Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2022; 6(5): 473-478
Published online October 25, 2022 https://doi.org/10.3807/COPP.2022.6.5.473
Copyright © Optical Society of Korea.
Li Jie Li^{1}, Wen Bo Jing^{1} , Wen Shen^{2}, Yue Weng^{3}, Bing Kun Huang^{1}, Xuan Feng^{1}
Corresponding author: *wenbojing@cust.edu.cn, ORCID 0000-0001-9088-3895
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
To overcome the capture-range problem in phase-diversity phase retrieval (PDPR), a geometricaloptics initial-estimate method is proposed to avoid a local minimum and to improve the accuracy of laser-wavefront measurement. We calculate the low-order aberrations through the geometrical-optics model, which is based on the two spot images in the propagation path of the laser, and provide it as a starting guess for the PDPR algorithm. Simulations show that this improves the accuracy of wavefront recovery by 62.17% compared to other initial values, and the iteration time with our method is reduced by 28.96%. That is, this approach can solve the capture-range problem.
Keywords: Capture range, Geometric optics, Laser wavefront, Phase diversity phase retrieval
OCIS codes: (080.1010) Aberrations (global); (100.5070) Phase retrieval; (120.5050) Phase measurement
Laser-wavefront measurement is essential in the quality analysis of laser beams [1]. According to the actual measurement, the laser wavefront in general is highly influenced by tilt, defocus, and astigmatism. The phase-diversity phase retrieval (PDPR) [2, 3] algorithm is one of the critical methods of wavefront detection, using a nonlinear optimization algorithm to try to match the collected laser-spot image to a calculated image based on scalar diffraction propagation.
Nonlinear optimization requires reasonable starting guesses. An excellent initial estimate improves the accuracy of the PDPR algorithm. Currently, the starting guesses are usually set to zero or random values [4, 5]. However, this leads to an issue known as the “Capture-range problem”: The nonlinear optimization algorithm falls into a local minimum during iteration when the starting guesses for the wavefront are too far from the ground-truth wavefront [4, 6]. Turman [7] proposed a geometrical-optics method to obtain wavefront tilt data for the phase retrieval (PR) algorithm, and it was modified by Jurling [4, 8, 9] and Carlisle [10]. The method improves the capture range when segment tilt errors are large. Paine [6, 11], Cao
In this paper, we propose a geometrical-optics initial-estimate method to improve the capture-range problem in the normal PDPR algorithm for laser-wavefront detection. Two spot images in the laser’s optical path are collected to build geometrical-optics theoretical models. We calculate the low-order aberration based on the models and set it as the initial estimate for the PDPR algorithm. The proposed method provides a good idea for selecting the starting guesses in the laser-wavefront restoration using the PDPR method, and has strong practicability.
We use the Zernike polynomials of Fringe [20] to represent the laser wavefront. The low-order Zernike terms are extremely important in wavefront aberration. In this section, we present two geometrical-optics theoretical models to calculate the low-order aberrations of the laser spot. These aberrations include
We take
where θ is the angle between the ray and the
and
The defocus and astigmatism terms share a geometrical-optics model for derivation and calculation. The astigmatism term is divided into separate terms for 0° astigmatism and 45° astigmatism. The term of highest degree in the Zernike polynomial for defocus and astigmatism is quadratic. So, we use the quadratic model.
We take the derivation of the defocus term as an example to explain the model shown in Fig. 2. The light spot propagates from Fig. 2(a) to Fig. 2(b). The spot radii in Fig. 2(a) and Fig. 2(b) are different due to wavefront distortion. The light is emitted from point
where
According to Eq. 4, the normal vector at point (
According to geometrical optics and the geometric relationships in Fig. 2,
where ∆
So, we obtain
We denote the 0°-astigmatism term as
The calculation method for the 45°-astigmatism term has one more step than the calculation method for the 0°-astigmatism term. We need to rotate the two light-spot images counter-clockwise by 45 degrees in advance when calculating the 45°-astigmatism term, and the subsequent calculation steps are the same.
In this section, we experiment with initial estimates for the simulation. The experiment aims to verify the accuracy of the theory presented in the second section.
The principle of the simulation experiment is shown in Fig. 3. We simulate a Gaussian laser spot’s intensity image
The radius of the
As shown in Fig. 4, the maximum error did not exceed 0.03 λ. The calculated low-order aberrations were closed to the true value, verifying the proposed method’s correctness. In addition, the reason for the large defocus aberration may be that the shape of the laser wavefront is approximated as spherical when calculating the defocus aberration.
After that, we apply the initial value predicted by the proposed method to the PDPR algorithm. Additionally, the accuracy of the wavefront sensors is no less than 0.01, so it is inappropriate to use the value detected by the sensors as the true value. Therefore, simulation experiments are carried out to verify whether the proposed method can improve the PDPR algorithm.
As shown in Fig. 5,
Traditionally, zeros or random numbers are generally used as the initial values for the PDPR algorithm. In this article, we use the values calculated by the method as the initial values for PDPR.
The values for λ, ∆
As shown in Fig. 6, the error for initial iteration value of the LBFGS algorithm predicted by the proposed method (IP) is closer to 0 in the noisy environment. We compare the RMS error of the restored wavefront obtained from different initial values and the ground-truth wavefront to verify the influence of different initial values on the wavefront. The Zernike coefficients from 2^{nd} to 15^{th} orders are combined to yield the RMS value. Table 1 shows the mean RMS error for 100 experiments. For a more intuitive comparison, Fig. 7 shows the wavefront map and residuals map in one of the experiments under the background noise of the Phaseview Beam Wave 500’s camera.
TABLE 1 System performance for different initial values in a noisy environment
Initial Value | RMS Residual Error ( | Total Time(s) | ||
---|---|---|---|---|
Max | Min | Mean | ||
IP | 0.0588 | 0.0065 | 0.0304 | 19.657 |
IZ | 0.1319 | 0.0129 | 0.0516 | 24.143 |
IR | 0.1802 | 0.0057 | 0.0470 | 26.557 |
From Table 1, it can be seen that the RMS error for IP was smaller than those for initial iteration value of the LBFGS algorithm is zero (IZ) and initial iteration value of the LBFGS algorithm is random (IR). The iteration time for IP is reduced by about 28.96% (5.69 seconds) compared to the other initial values. As shown in Fig. 7, the wavefront maps generated by different initial values in Figs. 7(b)–7(d) are similar to the ground-truth wavefront map in Fig. 7(a). However, we can find that our method achieves smaller residuals than others, by comparing to Figs. 7(e)–7(g). Therefore, the proposed method can obviously improve the accuracy of wavefront reconstruction.
In this paper, two geometric models were constructed to calculate the low-order aberration of a laser beam’s wavefront, which provided a high-accuracy initial value for the PDPR method. We conducted simulation experiments. In noise-free simulations, the maximum errors between the calculated low-order aberrations and the true value were less than 0.03 λ. In a noisy environment, the wavefront with the initial value calculated by our method was closer to the ground-truth wavefront, compared to initial values of zero or a random number. Also, the iteration time with our method was reduced by 28.96%. The proposed method was proven to solve the capture-range problem.
The authors declare no conflict of interest.
All of the data have been listed in this work; no extra data were generated in the presented research.
111 project of China (D21009); Ministry of Science and Technology of the People’s Republic of China (2018YFB 1107600); Jilin Scientific and Technological Development Program (20160204009GX, 20170204014GX).
Curr. Opt. Photon. 2022; 6(5): 473-478
Published online October 25, 2022 https://doi.org/10.3807/COPP.2022.6.5.473
Copyright © Optical Society of Korea.
Li Jie Li^{1}, Wen Bo Jing^{1} , Wen Shen^{2}, Yue Weng^{3}, Bing Kun Huang^{1}, Xuan Feng^{1}
^{1}School of Opto-electronic Engineering, Changchun University of Science and Technology, Jilin 130022, China
^{2}Department of Management Engineering, Jilin Communications Polytechnic, Jilin 130012, China
^{3}Chengdu Branch of Software Platform R&D Department, Dahua Technology, Sichuan 610095, China
Correspondence to:*wenbojing@cust.edu.cn, ORCID 0000-0001-9088-3895
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
To overcome the capture-range problem in phase-diversity phase retrieval (PDPR), a geometricaloptics initial-estimate method is proposed to avoid a local minimum and to improve the accuracy of laser-wavefront measurement. We calculate the low-order aberrations through the geometrical-optics model, which is based on the two spot images in the propagation path of the laser, and provide it as a starting guess for the PDPR algorithm. Simulations show that this improves the accuracy of wavefront recovery by 62.17% compared to other initial values, and the iteration time with our method is reduced by 28.96%. That is, this approach can solve the capture-range problem.
Keywords: Capture range, Geometric optics, Laser wavefront, Phase diversity phase retrieval
Laser-wavefront measurement is essential in the quality analysis of laser beams [1]. According to the actual measurement, the laser wavefront in general is highly influenced by tilt, defocus, and astigmatism. The phase-diversity phase retrieval (PDPR) [2, 3] algorithm is one of the critical methods of wavefront detection, using a nonlinear optimization algorithm to try to match the collected laser-spot image to a calculated image based on scalar diffraction propagation.
Nonlinear optimization requires reasonable starting guesses. An excellent initial estimate improves the accuracy of the PDPR algorithm. Currently, the starting guesses are usually set to zero or random values [4, 5]. However, this leads to an issue known as the “Capture-range problem”: The nonlinear optimization algorithm falls into a local minimum during iteration when the starting guesses for the wavefront are too far from the ground-truth wavefront [4, 6]. Turman [7] proposed a geometrical-optics method to obtain wavefront tilt data for the phase retrieval (PR) algorithm, and it was modified by Jurling [4, 8, 9] and Carlisle [10]. The method improves the capture range when segment tilt errors are large. Paine [6, 11], Cao
In this paper, we propose a geometrical-optics initial-estimate method to improve the capture-range problem in the normal PDPR algorithm for laser-wavefront detection. Two spot images in the laser’s optical path are collected to build geometrical-optics theoretical models. We calculate the low-order aberration based on the models and set it as the initial estimate for the PDPR algorithm. The proposed method provides a good idea for selecting the starting guesses in the laser-wavefront restoration using the PDPR method, and has strong practicability.
We use the Zernike polynomials of Fringe [20] to represent the laser wavefront. The low-order Zernike terms are extremely important in wavefront aberration. In this section, we present two geometrical-optics theoretical models to calculate the low-order aberrations of the laser spot. These aberrations include
We take
where θ is the angle between the ray and the
and
The defocus and astigmatism terms share a geometrical-optics model for derivation and calculation. The astigmatism term is divided into separate terms for 0° astigmatism and 45° astigmatism. The term of highest degree in the Zernike polynomial for defocus and astigmatism is quadratic. So, we use the quadratic model.
We take the derivation of the defocus term as an example to explain the model shown in Fig. 2. The light spot propagates from Fig. 2(a) to Fig. 2(b). The spot radii in Fig. 2(a) and Fig. 2(b) are different due to wavefront distortion. The light is emitted from point
where
According to Eq. 4, the normal vector at point (
According to geometrical optics and the geometric relationships in Fig. 2,
where ∆
So, we obtain
We denote the 0°-astigmatism term as
The calculation method for the 45°-astigmatism term has one more step than the calculation method for the 0°-astigmatism term. We need to rotate the two light-spot images counter-clockwise by 45 degrees in advance when calculating the 45°-astigmatism term, and the subsequent calculation steps are the same.
In this section, we experiment with initial estimates for the simulation. The experiment aims to verify the accuracy of the theory presented in the second section.
The principle of the simulation experiment is shown in Fig. 3. We simulate a Gaussian laser spot’s intensity image
The radius of the
As shown in Fig. 4, the maximum error did not exceed 0.03 λ. The calculated low-order aberrations were closed to the true value, verifying the proposed method’s correctness. In addition, the reason for the large defocus aberration may be that the shape of the laser wavefront is approximated as spherical when calculating the defocus aberration.
After that, we apply the initial value predicted by the proposed method to the PDPR algorithm. Additionally, the accuracy of the wavefront sensors is no less than 0.01, so it is inappropriate to use the value detected by the sensors as the true value. Therefore, simulation experiments are carried out to verify whether the proposed method can improve the PDPR algorithm.
As shown in Fig. 5,
Traditionally, zeros or random numbers are generally used as the initial values for the PDPR algorithm. In this article, we use the values calculated by the method as the initial values for PDPR.
The values for λ, ∆
As shown in Fig. 6, the error for initial iteration value of the LBFGS algorithm predicted by the proposed method (IP) is closer to 0 in the noisy environment. We compare the RMS error of the restored wavefront obtained from different initial values and the ground-truth wavefront to verify the influence of different initial values on the wavefront. The Zernike coefficients from 2^{nd} to 15^{th} orders are combined to yield the RMS value. Table 1 shows the mean RMS error for 100 experiments. For a more intuitive comparison, Fig. 7 shows the wavefront map and residuals map in one of the experiments under the background noise of the Phaseview Beam Wave 500’s camera.
TABLE 1. System performance for different initial values in a noisy environment.
Initial Value | RMS Residual Error ( | Total Time(s) | ||
---|---|---|---|---|
Max | Min | Mean | ||
IP | 0.0588 | 0.0065 | 0.0304 | 19.657 |
IZ | 0.1319 | 0.0129 | 0.0516 | 24.143 |
IR | 0.1802 | 0.0057 | 0.0470 | 26.557 |
From Table 1, it can be seen that the RMS error for IP was smaller than those for initial iteration value of the LBFGS algorithm is zero (IZ) and initial iteration value of the LBFGS algorithm is random (IR). The iteration time for IP is reduced by about 28.96% (5.69 seconds) compared to the other initial values. As shown in Fig. 7, the wavefront maps generated by different initial values in Figs. 7(b)–7(d) are similar to the ground-truth wavefront map in Fig. 7(a). However, we can find that our method achieves smaller residuals than others, by comparing to Figs. 7(e)–7(g). Therefore, the proposed method can obviously improve the accuracy of wavefront reconstruction.
In this paper, two geometric models were constructed to calculate the low-order aberration of a laser beam’s wavefront, which provided a high-accuracy initial value for the PDPR method. We conducted simulation experiments. In noise-free simulations, the maximum errors between the calculated low-order aberrations and the true value were less than 0.03 λ. In a noisy environment, the wavefront with the initial value calculated by our method was closer to the ground-truth wavefront, compared to initial values of zero or a random number. Also, the iteration time with our method was reduced by 28.96%. The proposed method was proven to solve the capture-range problem.
The authors declare no conflict of interest.
All of the data have been listed in this work; no extra data were generated in the presented research.
111 project of China (D21009); Ministry of Science and Technology of the People’s Republic of China (2018YFB 1107600); Jilin Scientific and Technological Development Program (20160204009GX, 20170204014GX).
TABLE 1 System performance for different initial values in a noisy environment
Initial Value | RMS Residual Error ( | Total Time(s) | ||
---|---|---|---|---|
Max | Min | Mean | ||
IP | 0.0588 | 0.0065 | 0.0304 | 19.657 |
IZ | 0.1319 | 0.0129 | 0.0516 | 24.143 |
IR | 0.1802 | 0.0057 | 0.0470 | 26.557 |