## Abstract

A paraxial solution to the coaxial holographic storage algorithm is proposed based on the scalar diffraction theory and a VOHIL model (volume hologram being an integrator of the lights emitted from elementary light sources), which can give insight into the optical characteristics of the collinear holographic storage system in an effective way, including the point spread function and shift selectivity. The paraxial solution shows that the reference pattern is the key issue in the point spread function. Thus, the bit error rate of the system can be improved by changing the reference pattern. The proposed solution will be useful in the design of a new reference pattern to perform a high-quality readout pattern in the coaxial holographic storage system.

©2007 Optical Society of America

## 1. Introduction

Volume holography has been regarded as the most likely option for next-generation data storage since volume holographic storage performs not only a high data transfer rate with optical parallelism but also a theoretical storage capacity as high as *V*/*λ*
^{3} [1–4], which enables an extra one-dimensional extension of storage capacity than a traditional optical disc. Recently, among most holographic storage algorithms of the off-axis type, a holographic versatile disc system using a collinear algorithm has been proposed and demonstrated [5]. The proposed collinear algorithm is a coaxially aligned optical structure for signal and reference beams, which are encoded simultaneously by the same spatial light modulator (SLM) and that interfere with each other in the recording medium through a single objective lens. In the reading process, only the reference part is displayed on the SLM, and the reconstructed beam is reflected through the same lens and is received by a CMOS sensor. The system shows a large storage capacity, high transfer rate, short access time, and also is compatible with existing disc storage systems such as CDs and DVDs [6–8]. Also, uniform shift selectivity in both radial and tangential directions and a fairly large wavelength shift and tilt tolerance are proposed [9]. In figuring out the system performance theoretically, the Bragg degeneracy with different reference patterns recently has been shown as a function of the size of pixel response [10,11]. These papers presented a very good model for describing the optical performance of the collinear holographic storage system; however, a more detailed and clearer equation for describing optical characteristics and performance is still desired. In addition, some studies have presented computer simulations of the volume hologram and light diffraction on the CCD plane. [6,12,13] This is a way to figure out the light pattern as a function of the recorded volume hologram; however, the effect of the reference pattern on diffraction of light cannot be observed simply. In this paper, we develop a paraxial approximating analytic solution for the collinear system based on the scalar diffraction theory and the VOHIL model [14] to give clear insight into the algorithm. Based on this analytic solution, we obtain some important properties of the system, including a no thickness-dependent multiplexing capacity, and we design new reference patterns for a narrower point spread function to obtain a larger signal-to-noise ratio (SNR) in order to improve optical performance.

## 2. Optical modeling

The geometrical structure of the storage system for theoretical modeling is shown in Figs. 1. In the model, the mirror on the back surface of the holographic disc in the real system is replaced by a disc of double-thickness in a modeled transmission algorithm to replace the reflection algorithm in the real system. The optical field across the recording medium can be described as [15]

where ⊗ denotes the convolution operation, *λ* is the wavelength, *k* is the wave number, *f* is the focal length of lenses, Δ*z* is the distance deviated from the focal plane within the volume of the recording medium, *u* and *v* are the lateral coordinates of the recording medium, and *U _{f}*, the Fourier transform of the input wavefront

*U*coming from the SLM, can be expressed as

_{i}Based on Eqs. (1) and (2), for a signal *U _{s}* encoded by the SLM, the optical field

*S(u,v,Δz)*on the recording medium can be expressed as

$$=\frac{\mathrm{exp}\left(\mathrm{jk}\left(2f+\Delta z\right)\right)}{j\lambda f}\Im \left\{{U}_{s}(x,y)\mathrm{exp}(-j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({x}^{2}+{y}^{2}\right))\right\},$$

where ℑ denotes the optical Fourier transform by the lens. Similarly, *R(u,v,Δz)* and *P(u,v,Δz)* standing for the optical field of reference beam and probing beam, respectively, can be expressed as

and

where *U _{r}* and

*U*stand for the optical fields of the reference and reading light encoded by the SLM, respectively. In the writing process, the interference of the signal beam (S) and the reference beam (R) can be expressed as

_{p}In the reading process, we use the reading beam (*P*) to probe the hologram, and the diffraction can be described simply as

The diffracted optical field *U _{df}* for each plane across the volume hologram located at the plane of the back front plane of the second lens in the model can be expressed as

When each diffracted wave propagates to the detector, the optical field is the optical Fourier transform of *U _{df}* and can be expressed as

$$\otimes \left[{U}_{r}^{*}(\xi ,\eta )\mathrm{exp}\left(j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi}^{2}+{\eta}^{2}\right)\right)\right]\otimes \left[{U}_{s}(-\xi ,-\eta )\mathrm{exp}(-j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi}^{2}+{\eta}^{2}\right))\right].$$

According to the VOHIL model [14], we obtain the diffracted wave *U _{det}* from the whole volume of the hologram with a thickness of

*T*:

Equation (10) shows that the diffraction field on the output plane is the summation of the field in each layer of the recording medium, and it is also a function of the convolution among the reading, signal, and complex conjugate of the reference lights. Eq. (10) is still not simple in figuring out the characteristic of the collinear algorithm. Next, we will analyze the point spread function based on Eq. (10) and provide a clearer physics insight to describe the algorithm.

## 3. Calculation of point spread response

First of all, we will start at the calculation of the readout point spread function (PSF). Thus, we set the signal to be a point source so the signal is expressed as *δ(ξ,η)* and the PSF on the output plane is written as

where *psfz* denotes the PSF by the diffraction from each layer of the hologram, and is expressed as

Since the reading pattern is always the same as the reference pattern, the convolution in Eq. (12) becomes the autocorrelation of the reference pattern with a phase term (or the reading pattern). Then from Eq. (11) we find that the PSF is a result of the integration of the *psfz* multiplied by a quadratic phase term across the whole volume of the hologram. Consequently, a well-designed reference pattern could reduce the PSF. We use four different reference patterns shown in Fig. 2, and the signal is a delta function. The simulation result of the intensity distribution in x direction for the corresponding reference patterns is shown in Fig. 3, where λ=532nm, T=0.6mm, the pixel size is 13×13 µm, the fill factor of each pixel is 71.6%, and the size of the DMD is 4.4×4.4 mm. In the calculation, the effective focal length is set as 7.5 mm instead of 5 mm, and the effective wavelength is 532 nm/1.5 instead of 532 nm, in order to simplify the calculation of the refraction on the boundary by the air to the holographic disc. We find that the calculated intensity of the PSF of the horizontal-line reference pattern is far wider than what of the vertical-line reference pattern when we check the PSF along x direction. Similarly, the PSF of the vertical-line reference pattern is far wider than what of the horizontal-line reference pattern when we check the PSF along y direction. Besides, we find that the radial-line pattern proposed by Shimura et al. indeed performs a narrower PSF.

Based on Eqs. (11) and (12), we understand that a pattern that performs a narrow autocorrelation pattern may cause a narrow PSF of the collinear algorithm. Therefore, we propose several different reference patterns in Fig. 4. The four patterns are all whirl structures but with different density and curvature. Also, to enlarge the intensity of the reference pattern, the outer boundary of the proposed patterns is a square. The calculation of intensity of the PSFs is shown in Fig. 5, where we find a similar PSF but smaller side lobes can be obtained.

To further check the performance of the reference patterns, we propose two new patterns; one is an enlarged radial pattern and the other is a cross-whirl pattern. We simulate the diffracted pattern and calculate the corresponding SNR on the output plane. The SNR is described as

where *m _{1}* and

*m*are the average levels of the ON pixels and the OFF pixels, respectively, and

_{0}*σ*

^{2}

_{1}and

*σ*

^{2}

_{0}are the level variances corresponding to the ON and OFF pixels, respectively. Here, except for an extended radial pattern, we propose a cross-whirl pattern that is based on the result in Fig. 5 and is radial symmetry. The simulated diffracted patterns are shown in Fig. 6, and the SNRs for the radial pattern, extended radial pattern, and cross-whirl pattern are 2.314, 3.272, and 3.601, respectively. It means that the new designed reference patterns based on the paraxial formula of the PSF obtain better performance in the diffraction pattern.

## 4. Analysis of shift selectivity

In the following analysis, we consider the diffraction pattern when the disk rotates. The small amount of rotation of the disc may be regarded as a linear displacement where *Δu* is the u axis of the recording medium. The optical field in the detector can be expressed as

$$\frac{\mathrm{exp}\left(\mathrm{jk}4f\right)}{{\left(\lambda f\right)}^{2}}\int \mathrm{exp}(-j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi}^{2}+{\eta}^{2}\right))\left\{\begin{array}{c}\left[{U}_{p}(-\xi ,-\eta )\mathrm{exp}\left(j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi}^{2}+{\eta}^{2}\right)\right)\right]\\ \otimes \left[\mathrm{exp}\left(\frac{-j2\pi}{\lambda f}\left(\Delta u\xi +\Delta v\eta \right)\right){U}_{R}{(\xi ,\eta )}^{*}\mathrm{exp}(-j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi}^{2}+{\eta}^{2}\right))\right]\\ \otimes \left[\mathrm{exp}\left(\frac{-j2\pi}{\lambda f}\left(\Delta u\xi +\Delta v\eta \right)\right){U}_{s}(-\xi ,-\eta )\mathrm{exp}\left(j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi}^{2}+{\eta}^{2}\right)\right)\right]\end{array}\right\}d\Delta z.$$

Eq. (14) can be written in a different but clearer form thusly:

$$\frac{2T\mathrm{exp}\left(\mathrm{jk}4f\right)}{{\left(\lambda f\right)}^{2}}\mathrm{exp}(-j\frac{2\pi}{\lambda f}\left(\Delta u.\xi +\Delta v.\eta \right))\underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\left\{\begin{array}{c}{U}_{S}(-\xi +{\xi}_{2},-\eta +{\eta}_{2}){U}_{R}^{*}({\xi}_{2}-{\xi}_{1},{\eta}_{2}-{\eta}_{1})\\ {U}_{P}(-{\xi}_{1},-{\eta}_{1})\mathrm{exp}(-j\frac{2\pi}{\lambda f}\left(\Delta u\xb7{\xi}_{1}+\Delta v\xb7{\eta}_{1}\right))\\ \mathrm{sin}c\left[\frac{2T}{\lambda {f}^{2}}\left(\xi \xb7{\xi}_{2}+\eta \xb7{\eta}_{2}-{\xi}_{2}{\xi}_{1}-{\eta}_{2}\xb7{\eta}_{1}\right)\right]\end{array}\right\}d{\eta}_{1}d{\eta}_{2}d{\zeta}_{1}d{\xi}_{2},$$

where *ξ _{1}, η_{1}, ξ_{2}*, and

*η*

_{2}are the variables used to perform the convolution. The simulation of the shift sensitivity,

*i.e*., the relative diffraction intensity at the center of the detector vs. the displacement based on Eq. (15), is shown in Fig. 7 where the signal is a point source located at the center of the signal plane. We find that the shift selectivity is the same for the thickness of the recording medium at a range from 200 to 2000 µm. We can conclude that the shift selectivity is independent of the thickness of the recording medium. This characteristic can be easily understood when we check Eq. (15). When

*U*equal to

_{s}(-ξ+ξ_{2},-η+η_{2})*δ(-ξ+ξ*and

_{2},-η+η_{2})*ξ=η*=0, the sinc function is equal to 1 and the shift sensitivity is independent of thickness. However, the thickness still has an effect on the point spread function, as shown in Fig. 8, where the reference pattern is the radial one as shown in Fig. 6(a). We may conclude that a thicker hologram performs a narrower point spread function.

## 4. Summary

In this paper, we have proposed an optical model to derive paraxial equations to describe the PSF of the collinear holographic storage structure. Based on the equations, we conclude that the reference pattern is a key issue to determine the PSF. Thus, we propose several whirl structures and obtain similar PSFs, but with suppressed side lobes in comparison with those having a radial pattern. Also, we propose an extended radial pattern to have more reading power and perform a similar PSF. In the further simulation of the SNR of the readout pattern, the extended radial pattern and newly proposed cross-whirl pattern both perform a higher SNR than that of the radial pattern. In regard to the shift selectivity, the derived equations predict that the PSF is independent of the thickness of the holographic disc, but a thicker disc can perform a narrower PSF. The optical characteristics can be understood easily with the proposed model and equations, and more effective reference patterns can be proposed through the calculation with use of the proposed equations.

Acknowledgement

This study was sponsored by the Ministry of Economic Affairs of China under grant 95-EC-17-A-07-S1-011 and the National Science Council under grant NSC 96-2221-E-008-031. The authors thank S. H. Lin, J. Y. Cheng, and T. H. Yang for their comments on the study.

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