Fast non-interferometric iterative phase retrieval for holographic data storage

Xiao Lin; Yong Huang; Tsutomu Shimura; Ryushi Fujimura; Yoshito Tanaka; Masao Endo; Hajimu Nishimoto; Jinpeng Liu; Yang Li; Ying Liu; Xiaodi Tan

doi:10.1364/OE.25.030905

1. Introduction

Storage demand is rising rapidly in the era of big data. Holographic data storage has been widely known as one of most potential next generation storage techniques due to its large capacity and high data transfer rate [1–3]. Nevertheless, the capacity of current amplitude modulation holographic data storage system is becoming limited to meet the increased storage demand [4,5].

To increase the storage capacity, phase encoding is proposed which has larger capacity than current amplitude encoding. Phase encoding has several advantages. First, it is easy to implement by using a phase-modulated spatial light modulator (SLM). Second, it has a much larger code rate of than that of amplitude code [6,7]. Third, phase modulation can improve reconstructed data signal to noise ratio due to the homogeneous recording beam intensity distribution in the material.

One challenging aspect of phase encoding is phase retrieval as phase information cannot be detected directly by detectors such as CCD or CMOS. Usually, interferometry method is used to transform phase information to detectable intensity signal by CCD or CMOS [8]. However, interferometric results are vulnerable to environmental disturbance. Even tiny vibrations can cause undesirable side effects. What’s more, there are phase ambiguity issues when two phases are of the same difference relative to the reference phase as they will generate same interferometric intensity. Phase shifting method was proposed to resolve the phase ambiguity issue. Usually, at least two-step phase shifting should be used to get the accurate phase information [9–11]. However, multiple operations will reduce data transfer rate and increase the error possibility.

Fast non-interferometric phase retrieval is thus very important for phase-encoded holographic data storage and other phase based applications due to its advantage of easy implementation, simple system setup and robust noise tolerance.

There are several non-interferometry phase retrieval methods such as the ptychographical iterative engine (PIE) algorithm, the transport of intensity equation (TIE) algorithm, iterative Fourier transform (IFT) algorithm and so on [12–16]. Among them both PIE and TIE require multiple operations and their corresponding system setups are relatively complex.

IFT algorithm proposed by R.W. Gerchberg and W. O. Saxton in 1970s and modified by James R. Fienup in 1980s is an iterative single operation method [16–18]. The IFT algorithm and its derivative algorithms have been used in many fields such as digital encryption, image resolution enhancement, wavefront shaping and so on [19–22]. Usually, hundreds of iterations are required for traditional IFT algorithms to reconstruct relative faithful phase information, which casts a heavy burden on computation time and will reduce the data transfer rate eventually. Under the circumstances where prior phase information can be utilized, convergence can be speeded up. Availability of prior information is usually application dependent. In holographic data storage, certain positions of the data page are used as the mark or calibration points. Their phase values are fixed and known. We call this type of data as embedded data. Portion of embedded data can be controlled during the encoding process.

Here we present an iterative non-interferometric phase retrieval method for 4-level phase encoded holographic data storage. The method is based on IFT algorithm with known portion of the encoded phase data as prior information. These known prior phase data are used to realize accurate and quick phase retrieval.

Only one single intensity image of the Fourier transform of reconstructed beam is captured by CCD as the input for the phase retrieval algorithm. Since beam intensity at the Fourier plane of reconstructed beam is more concentrated than reconstructed beam itself, the requirement of diffractive efficiency of the recording media is reduced. Therefore, both dynamic range of recording media and data storage density can be increased with our method.

We evaluated our method by recording a test image in the media based on 4-level phase encoding and then reconstructed the test image. Experimental results show that we can achieve less than 5% bit-error-rate after only 10 iterations, which is very fast and accurate.

2. Theory and method

Illustration of non-interferometric system for phase retrieval is shown in Fig. 1. Original phase data page with certain portion of embedded data is uploaded on the SLM. Original phase data as the signal beam will interference with another reference beam and be recorded in the media in the recoding process. In the reading process, reconstructed beam is read out by the same reference beam. Rectangular with red dotted line in the Fig. 1 denotes the reconstructed beam which is at the front focal plane of the lens. The CCD is placed on the back focal plane of the lens, capturing the intensity distribution. After intensity distribution is captured, remaining phase retrieval processes are all performed in the computer. The aim is to retrieve the rest unknown phase data in the original phase data using captured intensity distribution and embedded data constraints in the IFT algorithm.

Fig. 1 The diagram of non-interferometric system for phase retrieval.

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At the beginning, an initial guess phase distribution $φ_{0}$ and assign $φ_{0}$ to $φ_{n}$ , where n = 1,2,… denotes iteration number. Then we get a complex amplitude distribution $U_{n}$ in the object domain as shown in Eq. (1),

U_{n} = \exp (i \cdot φ_{n})

We can get a complex amplitude distribution $V_{n}$ in the Fourier domain after Fourier transform as shown in Eq. (2),

V_{n} = F {U_{n}} = | A_{n} | \cdot \exp (i \cdot ϕ_{n})

where

F

denotes Fourier transform operator,

A_{n}

and

ϕ_{n}

denote amplitude and phase respectively.

Then we use the square root of captured intensity to replace current amplitude and we get a new distribution $V_{n}^{'}$ as shown in Eq. (3),

V_{n}^{'} = | \sqrt{I_{0}} | \cdot \exp (i \cdot ϕ_{n})

where

I_{0}

denotes the intensity distribution image captured by CCD.

After inverse Fourier transform, the complex amplitude distribution $U_{n}^{'}$ is got in the object domain as shown in Eq. (4),

U_{n}^{'} = F^{- 1} {V_{n}^{'}} = | A_{n}^{'} | \cdot \exp (i \cdot ϕ_{n}^{'})

where

A_{n}^{'}

and

ϕ_{n}^{'}

denote new amplitude and phase respectively.

Here, we use phase-only and embedded phase data as constraint conditions to correct the complex amplitude distribution. Therefore, amplitude is all normalized to 1. The phase of unknown position is kept and the phase of embedded position is replaced by embedded phase data. So we get a new distribution $U_{n}^{''}$ as shown in Eq. (5),

U_{n}^{''} = \exp (i \cdot ϕ_{n}^{''})

where

ϕ_{n}^{''}

is the new phase distribution. Then

ϕ_{n}^{''}

will be the new guess

φ_{n + 1}

.

Next we compute the new intensity distribution according to Eq. (6) and Eq. (7),

V_{n}^{''} = F {U_{n}^{''}}

I_{n} = V_{n}^{''} \cdot V_{n}^{'' *}

where

V_{n}^{'' *}

is the conjugation of

V_{n}^{''}

.

We compute the intensity error rate $E_{n}$ using Eq. (8), and the difference between two adjacent intensity error rate $Δ E$ using Eq. (9). We set a threshold ɛ as a stop criteria if $Δ E$ falls below ɛ. In the experiment, ε is usually set to the order of 10⁻⁴.

E_{n} = \frac{\sum (| I_{0} - I_{n} |)}{\sum I_{0}}

Δ E = E_{n} - E_{n - 1}

3. Results and discussion

We performed the simulation study first and then validated our method experimentally. The proportion of the embedded phase data is a crucial factor for the IFT algorithm. The higher the proportion of embedded phase data, the faster the phase retrieval convergence speed and the more accurate the phase retrieval result. However, the higher proportion of embedded phase data means lower code rate. The calculation formula of code rate (CR) is shown in Eq. (10).

C R = (1 - p) \cdot \log_{2} N

where p denotes the proportion of embedded data and N denotes the number of phase level.

Therefore, we need to balance the code rate and proportion of embedded data. We simulated several phase retrieval results with different proportions of embedded data from 30% to 70%, which is shown in Fig. 2. The input on the object domain is a random 4-level phase distribution with a size of 64 × 64 data. Because we need a block of 8 × 8 pixels to denote one phase data to guarantee full sampling the intensity distribution at the Fourier plane due to the limited physical size of CCD in the experiment, we used same parameters with an experiment in the simulation. The threshold ɛ is set to 5 × 10⁻⁴. After iterative process stopped, we calculated the number of iterations, phase errors and code rate for different embedded data shown in Table 1.

Fig. 2 Convergent curves with different proportions of embedded phase data with threshold ɛ of 5 × 10⁻⁴.

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Table 1. Simulation results for 4-level phase and 8-level phase encoding for different proportions of embedded data

View Table

Usually, it is desirable to use more phase levels to increase the system code rate. However, more phase levels will put a higher demand for the phase retrieval algorithm. In Table 1 we also compared the simulation results for 4 and 8 levels phase encoding respectively under different proportions of embedded data. From Table 1, we can find that although the code rate of 8-level phase is higher, its phase retrieval error is also higher than 4-level phase encoding if same iteration number and proportion of embedded data are used.

Due to the requirement of high data transfer rate, we believe that iteration number should be around 10. Phase error before error correction should be smaller than 0.01 in the simulation. At the same time, we should choose code rate type as high as possible. Therefore, we think 50% embedded data type is most suitable.

The dynamic range of CCD device is an important parameter in our method. Higher dynamic range can offer more accurate phase retrieval result. Figure 3 shows phase retrieval results using cameras with 8-bit, 10-bit and 12-bit dynamic ranges respectively in the simulation. We can see that the reconstructed phase becomes more accurate with increased dynamic range. Nevertheless, result of 8-bit CCD camera is already of high fidelity to original phase with very small amount of error that can be corrected using post-processing method. In our experiment, we used a CCD camera with 8-bit dynamic range to validate our ideas.

Fig. 3 Phase retrieval results using cameras with different dynamic ranges (8-bit, 10-bit, 12-bit) in the simulation.

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In addition, we also found out that different distribution patterns of embedded phase data also have an influence on phase retrieval result. Here we use 4-level phase encoding and 50% embedded phase data as an example. Two embedded phase patterns are studied: regular pattern shown in Fig. 4(a) and random pattern shown in Fig. 4(b), where color squares denote embedded phase data and black parts denote unknown phase data.

Fig. 4 Different embedded phase patterns and their intensity distribution in the Fourier plane. (a) Regular pattern. (b) Random pattern. (c) Intensity in the Fourier plane of regular pattern. (d) Intensity in the Fourier plane of random pattern.

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If embedded phase pattern is regular, its corresponding intensity distribution in the Fourier plane shown in Fig. 4(c) shows spike features. If embedded phase pattern is random relatively, its corresponding intensity distribution in the Fourier plane looks also random shown in Fig. 4(d). Obviously, the intensity distribution of the latter is more uniform. Usually, the phase distribution of unknown part is closer to random style even though the phase code should be ruled by the coding table. The intensity distribution of the unknown phase part is also uniform in the Fourier plane. Therefore, random embedded phase pattern owns higher efficiency of convolution with unknown phase data. In the simulation, random embedded phase pattern retrieved phase much more quickly and accurately. 12 iterations are needed when we use random embedded phase pattern and the phase error rate is 0.37%. 20 iterations are needed when we use regular embedded phase pattern and the phase error is 35.6%. So in the experiment we first generate an embedded checkerboard phase pattern with totally random phase distribution which can achieve random intensity distribution at the Fourier plane. The advantage of using checkerboard pattern is that we don’t have to record the embedded data positions during encoding and decoding process. We can directly know embedded data positions and unknown data positions by the pattern itself. Then we will use this same phase pattern for all the data pages.

We set up an off-axis holographic data storage system shown in Fig. 5 for the experiment. In the writing process, the shutter was open. Reference beam and signal beam interfered with each other and were recorded in the media. In the reading process, the shutter was close. Only reference beam illuminates the media and reconstructed beam was read out.

Fig. 5 The diagram of optical setup. SF: spatial filter, HWP: half wave plate, BS: beam splitter, $L_{1} ~ L_{6}$ : lens ( $L_{1} = 300 mm$ , $L_{2} ~ L_{5} = 150 mm$ , $L_{6} = 75 mm$ ), SLM: spatial light modulator. The media is Irgacure 784-doped PMMA photopolymer which thickness is 1.5mm.

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The intensity distribution of Fourier transform of the reconstructed beam was captured by the CCD with exposure time set at 1ms. The wavelength of the laser is 532nm and its power is 200mw. The SLM is Holoeye-PLUTO-VIS with a pixel pitch of 8μm and resolution of 1920 × 1080 pixels. The CCD is Sony-ICX445 and pixel pitch of CCD is 3.75μm. The media is Irgacure 784-doped PMMA photopolymer reported by our group [23]. Diffraction efficiency of this material will increase with increased exposure time before reaching a maximum of 50%. The size of media is 40mm × 32mm and the thickness of media is 1.5mm. The exposure time of a single record was 30 seconds.

If we image the reconstructed beam at the object plane, 7 minutes exposure time was required to get a clear image. Since beam intensity at the Fourier plane is more concentrated to the low frequency region than the reconstruction beam itself, detection at the Fourier plane is of high SNR and requires less reconstructed beam intensity. This can effectively reduce the storage system requirement for recording material diffraction efficiency, which means less exposure time, faster recording time, potential more numbers of multiplexing and larger capacity for the holographic data storage system. One thing needs to be pointed out is that imaging at the Fourier plane requires higher dynamic range of the CCD camera, which can be satisfied in the future if necessary.

Our aim is to record and reconstruct a gray-scale test image shown in Fig. 6 using phase-modulated holographic data storage system. Figure 7 shows the process of encoding gray image to phase patterns. First, we divide the gray image to multiple blocks. We used 4-level phase as code state and 2 × 2 phase pixels to encode one gray value. We make 50% random embedded phase data whose arrangement is checkerboard style. Therefore, every block is encoded to one 64 × 64 phase data distribution based on the coding table combining half embedded phase data and half encoded phase data. In the experiment, we used a block of 8 × 8 pixels to denote one phase data to guarantee enough frequency collection range at the Fourier plane due to the limited physical size of CCD. If a block of 4 × 4 pixels or 2 × 2 pixels to denote one phase data is required, shorter focal lens and large size CCD are needed. Finally, 512 × 512 pixels phase pattern loaded on the SLM is what we record in the media.

Fig. 6 Test image to be recorded with size of 128 × 128 pixels.

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Fig. 7 Illustration of the process of encoding gray image to phase patterns to be recorded. Image of 128 × 128 pixels is divided into sub-blocks of 32 × 16 pixels first. Then encoding phase data with 64 × 32 pixels are generated as 4 phase pixels are used to encode a single image gray value. A fixed known embedded phase data page with 64 × 32 pixels is combined with the encoding phase data to form the 64 × 64 phase pattern. Then every pixel of the 64 × 64 phase pattern is enlarged by a 8 × 8 block to form the final 512 × 512 phase pattern uploaded on the SLM. Combination rules of the encoding phase and embedded phase is illustrated in the dashed line rectangular. (Pixels with north-east direction texture represent embedded phase data positions and pixels with north-west direction texture represent encoding phase data positions.)

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Second, we captured the intensity distribution of Fourier transform of the reconstructed beam shown in Fig. 8. The size of the CCD we used is 900 × 900 pixels. Then we gave an initial guess phase pattern shown in Fig. 9. It is clear to see a checkerboard style embedded phase pattern and random unknown phase. Embedded data own uniform phase value in the block of 8 × 8 pixels and random unknown data own different random phase values in the block of 8 × 8 pixels.

Fig. 8 Intensity distribution at the Fourier plane of the reconstructed beam.

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Fig. 9 Initial guess of phase pattern.

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Third, we used IFT algorithm with embedded phase data to retrieve unknown phase. The result of resolved phase after just 10 iterations is shown in Fig. 10(b). Compared with original phase pattern shown in Fig. 10(a), we can see clearly a general faithful reconstruction and certain spots artifact laid on top of the image. Phase error distribution between original phase pattern and resolved phase pattern is shown in Fig. 10(c). The phase data error was calculated to be 4.7%, which can be corrected by using BCH error correcting code in the future [24, 25].

Fig. 10 Phase pattern and corresponding gray image: (a) Original phase and gray image, (b) Resolved phase and gray image after 10 iterations, (c) Phase error distribution between original phase pattern and resolved phase pattern.

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The retrieval results of the whole image are shown in Fig. 11. Original image, reconstructed image based on initial guess phase and resolved phase after 10 iterations are shown in Figs. 11(a)-(c) respectively. If we use error correcting code on the phase pattern, 4.7% phase error can be corrected and there will be 0 data error in the gray image after decoding.

Fig. 11 Image reconstruction comparison: (a) Original image, (b) Reconstructed image based on initial guess, (c) Reconstructed image based on resolved phase after 10 iterations.

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4. Summary

We proposed a single-shot fast non-interferometric iterative phase retrieval method using embedded as prior information in the holographic data storage system based on IFT algorithm. Only 10 iterations are needed which is fast enough to meet the requirement of data readout rate in the holographic data storage. In the simulation, we found that random distribution of embedded data helps to accelerate the phase retrieval process and to improve the accuracy. In the experiment, we demonstrated a faithful reconstruction of a test image using the method proposed above with only 4.7% phase error rate, which can be corrected with additional error correction code in the future. For the phase retrieval, not only zero-order but also high orders information of the Fourier transform of reconstruction beam were used. Future work on multiplexing of holograms should address interferences caused by high orders information of the Fourier transform of reconstruction beam. We believe our method cannot only make phase-modulated holographic data storage system more stable and simple, but also increase storage density.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61475019 and 61205053).

References and links

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Embedded data	4-level phase			8-level phase
Embedded data	Iterations	Phase error	Code rate	Iterations	Phase error	Code rate
70%	9	0.0009	0.6	10	0.0364	0.9
60%	11	0.0029	0.8	11	0.0505	1.2
50%	12	0.0037	1	14	0.0681	1.5
40%	17	0.0066	1.2	23	0.0947	1.8
30%	40	0.0195	1.4	44	0.1226	2.1

Fast non-interferometric iterative phase retrieval for holographic data storage

Abstract

1. Introduction

2. Theory and method

3. Results and discussion

4. Summary

Funding

References and links

Cited By

Figures (11)

Tables (1)

Equations (10)

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