We used a 32 × 32 signal phase pattern randomly with 4-level phase (0, π/2, π, 3π/2), which contains 50% embedded data shown in Fig. 6(a), and each data was displayed in a 4 × 4 pixel block on the SLM with a pixel pitch of 20 μm. The pixel size of the CCD is 5.86 μm. According to the Nyquist sampling theorem, we chose two Nyquist size frequencies in the Fourier intensity shown in Fig. 6(b) for phase retrieval.
Fig.6 (a) Signal phase pattern and (b) Fourier intensity with two Nyquist size frequencies |
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According to the flow shown in Fig. 5, we obtained new embedded data. To distinguish the previously embedded data from the new embedded data, they were named as the assigned embedded data and the iterative embedded data, respectively. The difference in the phase patterns between these two embedded data is shown in Fig. 7. Compared with the assigned embedded data, the iterative embedded data are more complex and finer because space complexity provides more high-frequency components.
Fig.7 Schematic diagram of phase pattern of object plane. (a) Unknown data. (b) Assigned embedded data. (c) Assigned signal phase pattern. (d) Iterative embedded data. (e) Iterative signal phase pattern |
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Fig.8 Fourier intensity distributions corresponding to (a) assigned embedded data and (c) iterative embedded data, (b) and (d) are the enlargement of red boxes in (a) and (c), respectively |
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Figure 8 shows the Fourier intensity distributions corresponding to the assigned embedded and iterative embedded data. It is evident that the details of the high-frequency information in Fig. 8(d) are more abundant than those in Fig. 8(b). Following this, we also evaluated the intensity of the high- and low-frequency signals in the Fourier plane from two perspectives. It was found that when the high-frequency intensity increased, the ratio of the high-frequency intensity to the total intensity increased; whereas, when the low-frequency intensity decreased, it resulted in a decrease in the ratio of the low-frequency intensity to the total intensity.
One Nyquist size-frequency is denoted as , and we obtained two Nyquist size frequencies in the Fourier intensity, as shown in Fig. 9(a). Further, we calculated the average intensity of the two Nyquist size frequencies to be .
Furthermore, we calculated the intensity of the high-frequency information. The size of the yellow box gradually reduced from 1.9 to 1.5, and to express our choice of ideas more vividly; we only chose three yellow boxes of different sizes as the explanation. The yellow boxes in Figs. 9(b)–9(d) show three different size windows: 1.9, 1.7, and 1.5. Only the values outside these windows were retained, and thereafter the average intensity was calculated.
In contrast, we calculated the intensity of the low-frequency information. The size of the red box was gradually increased from 0.1 to 0.5, and similarly, three windows were selected as an illustration. The red boxes in Figs. 9(e)–9(g) are three different size windows of 0.1, 0.3, and 0.5. Similarly, only the values inside these windows were retained, followed by the calculation of the average intensity .
We calculated the proportional intensity coefficients and to evaluate the intensity distribution of the high- and low-frequency regions according to Eq. (6).
Fig.9 Evaluate the intensity of the high-frequency and low-frequency. (a) 2w. (b) 1.9w. (c) 1.7w. (d) 1.5w. (e) 0.1w. (f) 0.3w. (g) 0.5w |
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In the afore described embedded data algorithm, by controlling the number of iterations, we can obtain different embedded data envelope distributions corresponding to different iterative embedded data patterns, and thereby obtain a series of overall Fourier intensity distributions composed of different iterative embedded data patterns along with the same unknown data pattern. In Fig. 10, we selected three cases as illustrations.
In case 1, iterative embedded data are used to reduce the high-frequency Fourier intensity. Whereas in case 2, the high-frequency intensity was slightly greater than that in case 1 by using the assigned embedded data. Finally, in case 3, the iterative embedded data was changed to render the high-frequency of the Fourier intensity slightly greater than that in case 2. Figures 10(a) and 10(b) show the proportional intensity coefficient curves corresponding to the calculation results of and in Eq. (6), respectively. A high value of along with a low value of indicates greater high-frequency information intensity, and vice-versa. and represent the same conclusions. However, the differences among the values were small, whereas the differences among the values were larger. The latter is more conducive to distinguishing different Fourier intensities; thus, we chose under 0.1 times the Nyquist size and used it to express the following calculation.
Fig.10 Proportional intensity coefficients under different Nyquist spectrum. (a) E1. (b) E2 |
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In the absence of noise, the assigned embedded and iterative embedded data are approximately the same for the phase retrieval because the SNR of high-frequency intensity does not decrease without noise, therefore strengthening of the high-frequency intensity is not required; thus, we only discuss the impact of the two forms of embedded data on phase retrieval in the presence of noise.
Noise exists in every process in the HDS system and deteriorates the phase retrieval directly. To simulate the influence of the experimental noise in the simulation, we added a certain amount of Gaussian model noise to the Fourier intensities formed by the assigned embedded data and iterative embedded data methods. The Gaussian model noise is shown in Fig. 3(a).
Thereafter, we used the IFT algorithm to calculate the bit error rate (BER) after 10 iterations with different Fourier intensities and determined the value corresponding to the minimum BER as shown in Fig. 11, and thus we determined the corresponding embedded data pattern at this time. Smaller the value, lower the low-frequency and higher the high-frequency. In Fig. 11, the black curve and red dot represent the phase-retrieval results corresponding to different iterative embedded and assigned embedded data patterns, respectively. When was 6.00, the BER of the iterative embedded data method was 0.033, while for of 8.31, and the BER was 0.064 in the case of the assigned embedded data method. It can be seen that phase retrieval is better when using the iterative embedded data method.
In Fig. 12, we select two cases where the value of the assigned embedded data is 8.31, and the value of the iterative embedded data is 6.00, and compare the corresponding BER curves. When we use the iterative embedded data method, the phase retrieval is better because the value of the iterative embedded data method is lower than that of the assigned embedded method, which implies that the high-frequency intensity in the iterative embedded data method is higher. Under the same noise, the SNR of the assigned embedded data method was 6.0, while that of the iterative embedded data method was 6.6. Therefore, the convergence speed of the BER using the iterative embedded data method was faster.
Fig.12 Phase-retrieval results of simulation in different embedded data |
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The experimental setup of the non-interference phase-retrieval system is shown in Fig. 13.
Fig.13 Experimental setup. SF: spatial filter, HWP: half-wave plate, BS: beam splitter, SLM: spatial light modulator, L1−L6: lens (L1 = 300 mm, L2−L5 = 150 mm, L6 = 75 mm) |
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We used the same parameters as those in the simulation. The wavelength of the laser was 532 nm, and its power was 300 mW. The SLM is a HAMAMATSU X10468-04 with a pixel pitch of 20 μm and a resolution of 792 × 600 pixels. The CCD is Thorlabs' DCC3260M, with pixel pitch of 5.86 μm, and resolution of 1936 × 1216 pixels. The media used was a PMMA photopolymer. The phase pattern was 32 × 32 phase data based on the combination of half embedded and half unknown phase data, and we used a block of 4 × 4 pixels to denote one-phase data. Finally, a 128 × 128 pixel phase pattern was loaded onto the SLM.
In the experiment, the Fourier intensity distributions corresponding to the assigned embedded data method and the iterative embedded data method are shown in Fig. 14.
Fig.14 Fourier intensity distribution in different embedded data. (a) Iterative embedded data. (d) Assigned embedded data. (c) and (f) are part of the high-frequency detail information in (b) and (e), respectively |
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It was experimentally verified that different intensity distributions were captured on the CCD. It is evident that the high-frequency detail information in Fig. 14(c) is more abundant. The phase-retrieval results corresponding to the assigned embedded data and the iterative embedded data are shown in Fig. 15.
Fig.15 Phase-retrieval results using different embedded data in the experiment |
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In Fig. 15, after ten iterations, the iterative embedded data method can reduce the BER by approximately a factor of one. This proves that our proposed method of using embedded data to enhance the high-frequency intensity aids in improving phase retrieval.