Iterative Correction Method for Return Error in Spherical and Planar Measurements Using Fizeau Interferometer

Abstract

The present invention discloses an iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer. The method comprises the following steps: first, determining the order of the return error according to the type of the measured object and determining the number q of the sensitive factor matrix; changing the posture of the measured object q times and measuring the phase result each time; then, calculating the tilt amount in the result of the measured phase result, substituting the result into the equation group to calculate each sensitive factor matrix and the corrected phase; then calculating the local tilt amount caused by the surface shape in the x and y directions according to the corrected phase, updating the tilt amount, and re-substituting the result into the equation group to calculate each sensitive factor matrix and the corrected phase; repeating the above process until the iteration converges to obtain the sensitive factor matrix and the corrected phase result. The present invention can correct the return error caused by non-zero position measurement in spherical and planar measurement using a Fizeau interferometer, without the need to model the internal structure of the interferometer, and is simple, efficient, and has a wide range of applications.

Description

Technical Field

[0001] The invention belongs to the technical field of optical interference measurement, in particular to an iterative correction method for return error in spherical and plane measurement using a Fizeau interferometer. Background Art

[0002] When the interferometer operates in the zero-order fringe state, the optical paths of the reference light and the test light are consistent, and there will be no return error. However, it is difficult to ensure that the interferometer operates in the zero-order fringe state. For this reason, in the interferometer, the calculation of the surface shape using a non-zero interference pattern will introduce additional errors. For high-precision plane Fizeau measurements, it is necessary to consider the return error proportional to the beam wavefront slope and the cavity geometry. This error exists for both spherical and plane measurements, and when there is a surface deviation between the measured surface and the ideal surface shape, the test light cannot accurately move along the origin path, which will cause a test error. In order to ensure measurement accuracy, it is necessary to correct the return error in the measured phase result to improve the measurement precision and accuracy of the interferometer.

[0003] For example, Kinnstaetter (K. Kinnstaetter, A.W. Lohmann, J. Schwider, and N. Streibl, "Accuracy of phase shifting interferometry," Appl. Opt. 27, 5082 (1988).) discussed the accuracy of interferometers and pointed out that aberrations in interferometers can cause fringe distortion. Liu Dong (D. Liu, Y. Yang, C. Tian, ​​Y. Luo, and L. Wang, "Practical methods for retrace error correction in nonnull aspheric testing," 11 (2009).) divided retrace error into two forms: retrace device error and retrace path error. The retrace path error is further divided into retrace phase error and retrace coordinate error. He discussed the principle of error problems in aspheric non-null position detection systems from the perspective of ray tracing. Evans (CJ Evans and J.B. Bryan, "Compensation for Errors Introduced by Nonzero Fringe Densities in Phase-Measuring Interferometers," CIRP Annals 42, 577–580 (1993).) proposed a compensation method to eliminate the return error caused by slight tilt and used the relationship between tilt angle and return error to compensate for the return error. However, this method was not explained in detail and could not calculate the return phase error. Huang (C. Huang, "Propagation errors in precision Fizeau interferometry," Appl. Opt. 32, 7016 (1993).) applied the fourth-order aberration formula to study the return error and discussed various combinations of imaging system aberrations and the surface shape of the test plate.DESilin (DESilin, I.E. Kozhevatov, E.H. Kulikova, A.V. Pigasin, S.B. Speransky, S.A. Bel'kov, D.I. Derkach, D.I. Lobachev, and E.C. Hernov, "The Design and Characteristics of a 630-mm Phase-Shifting Interferometer," Instrum Exp Tech 61, 393–399 (2018).) compensated for the return error in a Φ630 mm phase-shifting interferometer by treating it as a linear equation related to the tilt angle. He Yiwei (H. Yiwei, X. Hou, Q. Haiyang, and W. Song, "Retrace error reconstruction based on point characteristic function," Opt. Express 23, 28216 (2015).) used a point characteristic function to simulate the wavefront of a Fizeau interferometer and used the PCF method to study the return error with and without component errors. Gappinger, RO (RO Gappinger and JEGreivenkamp, ​​"Iterative reverse optimization procedure for calibration of aspheric wave-front measurements on a nonnull interferometer," Appl. Opt. 43, 5152 (2004).) proposed using system modeling and ray tracing to iteratively measure system performance and adjust alignment accordingly to reduce the error of the optical system. This requires inputting optical component data into the model, which is computationally complex and the model will change with the interferometer structure, resulting in low reliability of the calculation results. Summary of the Invention

[0004] The object of the present invention is to provide a method for correcting the return error in plane measurement and spherical measurement of a Fizeau interferometer, which can realize the correction of the return error in plane measurement and spherical measurement of the Fizeau interferometer and improve the measurement precision and accuracy of the interferometer.

[0005] The technical solution for achieving the purpose of the present invention is: an iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer, comprising the following steps:

[0006] Step 1: Determine the order of the return error according to the type of the device under test and determine the sensitive factor matrix number q;

[0007] Step 2: Change the posture of the device under test and repeat 3q times to obtain 3q measurement results. Then, calculate the corresponding tilt value in each result based on the 3q measured phase data.

[0008] Step 3: Substitute the tilt parameters into the sensitivity factor linear equations to calculate the corresponding sensitivity factor matrix and the correction phase result;

[0009] Step 4: Calculate the tilt caused by the local surface shape based on the calculated phase result, then update the tilt parameter, re-substitute it into the sensitive factor linear equation group, calculate the new sensitive factor matrix and correct the phase;

[0010] Step 5: Obtain the convergence judgment function value based on the phase results before and after the calculation. If the convergence threshold is not met, continue to step 4. Otherwise, end the iterative loop and output the return error correction result and the corresponding sensitivity factor matrix.

[0011] Compared with the prior art, the present invention has the following significant advantages: (1) by establishing a relationship between the inclination angle and the return error, the return error can be intuitively calculated from the inclination angle; (2) the return device error in the return error can be corrected by iterative calculation; (3) there is no need to model the internal structure of the interferometer, the method is simple and efficient, does not require any additional auxiliary hardware, and is applicable to most Fizeau interferometers. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 The present invention provides a flow chart of an iterative correction method for return error in spherical and planar measurements using a Fizeau interferometer.

[0013] Figure 2 These are three phase result diagrams collected in the planar measurement in an embodiment of the present invention, where (a) is the phase diagram of the first tilt measurement of the device under test, (b) is the phase diagram of the second tilt measurement of the device under test, and (c) is the phase diagram of the third tilt measurement of the device under test.

[0014] Figure 3 Figures 1 and 2 show the phase results after return error correction, the error-free phase of zero-position measurement, and the relative residual diagram in an embodiment of the present invention. (a) shows the phase result after return error correction, (b) shows the error-free phase of zero-position measurement, and (c) shows the relative residual diagram.

[0015] Figure 4These are six phase result diagrams collected in spherical surface measurements in an embodiment of the present invention, where (a) is the phase diagram measured when the device under test is tilted for the first time, (b) is the phase diagram measured when the device under test is tilted for the second time, (c) is the phase diagram measured when the device under test is tilted for the third time, (d) is the phase diagram measured when the device under test is tilted for the fourth time, (e) is the phase diagram measured when the device under test is tilted for the fifth time, and (f) is the phase diagram measured when the device under test is tilted for the sixth time.

[0016] Figure 5 Figures 1 and 2 show the phase results after return error correction, the error-free phase of zero-position measurement, and the relative residual diagram in an embodiment of the present invention. (a) shows the phase result after return error correction, (b) shows the error-free phase of zero-position measurement, and (c) shows the relative residual diagram. DETAILED DESCRIPTION

[0017] The present invention provides an iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer, comprising the following steps:

[0018] Step 1: Determine the order of the return error according to the type of the device under test and determine the sensitive factor matrix number q;

[0019] Step 2: Change the posture of the device under test and repeat 3q times to obtain 3q measurement results. Then, calculate the corresponding tilt value in each result based on the 3q measured phase data.

[0020] Step 3: Substitute the tilt parameters into the sensitivity factor linear equations to calculate the corresponding sensitivity factor matrix and the correction phase result;

[0021] Step 4: Calculate the tilt caused by the local surface shape based on the calculated phase result, then update the tilt parameter, re-substitute it into the sensitive factor linear equation group, calculate the new sensitive factor matrix and correct the phase;

[0022] Step 5: Obtain the convergence judgment function value based on the phase results before and after the calculation. If the convergence threshold is not met, continue to step 4. Otherwise, end the iterative loop and output the return error correction result and the corresponding sensitivity factor matrix.

[0023] Furthermore, in step 1, the order of the return error is determined according to the type of the device under test, and the number of sensitive factor matrices q is determined, as follows:

[0024] In plane measurement, the number of sensitive factor matrices q=1; in spherical measurement, the number of sensitive factor matrices q=2.

[0025] Furthermore, in plane measurement, the number of sensitive factor matrices q=1; in spherical measurement, the number of sensitive factor matrices q=2, as follows:

[0026] In plane measurement, the return error is the wavefront error caused by the tilt of the measured object, so:

[0027]

[0028] Where W(ρ,θ,x0) is the wavefront quantity, ΔW n is the collimated beam wavefront in the interferometer, x0 is the wavefront coordinate, Δx is the offset in the x direction, W 222 is the astigmatism coefficient, ρ is the corresponding polar diameter in polar coordinates, θ is the polar angle in polar coordinates, W 131 is the coma coefficient, W 220 is the field curvature coefficient, W 311 is the distortion coefficient;

[0029] According to formula (1), the wavefront error introduced by the tilt of the measured object is approximately linearly related to the field of view angle of the imaging point position. Therefore, the return error in plane measurement is caused by the additional aberration introduced by the optical elements inside the interferometer for the tilted beam. Different aberrations are linearly related to the tilt of the measured object, and the sensitivity factor matrix number q = 1;

[0030] In spherical surface measurement, the return error also includes the return error caused by the inconsistent optical path length of the reflected light beam between the reference part and the test part, so:

[0031] OPD≈α 2 R1((ρ+1) / ρ) (2)

[0032] Wherein, OPD is the return error caused by the inconsistency of the optical path of the reflected beam between the reference piece and the test piece, α is the tilt angle of the test wavefront, R1 is the radius of the reference spherical lens, R2 is the radius of the test spherical lens, and ρ = R2 / R1;

[0033] Therefore, in spherical surface measurement, the return error is not only related to the radius of the measured object, but also has an approximately linear relationship with the square of the wavefront tilt, and the sensitivity factor matrix number q=2.

[0034] Furthermore, the posture of the device under test is changed as described in step 2, and the measurement is repeated 3q times to obtain 3q measurement results. Then, the corresponding tilt amount in each result is calculated based on the 3q measured phase data, as follows:

[0035] Step 2.1: Change the posture of the measured object, repeat the measurement 3q times, and obtain 3q measurement results;

[0036] Step 2.2: For the results of 3q measurements, calculate the inclination angle θ of the test piece in the x and y directions. xp ,θ yp calculate:

[0037] θ xp =θgxp_p +θ gxt_p

[0038] θ yp =θ gyp_p +θ gyt_p

[0039] Where θ gxt_p ,θ gyt_p is the tilt caused by the global tilt term in the measurement result, corresponding to the tilt in the Zernike coefficient in the measurement result; θ gxp_p ,θ gyp_p The tilt caused by global defocus in the measurement results as the measured object tilts, θ gxp_p =θ gyp_p =0;

[0040] In spherical measurement,

[0041]

[0042] Where a is the coefficient corresponding to the defocus term in the Zernike coefficient, x and y are the normalized spatial coordinates of the device under test, D is the aperture of the device under test, and λ is the operating wavelength of the interferometer.

[0043] Furthermore, the posture of the device under test is changed as described in step 2.1, and the measurement is repeated 3q times to obtain 3q measurement results, which are as follows:

[0044] To change the posture of the measured object, it is necessary to make requirements for tilting in the x and y directions; in order to ensure the independence of the equation group, the direction of the interference fringes during measurement must be consistent during 3q measurements.

[0045] Furthermore, the tilt parameters described in step 3 are substituted into the sensitivity factor linear equations to calculate the corresponding sensitivity factor matrix and the correction phase result, as follows:

[0046] Substitute the phase results measured under different postures and the corresponding calculated inclination angles into the equations between the return error and the inclination angle of the device under test:

[0047]

[0048] In the formula The measured surface shape after return error correction, when n and m are not all 0, is the sensitivity factor matrix of the interferometer system, θ xp ,θ yp is the global tilt in the measurement result, including the tilt caused by the local area surface shape; q is the last power series of the tilt angle in the black box model. In plane measurement, q = 1. The sensitive factor matrix in plane measurement is C x1 、C y1, respectively represent the corresponding coefficient matrix under different inclination relationships in formula (4), a total of 2 items; in spherical measurement, q = 2, the sensitivity factor matrix in spherical measurement is C x1 、C y1 、C xy 、 They represent the corresponding coefficient matrices under different inclination angles in formula (4), with a total of 5 items;

[0049] According to formula (4) and the least squares criterion, the corresponding sensitivity factor matrix and the phase C after return error correction are calculated. 11 .

[0050] Furthermore, according to the calculated phase result in step 4, the tilt amount caused by the local surface shape is calculated, and then the tilt parameter is updated and re-substituted into the sensitive factor linear equation group to calculate the new sensitive factor matrix and the correction phase, as follows:

[0051] Step 4.1, based on the result of calculating the correction phase, use the tilt correction formula to update the tilt parameters;

[0052] Step 4.2: Substitute the updated tilt angle into equation (4) to recalculate the sensitivity factor matrix and the phase after return error correction.

[0053] Furthermore, the tilt correction formula described in step 4.1 is as follows:

[0054]

[0055] In the formula is the global tilt of the calculated phase after k iterations, c is the calibration coefficient, which represents the actual measurement size corresponding to a single pixel, λ is the interferometer working wavelength, x and y are the spatial coordinates, is the error-free surface shape after eliminating the overall tilt calculated in the k-1th iteration; the first iteration uses the return error correction result calculated in step 3, θ gxt_p ,θ gyt_p is the tilt caused by the global tilt term in the measurement result; θ gxp_p ,θ gyp_p The tilt in the measurement result is caused by the global defocus that changes with the tilt of the device under test.

[0056] Furthermore, the convergence judgment function value is obtained according to the phase results before and after the calculation in step 5. If the convergence threshold is not met, step 4 is continued. Otherwise, the iterative loop is terminated, and the return error correction result and the corresponding sensitivity factor matrix are output, as follows:

[0057] Step 5.1: Calculate the convergence judgment function based on the relative error between the phase calculated in the new round and the phase calculated in the previous round;

[0058] Step 5.2: If the convergence judgment function does not meet the convergence critical threshold, return to step 4, substitute the phase result into the tilt angle update formula again, and perform the next round of iterative calculation; if the convergence judgment function meets the convergence critical threshold, end the iterative loop and output the phase result after return error correction.

[0059] Furthermore, the convergence judgment function described in step 5.1 is formulated as follows:

[0060]

[0061] In the formula, ∑ is the summation function, || ||2 is the two-norm function, Calculate the surface shape that eliminates the return error for the kth iteration, is the surface shape after the k-1th iteration to eliminate the back-pass error, ε is the convergence critical threshold, which is 10 -2 .

[0062] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0063] Example

[0064] Combine Figure 1 The iterative correction method for the return error in the spherical and plane measurement of the Fizeau interferometer in this embodiment includes the following steps:

[0065] Step 1: Determine the order of the return error according to the type of the device under test and determine the sensitivity factor matrix number q, as follows:

[0066] In plane measurement, the return error is the wavefront error caused by the tilt of the measured object, so:

[0067]

[0068] Where W(ρ,θ,x0) is the wavefront quantity, ΔW n The collimated beam wavefront in the interferometer, x0 is the wavefront coordinate, Δx is the offset in the x direction, W 222 is the astigmatism coefficient, ρ is the corresponding polar diameter in polar coordinates, θ is the polar angle in polar coordinates, W 131 is the coma coefficient, W 220 is the field curvature coefficient, W 311 is the distortion coefficient.

[0069] According to formula (1), the wavefront error introduced by the tilt of the measured object is approximately linearly related to the field of view angle of the imaging point position. Therefore, the return error in plane measurement is caused by the additional aberration introduced by the optical elements inside the interferometer for the tilted beam. Different aberrations are linearly related to the tilt of the measured object. The sensitivity factor matrix number q = 1;

[0070] In spherical surface measurement, the return error also includes the return error caused by the inconsistent optical path length of the reflected light beam between the reference part and the test part, so:

[0071] OPD≈α 2 R1((ρ+1) / ρ) (2)

[0072] Where OPD is the return error caused by the inconsistency of the optical path of the reflected beam between the reference piece and the test piece, α is the test wavefront tilt angle, R1 is the radius of the reference spherical lens, R2 is the radius of the test spherical lens, and ρ = R2 / R1;

[0073] Therefore, in spherical surface measurement, the return error is not only related to the radius of the measured object, but also has an approximately linear relationship with the square of the wavefront tilt, and the sensitivity factor matrix number q=2.

[0074] Step 2: Change the posture of the device under test and repeat the measurement 3q times to obtain 3q measurement results. Then, based on the 3q measured phase data, calculate the corresponding tilt in each measurement result, as follows:

[0075] Step 2.1: Change the posture of the measured object and repeat the measurement 3q times to obtain 3q measurement results. To ensure the independence of the equation system, the direction of the interference fringes during measurement must be consistent in the 3q measurements.

[0076] Step 2.2: For the results of 3q measurements, calculate the inclination angle θ of the test piece in the x and y directions. xp ,θ yp calculate:

[0077] θ xp =θ gxp_p +θ gxt_p

[0078] θ yp =θ gyp_p +θ gyt_p

[0079] Where θ gxt_p ,θ gyt_p is the tilt caused by the global tilt term in the measurement result, corresponding to the tilt in the Zernike coefficient in the measurement result; θ gxp_p ,θ gyp_p The tilt caused by global defocus in the measurement results as the measured object tilts, θ gxp_p =θ gyp_p =0;

[0080] In spherical measurement,

[0081]

[0082] Where a is the coefficient corresponding to the defocus term in the Zernike coefficient, x and y are the normalized spatial coordinates of the device under test, D is the aperture of the device under test, and λ is the operating wavelength of the interferometer.

[0083] Step 3: Substitute the tilt parameters into the sensitivity factor linear equations to calculate the corresponding sensitivity factor matrix and the correction phase results, as follows:

[0084] like Figure 2 As shown in (a), (b), and (c), the phase results and corresponding tilt angles measured at different postures are substituted into the equations between the return error and the tilt angle of the device under test:

[0085]

[0086] In the formula The measured surface shape after return error correction, when n and m are not all 0, is the sensitivity factor matrix of the interferometer system, θ xp ,θ yp is the global tilt in the measurement result, including the tilt caused by the local area surface shape; q is the last power series of the tilt angle in the black box model. In plane measurement, q = 1. The sensitive factor matrix in plane measurement is C x1 、C y1 , respectively represent the corresponding coefficient matrix under different inclination relationships in formula (4), a total of 2 items; in spherical measurement, q = 2, the sensitivity factor matrix in spherical measurement is C x1 、C y1 、C xy 、 They represent the corresponding coefficient matrices under different inclination angles in formula (4), with a total of 5 items;

[0087] According to formula (4) and the least squares criterion, the corresponding sensitivity factor matrix and the phase C after return error correction are calculated. 11 .

[0088] Step 4: Based on the calculated phase results, calculate the tilt caused by the local surface shape, then update the tilt parameter, re-substitute it into the sensitive factor linear equation group, calculate the new sensitive factor matrix and correct the phase, as follows:

[0089] Step 4.1: Based on the result of calculating the correction phase, use the tilt correction formula to update the tilt parameters. The formula is as follows:

[0090]

[0091] In the formula is the global tilt of the calculated phase after k iterations, c is the calibration coefficient, which represents the actual measurement size corresponding to a single pixel, λ is the interferometer working wavelength, x and y are the spatial coordinates, is the error-free surface shape after eliminating the overall tilt calculated in the k-1th iteration; the first iteration uses the return error correction result calculated in step 3, θ gxt_p ,θ gyt_p is the tilt caused by the global tilt term in the measurement result; θ gxp_p ,θ gyp_p The tilt in the measurement result is caused by the global defocus that changes with the tilt of the device under test.

[0092] Step 4.2: Substitute the updated tilt angle into equation (4) to recalculate the sensitivity factor matrix and the phase after return error correction.

[0093] Step 5: Obtain the convergence judgment function value based on the phase results before and after the calculation. If the convergence threshold is not met, continue to step 4. Otherwise, end the iterative loop and output the return error correction result and the corresponding sensitivity factor matrix, as follows:

[0094] Step 5.1: Calculate the convergence judgment function based on the relative error between the phase calculated in the new round and the phase calculated in the previous round. The formula is:

[0095]

[0096] In the formula, ∑ is the summation function, || ||2 is the two-norm function, Calculate the surface shape that eliminates the return error for the kth iteration, is the surface shape after the k-1th iteration to eliminate the back-pass error, ε is the convergence critical threshold, which is 10 -2 .

[0097] Step 5.2: If the convergence judgment function does not meet the convergence critical threshold, return to step 4, substitute the phase result into the tilt angle update formula again, and perform the next round of iterative calculation; if the convergence judgment function meets the convergence critical threshold, the iterative loop ends and the phase result after return error correction is output, such as Figure 3 As shown in (a), (b) and (c).

[0098] This embodiment adopts the iterative correction method of the return error in the spherical and plane measurement of the Fizeau interferometer of the present invention to correct the collection results in the spherical measurement. Figure 4 (a), (b), (c), (d), (e), and (f) are the six phase result images collected in the spherical measurement. Figure 5 (a), (b), and (c) are the phase results after return error correction, the error-free phase of zero-position measurement, and the relative residual diagram.

[0099] according to Figure 4 、 Figure 5It can be seen that the present invention uses a black box model to calculate the return error in spherical and plane interferometry measurements. It does not require modeling the interferometer for ray tracing, and the iteration speed is fast. By substituting the local tilt into the iteration process, the return device error in the return error can be corrected.

Claims

1. An iterative correction method for return error in spherical and planar measurements using a Fizeau interferometer, characterized in that: The following steps are involved: Step 1: Determine the order of the return error according to the type of the device under test and determine the sensitive factor matrix number q; Step 2: Change the posture of the device under test and repeat 3q times to obtain 3q measurement results. Then, calculate the corresponding tilt angle in each result based on the 3q measured phase data. Step 3: Substitute the tilt angle into the linear equations of the sensitivity factors to calculate the corresponding sensitivity factor matrix and the correction phase result; Step 4: Calculate the tilt angle caused by the local surface shape based on the calculated phase result, then update the tilt angle and re-substitute it into the sensitive factor linear equation group to calculate the new sensitive factor matrix and the corrected phase; Step 5: Obtain the convergence judgment function value based on the phase results before and after the calculation. If the convergence threshold is not met, continue to step 4. Otherwise, end the iterative loop and output the return error correction result and the corresponding sensitivity factor matrix; In step 1, the order of the return error is determined according to the type of the device under test, and the number of sensitive factor matrices q is determined as follows: In plane measurement, the number of sensitive factor matrices q = 1; in spherical measurement, the number of sensitive factor matrices q = 2; In plane measurement, the number of sensitive factor matrices q = 1; in spherical measurement, the number of sensitive factor matrices q = 2, as follows: In plane measurement, the return error is the wavefront error caused by the tilt angle of the measured object, so: Where W(ρ,θ,x0) is the wavefront quantity, ΔW n is the collimated beam wavefront in the interferometer, x0 is the wavefront coordinate, Δx is the offset in the x direction, W 222 is the astigmatism coefficient, ρ is the corresponding polar diameter in polar coordinates, θ is the polar angle in polar coordinates, W 131 is the coma coefficient, W 220 is the field curvature coefficient, W 311 is the distortion coefficient; According to formula (1), the wavefront error introduced by the tilt angle of the measured object is approximately linearly related to the field of view angle of the imaging point position. Therefore, the return error in plane measurement is caused by the additional aberration introduced by the optical elements inside the interferometer to the tilted beam. Different aberrations are linearly related to the tilt angle of the measured object, and the sensitivity factor matrix number q = 1; In spherical surface measurement, the return error also includes the return error caused by the inconsistent optical path length of the reflected light beam between the reference part and the test part, so: OPD≈α 2 R1((ρ+1) / ρ)(2)where OPD is the return error caused by the inconsistency of the optical path of the reflected beam between the reference piece and the test piece, α is the test wavefront tilt angle, R1 is the radius of the reference spherical lens, R2 is the radius of the test spherical lens, and ρ=R2 / R1; Therefore, in spherical surface measurement, the return error is not only related to the radius of the measured object, but also has an approximately linear relationship with the square of the wavefront tilt angle, and the sensitivity factor matrix number q=2.

2. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 1, characterized in that: Change the posture of the device under test as described in step 2, repeat the measurement 3q times, obtain 3q measurement results, and then calculate the corresponding tilt angle in each result based on the 3q measured phase data, as follows: Step 2.1: Change the posture of the measured object, repeat the measurement 3q times, and obtain 3q measurement results; Step 2.2: For the results of 3q measurements, calculate the inclination angle θ of the test piece in the x and y directions. xp ,θ yp calculate: i xp =θ gxp_p +θ gxt_p i yp =θ gyp_p +θ gyt_p Where θ gxt_p ,θ gyt_p is the tilt angle caused by the global tilt angle in the measurement result, corresponding to the tilt angle in the Zernike coefficient in the measurement result; θ gxp_p ,θ gyp_p The tilt angle caused by the global defocus in the measurement results changes with the tilt angle of the measured object. In plane measurement, θ gxp_p =θ gyp_p =0; In spherical measurement, Where a is the coefficient corresponding to the defocus term in the Zernike coefficient, x and y are the normalized spatial coordinates of the device under test, D is the aperture of the device under test, and λ is the operating wavelength of the interferometer.

3. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 2, characterized in that: Change the posture of the DUT as described in step 2.1, repeat the measurement 3q times, and obtain 3q measurement results, as follows: Changing the posture of the measured object requires requirements for the tilt angles in the x and y directions; in order to ensure the independence of the equation group, the direction of the interference fringes during measurement must be consistent during 3q measurements.

4. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 1, characterized in that: Substitute the tilt angle into the sensitivity factor linear equations described in step 3 to calculate the corresponding sensitivity factor matrix and the correction phase result, as follows: Substitute the phase results measured under different postures and the corresponding calculated inclination angles into the equations between the return error and the inclination angle of the device under test: In the formula The measured surface shape after return error correction, when n and m are not all 0, is the sensitivity factor matrix of the interferometer system, θ xp ,θ yp is the global tilt angle in the measurement result, including the tilt angle caused by the local area surface shape; q is the last power series of the tilt angle in the black box model. In plane measurement, q = 1. The sensitive factor matrix in plane measurement is They represent the corresponding coefficient matrices under different inclination angles in equation (4), which are 2 items in total. In spherical measurement, q = 2, and the sensitivity factor matrix in spherical measurement is They represent the corresponding coefficient matrices under different inclination angles in formula (4), with a total of 5 items; According to formula (4) and the least squares criterion, the corresponding sensitivity factor matrix and the phase C after return error correction are calculated. 11 .

5. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 4, characterized in that: According to the calculated phase result described in step 4, the tilt angle caused by the local surface shape is calculated, and then the tilt angle is updated and re-substituted into the sensitive factor linear equation group to calculate the new sensitive factor matrix and the corrected phase, as follows: Step 4.1, based on the result of calculating the correction phase, use the tilt correction formula to update the tilt parameters; Step 4.2: Substitute the updated tilt angle into equation (4) to recalculate the sensitivity factor matrix and the phase after return error correction.

6. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 5, characterized in that: The inclination correction formula described in step 4.1 is as follows: In the formula is the global tilt angle of the calculated phase after k iterations, c is the calibration coefficient, which represents the actual measurement size corresponding to a single pixel, λ is the interferometer working wavelength, x and y are the spatial coordinates, is the error-free surface shape after eliminating the overall tilt angle calculated in the k-1th iteration; the first iteration uses the return error correction result calculated in step 3, θ gxt_p ,θ gyt_p is the tilt angle caused by the global tilt angle in the measurement result; θ gxp_p ,θ gyp_p The tilt angle in the measurement result is caused by the global defocus that changes with the tilt angle of the test piece.

7. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 1, characterized in that: The convergence judgment function value is obtained according to the phase results before and after the calculation in step 5. If the convergence threshold is not met, continue to step 4. Otherwise, the iterative loop is terminated and the return error correction result and the corresponding sensitivity factor matrix are output. The details are as follows: Step 5.1: Calculate the convergence judgment function based on the relative error between the phase calculated in the new round and the phase calculated in the previous round; Step 5.2: If the convergence judgment function does not meet the convergence critical threshold, return to step 4, substitute the phase result into the tilt angle update formula again, and perform the next round of iterative calculation; if the convergence judgment function meets the convergence critical threshold, end the iterative loop and output the phase result after return error correction.

8. The iterative correction method for return error in spherical and planar measurement using a Fizeau interferometer according to claim 7, characterized in that: The convergence judgment function described in step 5.1 is: In the formula, ∑ is the summation function, || ||2 is the two-norm function, is the surface shape that eliminates the return error in the kth iteration calculation, ε is the convergence critical threshold, and its value is 10 -2 .

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