A Phase Shifter Calibration Method Based on Joint Search in Spatial and Frequency Domains

Abstract

The present invention relates to a phase shifter calibration method based on joint spatio-frequency domain search, comprising the following steps: modeling the phase shifter to be calibrated as a plane jointly supported and driven by a plurality of PZTs, determining the variation relationship between the phase shift values corresponding to each PZT and the voltage, and constructing the coefficients corresponding to the variation relationships of all PZTs into a matrix model; determining the interference pattern serving as a reference and the position of its first-order spectrum, and all subsequent spectral transforms are CZT transforms centered on the position of the first-order spectrum; calibrating the tilt coefficients in the matrix model by using a frequency domain search method; calibrating the position coefficients in the matrix model by using a spatial domain search method; and obtaining the calibrated matrix model as the calibration result of the phase shifter. Compared with the prior art, the present invention effectively reduces the phase shift error of the phase shifter and significantly improves the accuracy and stability of the phase shifter.

Description

Technical Field

[0001] The present invention relates to the technical field of phase shifter calibration, and in particular to a phase shifter calibration method based on joint search in spatial and frequency domains. Background Art

[0002] Phase-shifting interferometry (PSI) can accurately detect parameters such as the surface topography, thickness, and refractive index of large-aperture optical elements by measuring the interference fringe changes caused by the optical path change between the reference light and the measurement light, and plays an important role in the manufacturing of advanced optical devices such as astronomical telescopes and high-energy laser systems. PSI usually uses a phase shifter to move the reference mirror to control the optical path difference between the reference light and the measurement light. As the driving component of the phase shifter, the positioning accuracy of the piezoelectric ceramic (PZT) directly affects the accuracy and reliability of the measurement. Especially for the measurement of large-aperture optical elements, PSI requires multiple PZTs to cooperate to control the pose of the reference mirror. Once there is a deviation in the driving of a certain PZT, a large yaw will be introduced during the movement of the reference mirror, resulting in a significant reduction in the measurement accuracy of PSI. Therefore, the high-precision calibration of multi-PZT phase shifters has always been a concern in the field of PSI and is of great significance for the detection of parameters of high-precision large-aperture optical elements.

[0003] The purpose of calibrating a multi-channel piezoelectric phase shifter is to establish an accurate relationship between the applied voltage and the displacement of the reference mirror pushed by the phase shifter, so as to accurately control the voltage to drive the PZT and obtain an ideal interference pattern. The ideas for calibrating the entire phase shifter include single-channel discrete calibration and multi-channel joint calibration. In single-channel discrete calibration, only one PZT is driven at a time, and the other PZTs remain stationary, obtaining the relationship between the displacement of the reference mirror pushed by this PZT and the voltage value. Repeat this process multiple times until the characteristic curves of all PZTs are calibrated. This calibration strategy is generally analyzed based on two-dimensional fast Fourier transform (2-D FFT) or damped least squares (DLS). However, in practical applications, in order to effectively decouple torque and eccentric force, PZTs mostly have a ball-head structure. This ball-head structure causes the contact points of one PZT with the phase shifter to be different in the two states of working alone and working in combination with other PZTs, resulting in inaccurate calibration of the PZT characteristic curve, unable to obtain the accurate relationship between the displacement of the reference mirror and the voltage value, and it is difficult to adapt to the scenario of multi-channel PZT driving.

[0004] In the case where all PZTs are involved in the work, the multi-channel joint calibration can calibrate the relationship curve between the displacement of the reference mirror and the voltages applied to all channels of PZTs at one time. In the literature "Nonlinear calibrating for phase-shifting adapter with three PZTs" (Yu, Zhu, Jingbang, et al. Microwave and Optical Technology Letters, 1999), based on the position distribution of PZTs on the phase shifter end face, the functional expression of the phase shifter plane is fitted, and then the relationship between the offset of the reference mirror plane and the voltage value is obtained by using 2-D FFT. This method does not consider that the displacement of the PZT is not necessarily equal to the displacement of the reference mirror, and has a high requirement for the fitting accuracy of the reference mirror plane. In the literature "Chou HC. Self-calibration of a phase-shifting adapter for Fizeau interferometers" (Ouyang Y, Ou-Yang M, Chou H C. Optical Review, 2009, 16(4):495-499), it is proposed to find the projection position of each PZT on the interference surface by using the slope change of the interference fringe pattern, and find the corresponding voltage of the reference mirror displacement according to the relationship between the fringe angle and the PZT projection position. This method is complex to operate and requires keeping the interference fringes parallel to the connection line of the projections of a certain two PZTs all the time. In the literature "Design and calibration of a piezoelectric actuator for interferometric applications" (Bruno L, Poggialini A, Felice G. Optics & Lasers in Engineering, 2007, 45(12):1148-1156), the interference pattern is segmented into sub-images, two-period phase shifting is performed, and the frequency sine fitting of the sub-images is carried out to determine the definite relationship between the reference mirror displacement and the voltage. This method fits according to the frequency of a single pixel point, is extremely sensitive to interference such as noise, and has poor robustness. Summary of the Invention

[0005] The purpose of the present invention is to overcome the defects of the above-mentioned existing technologies and provide a phase shifter calibration method based on joint search in the spatial-frequency domain to improve the phase shifting accuracy and stability of a multi-channel phase shifter.

[0006] The purpose of the present invention can be achieved by the following technical solutions:

[0007] A phase shifter calibration method based on joint search in the spatial-frequency domain includes the following steps:

[0008] Model the phase shifter to be calibrated as a plane supported and driven by multiple PZTs, determine the relationship between the phase shift values corresponding to each PZT and the voltage, and construct the coefficients corresponding to the variation relationships of all PZTs into a matrix model;

[0009] Determine the interference pattern to be used as a reference and the position of its first-order spectrum. All subsequent spectrum transformations are CZT transformations centered on the position of the first-order spectrum;

[0010] Calibrate the tilt coefficient in the matrix model using a frequency-domain search method; calibrate the position coefficient in the matrix model using a spatial-domain search method; obtain the calibrated matrix model as the calibration result of the phase shifter.

[0011] Furthermore, the expression for the relationship between the phase shift value corresponding to the PZT and the voltage is:

[0012] δ i =a i V 2 +b i V

[0013] where δ i is the phase shift value corresponding to the i-th PZT, a i is the quadratic term coefficient, b i is the linear term coefficient, and V is the voltage;

[0014] Construct the matrix model from the quadratic term coefficients and linear term coefficients in the relationship between the phase shift values corresponding to each PZT and the voltage. This matrix model includes the tilt coefficient C ∈ R 1×2n and the position coefficient β, where 2n is the number of quadratic term coefficients and linear term coefficients, and β is an unknown constant.

[0015] Furthermore, the frequency-domain search method is the simulated annealing algorithm. The processing process of this simulated annealing algorithm includes the following steps:

[0016] S101: Set the position coefficient in the matrix model to 1, initialize the tilt coefficient, and set the phase shift value;

[0017] S102: Calculate the voltage corresponding to each PZT according to the relationship between the phase shift value corresponding to the PZT and the voltage, and thus input it into the control system of the phase shifter to drive the reference mirror in the phase shifter to move, obtain the phase-shifted interference pattern, calculate the corresponding spectrum based on the phase-shifted interference pattern and the reference interference pattern, and thus calculate the frequency-domain search loss function to guide the update of the tilt coefficient;

[0018] S103: Repeat step S102 until the preset frequency-domain search cut-off condition is reached, and output the search result of the tilt coefficient.

[0019] Further, the frequency domain search termination condition includes that the calculated frequency domain search loss function is less than a preset frequency domain search tolerance or reaches a preset maximum number of iterations for frequency domain search.

[0020] Further, by performing a Chirp-Z transform on the phase-shifted interference pattern and the reference interference pattern, the corresponding spectrum is calculated.

[0021] Further, the spatial domain search method includes the following steps:

[0022] S201: Set the phase shift value to 2π, obtain the tilt coefficient obtained by the frequency domain search method, initialize the position coefficient, and set the solution space of the position coefficient.

[0023] S202: Calculate the voltages corresponding to each PZT according to the relationship between the phase shift value corresponding to the PZT and the voltage, and thus input them into the control system of the phase shifter to drive the reference mirror to move, obtaining the phase-shifted interference pattern; compare the similarity between the interference patterns obtained after two adjacent phase shifts as the spatial domain search loss function.

[0024] S203: Repeat step S202, traverse and calculate the spatial domain search loss corresponding to each value in the solution space of the position coefficient, and select the value in the solution space where the spatial domain search loss is less than a preset spatial domain search tolerance and is the smallest as the spatial domain search result.

[0025] Further, the method further includes performing error analysis on the frequency domain search method and the spatial domain search method respectively, so as to obtain the overall error to determine whether the calibration result of the phase shifter meets the error condition.

[0026] Further, the error analysis process of the frequency domain search method includes:

[0027] Perform a 2D-FFT on the interference pattern obtained by using the calibrated matrix model to obtain a spectrum of size N×N, and obtain the corresponding normalized spatial angular frequency resolution Δω.

[0028] Take the position of the first-order spectrum in the interference pattern obtained by using the calibrated matrix model as the center and perform a Chirp-Z transform on a region of size T×T to obtain a spectrum diagram of size M×M, and calculate the corresponding normalized spatial angular frequency resolution Δω'.

[0029] Let Δf be the spatial frequency error and f_max be the maximum range of the spatial frequency. Since the ratio of the spatial frequency error to the maximum range of the spatial frequency is equal to the ratio of the corresponding normalized angular frequency, and f_max is inversely proportional to the pixel size Δd, we can obtain:

[0030]

[0031] In the formula, ε 1 is the allowable tolerance for frequency-domain search;

[0032] Assuming that the error is all manifested in the x direction, we get:

[0033] δ(x,y) = 2πΔfx + δ 0

[0034] In the formula, δ 0 is the ideal phase shift value;

[0035] Under the allowable tolerance ε 1 for frequency-domain search, calculate the corresponding maximum phase error as Let The calculation expression for the magnitude of the error 1 / p is obtained as:

[0036]

[0037] Furthermore, the error analysis process of the spatial-domain search method includes:

[0038] Rewrite the phase shift value δ of the phase shifter as a constant:

[0039] δ = δ 0 + α

[0040] In the formula, δ 0 is the ideal phase shift value 2π, and α is the maximum phase shift error corresponding to the allowable tolerance ε 2 for spatial-domain search;

[0041] Rewrite the mean square error calculation formula for the spatial-domain search method as:

[0042]

[0043] In the formula, I 1 , I 0 are two interference patterns taken before and after using the calibrated matrix model;

[0044] Let MSE[I 1 (x,y), I 0 (x,y)] < ε 2 , then

[0045]

[0046] Furthermore, the calculation expression for the overall error is:

[0047]

[0048] In the formula, E is the overall error.

[0049] Compared with the prior art, the present invention has the following advantages:

[0050] (1) Considering the coupling characteristics between PZTs of each channel, the present invention introduces a matrix model C ∈ R 1×2n , to effectively characterize the characteristic curve of the phase shifter, providing a basis for subsequent parameter solving; and designs a joint spatio-frequency domain search and calibration method. This method utilizes the frequency domain characteristics and realizes the high-precision global optimal search for the tilt coefficient of the phase shifter through a high-dimensional extended simulated annealing algorithm and a spatial carrier CZT technology; utilizes the spatial domain characteristics and combines with a similarity evaluation method to solve the position coefficient, overcoming the problem of relying on the prior of the PZT position distribution and avoiding the positioning error.

[0051] (2) The simulation and experimental results show that the method of the present invention effectively reduces the phase shift error of the phase shifter. After calibration, the linear error and tilt error of the phase shifter are reduced to 3.53% and 2.00% respectively, and the system error is reduced by 10 times. The four-step phase shift phase calculation results show that the method of this paper significantly improves the accuracy and stability of the phase shifter.

[0052] (3) The error analysis of the frequency domain search method and the spatial domain search method converts the abstract frequency domain tolerance into the maximum tilt phase error of the phase shifter and converts the spatial domain tolerance into the phase shift position error, making the error index more intuitive, helping to identify and analyze the error sources, facilitating actual operations in the system debugging and calibration process, and providing important support for the optimal design and stable operation of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 is a schematic flow chart of a phase shifter calibration method based on joint spatio-frequency domain search provided in an embodiment of the present invention;

[0054] Figure 2 is a schematic data flow chart of a phase shifter calibration method based on joint spatio-frequency domain search provided in an embodiment of the present invention;

[0055] Figure 3 is a schematic flow chart of a frequency domain search using a simulated annealing algorithm provided in an embodiment of the present invention;

[0056] Figure 4 is a schematic flow chart of a spatial domain search method provided in an embodiment of the present invention;

[0057] Figure 5 is a schematic diagram of the phase shift error of ten cycles provided in an embodiment of the present invention;

[0058] Figure 6 is a schematic diagram of the relationship curve between the S-phase shift times and the twenty-step phase shift provided in an embodiment of the present invention;

[0059] Figure 7 Schematic diagram of a Fizeau interference optical path set up in an embodiment of the present invention;

[0060] Figure 8 Schematic diagram of the four-step phase-shifting calculation result before calibration in an embodiment of the present invention;

[0061] Figure 9 Schematic diagram of the four-step phase-shifting calculation result after calibration in an embodiment of the present invention. Detailed implementation manners

[0062] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. The components of the embodiments of the present invention usually described and illustrated in the accompanying drawings here can be arranged and designed in various different configurations.

[0063] Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed present invention, but merely represents selected embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts fall within the scope of protection of the present invention.

[0064] It should be noted that similar reference numerals and letters denote similar items in the following drawings. Therefore, once an item is defined in one drawing, it does not need to be further defined and explained in subsequent drawings.

[0065] Embodiment 1

[0066] As shown in Figure 1 and Figure 2 , this embodiment provides a phase shifter calibration method based on joint spatial-frequency domain search, including the following steps:

[0067] S1: Model the phase shifter to be calibrated as a plane jointly supported and driven by multiple PZTs, determine the relationship between the phase shift values corresponding to each PZT and the voltage, and construct the coefficients corresponding to the variation relationships of all PZTs into a matrix model βC, where C ∈ R 1×2n , representing the non-linear parameter of the phase shifter, i.e., the tilt coefficient; β is an unknown constant, representing the linear parameter of the phase shifter, i.e., the position coefficient;

[0068] S2: Do not apply voltage to all PZT channels, i.e., V 1 =... = V n = 0, record the interference pattern I 0 (x, y) as a reference; for I 0Perform 2D-FFT on (x, y) to determine the position of a primary spectrum. All subsequent spectrum transformations are CZT transformations of the region centered on ;

[0069] S3: Use the frequency-domain search algorithm to search for the coefficient matrix C, and preliminarily calibrate the nonlinear curve of the phase shifter. The search result is denoted as

[0070] S4: Use the spatial-domain search algorithm to search for the unknown constant β, and calibrate the linear error between the curve obtained in step S3 and the actual characteristic curve of the phase shifter. The search result is denoted as

[0071] S5: Output the search result Obtain the calibrated matrix model as the calibration result of the phase shifter.

[0072] The process of constructing the matrix model βC in step S1 is specifically as follows:

[0073] Limited by the characteristics of PZT, when the externally applied driving voltage increases at equal intervals, its deformation will increase nonlinearly. Therefore, the displacement of the entire phase shifter pushing the reference mirror is also nonlinear. The calibration of the phase shifter is to solve the nonlinear driving voltages corresponding to all channels of PZT by measuring the voltage-displacement change relationship data, so that the phase shifter can be displaced at equal intervals.

[0074] First, model the phase shifter as a plane supported and driven by multiple PZTs, and represent the phase shift value by a quadratic equation. Since the concept of phase shift is relative, for simplicity of calculation, the initial phase shift value is defined as zero. Assume that the phase shifter is driven by n-channel PZTs, then the phase shift value δ i corresponding to the i-th PZT changes with the voltage V as follows:

[0075] δ i = a i V 2 + b i V (1)

[0076] In the formula, δ i is the phase shift value corresponding to the i-th PZT, a i is the quadratic term coefficient, b i is the linear term coefficient, and V is the voltage;

[0077] The ultimate goal of this solution is to find all the coefficients a and b, with a total of 2n, so as to drive the phase shifter to generate an ideal phase shift. Since the movement of each PZT will change the contact positions of other PZTs with the phase shifter and affect the attitude of the phase shifter, that is, the "coupled movement" of each part of the phase shifter, it is necessary to jointly solve the coefficients. Therefore, in this embodiment, the coefficients to be solved are constructed into a matrix model βC, where C ∈ R1×2n , is the tilt coefficient; β is an unknown constant and is the position coefficient.

[0078] The frequency-domain search technique in step S3 is specifically described as follows:

[0079] The frequency-domain search technique of this embodiment finds the optimal solution through a high-dimensional extended simulated annealing algorithm, and calculates the loss function by the spatial carrier CZT technique as the basis for updating the current solution.

[0080] Since there may be deviations in the installation position of the optical elements in the optical path each time, and the different ambient temperature and humidity of the phase shifter will change the PZT deformation, resulting in the calibration target not being a fixed value. Then the solution of the parameter model is actually to search for the optimal solution or a solution close to the optimal solution under the given constraints. Therefore, this embodiment conducts a global optimal parameter search based on the high-dimensional extension of the simulated annealing algorithm.

[0081] The algorithm flow is as Figure 3 shown, and the specific process is as follows:

[0082] S101: Set the position coefficient in the matrix model to 1, initialize the tilt coefficient C 0 , and set the phase shift value to δ;

[0083] S102: Calculate the voltage V corresponding to each PZT according to the relationship between the phase shift value corresponding to the PZT and the voltage i (i = 1, 2,..., n), and thus input it into the control system of the phase shifter to drive the reference mirror in the phase shifter to move. Obtain the interferogram I 1 (x, y) after phase shift. Based on the interferogram I 1 after phase shift and the reference interferogram I 0 , calculate the corresponding spectrum, and thus calculate the frequency-domain search loss function to guide the update of the tilt coefficient;

[0084] When updating the solution, the random value is a matrix composed of 2n random numbers, and there is no limit on n, so this method can be applied to different numbers of PZT channels. At the same time, the perturbation is related to the temperature of the current outer loop. As the loop temperature decreases, the new solution will gradually converge to the global optimal solution.

[0085] The calculation principle of the loss function is as follows:

[0086]

[0087] Among them, k x0 , k y0 is the spatial carrier frequency.

[0088] If the reference mirror tilts during phase shift, then δ is a variable with respect to the spatial coordinates (x, y):

[0089] δ(x,y) = k x x + k y y + α (3)

[0090] k x ,k y , where α is a variable with respect to voltage V.

[0091] Performing two-dimensional FFT on Equation (3) gives:

[0092]

[0093] where ~ represents the result of the spectral transform.

[0094] It can be seen from this that the change in the interference Figure 1 level spectral position corresponds to the tilt degree of the reference mirror movement. Detecting this value can be used to determine whether there is a phase-shifting tilt. The designed loss function is:

[0095]

[0096] where I is the number of interferograms sampled in one iteration.

[0097] This places extremely high requirements on the resolution of the spectrogram. Therefore, this paper introduces the Chirp-Z transform (CZT). While obtaining a refined spectrum in the first-order spectrum domain, it does not need to consider other parts of the frequency band, greatly reducing the amount of data processing and thus significantly reducing the processing time.

[0098] S103: Repeat step S102 until the frequency-domain search loss function loss 1 is less than the preset frequency-domain search tolerance ε 1 or reaches the preset maximum number of iterations for the frequency-domain search, and output the search result of the tilt coefficient.

[0099] The above description further refines the frequency-domain search process. Utilizing the frequency-domain characteristics, through high-dimensional expansion of the simulated annealing algorithm and in cooperation with the designed spatial carrier CZT technology, a global optimal search for the phase shifter tilt coefficient is achieved. During the search, the 2n coefficients to be solved are randomly updated. A series of fringe patterns are obtained using the spatial carrier, the spectrum is refined by performing the chirp-z transform, the change in the position of the first-order spectrum is calculated with high precision, and the tilt during the movement of the phase shifter is quantitatively detected. New solutions are accepted based on the metropolis criterion to ensure the global optimality of the search solution. The frequency-domain search obtains the preliminary calibration coefficients of the phase shifter. After calibration, the phase-shifting tilt error of the phase shifter is basically eliminated.

[0100] The specific description of the spatial-domain search technology in step S4 is as follows:

[0101] Based on the coefficient matrix obtained from the frequency-domain search algorithm For a computational voltage-driven phase shifter, it can be considered that the movement of the reference mirror hardly tilts. However, there is still a linear error between the actual displacement of the reference mirror and the preset value at this time, which is β. The β value is found through the spatial domain search algorithm. The process of the spatial domain search algorithm is as Figure 4 shown and includes the following steps:

[0102] S201: Set the phase shift value to 2π, obtain the tilt coefficient obtained by the frequency domain search method, initialize the position coefficient, and set the solution space of the position coefficient;

[0103] S202: Then, according to the theory, the interference fringe pattern after phase shift is the same as that before phase shift in Equation (2). By comparing the similarity of the interference fringe patterns, the phase shift accuracy can be calibrated.

[0104] Calculate the voltages corresponding to each PZT according to the relationship between the phase shift value corresponding to the PZT and the voltage, and then input them into the control system of the phase shifter to drive the reference mirror to move and obtain the interference pattern after phase shift; compare the similarity between the interference patterns obtained after two adjacent phase shifts as the spatial domain search loss function;

[0105] Since the interference fringe pattern contains relatively simple elements, the mean squared error (MSE) between the patterns can be calculated to know their similarity degree. Utilize this property to design the spatial domain search loss function loss 2 :

[0106]

[0107] S203: Repeat step S202, traverse and calculate the spatial domain search loss corresponding to each value in the solution space of the position coefficient, and select the value in the solution space where the spatial domain search loss loss 2 is less than the preset spatial domain search tolerance ε 2 and the spatial domain search loss loss 2 is the smallest as the spatial domain search result.

[0108] The above description details the spatial domain search process. Utilize the spatial domain characteristics, perform periodic phase shift to obtain a series of similar interference fringe patterns, and quantify the phase shift linear error by calculating the root mean square error (MSE) between the fringe patterns. Traverse the solution space to search for the optimal linear coefficient. After the spatial domain search, the linear error of the phase shifter is basically eliminated.

[0109] As a preferred implementation, the above method further includes S6: respectively perform error analysis on the frequency domain search method and the spatial domain search method to obtain the overall error, so as to determine whether the calibration result of the phase shifter meets the error conditions.

[0110] That is, for the design method itself, the influence of the tolerance is analyzed. The spectral displacement of the interference pattern is the tolerance. Substitute the tolerance into the analysis and calculate the maximum error. Assume that the size of the interference pattern sampled using the calibrated matrix model is N×N.

[0111] Specifically,

[0112] 1.1. Frequency domain search error

[0113] For the error analysis of the frequency domain search method, it includes the following steps:

[0114] As can be seen from the previous text, the spectral displacement of the corrected interference pattern corresponds to the setting of the tolerance ε 1 Substitute the value of ε 1 into the analysis, that is, consider the situation when the error is the largest.

[0115] Perform a 2D-FFT on the interference pattern to obtain a spectrum of size N×N, and obtain the corresponding normalized spatial angular frequency resolution Δω.

[0116] Take the position of the first-order spectrum in the interference pattern as the center and perform a Chirp-Z transform on a region of size T×T to obtain a spectrum of size M×M, and calculate the corresponding normalized spatial angular frequency resolution Δω′:

[0117]

[0118] Let Δf be the spatial frequency error and f_max be the maximum range of the spatial frequency. The ratio of the spatial frequency error to the maximum range of the spatial frequency is equal to the ratio of the corresponding normalized angular frequency:

[0119]

[0120] Also, since f_max is inversely proportional to the pixel size Δd, it can be obtained that:

[0121]

[0122] where Δf is the spatial frequency error and f_max is the maximum range of the spatial frequency.

[0123] In order to explore the maximum phase fluctuation introduced by the tolerance ε 1 subtract the two interference patterns taken before and after:

[0124]

[0125] Since the direction of the error in the frequency space does not affect the analysis, only the magnitude of the error needs to be concerned, so assume that the error is all manifested in the x direction to simplify the analysis:

[0126] δ(x,y) = 2πΔfx + δ0 (11)

[0127] where δ 0 is the ideal phase shift value.

[0128] Let the tolerance be ε 1 The corresponding maximum phase error is

[0129] Let The magnitude of the error 1 / p can be obtained as:

[0130]

[0131] 1.2 Spatial search error

[0132] The error analysis of the spatial search method includes the following steps:

[0133] Since the frequency domain search calibrates the tilt error when the reference mirror moves, δ(x,y,V) does not depend on the spatial coordinates (x,y) and is a constant related only to V, which can be written as δ(V).

[0134] δ(V) = δ 0 + α (13)

[0135] where δ 0 is the ideal phase shift value of 2π, and α is the maximum phase shift error corresponding to the tolerance ε 2 For the sake of simplicity of expression, (x,y) is omitted. Since α is very small,

[0136]

[0137] When the interference pattern contains enough fringes and speckle noise, then the above equation can be rewritten as:

[0138]

[0139] If the intensity distribution range of the interference pattern is 0 - 255, take B(x,y) = 128. Let MSE[I 1 (x,y),I(x,y)] < ε 0 then 2

[0140]

[0141] Let The following can be obtained:

[0142]

[0143] ​1.3. Overall error

[0144] For the overall error, since the frequency-domain search algorithm and the spatial-domain search algorithm belong to a cascaded structure in the overall algorithm block diagram, the algorithm system error is:

[0145]

[0146] Substitute T = 21, M = 4096, ε 1 = 1, ε 2 = 0.1, and the maximum theoretical error is obtained as 0.99%.

[0147] The above scheme is experimentally verified below.

[0148] 2.1. Simulation experiment

[0149] 2.1.1. Simulation results and analysis of ten-cycle phase shift

[0150] The characteristic curve of a general phase shifter can be expressed by a quadratic function. However, in the simulation, considering the extremely poor linearity of the phase shifter characteristic curve, for the case of a cubic function, the true value C ∈ R 1×9 . Calibrated by the method designed in the present invention, a quadratic function is used for fitting during calibration, and it is assumed that C ∈ R 1×6 . Using the calibration results, ten phase shifts are performed, each moving one cycle. Calculate the error between the plane obtained by each phase shift and the true value plane. It can be seen from the simulation results that the phase shift error for ten cycles is less than 0.5%, as Figure 5 shown, which is consistent with the error calculation results, proving the effectiveness of the calibration technology theory designed in the present invention.

[0151] 2.1.2. Simulation results and analysis of twenty-step phase shift

[0152] In the Fizeau interference optical path, the phase shift amount is twice the displacement amount of the phase shifter, so the ideal phase shift frequency for twenty-step phase shift is equal to 0.1 Hz. Using the calibration results, twenty-step phase shift is performed, and the curve of the relationship between S and the number of phase shifts is as Figure 6 shown. It can be calculated that both the position coefficient error and the tilt coefficient error approach zero, which is consistent with the theoretical error.

[0153] 2.2. Actual measurement experiment

[0154] 2.2.1. Experimental platform and configuration

[0155] Since even a small air disturbance will cause the interference fringes to shake, and the PZT is very sensitive to temperature changes, the experiment is carried out in a closed environment, controlling the constant temperature at 25 °C and the constant humidity at 40%. The entire optical path is built on an air-bearing platform to isolate vibrations.

[0156] The Fizeau interference optical path built in the experiment is as Figure 7 shown.

[0157] The laser used in the experiment is the LASOS RTK 40200TS with a wavelength of 639.77 nm. Both the reference mirror and the measured mirror are standard plano-convex lenses with a reflectivity of 5% and a diameter of 10 cm. The camera is a Basler industrial camera, model acA1920-155um, with a pixel size of 5.86×5.86 μm. The phase shifter is the CoreTomorrow P77 series open-loop piezoelectric nano-positioning platform, and the controller is the supporting E20 series modular controller.

[0158] 2.2.2. Results and Analysis of the Parameter Model Repeatability Experiment

[0159] The characteristic curve of the phase shifter was calibrated ten times repeatedly to calculate the repeatability of the search results.

[0160] Table 1 Experimental Results of the Repeatability Error of Each Parameter

[0161]

[0162] As can be seen from Table 1, the repeatability of the ten experiments is good. However, relatively speaking, the repeatability of the linear term coefficient is better than that of the quadratic term coefficient. The reason for this analysis is that the linearity of the calibrated curve interval selected is good, and the proportion of the linear term coefficient is large, so the range of acceptable solutions for the quadratic term is larger.

[0163] 2.2.3. Results and Analysis of the Twenty-Step Phase Shifting Experiment

[0164] For the twenty-step phase shifting, the ideal phase shift angular frequency w = 2πf = 2π / 20×2 = π / 10, and the ideal phase shift frequency f = 0.1. Using a calibrated coefficient matrix, twenty-step phase shifting was performed, and the interference fringe patterns for three cycles were recorded. Ten groups were repeated, and the data was processed and analyzed.

[0165] (1) Verification of the Calibration Effect of the Linear Coefficient

[0166] Using the spatial carrier FFT technique, the frequency of the entire interference pattern was fitted. The difference between the fitted frequency of the entire interference pattern and the ideal phase shift frequency characterizes the linear error index of the phase shifter.

[0167] The spatial carrier FFT technique subtracts the phase-shifted interference pattern I 1 (x,y) from the reference interference pattern I 0 (x,y), squares the difference, and takes the mean value, denoted as S. When the phase shift is accurate, δ only varies with the voltage V and is independent of the spatial coordinates (x,y). The S-V curve is an ideal sine curve; otherwise, the frequency of the S-V curve will change disorderly.

[0168] Table 2 Experimental Results of the Spatial Carrier FFT of the Entire Interference Pattern

[0169]

[0170] As can be seen from Table 2, the measured linear error of the phase shifter is between -2.34% and 3.52%. The analysis shows that the calibration accuracy is mainly limited by the inherent error of 15% - 20% of the open-loop phase shifter. Before calibration, the measured phase shift frequency was 0.0781 and the error was -21.88%. This indicates that the systematic error of the open-loop phase shifter has been reduced by a factor of 10, and the calibration is effective.

[0171] (2) Verification of the calibration effect of the tilt coefficient

[0172] The interference pattern is divided into 16 sub-images of 4*4, and the frequency of each sub-image is fitted using the spatial carrier FFT technique. The maximum error between the fitted frequencies of each sub-image characterizes the tilt angle index of the phase shifter.

[0173] Table 3 Experimental results of spatial carrier FFT for sub-interference patterns

[0174]

[0175]

[0176] As can be seen from the data in Table 3, the measured tilt error of the phase shifter is about 2.00%. When using a phase shifter not calibrated by the algorithm of the present invention for phase shifting, the maximum phase shift frequency is 0.0820, the minimum phase shift frequency is 0.0684, and the error is 13.60%. Therefore, the calibration algorithm of the present invention also effectively reduces the tilt error of the open-loop phase shifter, and the calibration is effective.

[0177] (3) Experimental results and analysis of random errors

[0178] Some characteristic points in the interference pattern are selected, and the frequencies of these points are fitted using the time carrier FFT technique. The difference from the ideal frequency represents the random error at the characteristic points.

[0179] For the time carrier FFT technique, a pixel point is selected for observation, and the driving voltage is equally divided and increased. The data of the light intensity change at this point is recorded. When the phase shifter drives the reference mirror to move, multiple interference fringes will sweep across this point in sequence. Therefore, the relationship between the light intensity at this point and the applied voltage is a cosine curve. To ensure good repeatability of the displacement measurement result and avoid the influence of noise, generally at least 3 characteristic points are selected.

[0180] The size of the experimental image is 500×500, and three characteristic points with pixel coordinates (125, 125), (250, 250), and (375, 375) are selected for analysis. As can be seen from the results in Table 4, the perturbation is relatively severe. Since the experimental environment was strictly controlled during the experiment of the present invention, after analysis, the largest source of this perturbation should be the noise of the amplifier in the phase shifter controller.

[0181] Table 4 Experimental results of time carrier FFT for characteristic points

[0182]

[0183]

[0184] 2.2.4, Results and Analysis of Four-Step Phase Shifting Experiment

[0185] The interferograms before and after phase-shifting calibration are solved for the phase using the standard four-step phase-shifting algorithm. If the phase shifting is accurate, the result of the standard four-step phase-shifting solution will be a smooth and continuous plane (both the reference mirror and the measured mirror used in the experiment are standard planes); otherwise, the solved phase will contain carrier fringes.

[0186] As Figure 8 and Figure 9 shown, the phase diagram calculated by the four-step phase shifting before calibration contains obvious carrier fringes, while the calculation result after calibration is a smooth plane, indicating that the calibration result is accurate.

[0187] The preferred specific embodiments of the present invention have been described in detail above. It should be understood that those of ordinary skill in the art can make many modifications and variations based on the concept of the present invention without creative work. Therefore, all technical solutions that can be obtained by those skilled in the art in this technical field based on the concept of the present invention through logical analysis, reasoning, or limited experiments on the basis of the prior art should fall within the protection scope determined by the claims.

Claims

1. A phase shifter calibration method based on space-frequency domain joint search, characterized in that: The following steps are involved: The phase shifter to be calibrated is modeled as a plane supported and driven by multiple PZTs, the relationship between the phase shift value corresponding to each PZT and the voltage is determined, and the coefficients corresponding to the change relationship of all PZTs are constructed as a matrix model; Determine the interference pattern and its first-order spectrum position as a reference, and all subsequent spectrum transformations are CZT transformations centered on the first-order spectrum position; The tilt coefficient in the matrix model is calibrated by using a frequency domain search method; the position coefficient in the matrix model is calibrated by using a spatial domain search method; and the calibrated matrix model is obtained as a calibration result of the phase shifter; The frequency domain search method is a simulated annealing algorithm, and the processing process of the simulated annealing algorithm includes the following steps: S101: setting the position coefficient in the matrix model to 1, initializing the tilt coefficient, and setting the phase shift value; S102: Calculate the voltage corresponding to each PZT according to the relationship between the phase shift value corresponding to the PZT and the voltage, and input it into the control system of the phase shifter to drive the reference mirror in the phase shifter to move, and obtain the interference pattern after the phase shift. Based on the interference pattern after the phase shift and the reference interference pattern, calculate the corresponding spectrum, and thus calculate the frequency domain search loss function to guide the update of the tilt coefficient; S103: repeating step S102 until a preset frequency domain search cutoff condition is reached, and outputting a search result of the tilt coefficient; The airspace search method comprises the following steps: S201: setting the phase shift value to 2π, obtaining the tilt coefficient obtained by the frequency domain search method, initializing the position coefficient, and setting the solution space of the position coefficient; S202: Calculate the voltage corresponding to each PZT according to the relationship between the phase shift value corresponding to the PZT and the voltage, and input it into the control system of the phase shifter to drive the reference mirror to move and obtain the interference pattern after the phase shift; compare the similarity between the interference patterns obtained after two adjacent phase shifts as the spatial domain search loss function; S203: Repeat step S202, traverse the spatial search loss corresponding to each value in the solution space of the calculated position coefficient, and select the minimum value in the solution space whose spatial search loss is less than the preset spatial search tolerance as the spatial search result.

2. The phase shifter calibration method based on space-frequency domain joint search according to claim 1, characterized in that: The expression of the relationship between the phase shift value corresponding to the PZT and the voltage is: δ i =a i V 2 +b i V In the formula, δ i is the phase shift value corresponding to the i-th PZT, a i is the coefficient of the quadratic term, b i is the coefficient of the first-order term, V is the voltage; The quadratic term coefficient and the linear term coefficient in the relationship between the phase shift value corresponding to each PZT and the voltage change are used to form the matrix model, which includes the tilt coefficient C∈R 1×2n and the position coefficient β, where 2n is the number of quadratic and linear coefficients, and β is an unknown constant.

3. The phase shifter calibration method based on space-frequency domain joint search according to claim 1, characterized in that: The frequency domain search cutoff condition includes that the calculated frequency domain search loss function is less than a preset frequency domain search tolerance or reaches a preset maximum number of frequency domain search iterations.

4. The phase shifter calibration method based on space-frequency domain joint search according to claim 1, characterized in that: The corresponding spectrum is calculated by performing Chirp-Z transform on the phase-shifted interference pattern and the reference interference pattern.

5. The phase shifter calibration method based on space-frequency domain joint search according to claim 1, characterized in that: The method further includes performing error analysis on the frequency domain search method and the space domain search method respectively, thereby obtaining an overall error to determine whether the calibration result of the phase shifter meets the error condition.

6. The phase shifter calibration method based on space-frequency domain joint search according to claim 5, characterized in that: The error analysis process of the frequency domain search method includes: Perform 2D-FFT on the interference pattern obtained by the calibrated matrix model to obtain a spectrum of size N×N, and obtain the corresponding normalized spatial angular frequency resolution Δω, Take the first-order spectrum position in the interference pattern obtained by the calibrated matrix model Perform Chirp-Z transform on the center and T×T area to obtain a spectrum of M×M size, and calculate the corresponding normalized spatial angular frequency resolution Δω ′ , Let Δf be the spatial frequency error, f_max be the maximum range of the spatial frequency, the ratio of the spatial frequency error to the maximum range of the spatial frequency is equal to the ratio of the corresponding normalized angular frequency, and f_max is inversely proportional to the pixel size Δd, we can get: Where, ε1 is the frequency domain search tolerance; Taking the error in the x direction, we get: δ(x,y)=2πΔfx+δ0 In the formula, δ0 is the ideal phase shift value; Under the frequency domain search tolerance ε1, the corresponding maximum phase error is calculated as make The calculation expression for the error 1 / p is:

7. The phase shifter calibration method based on space-frequency domain joint search according to claim 6, characterized in that: The error analysis process of the spatial search method includes: Rewrite the phase shift value δ of the phase shifter as a constant: δ=δ0+α Where δ0 is the ideal phase shift value 2π, and α is the maximum phase shift error corresponding to the spatial search tolerance ε2; The mean square error calculation formula of the spatial search method is rewritten as: Where I1 and I0 are two interference images taken before and after using the calibrated matrix model; Let MSE[I1(x,y),I0(x,y)]<ε2, but 8. The phase shifter calibration method based on space-frequency domain joint search according to claim 7, characterized in that: The calculation expression of the overall error is: Where E is the overall error.

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