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Abstract
Traditional methods of coherent diffraction imaging using random masks result in an insufficient difference between the diffraction patterns, making it challenging to form a strong amplitude constraint, causing significant speckle noise in the measurement results. Hence, this study proposes an optimized mask design method combining random and Fresnel masks. Increasing the difference between diffraction intensity patterns enhances the amplitude constraint, suppresses the speckle noise effectively, and improves the phase recovery accuracy. The numerical distribution of the modulation masks is optimized by adjusting the combination ratio of the two mask modes. The simulation and physical experiments show that the reconstruction results of PSNR and SSIM using the proposed method are higher than those using random masks, and the speckle noises are effectively reduced.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Because the frequency of light is significantly high, photoelectric detection equipment, such as Charge Coupled Devices (CCD), are generally sensitive to amplitude or intensity information only and cannot record the phase of the wavefronts directly. However, the phase contains significant contour and structural information of objects; hence, the problem of phase measurement must be solved. Phase retrieval (PR) technology is used to retrieve phases from a series of intensity patterns and has been applied in 3D imaging [1], scattering imaging [2], biomedicine [3], and other fields.
The classical Gerchberg–Saxton (GS) algorithm is a pioneering work in the field of phase retrieval. By imposing spatial and Fourier amplitude constraints on light waves, phase recovery is achieved by alternately iterating in the two constraint sets [4]. To solve the problem of slow iteration speed and inconsistent results of the GS algorithm, Fienup [5] proposed using positive constraints to accelerate the iterative convergence, but the problem of premature convergence remained. S. Marchisini [6] summarized and evaluated the iterative projection algorithm. Recently, methods based on sparse representation and compressed sensing have been proposed to improve the speed of recovering image phases [7–11]. In addition, PhaseLift [12], PhaseCut [13], and other algorithms that introduce convex relaxation techniques and relax rank constraints by trace norm turn the PR problem into a convex programming problem; however, they have significantly high computational load and increased complexity.
Another way to solve the problem of iteration stagnation is to conduct multiple intensity measurements. An example is recording diffraction patterns [14–18] in different planes using multiple wavelengths of illumination beams [19,20] and designing specific forms of phase masks [21–23] or amplitude masks [24–28] for modulation. A spatial light modulator (SLM) can achieve pure phase modulation with a high refresh rate, providing significant convenience for efficient wavefront modulation, and is widely used. The wavefront to be measured can be encoded several times simply by loading the designed grayscale pattern on the SLM. Moreover, in a fixed plane, the camera can be used to obtain various intensity figures without other mechanical adjustments. Therefore, it becomes an effective means of wavefront sensing and phase recovery. Random phases can destroy the symmetry of the image and enrich the information of diffraction patterns, so they are often used as phase masks for phase recovery [29–32]. It has been proved that a high-resolution wavefront imaging sensor based on SLM can accurately estimate the phase using random mask modulation and achieve a resolution of more than 10 million pixels to measure the light field of height change [30]. Moreover, vortex phase masks [33] are also proposed to improve the phase recovery accuracy through amplitude constraint and additional topology load constraint, but there was no discussion on the impact of the size of the vortex phase topology load and the amplitude or intensity distribution on the recovery efficiency. In [34], a comparison was made between four types of coding aperture, namely random uniform, random blue noise, Hadamard, and DFT-based. And the study found that the use of a random blue noise coding aperture requires fewer measurements to achieve the maximum achievable PSNR compared to the other methods. [35] proposed a phase recovery method utilizing a green noise mask and compared it with random white noise and blue noise masks. The research finding showed that by capturing diffraction patterns containing more intermediate frequency information in Fourier domain, the loss of high-frequency information during the phase recovery of white noise masks can be avoided. The limitations of using random white noise and blue noise masks were further highlighted from the perspective of insufficient utilization of information.
In practical measurements, more masks are often required to achieve iterative convergence due to the beam alignment error, environmental noise interference, and modulation error of an SLM. The multi-intensity measurement method can reduce the premature stagnation of the phase recovery iteration because various diffraction patterns with significant differences enhance the amplitude constraint. Therefore, a mask design should aim at increasing the diversity of diffraction patterns. In addition, the number of masks must be reduced to reduce the acquisition time and maximize recovery efficiency. Although the random mask modulation method can finely modulate the measured wavefront, the average intensity between the diffraction patterns is basically the same, and there is no apparent local feature to enhance the constraint.
Hence, this study proposes an optimized mask design method by combining random and Fresnel phases to solve the above problem. In the coherent diffraction imaging system, the Fresnel phase modulation mask can realize the function of the lens, that is, scaling the wavefront to be measured in different proportions, which will significantly change the intensity distribution between the diffraction patterns. This implies that the spectral information will be focused on the central region of the diffraction pattern, in line with the function suggested in [35] that the green noise mask can effectively gather spectral data. However, what distinguishes it is that the alteration of Fresnel phase focus creates frequency information that varies significantly, enabling the full exploitation of diffraction data and enhancing structural similarity between diffraction patterns. As a result, the amplitude replacement step in the phase recovery algorithm produces less change than when using random masks. In combination with the rich details generated by random mask modulation, the purpose of enhancing the amplitude constraint is achieved. From the perspective of mask optimization, we can reconstruct a high-precision wavefront by collecting only four diffraction patterns without additional prior information.
2. Methodology
The method of realizing phase retrieval by multi-intensity measurement based on phase modulation is shown in Fig. 1. The wavefront carrying the surface information of the unknown object is modulated by phase masks successively loaded on the SLM, and the diffraction patterns are collected by the CCD. The wavefront measurement can be realized by the improved GS algorithm.
Fig. 1. Schematic diagram of phase retrieval by multi-intensity measurement based on phase modulation.
The first step is to design the phase masks. We will start by elaborating on our optimization approach based on the characteristics of random masks. It has been proven that Gaussian white noise masks are the guarantee of achieving high-precision phase reconstruction [36], as they can generate more random diffraction patterns, which is beneficial to phase recovery. However, we believe that the randomness of diffraction patterns is not the only characteristic required for achieving high-precision phase reconstruction. Due to the inherent randomness of diffraction patterns, their structural similarity is relatively low, despite being basically the same. For methods involving phase reconstruction through continuous amplitude constraints, this aspect is worth studying. This is because the difference between the estimated wavefront amplitude before and after the amplitude replacement step largely determines the “step size” of the iteration. The refinement of the iteration process determines the final quality of the reconstruction to a large extent. A larger “step size” means a higher possibility of skipping the optimal solution. By introducing the Fresnel phase, we aim to improve the amplitude replacement process by concentrating diffraction information to enhance the contrast diffraction patterns. And this is the starting point of our research.
Based on the analysis above, we set the scale factor $\alpha \in [0,1]$ to combine the Fresnel and random phases. Therefore, the phase matrix for the $k$th mask is generated by the following equation:
wherein, $\varphi _k^f$ and $\varphi _k^r$ represent the Fresnel and randomly distributed phases, respectively. Furthermore, $\varphi _k^r$ is randomly generated for each $\varphi _k^f$, and the focal length of $\varphi _k^f$ increases as k varies. Through these two conditions, we can ensure that there is a significant structural difference between the generated masks, ensuring a difference between the diffraction patterns, which will be discussed in the next section.After the phase masks are finished and converted into gray-scale images, which are applied to an SLM in turn, we collect the diffraction patterns by CCD and use the GS algorithm for wavefront construction. The specific process is as follows.
Modulated by ${\varphi ^{\prime}_k}$ in the SLM, the light field ${U_{i,1}}(x,y)\textrm{ = }{A_{i,1}}\exp ({j{\varphi_{i,1}}} )$ incident on the SLM surface is expressed as:
where subscript i represents the number of iterations, and ${A_{i,1}}$ is the amplitude of light field incident on the SLM surface. For each iteration, ${A_{i,1}}$ and ${\varphi _{i,1}}$ are updated. When the first iteration is performed, the measured wavefront is initialized to ${U_{1,1}}(x,y)\textrm{ = }\exp ({j{\varphi_{1,1}}} )$, where ${\varphi _{1,1}}$ is a random phase uniformly distributed between [0,2π].Next, the light field distribution ${U_{i,3}}$ of the CCD plane is obtained by the angular spectrum diffraction theory as
On this basis, a new estimated wavefront ${U_{i + 1,1}}$ is obtained by dividing the modulation effect of ${\varphi ^{\prime}_k}$.
Repeat the above process until all iterations are completed and ${U_{i + 1,1}}$ updated in the last iteration is considered as the reconstruction wavefront. The peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are used as the evaluation indices of recovery accuracy.
Here ${P_{tar}}$ and ${P_{rec}}$ are the target and reconstructed images, $M \times N$ is the resolution of object image, and b represents the number of image bits. ${\mu _{{P_{tat}}}}$ and $\sigma _{{P_{rec}}}^2$ represent the mean and variance of the target image, respectively, ${\sigma _{{P_{rec}}{P_{tar}}}}$ is the covariance between ${P_{tar}}$ and ${P_{rec}}$, and ${c_1}$ and ${c_2}$ are constants.
3. Numerical simulations
We first conducted numerical simulations to verify the effectiveness of the proposed method. The complex wavefront to be measured in the simulation experiment comprises two images with a resolution of 600 × 600, as shown in Fig. 2. The amplitude is normalized, and the phase is scaled within the range [-π/2, π/2]. The wavelength of the collimated beam is set to 633 nm, and the pixel size of the SLM and CCD is 12.8 × 12.8 µm. The object is placed on the SLM surface, and the distance between the SLM and CCD is set to 100 mm. The resolution of the modulation mask is 400 × 400. Random and designed masks are used to modulate the wavefront to be measured, and the improved GS algorithm is used for wavefront reconstruction; the number of iterations is set to 300.
Determining the number of masks is done by observing the reconstruction efficiency and accuracy of random masks with different quantities. Figure 3 shows the iterative curves of PSNR for amplitude and phase in wavefront reconstruction using random masks when $K = 3,4,5,6$. It can be seen that when $K = 3$, the PSNR value for reconstructed amplitude is over 30 dB lower than that when $K = 4,5,6$, and the iterative curves for both amplitude and phase have not yet converged after 300 iterations. When $K = 4$, the curve converges before 50 iterations and the reconstruction accuracy for both amplitude and phase does not differ much from when $K = 5,6$. To ensure high iteration efficiency in the following mask combination schemes, $K = 4$ is a suitable choice.
Fig. 3. PSNR iteration curve for (a) amplitude and (b) phase retrieval using random masks under different mask number K.
We first study the case of using Fresnel phase masks alone to reconstruct the wavefront. The purpose here is to roughly determine a suitable focal length intervals d of Fresnel phases since our method achieves better reconstruction performance through the combination of the Fresnel focal distance d and the scaling factor $\alpha$, and it should be noted that the optimal combination of d and $\alpha$ is not unique. To compare the difference among the diffraction patterns introduced by mask modulation, we define the structural similarity ${S_p} = ssim({I_p},{I_1})$ between the $p$th $(p = 2,\ldots 4)$ and the first diffraction patterns, which is used to directly reflect the change in the intensity distribution of the diffraction patterns. At the same time, ${S_p}$ also can serve as a visual evaluation parameter for observing the amplitude replacement step in the phase recovery algorithm. Figure 4 shows a set of Fresnel masks with focal length intervals of 40 mm and their corresponding intensity patterns. After the target wavefront is modulated by the Fresnel masks, the diffraction patterns show different sizes of convergence. When the focal length of the Fresnel mask is close to the diffraction distance, the diffraction patterns converge to a focus.
Fig. 4. (a1)–(a4) Fresnel phase masks with focal lengths of 60–180 mm, (b1)–(b4) corresponding diffraction patterns, (c1)–(c2) reconstruction results using random masks, (c3)–(c4) reconstruction results using Fresnel masks with focal length intervals of 40 mm.
We studied the performance of six groups of Fresnel phase masks with focal length intervals of 10–60 mm, and Fig. 5 shows the change in ${S_p}$. It is obvious that ${S_p}$ gradually decreases with the increased d between the Fresnel masks, which corresponds to the change in the PSNR iterative efficiency of each group of masks shown in Fig. 6. This is interpretable. A smaller value of ${S_p}$ implies a greater amplitude change before and after the amplitude replacement step in the phase recovery algorithm, which is similar to the “step size” change in gradient descent. When the “step size” is large, it often has a faster convergence rate. It is evident that the random masks correspond to smaller ${S_p}$, making its convergence efficiency higher while maintaining a lower reconstruction accuracy. On the other hand, the Fresnel masks generate larger ${S_p}$, which represents smaller “step size” in the phase search process. Therefore, it has a higher possibility of obtaining the optimal solution but sacrifices iteration efficiency at the same time.
Fig. 6. PSNR iteration curve for (a) amplitude and (b) phase retrieval using Fresnel and random masks.
From the reconstruction results shown in Figs. 4(c1)-(c4), the SSIM of amplitude and phase image using Fresnel masks alone is the same as that by random masks, and the primary difference is in the PSNR. It is clear that the convergence efficiency is improved with the decrease in ${S_p}$. Moreover, the SSIM values between the diffraction patterns modulated by random masks are the smallest, and the iterative curves converge first. The intensity images of the random masks are characterized by disordered patterns and low structural similarity precisely due to their randomness, and it is difficult to observe the structural characteristics of the target object from the diffraction patterns. Therefore, although it owns the advantage of high iterative efficiency, the performance of random masks is not the best in terms of the quality of the reconstruction results. As shown in Fig. 5, due to premature convergence, the phase recovery results using random masks remain lower than their PSNR of 36.7360 dB, although the PSNR of the amplitude recovered is relatively high, which is 66.9450 dB. The reconstruction results are demonstrated in Figs. 4(c1) and (c2).
On the contrary, with low iterative efficiency, the PSNR values of phase reconstruction results by Fresnel masks with focal length intervals of 20–60 mm are higher than that of random masks after approximately 300 iterations. Among them, the Fresnel masks with a focal interval of 40 mm have the best performance, as shown in Figs. 4(c3) and (c4). The PSNR value of the amplitude recovery result is the same as that of the random masks, and the PSNR value of the phase recovery result is 57.9100 dB, which is 21.1740dB higher than that of the random masks.
In addition, some curve jitter behaviors can be observed in Fig. 6, which can be explained by the changes of ${S_p}$ in Fig. 5. Firstly, the cause of curve jitter is caused by the amplitude replacement step in the phase recovery algorithm. When the amplitude change introduced by the amplitude replacement step is significantly different from the previous and subsequent iterations, it means that the magnitude of the amplitude change in each iteration is inconsistent, which in turn leads to curve jitter. As shown in Fig. 5, although ${S_p}$ corresponding to random masks are small, they are basically the same, which means that the amplitude difference generated by each iteration's amplitude replacement step is similar, so the curve is relatively smooth. However, for Fresnel masks, their corresponding ${S_p}$ show an overall decreasing trend and significant fluctuations as the focal distance interval increases, indicating an increase in the difference in diffraction pattern intensity. Therefore, we can observe the curve jitter phenomenon of Fresnel masks. Moreover, it is quite evident that the amplitude of curve jitter increases with the increase of focal distance interval, which is consistent with our analysis.
On the other hand, in the phase recovery algorithm, the change in amplitude is most directly reflected after performing the amplitude replacement step, and then a new wavefront is composed of the retained phase and the replaced amplitude to continue iteration. The change in phase is mainly reflected in multiplying with and removing the SLM loaded phase, which undergoes forward propagation, amplitude replacement, and reverse propagation. That is to say, the change in phase during the entire iteration process is only due to the amplitude replacement step and is not directly reflected, so the jitter of the phase iteration curve is not as obvious as that of the amplitude iteration curve.
The diffraction images generated by Fresnel phases show large pattern differences, primarily reflected in the intensity distribution and pattern size, providing a strong amplitude constraint for phase retrieval. Hence, it can be used to improve the reconstruction accuracy. However, their apparent drawback is their low iterative efficiency.
Considering the high convergence efficiency of the random phase, we choose to combine Fresnel masks with random masks with a focal interval of 40 mm and a set scale factor $\alpha$ for optimization. Figure 7 shows the combined masks, diffraction patterns, and reconstruction results under different $\alpha$. It can be seen from Figs. 7(b1)-(b4) that the wavefront to be measured is modulated by Fresnel masks to achieve different sizes of convergence. Meanwhile, due to the introduction of random phase modulation, the sharpness of the diffraction patterns decreases, and the edges are blurred. This will undoubtedly make the collected diffraction images more diverse and provide the possibility to improve iteration efficiency.
Fig. 7. (a1)–(a4) Combined masks with $\alpha = 0.3$, (b1)–(b4) corresponding diffraction patterns, reconstructed amplitude (c1), (c3) and phase (c2), (c4) using optimized masks with $\alpha = 0.3$ and $\alpha = 0.4$, respectively.
Figure 8 shows the change in SSIM between the diffraction patterns of the optimized masks under different $\alpha$. Before the random phases are added, as shown in Fig. 5, the overall SSIM is concentrated between 0.06–0.8. With the addition of the random phase, a significant decline in SSIM, distributed between 0.02–0.06, was observed. Moreover, the SSIM gradually decreases overall as $\alpha$ increases. Therefore, the combination with random masks successfully reduces the structural similarity between the diffraction images. Moreover, the PSNR iteration curve of optimized masks in Fig. 8 shows that the orderly reduction in the structural similarity caused by the change in $\alpha$ still corresponds to the improvement in iteration efficiency, which is consistent with the above analysis.
However, it can be further seen from Figs. 9(a) and (b) that, contrary to the iterative efficiency, the reconstruction accuracy does not comply with this rule, which shows a process of first increasing and then decreasing. When $\alpha \in [0,0.3]$, the PSNR value gradually increases, and it orderly decreases when $\alpha \in [0.3,0.6]$. When $\alpha = 0.3$, the PSNR of the amplitude and phase recovered simultaneously reaches the highest before 50 iterations, i.e., 69.1155 dB and 72.5675 dB, respectively, as shown in Figs. 7(c1) and (c2), which are approximately 5.6225 dB and 32.3975 dB, respectively, higher than that of the original method. Therefore, as seen from the simulation results, the method using optimized masks successfully improved the accuracy of wavefront reconstruction while maintaining high iteration convergence efficiency.
Fig. 9. PSNR iteration curve of (a) amplitude and (b) phase retrieval using optimized masks with different $\alpha$.
It should be noted that for phase retrieval methods that based on angular spectrum theory, their applicable diffraction distance is limited between SLM and CCD. Generally speaking, we prefer a measurement system that is compact in size and simple in structure. From this perspective, a small diffraction distance is beneficial. Therefore, we believe it is feasible to determine the reconstruction accuracy rules through a few simple sets of data. Although this method may not find the best solution, it is also close to it. And for different measurement systems, the selection of d is not fixed and can be determined based on the actual situation. By incorporating the adjustment feature of $\alpha$, the process of modifying frequency distribution of diffraction data can be done flexibly to fit different phase retrieval systems. This approach ensures achieving high-precision and high-efficiency requirements, which ultimately maximizes overall performance.
4. Experiments and discussion
The experimental setup for phase retrieval by multi-intensity measurement based on phase modulation is shown in Fig. 10. The 632.8 nm laser beam emitted by the He-Ne laser (Newport N-STP-912) forms a standard plane wave after collimation and beam expansion. After posting target grayscale images on SLM1 (LCOS-SLM X13138) with a resolution of 1024 × 1272 and a pixel size of 12.5 × 12.5 µm, the target wavefront is generated. The 4f filtering system consisting of lens L1 and L2 with focal length f = 300 mm images the wavefront onto the SLM2 surface, and the spatial filter FL selects the 1st order diffracted light. The designed modulation phase images are applied to SLM2. The 8-bit CCD camera (Prosilica GT5120) with a resolution of 5120 × 5120 and a pixel size of 4.5 × 4.5 µm records the modulated diffraction patterns sequentially.
Fig. 10. Experimental setup for phase retrieval by multi-intensity measurement based on phase modulation.
To determine the diffraction distance between SLM2 and CCD, the Fresnel lens phase is applied on the SLM2 surface, and the incident light is focused on the CCD surface by adjusting the focal length. Simultaneously, the phase of the blazed grating with a period of ${d_1}$ is superimposed. The distance ${d_2}$ between ± 1 level of diffracted light is calculated. The distance z between SLM2 and CCD can be calculated according to the grating equation [37]:
After the Fresnel lens phase with a focal length of z is applied to the SLM2 surface, the focus is successively moved to the four theoretical vertices of the diffraction plane. The coordinates of the four focal spots in the diffraction plane are determined by calculating their centroids. Subsequently, the X and Y direction coordinates are averaged to divide the diffraction area from the collected figures. Zero padding is used to avoid aliasing so that the selected diffraction area is slightly larger than the actual modulation area, as shown in Fig. 11. In addition, to reduce the cross-talk effect, the mask should be locally smooth. Hence, we introduced Gaussian blur on the phase masks to smoothen the phase.
To generalize the experiment, in addition to the two images used in our simulations, we selected one gray image of “Goldhill” pattern for phase retrieval. The pictures with a pixel size of 400 × 400 to be tested are scaled within the range of [0,2π] and expanded to a size of 1024 × 1272 by zero padding and are subsequently loaded into SLM1. It is approximately considered that the wavefront to be measured coincides with the SLM2 surface after the 4f system. The distance between SLM2 and CCD is measured as approximately 98.6 mm. Since the CCD pixel size is smaller than the SLM pixel size, the acquired diffraction images are downsampled to 600 × 600 for angular spectrum theory.
The experimental results of wavefront reconstruction are shown in Fig. 12. We will analyze and compare the advantages of the proposed method in terms of phase and amplitude. First, the phase recovered results are analyzed. Figures 12 (a1)–(a3) and (b1)–(b3) show the results of phase reconstruction using random and proposed masks, respectively. Intuitively, the phase image reconstructed by random masks is fuzzy and full of speckle noise. Therefore, only the overall outline can be seen clearly, and the phase details are difficult to distinguish. Table 1 shows the reconstruction performance of phase images with proposed masks and original masks. The PSNR value of the phase picture reconstructed is generally less than 15 dB and the PSNR of the “Goldhill” pattern is the lowest, 11.8230 dB. The SSIM is also low and concentrated in the range of 0.29-0.34. Meanwhile, the phase results recovered by our proposed method achieved greater clarity. Compared with the original method, the PSNR increased by 4 dB on average, and the SSIM increased by nearly twice. It is worth noting that our method reduces a lot of speckle noise in the restored image, making the details of the image clearer and the restored result closer to the original image.
Fig. 12. Reconstructed phase (a1)-(a3), (b1)-(b3) and amplitude (c1)-(c3), (d1)-(d3) distribution of phase patterns with original and proposed method, respectively.
Table 1. Reconstruction performance of phase images with proposed masks and original masks
In addition, we illustrate the advantages of the proposed method from the amplitude reconstruction results. The beam is adjusted to be close to the form of a plane wave before its incidence on SLM; hence its amplitude matrix is close to constant, and the matrix data fluctuates slightly. After the phase-only SLM modulation, the amplitude can be considered approximately unchanged. Since the variance can reflect the degree of data deviation from the average value, we use it to judge whether the amplitude reconstruction is good or bad. The smaller the variance of the amplitude image matrix, the better the reconstruction. Figures 12(c1)-(c3) and (d1)-(d3) show the amplitude images restored by the original and proposed methods, respectively. From the intuitive sense, the amplitude pictures recovered by random masks are messy, with many spots of different sizes, and the variance distribution is between 0.11–0.14 as shown in Table 1. It can be judged that the numerical distribution is relatively disordered with large fluctuations; hence, the amplitude reconstruction quality is not high. The amplitude reconstructed by our method is smooth, the data distribution is relatively concentrated, and the average variance is 0.0344 less than the original method. Hence, the method proposed in this paper effectively improves the recovery quality of phase images. This is due to the optimized combination of Fresnel and random phases. That is, the Fresnel phase can segment the contour of the object image, while the random phase can fine modulate the details. In addition, the structural similarity between the diffraction intensity diagrams gradually increases, forming a strong amplitude constraint in the phase retrieval process. Therefore, the combination of the two effectively suppresses speckle noise and improves the quality of wavefront reconstruction.
To further prove the speckle noise suppression effect of the proposed method in wavefront measurement, we measured the vortex phase of different topological loads, and the experimental results are shown in Fig. 13.
Fig. 13. Reconstructed phase (b1)-(b3), (c1)-(c3) and amplitude (d1)-(d3), (e1)-(e3) distribution of target vortex phase (a1)-(a3) with original and proposed method, respectively.
Similar to the analysis of the above experiment, we demonstrate the effectiveness of the optimized method from the phase and amplitude of reconstruction. Figures 13 (b1)–(b3) and (c1)–(c3) show the vortex phases recovered using the original and proposed masks, respectively. And the corresponding experimental results are shown in Table 2. Compared with the target images, the reconstructed results of the optimized method are better than those of the original method, whether PSNR or SSIM. In addition, we know that the number of topological charges is determined by the number of phases changing from 0 to 2π, and there will be positions with phases of 0 and 2π at the exact times marked by blue, green, and red lines, as shown in Fig. 13. Hence, the quality of reconstruction results can be judged by the uniformity of phase changes and the coincidence of the actual and theoretical positions. Next, we will analyze them in detail.
Table 2. Reconstruction performance of L-G phase with proposed masks and original masks
Figure 13(a1) displays the Laguerre–Gaussian (LG) phase with radial index $p = 10$, azimuthal index $m = 0$, and beam size $s = 0$mm, compared with which four positions marked in blue of the reconstruction by random masks are significantly different from the target positions. Take the area divided by the blue circle ${c_1}$ as an example, which is divided by ${l_1}$ into two parts. The prominent feature of ${c_1}$ area is that the phase value of the upper part gradually increases and that of the lower part decreases. However, in Fig. 13 (b1), the position of the line ${l_1}$ is quite different from the actual phase and does not have a phase step. Conversely, the situation in Fig. 13 (c1) is consistent with that in Fig. 13 (a1), and the phase distribution of the recovered image is closer to the target image. Hence, the reconstruction result of the proposed method is better than that of the original method.
Additionally, the reconstructions of object LG phases with $p = 10$, $m = 0$, $s = 2$mm and $p = 20$, $m = 0$, $s = 0$, as shown in Figs. 13 (a2) and (a3), are discussed. Similar to the above analysis, one can find the random masks’ performance in recovering vortex phases. For example, in the ${c_2}$ region shown in Fig. 13 (b2) and the ${c_3}$ region shown in Fig. 13 (b3), the actual phase mutation position does not coincide with the theoretical phase transition boundary. There are many errors marked by dotted lines, and only a few of them are selected to illustrate. Moreover, the images reconstructed are still full of speckle noise with a PSNR of 7.33–8.04 dB and an SSIM of 0.14–0.24, making the image definition low. This is apparent at the transition positions, severely affecting the reconstruction accuracy.
Conversely, as demonstrated in Figs. 13 (c2) and (c3), no matter how the parameters of the LG beam change, using the proposed method, the phase change is more uniform and smooth, the speckle noise is significantly reduced, and the phase value is also closer to the theoretical value. Meanwhile, except for a few places with errors, which are still smaller than that of the original method, the actual measured boundaries of the phase mutation coincide with the theoretical dotted lines to a higher extent than the original masks. The reconstructed amplitude images are analyzed. Figures 13 (d1)–(d3) and (e1)–(e3) display the recovered amplitudes and Table 2 shows the variance of each amplitude matrix. Through comparison, it can be found that the variance of the amplitude matrix reconstructed by the proposed method is overall smaller than that of the method by random masks, indicating that the amplitude data fluctuates little, and the amplitude image looks not too cluttered. Therefore, the method proposed in this paper achieves higher reconstruction accuracy.
5. Summary
In conclusion, an optimized design method of phase masks for phase retrieval is proposed in this paper. By designing the combination of random and Fresnel phases, the quality of wavefront reconstruction is improved, ensuring the same recovery efficiency as random masks by optimizing the combination scale factor. This optimized method modulates the unknown wavefront by considering roughness and fineness and further increases the difference in the intensity distribution of the diffraction patterns, thus effectively enhancing the amplitude constraint in the phase recovery algorithm. Based on this, the speckle noises in the actual measurement process are effectively suppressed, and the accuracy of wavefront reconstruction is improved.
Moreover, the LG vortex beams with different parameters are measured in this study, and the experimental results show that the method proposed can be used to recover the vortex beam accurately. Hence, the proposed method also provides an effective means for vortex detection. It can be expected that the proposed method has application prospects in the field of multi-intensity phase recovery and coherent modulation imaging.
Funding
National Natural Science Foundation of China (51875447).
Acknowledgments
We appreciate the reviewers for their constructive comments to improve the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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