1 February 2009 Point spread function by random phase reference in collinear holographic storage
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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

Collinear holographic storage (CHS) was proposed by Horimai1 in 2003. This system intrinsically has many advantages including compatible with existing disk storage systems2, 3, 4 such as CDs and DVDs. In addition, uniform shift selectivity in both radial and tangential directions and a fairly large wavelength shift and tilt tolerance are observed.5 However, the point spread function (PSF) of the proposed CHS is too wide spreading and as a result the diffracted signal is blurred, so the storage capacity in one page is limited. This restriction was not removed until the radial line reference pattern was proposed3 in 2005. One of the disadvantages of the radial line reference is that the energy is wasted due to the low transmitting ratio in area in the reference pattern. The dc peak with extremely high energy in the focal point also causes saturation of the material and decreases the quality of the hologram. Introducing defocus of the object lens is a way to blur the dc peak to avoid this overexposure effect. However, for a CHS system, the reference beam and the signal beam interfere only around the focal plane. Defocus will decrease the interference region inside the media and inevitably decrease the effective dynamic range of the medium. A different approach is to use random binary phase (RBP) mask or random phase (RP) mask to make the spatial spectrum wider and suppress the dc component in the Fourier plane.6, 7 This method will not decrease the interference region inside the media and is more suitable for solving the problem of dc peak. In this paper, we calculate the PSF of the collinear holographic storage system when reference pattern are RBP modulation and RP modulation. The simulation results show that with proper arrangement both the RBP mask and the RP mask can achieve extremely sharp PSFs.

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

PSF(ξ,η)=exp(jk4f)(λf)2TTexp[jπΔzλf2(ξ2+η2)][δ(ξ,η)psfz(ξ,η,Δz)]dΔz,
where ξ and η are the lateral coordinates, Δz is the longitudinal deviation from the center of the disk, T is the thickness of the disk, k is the wave number, λ is the effective wavelength, f is the effective focal length of the Fourier transform lens, δ(ξ,η) 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

psfz={Up(ξ,η)exp[jπΔzλf2(ξ2+η2)]}{Ur*(ξ,η)exp[jπΔzλf2(ξ2+η2)]},
where Up(ξ,η) and Ur(ξ,η) 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.

020501_1_1.jpg

We use the three different reference patterns shown in Fig. 2, where Fig. 2 is the radial pattern with line width of 13.68μm , Fig. 2 is the RBP mask with the modulation pitch of 13.68×13.68μm , and Fig. 2 is the RP mask with the modulation pitch of 13.68×13.68μm . The simulation result of the intensity distribution in the x direction for the corresponding reference patterns is shown in Fig. 3, where T=0.6mm , and the size of the spatial light modulator (SLM) is 4.4×4.4mm . The pitch between each pixel in the SLM is 13.68μm . In the calculation, the effective focal length is set to 7.5mm and the effective wavelength is 532nm1.5 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.

020501_1_2.jpg

Fig. 3

PSF with different kinds of modulation.

020501_1_3.jpg

Then we change the modulation pitch from 13.68to1.05μm 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 13.68μm , the first zero point of the PSF locates at 1.05, 2.10, 3.15, and 13.68μm , respectively. This shows that the PSF can be improved by just narrowing the modulation pitch. The width of the PSF can be shortened to 2.1μm when the modulation pitch is 1.05μm .

Fig. 4

PSF for RBP and RP modulation with different pitches.

020501_1_4.jpg

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 2.1μm when the modulation pitch of the RBP mask or the RP mask is as small as 1μm .

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.

References

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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
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