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  Pseudo -Shear Interferometry Peter B. Keenan Spectra- Physics, Inc., Optics Division 1250 W. Middlefied Road, Mountain View, California 94042 Abstract Pseudo -Shear Interferometry (PSI) is a technique for obtaining enhanced, absolute ac-curacy in interferometric measurements. Whereas conventional interferometry yields measure- ments limited in accuracy by the optical system and by the reference surface, Pseudo -ShearInterferometry provides the capability for measurement accuracy which substantially surpas- ses it. The technique consists of a regimen for data taking and a mathematical procedure for analyzing the resulting data. Introduction A digital, phase- measuring interferometer provides sets of measurements on equal- spaced array of points. Between measurements, a translation is made of the optic being measured.The translation is adjusted to correspond to the sampling interval of the phase- measuring interferometer. Differences between the measurement sets are then taken numerically in a computer. These differences comprise a pseudo -shear interferogram.Further processing of these difference sets is then required to separate the deterministic variation, the syste- matic error, and the random error.The analysis of the data sets ultimately yields the following: 1. Absolute surface measurements, 2. Absolute interferometer calibration, and 3. An estimate of the random errors. The accuracy requirements for interferometric testing have become increasingly severe in the past few years. There are two primary reasons for this trend.First, fringe scanninginterferometers have become commercially available. These interferometers provide impres- sive levels of measurement resolution and accuracy.Second, there are increasing numbers of complex optical systems being attempted which contain large numbers of surfaces, systems which require a transmitted wavefront error of a fraction of a wavelength.Thus, the re- quirement for single surface measuring accuracy has continued to pressure the state of the art. Although digital interferometers have provided the user with a measurement resolution on the order of 1 /1000 wave, there remains the unavoidable fact that all interferometric mea-surements are comparisons.Each value is a path difference.Some method is required to separate the component of the path difference due to the optic under test from the compo-nent of the path difference which is due to surface variation in other elements of the sys- tem. The procedure described here has somewhat the flavor of lateral shear interferometry.The processing includes obtaining the differences between successive measurements with a lateral shift of the optic. The term pseudo is used because the comparison of the twowavefronts is affected in a computer rather than between the wavefronts themselves. The pseudo -shear interferogram may be processed to yield the actual profile of the opticand the systematic errors of the instrument. This process does not require the use of a reference standard of known accuracy.Elimination of the requirement for a reference stan- dard simplifies the testing and reduces the errors introduced by a reference which has lost its figure due to thermal or mechanical stress.It may be that the reference is not cer-tified to the level of accuracy required for the test.Two position PSI Conceptually, the method requires placing the test piece on a fixture that may be tran- slated laterally. After taking a first measurement, the part is translated and a second measurement is made. An analysis is performed on the results of the two measurements to separate the component which moved from the component which remained fixed.The measurement geometry is shown in Figure 1.At each position, the measurement yieldsthe distance between the surfaces.The measurements, Mi, are made at equal- spaced positions. 2 Pseudo-Shear Interferometry Peter B KeenanSpectra-Physics, Inc., Optics Division 1250 W Middlefied Road, Mountain View, California 94042Abstract Pseudo-Shear Interferometry PSI) is a technique  for obtaining enhanced, absolute ac curacy in interferometric measurements. Whereas conventional interferometry yields measure ments limited in  accuracy by the optical system and by the reference surface, Pseudo-Shear Interferometry provides the capability for measurement accuracy which substantially surpas  ses it The technique consists of a regimen for data taking and a mathematical procedure for analyzing the resulting data. Introduction A digital, phase-measuring interferometer provides sets of  measurements on equal-spaced array of  points. Between measurements, a translation is made of the optic being measured. The translation is adjusted to correspond to the sampling interval of the phase-measuring interferometer. Differences between the  measurement sets are then taken  numerically in a computer. These differences comprise a pseudo-shear interferogram. Further processing of these difference sets is then required to separate the deterministic variation, the syste matic error, and the random error.The analysis of the data sets ultimately yields the following: 1 Absolute surface measurements, 2 Absolute interferometer calibration, and 3  An estimate of the  random errors.The accuracy requirements for interferometric testing have become increasingly  severe in the past  few years. There are two primary reasons for this trend. First, fringe scanning interferometers have become commercially available. These interferometers provide impres sive levels of measurement resolution and accuracy. Second,  there are increasing numbers of complex optical systems being attempted which contain large  numbers of surfaces, systems which require a transmitted wavefront error of a fraction of a wavelength. Thus, the re quirement for single surface measuring accuracy has continued to pressure the state of the art. Although digital interferometers have provided the user with a  measurement resolution on the order of 1/1000 wave, there remains the unavoidable fact that all interferometric mea surements are comparisons. Each value is a path difference. Some method is required to  separate the component of the path difference  due to the optic under test from the compo nent of the path difference which is due to surface variation in other elements of the sys tem. The procedure described here has somewhat the flavor of lateral shear interferometry. The processing includes obtaining  the differences between successive measurements with  a lateral shift of the optic.  The term pseudo is used because the comparison of the two wavefronts  is affected in a computer rather than between the wavefronts themselves.The pseudo-shear interferogram may be processed  to  yield the actual profile of the optic and the systematic errors of the instrument. This process does not require the use of a reference standard of known  accuracy. Elimination of the requirement  for a reference stan dard simplifies the  testing  and reduces the errors  introduced by a reference which has lost its figure due to thermal or mechanical stress. It may be that the reference is  not cer tified to the level of accuracy required for the test. Two position PSI Conceptually, the method requires placing the test piece on  a fixture that may be tran slated laterally. After taking a first measurement, the part is translated and a second measurement is made. An analysis is performed on the results of  the two measurements to separate the component which moved from the component which remained fixed.The measurement geometry is shown in Figure 1 At each position, the measurement yields the distance between the surfaces. The measurements, Mi are made at equal-spaced positions. Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/04/2014 Terms of Use: http://spiedl.org/terms  Also shown in Figure 1 is a mathematical axis for describing the two surfaces. The surface of the test optic is described by its distance, Ti, from the axis.Likewise, the distancefrom the axis to the reference surface is given by Ri.Each measurement is simply a difference, Ti - Ri.In conventional interferometry it isonly this difference which can be determined and not the individual values of Ri and Ti.Consider in more detail this two measurement scheme.The expected values that would be obtained are shown in Figure 2. In the left hand column are the measurement values foreach position of the initial configuration.The right hand column shows the expected mea-surement values after the test optic has been translated.Note that the reference values,Ri, stay fixed with respect to the measurement positions. The difference between the items in the two columns is referred to as a Pseudo -Shear In- terferogram. In Figure 3, the group of equations that result is shown.This set is readilysolved for the values of Ti after choosing an arbitrary value for T one. After obtaining the values for each Ti, thé equations of Figure 2 may be solved for each Ri.This two position technique is simple and readily produces a set of values for both Ti, and Ri. However, there is no provision for including additional data nor is there provision for error estimation. These deficiencies are alleviated in the following multiposition analysis. Multiposition PSI Combining the results of three or more measurements with lateral shifts makes it possible to extend the previous analysis and simultaneously include a least- squares error reduction. This provides a means to estimate the measurement error as well as the best estimate of theprofile of the test optic and the reference surface.The analysis presented will be for the case of three measurements made with equal- spaced translations as represented in Figure 4. Each column corresponds to one of the measurement sets; each being a translation of one unit from the next. Also appearing in these equations is an error term, E. It is assumed that there is an independent, random error associated with each value.There are two independent sets of differences which may be obtained from these; they are shown in Figure 5. Note that D12 and D23 are both estimates of the same quantity and com- parable pairs comprise the lists. If additional measurements are made, they can be easily included at this point. Because the number of unknowns, including the error terms, exceeds the number of equa-tions, this set is solved in the least square sense.Figure 6 shows the derivation of thenormal equations from the conditions for minimizing the sum of the squares of the errors.Substitution and evaluation of the partial derivatives gives the normal equations. The nor- - mal equations may be displayed in somewhat more compact form by the use of matrices. Addi- tionally, display of the matrix for a specific size array shows the starting and ending values explicitly. Figure 7 shows the matrix equation for the case of five positions on the test optic. The matrix on the left is a symetric band matrix.All diagonal elements are four except the last. This same form prevails for any matrix size greater than two.The column vectoron the right maintains the pattern shown for arrays of any size.This matrix may be readily inverted to solve for the profile of the test optic, Ti. Ob- taining the values for Ti might be considered the end of the problem.However, for most ac-curate work, we are required to determine an error estimate for the measurements.Returning to Figure 4 for a moment, it will be noted the differences might be arranged toeliminate Ti and thus be used to determine Ri.The resulting differences are shown in Figure 8. These are shifted differences and yield PSI data for the interferometer. Application of the least- square methodology to this set of equations produces the matrix equation for de-termination of the reference as shown in Figure 9.Again, the patterns of the components of this equation is shown for dimension five by five. Extension to matrices of other sizes is readily accomplished.The solution for the values of Ri is straight forward. The solutions of these two matrix equations have given values for Ri and Ti. Substitu- tion of these values into the equations of Figure 4 permit determination of the errors as- sociated with each value. From the enumerated values, error statistics can be developed inorder to place confidence bounds on the absolute measurements. 3 Also shown in Figure 1 is a mathematical axis for describing the two surfaces. The surface of the test optic is described by its distance, Ti, from the axis.  Likewise, the distance from the axis to  the reference surface is  given by Ri. Each  measurement is simply a difference,  Ti - Ri. In conventional interferometry it is only this difference which can be determined and not the individual values of Ri and Ti. Consider  in more detail this two  measurement scheme. The expected values that would be obtained are shown  in Figure 2. In the left hand column are the measurement values for each position of the initial configuration. The right hand column shows the expected mea surement values after the test optic has been translated. Note that the reference values, Ri stay fixed with respect to  the measurement positions. The difference between  the items in the two columns is referred to as a Pseudo-Shear In terf erogram. In Figure 3 the group of equations that result is shown. This set is readily solved for the values  of Ti after choosing an arbitrary value for T one. After obtaining the values  for each Ti, the equations of Figure 2 may be solved for each Ri. This two position technique is simple and readily produces a set of values for both Ti and Ri. However, there is no provision for including additional data nor is there provision for error estimation. These deficiencies are alleviated in the following multiposition analysis. Multiposition PSI Combining the results of three or more measurements with lateral shifts makes it possible to extend the  previous analysis and simultaneously include a least-squares error reduction. This provides a means to estimate the  measurement error as well as the best estimate of the profile  of the test optic and the reference surface. The analysis presented will be for the case of three measurements made with equal-spaced translations as represented in Figure 4. Each column corresponds to one of the measurement sets; each being a translation of one unit from the next.  Also  appearing in these equations is an error term, E. It is assumed that there is an independent, random error associated with each value. There are two independent sets of differences which may be obtained from these; they are shown in Figure 5. Note that D12 and D23 are both estimates of the same quantity and com parable pairs comprise the lists. If additional measurements are made, they can be easily included at this point. Because the  number of unknowns, including the error terms, exceeds the number of equa tions, this set is solved in the least  square sense. Figure 6 shows the derivation of the normal equations from the conditions for minimizing the sum of the squares of the errors. Substitution and evaluation of the partial derivatives gives the normal equations. The nor mal equations may be displayed in somewhat more compact form by the use of matrices. Addi tionally, display of the matrix  for a specific size array shows the starting and ending values explicitly. Figure 7 shows the matrix equation for the case of five positions on the test optic. The matrix on the left is a symetric band matrix. All diagonal elements are four except the last. This same form prevails for any matrix size greater than two. The column vector on the right maintains the pattern shown for arrays of any size. This matrix may be readily inverted to solve for the profile  of the test optic, Ti. Ob taining the values for Ti might  be considered the end of the problem. However,  for most ac curate work, we are required to determine an error estimate for the measurements.Returning to Figure 4 for a moment, it will be noted the differences might be arranged to eliminate Ti and thus be used to determine Ri. The resulting differences are shown in  Figure 8. These are shifted differences and yield PSI data for the interferometer. Application of the least-square methodology to this  set of equations produces the matrix equation for de termination of the reference as shown in Figure 9. Again,  the patterns of the components of this equation is shown for dimension five by five. Extension  to matrices of other sizes is readily accomplished. The solution for the values of Ri is straight forward. The solutions of these two matrix equations have given values for Ri and Ti. Substitu tion of these values into the equations of Figure 4 permit determination of the errors as sociated with each  value. From the enumerated values, error statistics can be developed in order to place confidence bounds on the absolute measurements. Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/04/2014 Terms of Use: http://spiedl.org/terms  Summary A technique for processing interferometric data has been described which allows absolute measurements to be obtained without reference to a standard. The Pseudo -Shear Interferogram is obtained by digital processing of a sequence of measurements with lateral shifts between them. The algorithm has provision for including additional data and for error analysis.Surface profile of Test optic M1 M2 M3 M4M5 Surface profile of Reference Figure 1. Measurement geometry. At each position, themeasurement yields the distance between the surfaces.The measurements, Mi, are made at equally spaced positions.MEASUREMENT (Initial) M11= Tl- R1 1 MEASUREMENT 2 (Translated)M12=R2- R2M22= T1- R2 M13=T3- R3M23= T2- R3 M14=T4- R4M24= T3- R4Figure 2. Measurements before and after translation of test optic. 4 Summary A technique for processing interferometric data has been described which allows absolute measurements to be obtained without reference to standard. The Pseudo-Shear Interferogram is obtained by digital processing of sequence of measurements with lateral shifts between them. The  algorithm has provision for including additional data and for error analysis. Surface profile of Test optic Ml M2 M3 M4 M5 Surface profile of Reference Figure 1 Measurement geometry. At each position, the measurement yields the distance between the  surfaces. The measurements, Mi, are made  at equally spaced positions. MEASUREMENT 1 Initial) MEASUREMENT 2 Translated)Mll= Tl- Rl M12= R2- R2 M13= T3- R3 M14= T4- R4 M22= Tl- R2 M23= T2- R3 M24= T3-  R4 Figure Measurements before and after  translation of test optic. Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/04/2014 Terms of Use: http://spiedl.org/terms  M12 - M22 = T2 - T1M13 - M23 = T3 - T2M14 - M24 = T4 - T3M15 - M25 = T5 - T4 Figure 3. Measurement differences. This set of equations is readily solved for the values of Ti. MEASUREMENT 1 (Initial) M11= T1- R1+ EllM12= T2- R2+ El2M13= T3- R3+ E13M14= T4- R4+ El4M15= T5- R5+ E15 MEASUREMENT 2 MEASUREMENT 3 (1st translation)(2nd translation)M22= T1- R2+ E22M23= T2- R3+ E23M24= T3- R4+ E24M25= T4- R5+ E25 M33= Ti- R3+ E33M34= T2- R4+ E34M35= T3- R5+ F35 Figure 4. Measurements before and after each translation. Error terms are included. FIRST DIFFERENCESD12 = M12 - M22 = T2 - Ti + E112D13 = M13 - M23 = T3 - T2 + E113D14 = M14 - M24 = T4 - T3 + E114SECOND DIFFERENCESD23 = M23 - M33 = T2 - Tl + E223D24 = M24 - M34 = T3 - T2 + E224D25 = M25 - M35 = T4 - T3 + E225 Figure 5. Measurement differences. These arerepeated estimates of the same quantities. 5 M12 M22 = T2 - Tl M13 M23 = T3 - T2 M14 M24 = T4 T3 M15 M25 = T5 T4 Figure 3 Measurement differences. This set of  equations is  readily solved for the values of Ti. ME SUREMENT  1 ME SUREMENT 2 ME SUREMENT 3 (Initial) (1st  translation) (2nd  translation Mll= Tl- R1+ Ell M12= T2- R2+ E12  M22= Tl- R2+ E22 M13=  T3- R3 + E13 M23 T2- R3+ E23 M33= Tl-  R3 + E33 M14= T4- R4+ E14 M24= T3- R4+ E24 M34= T2- R4+ E34 M15=  T5- R5+ E15 M25= T4- R5+ E25 M35= T3- R5+ E35 Figure 4. Measurements before and after each translation. rror terms are included. FIRST DIFFERENCES D12 = M12 M22 = T2 Tl + E112 D13 = M13 M23 = T3 T2'+ E113 D14 =  M14 - M24 = T4  - T3 + E114 SECOND DIFFERENCES D23 = M23 -  M33 = T2 - Tl £223 D24 = M24 M34 = T3 T2 +  E224D25 =  M25 - M35 - T4 T3 + E225 Figure 5 Measurement differences. These are repeated estimates  of  the same quantities. Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/04/2014 Terms of Use: http://spiedl.org/terms
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