Abstract:
In general, in one aspect, the invention features an apparatus that includes an interferometer having a main cavity and an auxiliary reference surface, the main cavity including a partially reflective surface defining a primary reference surface and a test surface. The interferometer is configured to direct a primary portion of input electromagnetic radiation to the main cavity and an auxiliary portion of the input electromagnetic radiation to reflect from the auxiliary reference surface, wherein a first portion of the primary portion in the main cavity reflects from the primary reference surface and a second portion of the primary portion in the main cavity passes through the primary reference surface and reflects from the test surface. The interferometer is further configured to direct the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference surface to a multi-element detector to interfere with one another to form an interference pattern. The auxiliary reference surface is tilted so that the paths of the electromagnetic radiation reflected from the primary reference surface and auxiliary reference are non-parallel at the multi-element detector and the auxiliary reference surface is in the path of the primary portion of the electromagnetic radiation.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     Under 35 USC § 120, this application is a continuation in part application and claims the benefit of U.S. patent application Ser. No. 11/079,946, entitled “INTERFEROMETRY SYSTEMS AND METHODS,” filed on Mar. 15, 2005, which claims priority under 35 USC §119(e)(1) to Provisional Patent Application No. 60/553,312, entitled “METHOD AND APPARATUS FOR INTERFEROMETRIC PROFILING WITH REDUCED SENSITIVITY TO ENVIRONMENTAL EFFECTS,” filed on Mar. 15, 2004. The entire contents of U.S. patent application Ser. No. 11/079,946 and Provisional Patent Application No. 60/553,312 are incorporated by reference herein. 
    
    
     BACKGROUND 
     Interferometric optical techniques are widely used to measure surface profiles of precision optical components. 
     For example, to measure the surface profile of a test surface, one can use an interferometer to combine a test wavefront reflected from the test surface with a reference wavefront reflected from a reference surface to form an optical interference pattern. Spatial variations in the intensity profile of the optical interference pattern correspond to phase differences between the combined test and reference wavefronts caused by variations in the profile of the test surface relative to the reference surface. Phase-shifting interferometry (PSI) can be used to accurately determine the phase differences and the corresponding profile of the test surface. The surface profile measurement of the test surface is relative to the surface profile of the reference surface, which is assumed to be perfect (e.g., flat) or known within the tolerances of the measurement. 
     With PSI, the optical interference pattern is recorded for each of multiple phase-shifts between the reference and test wavefronts to produce a series of optical interference patterns that span, for example, at least a half cycle of optical interference (e.g., from constructive, to destructive interference). The optical interference patterns define a series of intensity values for each spatial location of the pattern, wherein each series of intensity values has a sinusoidal dependence on the phase-shifts with a phase-offset equal to the phase difference between the combined test and reference wavefronts for that spatial location. Using numerical techniques known in the art, the phase-offset for each spatial location is extracted from the sinusoidal dependence of the intensity values to provide a profile of the test surface relative the reference surface. Such numerical techniques are generally referred to as phase-shifting algorithms. 
     The phase-shifts in PSI can be produced by changing the optical path length from the measurement surface to the interferometer relative to the optical path length from the reference surface to the interferometer. For example, the reference surface can be moved relative to the measurement surface. Alternatively, the phase-shifts can be introduced for a constant, non-zero optical path difference by changing the wavelength of the measurement and reference wavefronts. The latter application is known as wavelength tuning PSI and is described, e.g., in U.S. Pat. No. 4,594,003 to G. E. Sommargren. 
     One type of interferometer that is often used for characterizing a surface of a test object is a Fizeau interferometer. In many embodiments, phase shifting for object surface profiling proceeds by mechanical translation of the reference surface or by wavelength tuning, during which time a computer captures successive frames of an interference pattern at a detector for later analysis. 
     In a number of situations, it can be attractive to profile surface without temporal modulation of the Fizeau interference pattern, for example, to accommodate high-speed measurements of dynamically actuated parts. Although a variety of such techniques exist for Twymann-Green interferometer geometries, including for example spatial phase shifting or phase shifting based on polarization, these techniques typically require separating the reference and object beam reflections. However, the common-path characteristics of a large-aperture Fizeau interferometer can make it difficult to separate the reference and object beam reflections spatially or by polarization. 
     SUMMARY 
     In general, in a first aspect, the invention features an apparatus that includes an interferometer having a main cavity and an auxiliary reference surface, the main cavity including a partially reflective surface defining a primary reference surface and a test surface. The interferometer is configured to direct a primary portion of input electromagnetic radiation to the main cavity and an auxiliary portion of the input electromagnetic radiation to reflect from the auxiliary reference surface, wherein a first portion of the primary portion in the main cavity reflects from the primary reference surface and a second portion of the primary portion in the main cavity passes through the primary reference surface and reflects from the test surface. The interferometer is further configured to direct the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference surface to a multi-element detector to interfere with one another to form an interference pattern. The auxiliary reference surface is tilted so that the paths of the electromagnetic radiation reflected from the primary reference surface and auxiliary reference are non-parallel at the multi-element detector and the auxiliary reference surface is in the path of the primary portion of the electromagnetic radiation. 
     Embodiments of the apparatus may include one or more of the following features and/or features of other aspects. For example, the auxiliary reference surface may transmit the primary portion of the electromagnetic radiation. The main cavity can define a Fizeau cavity. There may be no beam shaping optics in the beam path of second portion between the primary reference surface and the test surface. 
     In general, in another aspect, the invention features an apparatus that includes an interferometer having a main cavity and an auxiliary reference surface, the main cavity including a partially reflective surface defining a primary reference surface and a test surface. The interferometer is configured to direct input electromagnetic radiation to reflect from the auxiliary reference surface, to reflect from the primary reference surface and to reflect from the test surface. The interferometer is further configured to direct the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference surface to a multi-element detector to interfere with one another to form an interference pattern. The auxiliary reference surface is further configured to introduce spatial carrier fringes in the interference pattern at the multi-element detector. 
     Embodiments of the apparatus may include one or more of the following features and/or features of other aspects. For example, the auxiliary reference surface can be positioned in the path of the input electromagnetic radiation directed to the primary reference surface. 
     The apparatus can include a beam splitter configured to split the input electromagnetic radiation into an auxiliary portion and a primary portion, wherein the beam splitter directs the auxiliary portion along a path to the auxiliary reference surface and the beam splitter directs the primary portion along a path to the primary reference surface. 
     The auxiliary reference surface can be a partially transmissive surface. Alternatively, the auxiliary reference surface can be a reflective surface. The auxiliary reference surface can be a planar surface. Alternatively, or additionally, the primary reference surface can be a planar surface. In embodiments where both the primary and auxiliary reference surfaces are planar surfaces, the auxiliary reference surface and the primary reference surface can be non-parallel. 
     In some embodiments, the primary reference surface and the auxiliary reference surface are opposing surfaces of an optical component. The optical component can be a transmissive wedge. 
     The main cavity can define a Fizeau cavity. 
     The auxiliary reference surface can be configured to transmit the input electromagnetic radiation directed to reflect from the primary reference surface. The primary reference surface can be configured to reflect input electromagnetic radiation along a first path and the auxiliary reference surface is configured to reflect input electromagnetic radiation along a second path, where the second path overlaps the first path and is non-parallel to the first path. 
     In some embodiments, there are no beam shaping optics in the beam path of second portion between the primary reference surface and the test surface. 
     The apparatus can include a means for selectively preventing electromagnetic radiation from the test surface from reaching the detector. 
     The apparatus can include the multi-element detector and an electronic controller, wherein the electronic controller is configured to determine surface profile information about the test surface based on the interference pattern. The electronic controller can be configured to determine surface profile information about the test surface based on the interference pattern formed by the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference, and a second interference pattern formed by electromagnetic radiation reflected from the primary reference surface and the auxiliary reference, with no electromagnetic radiation reflected from the test surface reaching the detector. 
     The auxiliary reference surface can be tilted relative to an optical axis of the interferometer to form the spatial carrier fringes in the interference pattern. The apparatus can include a quadrature phase detection system including the multi-element detector. The interferometer can include a fold optic configured to allow the primary reference surface to be upward facing and part of a mount configured to support the test surface. In some embodiments, the auxiliary reference is coupled to a transducer configured to vary an optical path length of electromagnetic radiation reflected from the auxiliary reference to the detector. 
     In general, in a further aspect, the invention features an apparatus that includes a multi-element detector, an interferometer comprising a primary reference surface and a test surface, the interferometer being configured to interfere electromagnetic radiation reflected from the primary reference surface and electromagnetic radiation reflected from the test surface to form an interference pattern at the multi-element detector, and a means for generating spatial carrier fringes in the interference pattern at the multi-element detector. 
     Embodiments of the apparatus may include one or more of the following features and/or features of other aspects. For example, the apparatus can further include an electronic controller configured to determine surface profile information about the test surface based on the interference pattern and the carrier fringes. The means for generating the carrier fringes can include an auxiliary reference mirror configured to direct auxiliary electromagnetic radiation along a path to the multi-element detector, wherein the auxiliary electromagnetic radiation interferes with the electromagnetic radiation reflected from the primary reference surface and the test surface to form the carrier fringes in the interference pattern. The interferometer can be a Fizeau interferometer. 
     In general, in another aspect, the invention features apparatus that include an interferometer having a main cavity and an auxiliary reference surface, the main cavity including a primary reference surface and a test surface. The interferometer is configured to direct a primary portion of input electromagnetic radiation to the main cavity and an auxiliary portion of the input electromagnetic radiation to reflect from the auxiliary reference surface, wherein a first portion of the primary portion in the main cavity reflects from the primary reference surface and a second portion of the primary portion in the main cavity reflects from the test surface. The interferometer is further configured to direct the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference to a multi-element detector to interfere with one another to form an interference pattern. 
     In general, in another aspect, the invention features methods that include directing a primary portion of input electromagnetic radiation to a main cavity of an interferometer, wherein a first portion of the primary portion in the main cavity reflects from a primary reference surface of the main cavity and a second portion of the primary portion in the main cavity reflects from a test surface. The methods further include directing an auxiliary portion of the input electromagnetic radiation to reflect from an auxiliary reference surface of the interferometer, and directing the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference surface to a multi-element detector to interfere with one another forming an interference pattern. 
     Embodiments of the apparatus and/or methods can include one or more of the following features. 
     The methods can include determining surface profile information about the test surface based on the interference pattern formed by the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference, and a second interference pattern formed by electromagnetic radiation reflected from the primary reference surface and the auxiliary reference, with no electromagnetic radiation reflected from the test surface reaching the detector. 
     In embodiments, the primary reference surface can be a partially reflective surface. In some embodiments, the second portion of the primary portion in the main cavity passes through the primary reference surface and reflects from the test surface. The main cavity can define a Fizeau cavity. There can be beam shaping optics in the beam path of second portion between the primary reference surface and the test surface. 
     The surface area of the auxiliary reference surface can be smaller than that of the primary reference surface. The auxiliary reference surface can be flat and the primary reference surface can be curved. The apparatus can further include a means for selectively preventing electromagnetic radiation from the test surface from reaching the detector. For example, the apparatus can include an aperture between the primary reference surface and the test surface. 
     The apparatus can further include the multi-element detector and an electronic controller, wherein the electronic controller is configured to determine surface profile information about the test surface based on the interference pattern. The electronic controller can be configured to determine surface profile information about the test surface based on the interference pattern formed by the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference, and a second interference pattern formed by electromagnetic radiation reflected from the primary reference surface and the auxiliary reference, with electromagnetic radiation from the test surface being prevented from the reaching the detector. The auxiliary reference surface can be tilted relative to an optical axis of the interferometer to form spatial carrier fringes in the interference pattern. The apparatus can include a quadrature phase detection system including the multi-element detector. In some embodiments, the auxiliary reference is mounted on a transducer configured to vary an optical path length to the multi-element detector for electromagnetic radiation reflected from the auxiliary reference. The interferometer can include a fold optic (e.g., a mirror or a prism) configured to allow the primary reference surface to be upward facing and part of a mount configured to support the test surface. The interferometer can include a beam splitter (e.g., a non-polarizing beam splitter or a polarizing beam splitter) for separating the primary portion of the input electromagnetic radiation from the auxiliary portion of the input electromagnetic radiation. The interferometer can include one or more imaging optics for imaging the test surface onto the multi-element detector. The apparatus can further include a source for the input electromagnetic radiation (e.g., a laser). 
     The interferometer can be a Fizeau interferometer, a Michelson interferometer, a Linnik interferometer, or a Mirau interferometer. 
     In certain aspects, the methods are implemented by an embodiment of the apparatus. 
     Among other advantages, embodiments of the apparatus and methods can provide Fizeau interferometers that are mechanically stable and relatively insensitive to environmental sources of uncertainty, such as vibrations, which can cause surfaces in the interferometer to move while the interferometer is being used to make measurements on a test part, or between testing different parts. The stability and relative insensitivity to environmental effects result in interferometry systems that are reliable and accurate. 
     Furthermore, embodiments include interferometry systems than can perform surface profiling measurements based on a single frame of an interference pattern, rather than multiple frames that are commonly required in systems that use phase shifting techniques. Single frame measurements can be performed more rapidly than multiple frame techniques. Furthermore, single frame techniques can eliminate the need for moving parts (e.g., transducers for phase shifting), thereby reducing the cost of interferometry systems. 
     Single frame measurements using Fizeau interferometers can be implemented without any optical components between the reference surface and test surface. Accordingly, sources of error in measurements made using the Fizeau interferometers are reduced compared to systems that include optical components in the measurement beam path. 
     Aspects of the invention can be implemented using relatively simple alterations to commercially available interferometry systems. For example, commercially available Fizeau interferometer systems can be readily adapted to include an auxiliary mirror using relatively few additional components, and those components can be relatively inexpensive. 
     In some embodiments, the apparatus and methods can include mechanical phase-shifting interferometer systems having improved properties compared to conventional phase-shifting systems. For example, Fizeau interferometer systems can include an auxiliary mirror mounted on a transducer, so that the mechanical phase shifting is performed outside of the Fizeau cavity. In large aperture systems, this can be particularly advantageous since the auxiliary mirror can be much smaller in size than the reference mirror, requiring a less complicated transducer system for phase shifting. 
     Including an auxiliary mirror in a Fizeau interferometer system can allow for relatively easy implementation of quadrature phase measurements using polarization. This allows beam polarization to be managed way from the Fizeau cavity, allowing such phase measurements to be made without including any additional components (e.g., a quarter wave plate) between the reference and test surfaces. 
     Auxiliary mirrors also provide a convenient way to measure phases using carrier fringe techniques. In certain embodiments, for example, carrier fringes can be introduced into an interference pattern generated using a Fizeau interferometer by tilting the auxiliary mirror relative to the beam path, rather than the reference mirror. Tilting the auxiliary mirror also allows one to make measurements with the reference mirror orientated on the null position. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram showing an embodiment of an interferometry assembly that includes a Fizeau interferometer augmented by an auxiliary reference mirror. 
         FIG. 2  is a diagram showing the interferometry assembly from  FIG. 1  modified to have a narrow annular aperture, blocking the view of the object and therefore preserving a portion of the single-surface reflection from the Fizeau reference. 
         FIG. 3  is a diagram showing an embodiment of an interferometry assembly that includes a spherical Fizeau cavity and an auxiliary reference mirror. 
         FIG. 4  is a diagram showing an embodiment of an interferometry assembly that includes a Fizeau interferometer augmented by an auxiliary reference mirror configured for instantaneous quadrature phase measurement using polarization. 
         FIG. 5  is a diagram showing an embodiment of an interferometry assembly that includes a Fizeau interferometer augmented by an auxiliary reference mirror configured for phase measurement using mechanical phase shifting of the auxiliary reference mirror. 
         FIG. 6  is a diagram showing an embodiment of an interferometry assembly that includes an upward-looking Fizeau interferometer augmented by an auxiliary reference. 
         FIG. 7  is a diagram showing another embodiment of an interferometry assembly that includes an upward-looking Fizeau interferometer augmented by an auxiliary reference. 
     
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Embodiments described below include Fizeau interferometers that are augmented by the inclusion of an auxiliary reference mirror. The auxiliary reference mirrors allow a variety of interferometry techniques to be implemented using a Fizeau interferometer without having to introduce any additional components between the Fizeau&#39;s reference and test surfaces or altering the alignment between these surfaces. For example, the auxiliary mirror can be configured to introduce carrier fringes into a Fizeau interference pattern, enabling characterization of a test surface from a single interferogram frame formed using the test surface. Furthermore, as another example, the auxiliary mirror enables quadrature phase measurements using polarization to be implemented in a Fizeau interferometer relatively easily and without requiring additional optical components between the reference and test surfaces of the Fizeau interferometer. Since any optical components in the measurement beam path are sources of first order errors in measurements made using a Fizeau interferometer, using an auxiliary mirror can allow these techniques to be implemented without any substantial loss in system accuracy. 
     Auxiliary reference mirrors can also be used to implement mechanical phase shifting in a Fizeau interferometer without moving either of the reference or test surfaces. The environmental stability associated with the stationary Fizeau cavity can enhance the system&#39;s accuracy and repeatability. 
     Referring to  FIG. 1 , interferometry system  100  is adapted to measure the optical interference produced by reflections from a reference surface  141  of a reference optic  140 , a test surface  191  of a test object  190  (e.g., an optical flat), and a surface  151  of an auxiliary reference mirror  150 , which may be substantially smaller than reference optic  140 . Reference surface  141  and test surface  191  define a Fizeau cavity, labeled as cavity  101  in  FIG. 1 . System  100  includes a mount  192  for supporting test object  190  relative to reference optic  150 . 
     Interferometry system  100  also includes a beam splitter  120  that separates a beam from a light source (e.g., a laser diode, HeNe laser or the like) into two component beams, corresponding to a reflected component and a transmitted components of the input beam. The reflected component is directed towards Fizeau cavity  101 , while the transmitted components is directed towards auxiliary reference mirror  150 . Beam splitter  120  also combines light reflected from Fizeau cavity  101  and from auxiliary reference mirror  150 , and directs the combined light beam to a pixelated detector  160  (e.g., a CCD camera). 
     As depicted in  FIG. 1 , interferometry system  100  also includes several optical components, including a collimating lens  115  that collimates diverging light from source  110  before the light is incident on beam splitter  120 . Interferometry system  100  also includes beam shaping optics, namely lenses  130  and  132 , which expand the light reflected by beam splitter  120  prior to contacting reference optic  140 . In general, interferometry systems  100  can include further optical components in addition to collimating lens  115  and beam shaping lenses  130  and  132 . 
     During operation, source  110  illuminates beam splitter  120 , which transmits a portion of the illumination to reflect from auxiliary reference mirror surface  151 , and reflects a portion of the illumination towards Fizeau cavity  101 . This illumination is partly transmitted by reference optic  140  and reflects from surface  191  of test object  190 . In addition, a portion of the illumination incident on reference optic  140  from beam splitter  120  is reflected by reference surface  141 . Illumination reflected from reference surface  141  and from test object surface  191  propagate along a common path from Fizeau cavity  101  back through beam splitter  120  onto detector  160 . In addition, beam splitter reflects a portion of illumination reflected from surface  151  of auxiliary reference mirror  150  towards detector  160 . Wavefronts incident on detector  160  from mirror  150 , reference surface  141 , and test object surface  191  interfere, generating a pattern of fringes of varying intensity. 
     Interferometry system  100  includes an electronic controller  180 , which is in communication with detector  160 . Electronic controller  180  includes a frame grabber for storing images detected by detector  160 . Electronic controller  180  analyzes the images stored by the frame grabber, and provides a user with information about test surface  191  based on the analysis. 
     Auxiliary reference mirror  150  is nominally flat and is oriented so that mirror surface  151  is tilted with respect to the path of illumination reflected from reference surface  141  to introduce carrier fringes to the interference pattern formed by the measurement and reference wavefronts at detector  160 . In general, the tilt is sufficient to introduce multiple fringes into an interference pattern formed by wavefronts reflected from auxiliary reference mirror  150  and wavefronts reflected from reference surface  141 . In some embodiments, auxiliary reference mirror  150  is oriented so that the carrier fringes have a period corresponding to about three or more detector pixels at detector  160  (e.g., about five or more, about eight or more, about 10 or more, about 20 or more detector pixels). Auxiliary reference mirror  150  can be mounted on an adjustable mount to facilitate easy adjustment of the auxiliary mirror&#39;s orientation. 
     Determining a surface profile of test surface  191  using interferometry system  100  involves a two-step process: one interferometer measurement without the test object  191  and one measurement with the test object. For each measurement, electronic controller  180  determines an interference phase, θ, and a signal modulation, M, for the image data at each detector pixel. 
     One way to extract phase information from each measurement is to perform a spatial Fourier transform of the interference pattern and subsequently identify the carrier fringe frequency with a digital filter in the frequency domain. An inverse transform of the filtered spectrum provides the phase θ and signal modulation M information at each pixel. Phase extraction methods using Fourier transforms are described, for example, by W. Macy in “Two Dimensional Fringe Pattern Analysis,”  Appl. Opt.,  22, pp. 3898-3901 (1983), the entire contents of which are hereby incorporated by reference. 
     The surface height profile for test surface  141  is related to the phase information for each measurement based on the following derivation. In  FIG. 1 , when test optic  140  is not included in the system, the system measures the complex reflectivity r 2 t 2  with respect to r 1 , where r 1  is the effective amplitude reflectivity of the auxiliary reference, r 2  is the internal amplitude reflectivity of reference surface  141 , and t is the effect of transmission through the beam shaping optics and through the substrate of the reference optic  140 . 
     Setting aside irrelevant overall constants related to cavity length and phase change on reflection effects, the measured phase at each pixel is given by
 
θ A =θ′ 2 −θ 1   (1)
 
where
 
θ 1 =arg( r   1 )  (2)
 
θ′ 2 =arg( t   2   r   2 )  (3)
 
Here
 
 r   1 =√{square root over ( R   1 )}exp( iθ   1 )  (4)
 
 r   2 =√{square root over ( R   2 )}exp( iθ   2 )  (5)
 
 t=√{square root over (T)} exp( iθ   t )  (6)
 
and therefore
 
θ′ 2 =2θ t +θ 2   (7)
 
The phase values relate to surface heights according to
 
θ 1 =2kh 1   (8)
 
θ 2 =2kh 2   (9)
 
for
 
 k= 2π/λ  (10)
 
and λ is the source wavelength.
 
     The measured signal modulation can be expressed as follows:
 
 M   A =2V T√{square root over (R 1   R   2 )}  (11)
 
where V is a fringe visibility coefficient, assumed to be independent of field position and constant over time.
 
     For the interference pattern acquired with test object  190  included, the ray tracing is common path for light reflected from test surface  191  and reference surface  141 , with a phase measurement for each pixel now given by
 
θ B =θ′ Z −θ 1   (12)
 
where
 
θ′ Z =2θ t +θ Z   (13)
 
and the phase θ Z  returned by the Fizeau cavity itself is
 
θ Z =arg( z )  (14)
 
For
 
 z =( r   2   +r   3 ).  (15)
 
The measured signal modulation is
 
 M   B =2 V T √{square root over (R 1   Z )}  (16)
 
where Z=|z| 2 .
 
     The phase term, θ 3 , corresponding to the surface profile of test surface  191  at each pixel, is determined from the measured phases and signal modulations as follows. The complex reflectivity, z, of Fizeau cavity  101  can be calculated for each pixel from the measured data from: 
                   z   =         M   B       2   ⁢   VT   ⁢       R   1           ⁢       exp   ⁢           [     ⅈ   ⁡     (       θ   B     -     θ   A     +     θ   2       )       ]     .               (   17   )               
The reflectivity of reference surface  141  is from Eq. (11) and Eq.(5)
 
                     r   2     =         M   A       2   ⁢   VT   ⁢       R   1           ⁢   exp   ⁢           ⁢       (     ⅈ   ⁢           ⁢     θ   2       )     .               (   18   )               
Therefore, since from Eq. (15)
 
                       r   3     =     z   -     r   2         ,     
     ⁢     one   ⁢           ⁢   has             (   19   )                 r   3     =           exp   ⁢           ⁢     (     ⅈ   ⁢           ⁢     θ   2       )         2   ⁢   VT   ⁢       R   1           ⁡     [         M   B     ⁢   exp   ⁢           ⁢     (       ⅈ   ⁢           ⁢     θ   B       -     ⅈ   ⁢           ⁢     θ   A         )       -     M   A       ]       .             (   20   )               
the complex argument of which is the desired phase:
 
     
       
         
           
             
               
                 
                   
                     
                       θ 
                       
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                     = 
                     
                       
                         arg 
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 M 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   B 
                                 
                               
                               ⁢ 
                               
                                 exp 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ⅈθ 
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                       
                                     
                                     - 
                                     
                                       ⅈθ 
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         A 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               M 
                               
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                               
                             
                           
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                       + 
                       
                         θ 
                         
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Where the profile of reference surface, h 2 , is known, equation (21) allows one to determine the surface profile of test surface  191  independent of the effects t of the optical system. 
     In certain embodiments, it may be desirable to reduce the reflectivity of reference surface  141 . In such instances, an antireflection (AR) coating can be applied to reference surface  141 . This reduces the finesse of Fizeau cavity  101 , which can reduce the effect of errors associated with instability of the optical path between reference surface  141  and auxiliary mirror surface  151 . For example, the technique as described above is sensitive to the drift of the “internal” cavity formed by reference surface  141  and auxiliary mirror reference surface  151  than may occur between the first and the second measurements, potentially introducing measurement errors in unstable environments. The sensitivity is generally greatest for high-finesse fringes between r 2  and r 3 . The magnitude of the print through declines with the finesse of Fizeau cavity  101 , which is reduced when an AR coating is applied to reference surface  141 . For example, if R 2 =0.5% and R 3 =4%, the resulting errors for an internal cavity (i.e., the cavity formed by reference surface  141  and auxiliary mirror surface  151 ) drift are reduced by a factor of three. 
     Alternatively, or additionally, interferometer system  100  can be designed to be sufficiently rigid to substantially prevent any internal cavity drift between first and second measurements associated with characterizing a part. 
     In certain embodiments, the stability of the internal cavity can be monitored between measurements by including a portion of the interferometer system that monitors the interference pattern associated with the internal cavity regardless of whether or not a test object is positioned in the system. For example, an annulus can be provided in the field of view of detector  160  that examines only reference surface  141 , and excludes the test surface  191  reflection, r 3 . 
     Referring to  FIG. 2 , in some embodiments, an annulus can be provided by including an aperture  210  within Fizeau cavity  101 . For example, if test object  190  is larger than the system&#39;s field of view, aperture  210  excludes reflections from test surface  151  by masking portions of the test surface. 
     Including such an annulus in interferometer  100  allows one to determine the relative tip, tilt and piston of the internal cavity every time a measurement is made. The measured drift values modify the Eq. (21) to
 
θ 3 =arg[ M   B  exp( iθ   B   −iθ   A   +iθ   drift )− M   A ]+θ 2   (22)
 
where θ drift  is the phase change at a pixel associated with the drift in the internal cavity between measurements.
 
     Referring to  FIG. 3 , an interferometry system  300  can be adapted for profiling a curved test surface of a test object  390 . Interferometry system  300  includes a reference object  310  that has a curved (e.g., spherical) reference surface  311 . Reference object  310  and test object  390  defines a Fizeau cavity  301 . The interferometry system&#39;s beam shaping optics also includes an additional lens  320  which focuses illumination from beam splitter  120  so that it is nominally normally incident on the curved surface of test object  390 . Aside from the different geometry of the Fizeau cavity, the operation of interferometry system  300  is the same as interferometry system  100 . 
     While in the preceding embodiments phase information is determined using carrier fringes, in general, other techniques can also be used. For example, in some embodiments phase measurements can be obtained from a single interference pattern using polarization to provide phase quadrature between orthogonal polarizations in the interference pattern. Referring to  FIG. 4 , an example of an interferometry system adapted for phase quadrature measurements using polarization is interferometry system  400 . System  400  includes a polarizing beam splitter (PBS)  420  which splits incident illumination from source  110  (not shown in  FIG. 4 ) into component beams having orthogonal linear polarization states. One beam reflects from surface  151  of auxiliary reference mirror  150 , while the other reflects from the reference and test surface of Fizeau cavity  101 . Quarter wave plates  410  are positioned in the path of each beam so that the polarization state of each beam is rotated by 90 degrees. This ensures that the beam originally transmitted by PBS  420  is now reflected, and the originally reflected beam, now reflected from Fizeau cavity  101 , is transmitted by the PBS. 
     PBS  420  directs illumination reflected from auxiliary reference mirror  150  and Fizeau cavity  101  towards a detector assembly  401  that includes pixelated detectors  460  and  470 , and non-polarizing beam splitter  435 . Detector assembly  401  also includes lenses  430 ,  440 , and  445 , which serve to focus and collimate the illumination from PBS  420 . Beam splitter  435  reflects a portion of the incoming illumination from PBS  420  toward detector  470 , and transmits a portion towards detector  460 . A linear polarizer is positioned between each detector and beam splitter  435 . In particular, polarizer  462  is positioned in front of detector  460 , and polarizer  472  is positioned in front of detector  470 . Both polarizers are oriented at 45 degrees with respect to the transmission axis of PBS  420 , ensuring that both detectors sample illumination reflected from both the auxiliary reference mirror and the Fizeau cavity. A quarter waveplate is positioned between polarizer  472  and beam splitter  435 , and is oriented to introduce a 90 degree phase shift into the polarization state transmitted by polarizer  472 . Accordingly, the interference phase of the interference pattern at each detector pixel for one detector is offset by 90 degrees relative to the phase of the interference pattern at the corresponding pixel of the other detector. 
     Signals in quadrature allow for rapid measurement of modulation and phase assuming that the constant intensity offset I DC  is known. Thus if the measured intensity without the object for one camera is
 
 g   A   =I   A   DC   +M   A  cos(θ A )  (23)
 
then for the other camera in quadrature the intensity is
 
                       g   A   quad     =       I   A   DC     +       M   A     ⁢   sin   ⁢           ⁢     (     θ   A     )           ⁢     
     ⁢   then           (   24   )                 tan   ⁢           ⁢     (     θ   A     )       =         g   A   quad     -     I   A   DC           g   A     -     I   A   DC                 (   25   )                 M   A   2     =         (       g   A   quad     -     I   A   DC       )     2     +       (       g   A     -     I   A   DC       )     2               (   26   )               
To determine I A   DC , one can make a separate measurement with the auxiliary reference either highly tilted or in rapid movement so as to average out the modulation terms M A  cos(θ A ) and M A  sin(θ A ). In an alternative embodiment, the addition of at least one more camera with an additional phase shift provides sufficient information to solve for modulation, average intensity and phase simultaneously, as described by R. Smythe and R. Moore, Proc. Soc. Phot. Opt. Eng. 429, 16 (1983).
 
     In further embodiments, polarization-encoded reference and measurement beams project multiple images at various phase shifts onto a single camera detector, in a manner similar to that described by Kujawinska et al. in the Chapter “Spatial phase measurement methods,” in  Interferogram Analysis: Digital Fringe Pattern Measurement Techniques , edited by David W. Robinson and Graeme T. Reid, Institute of Physics Publishing, Philadelphia, Pa. (May 1, 1993), the entire contents of which are hereby incorporated by reference, and by James Millerd, et al., in “Pixelated Phase-Mask Dynamic Interferometer,” Proc. SPIE, Vol.5531 (2004), the entire contents of which are hereby incorporated by reference. 
     In some embodiments, mechanical phase shifting can be used to determine phase information in interferometry systems that include auxiliary mirrors. For example, referring to  FIG. 5 , in an interferometry system  500 , auxiliary mirror  150  is mounted on a transducer  510  (e.g., a piezoelectric transducer) which varies the optical path length between auxiliary reference surface  151  and beam splitter  120 . Transducer  510  is controlled by signals from electronic controller  180 , which synchronizes data acquisition at detector  160  with the motion of the auxiliary reference mirror. Transducer  510  displaces auxiliary reference mirror  150  smoothly over a range of approximately one wavelength, thereby introducing a sequence of N phase shift increments α while detector  160  records corresponding N intensity values g j  for j=0 . . . N−1 for each image pixel, as described, e.g., in Encyclopedia of Optics, vol. 3 (2004, Wiley-VCH Publishers, Weinheim) pp. 2100-2101. For example, for N=7 and phase shift increments α=π/2, the phase is given by 
                     tan   ⁢           ⁢     (   θ   )       =         7   ⁢     (       g   2     -     g   4       )       -     (       g   0     -     g   6       )             -   4     ⁢     (       g   1     +     g   5       )       +     8   ⁢     g   3                   (   27   )               
and the signal modulation is given by
 
                     M   2     =           [       7   ⁢     (       g   2     -     g   4       )       -     (       g   0     ⁢     g   6       )       ]     2     +       [       8   ⁢     g   3       -     4   ⁢     (       g   1     +     g   5       )         ]     2         16   2               (   28   )               
Alternatively, or additionally, other methods of determining interference phase and modulation with the aid of an auxiliary reference mirror can be used.
 
     In the embodiments described above, the aperture of the Fizeau cavity is sufficiently large to capture reflections from the entire test surface. This is achieved, in part, by using a large reference optic (e.g., the size of reference surface  141  is the same size or larger than the size of test surface  191 ). Indeed, where an interferometry system is designed for characterizing large test parts, the corresponding reference optic can be large and heavy. Mechanical phase shifting by translating the small auxiliary mirror can be advantageous since the size of the Fizeau aperture does not place the same size requirements on the auxiliary mirror as it does for the reference optic  140 . Accordingly, the transducer can be smaller and simpler compared to a transducer for reference optic  140 . Furthermore, due to the physical distance between the auxiliary reference and the Fizeau cavity, the auxiliary reference mirror can be physically isolated more effectively from the test object compared to reference object  140 , providing better stability of the Fizeau cavity during data acquisition. 
     Referring to  FIG. 6 , in some embodiments, interferometry systems can be configured to be upward-looking, for easy test part handling. Interferometry system  600 , for example, includes a fold mirror  610  between beam splitter  120  and the reference object  640  and test part  690 . fold mirror  610  directs illumination from beam splitter  120  along a vertical path to illuminate reference optic  640  which lies in a horizontal plane. Test part  690  can be easily mounted with test surface  691  relative to reference surface  641  by laying the test part horizontally on a mount  692  that maintains a small amount of separation between test surface  691  and reference surface  641 . Since gravity automatically aligns the test part to reference object  640 , tip and tilt alignments can be eliminated for all flat parts, regardless of size, in this configuration. 
     In the foregoing embodiments, the path of light between to the auxiliary reference surface and the primary reference surface are located along different paths. More generally, however, other configurations can also be used. In some embodiments, an interferometry system can include an auxiliary reference surface that is positioned in the path of light from the source to reference surface. For example, referring to  FIG. 7 , an interferometry system  700  includes a transmissive optical component  720  whose opposing surfaces  730  and  740  serve as the primary reference surface and auxiliary reference surface, respectively. A test object  710  is positioned with a test surface  711  relative to primary reference surface  730 . 
     During operation, auxiliary reference surface  740  reflects a portion of the light from source  110  incident thereon. Primary reference surface  730  reflects a portion of the light transmitted by the auxiliary reference surface. The light transmitted by transmissive optical component  720  is incident on test surface  711  and at least a portion reflects therefrom. Light reflected from the primary reference surface  730  and test surface  711  is transmitted again by transmissive optical component  720  and overlaps with the light reflected from auxiliary reference surface  740 . The overlapping light results in an interference pattern at detector  160 . 
     Transmissive optical component  720  is a wedge element. The wedge element transmits at least some of the incident light from the source. For example, the wedge element can transmit about 50% or more (e.g., about 60% or more, about 70% or more, about 80% or more) of light at the system&#39;s operational wavelength normally incident thereon. 
     The wedge angle of transmissive optical component  720  can vary as desired. Typically, the wedge angle is selected based on the desired carrier fringe frequency in the detected interference pattern. The wedge angle should be sufficiently small so that the light reflected from auxiliary reference surface overlaps the light from primary reference surface at detector  160 . Typically, the wedge angle is selected to introduce an appropriate number of carrier fringes at the detector, consistent with the detector resolution. In some embodiments, the wedge angle is in a range from 0.1° to about 10°. 
     Auxiliary reference surface  740  may be the same general size and shape as primary reference surface  730 . 
     In general, the thickness of transmissive optical component  720  can vary as desired. In certain embodiments, the thickness of the component can be selected so that the optical path length between the auxiliary reference surface is a certain amount (e.g., different from the optical path length of the primary cavity). In the present embodiments, the thickness of transmissive optical component  720  determines the distance between the auxiliary reference surface and the primary reference surface. In certain embodiments, auxiliary reference surface  740  is positioned relatively close to primary reference surface  730 . For example, auxiliary reference surface  740  can be about 5 cm or less (e.g., about 3 cm or less, about 2 cm or less, about 1 cm or less, about 0.5 cm or less) from primary reference surface  730 . 
     The reflectivity of auxiliary reference surface  740  can vary as desired. Generally, transmissive optical component  720  is designed so that auxiliary reference surface  740  has a reflectivity that provides suitable contrast of the carrier fringes in the interference pattern at the detector. Typically, the reflectivity of auxiliary reference surface  740  is the same as or greater than the reflectivity of primary reference surface  730 . In some cases, the reflectivity of the auxiliary and primary reference surfaces are relatively high (e.g., about 50% or more). Alternatively, their reflectivities can be relatively low (e.g., about 10% or less). In certain embodiments, the reflectivity of auxiliary reference surface  740  is relatively high, while the reflectivity of primary reference surface  730  is relatively low. The measurement can become increasingly sensitive to changes in the distance between the auxiliary and primary reference surfaces as their reflectivities become closer. Accordingly, in certain embodiments, it may be advantageous for their reflectivities to be different. 
     In some embodiments, one or both of auxiliary reference surface  740  and primary reference surface  730  can include an optical coating, such as a coating that decreases the reflectivity of the surface (e.g., an antireflection coating) or a coating that increases the reflectivity of the surface (e.g., a high-index coating). 
     Although auxiliary reference surface  740  and primary reference surface  730  are opposing surfaces of a common component in interferometry system  700 , in general, the auxiliary reference surface and the primary reference surface can be surfaces of different components. For example, the auxiliary reference surface can be a surface of a first transmissive optical flat (e.g., a glass flat with parallel opposing surfaces) while the primary reference surface is a surface of a second transmissive optical flat where the flats are oriented so that their surfaces are non-parallel. 
     The component that includes the auxiliary reference surface can be mounted in an adjustable mount (e.g., manually adjustable or adjustable using an electronically controlled actuator). For example, the mount can be used to adjust the distance between the auxiliary reference surface and the primary reference surface. Alternatively, or additionally, the mount can be used to adjust the orientation of the auxiliary reference surface with respect to the primary reference surface. In some embodiments, the mount can include piezoelectric actuators that are configured to effect phase shifting of the carrier fringes, enabling phase shifting techniques to be used to profile the surface of the test object. 
     While the foregoing embodiments involve the use of an auxiliary mirror in conjunction with a Fizeau interferometer, in general, the phase and magnitude techniques using an auxiliary mirror can be used with other types of interferometers as well. For example, the phase and magnitude techniques using an auxiliary mirror can be adapted for use with a conventional interference microscope objective, such as a Mirau, Michelson or Linnik. Conventional interference microscopes can be adapted to include an auxiliary reference mirror with relatively little modification of the standard geometry. 
     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.