Method of compensating for pixel migration in volume holographic optical disc (VHOD)

Yeh-Wei Yu; Chih-Yuan Cheng; Tun-Chien Teng; Cheng-Hsien Chen; Shiuan-Huei Lin; Bo-Rong Wu; Che-Chih Hsu; Yi-Jiun Chen; Xuan-Hao Lee; Chi-Yu Wu; Ching-Cherng Sun

doi:10.1364/OE.20.020863

1. Introduction

In this era of information proliferation, the demands for efficient data back-up are growing quickly. This growth is driven by the rapid progress in digital technologies, e.g., super computers, global servers, and cloud computing. In addition, huge piles of data are produced by new imaging technologies such as high definition and three-dimensional imaging used by media companies and magnetic resonance imaging (MRI), computed tomography (CT), ultrasonic imaging (USI), and computed radiography (CR) used in hospitals [1–5]. The storage of these data presents many challenges. Generally, the relevant archives must be resistant to corruption, retrievable and operable at all times, expandable, and easy to store and manage. In addition, damaged or lost data must be recoverable, and the related hardware and software must be energy efficient and compact. Media companies and hospitals often have the further requirement that the databases be un-rewritable and unerasable.

Library-style tape storage systems have been used for archive [6]. However, the serial access of this method makes it too inefficient for many modern applications. On the other hand, the hard disk archive consuming too much electrical energy becomes heavy loading of big company. The best modern alternative is the volume holography storage system, which has a theoretical storage capacity as high as V/λ³, with V being the recording volume and λ the wavelength of the recording light source [7]. The capacity of a volume holographic storage system is reported to be as high as 100 bits/μm² [8, 9]. Its densities are greater than those of even the latest 32 nm process dynamic random access memory (DRAM) [10], and because two-dimensional data arrays are recorded and retrieved at the speed of light, the page-based technique can transfer data at rates measured in GB/s [11]. Furthermore, multiphoton photorefractive processes and photorefractive polymers promise the rewritable-applications in the future [12–18].

The angular multiplexing holographic storage systems available commercially have high storage capacities and high transfer rates [19], but their mechanisms are intrinsically complex. Conversely, the shifting multiplexing of the spherical reference light in the so-called volume holographic optical disc (VHOD) system requires only that the disc be rotated [20, 21], simplifying and further compacting the mechanism. Nonetheless, two factors limit the application of VHOD. First, radial oriented inter-page cross-talk caused by Bragg degeneracy limits storage capacity (although this problem has recently been shown to be mitigated by introducing a spatial filter in the reference beam [22]). Second, the readout signal of the VHOD shifts when the disc rotates, resulting in extra inter-pixel crosstalk that decreases the signal-to-noise ratio (SNR). This study proposes a method for addressing these problems.

2. Theory

Figure 1 shows the structure of the VHOD system. During the writing process, the signal beam consists of a convergent spherical wave that illuminates the spatial light modulator (SLM) and focuses on the disc. The interference between a divergent spherical reference (R) and the signal beam (S) is recorded. In the readout process, the conjugation of the reference (R*) serves as the reading light incident on the disc. Thus, the conjugation of the signal (S*) is diffracted and imaged on the CCD. Although simpler than traditional shifting multiplexing, by retaining the spherical-wave reference beam and recording the Fourier spectrum of the signal as before, VHOD suffers from the same problems of limited storage capacity and decreased SNR. Figure 2 (Media 1) shows a contrast enhanced image captured by the CCD in the reading process. The shift of the disc not only causes a corresponding shift of the diffracting signal, thus degrading the captured signal, but also leads to critical deviation tolerance. To solve the problem of pixel migration, we start from the theoretical model based on the VOHIL model and Fresnel diffraction [23, 24]. A schematic diagram of VHOD is shown in Fig. 3 .

Fig. 1 Schematic outline of VHOD. (a) Writing process. (b) Reading process.

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Fig. 2 Contrast enhanced image captured by the CCD in the reading process (Media 1).

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Fig. 3 Schematic diagram of VHOD. (a) Writing process. (b) Reading process.

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The reference distribution in the disc is a divergent spherical wave. By paraxial approximation and coordinate rotation, the reference can be expressed as

R_{m} (x_{m}, y_{m}, z_{m}) = \frac{1}{r} \exp {i k (\frac{x_{m}^{2} \cdot \cos^{2} θ + z_{m}^{2} \cdot \sin^{2} θ - x_{m} \cdot z_{m} \sin 2 θ + y_{m}^{2}}{2 r} + r)},

where (x_m, y_m, z_m) is the coordinate attached on the disc, k is the wave number,

θ

is the incident angle of the reference beam, and r is the distance from a given position inside the disc to the focus plane of the reference. The derivation of r is as follows:

r = r_{0} + x_{m} \sin θ + z_{m} \cos θ,

where r₀ is the distance between the disc and the point source of the reference beam. The signal distribution inside the recording disc is

\begin{array}{l} S_{m} (x_{m}, y_{m}, z_{m}) = \\ \frac{\exp (i k (z_{0} + z_{m}))}{i λ (z_{0} + z_{m})} \exp (i π \frac{x_{m}^{2} + y_{m}^{2}}{λ (z_{0} + z_{m})}) \iint S_{0} (x_{s}, y_{s}) \exp (i π \frac{x_{s}^{2} + y_{s}^{2} - 2 x_{s} x_{m} - 2 y_{s} y_{m}}{λ (z_{0} + z_{m})}) d x_{s} d y_{s} . \end{array}

where S₀ is the field distribution after modulation of the focusing signal beam by SLM, (x_s, y_s) are the coordinates attached on the SLM or CCD,

z_{0}

is the distance between the disc and the SLM, and λ is the wave length.

The interfering fringe of the signal and the reference is

G_{m} = {| R_{m} |}^{2} + {| S_{m} |}^{2} + R_{m}^{*} S_{m} + S_{m}^{*} R_{m},

Because the probing beam is the phase conjugation of the reference, only the last term is retrieved by the system, and we only consider

G_{m} = S_{m}^{*} R_{m} .

The phase conjugation of the reference reads the grating when the disc shifts (Δx, Δy), and the amplitude of the signal diffracting from each position inside the disc is

D_{m} (x_{m}, y_{m}, z_{m}) = c i r c (\frac{\sqrt{x_{m}^{2} + y_{m}^{2}}}{w}) R_{m}^{*} (x_{m}, y_{m}, z_{m}) G_{m} (x_{m} - Δ x, y_{m} - Δ y, z_{m}),

where

c i r c (\frac{\sqrt{x_{m}^{2} + y_{m}^{2}}}{w})

stands for the system pupil with radius w. The amplitude of the diffracting signal propagating to the CCD is calculated by the Fresnel diffraction.

D_{s} (x_{s}, y_{s}, z_{m}) = \exp (i \frac{π (x_{s}^{2} + y_{s}^{2})}{λ (z_{0} + z_{m})}) \iint D_{m} (x_{s}, y_{s}, z_{m}) \exp (i π \frac{x_{m}^{2} + y_{m}^{2} - 2 x_{s} x_{m} - 2 y_{s} y_{m}}{λ (z_{0} + z_{m})}) d x_{m} d y_{m},

To obtain an analytical solution, we use the property

r_{0} > > x_{m} \sin θ .

The parameter r approximates as follows:

r \approx r_{0} + z_{m} \cos θ .

Therefore, we obtain the amplitude of the diffractive signal from depth z_m, as follows:

D_{s} (x_{s}, y_{s}, z_{m}) = F \cdot (S_{p} \otimes P S F \otimes D e l t a),

where

F (x_{s}, y_{s}, z_{m}) = \exp (\frac{i π}{λ} (\frac{x_{s}^{2} + y_{s}^{2} - Δ x^{2} - Δ y^{2}}{z_{0} + z_{m}} + \frac{Δ x^{2} \cos^{2} θ + Δ y^{2} + Δ x \cdot z_{m} \sin 2 θ}{r_{0} + z_{m} \cos θ} - 2 Δ x \sin θ)),

S_{p} (x_{s}, y_{s}, z_{m}) = S_{0}^{*} (x_{s}, y_{s}) \exp (- i π \frac{x_{s}^{2} + y_{s}^{2} + 2 Δ x x_{s} + 2 Δ y y_{s}}{λ (z_{0} + z_{m})}),

P S F (x_{s}, y_{s}, z_{m}) = \frac{λ (z_{0} + z_{m}) J_{1} (2 π w \frac{\sqrt{x_{s}^{2} + y_{s}^{2}}}{λ (z_{0} + z_{m})})}{π w \sqrt{x_{s}^{2} + y_{s}^{2}}},

and

D e l t a (x_{s}, y_{s}, z_{m}) = δ (x_{s} - Δ x (1 - \frac{(z_{0} + z_{m}) \cos^{2} θ}{r_{0} + z_{m} \cos θ}), y_{s} - Δ y (1 - \frac{(z_{0} + z_{m})}{r_{0} + z_{m} \cos θ})) .

Each function has a unique physical property. The F function is a phase distribution; the S_p function is the mathematical product of a signal distribution and a phase distribution; PSF is a point spread function caused by the system pupil; and the Delta function describes the shift of the diffracting pattern. The Delta function is the key factor in the present analysis. When the disc rotates, the tangential shift (Δx) is greater than the radial shift (Δy). Therefore, if

\frac{r_{0}}{z_{0}} = \cos^{2} θ,

Δx is multiplied by a number approaching zero in Eq. (14). In other words, the shift of the disc causes the smallest possible shift of the diffracting pattern. We use K to mean r₀/z₀:

K = \frac{r_{0}}{z_{0}} .

Described in physical terms, the solution of pixel migration is effectuated through displacement compensation. When the disc shifts, there is a corresponding positive shift of the recording position and a corresponding negative shift of the diffracting image. When the distance ratio equals cos²θ, the positive and negative shifts compensate for each other. Thus, the pixel migration is resolved.

3. Simulation

Using Eq. (7), a simulation can be conducted based on the Fresnel diffraction and the VOHIL theorem (the approximation in Eq. (9) is not used). By integrating D_s along depth z_m, we obtain the intensity distribution in the CCD plane.

I_{C C D} (x_{s}, y_{s}) = {| \int_{\frac{-T}{2}}^{\frac{T}{2}} D_{s} (x_{s}, y_{s}, z_{m}) d z_{m} |}^{2} .

In the simulation, the thickness of the disc is 1.8 mm and the refractive index is 1.5. The wavelength inside the disc is 0.532/1.5 μm. The effective incident angle of the reference is 15°. The effective distance, r₀, is kept at 40 mm to keep the same shift selectivity. z₀ is 20 mm for K = 2 and 42.9 mm for K = cos²θ. The signal in the SLM is 50 × 50 pixels with randomly distributed 0’s and 1’s. The pixel-number ratio between 0’s and 1’s is 1:1. The pixel size of

S N R = 20 \cdot \log [\frac{(m_{1} - v_{t h})}{σ_{1}}],

where m₁ and m₀ are the means of the distribution of 1’s and 0’s, respectively [25, 26]. σ₁ and σ₀ are the standard deviations of the distribution of 1’s and 0’s, respectively. v_th is the threshold value between the 1’s and the 0’s, and is calculated as follows:

v_{t h} = \frac{σ_{1} m_{0} + σ_{0} m_{1}}{σ_{1} + σ_{0}},

Figure 5 shows the SNR of the diffracting signal plotted against the disc rotation. The blue and red curves stand for K = 2 and K = cos²θ, respectively. With the disc in the initial position in the writing process, the blue curve is higher than the red curve because the smaller z₀ yields a greater point spread function. However, as the disc falls out of precise alignment, the blue curve drops considerably faster than the red curve. In other words, pixel migration has been avoided at the price of a slight decrease in SNR.

Fig. 4 Diffracted signal captured by the CCD. (a) K = 2 (Media 2). (b) K = cos²θ(Media 3). both the SLM and the CCD is 14 × 14 μm². Figure 4(a) (Media 2) and Fig. 4 (b) (Media 3) show the diffracting signal captured by the CCD when K = 2 and K = cos²θ, respectively. As expected, the diffracting signal in Fig. 4(a) (Media 2) shifts, but that in Fig. 4 (b) (Media 3) remains steady. Figure 4 (b) (Media 3) consequently shows greater image quality when the disc rotates. The SNRs were calculated according to the following formula:

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Fig. 5 SNR of diffracting signal depending on disc rotation.

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4. Experiment

Figure 6 shows the experimental apparatus used to demonstrate this technique. Its setup is the same as that shown in Fig. 1, except that an additional CCD B is used to capture diffracting signals (S). The light source is a Verdi-laser with a wavelength of 0.532 μm. The signal beam is a convergent spherical wave illuminating the SLM and focusing on the disc. The disc is positioned near the focus of the signal beam but deviates slightly from the focus depth to avoid a strong DC term.

Fig. 6 The experimental setup.

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The recording medium is a PQ/PMMA photopolymer 1.8 mm in thickness. The refractive index of the PQ/PMMA is 1.5. The signal beam normally strikes the disc with a distance of 6.8 cm between the SLM and the disc. Because of the refractive boundary, the effective distance inside the disc is 10.2 cm. The reference beam is a divergent spherical wave with an incident angle of 22.8°, which corresponds to 15° inside the disc. The distance from the divergent point of the reference to the recording region on the disc is 6.4 cm. From the geometric schematic diagram shown in Fig. 7 , we obtain

Fig. 7 Schematic diagram for position of effective light source.

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r_{a i r} \sin ({23.3}^{\circ}) = r_{d i s k} \sin (15^{\circ}) .

The resulting effective distance inside the disc (r_disc) is 9.5 cm. The K factor thus equals cos²θ and fits Eq. (15). In the readout process, the conjugation of the reference (R*) serves as the reading light. Thus, the conjugation of the signal (S*) is diffracted and imaged on CCD A. The contrast enhanced image in Fig. 8 (Media 4) shows that when the disc shifts, the diffracting image dims but remains in the same position. Compared to Fig. 2 (Media 1), the improvement is evident.

Fig. 8 Contrast enhanced image captured by the CCD in the reading process (Media4).

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5. Application

Using an optimal value for K, the signal can be read even when the disc continues to rotate. In the simulation, we accumulated the signal with the disc rotating from 0 rad to 10⁻³ rad (Fig. 9 ). Figure 9(a) (Media 5) and Fig. 9(b) (Media 6) show the accumulated signals when K = 2 and K = cos²θ, respectively, and show how the image quality in Fig. 9(b) (Media 6) remains relatively unchanged as the disc rotates. Figure 10 shows the SNR plotted against disc rotation. The horizontal axis is the accumulation distance, and the blue and red curves stand for K = 2 and K = cos²θ, respectively. Significantly, the red curve remains above 10 throughout the range of accumulated distance from 0 rad to 10⁻³ rad, reflecting the degree to which the data transfer rate is enhanced by the ability to read the signal while the disc is rotating.

Fig. 9 Accumulated signal with disc rotating from 0 rad to 10⁻³ rad. (a) K = 2 (Media 5). (b) K = cos²θ (Media 6).

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Fig. 10 SNR depending on disc rotation.

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6. Analysis for SLM with wide area

Equation (15) is derived out based on paraxial condition. Nevertheless, the system is composed of a SLM with wide area and a reference with a divergent beam. So, the geometric relationship described by Eq. (15) can’t be satisfied for signal in any position of the SLM. Here, we analyze the applicability of compensating pixel migration for signal in different locations. The Rayleigh-Sommerfeld diffraction theory is used in order to simulate light propagating at large angle [27]. Figure 11 shows the schematic diagram for both K = cos²θ and K = 2. The parameters are the same as other simulations in this paper except the dimension of the SLM. The width of SLM in x direction is 28mm, corresponding to 2000 pixel numbers. θ_s is the inclined angle of the light when it propagates from the signal to the focus point. The simulation results for SNR depending on disc rotation among different θ_s range are shown in Fig. 12 . In each case, 100 × 10 pixels are used as signal for K = cos²θ, and pixels inside the same range of θ_s are used as signal for K = 2. The simulation results show the SNR drops down for large θ_s. Even so, the performance for K = cos²θ is always better than K = 2.

Fig. 11 Schematic diagram for (a) K = cos²θ and (b) K = 2.

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Fig. 12 SNR depending on disc rotation forθ_s in different range. (a) 0°<θ_s<1.9°. (b) 1.9°<θ_s <3.7°. (c) 3.7°<θ_s<5.6°. (d) 5.6°<θ_s<7.4°. (e) 7.4°<θ_s<9.3°. (f) 9.3°<θ_s<11.1°. (g) 11.1°<θ_s <12.9°. (h) 12.9°<θ_s<14.6°. (i) 11.1°<θ_s <12.9°. (j) 12.9°<θ_s<14.6°.

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7. Conclusion

VHOD storage is simpler and more compact than angular multiplexing holographic storage. However, disc rotation typically causes pixel migration and a consequent decline in signal quality. This study proposed a geometrical algorithm to avoid pixel migration during the reading process. By making a paraxial approximation of VHOD, a geometrical distance ratio, K, can be calculated. When K equals cos²θ, the system achieves displacement compensation, and the pixel migration is compensated for. This technique was demonstrated in a simulation video in which the SNR was used to assess diffracting signal quality. The video shows that with the disc in the initial writing position, there is no improvement at K = cos²θ because the relatively small z₀ induces a greater point spread function. However, when the disc is not aligned precisely, the SNR for K = 2 drops substantially faster than that for K = cos²θ, indicating that the problem of pixel migration has been solved at the price of a slight decrease in SNR. An experiment to demonstrate the technique showed that when the disc shifted at K = cos²θ, the diffracting image dimmed but remained in the same position. Compared to the results obtained with K = 2, the progress is evident. Thus, the technique of reading signals during disc rotation enhances the data transfer rate significantly. In the end, we check the performance of the system for signal with large size by Rayleigh-Sommerfeld diffraction theory. Although the simulation results show the SNR drops down for large θ_s, the performance for K = cos²θ is always better than K = 2.

Acknowledgment

This study was sponsored by the National Science Council of the Republic of China with the contracts of no. 97-2221-E-008-025-MY3, 99-2623-E-008-002-ET and NSC100-3113-E-008-001.

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Method of compensating for pixel migration in volume holographic optical disc (VHOD)

Abstract

1. Introduction

2. Theory

3. Simulation

4. Experiment

5. Application

6. Analysis for SLM with wide area

7. Conclusion

Acknowledgment

References and links

Supplementary Material (6)

Cited By

Figures (12)

Equations (20)

Optics Express