## Abstract

Digital super-resolution holographic data storage based on Hermitian symmetry is proposed to store digital data in a tiny area of a medium. In general, reducing a recording area with an aperture leads to the improvement in the storage capacity of holographic data storage. Conventional holographic data storage systems however have a limitation in reducing a recording area. This limitation is called a Nyquist size. Unlike the conventional systems, our proposed system can overcome the limitation with the help of a digital holographic technique and digital signal processing. Experimental result shows that the proposed system can record and retrieve a hologram in a smaller area than the Nyquist size on the basis of Hermitian symmetry.

© 2017 Optical Society of America

## 1. Introduction

Holographic data storage has been promising for a next-generation technology that realizes huge data capacity owing to multiplexing and page-wise recording [1, 2]. In a typical holographic storage system shown in Fig. 1, a spatial light modulator (SLM) generates a signal beam according to a data page. A signal beam is Fourier transformed by a lens and interferes with a reference beam, and a hologram is recorded in a recording medium. In general, to avoid unnecessary illumination on a medium and to improve a recording density, a recording area is restricted with a square aperture [2]. During retrieving process, a reference beam is incident on the recording medium under the same conditions as used during recording. As a result, a signal beam is reconstructed from a recording medium. The intensity distribution of a reconstructed signal beam is captured with an image sensor. A captured data page is decoded, and an original digital data can be retrieved. When subsequent data pages are recorded, the optical property of a reference beam [2–11] and/or the position of a recording medium [2, 12–17] are changed. This allows the multiplexing of holograms. A large number of holograms are superimposed on the same volume of a recording medium, which results in high recording density. To further increase the recording density, multilevel recording is an attractive approach. By encoding digital data onto gray levels of amplitude values [18] and/or phase values [19–21] of a beam, each pixel in a data page carries multilevel information.

In addition to the above recording techniques, reducing a recording area with a square aperture is also an effective approach for improving recording density. However, data error is unavoidable when a recording size is smaller than a Nyquist size [2]. The Nyquist size *w* is defined as

*f*,

*λ*, and

*d*denote the focal length of a lens, the wavelength of a light source, and the length of ON and OFF cells of a data page, respectively. A holographic data storage system is the same as a 4

*f*imaging system although there are a reference beam and a recording medium. This means that reducing an aperture size corresponds to low-pass filtering to a data page. The use of a small aperture leads to a blurred data page on an image sensor plane due to the lack of high spatial frequency components [22]. When the aperture size is smaller than the Nyquist size, the fundamental frequency component of a data page is lost, and data error is unavoidable. Conventional holographic data storage systems, therefore, have the limitation in reducing a recording area.

In this paper, we propose digital super-resolution holographic data storage based on Hermitian symmetry to overcome the limitation of the conventional holographic data storage systems. To overcome the limitation, the introduction of super-resolution techniques [23–25] might be useful because high-frequency components can be restored from a restricted spectrum. However, super-resolution techniques generally require to acquire multiple low-resolution images, to solve an ill-posed problem under some assumptions, or to decrease the field-of-view of an image. These requirements decrease the data transfer rate in holographic data storage. Instead of using the above super-resolution techniques, we make use of the symmetry property of the Fourier transform for overcoming the limitation in holographic data storage. When a signal beam is a two-dimensional real function, its Fourier spectrum has symmetry property that is called Hermitian symmetry [26]. Even though half the Fourier spectrum of a signal beam is removed with an aperture, the removed component can be restored on the basis of the symmetry property. Owing to the property, the proposed system is capable of recording and retrieving a hologram of a data page in a smaller area than the Nyquist size with the help of digital signal processing.

This paper is organized as follows. In Section 2, we describe the principle of digital super-resolution holographic data storage. An optical setup and the procedure of digital signal processing for the proposed method are presented. In Section 3, a restoration process based on the symmetry property of the Fourier transform is numerically investigated. In Section 4, we experimentally demonstrate the proposed system. A single data page is recorded and retrieved in a smaller area than the Nyquist size. Finally, we provide our conclusion in Section 5.

## 2. Digital super-resolution holographic data storage

Figure 2 shows the schematic of digital super-resolution holographic data storage based on Hermitian symmetry. During recording process, the amplitude distribution of a plane wave is modulated with an SLM according to a data page with binary amplitude values 0 and 1. Moreover, a random phase mask that has binary phase values 0 and *π* modulates the phase distribution of the modulated beam to suppress the dc component of a spectrum and to prevent unnecessary consumption of the dynamic range of a recording medium [2]. The resulting beam serves as a signal beam. The complex amplitude distribution of a signal beam *s*(*x*, *y*) is given by

*a*

_{page}(

*x*,

*y*) and

*ϕ*

_{rand}(

*x*,

*y*) denote a data page and a random phase mask, respectively. The signal beam consists of ternary real values, −1, 0, and 1, and is a two-dimensional real function. The generated signal beam is Fourier transformed by a lens and spatially filtered with an aperture. In conventional holographic data storage systems, a recording area is restricted by using an aperture with the size of

*cw*×

*cw*, where

*c*(≧ 1) is positive. As depicted in Fig. 1, increasing the value of

*c*improves the quality of the reconstructed image and decreases a recording density, and vice versa. In general,

*c*is set to values ranging from 1.1 to 1.3 to increase the tolerance of optics and to keep the recording area small [2]. In contrast to the conventional systems, we use an aperture with the size of

*cw*/2 ×

*cw*, as indicated in the inset of Fig. 2. This aperture function is given by, The aperture removes half the Fourier spectrum of a signal beam, and the filtered spectrum

*S*(

_{h}*u*,

*v*) is given by where

*S*(

*u*,

*v*) is the Fourier transform of a signal beam

*s*(

*x*,

*y*). Because the signal beam is a two-dimensional real function, the Fourier spectrum of a signal beam has symmetry property [26], where * denotes the complex conjugate. This property is known as Hermitian symmetry [26]. Owing to the symmetry property, the removed spectrum with the aperture can be restored by digital signal processing as described later. The filtered spectrum

*S*(

_{h}*u*,

*v*) is imaged on a recording medium through a 4

*f*imaging setup. In a recording medium, the filtered signal beam interferes with a reference beam, and thus a hologram is recorded.

During retrieving process, the hologram in the recording medium is illuminated by the reference beam, and thereby the filtered spectrum *S _{h}*(

*u*,

*v*) is reconstructed. The reconstructed beam is Fourier transformed again, and is incident on an image sensor. On the image sensor, the complex amplitude of the filtered signal beam

*s*(

_{h}*u*,

*v*) is the Fourier transform of

*S*(

_{h}*x*,

*y*):

*ℱ*[

*...*] and ⊗ denote the Fourier transform and the convolution operators, respectively. Note that in the following theoretical description we ignore the magnification and transposition of the coordinate due to the 4

*f*imaging system. Unlike conventional holographic data storage, the proposed system requires to detect the complex amplitude distribution of the reconstructed signal beam

*s*(

_{h}*x*,

*y*). An image sensor, however, is insensitive to the phase distribution of a beam. To detect the complex amplitude distribution, an additional reference beam is incident on an image sensor, and interferes with the reconstructed signal beam. As a result, an interference pattern, or a digital hologram, can be obtained. By using a Fourier fringe analysis [27] or phase-shifting techniques [28–30] makes it possible to detect the complex amplitude distribution of a filtered signal beam. To retrieve the original data page, the removed spectrum is restored from the detected complex amplitude

*s*(

_{h}*x*,

*y*) by following digital signal processing. By calculating the Fourier transform of the detected complex amplitude distribution, the right half of the Fourier spectrum, or

*S*(

*u*,

*v*)

*A*(

*u*,

*v*) is obtained. According to the symmetry property of Eq. (5), the left half of the Fourier spectrum that is removed with the aperture can be restored. By superimposing

*S*(

*u*,

*v*)

*A*(

*u*,

*v*) and its complex conjugate, a restored spectrum

*O*(

*u*,

*v*) is obtained:

*O*(

*u*,

*v*), the intensity of the restored signal beam is obtained as

*a*

_{page}(

*x*,

*y*) by thresholding of the intensity of Eq. (8).

If a random phase mask has multilevel phase values, a signal beam consists of not only real values but also imaginary values. In this case a Fourier spectrum does not satisfy Eq. (5), and it is impossible to implement the above mentioned restoration process.

## 3. Numerical simulation

In this Section, we carry out a numerical simulation to validate the restoration process based on the symmetry property of Eq. (5). Figure 3 shows a schematic of a numerical simulation. In this numerical simulation, we performed spatial frequency filtering to a signal beam. The effects of a recording medium and a reference beam are ignored. Figure 4 shows a signal beam and its Fourier spectrum in the numerical simulation. The complex amplitude distribution of the signal beam was defined by 128 × 128 pixels in an input plane. The amplitude and phase distributions of a signal beam correspond to a data page and a random phase mask, respectively. The data page is obtained by a 3:16 coding method [31]. In the coding method, a single symbol consists of three ON (bright) cells and thirteen OFF (dark) cells in 4 × 4 cells. The data page in Fig. 4(a) consists of 4 × 4 symbols. The random phase mask in Fig. 4(b) has binary phase values, 0 and *π*. The number of phase cells of *π* seem to be less than that of 0 in Fig. 4(b), and their possibility of occurrence do not seem to be the same. However, these numbers are almost the same in practice because there is no phase in the area where the amplitude value is 0. The signal beam therefore consists of ternary real values, −1, 0, and 1. By calculating the Fourier transform of the signal beam, the Fourier spectrum shown in Fig. 4(c) was obtained. Note that the signal beam was padded with zeros whose area consists of 1024 × 1024 pixels to numerically obtain the Fourier spectrum with high accuracy. The Fourier spectrum was spatially filtered with an aperture shown in Fig. 5(a). In this simulation, we set *c* = 1.3 for the aperture to keep the high tolerance of a practical holographic storage system. The aperture size is therefore 0.65*w* × 1.3*w*. This area is 0.845*w*^{2}, which is smaller than the Nyquist aperture *w*^{2}. With the aperture, high-frequency components and the left half of the Fourier spectrum were removed. Figure 5(b) shows the intensity distribution of the filtered signal beam in an output plane. It can be seen that the intensity distribution is blurred along the horizontal direction. The quality of intensity distribution was quantitatively evaluated by means of a symbol error rate (SER) which is defined as,

*E*

_{symbol}and

*N*

_{symbol}denote the number of error and the total symbols in a reconstructed image, respectively. The SER of Fig. 5(b) is 19%, and thus it is impossible to retrieve original data page. However, the original data page can be retrieved by the following restoration process based on the Hermitian symmetry.

Figure 5(c) shows the complex amplitude distribution of the filtered signal beam in the output plane. This complex amplitude distribution was numerically Fourier transformed, and thus the right half of the Fourier spectrum was obtained. As shown in Fig. 6(a), the left half spectrum was restored on the basis of Eq. (7). By calculating the inverse Fourier transform of the restored Fourier spectrum, complex amplitude distribution of a signal beam was obtained in the output plane. Figure 6(b) shows the intensity distribution of the restored signal beam. The SER of Fig. 6(b) is 0%. The restoration process completely suppressed the SER of 19% in Fig. 5(b).

In conventional holographic data storage systems, an aperture is generally centered at the optical axis as shown in Fig. 7(a). To compare SERs between the proposed and conventional systems, we calculated spatial frequency filtering with the aperture of Fig. 7(a). Figure 7(b) shows the intensity distribution of a filtered signal beam in the output plane. The SER is 25%, which is caused by the lack of fundamental frequency components of a data page. The numerical results suggests that the restoration process based on the symmetry property is effective for a signal beam in holographic data storage. The conventional holographic data storage systems have recorded the whole Fourier spectrum of a signal beam. However, recording half the Fourier spectrum is sufficient to retrieve an original data page. The proposed system therefore allows us to record and retrieve a data page using a smaller aperture than the Nyquist size. In the next Section, we investigate the feasibility of the proposed method by experimentally recording a hologram in a recording medium.

## 4. Experimental demonstration

In this Section, we aim to show that the proposed system can overcome the limitation of the Nyquist size. As described in Section 2, the proposed system requires to generate a signal beam with ternary real values, −1, 0, and 1. This can be achieved by using a single amplitude-only SLM and a binary random phase mask, as used in traditional holographic data storage systems [2, 6, 15]. However, the misalignment between an SLM and a binary random phase mask leads to the degradation in the quality of a signal beam. Alternatively, we use a phase hologram technique [32–34] for the sake of simplification of an experimental setup. The phase hologram technique allows the generation of the ternary real values using a single SLM. This makes an experimental setup simple and compact. Moreover, there is no degradation due to the misalignment between an SLM and a binary random phase mask. In addition to the modulation technique for a signal beam, it is necessary to detect the complex amplitude distribution of a retrieved signal beam for the restoration process. To achieve this requirement, we introduce an interferometer and an additional reference beam.

Figure 8 shows an experimental setup for demonstrating the proposed method. For recording a hologram in a recording medium, we applied coaxial holographic data recording [14, 15, 35]. In the coaxial holographic data recording, a signal beam and a ring-shaped reference beam are generated from a single SLM and propagate along a common optical path, and thus optical setup is simple and compact. Moreover, the optical setup is stable against vibration. These advantages and the introduction of the phase hologram technique allow us to demonstrate our idea in a simple and compact optical setup.

As a proof-of-principle experiment, we recorded and retrieved a single data page indicated in Fig. 4. Figure 8 shows an experimental setup for the demonstration. A laser diode with a wavelength of 532 nm was used as a light source. A collimated beam was obtained through a spatial filter and a lens. The phase distribution of the collimated beam was modulated with a phase-only SLM (X10468-01, Hamamatsu Photonics K. K.) which has 792 × 600 pixels with a pixel pitch of 20 *μ*m. The SLM displayed phase holograms that consists of an encoded phase pattern [21, 34] and a computer-generated reference pattern (CGRP) [10, 35], shown in Figs. 9(a) and 9(b), for generating signal and reference beams, respectively. While describing the experimental procedure, we briefly explain the design method and the function of the encoded phase pattern and the CGRP. The encoded phase pattern is used to generate a signal beam with ternary real values using a single phase-only SLM. For generating the signal beam *s*(*x*, *y*) = *a*_{page}(*x*, *y*) exp{*iϕ*_{rand}(*x*, *y*)}, an encoded phase pattern *ψ*(*x*, *y*) is designed on the basis of a phase hologram technique [32–34]:

*ϕ*

_{linear}(

*x*,

*y*) is linear phase distribution with spatial carrier frequencies

*f*

_{x1}and

*f*

_{y1}: The linear phase is introduced to extract a desired spectrum from the undesired components in a Fourier plane as described later. Figure 9(a) shows the phase hologram

*ψ*(

*x*,

*y*) for generating the signal beam shown in Fig. 4. When a phase-modulated beam exp{

*iψ*(

*x*,

*y*)} is Fourier transformed by a lens, desired and undesired spectra are separately obtained as shown in Fig. 10(a). Owing to the linear phase

*ϕ*

_{linear}(

*x*,

*y*), the spectrum of the desired signal beam

*s*(

*x*,

*y*) is centered at (

*f*

_{x1},

*f*

_{y1}) in the spatial frequency domain. It can be seen that the spectrum in high-frequency region is consistent with the numerical one shown in Fig. 4(c). In contrast, the low-frequency component centered at the origin is unnecessary. By extracting only the high-frequency component with an aperture and Fourier transforming with a lens, an optical field

*s′*(

*x*,

*y*) is obtained on the conjugate plane of the SLM:

*ϕ*

_{linear}(

*x*,

*y*), the desired signal beam

*s*(

*x*,

*y*) can be generated from a single phase-only SLM. In this experiment, the Fourier spectrum was actually filtered with an aperture indicted in the inset of Fig. 8. We used the aperture with

*c*= 1.3 for the sake of the consistency with the numerical simulation. The Nyquist size

*w*was approximately 222

*μ*m. Therefore the aperture size was approximately 144 × 288

*μ*m

^{2}. The aperture removed the left half of the spectrum as shown in Fig. 10(a). The filtered spectrum was Fourier transformed, and the filtered beam

*s′*(

_{h}*x*,

*y*) is given by

The filtered beam *s′ _{h}* (

*x*,

*y*) was Fourier transformed again, and was incident on a photopolymer material (Kyoeisha Chemical Co.) with a thickness of 400

*μ*m. At the same time, a reference beam from a CGRP was incident on the photopolymer material. Unlike the encoded phase pattern, the CGRP is designed by an optimized calculation, which allows us to control its Fourier power spectrum and improve light utilization efficiency [10, 35]. The detailed process for designing the CGRP is described in [35]. In this experiment, we designed the CGRP to make sure that the hologram of the signal beam is recorded within only the area of 0.65

*w*× 1.3

*w*. As shown in Fig. 10(b), the Fourier spectrum of a reference beam, therefore, is a rectangle that is the same as the aperture size. The spectrum was filtered with the aperture and imaged on the photopolymer material through a 4

*f*imaging system. As a result, the hologram between filtered signal and reference beams was recorded.

During retrieving process, a beam splitter divided a collimated beam into two beams. One was incident on the phase-only SLM to generate a reference beam for illuminating the hologram. The other is an additional reference beam for detecting the complex amplitude distribution of a retrieved signal beam. The reference beam generated from the phase-only SLM was incident on the photopolymer material, and thereby the filtered signal beam was reconstructed. The reconstructed signal beam was incident on a CCD camera which has 1280 × 960 pixels with a pixel pitch of 4.65 *μ*m. Figure 11(a) shows the intensity distribution of the retrieved signal beam obtained by the experiment. Similar to the numerical simulation, the retrieved data is blurred along the horizontal direction. The SER of Fig. 11(a) is 31%. As expected from the simulation result, data error is unavoidable without the restoration process. Note that the SER of the experimental result is higher than that of the simulation result. This difference is caused by the scattered beam and nonlinearity of a recording medium, the point spread function of the coaxial system [35], and the misalignment of the optics.

To restore the signal beam, the additional reference beam *p*(*x*, *y*) was incident on the CCD camera, and interfered with the reconstructed signal beam. The additional reference beam is given by

*f*

_{x2}and

*f*

_{y2}are spatial carrier frequencies. In the following, we describe the procedure of detecting complex amplitude using the Fourier fringe analysis [27]. The CCD camera captured the interference pattern, or a digital hologram:

*B*(

*x*,

*y*) = |

*s′*(

_{h}*x*,

*y*)|

^{2}+ |

*p*(

*x*,

*y*)|

^{2}is the zero order component. Substituting Eqs. (14) and (15) into Eq. (16) yields

*f*and

_{xs}*f*, the spectrum

_{ys}*S*(

_{h}*u*,

*v*) can be obtained. Figure 11(b) shows the intensity of the obtained spectrum

*S*(

_{h}*u*,

*v*). Note that the spectrum was not directly captured with a CCD camera. It was obtained by signal processing from the captured digital hologram. The left half is lost due to the spatial filtering with the aperture during recording. From the spectrum, the left half was restored on the basis of Eq. (7), which results in the restored spectrum shown in Fig. 11(d). By calculating the inverse Fourier transform of Fig. 11(d), the restored signal

*o*(

*x*,

*y*) can be obtained. Figure 11(c) shows the intensity distribution of

*o*(

*x*,

*y*). The SER of Fig. 11(c) is 0%, and thus the original data page can be retrieved without any error.

For comparison purposes, we subsequently evaluated the SER in conventional holographic data storage using the aperture with the size of 0.65*w* × 1.3*w*. For the conventional system, the aperture was set on the center of the Fourier spectrum of a signal beam. The phase-only SLM displayed only the encoded phase pattern for generating a signal beam alone. In this experiment, the recording medium was not used. Spatial frequency filtering was carried out in the experimental setup shown in Fig. 8. Figures 11(e) shows the intensity distribution of a low-pass filtered signal beam. The intensity distribution is horizontally blurred, and it is difficult to resolve some ON cells. To confirm the Fourier spectrum of the low-pass filtered signal beam, we replace a recording medium with a CCD camera. Figure 11(f) shows the intensity distribution of the Fourier spectrum. The horizontal component is lost due to the aperture. The SER of Fig. 11(e) is 19%. Unlike the experimental results in Figs. 11(a) and 11(c), there is no scattering and nonlinearity of a recording medium in the experimental result of Fig. 11(e). However, the data error is unavoidable due to the lack of the fundamental frequency components of a data page with the aperture. Note that the SER of the experimental result is larger than that of the simulation result shown in Fig. 7(b). This is caused by the misalignment of optical components and discrepancy of the aperture size.

The above experimental results show the feasibility of the proposed method. Although the recording area of the proposed system is the same as that of the conventional system, the proposed method can reconstruct a data page with twice the horizontal resolution of the conventional system by signal processing based on the Hermitian symmetry. It is noteworthy that the proposed method can record a hologram in a smaller area than the Nyquist size overcoming the limitation in the conventional system.

## 5. Conclusion

We have proposed and demonstrated digital super-resolution holographic data storage based on Hermitian symmetry. To the best of our knowledge, the recording of a hologram in a tiny area beyond the Nyquist size has not been discussed before due to the limitation of the Nyquist size. In this paper, we overcome the limitation of conventional holographic data storage for the first time. In our proposed system, half the Fourier spectrum of a signal beam is recorded in a medium, and the other half is restored with the digital signal processing on the basis of Hermitian symmetry. Actually the restoration process in the proposed system is not a super-resolution technique because it is based on the symmetry property of the Fourier transform. However, the proposed system can retrieve a data page with twice the horizontal resolution of the conventional system reducing a recording area. We therefore coin the proposed system digital super-resolution holographic data storage.

The digital super-resolution holographic data storage can achieve high areal density, as compared with conventional holographic data storage. However, the proposed method has also a limitation in reducing the recording area because the restoration process is based on the Hermitian symmetry. This means that it is impossible to use a smaller aperture than half the Nyquist aperture, or *w*/2 × *w*, in theory. Moreover, the restoration process is not applicable to a signal beam has multilevel phase values. To solve these problems, the introduction of super-resolution technique [23–25] might be useful. Additionally, it is important to adequately use a data coding method for roughly controlling the Fourier power spectrum of a signal beam [36,37] and to design the aperture shape for selectively recording the significant spectrum information of data pages [38,39].

Although we used the holographic data storage system with a phase hologram technique [21] for the proof-of-principle demonstration, the digital super-resolution holographic data storage can be realized in traditional holographic data storage systems using a single amplitude-only SLM and a binary random phase mask. The digital super-resolution holographic data storage might contribute to realize large storage capacity in holographic data storage.

## Funding

Japan Society for the Promotion of Science (JSPS) (15J11996).

## References and links

**1. **L. Dhar, K. Curtis, and T. Fäche, “Holographic data storage: Coming of age,” Nat. Photonics **2**, 403–405 (2008). [CrossRef]

**2. **K. Curtis, L. Dhar, A. Hill, W. Wilson, and M. Ayres, *Holographic Data Storage: From Theory to Practical Systems* (Wiley, 2010). [CrossRef]

**3. **C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. **85**, 171–176 (1991). [CrossRef]

**4. **G. A. Rakuljic, V. Leyva, and A. Yariv, “Optical data storage by using orthogonal wavelength-multiplexed volume holograms,” Opt. Lett. **17**(20), 1471–1473 (1992). [CrossRef] [PubMed]

**5. **C. C. Sun and W. C. Su, “Three-dimensional shifting selectivity of random phase encoding in volume holograms,” Appl. Opt. **40**(8), 1253–1260 (2001). [CrossRef]

**6. **N. Kinoshita, T. Muroi, N. Ishii, K. Kamijo, H. Kikuchi, N. Shimidzu, and O. Matoba, “Half-data-page insertion method for increasing recording density in angular multiplexing holographic memory,” Appl. Opt. **50**(16), 2361–2369 (2011). [CrossRef] [PubMed]

**7. **T. Ochiai, D. Barada, T. Fukuda, Y. Hayasaki, K. Kuroda, and T. Yatagai, “Angular multiplex recording of data pages by dual-channel polarization holography,” Opt. Lett. **38**(5), 748–750 (2013). [CrossRef] [PubMed]

**8. **L. Cao, J. Liu, J. Li, Q. He, and G. Jin, “Orthogonal reference pattern multiplexing for collinear holographic data storage,” Appl. Opt. **53**(1), 1–8 (2014). [CrossRef] [PubMed]

**9. **S. H. Lin, S. L. Cho, S. F. Chou, J. H. Lin, C. M. Lin, S. Chi, and K. Y. Hsu, “Volume polarization holographic recording in thick photopolymer for optical memory,” Opt. Express **22**(12), 14944–14957 (2014). [CrossRef] [PubMed]

**10. **T. Nobukawa, Y. Wani, and T. Nomura, “Multiplexed recording with uncorrelated computer-generated reference patterns in coaxial holographic data storage,” Opt. Lett. **40**(10), 2161–2164 (2015). [CrossRef] [PubMed]

**11. **T. Nobukawa and T. Nomura, “Multilayer recording holographic data storage using a varifocal lens generated with a kinoform,” Opt. Lett. **40**(23), 5419–5422 (2015). [CrossRef] [PubMed]

**12. **D. Psaltis, M. Levene, A. Pu, G. Barbastathis, and K. Curtis, “Holographic storage using shift multiplexing,” Opt. Lett. **20**(7), 782–784 (1995). [CrossRef] [PubMed]

**13. **K. Anderson and K. Curtis, “Polytopic multiplexing,” Opt. Lett. **29**(12), 1402–1404 (2004). [CrossRef] [PubMed]

**14. **H. Horimai, X. Tan, and J. Li, “Collinear holography,” Opt. Lett. **44**(13), 2575–2579 (2005).

**15. **K. Tanaka, M. Hara, K. Tokuyama, K. Hirooka, K. Ishioka, A. Fukumoto, and K. Watanabe, “Improved performance in coaxial holographic data recording,” Opt. Express **15**(24), 16196–16209 (2007). [CrossRef] [PubMed]

**16. **S. Yoshida, H. Kurata, S. Ozawa, K. Okubo, S. Horiuchi, Z. Ushiyama, M. Yamamoto, S. Koga, and A. Tanaka, “High-density holographic data storage using three-dimensional shift multiplexing with spherical reference wave,” Jpn. J. Appl. Phys. **52**, 09LD07 (2013). [CrossRef]

**17. **M. Sawada, N. Kinoshita, T. Muroi, M. Motohashi, and N. Saito, “Rotation spacing and multiplexing number in angle-peristrophic multiplexing holographic memory,” Jpn. J. Appl. Phys. **54**, 09MA03 (2015). [CrossRef]

**18. **G. W. Burr, G. Barking, H. Coufal, J. A. Hoffnagle, M. Jefferson, and M. A. Neifeld, “Gray-scale data pages for digital holographic data storage,” Opt. Lett. **23**(15), 1218–1220 (1998). [CrossRef]

**19. **G. Berger, M. Dietz, and C. Denz, “Hybrid multinary modulation codes for page-oriented holographic data storage,” J. Opt. A: Pure Appl. Opt. **10**, 115305 (2008). [CrossRef]

**20. **M. Bunsen, S. Umetsu, M. Takabayashi, and A. Okamoto, “Method of phase and amplitude modulation/demodulation using datapages with embedded phase-shift for holographic data storage,” Jpn. J. Appl. Phys. **52**, 09LD04 (2013). [CrossRef]

**21. **T. Nobukawa and T. Nomura, “Multilevel recording of complex amplitude data pages in a holographic data storage system using digital holography,” Opt. Express **24**(18), 21001–21011 (2016). [CrossRef] [PubMed]

**22. **M.-P. Bernal, G. W. Burr, H. Coufal, and M. Quintanilla, “Balancing interpixel cross talk and detector noise to optimize areal density in holographic storage systems,” Appl. Opt. **37**(23), 5377–5385 (1998). [CrossRef]

**23. **C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. **81**(17), 3143 (2002). [CrossRef]

**24. **S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. **20**(3), 21–36 (2003). [CrossRef]

**25. **J. P. Wilde, J. W. Goodman, Y. C. Eldar, and Y. Takashima, “Coherent superresolution imaging via grating-based illumination,” Appl. Opt. **56**(1), A79–A88 (2017). [CrossRef]

**26. **R. B. Bracewell, *The Fourier Transform and Its Applications*, 2nd ed. (McGraw-Hill, 1986).

**27. **M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

**28. **L. Martínez-León, M. Araiza-E, B. Javidi, P. Andrés, V. Climent, J. Lancis, and E. Tajahuerce, “Single-shot digital holography by use of the fractional Talbot effect,” Opt. Express **17**(15), 12900–12909 (2009). [CrossRef] [PubMed]

**29. **T. Tahara, K. Ito, M. Fujii, T. Kakue, Y. Shimozato, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Experimental demonstration of parallel two-step phase-shifting digital holography,” Opt. Express **18**(18), 18975–18980 (2010). [CrossRef] [PubMed]

**30. **M. Imbe and T. Nomura, “Selective calculation for the improvement of reconstructed images in single-exposure generalized phase-shifting digital holography,” Opt. Eng. **53**(4), 044102 (2014). [CrossRef]

**31. **B. M. King and M. A. Neifeld, “Sparse modulation coding for increased capacity in volume holographic storage,” Appl. Opt. **39**(35), 6681–6688 (2000). [CrossRef]

**32. **J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**(23), 5004–5013 (1999). [CrossRef]

**33. **V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A **24**(11), 3500–3507 (2007). [CrossRef]

**34. **T. Nobukawa and T. Nomura, “Linear phase encoding for holographic data storage with a single phase-only spatial light modulator,” Appl. Opt. **55**(10), 2565–2573 (2016). [CrossRef] [PubMed]

**35. **T. Nobukawa and T. Nomura, “Design of high-resolution and multilevel reference pattern for improvement of both light utilization efficiency and signal-to-noise ratio in coaxial holographic data storage,” Appl. Opt. **53**(17), 3773–3781 (2014). [CrossRef] [PubMed]

**36. **Y. Nakamura, H. Shimada, T. Ishii, H. Ishihara, M. Hosaka, and T. Hoshizawa, “High-density recording method with RLL coding for holographic memory system,” in *Nonlinear Optics*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMB5. [CrossRef]

**37. **S. Yoshida, Y. Takahata, S. Horiuchi, and M. Yamamoto, “Spatial run-length limited code for reduction of hologram size in holographic data storage,” Opt. Commun , **358**, 103–107 (2016). [CrossRef]

**38. **S. Yasuda, Y. Ogasawara, J. Minabe, K. Kawano, and K. Hayashi, “Homodyne readout on dc-removed coaxial holographic data storage,” Appl. Opt. **48**(36), 6851–6861 (2009). [CrossRef] [PubMed]

**39. **H. Gu, S. Yin, Q. Tan, L. Cao, Q. He, and G. Jin, “Improving signal-to-noise ratio by use of a cross-shaped aperture in the holographic data storage system,” Appl. Opt. **48**(32), 6234–6240 (2009). [CrossRef] [PubMed]