1
                            
                          
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    A
                    1
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        j
                        ⁡
                        
                          (
                          
                            
                              φ
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            +
                            
                              ζ
                              1
                            
                          
                          )
                        
                      
                    
                  
                
                +
                
                  
                    ɛ
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
                        j
                        ⁡
                        
                          (
                          
                            
                              -
                              
                                φ
                                ⁡
                                
                                  (
                                  t
                                  )
                                
                              
                            
                            +
                            
                              ζ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        117
        )
      
    
  




The effect of adding the cyclic error term Dψ−1(t)=ε−1e−j(−φ(t)+ζ−1) to the ideal signal A1e−j(φ(t)+ζ1) is shown in the phasor diagram of FIG. 7b. As the phase φ(t)+ζ1. of the ideal signal shifts through 2π radians, the phase −φ(t)+ζ−1 of the cyclic error term Dψ−1(t) shifts through −2π radians. The effect of the cyclic error term Dψ−1(t) is to change the shape of the ideal curve to a curve 704 that is an ellipse that has a major axis to minor axis ratio (or “ellipticity”) equal to the sum A1+ε−1 divided by the difference A1−ε−1. The direction of the major axis (relative to the real axis) is a function of both ζ1 and ζ−1 and occurs when φ(t)+ζ1 is equal to −φ(t)+ζ−1, which occurs when




  
    φ
    ⁡
    
      (
      t
      )
    
  
  =
  
    
      
        
          ζ
          
            -
            1
          
        
        -
        
          ζ
          1
        
      
      2
    
    .
  




Similarly, other cyclic error terms can be analyzed to determine their effect on a curve representing an ideal two-dimensional measurement signal over time. The CEC error estimator 144 uses selected cyclic error terms, depending on the level of compensation desired, to model a modified curve that represents the effects of errors on an ideal curve (corresponding to an ideal two-dimensional measurement signal over time).
For example, FIG. 8a shows a modified curve 800 representing the effects of cyclic error terms Dψ0 and Dψ−1(t) on the ideal curve 700. Various parameters of the modified curve 800 can be used to determine cyclic error basis functions and their coefficients and phase constants used for performing error compensation. In this example, the modified curve 800 is an ellipse shifted from the origin of the real and imaginary axes by the same distance ε0 and in the same direction ζ0 as the curve 702, and having the same major to minor axis ratio and major axis direction as the curve 704. For other combinations of cyclic error terms and/or other two-dimensional signals, the resulting curve may have other relevant parameters based on its shape, size, or position, for example. As described above, once the form of the modified curve is determined, a parameterized version of the modified curve can be fit to values of D(t). The fit curve can then be sampled to generate interpolated values of D(t) that have an adequate phase distribution for using the averaging function mean(x) in place of low pass filtering function Tn(x) in the calculations above. Referring to FIG. 8b, a curve 810 is fit (e.g., by varing the center, lengths of major and minor axes, and angle of major axis) to a set of updated values of the complex measurement singal D(t). In this example, the updated values do not have a uniform phase distribution over a 2π range. This curve 810 is fit, for example, using a minimum mean squared error criterion that minimizing the mean squared error between the values and the curve. For example, one error function whose average squared value can be minimized is the shortest distance between a given value and the curve 810. In this case, the curve would be fit more closely in the region with more densly spaced values than in the region having fewer values.
Referring to FIG. 8c, a curve 820 is fit to a reduced set of the updated values of the complex measurement signal D(t). In this example, the updated values used to perform the fit are selected to have a more uniform spacing. For example, the CEC error estimator 144 divides the values into sets according to the phase sector in which the values occur. A maximum of two representitive values are selected in each of eight evenly spaced sectors. In sectors having more than two values, the two values used can be selected according to predetermined criteria (e.g., maximizing the spacing between the selected values).
Using the coefficients derived using either the low pass filtering approach or the phase analysis approach, the processor 204 calculates compensating signal Dψ(t) as shown in FIG. 1e and Equation (114).




  
    
      
        
          
            
              
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                =
                
                  
                    C
                    
                      0
                      ⁢
                      R
                    
                  
                  +
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          R
                        
                      
                      
                        C
                        
                          1
                          ⁢
                          R
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        +
                        
                          (
                          
                            
                              C
                              
                                5
                                ⁢
                                R
                              
                            
                            
                              C
                              
                                1
                                ⁢
                                R
                              
                              2
                            
                          
                        
                      
                    
                  
                
              
              ⁢
              
                *
                )
              
            
            ⁢
            
              
                ∑
                2
              
              ⁢
              
                +
                
                  (
                  
                    
                      C
                      
                        6
                        ⁢
                        R
                      
                    
                    
                      C
                      
                        1
                        ⁢
                        R
                      
                      3
                    
                  
                
              
            
          
          ⁢
          
            *
            )
          
        
        ⁢
        
          ∑
          3
        
      
    
    
      
        (
        118
        )
      
    
  




This can be expanded using the most significant terms from the previously expressed equations (98), (113), (102), (107), and (109) for the CN terms, and equations (100), (101), and (105) for the ΣN terms to:




  
    
      
        
          
            
              
                
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ψ
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  =
                  
                    
                      
                        ɛ
                        0
                      
                      ⁡
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                ζ
                                0
                              
                              )
                            
                          
                          -
                          
                            jsin
                            ⁡
                            
                              (
                              
                                ζ
                                0
                              
                              )
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        
                          
                            
                              
                                
                                  A
                                  1
                                
                                ⁢
                                
                                  
                                    ɛ
                                    
                                      -
                                      1
                                    
                                  
                                  (
                                  
                                    
                                      cos
                                      ⁢
                                      
                                        (
                                        
                                          
                                            ζ
                                            1
                                          
                                          +
                                          
                                            ζ
                                            
                                              -
                                              1
                                            
                                          
                                        
                                        )
                                      
                                    
                                    -
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  jsin
                                  ⁢
                                  
                                    (
                                    
                                      
                                        ζ
                                        1
                                      
                                      +
                                      
                                        ζ
                                        
                                          -
                                          1
                                        
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                          
                        
                        
                          A
                          1
                          2
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            
                              
                                A
                                1
                              
                              (
                              
                                
                              
                              ⁢
                              
                                
                                  cos
                                  ⁢
                                  
                                    (
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        -
                                        φ
                                      
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        ζ
                                        1
                                      
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  
                                
                                -
                              
                            
                          
                        
                        
                          
                            
                              
                                jsin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                  
                                  ⁢
                                  
                                    
                                      -
                                      
                                        
                                      
                                      ⁢
                                      φ
                                    
                                    ⁢
                                    
                                      
                                    
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                
                              
                              )
                            
                          
                        
                      
                    
                    +
                    
                      (
                      
                        
                          
                            
                              
                                
                                  A
                                  1
                                  2
                                
                                ⁢
                                
                                  
                                    ɛ
                                    2
                                  
                                  (
                                  
                                    
                                      cos
                                      ⁢
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              ζ
                                              1
                                            
                                          
                                          -
                                          
                                            ζ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    -
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  jsin
                                  ⁢
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        
                                          ζ
                                          1
                                        
                                      
                                      -
                                      
                                        ζ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                          
                        
                        
                          A
                          1
                          4
                        
                      
                    
                  
                
                ⁢
                
                  *
                  )
                
              
              ⁢
              
                
                  
                    
                      
                        A
                        1
                        2
                      
                      (
                      
                        
                          cos
                          ⁢
                          
                            (
                            
                              
                                2
                                ⁢
                                φ
                              
                              +
                              
                                2
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                            
                            )
                          
                        
                        -
                      
                    
                  
                
                
                  
                    
                      
                        jsin
                        ⁢
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              2
                              ⁢
                              
                                ζ
                                1
                              
                            
                          
                          )
                        
                      
                      )
                    
                  
                
              
            
            +
            
              (
              
                
                  
                    
                      
                        
                          A
                          1
                          3
                        
                        ⁢
                        
                          
                            ɛ
                            3
                          
                          (
                          
                            
                              cos
                              ⁢
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      ζ
                                      1
                                    
                                  
                                  -
                                  
                                    ζ
                                    3
                                  
                                
                                )
                              
                            
                            -
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          jsin
                          ⁢
                          
                            (
                            
                              
                                3
                                ⁢
                                
                                  ζ
                                  1
                                
                              
                              -
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                        )
                      
                    
                  
                
                
                  A
                  1
                  6
                
              
            
          
          ⁢
          
            *
            )
          
        
        ⁢
        
          
            
              
                
                  A
                  1
                  3
                
                (
                
                  
                    cos
                    ⁢
                    
                      (
                      
                        
                          3
                          ⁢
                          φ
                        
                        +
                        
                          3
                          ⁢
                          
                            ζ
                            1
                          
                        
                      
                      )
                    
                  
                  -
                
              
            
          
          
            
              
                
                  jsin
                  ⁢
                  
                    (
                    
                      
                        3
                        ⁢
                        φ
                      
                      +
                      
                        3
                        ⁢
                        
                          ζ
                          1
                        
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        (
        119
        )
      
    
  




and simplified to:





  
    
      
        
          
            
              
                D
                ⁢
                
                  
                
                ⁢
                
                  ψ
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              =
              
              ⁢
              
                
                  ɛ
                  0
                
                ⁡
                
                  (
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          ζ
                          0
                        
                        )
                      
                    
                    -
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            ζ
                            0
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              +
              
              ⁢
              
                
                  ɛ
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            -
                            φ
                          
                          +
                          
                            ζ
                            
                              -
                              1
                            
                          
                        
                        )
                      
                    
                    -
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              -
                              φ
                            
                            +
                            
                              ζ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              +
              
              ⁢
              
                
                  ɛ
                  2
                
                ⁡
                
                  (
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            2
                            ⁢
                            φ
                          
                          +
                          
                            ζ
                            2
                          
                        
                        )
                      
                    
                    -
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              2
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              2
                            
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              +
              
              ⁢
              
                
                  
                    ɛ
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              3
                              ⁢
                              φ
                            
                            +
                            
                              ζ
                              3
                            
                          
                          )
                        
                      
                      -
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                3
                                ⁢
                                φ
                              
                              +
                              
                                ζ
                                3
                              
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        120
        )
      
    
  




Note that the calculated result in equation (116) is identical to Dψ(t) as described in equations (90) to (94).
Control signal 211 enables the output of Dψ(t) when conditions indicate that a valid compensation signal is present. Control signal 211 is generated by control circuitry (not shown) that may consider the speed of motion, the signal level, the cyclic error level, the duration of motion, and possible other conditions to determine the validity or usefulness of Dψ(t). It may also be desirable to enable each of the four cyclic error terms separately, for example if the ε3 term was found to be insignificant for a specific implementation, the calculation could be disabled to reduce noise contributed by the calculation. When a Dψ(t) term or output is disabled, zero is substituted.
The calculated Dψ(t) is subtracted from signal D(t) by subtractor 146, resulting in a measurement signal with substantially reduced cyclic errors.
Using either the DMS approach or the CMS approach, other techniques can be incorporated to compensate for some of the sources of cyclic errors. For example, the dominant error term is typically the unshifted cyclic error component, Sψ0, or equivalently Dψ0, which stays at constant phase and frequency regardless of stage motion. This term arises from the presence of both optical frequencies in either the reference arm or the measurement arm of the displacement measuring interferometer or both. This occurs, for example, if the optical frequencies of the light source are not perfectly separated into orthogonal linear polarization states.
Though the cyclic error compensation techniques are described using the double pass plane mirror interferometer by way of example, they can be applied to any two-frequency, displacement measuring interferometer in which the cyclic error term which does not experience Doppler shift is dominant.
Compensation for this unshifted term can be achieved by momentarily blocking the beam in the reference path of the interferometer and then in the measurement path of the interferometer at two different times, during, for example, interferometer installation; measuring and recording the complex amplitude of the measurement signal in the beam-blocked condition; adding the two different complex amplitudes so obtained to obtain a compensation coefficient representing the unshifted term that is present when neither beam is blocked; then using the complex coefficient to compensate for the unshifted term during operation of the electronic compensation techniques described herein with the automatic coefficient updating disabled.
The technique of beam blocking can either be manual or automated using, for example, motorized shutters in the reference and measurement paths of the interferometer which are controlled by the same computer that reads the complex amplitudes in the beam-blocked condition; calculates the sum; and stores the coefficient for use in the electronic compensation techniques described herein.
Finally, it is noted that, if desired, the accuracy of the cyclic error correction can further be improved to higher order in the amplitude of the cyclic error coefficients by iterating the compensation of the main signal S(t). In other words, for each subsequent iteration the compensated signal for the main signal is used to generate corresponding cyclic error basis functions and determine a higher order correction to each of the cyclic error coefficients.
Referring now to FIG. 2, a block diagram is shown for an Mth order digital filter used in the cyclic error compensation described above. The figure is in the “Direct Form I” representation standard to the digital signal processing community. The input discrete time series is x(n) and the output discrete time series is y(n). The z−1 operation indicates a one-sample delay. A time-domain representation of the filter takes the form:

y(n)=b0x(n)+b1x(n−1)+b2x(n−2)+ . . . +bMx(n−M)−a1y(n−1)−a2y(n−2)− . . . −aMy(n−M)  (121).

The coefficients ai and bi are selected to produce the desired properties for the filter. For the case of the Butterworth filter, the ai and bi coefficients are selected to produce the frequency filtering given by Equations 28, 29, 43, 44, 55, and 56. Furthermore, other embodiments of the cyclic error compensation may implement different low-pass filtering schemes to yield the coefficients of low-frequency terms. The Butterworth filter, and other low-pass digital filters, are well known in the art. See, for example: Oppenhiem, A. V., Schafer, R. W., and, J. R. Buck, “Discrete-Time Signal Processing”, Upper Saddle River, N.J.: Prentice Hall, 1999; and Proakis, J. G., and, D. G. Manolakis, “Digital Signal Processing: Principles, Algorithms, and Applications”, New York, N.Y.: MacMillan, 1992.

Depending on the embodiment, the compensation technique described above can be implemented in hardware or software, or a combination of both. The technique can be implemented in computer programs using standard programming techniques following the method and figures described herein. Program code is applied to input data to perform the functions described herein and generate output information. The output information is applied to one or more output devices such as the servo control system.
Each program may be implemented in a high level procedural or object oriented programming language to communicate with a computer system, or the programs can be implemented in assembly or machine language, if desired. In any case, the language can be a compiled or interpreted language. Moreover, the program can run on dedicated integrated circuits preprogrammed for that purpose.
Each such computer program may be stored on a storage medium or device (e.g., ROM or magnetic diskette) readable by a general or special purpose programmable computer, for configuring and operating the computer when the storage media or device is read by the computer to perform the procedures described herein. The computer program can also reside in cache or main memory during program execution. The compensation technique can also be implemented as a computer-readable storage medium, configured with a computer program, where the storage medium so configured causes a computer to operate in a specific and predefined manner to perform the functions described herein.
Now referring to FIG. 3, an interferometry system including a high stability plane mirror interferometer (HSPMI) 411 is shown for optical generating the main interference signal. The HSPMI 411 includes a polarization beam-splitter 330, a retroreflector 332, quarter wave phase retardation plates 334 and 336, and a plane mirror reference object 342. Input beam 422 is a two-component beam. The two components have different frequencies and are orthogonally plane polarized. The different frequencies can be produced in source 415, for example, by laser Zeeman splitting, by acousto-optical modulation, or internal to the laser using birefringent elements or the like. HSPMI 411 splits input beam 422 into two components. One component, shown as first and second pass measurement beams 322 and 324, reflects from measurement object 490 twice before exiting HSPMI 411. The other component, shown by first and second pass reference beams 328 and 327, reflect from reference mirror 342 twice before exiting HSPMI 411. The exiting beam components overlap and form output beam 423.
An electrical interference signal 352 is generated by the detection of output beam 423 in detector 420. Detector 420 includes a polarizer to mix the reference and measurement beam components of output beam 423 with respect to polarization. Electrical interference signal 352 contains a heterodyne interference signal corresponding to main interference signal S(t).
In further embodiments, the interferometry system may be different than that shown in FIG. 3. In particular, the cyclic error compensation technique is applicable to other types of interferometers as well. For example, the main interference signal S(t) may be produced by an interferometry system that may include any of single and/or multiple pass interferometers, passive interferometers, dynamic interferometers, and dispersion interferometers. Furthermore, the interferometry system may monitor one or more degrees of freedom, each of which may produce a corresponding main interference signal S(t), which may be compensated for cyclic errors as disclosed herein. Furthermore, the degree(s) of freedom monitored by the interferometry system may include any of changes in distance to a measurement object, changes in relative distance between two measurement objects, changes in the angular orientation of a measurement object, and changes in the direction of the input beam.
Examples of dynamic interferometers are described in U.S. patent application Ser. No. 10/226,591 filed Aug. 23, 2002 and entitled “DYNAMIC INTERFEROMETER CONTROLLING DIRECTION OF INPUT BEAM” by Henry A. Hill. Examples of passive zero shear interferometers are described in U.S. patent application Ser. No. 10/207,314, entitled “PASSIVE ZERO SHEAR INTERFEROMETERS,” filed Jul. 29, 2002, by Henry A. Hill. Examples of angular displacement interferometers are described in: U.S. patent application Ser. No. 10/226,591 entitled “DYNAMIC INTERFEROMETER CONTROLLING DIRECTION OF INPUT BEAM,” filed Aug. 23, 2002; U.S. Provisional Application 60/314,345 filed Aug. 22, 2001 and entitled “PASSIVE ZERO SHEAR INTERFEROMETERS USING ANGLE SENSITIVE BEAM-SPLITTERS,” both by Henry A. Hill, and U.S. patent application Ser. No. 10/272,034 entitled “INTERFEROMETERS FOR MEASURING CHANGES IN OPTICAL BEAM DIRECTION” and filed Oct. 15, 2002 by Henry A. Hill and Justin Kreuzer. Alternatively, or additionally, interferometry systems may include one or more differential angular displacement interferometers, examples of which are also described in U.S. patent application Ser. No. 10/272,034. Examples of interferometry systems for measuring more than one degree of freedom and for reducing beam shear are described in U.S. patent application Ser. No. 10/352,616 filed Jan. 28, 2003 and entitled “MULTIPLE-PASS INTERFEROMETRY” by Henry A. Hill and U.S. patent application Ser. No. 10/351,708 filed Jan. 27, 2003 and entitled “MULTI-AXIS INTERFEROMETER” by Henry A. Hill. Other forms of multiple pass interferometers are described in an article entitled “Differential interferometer arrangements for distance and angle measurements: Principles, advantages and applications” by C. Zanoni, VDI Berichte Nr. 749, 93-106 (1989). Examples of two-wavelength dispersion interferometers are described in U.S. Pat. No. 6,219,144 B1 entitled “APPARATUS AND METHOD FOR MEASURING THE REFRACTIVE INDEX AND OPTICAL PATH LENGTH EFFECTS OF AIR USING MULTIPLE-PASS INTERFEROMETRY” by Henry A. Hill, Peter de Groot, and Frank C. Demarest and U.S. Pat. No. 6,327,039 B1 by Peter de Groot, Henry A. Hill, and Frank C. Demarest.
Because of the cyclic error compensation, the interferometry systems described herein provide highly accurate measurements. Such systems can be especially useful in lithography applications used in fabricating large scale integrated circuits such as computer chips and the like. Lithography is the key technology driver for the semiconductor manufacturing industry. Overlay improvement is one of the five most difficult challenges down to and below 100 nm line widths (design rules), see, for example, the Semiconductor Industry Roadmap, p. 82 (1997).
Overlay depends directly on the performance, i.e., accuracy and precision, of the distance measuring interferometers used to position the wafer and reticle (or mask) stages. Since a lithography tool may produce $50-100 M/year of product, the economic value from improved performance distance measuring interferometers is substantial. Each 1% increase in yield of the lithography tool results in approximately $1 M/year economic benefit to the integrated circuit manufacturer and substantial competitive advantage to the lithography tool vendor.
The function of a lithography tool is to direct spatially patterned radiation onto a photoresist-coated wafer. The process involves determining which location of the wafer is to receive the radiation (alignment) and applying the radiation to the photoresist at that location (exposure).
To properly position the wafer, the wafer includes alignment marks on the wafer that can be measured by dedicated sensors. The measured positions of the alignment marks define the location of the wafer within the tool. This information, along with a specification of the desired patterning of the wafer surface, guides the alignment of the wafer relative to the spatially patterned radiation. Based on such information, a translatable stage supporting the photoresist-coated wafer moves the wafer such that the radiation will expose the correct location of the wafer.
During exposure, a radiation source illuminates a patterned reticle, which scatters the radiation to produce the spatially patterned radiation. The reticle is also referred to as a mask, and these terms are used interchangeably below. In the case of reduction lithography, a reduction lens collects the scattered radiation and forms a reduced image of the reticle pattern. Alternatively, in the case of proximity printing, the scattered radiation propagates a small distance (typically on the order of microns) before contacting the wafer to produce a 1:1 image of the reticle pattern. The radiation initiates photo-chemical processes in the resist that convert the radiation pattern into a latent image within the resist.
Interferometry systems are important components of the positioning mechanisms that control the position of the wafer and reticle, and register the reticle image on the wafer. If such interferometry systems include the features described above, the accuracy of distances measured by the systems increases as cyclic error contributions to the distance measurement are minimized.
In general, the lithography system, also referred to as an exposure system, typically includes an illumination system and a wafer positioning system. The illumination system includes a radiation source for providing radiation such as ultraviolet, visible, x-ray, electron, or ion radiation, and a reticle or mask for imparting the pattern to the radiation, thereby generating the spatially patterned radiation. In addition, for the case of reduction lithography, the illumination system can include a lens assembly for imaging the spatially patterned radiation onto the wafer. The imaged radiation exposes resist coated onto the wafer. The illumination system also includes a mask stage for supporting the mask and a positioning system for adjusting the position of the mask stage relative to the radiation directed through the mask. The wafer positioning system includes a wafer stage for supporting the wafer and a positioning system for adjusting the position of the wafer stage relative to the imaged radiation. Fabrication of integrated circuits can include multiple exposing steps. For a general reference on lithography, see, for example, J. R. Sheats and B. W. Smith, in Microlithography: Science and Technology (Marcel Dekker, Inc., New York, 1998), the contents of which is incorporated herein by reference.
Interferometry systems described above can be used to precisely measure the positions of each of the wafer stage and mask stage relative to other components of the exposure system, such as the lens assembly, radiation source, or support structure. In such cases, the interferometry system can be attached to a stationary structure and the measurement object attached to a movable element such as one of the mask and wafer stages. Alternatively, the situation can be reversed, with the interferometry system attached to a movable object and the measurement object attached to a stationary object.
More generally, such interferometry systems can be used to measure the position of any one component of the exposure system relative to any other component of the exposure system, in which the interferometry system is attached to, or supported by, one of the components and the measurement object is attached, or is supported by the other of the components.
An example of a lithography scanner 1100 using an interferometry system 1126 is shown in FIG. 4. The interferometry system is used to precisely measure the position of a wafer (not shown) within an exposure system. Here, stage 1122 is used to position and support the wafer relative to an exposure station. Scanner 1100 includes a frame 1102, which carries other support structures and various components carried on those structures. An exposure base 1104 has mounted on top of it a lens housing 1106 atop of which is mounted a reticle or mask stage 1116, which is used to support a reticle or mask. A positioning system for positioning the mask relative to the exposure station is indicated schematically by element 1117. Positioning system 1117 can include, e.g., piezoelectric transducer elements and corresponding control electronics. Although, it is not included in this described embodiment, one or more of the interferometry systems described above can also be used to precisely measure the position of the mask stage as well as other moveable elements whose position must be accurately monitored in processes for fabricating lithographic structures (see supra Sheats and Smith Microlithography: Science and Technology).
Suspended below exposure base 1104 is a support base 1113 that carries wafer stage 1122. Stage 1122 includes a plane mirror 1128 for reflecting a measurement beam 1154 directed to the stage by interferometry system 1126. A positioning system for positioning stage 1122 relative to interferometry system 1126 is indicated schematically by element 1119. Positioning system 1119 can include, e.g., piezoelectric transducer elements and corresponding control electronics. The measurement beam reflects back to the interferometry system, which is mounted on exposure base 1104. The interferometry system can be any of the embodiments described previously.
During operation, a radiation beam 1110, e.g., an ultraviolet (UV) beam from a UV laser (not shown), passes through a beam shaping optics assembly 1112 and travels downward after reflecting from mirror 1114. Thereafter, the radiation beam passes through a mask (not shown) carried by mask stage 1116. The mask (not shown) is imaged onto a wafer (not shown) on wafer stage 1122 via a lens assembly 1108 carried in a lens housing 1106. Base 1104 and the various components supported by it are isolated from environmental vibrations by a damping system depicted by spring 1120.
In other embodiments of the lithographic scanner, one or more of the interferometry systems described previously can be used to measure distance along multiple axes and angles associated for example with, but not limited to, the wafer and reticle (or mask) stages. Also, rather than a UV laser beam, other beams can be used to expose the wafer including, e.g., x-ray beams, electron beams, ion beams, and visible optical beams.
In some embodiments, the lithographic scanner can include what is known in the art as a column reference. In such embodiments, the interferometry system 1126 directs the reference beam (not shown) along an external reference path that contacts a reference mirror (not shown) mounted on some structure that directs the radiation beam, e.g., lens housing 1106. The reference mirror reflects the reference beam back to the interferometry system. The interference signal produce by interferometry system 1126 when combining measurement beam 1154 reflected from stage 1122 and the reference beam reflected from a reference mirror mounted on the lens housing 1106 indicates changes in the position of the stage relative to the radiation beam. Furthermore, in other embodiments the interferometry system 1126 can be positioned to measure changes in the position of reticle (or mask) stage 1116 or other movable components of the scanner system. Finally, the interferometry systems can be used in a similar fashion with lithography systems involving steppers, in addition to, or rather than, scanners.
As is well known in the art, lithography is a critical part of manufacturing methods for making semiconducting devices. For example, U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods. These steps are described below with reference to FIGS. 5a and 5b. FIG. 5a is a flow chart of the sequence of manufacturing a semiconductor device such as a semiconductor chip (e.g., IC or LSI), a liquid crystal panel or a CCD. Step 1151 is a design process for designing the circuit of a semiconductor device. Step 1152 is a process for manufacturing a mask on the basis of the circuit pattern design. Step 1153 is a process for manufacturing a wafer by using a material such as silicon.
Step 1154 is a wafer process which is called a pre-process wherein, by using the so prepared mask and wafer, circuits are formed on the wafer through lithography. To form circuits on the wafer that correspond with sufficient spatial resolution those patterns on the mask, interferometric positioning of the lithography tool relative the wafer is necessary. The interferometry methods and systems described herein can be especially useful to improve the effectiveness of the lithography used in the wafer process.
Step 1155 is an assembling step, which is called a post-process wherein the wafer processed by step 1154 is formed into semiconductor chips. This step includes assembling (dicing and bonding) and packaging (chip sealing). Step 1156 is an inspection step wherein operability check, durability check and so on of the semiconductor devices produced by step 1155 are carried out. With these processes, semiconductor devices are finished and they are shipped (step 1157).
FIG. 5b is a flow chart showing details of the wafer process. Step 1161 is an oxidation process for oxidizing the surface of a wafer. Step 1162 is a CVD process for forming an insulating film on the wafer surface. Step 1163 is an electrode forming process for forming electrodes on the wafer by vapor deposition. Step 1164 is an ion implanting process for implanting ions to the wafer. Step 1165 is a resist process for applying a resist (photosensitive material) to the wafer. Step 1166 is an exposure process for printing, by exposure (i.e., lithography), the circuit pattern of the mask on the wafer through the exposure apparatus described above. Once again, as described above, the use of the interferometry systems and methods described herein improve the accuracy and resolution of such lithography steps.
Step 1167 is a developing process for developing the exposed wafer. Step 1168 is an etching process for removing portions other than the developed resist image. Step 1169 is a resist separation process for separating the resist material remaining on the wafer after being subjected to the etching process. By repeating these processes, circuit patterns are formed and superimposed on the wafer.
The interferometry systems described above can also be used in other applications in which the relative position of an object needs to be measured precisely. For example, in applications in which a write beam such as a laser, x-ray, ion, or electron beam, marks a pattern onto a substrate as either the substrate or beam moves, the interferometry systems can be used to measure the relative movement between the substrate and write beam.
As an example, a schematic of a beam writing system 1200 is shown in FIG. 6. A source 1210 generates a write beam 1212, and a beam focusing assembly 1214 directs the radiation beam to a substrate 1216 supported by a movable stage 1218. To determine the relative position of the stage, an interferometry system 1220 directs a reference beam 1222 to a mirror 1224 mounted on beam focusing assembly 1214 and a measurement beam 1226 to a mirror 1228 mounted on stage 1218. Since the reference beam contacts a mirror mounted on the beam focusing assembly, the beam writing system is an example of a system that uses a column reference. Interferometry system 1220 can be any of the interferometry systems described previously. Changes in the position measured by the interferometry system correspond to changes in the relative position of write beam 1212 on substrate 1216. Interferometry system 1220 sends a measurement signal 1232 to controller 1230 that is indicative of the relative position of write beam 1212 on substrate 1216. Controller 1230 sends an output signal 1234 to a base 1236 that supports and positions stage 1218. In addition, controller 1230 sends a signal 1238 to source 1210 to vary the intensity of, or block, write beam 1212 so that the write beam contacts the substrate with an intensity sufficient to cause photophysical or photochemical change only at selected positions of the substrate.
Furthermore, in some embodiments, controller 1230 can cause beam focusing assembly 1214 to scan the write beam over a region of the substrate, e.g., using signal 1244. As a result, controller 1230 directs the other components of the system to pattern the substrate. The patterning is typically based on an electronic design pattern stored in the controller. In some applications the write beam patterns a resist coated on the substrate and in other applications the write beam directly patterns, e.g., etches, the substrate.
An important application of such a system is the fabrication of masks and reticles used in the lithography methods described previously. For example, to fabricate a lithography mask an electron beam can be used to pattern a chromium-coated glass substrate. In such cases where the write beam is an electron beam, the beam writing system encloses the electron beam path in a vacuum. Also, in cases where the write beam is, e.g., an electron or ion beam, the beam focusing assembly includes electric field generators such as quadrapole lenses for focusing and directing the charged particles onto the substrate under vacuum. In other cases where the write beam is a radiation beam, e.g., x-ray, UV, or visible radiation, the beam focusing assembly includes corresponding optics and for focusing and directing the radiation to the substrate.
A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.

## Claims

1. A method comprising:
providing an interference signal S(t) from interference between two beams directed along different paths, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time; and
providing an error signal to reduce errors in an estimate of {tilde over (L)}(t), the error signal being derived at least in part based on one or more collective properties of a distribution of multi-dimensional values, at least one of the multi-dimensional values in the distribution being generated from a plurality of samples of the signal S(t);
wherein one of the paths is associated with a position of a reference object and the other path is associated with a position of a moveable measurement object;
wherein the position of the moveable object is controlled by a servo system having a bandwidth and fS, the samples of the signal S(t) from which the multi-dimensional values are calculated correspond to sample times at which the velocity v of the moveable measurement object relative to the reference object divided by a wavelength λ of the beams is less than ten times the bandwidth fS; and
wherein the servo system controls the position of the measurement object based on the signal S(t) and the error signal.

2. The method of claim 1, wherein the two beams have different frequencies.

3. The method of claim 1, wherein providing the error signal comprises providing one or more coefficients representative of one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A1 cos(ωRt+φ(t)+ζ1), where A1 and ζ1 are constants, ωR is an angular frequency difference between the two beams before being directed along the different paths, and φ(t)=nk{tilde over (L)}(t), with k=2π/λ and λ equal to a wavelength for the beams.

4. The method of claim 3, wherein the deviation can be expressed as




  
    
      ∑
      
        m
        ,
        p
      
    
    ⁢
    
      
    
    ⁢
    
      
        A
        
          m
          ,
          p
        
      
      ⁢
      
        cos
        ⁡
        
          (
          
            
              
                ω
                R
              
              ⁢
              t
            
            +
            
              
                m
                p
              
              ⁢
              
                φ
                ⁡
                
                  (
                  t
                  )
                
              
            
            +
            
              ζ
              
                m
                ,
                p
              
            
          
          )
        
      
    
  
  ,




where p=1, 2, 3 . . . , and m is any integer not equal to p, and where the provided coefficients comprise information corresponding to at least some of Am,p and ζm,p.

5. The method of claim 3, wherein the coefficients are derived at least in part based on the one or more collective properties of the distribution of multi-dimensional values.

6. The method of claim 5, wherein the error signal is generated from the coefficients and one or more error basis functions derived at least in part from a plurality of samples of the signal S(t).

7. The method of claim 6, wherein each of the error basis functions is derived at least in part from a linear combination of samples of the signal S(t).

8. The method of claim 6, wherein each of the error basis functions corresponds to a function that includes one or more leading sinusoidal terms having a time-varying argument that corresponds to a time-varying argument of an error term that represents a portion of the deviation of S(t) from the ideal expression.

9. The method of claim 1, wherein the multi-dimensional values comprise two-dimensional values.

10. The method of claim 9, wherein the one or more collective properties comprise one or more properties associated with a curve fit to at least some of the two-dimensional values in the distribution.

11. The method of claim 10, wherein the one or more properties associated with the curve comprise values interpolated from the curve.

12. The method of claim 10, wherein the curve comprises an ellipse.

13. The method of claim 12, wherein the one or more properties associated with the curve comprise one or more of a center location, an ellipticity, and an angle of a major or minor axis.

14. The method of claim 9, wherein each of the two-dimensional values is represented as a complex number.

15. The method of claim 14, wherein the complex number comprises a linear combination of samples of the signal S(t).

16. The method of claim 15, wherein the linear combination comprises a transform signal Dq (t1) based on values of the signal S(t), for a frequency value determined by q, updated at a time t1.

17. The method of claim 16, wherein the transform signal Dq (t1) comprises a sample of a discrete Fourier transform of the signal S(t).

18. The method of claim 17, wherein




  
    
      
        D
        q
      
      ⁡
      
        (
        
          t
          1
        
        )
      
    
    =
    
      
        ∑
        
          n
          =
          0
        
        
          N
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          W
          n
        
        ⁢
        
          S
          ⁡
          
            (
            
              
                n
                ⁢
                
                  
                
                ⁢
                τ
              
              +
              
                t
                1
              
            
            )
          
        
        ⁢
        exp
        ⁢
        
          {
          
            -
            
              j
              (
              
                2
                ⁢
                
                  π
                  (
                  
                    n
                    -
                    
                      
                        N
                        -
                        1
                      
                      2
                    
                  
                  )
                
                ⁢
                q
                ⁢
                
                  /
                
                ⁢
                N
              
              )
            
          
          }
        
      
    
  
  ,




where τ=1/fS, and Wn is a window function centered at




  n
  =
  
    
      
        N
        -
        1
      
      2
    
    .

19. The method of claim 14, wherein the complex number is represented as a phase and a magnitude.

20. The method of claim 19, wherein at least two phases associated with respective values in the distribution are separated from each other by at least 90 degrees.

21. The method of claim 19, wherein the spacing between the phases of the values in the distribution varies by less than 20%.

22. The method of claim 1, wherein the one or more collective properties of the distribution of multi-dimensional values comprise a statistical average derived from the multi-dimensional values in the distribution.

23. The method of claim 1, the distribution of multi-dimensional values includes at least 20 values.

24. The method of claim 1, wherein reducing errors in the estimate of {tilde over (L)}(t) comprises deriving the estimate of {tilde over (L)}(t) from a difference between the error signal and a discrete Fourier transform of samples of S(t).

25. The method of claim 1, further comprising directing the two beams along the different paths and measuring the interference signal S(t).

26. The method of claim 25, wherein at least one of the beams is directed to reflect from a movable measurement object before producing the interference signal S(t).

27. The method of claim 26, wherein the beam directed to reflect from the movable measurement object reflects from the measurement object multiple times before producing the interference signal S(t).

28. The method of claim 26, wherein the beams are directed to reflect from different locations of the movable measurement object before producing the interference signal S(t).

29. The method of claim 1, wherein the errors correspond to spurious beam paths.

30. A lithography method for use in fabricating integrated circuits on a wafer, the method comprising:
supporting the wafer on a moveable stage;
imaging spatially patterned radiation onto the wafer;
adjusting the position of the stage; and
monitoring the position of the stage using an interferometry system, wherein monitoring the position of the stage comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the stage using the method of claim 1.

31. A method for fabricating integrated circuits, the method comprising:
applying a resist to a wafer;
forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method of claim 30; and
producing an integrated circuit from the wafer.

32. A lithography method for use in the fabrication of integrated circuits, the method comprising:
directing input radiation through a mask to produce spatially patterned radiation;
positioning the mask relative to the input radiation;
monitoring the position of the mask relative to the input radiation using an interferometry system, wherein monitoring the position of the mask comprises reducing errors in an estimate of a physical path difference associated with the position of the mask using the method of claim 1; and
imaging the spatially patterned radiation onto a wafer.

33. A method for fabricating integrated circuits, the method comprising: applying a resist to a wafer;
forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method of claim 32; and
producing an integrated circuit from the wafer.

34. A lithography method for fabricating integrated circuits on a wafer, the method comprising:
positioning a first component of a lithography system relative to a second component of a lithography system to expose the wafer to spatially patterned radiation; and
monitoring the position of the first component relative to the second component using an interferometry system, wherein monitoring the position of the first component comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the first component using the method of claim 1.

35. A method for fabricating integrated circuits, the method comprising:
applying a resist to a wafer;
forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method of claim 34; and
producing an integrated circuit from the wafer.

36. A method for fabricating a lithography mask, the method comprising:
directing a write beam to a substrate to pattern the substrate;
positioning the substrate relative to the write beam; and
monitoring the position of the substrate relative to the write beam using an interferometry system, wherein monitoring the position of the substrate comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the substrate using the method of claim 1.

37. A computer readable medium storing instructions that cause a processor to perform the method of claim 1.

38. An apparatus comprising:
an interferometry system, which during operation directs two beams along different paths and provides an interference signal S(t) from interference between the two beams, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time; and
an electronic processor, which during operation receives the interference signal S(t) from the interferometry system and provides an error signal to reduce errors in an estimate of {tilde over (L)}(t), the error signal being derived at least in part based on one or more collective properties of a distribution of multi-dimensional values, at least one of the multi-dimensional values in the distribution being generated from a plurality of samples of the signal;
wherein one of the paths is associated with a position of a reference object and the other path is associated with a position of a moveable measurement object;
wherein the position of the moveable object is controlled by a servo system having a bandwidth fS, and the samples of the signal S(t) from which the multi-dimensional values are calculated correspond to sample times at which the velocity v of the moveable measurement object relative to the reference object divided by a wavelength λ of the beams is less than ten times the bandwidth fS; and wherein the servo system controls the position of the measurement object based on the signal S(t) and the error signal S(t).

39. A lithography system for use in fabricating integrated circuits on a wafer, the system comprising:
a stage for supporting the wafer;
an illumination system for imaging spatially patterned radiation onto the wafer;
a positioning system for adjusting the position of the stage relative to the imaged radiation; and
the apparatus of claim 38 for monitoring the position of the wafer relative to the imaged radiation.

40. A method for fabricating integrated circuits, the method comprising:
applying a resist to a wafer;
forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system of claim 39; and
producing an integrated circuit from the wafer.

41. A lithography system for use in fabricating integrated circuits on a wafer, the system comprising:
a stage for supporting the wafer; and
an illumination system including a radiation source, a mask, a positioning system, a lens assembly, and the apparatus of claim 38, wherein during operation the source directs radiation through the mask to produce spatially patterned radiation, the positioning system adjusts the position of the mask relative to the radiation from the source, the lens assembly images the spatially patterned radiation onto the wafer, and the apparatus monitors the position of the mask relative to the radiation from the source.

42. A method for fabricating integrated circuits, the method comprising:
applying a resist to a wafer;
forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system of claim 41; and
producing an integrated circuit from the wafer.

43. A beam writing system for use in fabricating a lithography mask, the system comprising:
a source providing a write beam to pattern a substrate;
a stage supporting the substrate;
a beam directing assembly for delivering the write beam to the substrate;
a positioning system for positioning the stage and beam directing assembly relative one another; and
the apparatus of claim 38 for monitoring the position of the stage relative to the beam directing assembly.

44. A method for fabricating a lithography mask comprising:
directing a beam to a substrate using the beam writing system of claim 43;
varying the intensity or the position of the beam at the substrate to form a pattern in the substrate; and
forming the lithography mask from the patterned substrate.