A high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method and apparatus

Abstract

This invention discloses a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry measurement method and device, belonging to the field of optical precision measurement technology. The implementation method of this invention is as follows: initial values ​​of the phase shift are obtained through principal component analysis; initial values ​​for the iterative algorithm are obtained by calculating the background light intensity and modulation parameters of the interferogram; an environmental vibration error model is established to improve the suppression capability of noise such as vibration in the measured environment; and an iterative linear regression algorithm is used in conjunction with the initial estimation of the measured surface shape, iteratively solving for spatial and temporal variables respectively to improve the system's response speed and measurement accuracy. This invention features fast convergence and strong robustness. By obtaining the initial values ​​of the measured surface shape, as well as the modulation and background noise, and constructing a noise model that varies with time, environmental noise is suppressed, achieving high-precision measurement of the surface shape parameters of large-aperture optical elements.

Description

Technical Field

[0001] This invention relates to a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method and apparatus, belonging to the field of optical precision measurement technology. Background Technology

[0002] Phase-shifting interferometry (PSI) is widely used in high-precision optical systems such as astronomical telescopes, inertial confinement fusion devices, and semiconductor wafer inspection due to its unique high-precision measurement capabilities and ability to eliminate background interference. As one of the most important components of PSI, the performance of the phase-shifting interferometry algorithm directly determines the measurement accuracy under the same hardware and environmental conditions. Traditional fixed-step-size algorithms and classic random phase-shifting interferometry algorithms both assume that the phase-shifting error is uniform within the pupil plane, i.e., only "position phase-shifting error" exists. However, in actual measurements, various factors such as environmental vibrations will inevitably cause the phase-shifting plane to tilt, thus introducing "tilt phase-shifting error." Therefore, neither fixed-step-size nor random phase-shifting algorithms can eliminate the impact of tilt phase-shifting error on measurement accuracy. This leads to the need for further improvement in the measurement accuracy of current phase-shifting interferometry, and the continued necessity of expensive precision phase shifters to ensure measurement accuracy.

[0003] In fact, improving the anti-vibration phase-shifting error performance of phase-shifting interferometry algorithms has become key to further improving the accuracy of phase-shifting interferometry measurements and completely eliminating the need for expensive high-precision phase shifters. To this end, various anti-tilt phase-shifting interferometry algorithms have been proposed, which can be divided into two categories based on whether an iterative process is required. One category consists of non-iterative algorithms, which have faster computation speeds but generally lower accuracy. For example, Oleg Soloviev et al. and Li et al. proposed using the zero-crossing points generated by subtracting phase-shifting interferograms to solve for the tilt phase-shifting parameters and then extract the phase distribution. However, this method requires a large tilt phase-shifting error to significantly change the fringe direction; otherwise, it cannot be calculated and therefore cannot handle small-amplitude tilt errors. Maciek Wielgus et al. proposed a two-frame algorithm that can handle both large and small tilt phase-shifting errors, but its nonlinear minimization method makes the algorithm quite complex. Liu et al. divided the tilt amplitude into three cases based on size and used corresponding algorithms for calculation. This method has the ability to handle large-range tilt phase-shifting errors, but its classification and processing process makes it difficult to automatically handle various situations that may occur in actual measurements.

[0004] On the other hand, iterative algorithms offer high computational accuracy but are slow. Chen et al. and L. Deck transformed the transcendental equations containing tilt phase shift coefficients into a linear form using a first-order Taylor expansion, and then calculated the phase distribution using a linear least squares iterative method. Liu et al. proposed a three-step iterative method that iteratively calculates the tilt phase shift coefficients in two directions separately, ultimately extracting the phase distribution. These algorithms require high accuracy for the initial iteration values; excessive phase shift errors significantly reduce the accuracy and speed of the algorithms, and may even prevent convergence to the correct result. In summary, existing methods struggle to simultaneously meet the requirement of high accuracy and generally cannot handle large-scale phase shift and tilt errors. Summary of the Invention

[0005] To address the limitations of existing methods in simultaneously meeting the requirements of high precision and high speed, as well as their weak robustness and inability to handle large-scale phase shift and tilt errors, this invention aims to provide a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method and apparatus. The method involves obtaining initial measured surface parameters through principal component analysis; acquiring initial values ​​of the phase shift based on these parameters; calculating the background light intensity and modulation depth of the interferogram to obtain initial values ​​for the iterative algorithm, thereby improving the surface shape accuracy; establishing an environmental vibration error model to enhance the suppression of vibration noise in the measured environment; and combining the initial estimation of the measured surface shape with an iterative linear regression algorithm, iteratively solving for spatial and temporal variables separately. This approach features fast convergence, strong robustness, effectively suppressing environmental vibrations, and improving the device's response speed and measurement accuracy.

[0006] The objective of this invention is achieved through the following technical solution.

[0007] This invention discloses a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method and apparatus, comprising the following steps:

[0008] Step 1: Acquire the interference fringe pattern of the surface under test using a large-aperture interferometer, obtain the phase distribution of the surface under test through principal component analysis, and calculate the true initial value of the phase shift.

[0009] The beam emitted from the interferometer main unit is reflected successively by the surfaces of the reference mirror TF and the standard mirror RF, forming a reference beam and a measurement beam. The reference beam and the measurement beam return along the original optical axis of the interferometer system. When the two beams interfere, interference fringes are formed. The intensity of the interference pattern, which includes environmental vibration noise, is represented as follows:

[0010]

[0011] Where A(x,y) represents the background light intensity of the interference fringes; V(x,y) represents the modulation degree of the interference fringes; This represents the wavefront phase distribution of the measurement light reflected back from the RF source; δ(x,y,t) represents the phase shift, which is caused by environmental vibrations tilting the phase shift plane, as shown below:

[0012] δ(x,y,t)=α n x+β n y+γ n (2)

[0013] Where α n ,β n and γ n These represent the tilt phase shift coefficients and translation phase shifts in the x and y directions, respectively, caused by environmental vibrations. The phase distribution is rapidly calculated using principal component analysis.

[0014] In formula (1), the background light intensity and modulation do not change with pixels, i.e., A m (x,y)=a m B m (x,y)=b m , where a m and b m These are two pixel-independent constants, expressed as:

[0015]

[0016] Where C m (t0)=b m cosδ m (t0), S m (t0)=-b m sinδ m (t0); Since formula (3) applies to C m and S m It is linear; S can be obtained using the linear least squares method. m (t0) and C m (t0), then the true initial value of the phase shift δ(t0) can be obtained by the arctangent formula:

[0017]

[0018] The phase shift calculated by formula (4) is bound within the range of (-π, π]; the unbound criterion SD = δ is defined. m (n)-δ m (n-1), when |SD|<π, it is considered that δ(t0) is not enclosed;

[0019] Step 2: Obtain the true background light intensity and modulation index using the interference pattern obtained in Step 1;

[0020] Based on the obtained interferogram, the background light intensity A(x,y) and modulation index V(x,y) are extracted from the interferogram and expressed as:

[0021]

[0022] Where I(x,y,t) is the light intensity value of the interference pattern obtained by interferometry of the measured surface.

[0023] Step 3: Noise suppression is achieved by establishing an error model for environmental vibration and noise;

[0024] Due to environmental vibrations, the measurement cavity formed by the tested mirror and the reference mirror changes in real time; a noise model expression for the cavity structure is established:

[0025]

[0026] Where τ(t) is the phase shift that varies with time; α(t) and β(t) are the tilt coefficients of the reference surface and the measured surface, respectively; and ρ(x) is the phase shift coefficient that varies with time. 2 =x 2 +y 2 Where c is the phase curvature, and c = 0 when the measured surface is a plane; t0 represents the time of acquiring the first interferogram. The new estimated value is then obtained as:

[0027] δ n (t)=δ(t)+δ'(t)

[0028] α n (t)=α(t)+α'(t) (7)

[0029] β n (t)=β(t)+β'(t)

[0030] Where the subscript δ n (t), α n (t), β n (t) represents the new estimate; δ'(t), α'(t), and β'(t) represent the deviation values;

[0031] Substituting equation (7) into equation (1), we obtain the expression for the actual light intensity affected by the environment:

[0032]

[0033] in,

[0034]

[0035] K represents the iteration order, k represents the number of iterations, k = 1, 2, ..., K; g represents the square root of the product of the reflectivity of the tested mirror and the reference mirror, characterizing the reflection quality of the cavity. This value is greatly affected by the environment and is updated as the number of iterations increases; the evaluation function of the iteration is expressed as:

[0036]

[0037] Where M represents the number of phase shifts; noise suppression is achieved through iterative calculation by updating time-related variables δ(t), α(t), and β(t).

[0038] Step 4: Based on the background light intensity and modulation index obtained in Step 2 and the cavity noise model established in Step 3, the phase-shifting plane is iteratively fitted using an iterative linear regression algorithm to extract the phase to be measured.

[0039] Substituting the background light intensity A(x,y) and modulation index V(x,y) obtained in step two and the δ(x,t) obtained in step three into the iterative linear regression algorithm, we can express the following:

[0040]

[0041] in,

[0042]

[0043] During phase-shifting interferometry, the corresponding interferogram is obtained based on the number of phase shifts M, and the iterative parameter g is updated and optimized accordingly.

[0044]

[0045] The decision to continue iterative calculation is based on whether the iterative evaluation function χ(t) is below 0.1%. If the threshold condition is not met, it is passed to steps three and four for further iterative updates, continuously updating the g value until it converges to within the threshold of the evaluation function; thus, the wrapping phase of the measured surface shape is obtained.

[0046] This invention discloses a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry measurement device, which is based on the large-aperture high-precision phase-shifting interferometry vibration-resistant measurement method. The large-aperture high-precision phase-shifting interferometry vibration-resistant measurement device includes a laser, a beam splitter, a small-aperture collimating lens, a beam expander, a large-aperture collimating lens, a precision adjustment frame for a reference mirror, a reference mirror, a standard mirror under test, a precision adjustment frame for the standard mirror under test, a beam splitter prism, an imaging lens group, and a CCD camera. The point light source emitted from the point light source passes through the beam splitter and then through the small-aperture collimating lens to obtain parallel light of a preset aperture. After passing through the beam expander group and the large-aperture collimating lens, it obtains parallel light of a preset enlarged aperture. One beam is reflected by the rear surface of the reference mirror and then passes through the large-aperture collimating lens again. A reference beam is formed by reflecting off a beam splitter after passing through a collimating mirror, a beam expander group, and a small-aperture collimating mirror beam splitter. Another beam passes through the reference mirror, is reflected off the front surface of the mirror under test, and then passes through the reference mirror, a large-aperture collimating mirror, a beam expander group, and a small-aperture collimating mirror beam splitter again before being reflected off the beam splitter to form the measurement beam. When the reference beam and the measurement beam coherently interfere, the interference pattern of the measured surface is acquired by an interferometer. High-precision phase calculation is obtained based on the large-aperture high-precision phase-shifting interferometry vibration-resistant measurement method, achieving high-precision surface fitting.

[0047] Preferably, the point light emitted from the point light source passes through a beam splitter and then through a small-aperture collimating lens to obtain parallel light with an aperture of 100mm. After passing through a beam expander group and a large-aperture collimating lens, it obtains parallel light with an aperture of 820mm.

[0048] Beneficial effects:

[0049] 1. The present invention discloses a high-precision vibration-resistant large-aperture mechanical phase-shifting interferometry method and device, which obtains the initial phase distribution of the measured surface shape through principal component analysis technology; and extracts the initial value of the phase shift amount according to optical interference theory, which helps to solve the interferometer surface shape with high precision.

[0050] 2. The present invention discloses a high-precision anti-vibration large-aperture mechanical phase-shifting interferometry method and device, which calculates the wrapped phase value from noise fringes through an iterative algorithm, ensuring that the interferometer can be used under conditions of significant environmental interference, calculates the true phase and true surface shape of the mirror under test, realizes high-precision interferometric measurement of the interferometer, can suppress the interference of environmental noise, and improve the surface shape measurement accuracy in harsh environments.

[0051] 3. This invention discloses a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method and apparatus. It has low environmental requirements and can achieve high-precision measurement of the surface parameters of large-aperture optical elements. Through accurate acquisition of initial values, it offers advantages such as fast iteration speed and high measurement accuracy. It avoids the interference problems of existing interferometer measurements due to environmental factors. This invention is applicable not only to large-aperture interferometry systems but also to small-aperture interferometers; it provides a new method and means for achieving high-precision, interference-resistant interferometry measurements with large-aperture interferometers. Attached Figure Description

[0052] Figure 1 This is a flowchart of a high-precision, vibration-resistant, large-diameter mechanical phase-shifting interferometry method according to the present invention.

[0053] Figure 2 This is a diagram of a high-precision, vibration-resistant, large-diameter mechanical phase-shifting interferometry method and device according to the present invention;

[0054] Among them: 1-Laser, 2-Beam splitter, 3-Small aperture collimator, 4-Beam expander, 5-Large aperture collimator, 6-Precision adjustment frame for reference mirror, 7-Reference mirror TF, 8-Standard test mirror RF, 9-Precision adjustment frame for standard test mirror, 10-Beam splitter prism, 11-Imaging lens group, 12-CCD camera, 13-Interference fringe pattern, 14-Wrapped phase pattern.

[0055] Figure 3 The result diagram of a high-precision, vibration-resistant, large-diameter mechanical phase-shifting interferometry measurement method of the present invention. Detailed Implementation

[0056] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0057] Example

[0058] like Figure 2 As shown in this embodiment, a high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry device includes a laser 1, a beam splitter 2, a small-aperture collimating lens 3, a beam expander 4, a large-aperture collimating lens 5, a precision adjustment frame for a reference mirror 6, a reference mirror TF7, a standard test mirror 8, a precision adjustment frame for the standard test mirror 9, a beam splitter 10, an imaging lens group 11, and a CCD camera 12. The laser beam emitted from the laser source 1 passes through the beam splitter 2 and is collimated into a parallel beam by the collimating lens 3. It is then reflected by the surfaces of the reference mirror 7 and the standard test mirror 8, forming a reference beam and a measurement beam. The reference beam and the measurement beam return along the original optical axis of the interferometer system. After coherent interference, the two beams pass through the beam splitter 10 and are imaged in the CCD camera 12 by the imaging lens group 11. The resulting interference fringes 13 are used to calculate the wrapping phase using a large-aperture, high-precision phase-shifting interferometry vibration-resistant measurement method, thereby obtaining the true phase diagram of the measured surface shape.

[0059] like Figure 1 As shown in the figure, this embodiment discloses a high-precision vibration-resistant large-aperture mechanical phase-shifting interferometry method, and the specific implementation steps are as follows:

[0060] Step 1: Acquire the interference fringe pattern of the surface under test using a large-aperture interferometer, obtain the phase distribution of the surface under test through principal component analysis, and calculate the true initial value of the phase shift.

[0061] The beam emitted from the interferometer main unit is reflected successively by the surfaces of the reference mirror TF and the standard mirror RF, forming a reference beam and a measurement beam. The reference beam and the measurement beam return along the original optical axis of the interferometer system. When the two beams interfere, interference fringes are formed. The intensity of the interference pattern, which includes environmental vibration noise, is represented as follows:

[0062]

[0063] Where A(x,y) represents the background light intensity of the interference fringes; V(x,y) represents the modulation degree of the interference fringes; This represents the wavefront phase distribution of the measurement light reflected back from the RF source; δ(x,y,t) represents the phase shift, which is caused by environmental vibrations tilting the phase shift plane, as shown below:

[0064] δ(x,y,t)=α n x+β n y+γ n (2)

[0065] Where α n ,β n and γ n These represent the tilt phase shift coefficients and translation phase shifts in the x and y directions, respectively, caused by environmental vibrations. The phase distribution is rapidly calculated using principal component analysis.

[0066] In formula (1), the background light intensity and modulation do not change with pixels, i.e., A m (x,y)=a m B m (x,y)=b m , where a m and b m These are two pixel-independent constants, expressed as:

[0067]

[0068] Where C m (t0)=b m cosδ m (t0), S m (t0)=-b m sinδm (t0); Since formula (3) applies to C m and S m It is linear, and S can be obtained using the linear least squares method. m (t0) and C m (t0), then the true initial value of the phase shift δ(t0) can be obtained by the arctangent formula:

[0069]

[0070] The phase shift calculated by formula (4) is bound within the range of (-π, π]; the unbound criterion SD = δ is defined. m (n)-δ m (n-1), when |SD|<π, it is considered that δ(t0) is not enclosed;

[0071] Step 2: Obtain the true background light intensity and modulation index using the interference pattern obtained in Step 1;

[0072] Based on the obtained interferogram, the background light intensity A(x,y) and modulation index V(x,y) are extracted from the interferogram and expressed as:

[0073]

[0074] Where I(x,y,t) is the light intensity value of the interference pattern obtained by interferometry of the measured surface.

[0075] Step 3: Noise suppression is achieved by establishing an error model for environmental vibration and noise;

[0076] Due to environmental vibrations, the measurement cavity formed by the tested mirror and the reference mirror changes in real time; a noise model expression for the cavity structure is established:

[0077]

[0078] Where τ(t) is the phase shift that varies with time; α(t) and β(t) are the tilt coefficients of the reference surface and the measured surface, respectively; and ρ(x) is the phase shift coefficient that varies with time. 2 =x 2 +y 2 Where c is the phase curvature, and c = 0 when the measured surface is a plane; t0 represents the time of acquiring the first interferogram. The new estimated value is then obtained as:

[0079] δ n (t)=δ(t)+δ'(t)

[0080] α n (t)=α(t)+α'(t) (7)

[0081] β n (t)=β(t)+β'(t)

[0082] Where the subscript δ n (t), α n (t), β n (t) represents the new estimate; δ'(t), α'(t), and β'(t) represent the deviation values;

[0083] Substituting equation (7) into equation (1), we obtain the expression for the actual light intensity affected by the environment:

[0084]

[0085] in,

[0086]

[0087] K represents the iteration order; calculations show that K=3 satisfies the requirements. k represents the number of iterations, k=1,2,…K. g represents the square root of the product of the reflectivities of the tested mirror and the reference mirror, characterizing the reflection quality of the cavity. This value is greatly affected by the environment and is updated as the number of iterations increases. The evaluation function for the iteration is expressed as:

[0088]

[0089] Where M represents the number of phase shifts; noise suppression is achieved through iterative calculation by updating time-related variables δ(t), α(t), and β(t).

[0090] Step 4: Based on the background light intensity and modulation index obtained in Step 2 and the cavity noise model established in Step 3, the phase-shifting plane is iteratively fitted using an iterative linear regression algorithm to extract the phase to be measured.

[0091] Substituting the background light intensity A(x,y) and modulation index V(x,y) obtained in step two and the δ(x,t) obtained in step three into the iterative linear regression algorithm, we can express the following:

[0092]

[0093] in,

[0094]

[0095] During phase-shifting interferometry, the corresponding interferogram is obtained based on the number of phase shifts M, and the iterative parameter g is updated and optimized accordingly.

[0096]

[0097] The iterative calculation stops when the iterative evaluation function χ(t) = 0.1%. If the threshold condition is not met, it is substituted into steps three and four for further iterative updates, continuously updating the g value until it converges to within the threshold of the evaluation function; thus, the wrapping phase of the measured surface shape is obtained. When the ripple error caused by environmental vibration exceeds 30nm, traditional phase-shifting methods cannot solve the problem; however, this invention can solve the measured surface shape and effectively suppress the influence of environmental noise. Figure 3 As shown, the measured surface shape PV is 0.09λ.

[0098] The specific embodiments of the present invention have been described above with reference to the accompanying drawings. However, these descriptions should not be construed as limiting the scope of the present invention. The scope of protection of the present invention is defined by the appended claims. Any modifications based on the claims of the present invention are within the scope of protection of the present invention.

Claims

1. A high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method, characterized in that: Includes the following steps, Step 1: Acquire the interference fringe pattern of the surface under test using a large-aperture interferometer, obtain the phase distribution of the surface under test through principal component analysis, and calculate the true initial value of the phase shift. Step 2: Obtain the true background light intensity and modulation index using the interference fringe pattern obtained in Step 1; Step 3: Establish an error model for environmental vibration and noise, and perform noise suppression based on the error model. Step 4: Based on the background light intensity and modulation index obtained in Step 2, and combined with the environmental vibration and noise model established in Step 3, the phase-shifting plane is iteratively fitted using an iterative linear regression algorithm to extract the phase to be measured, thus realizing high-precision vibration-resistant large-diameter mechanical phase-shifting interferometry.

2. The high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method as described in claim 1, characterized in that: The first step is implemented as follows: The beam emitted from the interferometer main unit is reflected successively by the surfaces of the reference mirror TF and the standard mirror RF, forming a reference beam and a measurement beam. The reference beam and the measurement beam return along the original optical axis of the interferometer system. When the two beams interfere, interference fringes are formed. The intensity of the interference pattern, which includes environmental vibration noise, is represented as follows: Where A(x,y) represents the background light intensity of the interference fringes; V(x,y) represents the modulation degree of the interference fringes; This represents the wavefront phase distribution of the measurement light reflected back from the RF source; δ(x,y,t) represents the phase shift, which is caused by environmental vibrations tilting the phase shift plane, as shown below: δ(x,y,t)=α n x+b n y+γ n (2) Where α n ,β n and γ n These represent the tilt phase shift coefficients and translation phase shifts in the x and y directions, respectively, caused by environmental vibrations. The phase distribution is rapidly calculated using principal component analysis. In formula (1), the background light intensity and modulation do not change with pixels, i.e., A m (x,y)=a m B m (x,y)=b m , where a m and b m These are two pixel-independent constants, expressed as: Where C m (t0)=b m cosδ m (t0), S m (t0)=-b m sinδ m (t0); Since formula (3) applies to C m and S m It is linear; S can be obtained using the linear least squares method. m (t0) and C m (t0), then the true initial value of the phase shift δ(t0) can be obtained by the arctangent formula: The phase shift calculated by formula (4) is bound within the range of (-π, π]; the unbound criterion SD = δ is defined. m (n)-δ m (n-1), when |SD|<π, it is determined that δ(t0) is not wrapped.

3. The high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method as described in claim 2, characterized in that: Step two is the specific implementation method, Based on the obtained interferogram, the background light intensity A(x,y) and modulation index V(x,y) are extracted from the interferogram and expressed as: Where I(x,y,t) is the light intensity value of the interference pattern obtained by interferometry of the measured surface.

4. The high-precision, vibration-resistant, large-aperture mechanical phase-shifting interferometry method as described in claim 3, characterized in that: Step three is the specific implementation method as follows: Due to environmental vibrations, the measurement cavity formed by the tested mirror and the reference mirror changes in real time; a noise model expression for the cavity structure is established: Where τ(t) is the phase shift that varies with time; α(t) and β(t) are the tilt coefficients of the reference surface and the measured surface, respectively; and ρ(x) is the phase shift coefficient that varies with time. 2 =x 2 +y 2 c is the phase curvature; when the measured surface is a plane, c = 0; t0 represents the time for acquiring the first interferogram. The noise model of the cavity structure shown in Equation (6) is the error model of environmental vibration noise; The noise model of the cavity structure shown in Equation (6) yields a new estimate, expressed as follows: Where the subscript δ n (t), α n (t), β n (t) represents the new estimate; δ'(t), α'(t), and β'(t) represent the deviation values; Substituting equation (7) into equation (1), we obtain the expression for the actual light intensity affected by the environment: in, K represents the iteration order, k represents the number of iterations, k = 1, 2, ..., K; g represents the square root of the product of the reflectivity of the tested mirror and the reference mirror, characterizing the reflection quality of the cavity. This value is greatly affected by the environment and is updated as the number of iterations increases; the evaluation function of the iteration is expressed as: Where M represents the number of phase shifts; by updating the time-related variables δ(t), α(t), and β(t), the evaluation function is iteratively calculated to achieve noise suppression.

5. The high-precision vibration-resistant large-aperture mechanical phase-shifting interferometry method as described in claim 4, wherein step four is specifically implemented as follows: Substituting the background light intensity A(x,y) and modulation index V(x,y) obtained in step two and the δ(x,t) obtained in step three into the MPSI iterative linear regression algorithm, we can express the following: in, During phase-shifting interferometry, the corresponding interferogram is obtained based on the number of phase shifts M, and the iterative parameter g is updated and optimized accordingly. The decision to continue iterative calculation is based on whether the iterative evaluation function χ(t) is below 0.1%. If the threshold condition is not met, the iterative evaluation function χ(t) is substituted into steps three and four for further iterative updates, continuously updating the g value until it converges to within the evaluation function threshold; thus, the wrapping phase of the measured surface shape is obtained.

6. A high-precision, vibration-resistant, large-diameter mechanical phase-shifting interferometry device for implementing the method described in claims 1, 2, 3, 4, or 5, characterized in that: The system includes a laser, beam splitter, small-aperture collimating lens, beam expander, large-aperture collimating lens, precision adjustment mount for the reference lens, reference lens, standard test lens, precision adjustment mount for the standard test lens, beam splitter prism, imaging lens group, and CCD camera. The point light source emitted from the point source passes through the beam splitter and then through the small-aperture collimating lens to obtain parallel light of a preset aperture. After passing through the beam expander group and the large-aperture collimating lens, it obtains parallel light of a preset enlarged aperture. One beam is reflected by the rear surface of the reference lens and then passes again through the large-aperture collimating lens and beam expander. The reference beam is formed by the reflection of the beam splitter after passing through the mirror group, the small-aperture collimating mirror, and the beam expander group. The beam is then reflected by the beam splitter after passing through the reference mirror, the large-aperture collimating mirror, the beam expander group, and the small-aperture collimating mirror and beam splitter to form the measurement beam. When the reference beam and the measurement beam coherently interfere, the interference pattern of the measured surface is collected by the interferometer. High-precision phase calculation is obtained according to the large-aperture high-precision phase-shifting interferometry vibration-resistant measurement method, and high-precision surface fitting is achieved.

7. The high-precision, vibration-resistant, large-diameter mechanical phase-shifting interferometry device as described in claim 6, characterized in that: The point light emitted from the point light source passes through a beam splitter and then through a small-aperture collimating lens to obtain parallel light with an aperture of 100mm. After passing through a beam expander group and a large-aperture collimating lens, it becomes parallel light with an aperture of 820mm.

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