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Patent US7375823 - Interferometry systems and methods of using interferometry systems - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Sign in<nobr>Advanced Patent Search</nobr>PatentsIn general, in one aspect, the invention features methods that include interferometrically monitoring a distance between an interferometry assembly and a measurement object along each of three different measurement axes while moving the measurement object relative to the interferometry assembly. The...http://www.google.com/patents/US7375823?utm_source=gb-gplus-sharePatent US7375823 - Interferometry systems and methods of using interferometry systemsAdvanced Patent SearchPublication numberUS7375823 B2Publication typeGrantApplication numberUS 11/365,991Publication dateMay 20, 2008Filing dateMar 1, 2006Priority dateApr 22, 2004Fee statusPaidAlso published asUS20060187464, WO2007103082A2, WO2007103082A3Publication number11365991, 365991, US 7375823 B2, US 7375823B2, US-B2-7375823, US7375823 B2, US7375823B2InventorsGary Womack, Henry A. HillOriginal AssigneeZygo CorporationExport CitationBiBTeX, EndNote, RefManPatent Citations (83), Non-Patent Citations (9), Classifications (24), Legal Events (2) External Links: USPTO, USPTO Assignment, EspacenetInterferometry systems and methods of using interferometry systemsUS 7375823 B2Abstract In general, in one aspect, the invention features methods that include interferometrically monitoring a distance between an interferometry assembly and a measurement object along each of three different measurement axes while moving the measurement object relative to the interferometry assembly. The methods also include monitoring an orientation angle of the measurement object with respect to a rotation axis non-parallel to the three different measurement axes while the measurement object is moving, determining values of a parameter for different positions of the measurement object from the monitored distances, wherein for a given position the parameter is based on the distances of the measurement object along each of the three different measurement axes at the given position, and deriving information about a surface figure profile of the measurement object from a frequency transform of at least the parameter values. Deriving the information includes accounting for variations of the monitored orientation angle during the measurement object's motion.
CROSS-REFERENCE TO RELATED APPLICATIONS This application is a continuation-in-part application of and claims priority to U.S. application Ser. No. 11/112,681, entitled �INTERFEROMETRY SYSTEMS AND METHODS OF USING INTERFEROMETRY SYSTEMS,� filed on Apr. 22, 2005, now U.S. Pat. No. 7,280,224 which claims priority under 35 USC �119(e)(1) to Provisional Patent Application No. 60/564,448, entitled �MULTI-AXIS INTERFEROMETER AND DATA PROCESSING FOR MIRROR MAPPING,� filed on Apr. 22, 2004 and to Provisional Patent Application No. 60/644,898, entitled �MULTI-AXIS INTERFEROMETER AND DATA PROCESSING FOR MIRROR MAPPING,� filed on Jan. 19, 2005. The entire contents of U.S. application Ser. No. 11/112,681, Provisional Patent Application No. 60/564,448, and Provisional Patent Application No. 60/644,898 are hereby incorporated by reference.
TECHNICAL FIELD This disclosure relates to interferometry systems and methods of using interferometry systems.
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 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 2νnp/λ, where ν 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 phase of the measured interference signal correspond to changes in the relative position of the measurement object, e.g., a change in phase of 2π corresponds substantially to a distance change d of λ/(2np). Distance 2d is a round-trip distance change or the change in distance to and from a stage that includes the measurement object. In other words, the phase Φ, ideally, is directly proportional to d, and can be expressed as
Φ = 2 ⁢ pkd , ⁢ where ⁢ ⁢ k = 2 ⁢ ⁢ π ⁢ ⁢ n λ . ( 1 ) Unfortunately, the observable interference phase, {tilde over (Φ)}, is not always identically equal to phase Φ. Many interferometers include, for example, non-linearities such as �cyclic errors.� Cyclic errors can be expressed as contributions to the observable phase and/or the intensity of the measured interference signal and have a sinusoidal dependence on the change in for example optical path length 2pnd. In particular, a first order cyclic error in phase has for the example a sinusoidal dependence on (4πpnd)/λ and a second order cyclic error in phase has for the example a sinusoidal dependence on 2(4πpnd)/λ. Higher order cyclic errors can also be present as well as sub-harmonic cyclic errors and cyclic errors that have a sinusoidal dependence of other phase parameters of an interferometer system comprising detectors and signal processing electronics. Different techniques for quantifying such cyclic errors are described in commonly owned U.S. Pat. No. 6,137,574, U.S. Pat. No. 6,252,688, and U.S. Pat. No. 6,246,481 by Henry A. Hill.
A second source of non-cyclic errors is the effect of �beam shearing� of optical beams across interferometer elements and the lateral shearing of reference and measurement beams one with respect to the other. Beam shears can be caused, for example, by a change in direction of propagation of the input beam to an interferometer or a change in orientation of the object mirror in a double pass plane mirror interferometer such as a differential plane mirror interferometer (DPMD) or a high stability plane mirror interferometer (HSPMI).
SUMMARY In general, in one aspect, the invention features methods that include interferometrically monitoring a distance between an interferometry assembly and a measurement object along each of three different measurement axes while moving the measurement object relative to the interferometry assembly. The methods also include monitoring an orientation angle of the measurement object with respect to a rotation axis non-parallel to the three different measurement axes while the measurement object is moving, determining values of a parameter for different positions of the measurement object from the monitored distances, wherein for a given position the parameter is based on the distances of the measurement object along each of the three different measurement axes at the given position, and deriving information about a surface figure profile of the measurement object from a frequency transform of at least the parameter values. Deriving the information includes accounting for variations of the monitored orientation angle during the measurement object's motion.
x j = 1 2 ⁢ ( x j ′ + x 0 ′ ) , j = 1 , 2 ⁢ ⁢ and ⁢ ⁢ 3. ( 2 ) The difference between two linear displacements xi and xj, i≠j, referred to as a first difference parameter (FDP) is independent of x′0, i.e.,
x i - x j = 1 2 ⁢ ( x i ′ - x j ′ ) , ⁢ i , j = 1 , 2 , and ⁢ ⁢ 3 , ⁢ i ≠ j . ⁢ ( 3 ) Reference is made to FIG. 2 which is a diagrammatic perspective view of an interferometry system 15 that employs a pair of orthogonally arranged interferometers or interferometer subsystems by which the shape of on-stage mounted stage mirrors may be characterized in situ along one or more datum lines. As shown in FIG. 2, system 15 includes a stage 16 that can form part of a microlithography tool. Affixed to stage 16 is a plane stage mirror 50 having a y-z reflective surface 51 elongated in the y-direction.
SDP ⁢ ⁢ ( y ) ≡ ( x 2 - x 1 ) - b 2 b 3 - b 2 ⁢ ( x 3 - x 2 ) ( 4 ) or SDP ⁢ ⁢ ( y ) = ( x 2 - x 1 ) - η ⁢ ⁢ ( x 3 - x 2 ) ( 5 ) where η ≡ b 2 b 3 - b 2 . ( 6 ) Note that SDP can be written in terms of single pass displacements using Equation (3), i.e.,
SDP ⁢ ⁢ ( y ) = 1 2 ⁡ [ ( x 2 ′ - x 1 ′ ) - η ⁢ ⁢ ( x 3 ′ - x 2 ′ ) ] . ( 7 ) Of course, corresponding equations apply for a y-axis stage mirror.
SDP 1 e ⁡ ( y , ϑ z ) ≡ ⁢ 1 2 ⁢ { [ x 2 ′ ⁡ ( y ) - x 1 ′ ⁡ ( y ) ] + ⁢ η ⁢ [ x 2 ′ ⁡ ( y ) - x 1 ′ ⁡ ( y - L + 2 ⁢ b 3 ) ] } - ⁢ x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP 1 e ⁡ ( y , ϑ z = 0 ) ] , ⁢ 1 2 - 2 ⁢ ⁢ ( b 3 - b 2 ) L ≤ y L ≤ 1 2 , ( 9 ) SDP 2 e ⁡ ( y , ϑ z ) ≡ ⁢ - 1 2 ⁢ { [ x 3 ′ ⁡ ( y + L - 2 ⁢ b 3 ) - x 2 ′ ⁡ ( y ) ] + η ⁢ [ x 3 ′ ⁡ ( y ) - x 2 ′ ⁡ ( y ) ] } - ⁢ x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP 2 e ⁡ ( y , ϑ z = 0 ) ] , - ⁢ 1 2 ≤ y L ≤ - 1 2 + 2 ⁢ b 2 2 . ( 10 ) where the last term in Equations (9) and (10) is a third order term with an origin in a second order geometric term such as described in commonly owned U.S. patent application Ser. No. 10/347,271 entitled �COMPENSATION FOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETERS� and U.S. patent application Ser. No. 10/872,304 entitled �COMPENSATION FOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETER METROLOGY SYSTEMS,� both of which are by Henry A. Hill and both of which are incorporated herein in their entirety by reference.
SDP 1 e ⁡ ( y , ϑ z ) ≡ ⁢ 1 2 ⁢ ⁢ η ⁢ ( 1 + η ) ⁡ [ x 2 ′ ⁡ ( y , t 1 ) - x 1 ′ ⁡ ( y , t 1 ) ] + { [ x 3 ′ ⁡ ( y - 2 ⁢ b 3 , t 2 ) - x 1 ′ ⁡ ( y - 2 ⁢ b 3 , t 2 ) ] + [ x 3 ′ ⁡ ( y - 2 � 2 ⁢ b 3 , t 3 ) - x 1 ′ ⁡ ( y - 2 � 2 ⁢ b 3 , t 3 ) ] + ⋮ [ x 3 ′ ⁡ ( y - ( q - 1 ) ⁢ 2 ⁢ b 3 , t q ) - x 1 ′ ⁡ ( y - ( q - 1 ) ⁢ 2 ⁢ b 3 , t q ) ] } ⁢ ⁢ - x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP 1 e ⁡ ( y , ϑ z = 0 ) ] , ⁢ 1 2 - 2 ⁢ ⁢ ( b 3 - b 2 ) L ≤ y 2 ≤ 1 2 , ( 11 ) SDP 2 e ⁡ ( y , ϑ z ) = ⁢ - 1 2 ⁢ ⁢ ( 1 + η ) ⁡ [ x 3 ′ ⁡ ( y , t 1 ) - x 2 ′ ⁡ ( y , t 1 ) ] + { [ x 3 ′ ⁡ ( y + L - 2 ⁢ b 3 , t 2 ) - x 1 ′ ⁡ ( y + L - 2 ⁢ b 3 , t 2 ) ] + [ x 3 ′ ⁡ ( y + L - 2 � 2 ⁢ b 3 , t 3 ) - x 1 ′ ⁡ ( y + L - 2 � 2 ⁢ b 3 , t 3 ) ] + ⋮ [ x 3 ′ ⁡ ( y + L - ( q - 1 ) ⁢ 2 ⁢ b 3 , t q ) - x 1 ′ ⁡ ( y + L - ( q - 1 ) ⁢ 2 ⁢ b 3 , t q ) ] } ⁢ ⁢ - x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP 2 e ⁡ ( y , ϑ z = 0 ) ] , ⁢ - 1 2 ≤ y L ≤ - 1 2 + 2 ⁢ b 2 L , ( 12 ) where ⁢ ⁢ t i ≠ t j , i ≠ j . FIG. 2 b illustrate the relationship of the domains for which SDP, SDP1 e, and SDP2 e are defined on mirror surface 51. Domain L corresponds to a portion of mirror surface 51 along a scan line 201 in the y-direction. The locations of the first pass beams on mirror surface 51 are shown for the mirror at position yi. Note that the domains in y for SDP,
SDP 1 e , and ⁢ ⁢ SDP 2 e , i . e . , - 1 2 + ( 2 ⁢ b 2 L ) ≤ y L ⁢ ≤ 1 2 - ( 2 ⁢ b 3 - 2 ⁢ b 2 L ) , ⁢ 1 2 - ( 2 ⁢ b 3 - 2 ⁢ b 2 L ) ≤ y L ≤ 1 2 , and ⁢ - 1 2 ≤ y L ≤ - 1 2 + 2 ⁢ b 2 L , are mutually exclusive and that the combined domains in y of the three domains cover the domain
ξ ⁢ ⁢ ( y ) = ⁢ ∑ m = 0 N ⁢ A m ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) + ⁢ ∑ m = 1 N ⁢ B m ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) , ⁢ - 1 2 ≤ y L ≤ 1 2 , ( 13 ) where N is an integer determined by consideration of the spatial frequencies that are to be included in the series representation. The term represented by A0 which is sometimes referred to as a �piston� type error is included in Equation (13) for completeness. An error of this type is equivalent to the effect of a displacement of the stage mirror in the direction orthogonal to the stage mirror surface and as such is not considered an intrinsic property of the surface figure error function. Using the definition of SDP given by Equation (7), the corresponding series for SDP is next written as
SDP ⁡ ( x , y , ϑ ⁢ z ) = 1 ⁢ 2 ⁢ ∑ m ⁢ = ⁢ 1 ⁢ N ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y ⁢ L ) � { A ⁢ m [ ( 1 + η ) - cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ ⁢ b ⁢ 2 ⁢ L ) - η ⁢ ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ - ⁢ 2 ⁢ ⁢ b ⁢ 2 ⁢ L ) ] + B m ⁡ [ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ b 2 L ) - η ⁢ ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ b 3 - 2 ⁢ b 2 L ) ] } + 1 2 ⁢ ∑ m ⁢ = ⁢ 1 ⁢ N ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y ⁢ L ) � { - A m ⁡ [ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ b 2 L ) - η ⁢ ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ b 3 - 2 ⁢ b 2 L ) ] + B m ⁡ [ ( 1 + η ) - cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ b 2 L ) - η ⁢ ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 2 ⁢ b 3 - 2 ⁢ b 2 L ) ] } - x ⁢ ⁢ ϑ ⁢ z ⁢ ∂ ∂ y ⁡ [ SDP ⁢ ( y , ϑ ⁢ z = 0 ) ] , ⁢ - 1 2 ⁢ ( 1 - 2 ⁢ b 2 L ) ≤ y L ≤ 1 2 ⁡ [ 1 - ( 2 ⁢ b 3 - 2 ⁢ b 2 L ) ] , ( 14 ) where x is a linear displacement of the stage mirror based on one or more of the linear displacements xi, i=1, 2, and/or 3, and θz is the angular orientation of the stage mirror in the x-y plane. The last term in Equation (14) is a third order term with an origin in a second order geometric term such as described in commonly owned U.S. patent application Ser. No. 10/347,271 entitled �COMPENSATION FOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETERS� and U.S. patent application Ser. No. 10/872,304 entitled �COMPENSATION FOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETER METROLOGY SYSTEMS,� both of which are by Henry A. Hill and both of which are incorporated herein in their entirety by reference.
SDP ⁡ ( x , y , ϑ ⁢ z ) = 1 ⁢ 2 ⁢ ∑ m ⁢ = ⁢ 1 ⁢ N ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y ⁢ L ) � { A ⁢ m [ ( 1 + η ) - cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ η 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) - η ⁢ ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 1 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] + B m ⁡ [ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ η 1 + η ⁢ 2 ⁢ b 3 L ) - η ⁢ ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 1 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] } + 1 2 ⁢ ∑ m ⁢ = ⁢ 1 ⁢ N ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y ⁢ L ) � { - A m ⁡ [ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ η 1 + η ⁢ 2 ⁢ b 3 L ) - η ⁢ ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 1 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] + B m ⁡ [ ( 1 + η ) - cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ η 1 + η ⁢ 2 ⁢ b 3 L ) - η ⁢ ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 1 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] ⁢ } - x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP ⁡ ( y , ϑ z = 0 ) ] , ⁢ - 1 2 ⁡ [ 1 - ( η 1 + η ) ⁢ 2 ⁢ b 3 L ] ≤ y ⁢ L ≤ 1 2 ⁡ [ 1 - ( 1 1 + η ) ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ] . ( 15 ) A contracted form of Equation (15) is obtained with the introduction of a complex transfer function T(m) having real and imaginary amplitudes of TRe and TIm, respectively, as
SDP ⁢ ( x , y , ϑ ⁢ z ) = ⁢ 1 ⁢ 2 ⁢ ∑ m ⁢ = ⁢ 1 ⁢ N ⁢ [ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y ⁢ L ) ⁢ A m ′ + sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y ⁢ L ) ⁢ B m ′ ] - x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP ⁡ ( y , ϑ z = 0 ) ] , - 1 2 ⁡ [ 1 - ( η 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] ≤ y ⁢ L ≤ 1 2 ⁡ [ 1 - ( 1 1 + η ) ⁢ ( 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] ⁢ ⁢ where ( 16 ) A m ′ = A m ⁢ T Re + B m ⁢ T Im ⁢ ⁢ and ( 17 ) B m ′ = - A m ⁢ T Im + B m ⁢ T Re , ⁢ with ( 18 ) T Re = [ ( 1 + η ) - cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ η 1 + η ⁢ 2 ⁢ b 3 L ) - η ⁢ ⁢ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 1 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] , ⁢ and ( 19 ) T Im = [ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ η 1 + η ⁢ 2 ⁢ b 3 L ) - η ⁢ ⁢ sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ 1 1 + η ⁢ 2 ⁢ ⁢ b ⁢ 3 ⁢ L ) ] . ( 20 ) The transfer function, T(m), relates the Fourier coefficients A′m and B′m of the SDP to the Fourier coefficients of ξ(y), Am and Bm. As is evident from Equations (19) and (20), T(m) depends on η and b3/L in addition to depending on spatial frequency (as indicated by dependence on m). The dependence on spatial frequency manifests as a varying sensitivity of the transfer function to spatial frequency, with zero sensitivity occurring at certain spatial frequencies. Lack of sensitivity at a spatial frequency means that the SDP parameter does not contain any information of the mirror surface for that frequency. Subsequent calculation of the function ξ(y) should make note of these low sensitivity frequencies and handle them appropriately.
η = 2 ⁢ n - 1 n , ⁢ ⁢ n = 2 , 3 , � ⁢ . ( 21 ) An η of the corresponding second set of η's is a subset of the first set of η's defined by Equation (21) for even values of n.
η = 2 ⁢ n + 1 n , ⁢ n = 2 , 3 , � ( 26 ) The corresponding values for Λ1, b2, and (2b3−b2) are
η = 2 ⁢ n � 1 2 ⁢ n , ⁢ n = 2 , 3 , � ⁢ . ( 30 ) The corresponding values for Λ1, b2, and (2b3−b2) are
T Beam = e - ( π 2 8 ) ⁢ ( 2 ⁢ s Λ ) 2 . ( 34 ) Consider for example the effect of a Gaussian beam profile with a 1/e2 diameter of 2s=5.0 mm. The attenuation effect of the spatial filtering will be 0.225, 0.089, and 0.028 for Λ=4.55, 3.57, and 2.94 mm, respectively, for the respective Fourier series component amplitudes.
SDP 1 e ⁡ ( x , y , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m2 ⁢ ⁢ π ⁢ y L ) ⁢ A m ′ + sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ B m ′ ] + η ⁢ ⁢ L 2 ⁢ ϑ z - x ⁢ ⁢ ϑ z ⁢ ∂ ∂ y ⁡ [ SDP 1 e ⁡ ( y , ϑ z = 0 ) ] , 1 2 ⁡ [ 1 - 2 ⁢ ( 1 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] ≤ y L ≤ 1 2 , ⁢ ( 35 ) SDP 1 e ⁡ ( x , y , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m2 ⁢ ⁢ π ⁢ y L ) ⁢ A m ′ + sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ B m ′ ] - L 2 ⁢ ϑ z - ϑ z ⁢ ∂ ∂ y ⁡ [ SDP 2 e ⁢ ( y , ϑ z = 0 ) ] , - 1 2 ≤ y L ≤ 1 2 ⁡ [ 1 - 2 ⁢ ( η 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] . ( 36 ) Note in Equations (35) and (36) that the constant term A0 is not present. The discussion of this feature relevant to the intrinsic surface figure error function is the same as the corresponding portion of the discussion related to the constant term A0 not being included in Equation (14).
SDP ⁡ ( x , y , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ A m ′ + sin ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ B m ′ ] - x ⁢ ⁢ ϑ _ z ⁢ ∂ ∂ y ⁡ [ SDP ⁡ ( y , ϑ z = 0 ) ] - E 1 - ( 1 + η ) ⁢ E 2 + η ⁢ ⁢ E 3 ⁢ ] , - 1 2 ⁡ [ 1 - 2 ⁢ ( η 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] ≤ y L ≤ 1 2 ⁡ [ 1 - 2 ⁢ ( 1 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] ; ( 37 ) SDP 1 e ⁡ ( x , y , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ A m ′ + sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ B m ′ ] - x ⁢ ⁢ ϑ _ z ⁢ ∂ ∂ y ⁡ [ SDP 1 e ⁡ ( y , ϑ z = 0 ) ] - ( 1 + η ⁢ ⁢ q ) ⁢ E 1 + ( 1 + η ) ⁢ E 2 + η ⁡ ( q - 1 ) ⁢ E 3 + η ⁢ ⁢ b 3 ⁢ { ϑ z ⁡ ( y ) + ϑ z ⁡ ( y - 2 ⁢ b 3 ) + � + ϑ z ⁡ [ y - 2 ⁢ ( q - 1 ) ⁢ b 3 ] } , 1 2 ⁡ [ 1 - 2 ⁢ ( 1 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] ≤ y L ≤ 1 2 ; and ( 38 ) SDP 2 e ⁡ ( x , y , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ A m ′ + sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y L ) ⁢ B m ′ ] - x ⁢ ⁢ ϑ _ z ⁢ ∂ ∂ y ⁡ [ SDP 2 e ⁢ ( y , ϑ z = 0 ) ] + ( q - 1 ) ⁢ E 1 + ( 1 + η ) ⁢ E 2 - ( q + η ) ⁢ E 3 + b 3 ⁢ { ϑ z ⁡ ( y ) + ϑ z ⁡ ( y + L - 2 ⁢ b 3 ) + ϑ z ⁡ ( y + L - 4 ⁢ b 3 ) + � + ϑ z ⁡ [ y + L - 2 ⁢ ( q - 1 ) ⁢ b 3 ] } , - 1 2 ≤ y L ≤ - 1 2 ⁡ [ 1 - 2 ⁢ ( η 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] ( 39 ) where fourth order of effects arising from the y dependence of θz(y) have been neglected in Equations (37), (38), and (39) and θ z represents the average value of θz(y) over the domain in y.
SDP 1 e ⁡ ( x , y 1 , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y 1 L ) ⁢ A m ′ + sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y 1 L ) ⁢ B m ′ ] - x ⁢ ⁢ ϑ _ z ⁢ ∂ ∂ y ⁡ [ SDP 1 e ⁡ ( y 1 , ϑ z = 0 ) ] - ( 1 + η ⁢ ⁢ q ) ⁢ E 1 + ( 1 + η ) ⁢ E 2 + η ⁢ ( q - 1 ) ⁢ E 3 + ⁢ η ⁢ ⁢ b 3 ⁢ { ϑ z ⁡ [ y 0 + L 2 - 2 ⁢ ( b 3 - b 2 ) ] + ϑ z ⁡ [ y 0 + L 2 - 2 ⁢ ( b 3 - b 2 ) - 2 ⁢ b 3 ] + � + ϑ z ⁡ [ y 0 + L 2 - 2 ⁢ ( b 3 - b 2 ) - 2 ⁢ ( q - 1 ) ⁢ b 3 ] } , ( 42 ) SDP 2 e ⁡ ( x , y 2 , ϑ z ) = 1 2 ⁢ ∑ m = 1 N ⁢ [ cos ⁢ ⁢ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y 2 L ) ⁢ A m ′ + sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ y 2 L ) ⁢ B m ′ ] - x ⁢ ⁢ ϑ _ z ⁢ ∂ ∂ y ⁡ [ SDP 2 e ⁡ ( y 2 , ϑ z = 0 ) ] + ( q - 1 ) ⁢ E 1 + ( 1 + η ) ⁢ E 2 - ( q + η ) ⁢ E 3 + b 3 ⁢ { ϑ z ⁡ [ y 0 - L 2 + 2 ⁢ b 2 ] + ϑ z ⁡ [ y 0 + L 2 + 2 ⁢ b 2 - 2 ⁢ b 3 ] + ϑ z ⁡ [ y 0 + L 2 + 2 ⁢ b 2 - 4 ⁢ b 3 ] + � + ϑ z ⁡ [ y 0 + L 2 + 2 ⁢ b 2 - 2 ⁢ ( q - 1 ) ⁢ b 3 ] } . ( 43 ) It is evident on examination of Equations (37), (42), and (43) that the contribution of the surface figure error function terms to SDP and SDP1 e are continuous at y1 and that the contribution of the surface figure error function terms to SDP and SDP2 e are continuous at y2. This is a very significant property which is subsequently used in a procedure to eliminate the effects of offset errors E1, E2, and E3 and the effects of the y dependence of θz(y).
[ SDP 1 e ⁡ ( x , y 1 , ϑ z ) - SDP ⁢ ( y 1 ) ] = - η ⁢ ⁢ q ⁡ ( E 1 - E 3 ) + ( ξ ⁡ ( y 0 + L 2 ) - ξ ⁡ ( y 0 - L 2 ) 2 ) + η ⁢ ⁢ b 3 ⁢ { ϑ z ( ⁢ y 0 ⁢ + ⁢ L 2 ⁢ + ⁢ 2 ⁢ ⁢ b 2 ⁢ - ⁢ 2 ⁢ ⁢ b 3 ) + ϑ z ⁡ ( y 0 ⁢ + ⁢ L 2 ⁢ + ⁢ 2 ⁢ ⁢ b 2 ⁢ - ⁢ 4 ⁢ ⁢ b 3 ) + � ⁢ + ⁢ ϑ z ⁡ ( y 0 ⁢ + ⁢ L 2 ⁢ + ⁢ 2 ⁢ ⁢ b 2 ⁢ - ⁢ 2 ⁢ ⁢ qb 3 ) } , ( 44 ) [ SDP ⁡ ( y 2 ) - SDP 2 e ⁡ ( x , y 2 , ϑ z ) ] = - q ⁢ ( E 1 - E 3 ) + ( ξ 3 ⁡ ( y 0 + L 2 ) - ξ 1 ⁡ ( y 0 - L 2 ) 2 ) + b 3 ⁢ { ϑ ⁢ z ⁡ ( y ⁢ 0 - L ⁢ 2 + 2 ⁢ b ⁢ 2 ) + ϑ ⁢ z ⁡ ( y ⁢ 0 + L ⁢ 2 + 2 ⁢ b ⁢ 2 - 2 ⁢ b ⁢ 3 ) + ϑ z ⁡ ( y 0 + L 2 + 2 ⁢ b 2 - 4 ⁢ b 3 ) + � + ϑ z ( y 0 + L 2 + 2 ⁢ b 2 - 2 ⁢ ( q - 1 ) ⁢ b 3 ] } . ( 45 ) Using the relationship L=2qb3 given by Equation (8), Equation (45) is written as
[ SDP ⁡ ( y 2 ) - SDP 2 e ⁡ ( x , y 2 , ϑ z ) ] = ⁢ - q ⁢ ( E 1 - E 3 ) + ⁢ ( ξ ⁡ ( y 0 + L 2 ) - ξ 1 ⁡ ( y 0 - L 2 ) 2 ) + ⁢ b 3 ⁢ { ϑ ⁢ z ⁡ ( y ⁢ 0 + L ⁢ 2 + 2 ⁢ b ⁢ 2 - 2 ⁢ b ⁢ 3 ) + ϑ z ⁡ ( y 0 + L 2 + 2 ⁢ b 2 - 4 ⁢ b 3 ) + � + ϑ z ( y 0 + L 2 + 2 ⁢ b 2 - 2 ⁢ qb 3 ] } . ( 46 ) The ratio of the value of the discontinuity D1 (y0) given Equation (44) and the value of the discontinuity D2 (y0) given by Equation (46) is thus equal to η, i.e.
D 1 ⁡ ( y 0 ) = η ⁢ ⁢ D 2 ⁡ ( y 0 ) ⁢ ⁢ where ( 47 ) D 1 ⁡ ( y 0 ) ≡ [ SDP 1 e ⁡ ( x , y 1 , ϑ z ) - SDP ⁡ ( y 1 , ϑ z ) ] , ( 48 ) D 2 ⁡ ( y 0 ) ≡ [ SDP ⁡ ( y 2 , ϑ z ) - SDP 2 e ⁡ ( x , y 2 , ϑ 2 ) ] . ( 49 ) Based on the foregoing mathematical development, a general procedure for determining a surface figure error function, ξ(y), without any prior knowledge of the surface figure is as follows. First, select the length of the mirror that is to be mapped by selecting a value of q. The length of the mirror is given by Equation (8). Orient the stage mirror so that θz is at or about zero and the distance x from the interferometer to the stage mirror being mapped is relatively small. The small distance can reduce the contribution to the geometric error correction term. While maintaining a nominal x distance to the stage mirror, acquire simultaneous values for x1, x2, and x3 while scanning the mirror in the y-direction and monitor the position of the mirror in the y-direction and changes in stage orientation during the scan. The x1, x2, and x3 values, the position of the mirror in the y-direction, and the changes in stage orientation can be stored to the electronic controller's memory or to disk. The stored data will be in the form of a 3�N array for the x1, x2, and x3 information, where N is the total number of samples across the mirror, a 1�N array for y position information and a 1�N array for changes in stage orientation information.
A S ″ = 4 ( L - 2 ⁢ b 3 ) � ⁢ ∫ ( - L 2 + 2 ⁢ b 2 ) [ L 2 - 2 ⁢ ( b 3 - b 2 ) ] ⁢ [ SDP ⁡ ( y , ϑ _ z = 0 ) - 〈 SDP ⁡ ( y , ϑ _ z = 0 ) 〉 ] ⁢ cos ⁢ ⁢ ( s ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) ⁢ ⁢ ⅆ y , s ≥ 1 , ( 50 ) B S ″ = 4 ( L - 2 ⁢ b 3 ) � ⁢ ∫ ( - L 2 + 2 ⁢ b 2 ) [ L 2 - 2 ⁢ ( b 3 - b 2 ) ] ⁢ [ SDP ⁢ ( y , ϑ _ z = 0 ) - 〈 SDP ⁢ ( y , ϑ _ z = 0 ) 〉 ] ⁢ ⁢ sin ⁢ ⁢ ( s ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) ⁢ ⁢ ⅆ y , s ≥ 1. ( 51 ) The integrals expressed by Equations (50) and (51) are evaluated using the series representation of SDP given by Equation (16) with the results
A q ′′ = 4 ( L - 2 ⁢ b 3 ) ⨯ ∫ ( - L 2 + 2 ⁢ b 2 ) [ L 2 - 2 ⁢ ( b 3 - b 2 ) ] ⁢ [ ∑ m = 1 N ⁢ A m ′ ⁢ cos ⁡ ( m ⁢ ⁢ 2 ⁢ π ⁢ ⁢ y L ) ⁢ cos ⁡ ( q ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) + ∑ m = 1 N ⁢ B m ′ ⁢ sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) ⁢ cos ⁡ ( q ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) ] ⁢ ⁢ ⅆ y , ⁢ q ≥ 1 , ⁢ ( 52 ) B q ′′ = 4 ( L - 2 ⁢ b 3 ) ⨯ ∫ ( - L 2 + 2 ⁢ b 2 ) [ L 2 - 2 ⁢ ( b 3 - b 2 ) ] ⁢ [ ∑ m = 1 N ⁢ A m ′ ⁢ cos ⁡ ( m ⁢ ⁢ 2 ⁢ π ⁢ ⁢ y L ) ⁢ sin ⁡ ( q ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) + ∑ m = 1 N ⁢ B m ′ ⁢ sin ⁡ ( m ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) ⁢ sin ⁡ ( q ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ y L ) ] ⁢ ⁢ ⅆ y ⁢ ⁢ q ≥ 1. ⁢ ( 53 ) where coefficients A′q and B′q are with respect to SDP(y, θ z=0) with <SDP(y, θ z=0)> subtracted. The evaluation of the integrals in Equation (52) is next performed with the results
A q ′′ = 1 ( 1 - 2 ⁢ b 3 L ) ⨯ { ∑ m = 1 N ⁢ A m ′ ⁡ ( ( 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } ) ) - ∑ m = 1 N ⁢ B m ′ ⁡ ( ( 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } - 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } ) ) } , ⁢ q ≥ 1 , ⁢ ( 54 ) A q ′′ = A q ′ + 1 ( 1 - 2 ⁢ b 3 L ) ⨯ { ∑ m = 1 , m ≠ q N ⁢ A m ′ ⁡ ( ( ( - 1 ) q + m + 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q + m ) ⁢ ( 4 ⁢ b 2 L ) } + ( - 1 ) q - m + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q - m ) ⁢ ( 4 ⁢ b 2 L ) } ) ) - ∑ m = 1 N ⁢ B m ′ ⁡ ( ( ( - 1 ) q - m ( q + m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 4 ⁢ b 2 L ] } - ( - 1 ) q - m ( q - m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q - m ) ⁢ ( 4 ⁢ b 2 L ) } ) ) } , ⁢ q ≥ 1 , ⁢ ( 55 ) A q ′′ = A q ′ + 1 ( 1 - 2 ⁢ b 3 L ) ⨯ { ∑ m = 1 , m ≠ q N ⁢ A m ′ ⁡ ( ( ( - 1 ) q + m + 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { sin ⁡ [ 2 ⁢ ⁢ π ⁡ ( q + m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] + sin ⁡ [ 2 ⁢ ⁢ π ⁡ ( q + m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } + ( - 1 ) q - m + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { sin ⁡ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] + sin ⁡ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) - ∑ m = 1 N ⁢ B m ′ ⁡ ( ( ( - 1 ) q - m ( q + m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ [ 2 ⁢ π ⁡ ( q + m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] - cos ⁡ [ 2 ⁢ ⁢ π ⁡ ( q + m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } - ( - 1 ) q - m ( q - m ) ⁢ 2 ⁢ π ⁢ { cos ⁡ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] - cos ⁢ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) } , ⁢ q ≥ 1 , ⁢ ( 56 ) A q ′′ = A q ′ + ( 2 ⁢ b 3 L ) [ 1 - 2 ⁢ b 3 L ] ⨯ { ∑ m = 1 , m ≠ q N ⁢ A m ′ ⁡ ( ( ( - 1 ) q + m + 1 ⁢ { sin ⁢ ⁢ c [ ⁢ π ⁡ ( q + m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ cos [ ⁢ π ⁡ ( q + m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } + ( - 1 ) q - m + 1 ⁢ { sin ⁢ ⁢ c [ ⁢ π ⁡ ( q - m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ cos [ ⁢ π ⁡ ( q - m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) - ∑ m = 1 N ⁢ B m ′ ⁡ ( ( ( - 1 ) q - m + 1 ⁢ { sin ⁢ ⁢ c ⁢ [ π ⁡ ( q + m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ sin [ ⁢ π ⁡ ( q + m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } - ( - 1 ) q - m + 1 ⁢ { sin ⁢ ⁢ c [ ⁢ π ⁡ ( q - m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ sin ⁢ [ π ⁡ ( q - m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) } , ⁢ q ≥ 1 , ⁢ ( 57 ) The evaluation of the integrals in Equation (53) is next performed with the results
B q ′′ = 1 ( 1 - 2 ⁢ b 3 L ) ⨯ { - ∑ m = 1 N ⁢ A m ′ ⁡ ( ( 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } ) ) + ∑ m = 1 N ⁢ B m ′ ⁡ ( ( - 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 1 - 4 ⁢ b 2 L ] } ) ) ⁢ } , ⁢ q ≥ 1 , ⁢ ( 58 ) B q ′′ = B q ′ + 1 ( 1 - 2 ⁢ b 3 L ) ⨯ { - ∑ m = 1 N ⁢ A m ′ ⁡ ( ( ( - 1 ) q + m ( q + m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 4 ⁢ b 2 L ] } - ( - 1 ) q - m ( q - m ) ⁢ 2 ⁢ π ⁢ { cos ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] - cos ⁢ ⁢ π ⁡ ( q - m ) ⁢ ( 4 ⁢ b 2 L ) } ) ) + ∑ m = 1 , m ≠ q N ⁢ B m ′ ⁡ ( ( ( - 1 ) q - m + 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q + m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q + m ) ⁢ ( 4 ⁢ b 2 L ) } + ( - 1 ) q - m + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { sin ⁢ ⁢ π ⁡ ( q - m ) ⁡ [ 4 ⁢ ( b 3 - b 2 ) L ] + sin ⁢ ⁢ π ⁡ ( q - m ) ⁢ ( 4 ⁢ b 2 L ) } ) ) } , ⁢ q ≥ 1 , ⁢ ( 59 ) B q ′′ = B q ′ + 1 ( 1 - 2 ⁢ b 3 L ) ⨯ ⁢ ⁢ { ∑ m = 1 N ⁢ A m ′ ⁡ ( ( ( - 1 ) q + m ( q + m ) ⁢ 2 ⁢ π ⁢ { cos ⁡ [ 2 ⁢ ⁢ π ⁡ ( q + m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] - cos ⁡ [ 2 ⁢ ⁢ π ⁡ ( q + m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } + ( - 1 ) q - m ( q - m ) ⁢ 2 ⁢ π ⁢ { cos ⁡ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] - cos ⁡ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) + ∑ m = 1 , m ≠ q N ⁢ B m ′ ⁡ ( ( - ( - 1 ) q - m + 1 ( q + m ) ⁢ 2 ⁢ π ⁢ { sin ⁡ [ 2 ⁢ π ⁡ ( q + m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] + sin ⁡ [ 2 ⁢ ⁢ π ⁡ ( q + m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } + ( - 1 ) q - m + 1 ( q - m ) ⁢ 2 ⁢ π ⁢ { sin ⁡ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ 1 ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] + sin ⁢ [ 2 ⁢ ⁢ π ⁡ ( q - m ) ⁢ η ( 1 + η ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) } , ⁢ ⁢ q ≥ 1. ⁢ ( 60 ) B q ′′ = B q ′ + ( 2 ⁢ b 3 L ) [ 1 ⁢ - ( 2 ⁢ b 3 L ) ] ⨯ { ∑ m = 1 N ⁢ A m ′ ⁡ ( ( ( - 1 ) q + m + 1 ⁢ { sin ⁢ ⁢ c [ ⁢ π ⁡ ( q + m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ sin ⁡ [ π ⁡ ( q + m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } + ( - 1 ) q - m + 1 ⁢ { sin ⁢ ⁢ c ⁡ [ π ⁡ ( q - m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ sin ⁡ [ π ⁡ ( q - m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) + ∑ m = 1 , m ≠ q N ⁢ B m ′ ⁡ ( ( - ( - 1 ) q + m + 1 ⁢ { sin ⁢ ⁢ c ⁢ [ π ⁡ ( q + m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ cos [ ⁢ π ⁡ ( q + m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } + ( - 1 ) q - m + 1 ⁢ { sin ⁢ ⁢ c ⁡ [ π ⁡ ( q - m ) ⁢ ( 2 ⁢ b 3 L ) ] ⨯ cos ⁡ [ π ⁡ ( q - m ) ⁢ ( η - 1 ) ( η + 1 ) ⁢ ( 2 ⁢ b 3 L ) ] } ) ) } , ⁢ q ≥ 1. ⁢ ( 61 ) The effects of non-orthogonality of functions cos (m2πy/L) and sin(m2πy/L) over the domain of y in SDP(y, θ z=0) are evident upon examination of Equations (57) and (61). The effects of non-orthogonality are represented by the differences (A″q−A′q) and (B″q−B′q) in Equations (57) and (61), respectively.
r ≅ ( 2 ⁢ b 3 L ) 1 - ( 2 ⁢ b 3 L ) ⁢ sin ⁢ ⁢ c ⁡ [ π ⁡ ( q - m ) ⁢ ( 2 ⁢ b 3 L ) ] . ( 62 ) The iterative procedure is a nontrivial step for which two solutions are given. The step is nontrivial since cos(m2πy/L) and sin(m2πy/L) are functions that are not orthogonal as a set over the domain of integration in y. One solution involves using a surface figure error function ξ obtained at an earlier time based on measured values of SDP, SDP1 e, and SDP2 e to compute values for the off-diagonal terms in Equations (57) and (61).
∑ n = 2 N ⁢ a n ⁢ y 0 n , a sum of Fourier terms
∑ m = 1 M ⁢ a m ⁢ cos ⁡ ( 2 ⁢ π ⁢ ⁢ m L ⁢ y 0 ) + b m ⁢ sin ⁡ ( 2 ⁢ π ⁢ ⁢ m L ⁢ y 0 ) , or another series. The coefficients in the series can be related analytically using a mapping function to corresponding coefficients in a series representing the errors in θz(y) due to relative scaling errors in the y -axis interferometer. Subsequently, during use of the interferometry system, the mapping function can be used to correct for the errors in θz(y).
x ~ 1 ⁡ ( y ) = d + ( ξ 1 ′ ⁡ ( y ) + ξ 0 ′ ⁡ ( y ) 2 ) + d nom ⁢ θ z ⁡ ( ∂ ξ 1 ′ ∂ y - ∂ ξ 0 ′ 3 ⁢ ∂ y ) , ( 63 ) where the lowest order terms dependent on the mirror surface figure error function have been shown. ξ′1 and ξ′0 refer to the surface figure error function values at positions x′1 and x′0, respectively, and dnom is a nominal displacement value used to calculate the geometric error term that occurs for non-zero θz.
x c ⁡ ( y ) = x ~ 1 ⁡ ( y ) - ( ξ 1 ⁡ ( y ) + ξ 0 ⁡ ( y ) 2 ) - d nom ⁢ θ z ⁡ ( ∂ ξ 1 ∂ y - ∂ ξ 0 ∂ y ) . ( 64 ) where the lowest order correction terms dependent on the mirror surface figure error function have been shown.
In embodiments, various other error compensation techniques can be used to reduce other sources of error in interferometer measurements. For example, cyclic errors that are present in the linear displacement measurements can be reduced (e.g., eliminated) and/or compensated by use of one of more techniques such as described in commonly owned U.S. patent application Ser. No. 10/097,365, entitled �CYCLIC ERROR REDUCTION IN AVERAGE INTERFEROMETRIC MEASUREMENTS,� and U.S. patent application Ser. No. 10/616,504 entitled �CYCLIC ERROR COMPENSATION IN INTERFEROMETRY SYSTEMS,� which claims priority to Provisional Patent Application No. 60/394,418 entitled �ELECTRONIC CYCLIC ERROR COMPENSATION FOR LOW SLEW RATES,� all of which are by Henry A. Hill and the contents of which are incorporated herein in their entirety by reference.
An example of another cyclic error compensation technique is described in commonly owned U.S. patent application Ser. No. 10/287,898 entitled �INTERFEROMETRlC CYCLIC ERROR COMPENSATION,� which claims priority to Provisional Patent Application No. 60/337,478 entitled �CYCLIC ERROR COMPENSATION AND RESOLUTION ENHANCEMENT,� by Henry A. Hill, the contents of which are incorporated herein in their entirety by reference.
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Applied Optics, 37:28, pp. 6696-6700, 1998.Classifications U.S. Classification356/500, 356/508, 356/511International ClassificationG01B9/02, G01B11/02, G03F7/20Cooperative ClassificationG01B2290/15, G01B9/0207, G01B2290/70, G01B9/02007, G01B9/02059, G01B9/02072, G01B9/02058, G01B9/02018, G01B9/02061, G01B9/02084, G01B9/02027, G03F9/7034, G03F9/7049, G03F7/70775European ClassificationG01B9/02, G03F7/70N12, G03F9/70B6L, G03F9/70DLegal EventsDateCodeEventDescriptionNov 21, 2011FPAYFee paymentYear of fee payment: 4May 3, 2006ASAssignmentOwner name: ZYGO CORPORATION, CONNECTICUTFree format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNORS:WOMACK, GARY;HILL, HENRY A.;REEL/FRAME:017577/0626;SIGNING DATES FROM 20060418 TO 20060420RotateOriginal ImageGoogle Home - Sitemap - USPTO Bulk Downloads - Privacy Policy - Terms of Service - About Google Patents - Send FeedbackData provided by IFI CLAIMS Patent Services©2012 Google