Discontinuous wavelength scanning interferometric phase recovery method

Abstract

The invention discloses a discontinuous wavelength scanning interferometric phase recovery method, and in depth resolution wavelength scanning, sampling of a discontinuous spectrum can be regarded as an observation process of a missing sample. On the basis, a discrete optimization model is established, and a missing sample is estimated by taking minimization of amplitude spectrum energy as a target. According to the method, interference signals lost in a spectral gap interval are predicted under the condition that a special hardware system is not needed to monitor a wave number sequence, so that phase jitter caused by discontinuous spectrums is eliminated, and accurate reconstruction of a depth resolution phase diagram is realized.

Description

Technical Field

[0001] The present invention relates to the technical field of interferometric phase retrieval, and particularly to a discontinuous wavelength scanning interferometric phase retrieval method. Background Art

[0002] In recent years, the quantification and evaluation of optical performance have gradually become a research hotspot, especially in the field of multi-surface measurement of optical lenses. This demand has given rise to a variety of measurement techniques, among which depth-resolved wavelength scanning interferometry has received extensive attention due to its unique advantages. Depth-resolved wavelength scanning interferometry uses interferometric phase detection technology to achieve multi-surface topology measurement. Its high sensitivity and full-field measurement characteristics make it a powerful tool for surface characterization. Currently, depth-resolved wavelength scanning interferometry has been successfully applied to the surface and thickness measurement of optical components, providing important support for optical performance evaluation.

[0003] The depth-resolved wavelength scanning interferometry technology accurately reconstructs the depth-resolved phase map by performing a Fourier transform on the interference pattern along the wavenumber axis. In this process, it is required that the laser emits a spectrum with continuously increasing or decreasing wavelengths. However, in practical applications, lasers often cannot guarantee the continuity of spectral output due to reasons such as mode hopping. This non-continuity of spectral output will cause phase jitter and generate abnormal side lobes in the Fourier transform analysis, thereby affecting the depth resolution and measurement accuracy. Specifically, the abnormal side lobes may undergo spectral aliasing with adjacent main lobes, making it difficult to distinguish the depth positions of adjacent layers and reducing the depth resolution ability of the system. Even if the abnormal side lobes do not cause aliasing, their large amplitudes will still interfere with the identification of the main lobe peak position, thereby reducing the measurement accuracy. Therefore, developing a signal processing algorithm that can effectively reconstruct the depth-resolved phase map under non-continuous laser spectral conditions is of great significance for improving measurement accuracy and system performance.

[0004] Random sampling Fourier transform and iterative adaptive algorithm are currently the two main algorithms for depth-resolved phase reconstruction under non-continuous spectral conditions.

[0005] The random sampling Fourier transform regards the discontinuous depth-resolved wavelength scanning interference signal as a non-uniform sampling process. In this process, the Fourier transform is redesigned to adapt to the non-uniform sampling of the wavenumber. By calculating the change in the wavelength sampling interval caused by phase jitter, the random sampling Fourier transform can effectively suppress abnormal side lobes and improve the accuracy of interferometric phase reconstruction.

[0006] The iterative adaptive algorithm is a spectral estimation algorithm developed in recent years. This algorithm introduces weighted least squares analysis and optimizes the spectral estimation by minimizing the error function between the collected interference signal and the inverse Fourier transform of the estimated interference spectrum. Under the sparsity of the weighted matrix, the iterative adaptive algorithm can effectively suppress the amplitude of abnormal side lobes.

[0007] In addition to the above algorithms driven by physical models, a conditional generative adversarial network is introduced to recover the amplitude-frequency diagram under non-continuous spectrum conditions. The conditional generative adversarial network is trained using a continuous spectrum light source with an ultra-wide bandwidth and tested with data under a light source with a spectral gap, so as to achieve the effect of removing sidelobe artifacts in the amplitude-frequency diagram.

[0008] The models of the above three algorithms all assume that the wave number sequence of the laser output is known, which requires designing a dedicated hardware system to obtain the spectral sequence of the laser output, thereby increasing the cost of the depth-resolved wavelength scanning interference system and having poor practicability and flexibility. Summary of the Invention

[0009] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a non-continuous wavelength scanning interference phase recovery method.

[0010] To achieve the above purpose, the technical solution provided by the present invention is as follows:

[0011] A non-continuous wavelength scanning interference phase recovery method includes:

[0012] S1. Build a Michelson interference optical path, and use a CCD industrial camera to photograph the interference light to obtain multiple interference patterns;

[0013] S2. Remove the DC signal interference to obtain a non-continuous wavelength scanning interference signal;

[0014] S3. Insert signal points to be predicted, that is, the missing number of interference signal points, at the discontinuous points of the non-continuous wavelength scanning interference signal;

[0015] S4. Calculate the autoregressive model of the non-continuous wavelength scanning interference signal including the coefficient matrix A;

[0016] S5. Decompose the coefficient matrix A of the autoregressive model so that it corresponds to the coefficient matrices of the missing interference signal and the non-continuous wavelength scanning interference signal respectively;

[0017] S6. Predict the missing interference signal;

[0018] S7. Combine the predicted missing interference signal with the non-continuous wavelength scanning interference signal to form a predicted depth-resolved wavelength scanning interference signal; subsequently, use Fourier transform to demodulate the predicted depth-resolved wavelength scanning interference signal and extract the corresponding amplitude spectrum;

[0019] S8. Obtain the amplitude spectrum with minimized energy leakage by minimizing the objective function to determine the optimal number of predicted signal points. If the convergence condition is satisfied, go to step S9; otherwise, return to step S3 to iteratively update the number of predicted signal points;

[0020] S9. Solve the wavelength scanning interference phase distribution for suppressing phase jitter;

[0021] Further, collect the discontinuous wavelength scanning interference signal I(n) through the depth-resolved wavelength scanning interference technique. The formula is as follows:

[0022]

[0023] In formula (1), R is the reference surface, M is the number of surfaces, p is the serial number of the measured surface, and β R is the reflected light intensity of the reference surface, and β p is the reflected light intensity of the surface corresponding to the serial number p. Λ Rp is the optical path difference between the reference surface R and the surface corresponding to the serial number p; the laser scanning wavenumber sequence k is discretized by points n = 1, 2, …, N1, N2, …, N J , and the initial value N0 is set to 0. Δk j and k1 respectively represent the bandwidth of the j-th continuous scanning wavenumber and the initial wavenumber. J represents the number of continuous scanning wavenumber bandwidths; Δ∏ j represents the wavelength jump, where the initial jump ΔΠ0 is set to 0; ε(x) is the step function, which has a value of 1 when x is greater than or equal to 0 and a value of 0 when x is less than 0; formula (1) shows that the wavelength jump will cause phase jitter to be introduced in the depth-resolved wavelength scanning interference measurement.

[0024] Further, the autoregressive model of the discontinuous wavelength scanning interference signal is as follows:

[0025]

[0026] In formula (2), Z is the depth-resolved wavelength scanning interference signal vector; γ represents the order of the autoregressive model, and the sequences a1, a2, …, a γ represent the coefficients of the autoregressive model; the vector e represents the background noise interference;

[0027] θ1, θ2, …, θ L represent the missing interference signals; I1, I2, …, I P represent the discontinuous wavelength scanning interference signals, where L is the number of points of the signal to be predicted, that is, the number of points of the missing interference signals, and P is the number of interference signals collected in the experiment; combine these collected signals and the missing signals together to form the depth-resolved wavelength scanning interference signal vector Z = {I1, I2, …, I g , θ1, θ2, …, θ L , I g+1 , I g+2 , …, I P}, where the subscript g represents the g-th sampling point where wavelength hopping occurs; the coefficient matrix A is a Toeplitz matrix, and the elements on each diagonal of this matrix are the same. The number of columns is P + L, and the number of rows is P + L - γ.

[0028] Further, after decomposing the coefficient matrix A, we get:

[0029] B·(θ1 θ2 … θ L ) T +C·(I1 I2 … I P ) T =e (3)

[0030] In Equation (3), both matrix B and matrix C are sub-block matrices of matrix A, corresponding to the coefficient matrices of the missing interference signals and the discontinuous wavelength scanning interference signals respectively; the superscript T represents the matrix transpose.

[0031] Further, the missing interference signals are predicted by the linear least squares method, as shown in the following equation:

[0032] (θ1 θ2 … θ L ) T =-(B T B) -1 ·B T ·C·(I1 I2… I P ) T (4).

[0033] Further, the predicted missing interference signals are combined with the discontinuous wavelength scanning interference signals to form the predicted depth-resolved wavelength scanning interference signals; subsequently, the Fourier transform is used to demodulate the predicted depth-resolved wavelength scanning interference signals to extract the corresponding amplitude spectrum, and the formula is as follows:

[0034]

[0035] In Equation (5), the left coefficient matrix is the Fourier transform matrix, where Q represents the total number of frequency-domain sampling points, the vector Y represents the amplitude spectrum after the Fourier transform of the interference signal, U is the integer domain, and i is the imaginary unit.

[0036] Further, the amplitude spectrum with minimized energy leakage is obtained by minimizing the objective function to determine the optimal number of predicted signal points, and the formula is as follows:

[0037]

[0038] In Equation (6), L * represents the optimal number of predictions, and argmin represents that the objective function reaches the minimum value.

[0039] Compared with the prior art, the principle and advantages of the present technical solution are as follows:

[0040] In depth-resolved wavelength scanning, the sampling of a discontinuous spectrum can be regarded as an observation process of missing samples. On this basis, the present technical solution estimates the missing samples by establishing a discrete optimization model with the goal of minimizing the amplitude spectrum energy. This enables the prediction of the lost interference signals within the spectral gap interval without the need for a dedicated hardware system to monitor the wavenumber sequence, thereby eliminating the phase jitter caused by the discontinuous spectrum and achieving the accurate reconstruction of the depth-resolved phase map. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the services required in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0042] Figure 1 It is a principle flowchart of a discontinuous wavelength scanning interference phase recovery method of the present invention (step S1 is missing);

[0043] Figure 2 It is the Michelson interference optical path built;

[0044] Figure 3 It is the interference phase map of the optical wedge;

[0045] Figure 4 It is the interference phase map of the resolution target. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0046] The present invention will be further described below in conjunction with specific embodiments:

[0047] As Figure 1 shown, a discontinuous wavelength scanning interference phase recovery method described in this embodiment includes the following steps:

[0048] S1. Build the Michelson interference optical path as Figure 2 shown; then start the power controller and set the operating current of the mode-hop laser to 55.20 mA; set the temperature controller and start it, so that the temperature of the mode-hop laser diode gradually rises from 25 °C to 40 °C to achieve the wavelength output tuning of the mode-hop laser; use a CCD industrial camera to photograph the interference light to obtain multiple interference patterns;

[0049] S2. Remove the DC signal interference, and collect the discontinuous wavelength scanning interference signal I(n) through the depth-resolved wavelength scanning interference technology. The formula is as follows:

[0050]

[0051] In formula (1), R is the reference surface, M is the number of surfaces, p is the serial number of the measured surface, and β R is the reflected light intensity of the reference surface, and β p is the reflected light intensity of the surface corresponding to the serial number p, and Λ Rp is the optical path difference between the reference surface R and the surface corresponding to the serial number p; the laser scanning wavenumber sequence k is discretized by n = 1, 2, …, N1, N2, …, N J points, and the initial value N0 is set to 0, and Δk j and k1 represent the bandwidth of the j-th continuous scanning wavenumber and the initial wavenumber respectively, and J represents the number of continuous scanning wavenumber bandwidths; Δ∏ j represents the wavelength jump, where the initial jump ΔΠ0 is set to 0; ε(x) is the step function, which is 1 when x is greater than or equal to 0 and 0 when x is less than 0; formula (1) shows that the wavelength jump will cause phase jitter to be introduced in depth-resolved wavelength scanning interferometric measurement; this interference will cause distortion of the interference spectrum, introduce abnormal side lobes, and make it impossible to correctly reconstruct the interference phase.

[0052] S3. Insert the signal points to be predicted, that is, the missing interference signal points, at the discontinuity of the non-continuous wavelength scanning interference signal;

[0053] S4. Since the depth-resolved wavelength scanning interference signal is periodic, an autoregressive model can be used to predict the missing interference signal between the wavelength jumps. For this purpose, calculate the autoregressive model of the non-continuous wavelength scanning interference signal including the coefficient matrix A, and the formula is as follows:

[0054]

[0055] In formula (2), Z is the depth-resolved wavelength scanning interference signal vector; γ represents the order of the autoregressive model, and the sequence a1, a2, …, a γ represents the coefficients of the autoregressive model; the vector e represents the background noise interference;

[0056] θ1, θ2, …, θ L represent the missing interference signals; I1, I2, …, I P represent the non-continuous wavelength scanning interference signals, where L is the number of signal points to be predicted, that is, the number of missing interference signal points, and P is the number of interference signal points collected in the experiment; combine these collected signals and the missing signals together to form the depth-resolved wavelength scanning interference signal vector Z = {I1, I2, …, I g , θ1, θ2, …, θ L , I g+1 , I g+2 , …, I P}, where the subscript g represents the g-th sampling point where wavelength hopping occurs; the coefficient matrix A is a Toeplitz matrix, and the elements on each diagonal of this matrix are the same. The number of columns is P + L, and the number of rows is P + L - γ.

[0057] S5. Decompose the coefficient matrix A of the autoregressive model so that it corresponds to the coefficient matrices of the missing interference signal and the discontinuous wavelength scanning interference signal respectively;

[0058] After decomposing the coefficient matrix A, we get:

[0059] B·(θ1 θ2 … θ L ) T +C·(I1 I2 … I P ) T =e (3)

[0060] In Equation (3), both matrix B and matrix C are sub-block matrices of matrix A, corresponding to the coefficient matrices of the missing interference signal and the discontinuous wavelength scanning interference signal respectively; the superscript T represents matrix transpose.

[0061] S6. Predict the missing interference signal by linear least squares method, as shown in the following equation:

[0062] (θ1 θ2 … θ L ) T =-(B T B) -1 ·B T ·C·(I1 I2… I P ) T (4)

[0063] Formula (4) successfully eliminates the phase jitter problem by predicting the continuous change of the missing interference signal before and after wavelength hopping.

[0064] S7. Combine the predicted missing interference signal and the discontinuous wavelength scanning interference signal to form the predicted depth-resolved wavelength scanning interference signal; then, perform Fourier transform on this signal to demodulate it and extract the corresponding amplitude spectrum. The formula is as follows:

[0065]

[0066] In Equation (5), the left coefficient matrix is the Fourier transform matrix, where Q represents the total number of frequency domain sampling points, the vector Y represents the amplitude spectrum after Fourier transform of the interference signal, U is the integer domain, and i is the imaginary unit.

[0067] S8. Obtain the amplitude spectrum with minimized energy leakage by minimizing the objective function to determine the optimal number of predicted signal points. If the convergence condition is satisfied, proceed to step S9; otherwise, return to step S3 and iteratively update the number of predicted signal points.

[0068] Among them, the amplitude spectrum with minimized energy leakage is obtained by minimizing the objective function to determine the optimal number of predicted signal points. The formula is as follows:

[0069]

[0070] In Equation (6), L * represents the optimal number of predictions, and argmin represents that the objective function reaches the minimum value.

[0071] S9. Solve the wavelength-scanned interference phase distribution for phase jitter suppression.

[0072] The following is an analysis in combination with the optical wedge phase diagram with an inclination angle of 6':

[0073] The optical wedge phase diagram with an inclination angle of 6' is as Figure 3 shown. Theoretically, the phase diagram should present a smooth distribution, reflecting the measurement results of the optical surface. However, under uncorrected discontinuous interference signals, traditional Fourier transform methods often produce phase fluctuation errors. These errors are mainly caused by abnormal side lobes, which will cause spectral crosstalk and then distort the phase frequency at the main lobe position. Due to the high sensitivity of depth-resolved wavelength-scanned interference to phase measurement, even a tiny phase distortion may cause significant phase errors. To solve this problem, the non-continuous wavelength-scanned interference phase recovery method based on the autoregressive model in the present invention optimizes the phase recovery process by effectively suppressing abnormal side lobes. This method can significantly improve the quality of the optical wedge phase diagram, making the calculation results of the wrapped phase diagram smoother and more accurate. Compared with the traditional method, the phase diagram after using the autoregressive model can better reflect the true measurement situation of the optical surface, avoid the interference of phase fluctuation errors, and improve the reconstruction accuracy of the interference phase.

[0074] Since the inclination angle of the optical wedge has been clearly calibrated in the parameter table provided by the manufacturer, the inclination angle value can be calculated by performing a plane fitting on the unwrapped phase diagram. This method determines the inclination angle by the angle between the normal vectors obtained by fitting the plane. The results are shown in Table 1. The data in the table show that the difference between the inclination angle measured by the non-continuous wavelength-scanned interference phase recovery method based on the autoregressive model proposed in the present invention and the true value of 6' is smaller, and the relative error is only 0.58%.

[0075] Table 1: Inclination angles of 6' optical wedges calculated using different methods

[0076] Method Inclination measurement result Relative error Traditional Fourier transform 5.7646′ 3.923％ Autoregressive model 6.0350′ 0.583％

[0077] In Figure 4 In the interference phase diagram of the resolution target board, due to the relatively serious abnormal side lobes, the traditional Fourier transform method cannot evaluate the phase at the correct position of the main lobe peak. Therefore, the pattern features of the USAF1951 resolution target board cannot be accurately displayed in the finally obtained wrapped phase diagram and unwrapped phase diagram. In addition, by adjusting the unwrapped phase diagram to be perpendicular to the optical axis, the visibility of the target pattern can be enhanced. This adjustment method is achieved by fitting a plane to the patternless area and subtracting the fitted plane from the unwrapped phase result. However, the target pattern is still invisible, indicating that the impact of non-continuous wavelength scanning on phase reconstruction is relatively serious. After adopting the non-continuous wavelength scanning interference phase recovery method based on the autoregressive model of the present invention, the target pattern is clearly presented in the wrapped phase diagram and the unwrapped phase diagram. In addition, the unwrapped phase diagram aligned with the optical axis completely restores the target pattern. The experimental results of the resolution target board measurement show that the non-continuous wavelength scanning interference phase recovery method based on the autoregressive model can effectively eliminate the interference of abnormal side lobes with high amplitudes, so as to achieve high-precision multi-surface perspective measurement under non-continuous wavelength scanning conditions.

[0078] The above-described embodiments are only the preferred embodiments of the present invention, and do not limit the scope of implementation of the present invention. Therefore, all changes made according to the shape and principle of the present invention should be covered within the protection scope of the present invention.

Claims

1. A non - continuous wavelength scanning interference phase recovery method, characterized in that, Including: S1. Set up the Michelson interference optical path, and capture the interference light through a CCD industrial camera to obtain multiple interference patterns; S2. Remove the DC signal interference to obtain the discontinuous wavelength scanning interference signal; S3. Insert the signal points to be predicted, that is, the missing interference signal points, at the discontinuous points of the discontinuous wavelength scanning interference signal; S4. Calculate the autoregressive model of the discontinuous wavelength scanning interference signal including the coefficient matrix A; S5. Decompose the coefficient matrix A of the autoregressive model so that it corresponds to the coefficient matrices of the missing interference signal and the discontinuous wavelength scanning interference signal respectively; S6. Predict the missing interference signal; S7. Combine the predicted missing interference signal with the discontinuous wavelength scanning interference signal to form the predicted depth-resolved wavelength scanning interference signal; Subsequently, perform Fourier transform on the predicted depth-resolved wavelength scanning interference signal to extract the corresponding amplitude spectrum; S8. Obtain the amplitude spectrum with minimized energy leakage by minimizing the objective function to determine the optimal number of predicted signal points. If the convergence condition is met, go to step S9; otherwise, return to step S3 to iteratively update the number of predicted signal points; S9. Solve the wavelength scanning interference phase distribution with phase jitter suppression.

2. The non - continuous wavelength scanning interference phase recovery method according to claim 1, characterized in that Collect the discontinuous wavelength scanning interference signal I(n) through the depth-resolved wavelength scanning interference technique. The formula is as follows: In Equation (1), R is the reference surface, M is the number of surfaces, p is the serial number of the surface to be measured, and β R is the reflected light intensity of the reference surface, and β p is the reflected light intensity of the surface corresponding to the serial number p. Λ Rp is the optical path difference between the reference surface R and the surface corresponding to the serial number p; the laser scanning wavenumber sequence k is discretized by points n = 1, 2, …, N1, N2, …, N J , and the initial value N0 is set to 0. Δk j and k1 respectively represent the bandwidth of the j-th consecutive scanning wavenumber and the initial wavenumber, and J represents the number of bandwidths of consecutive scanning wavenumbers; ΔΠ j represents the wavelength jump, where the initial jump ΔΠ0 is set to 0; ε(x) is a step function, which is 1 when x is greater than or equal to 0 and 0 when x is less than 0. Formula (1) indicates that wavelength hopping will cause phase jitter in the depth-resolved wavelength scanning interference measurement.

3. A non - continuous wavelength scanning interference phase recovery method according to claim 2, characterized in that The autoregressive model of the discontinuous wavelength scanning interference signal is as follows: In Equation (2), Z is the depth-resolved wavelength-scanned interference signal vector; γ represents the order of the autoregressive model, and the sequence a1, a2, …, a γ represents the coefficients of the autoregressive model; the vector e represents the background noise interference; θ1, θ2, …, θ L represent the missing interference signals; I1, I2, …, I P represent the non - continuous wavelength - scanned interference signals, where L is the number of points of the signal to be predicted, that is, the number of points of the missing interference signals, and P is the number of points of the interference signals collected in the experiment; combining these collected signals and the missing signals together forms a depth - resolved wavelength - scanned interference signal vector Z = {I1, I2, …, I g , θ1, θ2, …, θ L , I g+1 , I g+2 , …, I P}, where the subscript g represents the g - th sampling point where a wavelength jump occurs; the coefficient matrix A is a Toeplitz matrix, and the elements on each diagonal of this matrix are the same, with the number of columns being P + L and the number of rows being P + Lγ.

4. A non - continuous wavelength scanning interference phase recovery method according to claim 3, characterized in that After decomposing the coefficient matrix A, we get: B·(θ1 θ2 … θ L ) T +C·(I1 I2 … I P ) T =e (3) In formula (3), both matrix B and matrix C are sub-block matrices of matrix A, corresponding to the coefficient matrices of the missing interference signal and the discontinuous wavelength scanning interference signal respectively. The superscript T represents the matrix transpose.

5. A non - continuous wavelength scanning interference phase recovery method according to claim 4, characterized in that, Predict the missing interference signal by linear least squares method, as shown in the following formula: (θ1 θ2 … θ L ) T =-(B T B) -1 ·B T ·C·(I1 I2 … I P ) T (4).

6. A discontinuous wavelength scanning interference phase retrieval method according to claim 5, characterized in that Combine the predicted missing interference signal with the discontinuous wavelength scanning interference signal to form the predicted depth-resolved wavelength scanning interference signal. Subsequently, perform Fourier transform on this signal to extract the corresponding amplitude spectrum. The formula is as follows: In formula (5), the left coefficient matrix is the Fourier transform matrix, where Q represents the total number of frequency domain sampling points, vector Y represents the amplitude spectrum after Fourier transform of the interference signal, U is the integer domain, and i is the imaginary unit.

7. A method for non - continuous wavelength - scanning interference phase recovery according to claim 6, characterized in that, Obtain the amplitude spectrum with minimized energy leakage by minimizing the objective function to determine the optimal number of predicted signal points. The formula is as follows: L * = argmin{[Y(L)] T ·[Y(L)]} (6) In formula (6), L * represents the optimal number of prediction points, and argmin represents that the objective function reaches the minimum value.

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