June 15, 2008 / Vol. 33, No. 12 / OPTICS LETTERS 1297

Comparison of the angular selectivitycharacteristics for an off-axis holographic storage

system calculated by Born approximationand coupled-wave theory

Chih-Ming Lin1,* and Ludovic Angot2

1Department of Natural Science Education, National Taitung University, Taitung 950, Taiwan2Electronics and Optoelectronics Research Laboratories, Industrial Technology Research Institute (ITRI),

Hsinchu 310, Taiwan*Corresponding author: cmlin@nttu.edu.tw

Received January 22, 2008; revised April 22, 2008; accepted May 5, 2008;posted May 9, 2008 (Doc. ID 91826); published June 5, 2008

The first Born approximation is applied to calculate the angular selectivity for different positions on thereconstructed image as a function of the object beam’s optical axis angle �ob and reference beam angle �rw fora holographic data storage system that records the Fourier transform holograms in a medium with an infi-nite plane-wave reference beam. Results are compared with those calculated by the coupled-wave theory.© 2008 Optical Society of America

OCIS codes: 210.2860, 110.2990, 090.7330.

For holographic data storage (HDS) systems with anangular multiplexing configuration, the recording ca-pacity and the number of multiplexed pages at one lo-cation on the medium are closely related to the angu-lar selectivity. We use the first Born approximationmethod [1] to calculate the angular selectivity for theoff-axis HDS system of Fig. 1, with the assumption ofa weak coupling between the incident and diffractedwaves (modulation of refractive index �n muchsmaller than 1). We also compare and validate theseresults with the coupled-wave theory [2]: An exactapproach to compute the diffraction efficiency of asingle sinusoidal grating made by two infinite planewaves, without any constraint on the �n value.

The optical system with an angular multiplexingconfiguration is shown in Fig. 1. The objective beampath has a 4f configuration with the spatial lightmodulator (SLM), lens1, medium, lens2, and charge-coupled device (CCD) set apart from each other at thesame interval, the focal length f. The angle �ob be-tween the optical axis of the 4f configuration and thez axis represents the direction of the object beam.Angles �rw and �rr are respectively the incident direc-tions of the infinite plane-wave read and write refer-ence beams. For a symmetric incidence recording�ob=�rw. Angles and wavelength are considered in thematerial with a refractive index n=1.5. Using thefirst Born approximation [1], we can express the elec-tric field distribution U1�x1 ,y1� of the reconstructedimage on the CCD plane in terms of the electric fieldsof the object beam S�r�, the read and write referencebeams Err�r� and Erw�r�, the dielectric constantmodulation of the holographic grating ���r�, and thediffracted beam wave vector kd. This electric fielddistribution can be written as

U1�x1,y1� �� � � Err�r����r�exp�− ikd · r�d3r, �1�

0146-9592/08/121297-3/$15.00 ©

where

���r� � Erw* �r�S�r�, �2�

S�r� = c1� � dx0dy0U0�x0,y0�exp�− i��f�−12�y0y�

� exp�− i��f�−12�x0�z sin �ob + x cos �ob��

� exp�i�2�f2�−12��2f2 − x02 − y0

2��z cos �ob

− x sin �ob��, �3�

Erw�r� = exp�i�−12��x sin �rw + z cos �rw��, �4�

Err�r� = exp�i�−12��x sin �rr + z cos �rr��, �5�

Fig. 1. Holographic data storage system with an objectand a reference beam incident to the medium with oblique

angles.

2008 Optical Society of America

1298 OPTICS LETTERS / Vol. 33, No. 12 / June 15, 2008

exp�− ikd · r� = exp�− i�−12��f−1x1 cos �ob − �2f2�−1�2f2

− x12 − y1

2�sin �ob�x� � exp�− i�−12�y1y�

� exp�− i�−12��f−1x1 sin �ob

+ �2f2�−1�2f2 − x12 − y1

2�cos �ob�z�, �6�

U0�x0 ,y0� is the electric field amplitude distributionof the input pattern on the SLM plane, � is the wave-length of light in the medium, and c1 is a propagationterm. After simplification we get the intensity distri-bution on the CCD plane:

I1�x1,y1� � U12�x1,y1� � U0

2�x�,− y1�sinc2��−1T�f−1�x�

+ x1�sin �ob + �2f2�−1�x�2 − x12�cos �ob

+ cos �rw − cos �rr��, �7�

where T is the thickness of the holographic medium,sinc�x�=sin��x� / ��x�, and

x� = � ��x1 + f cot �ob�2 + 2f2 sin−1 �ob�sin �rw

− sin �rr��1/2 + f cot �ob. �8�

The sign before the bracket in Eq. (8) is negative for�ob0 and positive for �ob0. Equations (7) and (8)show that the sinc and x� functions are independentof y1. The first-order Taylor expansion of the argu-ment of the sinc function in Eq. (7) for ��=�rr−�rw�1 leads to

I1�x1,y1� � U02�x�,− y1�sinc2��−1T�� � �f cos �rw�x1

+ f cot �ob�−1�1 − f−1x1 cot �ob� + sin �rw��.

�9�

For x1=0, we can get the corresponding first nullvalue from Eq. (9) as

��null = � cos �ob�T sin��rw + �ob��−1, �10�

which makes I1�0,y1�=0 if ��=��null. The expressionin Eq. (10) is also the same as that calculated for theinterference grating of two infinite plane waves usingthe same Born approximation.

In Fig. 2 we show three angular sensitivity curvesfor x1=−5, 0, and 5 mm on the CCD image deducedfrom Eqs. (7) and (8). We assumed that U0�x ,y1�=1

Fig. 2. Sensitivity curves for x1=−5, 0, and 5 mm;

�ob=30°; �rw=0°; and ��=�rr−�rw.

for f=50 mm, �=354.7 nm (corresponding to 532 nmin air), �ob=30°, T=1 mm, �rw=0°, and the SLM sizematches the CCD size (100�100 pixels, 10�10 mm).Noncoincident intensity decays of the CCD image atdifferent x1 positions are found with an increase ofthe angular misalignment ��. The smallest ��nullvalues over the CCD image are found for x1=−5 mm.To illustrate the results shown in Fig. 2, in Fig. 3 weplot the intensity distribution of a reconstructed ran-dom binary pattern for �rr=0.028°. The intensity atx1=−5 mm is close to zero. In Fig. 4 we show the��null curves for x1=−5, 0, and 5 mm with �ob=30°and �rw=−15° –40°. The ��null values were obtainedby numerical interpolation. It should be noted thatSnell’s law and the possible spatial blocking of thereference beam by the object lens limit the angularrange inside the medium. We also found that the po-sitions at x1=−5 mm always have the smallest ��nullvalues, and the positions at x1=5 mm always havethe largest ��null values according to Eq. (7) if �rw−�ob. This implies that the write reference beam’s an-gular pitch when recording one page should be ad-equately chosen: It should be larger than the largest��null value to prevent cross talk between neighborpages; on the other hand, the angular misalignmentof the read reference beam should be smaller thanthe smallest ��null value to prevent a strong intensitydecrease in areas of the CCD image (cf. Fig. 3).

Fig. 3. CCD image for �rr=0.028° with �ob=30° and�rw=0°.

Fig. 4. First null values obtained from the Born approxi-

mation for x1=−5, 0, and 5 mm and �ob=30° (CCD plane).

June 15, 2008 / Vol. 33, No. 12 / OPTICS LETTERS 1299

In Fig. 5 we plot the ��null values of x1=0 deducedfrom Eq. (7) for �ob=20°, 30°, and 40° and comparethem with the ��null values numerically computedusing the diffraction efficiency equation�=�2 sinc2���2+ 2�1/2 /�� from the coupled-wavetheory [2], where �=��nT / �n��CRCS�1/2�; =�T / �2CS�; CR=cos �rr; CS=cos �rr− �K cos � /��;�=K cos��−�rr�− �K2� / �4���; �=2� /�; K and � arethe amplitude and angle of the grating vector, respec-tively, and can be deduced from the recording param-eter (�, �rw, and �ob); and �n is chosen to be 1�10−4.This �n value is arbitrarily chosen to frame with thedomain of validity of the Born approximation andcorresponds to a weak coupling between incident anddiffracted waves. Smaller values lead to similar re-sults, while larger values highlight the differencesbetween the two methods.

In Fig. 6 we compare the ��null values obtainedfrom the coupled-wave theory for different �n valueswith the results obtained using the Born approxima-tion for x1=0. When �n is larger than 1�10−4, thedifference of the ��null values becomes larger, and thebehavior is noticeably different at a high �rw angularrange. This shows the limits of the Born approxima-

Fig. 5. First null values obtained from the Born approxi-mation for x1=0 mm; �ob=20°, 30°, and 40°; and compared

with the results from coupled-wave theory.

tion to explain the behavior of some gratings andsome angular configurations.

In conclusion, this work aimed at comparing twotheoretical methods (first Born approximation andcoupled-wave theory) to compute the angular selec-tivity of a holographic data storage system in an an-gular multiplexing configuration. Having found thedomain of validity of the Born approximation, weused it to evaluate the reconstructed image intensitydistribution. We have quantitatively shown that thefirst Born approximation fails to predict an accurateangular selectivity for large modulations of therefractive index, a condition needed to obtain anincrease of the signal intensity. The approach of com-bining the two methods provides a new perspectivewhen analyzing a Fourier transform holographicsystem. In particular, the results presented can beuseful in the design and study of a HDS system.

References

1. C. Gu, P. Yeh, X. Yi, and J. Hong, in Holographic DataStorage, H. J. Coufal, D. Psaltis, and G. T. Sincerbox,eds. (Springer, 2000), p. 64.

Fig. 6. First null values obtained from the Born approxi-mation for x1=0 mm, �ob=30°, and from a coupled-wavetheory for �ob=30° and different �n values.

2. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).