## Abstract

A novel design for a volume holographic optical disc to reduce radial cross talk is proposed. By adopting a proper spatial filter, the radial cross talk can be reduced and the radial selectivity increases so that the multiplexing capacity can increase effectively. The theory and the corresponding experiment are demonstrated.

© 2006 Optical Society of America

## 1. Introduction

Volume holography has been extensively studied for its potential in optical storage as well as novel optical elements. Volume holographic storage is awarded theoretical storage capacity as high as *V*/*λ*
^{3} [1], which enables an extra one-dimensional extension of storage capacity than a traditional reflection-type optical disc. Owing to the progress in volume holographic storage, many holographic multiplexing techniques have been proposed with Bragg condition in spatial and temporal domains [2–3], including angle multiplexing [4–5], peristrophic multiplexing [6], angle and peristrophic multiplexing [7], shift multiplexing [8], wavelength multiplexing [9–10], orthogonal phase multiplexing [11], random multiplexing [12–15]. Besides, parallel processing in volume holography is also a key feature, which greatly increases the data access rate. Among the above algorithms, shifting multiplexing of a spherical reference light in a disc-type is regarded as the most potential one for practical application. In such a case, here so-called volume holographic optical disc (VHOD), cross talk is one of the main factors to limit the storage capacity. [16–18] Due to the feature of Bragg degeneracy, the cross talk in the radial direction, i.e., the direction of Bragg degeneracy, is much more serious than that in the tangential direction. Thus, reducing the radial cross talk may effectively increase the recording capacity. Such an effort will also benefit the sensing sensitivity in volume holographic spatial filter with spherical wave. [19–20] In this Letter, a novel and simple design for reducing radial cross talk in a VHOD is proposed, and both the theory and the experiment are demonstrated.

## 2. The principle

The VHOD is supposed that multiple transmission gratings are recorded across the disc. Each grating is built by the interference of a divergent spherical reference and a signal. In practical application of storage, the signal should be incident on the recording medium through an imaging system or 4-f system so that the readout plane is a conjugate plane of the input signal. For simplicity of analysis, the signal is a divergent spherical wave from a point source instead of an extended object in deriving the equations as shown in Fig. 1. Also, the reading light is assumed to be a convergent spherical light, which is a conjugate of the reference light, so the diffracted light is converged to the point source at the same location of the signal.

In paraxial approximation, the divergent spherical reference from a point located at (x_{r},y_{r},-s_{1}) and a signal light from the point at (α_{s}, y_{s}, -s_{2}) can be expressed respectively

where k is the wave number; A_{r} and A_{s} are the amplitudes; s1 and s2 are the distances from the recording plane to the reference and signal point respectively; α and β is the coordinates describing the plane for the signal and the diffracted light, which is on-plane rotating from (x,z) coordinate with an angle θ. In the readout process, a convergent spherical light is incident on the disc, and thus the field of the diffracted light can be expressed [21]

where the ℓ_{x}, ℓ_{y}, and ℓ_{z} are the dimensions of the corresponding hologram along the x, y, and z axes, respectively, and the recording area here is a square instead of a circle, for simplicity. (α^{d}, β_{d}) is the coordinates of the spot of the diffracted light; *E _{p}* is the reading light convergent to (x

_{p}, y

_{p}, -s

_{1}) and can be expressed

If we only considers the displacement of the reading light in the Bragg degeneracy direction (y direction in this case), (y_{p}-y_{r})/s_{1}= Δy_{s}/s_{2} and x_{p}-x_{r}= Δα_{s} =0 can be obtained due to momentum conservation [22]. Thus, Eq. (3) can be rewritten as

where

$$B\left(y\right)=\mathrm{sin}c\left[\frac{(\left(y-0.5\left(\Delta {y}_{s}+{y}_{\mathit{or}}\right)\right)\left(\Delta {y}_{s}+{y}_{\mathit{or}}\right)\mathrm{sin}\theta -{\alpha}_{\mathit{or}}\left({s}_{2}\mathrm{cos}\theta +0.5{\alpha}_{\mathit{or}}\mathrm{sin}\theta \right){\ell}_{x}}{\lambda {{s}_{2}}^{2}}\right],$$

$$C\left(y\right)=s\mathrm{inc}\left[\frac{{\ell}_{z}\left(y\left({s}_{1}\mathrm{cos}\theta \left(\Delta {y}_{s}+{y}_{\mathit{or}}\right)-{s}_{2}\Delta {y}_{s}\right)+{s}_{1}{s}_{2}{\alpha}_{\mathit{or}}\mathrm{sin}\theta +0.5{s}_{1}\left(\Delta {{y}_{s}}^{2}-\mathrm{cos}\theta ({\left(\Delta {y}_{s}+{y}_{\mathit{or}}\right)}^{2}+{{\alpha}_{\mathit{or}}}^{2}\right)\right)}{\mathit{\lambda}{s}_{1}{{s}_{2}}^{2}}\right],$$

and both (x_{r}, y_{r}) and (α_{s}, y_{s}) are set (0, 0). For the convenience to describe the coordinates of the diffracted light spot, α_{d} and y_{d} can be rewritten as Δα_{s}+α_{or} and Δy_{s}+y_{or} respectively, where Δα_{s} and Δy_{s} are the deviation of the center of the diffraction spot from the signal point and α_{or} and y_{or} are the coordinates originated at the center of the spot. For simplicity, the spatial parameters in the above equations are under the condition that the refractive index outside the recording medium is the same as that of the medium. These parameters can be easily transferred into ones in the air with a simple calculation based on refraction effect in practical application. The amplitude of the diffracted light at (0, Δy_{s}) depends on integration of the product of two sinc functions in Eq. (5). The sinc functions in the equation enable that the smaller y is, the slower the value of the sinc function decays as Δy_{s} increase. Therefore, we can find an important fact that the value of integration of the product of two sinc functions decays faster as Δy_{s} increases if the area of the hologram around y=0 is blocked as shown in Fig. 2. In other words, the radial (y-directional) cross talk can be greatly reduced by a properly designed spatial filter to block some area of the hologram around y=0. Though the equations derived above are based on the algorithm shown in Fig. 1, the equations are applicable to the case with 4f system if a little modification is made. When the signal light through a 4-f system, the signal light emerging from each point on the SLM can be regarded as the plane wave incident on the disc with a specific angle. After the similar steps of derivation, the modified Eq. (5) can be also easily obtained.

## 3. Verification of the optical model

In the experiment, for convenience, a cubic LiNbO_{3} crystal of 10mm×10mm×10mm with two spherical waves of 90°-incidence was used, as shown in Fig. 2, where s_{1}=7cm, s_{2}=11cm in the air and an Verdi laser of wavelength 0.532 μm was as the light source. In the experiment, the rectangle blocked area around y=0 is 50% of the total recording area. The experimental results of the Bragg selectivity including unblocked and blocked cases as well as the corresponding theoretical simulations are shown in Fig. 3. We can find that the selectivity in the blocked case is higher than that in the unblocked case and thus Eq. (5) can be used to predict the behavior.

To figure out the effect on the selectivity by the blocked area, we calculate the selectivity as a function of the blocked area, as shown in Fig. 4, where it can be found that the more the blocked area is, the more the radial cross talk reduces. However, the cross talk reduces little more when the blocked area is larger than 50% of the whole area. Besides, increase of blocked area changes effective numerical aperture (NA) in y direction, so the energy of the diffracted spot spreads outward. Thus the quality of the diffracted signal images becomes poor and cross talk and extra effects may be introduced.

In the following, we will discuss the diffraction of an extended object (a spatial light modulator with multiple pixels) instead of an ideal point signal. Here we discuss the cross talk between the pixels at the same location of different pages on the diffraction plane. The period of the pixel is 10 μm and the active region of each pixel is set 4μm×4μm for reducing inter-pixel cross talk. The dimensional parameters of each pixel hold for both SLM and CCD. The recording area is 10mm×10mm×2mm and s_{1}=s_{2}=4.5 cm, θ=45° in the air. For simplicity, now we consider a specific pixel in off state in a certain page, and the pixel at the same location of the other pages are all in on state. Assume the threshold of the on-state pixel is 50% of the Bragg-match diffracted intensity. The readout error does not occur when the diffracted intensity of this specific pixel in off state on the diffraction plane is less than 50% (-3 dB) of the intensity of an on-stated pixel in Bragg match. The intensity diffracted onto one CCD pixel is calculated to check whether it is in on or off state. Fig. 5 shows the simulation results, including blocked (25% blocked ratio for better image quality) and unblocked cases. We can find that the blocked case enables lower radial cross talk and the radial interval distance is about half of the unblocked case. In other words, the multiplexing capacity can be doubled.

It should be noted that the spatial filter proposed in this Letter may affect the quality of the readout image and cause higher inter-pixel cross talk within a page despite the inter-page cross talk in the radial direction is greatly reduced. However, such effect can be reduced by adjusting some parameters, such as pitches and/or filled factors of CCD and/or SLM, the shape, dimensions and transmittance function of the filter, and so on. Once inter-pixel cross talk is maintained around the original level, the corresponding bit error rate (BER) can be reasonably expected acceptable. Thus the storage density can increase without impairing BER.

## 4. Summary

In summary, we have proposed and theoretically and experimentally demonstrated a novel spatial filter to reduce the radial cross talk in a VHOD with a spherical reference light. The spatial filter is a stripe area around y=0 for each recording area. We simulate the cross talk with a point object as well as a pixel as the signal. Under the condition that the recording area is 10mm×10mm×2mm and s_{1}=s_{2}=4.5 cm, θ=45° in the air, we find that the multiplexing capacity can increase twice if the threshold value is set 50% of the Bragg-matched pixel in on state when the blocked area is 25% of the area for each recorded hologram.

## Acknowledgments

This study is sponsored by the National Science Council of the Republic of China with contract number NSC 2215-E-008-018.

## References and links

**1. **P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. **2**, 393–400 (1963). [CrossRef]

**2. **G. Barbastathis and D. Psaltis, “Volume holographic multiplexing methods”, in *Holographic Data Storage*, H. J. Coufal, D. Psaltis, and G. T. Sincerbox, Eds, Springer-Verlag, pp.21–22 (2000).

**3. **L. Hesselink, S. S. Orlov, and M. C. Bashaw, “Holographic data storage systems,” Proceedings of IEEE **92**, 1231–1280 (2004). [CrossRef]

**4. **E. N. Leith, A. Kozma, J. Marks, and N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. ,**5**, 1303–1311 (1966). [CrossRef] [PubMed]

**5. **G.W. Burr, F. H. Mok, and D. Psaltis, “Angle and space multiplexed storage using the 90ogeometry,” Opt. Commun. **117**, 49–55 (1995). [CrossRef]

**6. **K. Curtis, A. Pu, and D. Psaltis, “Method for holographic storage using peristrophic multiplexing,” Opt. Lett. **19**, 993–995 (1994). [CrossRef] [PubMed]

**7. **A. Pu and D. Psaltis, “High density recording in photopolymer-based holographic 3-D disks,” Appl. Opt. **35**, 2389–2398 (1996). [CrossRef] [PubMed]

**8. **G. Barbastathis, M. Levene, and D. Psaltis, “Shift multiplexing with spherical reference waves,” Appl. Opt. **35**, 2403–2417 (1996). [CrossRef] [PubMed]

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

**10. **S. Yin, H. Zhou, F. Zhao, M. Wen, Y. Zang, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode-laser,” Opt. Commun. **101**, 317–321 (1993). [CrossRef]

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

**12. **J. T. LaMacchia and D. L. White, “Coded multiple exposure holograms,” Appl. Opt. **7**, 91–94 (1968). [CrossRef] [PubMed]

**13. **J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Encrypted holographic data storage based on orthogonal-phase-code multiplexing,” Applied Optics **34**, 6012–6015 (1995). [CrossRef] [PubMed]

**14. **C. C. Sun, R. H. Tsou, W. Chang, J. Y. Chang, and M. W. Chang, “Random phase-coded multiplexing in LiNbO3 for volume hologram storage by using a ground-glass,” Opt. Quantum Electron. **28**, 1509–1520 (1996).

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

**16. **C. Gu, J. Hong, I. McMichael, and R. Saxena, “Cross-talk-limited storage capacity of volume holographic memory,” J. Opt. Soc. Am. A **9**, 1978–1983 (1992). [CrossRef]

**17. **X. Yi, S. Campbell, P. Yeh, and C. Gu, “Statistical analysis of cross-talk noise and storage capacity in volume holographic memory: image plane holograms,” Opt. Lett. **20**, 779–781 (1995). [CrossRef] [PubMed]

**18. **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**, 5377–5385 (1998). [CrossRef]

**19. **G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. **24**, 811–813 (1999). [CrossRef]

**20. **C. C. Sun, S. P. Yeh, Y. N. Lin, W. C. Su, and Y. Ouyang, “High Longitudinal Selectivity of Shifting Multiplexing in Volume Holograms,” Opt. Laser Tech. **34**, 523–526 (2002). [CrossRef]

**21. **C. C. Sun, “A simplified model for diffraction analysis of volume holograms,” Opt. Eng. **42**, 1184–1185 (2003). [CrossRef]

**22. **C. C. Sun, T. C. Teng, and Y. W. Yu, “Linearly optical imaging with optical holographic element,” Opt. Lett. **30**, 1132–1134 (2005). [CrossRef] [PubMed]