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Abstract
A collinear non-interferometric phase retrieval method for holographic data storage is proposed. The code rate of the collinear system can be increased by 2 times due to transferring the 50% embedded data from a signal beam to a reference beam. Because the reference beam should be always known both in the recording process and reading process, there is no extra material cost for saving the embedded data. Therefore, the storage density of collinear system is increased by about 2 times compared with previous off-axis non-interferometric phase retrieval systems. Besides, we can enhance the algorithm constraint to shorten iteration numbers by increasing the amplitude weight of the reference beam. In the experiment, we shortened the iteration number and the BER by about 3 times. We also find that the reference number can be reduced by using the collinear way and increasing the amplitude weight of reference. This law allows us to give more code positions to signal so that the data amount in one data page can be increased.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Data storage demand in the era of big data is becoming huge [1,2]. Holographic data storage has been known widely as one of most potential next generation storage technology due to its ultra-high density and ultra-fast data transfer rate [3]. Traditional holographic data storage uses amplitude modulation, however, its code rate is too low to raise storage density further [4]. Phase modulation system is paid attention recently due to its higher code rate and higher signal-noise-ratio (SNR) [5–7]. However, phase information cannot be captured directly by detectors. Usually, interferometric method is used to transform phase information to detectable intensity signal on the object domain [8]. However, interferometric results are vulnerable to environmental disturbance. Even tiny vibration can cause undesirable results. Usually, at least two-step phase shifting interferometry should be used to get the accurate phase information [9–11]. However, multiple operations will reduce data transfer rate and increase the error possibility. Therefore, interferometry is not a suitable solution for high-speed reading phase-modulated holographic data storage system. Non-interferometric method may be a solution due to its stable system [12].
There are several non-interferometry phase retrieval methods such as the ptychographical iterative engine (PIE) algorithm and the transport of intensity equation (TIE) algorithm [13–15]. Both PIE and TIE require multiple operations and their corresponding system setups are relatively complex. A single-shot stable non-interferometric phase retrieval method combining embedded phase data and iterative Fourier transform algorithm is proposed to realize accurate and quick phase retrieval in the holographic data storage system [16]. This method based on some traditional iterative Fourier transform algorithms proposed by Fienup [17–19]. Only intensity distribution of the reconstructed beam on the Fourier domain is required to retrieve phase so that the system is very simple. In the non-interferometric phase retrieval method, embedded data is very important which works as a strong constraint in the object domain. Higher proportion of embedded data is more benefit for fast and accurate phase retrieval, however, its meanwhile occupy positions of code to decrease the code rate. In the previous work, off-axis system was used where 50% embedded data have to be in the signal part to provide strong enough constraint for quick phase retrieval. In this paper, we proposed a collinear system transferring the 50% embedded data from signal beam to reference beam. The code rate is increased by 2 times compared with previous off-axis non-interferometric phase retrieval system because positions of code in the signal part is released. Because the reference beam should be always known both in the recording process and reading process, there is no extra material cost for saving embedded data. Besides, we can enhance algorithm constraint to shorten iterative numbers by increasing the weight of reference beam.
2. Theory and method
The simple illustration of the collinear non-interferometric phase retrieval system is shown in Fig. 1. Usually, in the collinear system, the signal and reference beam are put in the center and around respectively. But for a clearer explanation of relationship between the signal and reference beam, we set the signal and reference as same size matrix. In the SLM (spatial light modulator) which is phase-only one used to upload phase pattern, we put the signal beam at the left and reference beam at the right. We refer to the reference beam as embedded data. Both reference beam and signal beam are modulated by same phase-level. In the writing process, both reference beam and signal beam are uploaded on the SLM and they interfere with each other in the media to make a hologram. In the reading process, only reference beam is uploaded on the SLM to illuminate the hologram. Then a reconstruction beam will be diffracted and imaged clearly in the back focal plane of L2. Here we should use an aperture to promise both reference part and signal part passing through. Not only that, an attenuator should be used meanwhile to decay the intensity of reference part relatively because the intensity of reference part is much higher than that of signal part in the reconstruction beam. Finally, we capture the Fourier transform intensity of the reconstruction beam by CMOS (complementary metal oxide semiconductor). After that, we can use iterative Fourier transform algorithm to retrieve phase.
Fig. 1. The illustration of the collinear non-interferometric phase retrieval system. L1, L2 and L3 are lens.
Compared with off-axis system, collinear system allows reference part into the CMOS. Therefore, we can change the amplitude weight of reference part to change the spectrum distribution on the CMOS to further shorten iterative numbers of phase retrieval. The iterative Fourier transform algorithm in the collinear system is as follows. In the following formulas, items with bold mean two-dimensional matrix and items without bold mean constant. At the beginning, we set an initial guess complex amplitude distribution ${{\boldsymbol{U}}_{\boldsymbol{k}}}$ in the object domain as shown in Eq. (1),
We can get a complex amplitude distribution ${{\boldsymbol V}_{\boldsymbol k}}$ on the Fourier domain after Fourier transform as shown in Eq. (2),
Then we use the root square of intensity $\sqrt {\boldsymbol I} $ captured by the CMOS to replace the amplitude ${{\boldsymbol A}_{\boldsymbol k}}$, thus a new distribution ${\boldsymbol V}_{\boldsymbol k}^{\prime}$ can be obtained as shown in Eq. (3),
Next we should do the inverse Fourier transform and get a new complex amplitude distribution ${\boldsymbol U}_{\boldsymbol k}^{\prime}$ in the object domain as shown in Eq. (4),
In the object domain, we have some constraint conditions that the amplitude of signal beam and reference beam are 1 and ${A_w}$ respectively and the phase distribution of reference beam is ${{\boldsymbol \varphi }_{{\boldsymbol{ref}}}}$. Therefore, we can modify ${\boldsymbol U}_{\boldsymbol k}^{\prime}$ to ${\boldsymbol U}_{\boldsymbol k}^{\prime\prime}$ as shown in Eq. (5),
We can compute the intensity error rate ${E_k}$ as shown in Eq. (8) and the difference $\Delta E$ between two adjacent intensity error rates as shown in Eq. (9),
We set a threshold ${\varepsilon }$ as a stop criteria if $\Delta E$ falls below ${\varepsilon }$. In the simulation and experiment, we set ${\varepsilon }$=10−3. When the iteration stops, we believe ${\boldsymbol U}_{\boldsymbol k}^{\prime\prime}$ at present is phase retrieval result what we want.
3. Simulation and experimental results
In the simulation, we used parameters same to the experiment. We used a random distributed 32×16 phase data with 4-level phase 0, π/2, π, 3π/2 as the signal and used another random distribution as the reference. First, we make the intensity of signal and reference same which means the amplitude weight of reference ${A_w}$ is 1 at this time. The phase pattern is shown in Fig. 2(a). Every data will be displayed by a block of 4×4 pixels on the SLM where the pixel pitch is 20µm. After Fourier transform, we can get the Fourier intensity of the phase pattern and only two Nyquist size frequency in the Fourier intensity, as shown in Fig. 2(b), will be kept to do phase retrieval. The aim of using only two Nyquist size frequency is to raise the anti-noise ability. In the phase retrieval algorithm, we set an initial guess distribution at the beginning as shown in Fig. 2(c). There are two parts at the initial guess, random 4-level phase distribution in the signal part and the embedded data in the reference part. In our collinear system, the embedded data is exactly the reference data. According to phase retrieval calculation process Eqs. (1)–(9), after 15 iterations, $\Delta E$ is below the threshold ${\varepsilon }$. Then we get the phase retrieval result as shown in Fig. 2(d) where the BER (bit-error-rate) is 0.
Fig. 2. (a) Original phase pattern, (b) Fourier intensity with two Nyquist size, (c) initial guess distribution, (d) phase retrieval result.
Next, we simulated phase retrieval results with different amplitude weights of reference ${A_w}$ from 1 to 10 which means the amplitude of reference to that of signal are from 1:1 to 10:1. When we didn’t consider noise effect, the BER of these situations are all 0. For comparing the differences of these situations, we used the intensity error rate and iteration numbers as shown in Fig. 3 and phase retrieval original data without correction as shown in Fig. 4 to describe them.
Fig. 3. (a) Intensity error rate and (b) iteration numbers with different amplitude weights of reference ${A_w}$ under the situation without noise.
Fig. 4. Phase retrieval original data without correction with different amplitude weights of reference under the situation without noise.
In the Fig. 3, with the increasing of the amplitude weights of reference, the intensity error rate is becoming lower and the iteration number is becoming fewer. Iteration number from 15 to 6 is shortened by 2.5 times when we increase the amplitude weight of reference. That makes sense that increasing the amplitude weight of reference means enhancing the constraint condition because the reference is the embedded data in the collinear system.
In the Fig. 4, the horizontal coordinates and the vertical coordinates of all pictures are phase value of retrieval results without correction and numbers respectively. Because we use 4-level phase, there are 4 peaks. With the increasing of the amplitude weights of reference, the height of 4 peaks have the tendency to increase first and then decrease.
In the real system, noise always exist. Therefore, we simulate those situations above under the noise. We choose to add noise on the Fourier intensity. The gray value of every Fourier intensity distribution is normalized to 0-255. We add random pepper noise [-5,5] to every pixel on the Fourier intensity, and if the gray value is smaller than 0 or larger than 255, it will be set to 0 or 255 respectively. We get the intensity error rate curves, iteration numbers and BER curves as shown in Fig. 5 and phase retrieval original data without correction as shown in Fig. 6.
Fig. 5. (a) Intensity error rate and (b) iteration numbers and BER with different amplitude weights of reference ${A_w}$ under the situation with noise.
Fig. 6. Phase retrieval original data without correction with different amplitude weights of reference under the situation with noise.
In the Fig. 5, we can see same trend that when the amplitude weights of reference are increasing, the intensity error rates are becoming lower and the iteration number is becoming fewer sharply first and almost unchanged then. Iteration number from 23 to 6 is shortened by almost 4 times when we increase the amplitude weight of reference. However, the BER of these situations are not all 0 anymore. When the amplitude weight of reference is larger than 5, the BER appears and increases sharply. From the Fig. 6, it also can be seen that when the amplitude weight of reference is larger than 5, 4 peaks of phase value disappear gradually.
We can explain the reason that the BER appear and become larger when the amplitude weights of reference become larger under the noise. The complex amplitude of 4 kinds of phase patterns (${A_w} = 1$, ${A_w} = 2$, ${A_w} = 5$, and reference only) and their corresponding Fourier intensity distributions are shown in Fig. 7. When ${A_w} = 1$, frequency is complex relatively because frequency is affected by the signal and reference both. When ${A_w} = 2$, it is still seen partial features affected by the signal. However, when ${A_w} = 5$, it is hardly to see the features affected by the signal because at this time the amplitude of reference is strong enough to dominate the frequency where the frequency is similar to that of reference only. Therefore, even if we don’t consider noise, the BER will appear when the amplitude of reference is large enough. When noise exist, the SNR (signal-to-noise ratio) of signal is becoming lower with the increasing of the amplitude of reference, causing BER appear ahead.
Fig. 7. The complex amplitude of 4 kinds of phase patterns and their corresponding Fourier intensity distributions. In the signal part of the upper images, pattern distributions represent the phase code and brightness overall represents the amplitude. Ref: reference, Sig: signal.
The experimental setup of the collinear system is shown in Fig. 8. A parallel light with round spot illuminates the aperture A1 which shape is two same rectangular windows. The left window is the signal part and the right window is the reference part. The size of the rectangular window is 2.56mm×1.28 mm. The aperture A1 and phase-only SLM1 are at the same plane by using a 4-f system to promise displaying phase pattern uploaded on the SLM exactly. In the recording process, two rectangular windows are open. In the reading process, we use a black screen to block the window of signal part. The phase-only SLM1 and the amplitude-only SLM2 are also at the same plane by using a 4-f system. We use the SLM2 as the attenuator to adjust the intensity of signal part and reference part in the reading process. The aperture A2 is used to block redundant diffraction orders and only keep the 0-order after the SLM2. Then we can get the reconstruction beam at the plane with dot line. Due to the reconstruction beam is phase distribution, we use a Fourier lens to make the CMOS capture the Fourier intensity of the reconstruction beam.
Fig. 8. Experimental setup. BE: beam expander, P: linear polarizer, A1 and A2: aperture, HWP: half wave plate, BS: beam splitter, SLM1: phase-only spatial light modulator, SLM2: amplitude-only spatial light modulator as an attenuator, L1-L6 are 150 mm lens, L7 is 300 mm lens.
The wavelength of laser is 532 nm and the power is 300 mW. The media is Irgacure 784-doped PMMA photopolymer which thickness is 1.5 mm [20]. The phase-only SLM is X10468-04 by HAMAMATSU where the resolutions are 792×600 and pixel pitch is 20µm. The amplitude-only SLM is TSLM07U-A by CAS MICROSTAR where the resolutions are 1920×1080 and pixel pitch is 8.5µm. The CMOS is DCC3260M by Thorlabs where the resolutions are 1936×1216 and pixel pitch is 5.86µm. The phase pattern includes reference part 32×16 phase data and signal part 32×16 phase data. Every phase data is displayed by a block of 4×4 pixels on the SLM1. The Fourier lens L7 is 300 mm focal length. According to these parameters, we can calculate two Nyquist size frequency captured by 682×682 pixels on the CMOS.
The amplitude-only SLM2 as the attenuator is important to control the intensity ratio of reference to signal. In the experiment, we calibrate the intensity value curve with uploaded gray values on the amplitude-only SLM2 as shown in Fig. 9. The maximum extinction ratio is about 400:1. Besides, we cannot promise the amplitude weight of reference to be ideal integer value. The average intensity ratio deviation is about 1%.
Experimental results are shown in Fig. 10. From the intensity error rate curve shown in Fig. 10(a), the results agree with the expectation. In the Fig. 10(b), the iteration numbers are shortened with increasing of the amplitude weight of reference. The BER values become lower obviously when ${A_w} = 2$ and ${A_w} = 3$. When ${A_w}$ is larger than 5, the BER values become higher and higher. The experimental results prove that increasing the amplitude weight of reference (embedded data) can indeed shorten the iteration numbers which is unique advantage of the collinear system. Besides, this method can not only shorten the iteration numbers, but also decrease the BER at some situations. Compared with traditional condition ${A_w} = 1$, when ${A_w} = 2$, the BER is decreased by about 4.5 times from 0.0176 to 0.0039 and the iteration number is decreased by about 2.4 times from 26 to 11, and when ${A_w} = 3$, the BER is decreased by about 3 times from 0.0176 to 0.0059 and the iteration number is decreased by about 3.7 times from 26 to 7.
Fig. 10. Experimental results. (a) Intensity error rate and (b) iteration numbers and BER with different amplitude weights of reference ${A_w}$.
4. Discussions
Traditional off-axis system using phase retrieval algorithm needs 50% embedded data. Because the intensity of reference and signal are same, to some extent, it realized a balance between reference beam and signal beam. In our simulation and experiment of collinear system, though the reference beam and signal beam with same intensity interfere with each other in the media, in the phase retrieval process, increasing the amplitude of reference breaks out this balance actually. We wonder whether our method still works when the reference number is fewer than signal number. The reference number and signal number are the pixel number in the reference part and signal part respectively. The meaning of decreasing the reference number is that data amount in one data page can be more because there will be more positions for the signal in a certain size range. We compared 4 kinds of phase patterns with different reference numbers (1, 1/2, 1/4 and 1/8 of the original reference) as shown in Fig. 11 in the simulation. We give 1, $\sqrt 2 $, 2, 2$\sqrt 2 $ initial amplitude weight of reference to these four situations shown in Figs. 11(a)-(d) respectively to promise same initial intensity between reference and signal. Then we can use uniform amplitude weight of reference ${A_w}$ from 1-10 to evaluate phase retrieval results.
Fig. 11. Phase patterns with different reference numbers (a) 1, (b) 1/2, (c) 1/4 and (d) 1/8 of the original reference). Ref: reference.
When we didn’t consider the noise, the BER of all situations are 0. The iteration number curves are shown in Fig. 12. There is a similar trend that the iteration numbers are decreased with the increasing of the amplitude of reference. When we add random pepper noise [-5,5] to every pixel on the Fourier intensity, the iteration number and BER curves are shown in Fig. 13. The trend of the iteration number curves is no change. The trend of BER agrees with the simulation result shown in Fig. 5(b). These results prove that our method can support decreasing the reference number and we can always find some amplitude weights of reference to shorten iteration numbers without BER.
Fig. 12. Iteration numbers with different amplitude weights of reference ${A_w}$ under the situation without noise.
Fig. 13. (a) Iteration numbers and (b) BER with different amplitude weights of reference ${A_w}$ under the situation with noise.
5. Summary
A collinear non-interferometric phase retrieval method for holographic data storage is proposed. The code rate of collinear system can be increased by 2 times due to transferring the 50% embedded data from signal beam to reference beam. Because the reference beam should be always known both in the recording process and reading process, there is no extra material cost for saving embedded data. We can shorten iteration numbers by increasing the amplitude weight of reference beam. By the simulation and experiment, we find better results when the amplitude weight of reference is 2 or 3. In the experiment, we decreased the iteration number and the BER by both about 3 times. We also find that the reference number can be reduced by using the collinear way and increasing the amplitude weight of reference. This law allows us to give more code positions to signal so that the data amount in one data page can be increased. In near future, we need to demonstrate these situations of fewer reference numbers in the experiment to observe whether there will be effect in the recording process. We also need to keep analyzing whether there is a better position between the amplitude weight of reference 2 and 3.
Funding
National Key Research and Development Program of China (2018YFA0701800); Wuhan National Laboratory for Optoelectronics (2019WNLOKF007).
Disclosures
The authors declare no conflicts of interest.
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