Reduction of phase error on phase-only volume-holographic disc rotation with pre-processing by phase integral

Yeh-Wei Yu; Yuan-Cheng Chen; Kun-Hao Huang; Chih-Yuan Cheng; Tsung-Hsun Yang; Shiuan-Huei Lin; Ching-Cherng Sun; Ching-Cherng Sun; Ching-Cherng Sun

doi:10.1364/OE.399843

1. Introduction

Volume holographic storage (VHS) has been known one of the best ways to cold storage for archival information [1–7]. The advantages of VHS include large capacity, fast access rate, energy saving, long life and parallel processing [8,9]. In the general approach, the stored medium is in a disc shape [10–12]. The pickup head is used to read out the signal through diffraction from the volume hologram when the VHS disc is rotated or is displaced [13–16]. The stored or the readout signal are both two-dimensional images, which could contain amplitude and/or phase information [17–24]. Comparing with amplitude signal, phase-only signal takes the advantage of higher encoding capacity, more uniform focusing spot and higher diffraction efficiency [25–27]. Therefore, two-dimensional phase-only images are the major signals used in a high-capacity VHS. However, the retrieval of the phase signal is not as straight forward as that of amplitude signal, because a CMOS image sensor cannot distinguish the phase level of readout signal unless a reference light is used to form an interferogram with the phase signal [28–29]. The problem is that the phase of the reference light must be free of error, but this is not easy. A clever approach has been proposed to use a double-frequency-grating shearing interferometry (DFGSI) to retrieve the phase signal [30–31]. An obvious advantage is not only free of additional reference light, but also reduction of the slow-varying phase error induced in the readout process [32]. The shortage is that the resulting amplitude image is obtained from phase subtraction of two adjacent pixels along the shearing direction. The true signal retrieval of the phase signal needs pixel-by-pixel one-dimensional integral along the shearing direction. There is a possible risk when a phase error occurs at a certain pixel, and then the phase retrieval of the following pixels will be incorrect [33,34]. A robust mechanism to solve this problem not only in steady state but also in dynamic condition is highly demanded. In this paper, we first present the theoretical/experimental observation of an additional phase error when the VHS disc is rotated during the readout process, and then we propose a novel way to reduce the phase error in the readout process.

2. Additional phase error caused by displacement of VHS disc

The schematic diagram is shown in Fig. 1 [35]. The signal light is reflected from a spatial light modulator (SLM). The reference is a point source at the SLM plane, and becomes in a tilting plane wave to interfere with the signal light in the VHS disc. In the reading process, the original reference light serves as the reading light while the VHS disc is rotated. The diffracted light is directed to lens 2 and lens 3 and then is focused onto the DFGSI. The diffracted light from the DFGSI is imaged onto the CMOS image sensor and then will be retrieved to have the stored information.

Fig. 1. Schematic diagram of the phase retrieval system with DFGSI [35].

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The most important element in the system is the double-frequency-grating (DFG), which records two gratings, and the spatial frequency of one grating is slightly different from the other. The first-order diffraction light must be completely separated from the zero-order light. Thus the first-order diffraction lights from the two gratings will interfere with each other and form a shearing interferogram, which can be written

(1)$$\begin{aligned} &I({{\xi_3},{\eta_3}}) = \\ &{\left|\begin{array}{l} \textrm{exp}\left[ {i\varDelta {\phi_0} + \frac{{i\pi \varDelta z\textrm{cos}({\theta - {\theta_1}} )({\xi_3^2 + \eta_3^2} )}}{{\lambda f_4^2}} + \frac{{i2\pi \varDelta z\textrm{sin}({\theta - {\theta_1}} ){\xi_3}}}{{\lambda {f_4}}}} \right]{U_{im}}\left( {\frac{{{f_3}}}{{{f_4}}}[{{\xi_3} - {f_4}\textrm{sin}({{\theta_1} - \theta } )} ],\frac{{{f_3}}}{{{f_4}}}{\eta_3}} \right)\\ + \textrm{exp}\left[ {i\varDelta {\phi_x} + \frac{{i\pi \varDelta z\textrm{cos}({\theta - {\theta_2}} )({\xi_3^2 + \eta_3^2} )}}{{\lambda f_4^2}} + \frac{{i2\pi \varDelta z\textrm{sin}({\theta - {\theta_2}} ){\xi_3}}}{{\lambda {f_4}}}} \right]{U_{im}}\left( {\frac{{{f_3}}}{{{f_4}}}[{{\xi_3} - {f_4}\textrm{sin}({{\theta_2} - \theta } )} ],\frac{{{f_3}}}{{{f_4}}}{\eta_3}} \right) \end{array} \right|^2} \end{aligned}, $$

where, ξ₃ and η₃ are the lateral coordinate of the CMOS image sensor; z axis is the optical axis of lens₃; Δz is the shift of DFG along the z axis; θ is the angle from z axis to the optical axis of lens₄; θ₁ and θ₂ are the angles to the z axis of the first diffracted light and the second diffracted light, respectively; U_im is the readout image from the volume holographic storage system with wavefront aberration, located at the image plane; λ is the optical wavelength; f₃ and f₄ are the focal lengths of the lens₃ and lens₄, respectively; ΔΦ₀ is the phase difference between the two gratings. ΔΦ_x is the phase difference induced by lateral displacement of the DFG. In this setup, λ=532 nm, θ₁ - θ = 10.2°, θ₂ - θ = 10.25°, f₃=5 mm, f₄=10.5 mm, Δz=0, ΔΦ₀=0, and ΔΦ_x=π.

In the reading process, the design of the spatial frequency of the grating in DFGSI must coincide with the pixel pitch of the two-dimensional signal. As a result, the shearing interferometry will cause phase subtraction of the two adjacent pixels, and thus the image sensor can image the interferogram. For a (0, π) binary phase distribution signal, the result of phase subtraction must be 0 or π. Therefore, if the first column of the signal is known, we can integrate the phase difference along the shearing direction, to retrieve the original phase distribution from the interferogram, i.e., the decoded signal. The usage of DFGSI has several advantages, including removal of slow-varying phase error across the image plane, and free of additional reference light to retrieve the phase signal. However, the shortage is the final phase signal must be obtained by phase integral along the shearing direction for each raw of pixel. Once phase error occurs at a certain pixel, the decoded phase signal in the following pixels on the same raw will be incorrect, and cause breakdown in phase decoding.

In a VHS disc, owing to the strict Bragg condition, if the reference beam is not a plane wave, a slight position deviation in the reading process could cause serious Bragg mismatch. Therefore, an extremely short displacement range could be allowed for effective readout. Through theoretical calculation, we find that an additional phase error across the readout image is observed when the VHS disc is not at the exact original position. In order to analyze the additional phase error independently, and to avoid ambiguity caused by Bragg mismatch, we use a plane wave as the reference beam.

In the disc, the optical fields of the reference (A_R), signal (A_S) and reading light (A_P) are derived based on Fresnel diffraction

(2)$${A_{R = }}\frac{{{e^{jk({2{f_1} + \Delta {z_d}} )}}}}{{j\lambda {f_1}}}\textrm{F}\left\{ {{U_R}({x,y} )\textrm{exp}\left[ { - j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({{x^2} + {y^2}} )} \right]} \right\}, $$

(3)$${A_{S = }}\frac{{{e^{jk({2{f_1} + \Delta {z_d}} )}}}}{{j\lambda {f_1}}}\textrm{F}\left\{ {{U_R}({x,y} )\textrm{exp}\left[ { - j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({{x^2} + {y^2}} )} \right]} \right\}, $$

(4)$${A_{P = }}\frac{{{e^{jk({2{f_1} + \Delta {z_d}} )}}}}{{j\lambda {f_1}}}\textrm{F}\left\{ {{U_R}({x,y} )\textrm{exp}\left[ { - j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({{x^2} + {y^2}} )} \right]} \right\}, $$

where U_P, U_R and U_S is the reading, reference and signal lights, respectively, λ is the wave length, f₁ is the focal length of lens1, Δz_d is the distance deviated from the focal plane of Lens1, and $\textrm{F}$ is the Fourier transform operator. When the disc is shifted laterally, the recorded grating is convoluted with $\delta ({u - \Delta u,v - \Delta v} )$, where u and v are the lateral coordinates of the recording medium, Δu and Δv are the displacement of the recording medium. Therefore, the optical field diffracted from a certain layer at a specific depth is expressed

(5)$$\begin{aligned} &{U_z}({u - \Delta u,v - \Delta v,\Delta z} )\\ &= \frac{{{e^{jk({2f + \Delta z} )}}}}{{j{\lambda ^3}f_1^3}}\textrm{F}\left\{ {{U_P}({x,y} )\textrm{exp}\left[ { - j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({{x^2} + {y^2}} )} \right]} \right\}\delta ({u - \Delta u,v - \Delta v} )\\ &\otimes \left\{ {\textrm{F}{{\left\{ {{U_R}({x,y} )\textrm{exp}\left[ { - j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({{x^2} + {y^2}} )} \right]} \right\}}^\ast }\textrm{F}\left\{ {{U_S}({x,y} )\textrm{exp}\left[ { - j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({{x^2} + {y^2}} )} \right]} \right\}} \right\} \end{aligned},$$

The optical field from each depth is propagated to the image plane and is integrated along Δz_d. As a result, the diffracted light can be illustrated [15,36]

(6)$$\begin{aligned} &{U_{im}}({\xi ,\eta ,\Delta u,\Delta v} )= \\ &\frac{{exp ({jk4{f_1}} )}}{{{\lambda ^2}f_1^2}}\mathop \smallint \limits_{ - T}^T {\int\!\!\!\int \int\!\!\!\int }\left\{ {\begin{array}{c} {{U_P}({{\xi_2} - \xi ,{\eta_2} - \eta } )U_R^\ast ({{\xi_2} + {\xi_1},{\eta_2} + {\eta_1}} ){U_S}({{\xi_1},{\eta_1}} )}\\ {exp \left( {j\frac{{\pi \Delta {z_d}}}{{\lambda f_1^2}}({2{\xi_2}({\xi + {\xi_1}} )+ 2{\eta_2}({\eta + {\eta_1}} )} )} \right)}\\ {exp \left( { - j2\pi \left( {\frac{{\Delta u}}{{\lambda {f_1}}}{\xi_2},\frac{{\Delta v}}{{\lambda {f_1}}}{\eta_2}} \right)} \right)} \end{array}} \right\}d{\xi _2}d{\eta _2}d{\xi _1}d{\eta _1}\,d\Delta {z_d} \end{aligned}, $$

where T is the half thickness of the VHS disc, (ξ, η) is the coordinate on the image plane, (ξ₁, η₁) is the coordinate on the SLM, and (ξ₂, η₂) are the parameters induced by the convolution. In Eq. (6), the multiplication between U_p, U_R and U_S inside the integration indicates that U_im is a function of correlation of U_p and U_R and then convolution with U_S. Once the correlation between U_p and U_R is not a delta function, the reconstructed function U_im will be blurred. The first exponential term signifies the phase accumulation from different depths. The point in the image plane (ξ, η) = (-ξ₁, -η₁) is the Gaussian image point, and the first exponential term leads to a constructive interference. The second exponential term signifies the phase induced by disc shifting. It shows that an additional phase error will be induced when the VHS disc is not at the original location in the writing process. Such additional phase error is called phase error of disc displacement (simplified PEDD). That is to say, in the reading process of a VHS disc as shown in Fig. 1, an unexpected phase error occurs when the disc rotates. To simulate the optical system, we make U_P and U_R as point sources located at (2851 µm,0) in contrast to (0,0), the center of the SLM. The input signal U_s is generated from a frame with data size of 32400 bits. It is encoded by a LDPC (low density parity check) error correction code using additional 32400 parity bits, and then encoded by a “3/16” 2D sparse code. As a result, it is transformed to 6 pages of signal, and each page is composed of 19136 effective pixels. The diffraction image U_im at various VHS disc displacement is simulated using Eq. (6). Then, a random phase scattering noise is added. Its amplitude is in normal distribution when the average value and the standard deviation are both set 20% of the maximum diffraction signal. Then the diffracted phase-only signal interferes an additional plane wave to get an intensity distribution. Figure 2(a)-(c) show the simulation image from one of the six signal pages when the VHS disc is displaced 0 µm, 5 µm, and 10 µm. In the simulation, λ is 532 nm; f₁ is 4 mm; the refractive index of the VHS disc is 1.5; T is 1 mm; the pixel size of SLM is 5.76 µm; one effective pixel of the signal is composed of 4×4 SLM pixels, where the total SLM pixel number is 1080×1080. The straight fringes shown in Fig. 2 are caused by the PEDD when the VHS disc is rotated or lateral displaced. It shows the PEDD is a function of displacement, and could dramatically increase the bit error rate (BER). This finding is important to the system design of a VHS system when the recorded signal is phase only. Figure 2(d) - (f) show the experimental results of the interfering image on the image sensor, for the displacement 0 µm, 5 µm, and 10 µm. Although the image quality was degraded by the multi-reflection noise inside the optical system, the observed PEDD is similar to the simulation result.

Fig. 2. Simulation of the interferogram by the reconstructed phase signal and a plane wave, in different displacements of the VHS disc : (a) 0 µm, (b) 5µm, and (c) 10 µm. The corresponding experimental result for the displacements of the VHS disc are shown for displacement of (d) 0 µm, (e) 5µm, and (f) 10 µm. The bottom left pattern is an enlarged image inside the red square.

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Figure 3 shows the Schematic diagram of the experimental setup to demonstrate Eq. (6) and the PEDD effect. Figure 3(a) shows the writing process. The reference beam (in dark green) and the signal beam (in light green) was derived from the laser source (Verdi-G2 SLM, Coherent Inc.). The signal beam was phase modulated by the SLM (JD955B EDK, Jasper Display Corp., pixel number 1920 × 1080, pixel size 6.4 um), and then imaged to the image plane by the 4f system composed by lens 5 (focal length:200 mm) and lens 6 (focal length: 180 mm). The pixel size of the image of the SLM was reduced to 5.76 µm due to 0.9X magnification of the 4f system. The real image of the SLM at the image plane played the role of the SLM in Fig. 1. It was then focused to the VHS disc by lens 1 (M Plan APO HR 50X, Mitutoyo, f = 4 mm, NA = 0.75). The reference beam was focused to be a point source by lens 8, and was relayed to the image plane by lens 5 and lens 6. It was then transferred to a tilting plane wave by lens 1, to interfere with the signal beam inside the VHS disc. A PQ/PMMA plate inside the VHS disc with thickness of 1 mm, was used as the recording material to record the interference fringes [37,38]. Figure 3(b) shows the reading process. The reading beam (in dark green) and the interfering beam (in green dash line) were derived from the same laser source.

Fig. 3. The schematic diagram of the experimental setup for (a) writing process, and (b) reading process, where BS : beam splitter; PBS : polarization beam splitter; QWP : quarter wave plate.

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The reading beam was exactly the same as the reference beam, and was transferred to the VHS disc to generate a diffraction beam (in light green). The diffraction beam was reflected by the mirror behind the VHS disc, and was transform to an image at the image plane by lens 1. It was then relayed to the image sensor (UI-306xCP-M, IDS, pixel number 1936 × 1216, pixel size 5.86 µm) by lens 6 and lens 7 (focal length:200 mm). The interfering beam was the same as the signal beam, except the SLM modulated a blank image rather than an input signal. It was transferred to the image sensor to interfere with the phase-only diffracted signal. We shifted the VHS disc in the reading process, and found that the fringe number increased as the disc displacement increased. We calculated the phase gradient of the PEDD by counting interference fringe density. Figure 4 shows the comparison of phase gradient of the PEDD between the experiment and the simulation. Obviously the two curves are similar to each other. The slight difference is mainly caused by the noise of multi-reflected light and aberration of the optical system.

Fig. 4. Phase gradient of the PEDD. Red curve : the experimental result; blue curve : the simulation result.

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The PEDD could be partially eliminated by using DFGSI, because DFGSI can remove slow-varying phase variation through phase subtraction of the adjacent pixels [35]. Since 4×4 pixels form an effective pixel, we thus set θ-θ₁=10.2°, θ-θ₂=10.25° and f₄= 5 mm to enable image shearing with 4 SLM pixels. It thus makes sure that an effective pixel can interfere with its adjacent effective pixel. Figure 5 shows the simulation results when an appropriate DFGSI is applied to retrieve the phase-only signal. When Fig. 5(a) is used as the phase-only input signal of the holographic data storage system, we apply Eq. (6) to simulate U_im with different disc deviation. U_im is then inserted into Eq. (1) to calculate the DFGSI image on the image sensor shown in Fig. 5(b). In the decoding process, we first determine the threshold value by using the histogram of the image in Fig. 5(b), and then generate a (0, π) two-steps phase-difference distribution signal by comparing it to the threshold value. Then, we integrate the phase difference along the shearing direction, to get a decoded image, as shown in Fig. 5(c). When the displacement is large, the PEDD is too large to be eliminated. The simulation shows that once there is a phase error, the decoded signal will be incorrect through the following raw of pixels. A detailed simulation is shown in Fig. 6, where more error in the decoding process can be observed when the PEDD is getting larger with larger displacement. Because the larger displacement leads to a larger phase gradient of the PEDD. It thus degrades the ratio of the DFGSI image contrast over the scattering noise.

Fig. 5. Simulation result of the images in different system levels. (a) The original phase-only signal. (b) The readout signal through DFGSI of a VHS disc displaced 10 µm. (c) The decoded image through integral. The bottom right pattern is an enlarged pattern of the central image.

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Fig. 6. Simulation of the decoded phase signal for different displacement of the VHS disc with a DFGSI. (a) 0, (b) 5 µm, (c) 10 µm, and (d) 15 µm.

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3. Novel approach to reduce the phase error

Failure decoding with applying DFGSI is caused by a large PEDD in the readout process. The present of the PEDD could destroy the decoding process because integral through a raw of pixel is necessary. This is why the error in a certain pixel will cause decoding failure. It will be perfect if the decoding is free of the integral process but still keep the advantages using DFGSI. In order to do so, we propose a novel approach, where the stored signal is modified before storing in a way of pre-process of the signal. Figure 7 shows the principle of pre-process. For the original phase-only signal ${\emptyset _s}({{i_0},j} )$, the pre-process is to get the pre-integrated signal ∅︀_PI (i,j)

(7)$${\emptyset _{PI}}({i,j} )= \mathop \sum \nolimits_{{i_0} = 1}^{{i_0} = i - 1} {\emptyset _s}({{i_0},j} )+ j\pi , $$

where (i, j) is the pixel location. It integrates the original phase-only signal along the shearing direction. Besides, the initial phase is designed as crossly distributing between 0 and π along j direction, to eliminate the strong DC peak inside the VHS disc in the recording process. Figure 7(a) shows the original phase-only signal, and the modified phase-only signal is shown in Fig. 7(b). Because DFGSI will cause differential effect on the adjacent pixels, there is no need of additional process to obtain the decoding signal in the reading process. The major advantage is signal decoding is nothing with raw integral of the diffracted image. An error occurs at a certain pixel will have no effect on its neighbor pixel. We call this novel approach PI-DFGSI. When Fig. 7(b) is used as the phase-only input signal of the holographic data storage system, we apply Eq. (6) to simulate U_im with different disc deviations. U_im is then inserted to Eq. (1)) to calculate the interferogram on the image sensor. Figure 8 shows the simulation with the condition the same as that in Fig. 2 and Fig. 6. Obviously, the decoded signal is more accurate.

Fig. 7. Simulation result of the images in different system levels. (a) The original phase-only signal. (b) The modified phase-only signal. (c) The decoded image with using DFGSI.

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Fig. 8. Simulation of the decoded phase signal for different displacement of the VHS disc with use of PI-DFGSI. (a) 0, (b) 5 µm, (c) 10 µm, and (d) 15 µm. The bottom right pattern is an enlarged pattern of the central image.

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BER (bit error rate) is a useful factor to evaluate the system performance, and can be written

(8)$$BER = \frac{{{N_{err}}}}{{{N_t}}}, $$

where N_err is the total number of the bit error, and N_t is the total bit number. In the simulation, totally 6 encoded signal pages are used to obtain the simulation results in various displacements. In calculating BER of the raw data, the N_err is counted directly from the simulation image, and N_t is the total effective pixel number, i.e., 19136×6. A comparison of the calculated BER based on the raw data of diffracted signal with three different decoding methods is shown in Fig. 9. The blue line (w/o DFGSI) is to use a plane wave to form an interferogram as the decoded signal, the green line (w/ DFGSI) is to apply DFGSI and raw integral, and the red line (w/ PI-DFGSI) is to apply PI-DFGSI. Obviously, applying PI-DFGSI can effectively reduce the negative effect caused by the PEDD and take the advantages by using shearing interferometry in retrieval and decoding of the diffracted phase signal. The proposed approach will be useful in practical signal retrieval of a VHS disc in the readout process. The actual value of BER is shown in Table 1. The first part is the BER of the raw data, which has been plotted in the Fig. 9. The second part and the third part show that the BER can be further decreased by the process of “3/16” sparse code decoding and the LDPC decoding. It shows in the condition of PI-DFGSI after LDPC decoding, even the displacement is 15µm, there is no error bit be found from totally 32400 bits of the input data. It thus satisfies the practical requirement of a data storage system.

Fig. 9. A comparison of BER calculation for the raw data with three different decoding approaches. Note that the calculation of the BER doesn't consider the effect by the sparse code and the LDPC code.

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Table 1. The calculated BER though three processing schemes.

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4. Conclusion

In conclusion, we first present a study of theoretical observation of a displacement-dependent phase error across the diffracted signal when a VHS disc rotates during the reading process. An experiment is applied to demonstrate the PEDD effect and the optical model described in Eq. (6) as well. The PEDD makes negative effect to a phase-only signal and increases BER. Then we discuss use of double-frequency grating shearing interferometer and raw integral to decode the phase signal and the response to the PEDD. Although a DFGSI can eliminate slow-varying phase error, it is still not able to reduce BER when the PEDD occurs because the phase error will be accumulated through integrating pixel by pixel. Finally, based on the advantages and shortage of the DFGSI, we propose a novel approach, the PI-DFGSI, that is to modify the input signal by pre-integral of the phase for each pixel along the shearing direction. Then it is free of the integral process in the decoding process, but keeps the major advantages of the DFGSI approach. The simulation shows that the PI-DFGSI is an effective way to retrieve and decode a phase-only signal of a VHS disc, and performs advantages including free of additional reference light, removal of slow-varying phase error and reduction of the decoding error when the PEED occurs. The corresponding experimental measurement fits well the theoretical simulation and verify the validity of the theoretical finding of the phase error when the volume holographic disc rotates.

The proposed and demonstrated novel approach will be useful in reducing retrieved phase error and the BER as well from a phase-only volume holographic disc.

Funding

Ministry of Science and Technology, Taiwan (104-2221-E-008-073-MY3, 108-2221-E-008-084-MY3).

Disclosures

The authors declare no conﬂicts of interest.

References

1. J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume Holographic Storage and Retrieval of Digital Data,” Science 265(5173), 749–752 (1994). [CrossRef]

2. H. J. Coufal, D. Psaltis, and G. T. Sincerbox, eds., Holographic Data Storage (Springer-Verlag, 2000).

3. 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]

4. P. Liu, L. Wang, Y. Zhao, Z. Li, and X. Sun, “Holographic memory performances of titanocene dispersed poly (methyl methacrylate) photopolymer with different preparation conditions,” Opt. Mater. Express 8(6), 1441–1453 (2018). [CrossRef]

5. B. Shih, C. Chen, Y. Hsieh, T. Chung, and S. Lin, “Modeling the Diffraction Efficiency of Reflective-Type PQ-PMMA VBG Using Simplified Rate Equations,” IEEE Photonics J. 10(6), 1–7 (2018). [CrossRef]

6. P. Liu, X. Sun, Y. Zhao, and Z. Li, “Ultrafast volume holographic recording with exposure reciprocity matching for TI/PMMAs application,” Opt. Express 27(14), 19583–19595 (2019). [CrossRef]

7. Y. Liu, F. Fan, Y. Hong, A. Wu, J. Zang, G. Kang, X. Tan, and T. Shimura, “Volume holographic recording in Al nanoparticles dispersed phenanthrenequinone-doped poly(methyl methacrylate) photopolymer,” Nanotechnology 30(14), 145202 (2019). [CrossRef]

8. S. S. Orlov, W. Phillips, E. Bjornson, Y. Takashima, P. Sundaram, L. Hesselink, R. Okas, D. Kwan, and R. Snyder, “High-transfer-rate high-capacity holographic disk data-storage system,” Appl. Opt. 43(25), 4902–4914 (2004). [CrossRef]

9. L. Dhar, K. Curtis, M. Tackitt, M. Schilling, S. Campbell, W. Wilson, A. Hill, C. Boyd, N. Levinos, and A. Harris, “Holographic storage of multiple high-capacity digital data pages in thick photopolymer systems,” Opt. Lett. 23(21), 1710–1712 (1998). [CrossRef]

10. H. Horimai and X. Tan, “Holographic versatile disc system,” Proc. SPIE 5939, 593901 (2005). [CrossRef]

11. H. Horimai and X. Tan, “Collinear technology for a holographic versatile disk,” Appl. Opt. 45(5), 910–914 (2006). [CrossRef]

12. Y. W. Yu, C. Y. Cheng, T. C. Teng, C. H. Chen, S. H. Lin, B. R. Wu, C. C. Hsu, Y. J. Chen, X. H. Lee, C. Y. Wu, and C. C. Sun, “Method of compensating for pixel migration in volume holographic optical disc (VHOD),” Opt. Express 20(19), 20863–20873 (2012). [CrossRef]

13. T. Shimura, S. Ichimura, Y. Ashizuka, R. Fujimura, K. Kuroda, X. D. Tan, and H. Horimai, “Shift selectivity of the collinear holographic storage system,” Proc. SPIE 6282, 62820S (2007). [CrossRef]

14. T. Nobukawa and T. Nomura, “Shift multiplexing with a spherical wave in holographic data storage based on a computer-generated hologram,” Appl. Opt. 56(13), F31–F36 (2017). [CrossRef]

15. Y. W. Yu, T. C. Teng, S. C. Hsieh, C. Y. Cheng, and C. C. Sun, “Shifting selectivity of collinear volume holographic storage,” Opt. Commun. 283(20), 3895–3900 (2010). [CrossRef]

16. Y. W. Yu, C. M. Shu, C. C. Sun, P. K. Hsieh, and T. H. Yang, “Optical servo with high design freedom using spherical-wave Bragg degeneracy in a volume holographic optical element,” Opt. Express 27(24), 35512–35523 (2019). [CrossRef]

17. H. Horimai, X. Tan, and J. Li, “Collinear holography,” Appl. Opt. 44(13), 2575–2579 (2005). [CrossRef]

18. H. Horimai and X. Tan, “Advanced collinear holography,” Opt. Rev. 12(2), 90–92 (2005). [CrossRef]

19. T. Nobukawa and T. Nomura, “Correlation-Based Multiplexing of Complex Amplitude Data Pages in a Holographic Storage System Using Digital Holographic Techniques,” Polymers 9(12), 375 (2017). [CrossRef]

20. T. Shimura, S. Ichimura, R. Fujimura, K. Kuroda, X. Tan, and H. Horimai, “Analysis of a collinear holographic storage system: introduction of pixel spread function,” Opt. Lett. 31(9), 1208–1210 (2006). [CrossRef]

21. Y. W. Yu, C. Y. Chen, and C. C. Sun, “Increase of signal-to-noise ratio of a collinear holographic storage system with reference modulated by a ring lens array,” Opt. Lett. 35(8), 1130–1132 (2010). [CrossRef]

22. 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]

23. M. Hara, K. Tanaka, K. Tokuyama, M. Toishi, K. Hirooka, A. Fukumoto, and K. Watanabe, “Linear reproduction of a holographic storage channel using coherent addition of optical DC components,” Jpn. J. Appl. Phys. 47(7), 5885–5890 (2008). [CrossRef]

24. J. Liu, H. Horimai, X. Lin, Y. Huang, and X. Tan, “Phase modulated high density collinear holographic data storage system with phase-retrieval reference beam locking and orthogonal reference encoding,” Opt. Express 26(4), 3828–3838 (2018). [CrossRef]

25. 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]

26. X. Tan, O. Matoba, T. Shimura, and K. Kuroda, “Improvement in holographic storage capacity by use of double-random phase encryption,” Appl. Opt. 40(26), 4721–4727 (2001). [CrossRef]

27. B. Das, J. Joseph, and K. Singh, “Phase-image-based sparse-gray-level data pages for holographic data storage,” Appl. Opt. 48(28), 5240–5250 (2009). [CrossRef]

28. J. S. Jang and D. H. Shin, “Optical representation of binary data based on both intensity and phase modulation with a twisted-nematic liquid-crystal display for holographic digital data storage,” Opt. Lett. 26(22), 1797–1799 (2001). [CrossRef]

29. M. He, L. Cao, Q. Tan, Q. He, and G. Jin, “Novel phase detection method for a holographic data storage system using two interferograms,” J. Opt. A: Pure Appl. Opt. 11(6), 065705 (2009). [CrossRef]

30. J. C. Wyant, “Double Frequency Grating Lateral Shear Interferometer,” Appl. Opt. 12(9), 2057–2060 (1973). [CrossRef]

31. G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35(31), 6151–6161 (1996). [CrossRef]

32. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34(21), 4186–4195 (1995). [CrossRef]

33. G. W. Burr, J. Ashley, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, and B. Marcus, “Modulation coding for pixel-matched holographic data storage,” Opt. Lett. 22(9), 639–641 (1997). [CrossRef]

34. R. M. Shelby, J. A. Hoffnagle, G. W. Burr, C. M. Jefferson, M.-P. Bernal, H. Coufal, R. K. Grygier, H. Günther, R. M. Macfarlane, and G. T. Sincerbox, “Pixel-matched holographic data storage with megabit pages,” Opt. Lett. 22(19), 1509–1513 (1997). [CrossRef]

35. Y. W. Yu, S. Xiao, C. Y. Cheng, and C. C. Sun, “One-shot and aberration-tolerable homodyne detection for holographic storage readout through double-frequency grating-based lateral shearing interferometry,” Opt. Express 24(10), 10412–10423 (2016). [CrossRef]

36. Y. W. Yu, C. Y. Cheng, S. C. Hsieh, T. C. Teng, and C. C. Sun, “Point spread function by random phase reference in collinear holographic storage,” Opt. Eng. 48(2), 020501 (2009). [CrossRef]

37. K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental Characterization of Phenanthrenequinone- Doped Poly(methyl methacrylate) Photopolymer for Volume Holographic Storage,” Opt. Eng. 42(5), 1390–1396 (2003). [CrossRef]

38. S. H. Lin, Y. N. Hsiao, and K. Y. Hsu, “Preparation and Characterization of Irgacure 784 Doped Photopolymers for Holographic Data Storage at 532 nm,” J. Opt. A: Pure Appl. Opt. 11(2), 024012 (2009). [CrossRef]

		0um	2.5um	5um	7.5um	10um	12.5um	15um
W/O DFGSI	Raw data	0.00%	50.23%	49.88%	49.99%	49.85%	49.96%	49.92%
W DFGSI		4.28%	2.52%	5.74%	8.28%	18.90%	29.97%	31.10%
PI-DFGSI		0.00%	0.00%	0.00%	0.01%	0.01%	0.07%	0.49%
W/O DFGSI	“3/16” Saprse decode	0.00%	50.34%	50.50%	50.80%	50.22%	49.71%	50.71%
W DFGSI		7.06%	4.25%	9.92%	14.00%	31.47%	47.76%	49.98%
PI-DFGSI		0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.23%
W/O DFGSI	LDPC decode	0.00%	46.90%	46.85%	47.55%	46.46%	45.94%	47.25%
W DFGSI		0.58%	0.23%	0.94%	7.86%	33.58%	45.57%	46.33%
PI-DFGSI		0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%

Reduction of phase error on phase-only volume-holographic disc rotation with pre-processing by phase integral

Abstract

1. Introduction

2. Additional phase error caused by displacement of VHS disc

3. Novel approach to reduce the phase error

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (9)

Tables (1)

Equations (8)

Optics Express