1 February 2009 Point spread function by random phase reference in collinear holographic storage
Abstract
We use a paraxial approximating solution to calculate the point spread function of the collinear holographic storage system and show that the point spread function can be dramatically enhanced by the reference pattern with random binary phase modulation or random phase modulation.
Yu, Cheng, Hsieh, Teng, and Sun: Point spread function by random phase reference in collinear holographic storage

Figure 1 shows a schematic diagram of the storage system for theoretical modeling. In the model, the mirror on the back surface of the holographic disk in the real system is replaced by a double-thickness disk in a modeled transmission algorithm to replace the reflection algorithm in the real system. In previous research, we showed that the PSF for the collinear system can be written8 as

## 1

$\mathrm{PSF}\left(\xi ,\eta \right)=\frac{\mathrm{exp}\left(jk4f\right)}{{\left(\lambda f\right)}^{2}}{\int }_{-T}^{T}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left[\frac{j\pi \Delta z}{\lambda {f}^{2}}\left({\xi }^{2}+{\eta }^{2}\right)\right]\left[\delta \left(\xi ,\eta \right)\otimes \mathrm{psfz}\left(\xi ,\eta ,\Delta z\right)\right]\mathrm{d}\Delta z,$
where $\xi$ and $\eta$ are the lateral coordinates, $\Delta z$ is the longitudinal deviation from the center of the disk, $T$ is the thickness of the disk, $k$ is the wave number, $\lambda$ is the effective wavelength, $f$ is the effective focal length of the Fourier transform lens, $\delta \left(\xi ,\eta \right)$ is a delta function that stands for the point input of the signal, and psfz denotes the PSF by the diffraction from each layer of the hologram and is expressed as

## 2

$\mathrm{psfz}=\left\{{U}_{p}\left(-\xi ,-\eta \right)\mathrm{exp}\left[-j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi }^{2}+{\eta }^{2}\right)\right]\right\}\otimes \left\{{U}_{r}^{*}\left(\xi ,\eta \right)\mathrm{exp}\left[j\frac{\pi \Delta z}{\lambda {f}^{2}}\left({\xi }^{2}+{\eta }^{2}\right)\right]\right\},$
where ${U}_{p}\left(\xi ,\eta \right)$ and ${U}_{r}\left(\xi ,\eta \right)$ are the reading and the reference patterns, respectively. Since the reading pattern is always the same as the reference pattern, the convolution in Eq. 2 becomes the autocorrelation of the reference pattern with a phase term (or the reading pattern). Then from Eq. 1 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.

## Fig. 1

Transmission model of the collinear algorithm for (a) a writing process and (b) a reading process.

We use the three different reference patterns shown in Fig. 2, where Fig. 2 is the radial pattern with line width of $13.68\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ , Fig. 2 is the RBP mask with the modulation pitch of $13.68×13.68\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ , and Fig. 2 is the RP mask with the modulation pitch of $13.68×13.68\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ . The simulation result of the intensity distribution in the $x$ direction for the corresponding reference patterns is shown in Fig. 3, where $T=0.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , and the size of the spatial light modulator (SLM) is $4.4×4.4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The pitch between each pixel in the SLM is $13.68\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ . In the calculation, the effective focal length is set to $7.5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ and the effective wavelength is $532\phantom{\rule{0.3em}{0ex}}\mathrm{nm}∕1.5$ to simplify the calculation of the refraction on the boundary between the air and the holographic disk. We find that the first zero point of the PSF is the same in all three modulation cases. However, the sidelobe of the PSF can be dramatically reduced by RBP and RP modulation.

## Fig. 2

Three kinds of reference patterns.

## Fig. 3

PSF with different kinds of modulation.

Then we change the modulation pitch from $13.68\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}1.05\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ for both RBP and RP modulation. Figure 4 shows the simulation result. We find that the simulation result for RBP modulation is similar to that for RP modulation. When the modulation pitch is 1.05, 2.10, 3.15, and $13.68\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ , the first zero point of the PSF locates at 1.05, 2.10, 3.15, and $13.68\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ , respectively. This shows that the PSF can be improved by just narrowing the modulation pitch. The width of the PSF can be shortened to $2.1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ when the modulation pitch is $1.05\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ .

## Fig. 4

PSF for RBP and RP modulation with different pitches.

In summary, we proposed a theoretical analysis for calculating the PSF of a collinear holographic storage system when the reference pattern is RBP or RP modulation. The simulation results show that both RBP and RP modulation can be used to improve the PSF, and their performances are similar. In addition, we can shorten the width of the PSF to $2.1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ when the modulation pitch of the RBP mask or the RP mask is as small as $1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ .

## Acknowledgments

This study was sponsored by the Ministry of Economic Affairs of R.O.C. under Grant No. 95-EC-17-A-07-S1-011 and the National Science Council under Grant No. NSC 96-2221-E-008-031.

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© (2009) Society of Photo-Optical Instrumentation Engineers (SPIE)
Ye-Wei Yu, Chih-Yuan Cheng, Shu-Ching Hsieh, Tun-Chien Teng, Ching-Cherng Sun, "Point spread function by random phase reference in collinear holographic storage," Optical Engineering 48(2), 020501 (1 February 2009). https://doi.org/10.1117/1.3080725 . Submission:
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