Delta-correlated pseudorandom address codes for volume holographic storage

Optics & Laser Technology 32 (2000) 429–441

www.elsevier.com/locate/optlastec

Delta-correlated pseudorandom address codes for volume holographic storage Adil Lahrichia; ∗ , Valentin Morozovb a Fotonix,

b E-TEK

2810 Marine Street, Suite B, Boulder, CO 80303, USA Dynamics, 1885 Lundy Avenue, San Jose, CA 95131, USA

Abstract We solve Maxwell’s equations for the intensity pro le of holograms recorded in a photo-sensitive material. The solution shows that interpage crosstalk noise is absent when the address codes used to reference the holograms are delta-correlated instead of orthogonal. We also develop a technique by which pseudorandom codes, which are delta-correlated, can be generated. We demonstrate both numerically and experimentally that these pseudorandom codes yield good hologram reconstruction and bounded interpage crosstalk c 2000 Elsevier Science Ltd. All rights reserved. noise. Keywords: Volume holographic storage; Pseudorandom codes; Interpage crosstalk noise; Contrast ratio; Delta correlated codes; Holograms; Multiplexing

1. Introduction Volume holographic storage oers high bit storage density of the order of 1012 =cm3 and fast information processing at potential write speeds of 100 MB=s and read speeds of 1 GB=s. Volume holographic storage is based on multiplexing multiple holograms or pages of data in a volume material. A hologram is formed when an object encoded with a light pattern interferes with a reference beam within a volume material. Various techniques have been demonstrated for multiplexing holograms in volume materials [1–3], namely angular multiplexing [1], wavelength multiplexing [3], and spatial light modulation addressing [4 – 6]. For instance, multiple holograms are multiplexed in a thick volume material by varying the angle between the reference beam and the object beam for each hologram. Holograms are also multiplexed in a volume material by assigning a dierent wavelength to the reference beam for each hologram during recording. Spatial light modulation where the reference beam is encoded with a light pattern can also be used to multiplex holograms in a volume material. Holograms are referenced with dierent light patterns and a hologram reconstructs when the reference beam illumi∗ Corresponding author. Tel.: +1-303-417-9381; fax: +1-303-4170922. E-mail address: adil [email protected] (A. Lahrichi).

nates the material with the address code assigned to it during recording. Spatial light modulation multiplexing requires a modulating device such as a spatial light modulator (SLM) to encode the reference beam with an address. In spatial light modulation multiplexing the reference beam and the object beam are xed during recording and readout, which makes it very attractive for commercial systems. However, it suffers limitations due to the SLM [4, 7] and the codes used to reference the holograms [4, 5, 8–10]. The SLM introduces crosstalk noise due to its nite contrast ratio and its pixels’ dimensions and spacings. The hologram addresses also contribute interpage crosstalk noise, which originates when a given hologram address accesses other holograms during readout. Besides, the number of addresses available to reference all desired holograms must not be limited. The ability to nd address codes that yield good hologram reconstruction and eliminate interpage crosstalk noise will address a major limitation for spatial light modulation multiplexing. Initially, orthogonal codes were proposed [4] as a solution to minimize interpage crosstalk for spatial light modulation multiplexing. Yet, it was shown later that orthogonal address codes must be additionally symmetric or antisymmetric around the midpoint [8] to minimize interpage crosstalk noise. Needless to say that the number of orthogonal codes that can be generated from a nite element sequence is limited to n2 , where n is the number of elements in the sequence. Hence, it is essential to

c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 9 8 - 0

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A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

index. The SLM is a phase SLM comprising n × m pixels, which are a wide, b long, and separated by a center-tocenter distance d, where each pixel represents a bit in the address The SLM transmittance function is P pattern. P Ts (x; y)= n m Rect((x−nd)=a)((y−md)=b))e jnm , where nm represents the phase of a given pixel in the address pattern. nm is 0 when the pixel is ON and  when it is OFF. The diraction pattern of the lens-SLM aperture at the back of the crystal located at f+z away from the aperture in the Fresnel regime [11] is URef (x; y) = ZZ × Fig. 1. Conventional setup used for (a) recording, and (b) reading out holograms.

develop rigorous criteria for selecting address codes that eliminate interpage crosstalk and yield good hologram reconstruction. In this paper, we solve three-dimensional Maxwell’s equations for the intensity pro le of encoded holograms during readout. We show from the solution that interpage crosstalk noise is absent when the hologram addresses are delta-correlated instead of orthogonal. We also outline a technique by which delta-correlated codes (pseudorandom codes) can be generated in large sets. Further, we show numerically that these codes yield good hologram reconstruction and achieve xed interpage crosstalk noise. Finally, we demonstrate experimentally that when plane wave holograms are referenced with these pseudorandom address codes interpage crosstalk noise peaks are bounded. 2. Hologram recording and readout 2.1. Hologram recording Fig. 1(a) shows a conventional system that is used for recording holograms using spatial light modulation. An object beam and an encoded reference beam intersect at an angle  inside a volume photosensitive material to record a hologram. The SLM encodes the reference beam with a checker board pattern made of OFF and ON pixels, which serves as the hologram address. The object beam is chosen to be a plane wave to simplify the task of deriving a closed form solution for the intensity pro le of the hologram during readout. We treat the combination of the lens and the spatial light modulator as a plane aperture, whose transmittance function is T (x; y) = T‘ (x; y)Ts (x; y). The transmittance of 2 2 the lens is T‘ = e j knL
0 e−( j k=2f)(x +y ) , where k = 2= is the wave vector,  is the wavelength, f is the focal length of the lens,
0 its thickness, and nL its refractive



Ae j k(z+f) j knL
0 −( j k=2z0 )(x2 +y2 ) e e j(z + f) Ts exp(−(jk=z 0 )(xx0 + yy0 ))e−j d x0 dy0 ; (1)

where A is the amplitude of the monochromatic wave impinging on the lens, is equal to ((x02 + y02 )=2)(−1=f + 1=z),  is the area of the aperture, x0 and y0 are the Cartesian coordinates in the aperture plane, and x and y are the Cartesian coordinates at the crystal’s back face. Since

is small, it can be replaced by its mean  to obtain (see Appendix A) e j k(z+f) j k(x2 +y2 )=2(f+z) e j(f + z)     by ax sinc ×sinc (f + z) (f + z) P P −j kn d x=x(f+z) −j km dy=(f+z) jnm e e e : × 

Uref (x; y) = Aabe−j k

n

m

(2)

For a unit amplitude wave A is equal to 1 and for small z we have 1=(f+z) ≈ (1=f−z=f). The object beam is a unit monochromatic wave given by UObj (y; z)=e j k(y sin −z cos ) , where  is the angle between the reference and object beams. When the object beam interferes with the reference beam, an index modulation n is produced in the material   ax 2ab sinc n(x; y; z) ˙ (f + z) (f + z)   by p(x; y; z); (3) ×sinc (f + z) where

 n dx cos k z(1 + cos ) − y sin  − p(x; y; z) = f +z n m  2    x + y2 1 z nm m dy + − 2 : − f+z 2 f+z f k XX

In this analysis the SLM is considered ideal and does not contribute noise due to its nite ratio and pixel size and pitch. Besides, the amplitude and phase during recording and readout are taken to be the same. Energy transfer and phase redistributions, which result from the interaction of the object and reference beam inside the material

A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

are also neglected. It is also assumed that the interference pattern and index modulation in the material are in phase.

0

Ez (x; y; z) ˙

2.2. Holgram readout Fig. 1(b) shows the apparatus used to reconstruct a hologram that is recorded in the material. The encoded reference beam illuminates the material, while the object beam is blocked. The reconstructed hologram is imaged onto the detector via the imaging lens that is located at distance f in front of the crystal and f2 from the detector. The Helmholtz equation [12,13] −∇2 E + (∇(∇ · E)) = 2 k E, where =n20 is the medium permittivity and n0 is the medium index of refraction, gives the electric eld pro le of the reconstructed hologram that is diracted from the crystal. Assuming a small index modulation n=n0 1, we write n ≈ n(1+(n=n0 )). The material conductivity is also considered small and ideally zero, which makes ∇ · E = −E · ∇log , and the Helmholtz equation becomes   n n2 + 2 ∇2 E + k 2 n20 1 + 2 n0 n0    n = 0: (4) +2∇ E · ∇log n0 1 + n0 Since the index modulation n is small, the method of small perturbation [13] can be used. The total eld solution of the Helmholtz equation is set to E = E0 + E1 , where E1 is the rst-order component of the electric eld due to the refractive index modulation. E1 yields the pro le of the diracted eld at the exit of the crystal, i.e.     0 0 n n − 2k 2 E0 ; (5) ∇2 E1 + k 2 E1 = −2∇ E0 · ∇ n0 n0 whose solution is     Z   0 n 1 n 2 2∇ E0 · ∇ + 2k E0 E1 (R) = 4 v0 n0 n0

×e jpq q(x; y; z)k;

  n dx sin k z(1 + cos ) − y sin  − q(x; y; z) = f+z n m    1 z nm m d x (x2 + y2 ) + − 2 + : − f+z 2 f+z f k PP

The phases of the pixels of the address codes are indexed by n and m during recording and p and q during readout. When the hologram is read out with its corresponding code mn = pq , and when it is read out with a dierent code mn 6= pq . The coordinates at the back of the material are labeled by x0 and y0 and at the front of the material by x and y. When Ey and Ez are substituted in Eq. (6), the diracted eld at the exit of the material becomes 0

1 k 2 a2 b2 L j k 0 f j k 0 z (j k 0 =2z)(x2 +y2 ) e e e E1 (x; y; z) ˙ − 2 jn0 2 f2 z     (1 + cos ) (j k 0 L=2)(1+cos ) × e F sinc k 0 L 2 #!! 0 0 02 02 0 (x 2 + y 2 ) ∗ F(e( jk =2z)(x +y ) ) + 2 f ∗F(e−j k y sin  ) ∗ F( nm )·F( pq )(j + k sin )   0 0 kL (1 + cos ) + e−(j k L=2)(1+cos ) sinc 2

(6)

2a2 b2 k 2 j k(z+f) j k(x2 +y2 )=2(f+z) e e j2 f2     ax by 2 2 sinc ×sinc (f + z) (f + z) P P −j kp d x=x(f+z) −j kq dy=(f+z) e e ×

0

×F(e(j k =2z)(x

0

+y 2 )

0 0

) ∗ F(e j k y

sin 



) (9)

where PP n

m

0

e(j k =2)(x

×e jnm sinc

(7)

02

∗ ∗ F( nm ) ∗ F( pq )(j − k sin ) ;

nm (x; y) =

q

×e jpq p(x; y; z)j;

(8)

where

0

p

q

0 0

where R0 represents a variable vector ranging over the volume v0 of the medium and R represents the observation point. When the incoming plane wave impinging on the SLM is a transverse electromagnetic (TE) wave [12], dominant 0 components in 2∇(E0 · ∇n=n0 ) + 2k 2 (n=n0 )E0 , which occur in the y and z directions are Ey (x; y; z) ˙

2a2 b2 k 2 sin  j k(z+f) j k(x2 +y2 )=2(f+z) e e 2 f 2     ax by sinc2 ×sinc2 (f + z) (f + z) P P −j kp d x=(f+z) −j kq dy=(f+z) e e × p

0

e j k|R−R | 0 dv ; × |R − R0 |

431



02

0

+y 2 )(1=f−L=f2 ) −j k 0 (n d x0 +m dy0 )=f

ax0 f

e



 sinc

by0 f



and L is the crystal thickness. The electric eld is then transformed by the Fourier lens whose focal length is f2 before it reaches the detector

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A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

(see Fig. 1(b)). At the detector the intensity pro le is [11,14,15]   0 kL (1 + cos ) Idif (x; y) ˙ (1 + sin2 ) sinc2 2        D1 x D2 y −f sin  × sinc sinc ∗ f2 f2 b     PP f f nd md Rect Rect x− y− ∗ af2 a bf2 b n m   PP f pd Rect x+ ×e−jnm ∗ af2 a p q  2  qd f e jpq ; y+ (10) × Rect bf2 b where sinc(D1 x=f2 ) sinc(D2 y=f2 ) is the Fourier transform of the lens pupil function and D1 and D2 are the width and length of the imaging lens aperture (see Appendix B). 3. Interpretation of the intensity proÿle of the hologram When the SLM is ideal, Eq. (10) shows that the spectrum of the reconstructed image is a convolution of the Fourier transform of the aperture, the Fourier transform of the object, which is a plane wave, the address code used to record the hologram, and the address code used to read it out. When the codes for recording the hologram and reading it out are the same, i.e. nm = pq , the convolution of the codes is an autocorrelation, and when they are dierent i.e. nm 6= pq , it is a crosscorrelation. We deduce from Eq. (10) that interpage crosstalk noise originating from the address codes is absent when the pairwise crosscorrelation of the codes is identically zero. Eq. (10) also shows that a replica of the object forms when the autocorrelation of the record and readout codes is exactly a delta-dirac function. When it is the case, the intensity diracted to the detector is proportional to the Fourier transform of the object beam, which is also a delta-dirac function. When the pairwise crosscorrelation of the address codes in a set is zero and their autocorrelation is a delta-dirac function, the address codes are said to be delta-correlated. Consequently, interpage crosstalk noise is absent in the reconstructed images when the codes are delta-correlated instead of orthogonal as it was previously believed. 4. Hologram contrast ratio When the address codes used to reference holograms are not perfectly delta-correlated, interpage crosstalk noise will result. For instance, a given address code will access other holograms during readout. When the SLM does not contribute additional crosstalk noise, interpage crosstalk

noise will set the upper limit on the number of holograms that can be multiplexed in a volume material for a target signal-to-noise ratio. The amplitude of interpage crosstalk noise will depend on how highly delta-correlated the address codes are. Hence, the aim is to nd codes that make the crosscorrelation peaks very small. When aperture eects due to the lenses are ignored, the pro le of the reconstructed hologram will be ideally a delta-dirac function, which is the Fourier transform of the object beam (plane wave). Assuming that l holograms are multiplexed in a storage material, the interpage crosstalk noise amplitude is de ned by the parameter HCR l−1 . HCR l−1 is the peak intensity in the spectrum of the reconstructed hologram divided by the sum of the intensity of the maximum peak in the image spectrum when the hologram is read out with each of the l − 1 address codes, i.e. HCR l−1 =

∗ ∗ Cnm )2 (Cnm ; ∗ l [Cnm ∗ Cpq ]2l

(11)

where Cij is i; j Rect((fx − idf2 )=af2 ; (fy − jdf2 )=bf2 ) e−jij and Cij∗ is i; j Rect((fx + idf2 )=af2 ; (fy + jdf2 )=bf2 )e jij ; i and j index a pixel in the hologram address mask pattern, i; j represents a double summation over the number of pixels in the SLM, and Cnm 6= Cpq . Even though the noise power contributions from other holograms must be added incoherently, they were added coherently to assess the worst case scenario for interpage crosstalk noise. Thus, Eq. (11) gives the minimum number of plane wave holograms that can be multiplexed in a volume material for a predetermined HCR l−1 . 5. Pseudorandom codes Besides being delta-correlated, the address codes must be easy to generate and preferably binary so that an SLM can be used. In addition, they must be short to t a reasonable mask size so that no resolution problem occurs. Most importantly, the number of holograms that can be referenced must not be limited by the number of address codes available. Pseudorandom codes [16, 17] constitute a special class of random codes. Their autocorrelation properties mimic those of random codes with the added advantage that their crosscorrelation peaks are bounded [9, 10, 18–21]. They can be generated using the shift register method [16, 17]. In this technique an initial preselected sequence is input into an algorithm, which generates sequences with similar properties to the initial sequence. However, the shift register method presents the limitation that it relies on the initial input sequence. In our present work, pseudorandom address codes are generated as follows. A random generator is used to form a set of one-dimensional random sequences [22]. Two sequences are used to form a two-dimensional address mask that is separable in x and y.

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433

Fig. 2. (a) – (d) Normalized light intensity spectrum of a reconstructed plane wave hologram referenced with a 20 × 20 pixels random mask.

One sequence is laid on all the SLM rows and the other is laid over all the SLM columns. Similarly, other twodimensional addresses are generated using other sequences from the one-dimensional random set. Dierent addresses are assigned to dierent holograms. Next, each hologram is read out with its corresponding address code and with other address codes and HCR 1 is calculated. The results are then tabulated to extract the address codes that yield a xed HCR 1 . These address codes form a set of pseudorandom address codes whose maximum crosscorrelation peaks are bounded.

6. Numerical simulations and results Using the method outlined in the previous section a two-dimensional set of 20×20 bit random codes was generated. The SLM parameters were selected to be nd=a = md=b = 10, the focal lengths of the lenses L and L2 were set equal, i.e. f = f2 . Four distinct addresses were selected from the set to reference holograms. Fig. 2 shows the pro le of the reconstructed holograms. The address masks used for recording a hologram (left) and reading it out (right) shown at the top of each plot are identical.

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A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

Fig. 3. (a) – (d) Normalized light crosstalk noise spectrum when a plane wave hologram is read out with a dierent 20 × 20 random mask.

The black square in each hologram address mask represents an OFF pixel with a  phase state (bit 0) and the white square represents an ON pixel (bit 1) with phase zero. The pro le of the reconstructed holograms shown in Fig. 2, is marked by a narrow central lobe surrounded by lower amplitude peripheral lobes. The width of the central lobe, which is 2d=a in the x-direction and 2d=b in the y-direction, determines the resolution of the image. The peripheral lobes represent undesired light intensity. The pro le of the reconstructed holograms approaches that

of a delta-dirac function, which represents ideally the Fourier transform of the object beam. Fig. 3 displays the noise spectra when holograms are read out with dierent random address codes. The masks on the top of each plot represent the code used for recording the hologram (left) and the code used for readout (right). The spectra display peaks of variable amplitude, with the highest peak being the maximum noise intensity that is contributed to the recorded hologram during readout. For these four examples HCR 1 was equal to 12.5, 66.6, 40, 1250. For the other addresses in the random set

A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

435

Fig. 4. (a) – (d) Normalized light intensity spectrum of a reconstructed plane wave hologram referenced with a 20 × 20 pixels pseudorandom mask.

HCR 1 uctuated from a low of 1.9 to a high of 66.7. Similar results [14] were obtained with codes that are 10×10, 15×15, and 25×25 bits long. Based on these results, it is evident that purely random address codes are not feasible for mass storage since the amplitude of their crosscorrelation peaks cannot be predicted. Nevertheless, they still achieve good hologram reconstruction. A close analysis of the calculated values of HCR 1 for dierent 20 × 20 bits address codes showed that there are address codes for which HCR 1 is xed. These address codes, which are random in nature and yield bounded

crosscorrelation peaks, can be grouped to form a set of pseudorandom addresses. For illustration four 20 × 20 bit pseudorandom address codes were selected. Fig. 4 shows the reconstructed holograms referenced with these codes while Fig. 5 shows the noise peaks when holograms are readout with dierent address codes. The masks on the top of each plot show the address code used for recording the hologram (left) and the address code used to read it out (right). The plots show that the holograms mimic a delta-dirac function pro le and that the amplitude of their maximum noise peak is xed.

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A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

Fig. 5. (a) – (d) Normalized light crosstalk noise spectrum when a plane wave hologram is read out with a dierent 20 × 20 pseudorandom mask.

Fig. 6 shows a matrix of 15 dierent one-dimensional random sequences that are 15 bits long. The sequences occupy rows in the matrix and are labeled in ascending order. The sequence in the bottom row is referenced by the numeral 1 and the sequence in the top row by the numeral 15. The dark square in each sequence represents the bit zero and the white square represents the bit 1. Table 1 lists the reference number of two one-dimensional sequences used to form each two-dimensional record and readout address code. The sequence on the left column is laid on all the rows of the address mask (code x) and the sequence on the right is laid all over the columns of

the address mask (code y). The results in Table 1 show the random address codes (marked by a star) that yield a constant HCR 1 . Since the crosscorrelation peaks for pseudorandom address codes are bounded, the minimum number of holograms that can be multiplexed in a volume material to achieve a target HCR l−1 can be predicted. When the pseudorandom address codes are 10×10; 15×15; 20×20; 25× 25 bits, it was found [14] numerically that for HCR l−1 equal to 1 interpage crosstalk noise sets the minimum number of holograms that can be multiplexed in the material to 17, 30, 44, and 150, respectively. Note that the

A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

437

Table 1 Calculated hologram contrast ratio (HCR 1 ) for 15 × 15 bits random masks

Fig. 6. Matrix displaying 15 pixels long one-dimensional random codes (bottom row represents code 1 and top row represents code 15).

noise amplitudes were added coherently instead of incoherently. For pseudorandom addresses that are 15 × 15 bits, it was veri ed that over 30 addresses were available to address all the 30 holograms for HCR l−1 equal to 1 (Table 1). It must be also pointed out that there were 15 × 15 bits random address codes that yielded a value of HCR 1 larger than 30 (Table 1). However, only few pseudorandom codes were available to reference all the holograms for a target HCR l−1 of 1. 7. Experimental results Light from an argon laser ( = 514 nm), was split into plane waves, a reference beam (10 mW=cm2 ), which illuminates a 15×15 bits pattern etched on a 2×2 cm2 transparent mask, and an object beam (10 mW=cm2 ) which passes through a Fourier lens. The beams were made to interfere within a 1 × 2 × 0:2 cm3 and 0.015% iron-doped lithium niobate crystal at a narrow angle (few degrees) to minimize lens abberations that might aect the measurements. A total of four 15 × 15 bits random masks labeled C1 ; C2 ; C3 , and C4 and four pseudorandom masks labeled C1 ; C2 ; C3 , and C4 were fabricated for this experiment. A Fourier transform plane wave hologram was recorded in the crystal using a random mask Ci , where i = 1; 2; 3; 4 is the index number assigned to each mask in the set. Then, the object beam was blocked and the hologram was read out with the reference beam encoded with its corresponding address mask. During readout, the intensity of the reference beam was reduced drastically to minimize hologram erasure characteristic of photorefractive materials. The maximum intensity peak in the spectrum was

Recording code Code x Code y

Readout code Code x

Code y

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 9 10 11 12 13 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 6 6 12 12 4 6 9

2 3 4 5 6 7 8 9 10 10 11 12 13 14 15 7 7 7 7 7 7 7 7 7 7 7 2 3 6 13 14 15 15 15 15 15 15 15 15 15 15 15 15 15 7 7 7 7 7 7

2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 6 7 8 3 8 8 8 8 8 8 8 8 8 8 8 8 8 8

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 6 7 10 13 14 15 6 6 4 14 14 14 14

HCR 1 9.4 29:7∗ 1.9 9.4 29:7∗ 29:7∗ 29:7∗ 9.4 4.2 1.9 9.4 1.9 3.8 9.4 9.4 29:7∗ 10.7 7.4 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 16.7 29:7∗ 16.7 10.7 16.7 16.7 16.7 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 66:7 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗ 29:7∗

found by scanning the image spectrum horizontally with a movable iris. The diameter of the iris was many times smaller than the diameter of the diracted spot. Next, the hologram was read out using the other three address codes and the amplitude of the maximum peak in the diracted light spectrum minus the background light was measured.

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Table 2 Measured values of hologram contrast ratio (HCR 1 ) for plane waves referenced with random and pseudorandom codes Code type

C1=C2

Pseudorandom 7.74 Random 2.48

C1=C3

C1=C4

C3=C1

C3=C2

C3=C4

7.74 1.97

7.12 3.21

6.37 3.8

6.15 2.16

7.48 2.11

Fig. 7. Numerical and experimental hologram contrast ratio (HCR 1 ) as a function of the number of pixels in the one-dimensional pseudorandom sequence.

Upon completion of this procedure for the random masks, the experiments were duplicated for the pseudorandom masks. Table 2 lists the results of the previous experiments, where Ci represents the address code used for recording the hologram and Cj is the address code used for reading it out. The results show that when the address codes are purely random, the variations in HCR 1 are large with respect to the amplitude of HCR 1 . Thus, for purely random address codes the amplitude of HCR 1 depends on the address codes in the set of random addresses. In contrast, for pseudorandom codes the measured HCR 1 is approximately constant within a small marginal error of few percents. The impact of increasing the number of bits in the address code on HCR 1 was also investigated. Four sets of pseudorandom addresses, which comprised address codes that are 10 × 10; 15 × 15; 20 × 20, and 25 × 25 bits were formed and masks corresponding to each code were fabricated. The previous experiments were repeated for each pseudorandom set. It was found that HCR 1 remained approximately constant for all the address codes in each set, but increased when the number of bits in the codes were increased. The amplitude of HCR 1 was measured at 6, 7.3, 11.2, 21.8 when the codes were 10×10; 15×15; 20×20, and 25 × 25 bits, respectively. The results are plotted in Fig. 7 (lower curve) and are tted to the function 0:03n2 .

8. Discussion and analysis The two-dimensional pseudorandom codes that were generated in our work were based on one-dimensional codes. Therefore, the maximum amplitude of the autocorrelation peak that can be achieved with these address codes is n2 , where n is the number of bits in the one-dimensional sequence. When the codes are perfectly delta-correlated the maximum HCR 1 is n2 . This explains the observed increase in HCR 1 when the number of bits in the address code was increased. Physically, as the number of the bits in the sequences increases, the sequences become more random and as a result highly delta-correlated. This causes the autocorrelation peaks to increase. Consequently, the number of holograms that can be multiplexed in the material for a given HCR 1 also increases. Fig. 7 compares the values of HCR 1 predicted numerically to the measured HCR 1 values as a function of the number of bits in the one-dimensional sequences used to generate the two-dimensional address codes. Even though our experimental results con rmed our theoretical results, our measured values of HCR 1 were far below the calculated values. For instance, the upper curve is tted to the function 0:22n2 , while the lower curve is tted to 0:03n2 . Numerous factors contributed to this discrepancy. First, the masks that were fabricated for referencing the holograms used amplitude modulation instead of phase modulation. Phase modulation where OFF pixels assume a value of −1 and ON pixels a value of 1 is considered more energy ecient than amplitude modulation where ON pixels assume the values 0 and OFF pixels a value of 1. As a result, the crosscorrelation peaks will be higher in the case of amplitude modulation resulting in a decrease of HCR 1 . Moreover, the address masks have nite contrast ratio, which might produce errors in the address codes assigned to given holograms. When an error occurs in an address code, the amplitude of the crosscorrelation peaks will increase causing a decrease in HCR 1 . Thirdly, lens aberrations, aperture eects, light re ections, and light scattering were not accounted for in our numerical simulations. Finally, the setup used for demonstration was not optimized. Ideally, interpage crosstalk is absent in the reconstructed holograms, when holograms are referenced with deltacorrelated address codes. However, since in practice the codes are never perfectly delta-correlated interpage crosstalk noise originates during readout. Consequently, the success of spatial light modulation multiplexing will always rely on nding address codes that are highly delta-correlated. Further, for practical purposes generating large sets of pseudorandom codes must not be computer intensive. The pseudorandom addresses that were generated for our analysis reconstructed the holograms well and produced xed interpage crosstalk noise peaks. Nonetheless, extensive testing of two-dimensional random address codes was required to form the set of pseudorandom address

A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

codes. This is acceptable when few address codes are needed, but might not be feasible otherwise. Hence, it is mandatory to compare the performance of dierent methods used to generate large sets of pseudorandom address codes. Presently, a method where extensive testing of the address codes is not required is being investigated. For instance, few address codes that are highly delta-correlated are generated using our technique, then the address codes are input into the shift register algorithm to generate address codes with similar properties.

where 0

Ey1 ˙

0

×e( j k =2)(x ×e

PP p

q



mdy + f 

exp −jk

z 1− f

0





sinc2 e 

0

0

2



by f

2

0





PP n



m

z 1− f 

pdx f

pq − k

ax f

×e j k (z+f) e ( j k =2)(x



0

e j k (z+f)



 −

mn k

z 1− f



ndx f 

exp −jk

0





sinc2



by f



+y2 )(1=f−2z=f2 )

2

2

0

0

×e−j k =2(x +y )(1=f−2z=f ) e−j k z(1+cos ) e j k y sin       PP z ndx 1− × exp jk 0 f f n m    z mn mdy 1− − + f f k PP

When the focal length of the lens is much larger than the dimensions of the volume storage material, numerical simulations show that can be approximated with its  For instance, for an argon laser ( = 514 nm) median . a 1 cm × 1 cm × 1 mm crystal, and a 10 × 10 address mask, the amplitude of the index modulation varies by 4% while the index pro le remains unchanged when the term e−j is taken outside the integral in Eq. (1). Eq. (1) can thus be rewritten as

qdy + f

p

q



 exp −jk

z 1− f



0



pdx f

pq − k



z 1− f



 :

The following integrals can thus be solved, i.e. Z 0 0 e j k |R−R | 0 k 02 a2 b2 L j kf j k 0z (j k 0=2z)(x2 +y2 ) dv e e e Ey1 ˙ |R − R0 | jn0 2 f2 z v0 Z 0 0 ( j k 0L=2)(1+cos ) e−j k y sin  ×e

Ae j k(f+z) j knL
0 −(j k=2z0 )(x2 +y2 ) −j  e e e j(f + z)     PP x − nd y − md inm ; e ; ×F Rect a b n m (A.1)

0

02

02

0

0

×e(j k =2z)(x +y )   1 + cos  ×sinc k 0L 2  1 02 02 nm pq − 2 (x + y ) f

where F represents the Fourier transform of the aperture at the spatial frequencies fx = x=z 0 ; fy = y=z 0 .

0

×e ( j k =z)(xx +yy ) dx0 dy0 ; Z

When z=f is small, the diracted eld Ey in Eq. (7) can be rewritten in exponential form, i.e Ey = Ey1 + Ey2 ,



k 02 a2 b2 sinc2 jn0 2 f2

×

Appendix B. The diracted ÿeld at the detector



e

qdy + f Ey2 ˙

ax f

+y2 )(1=f−2z=f2 ) ( j k 0=2)(x2 +y2 )(1=f−2z=f2 )

z × 1− f ×



j k 0z(1+cos ) −j k 0y sin 

Appendix A. Approximation of by its median 

URef (x; y) ≈

2



9. Conclusions We solved Maxwell’s equations and derived the threedimensional intensity pro le of reconstructed plane wave holograms encoded with a checker board light pattern. The pro le of the reconstructed hologram showed that interpage crosstalk is absent when the address codes used to reference the holograms are delta-correlated instead of orthogonal. Numerical simulations and experiments also showed that holograms reconstruct well and yield bounded crosscorrelation peaks when they are referenced with deltacorrelated codes. Still further, it was shown that interpage crosstalk decreases when the number of pixels in the address codes increases. Finally, it must be pointed out that other techniques to generate highly delta-correlated address codes must be investigated to propel spatial light modulation multiplexing toward commercial use.

k 2 a2 b2 sinc2 jn0 2 f2

439

0

e j k|R−R | 0 k 02 a2 b2 L j kf j k 0z ( j k 0=2z)(x2 +y2 ) dv e e e Ey2 ˙ |R − R0 | n0 j2 f2 z v0 Z Z 0 0 −j k 0L(1+cos )=2 e+j k y sin  ×e

440

A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441 ( j k 0=2z)(x0 2+y0 2 )

×e

  0 kL (1 + cos ) sinc 2

0

0

0

∗ pq e( j k =z)(xx +yy ) dx0 dy0 ; × nm ∗ is the complex conjugate of nm . where nm R j k 0|R−R0 | The integral Ez e |R−R0 | dv0 is solved [14] similarly and the total diracted eld at the exit of the crystal becomes [15]

1 k 02 a2 b2 L j k 0f j k 0z ( j k 0=2z)(x2 +y2 ) e e e 2 jn0 2 f2 z     0 (1 + cos ) × e( j k L=2)(1+cos ) F sinc k 0L 2   0 02 02 0 0 (x0 2+y0 2 ) ∗ F(e(jk =2z)(x +y ) )∗F(e −jk y sin  ) + f2

E1 (R)˙−

−( j k 0L=2)(1+cos )

∗F( nm ) ∗ F( pq )(j + k sin ) + e  0  0 02 02 kL ×sinc (1 + cos ) F(e( jk =2z)(x +y ) 2

 0 0 ∗ ∗F(e j k y sin  ) ∗F( nm ) ∗ F( pq )(j − k sin ) : (B.1) After exiting the back face of the crystal, the diracted eld propagates a distance d0 before it traverses the imaging lens whose focal length is f2 . Then, it reaches the detector located at the focal length of the imaging lens to become [11,14,15] 0

0

0

2

2

E2 (x; y; f2 ) ˙ e j k f e j k L e j kf2 e( jk =2f2 )(x +y )    0 02 02 y x × P ; ∗ F(e ( j k =2d0 )(x +y ) ) f2 f2    (1 + cos ) 2 2 (jk 0L=2)(1+cos ) sinc k 0L ∗  f e 2  2 2 0 2 2 2 2 0 (x + y ) e ( jk f =2f2 d0 )(x +y ) e( jk fsin y=f2 ) + 2 f2     0 2 ax 2 by ( jk (f−L)=f22 )(x2+y2 ) sinc ×e sinc f2 f2 P P j2n d x=f2 j2m dy=f2 jnm e e e ×

where P(x; y) = D1 D2 sinc(D1 x=f2 ) sinc(D2 y=f2 ) is the Fourier transform of the lens pupil function P(x; y) = Rect(x=D1 ) Rect (y=D2 ). For d0 equal to f and when the crystal is not very thick, i.e. L is small, the diracted eld becomes 0

0

E2 (x; y; f2 ) ˙ −je2j k f e ( j kf2 ej k =2f2 )(x

2

+y2 ) −(j k 0d0 =2f22 )(x2 +y2 )

e

0

0 2

2

2

n

 ×sinc ∗F

m

PP p





a f2 q



b sinc f2



e( j2p d=f2 ) e( j2q d=f2 ) e jpq

 b (j + k sin ) f2   0 0 kL (1 + cos ) +e−( j k L=2)(1+cos ) sinc 2   0 y x ; ×P ∗ F(e−j k f sin y=f2 ) f2 f2  P P −( j2n d=f2 ) −( j2m d=f2 ) −jnm e e e ∗F ×sinc

n

 ×sinc ∗F



a f2

p



q



sinc

m

PP

×sinc



a f2

 sinc

b f2



e( j2p d=f2 ) e( j2q d=f2 ) e jpq

a f2





b sinc f2



 (j + k sin ) ; (B.3)

n m

×

PP p q

e

j2p d x=f2 j2q dy=f2 jpq

e

e



(j + k sin ) 

k 0L (1 + cos ) 2   2 ax ( jk 0f2 =2f22 d0 )(x2+y2 ) −jk 0f sin y=f2 ×e e sinc f2   by P P −j2n d x=f2 −j2m dy=f2 e e ×sinc2 f2 n m P P j2p d x=f2 j2q dy=f2 ×e−jnm e e p q 
× e jpq (j − k sin ) (B.2) 0

+ e−(j k L=2)(1+cos ) sinc

2

×[2 f24 e( j k L=2)(1+cos ) e( j k f =4f2 (f−L))(x +y )    f4 (x2 + y2 ) (1 + cos ) 0 + ×sinc k L 2 4f24 (f − L)2   0 y x ∗ F(e j k f sin =f2 ) ; ∗P f2 f2  P P ( j2n d=f2 ) ( j2m d=f2 ) jnm e e e ×F

2

2

2

where we used the property (x)=limN →∞ N 2 e−N (x +y ) . The comparison of the magnitude terms in E2 reveals that the second term is most relevant [14]. Hence, the diracted eld at the detector becomes [15] 0

0

E2 (x; y; f2 ) ˙ e2j k f e j kf2 e j(k =2f2 )(x

2

+y2 ) −( j k 0 d0 =2f22 )(x2 +y2 )



e



k 0L (1 + cos ) 2     D1 x D2 y ×sinc sinc f2 f2 0

×e(−j k L=2)(1+cos ) sinc

A. Lahrichi, V. Morozov / Optics & Laser Technology 32 (2000) 429–441

PP



f nd f f sin  ; x− y− af a bf b 2 2 n m   PP f pd md e−jnm ∗ ; Rect x+ − b af2 a p q  f qd e jpq (j − k sin ) y+ bf2 b ∗

Rect

and the light intensity at the detector is   0 kL 2 2 (1 + cos ) Idif (x; y) ˙ (1 + sin ) sinc 2        D2 y −f sin  D1 x sinc ∗ × sinc f2 f2 b   PP f nd Rect x− ∗ af a 2 n m   md f e−jnm y− ×Rect bf2 b   PP pd f ∗ Rect x− af2 a p q   2 qd f jpq e y+ (B.4) ×Rect bf2 b References [1] Mok FH. Angle-multiplexed of 5000 holograms in Lithium Niobate. Opt Lett 1993;18:915–7. [2] Anderson DZ, Lininger DM. Dynamic optical interconnects: volume holograms as two-port operators. Appl Opt 1987;6:5031–8. [3] Rakuljic GA, Leyva V, Yariv A. Optical data storage by using orthogonal wavelength-multiplexed volume holograms. Opt Lett 1992;17:1471–3. [4] Denz C, Pauliat G, Roosen G, Tschudi T. Potentialities and limitations of hologram multiplexing by using the phase-encoding technique. Appl Opt 1992;31:5700–5. [5] Denz C, Pauliat G, Roosen G, Tschudi T. Volume hologram multiplexing using a deterministic phases encoding method. Opt Commun 1991;85:171–6. [6] Pauliat G, Roosen G. New Advances in photorefractive holographic memories. Int J Opt Comput 1991;2:271–91. [7] Lahrichi A, Johnson K, Manilo E. Signal–to–noise limitations on the number of channels in holographic interconnection networks. J Opt Soc Am A 1992;9:749–54. [8] Hermanns A. Ph.D. thesis, Department of Physics, University of Colorado, Boulder, CO 80309, 1994. [9] Lahrichi A, Morozov V, Johnson K. Volume hologram storage using random patterns. OSA 1994 Annual Meeting, Dallas, Texas. [10] Lahrichi A, Morozov V, McKnight D, Mao C, Johnson K. Phase only encoding in 3-d holographic memory. OSA 1995 Annual Meeting, Portland, Oregon. [11] Goodman JW. Introduction to Fourier optics. New York: McGraw-Hill, 1968. p. 30 –280. [12] Ramo S, Whinnery JR, Van Duzer TY. Fields and waves in communication electronics. New York: Wiley, 1965. p. 270 –311. [13] Tatarski VI. Wave propagation in a turbulent medium. New York: McGraw-Hill, 1961. p. 6 –127.

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[14] Lahrichi A. Ph.D. thesis, Department of Electrical Engineering, University of Colorado, Boulder, CO 80309, 1995. [15] Papoulis A. Systems and transforms with applications in optics. New York: McGraw-Hill, 1968. p. 61–98, 297–358. [16] Lewis TG, Payne WH. Generalized feedback shift register pseudorandom number algorithm. J Assoc Comput Mach 1973;20:456–68. [17] Arvillias AC, Maritsas DG. Partitioning the period of a class of m-sequences and application to pseudorandom number generation. J Assoc Comput Mach 1978;25:675–86. [18] Morozov VN. Theory of holograms formed using a coded reference beam. Sov J Quantum Electron 1977;7:961–4. [19] Morozov VN. Associative parallel-search memory. Sov J Quantum Electron 1978;8:1–3. [20] Vasil’ev AA, Kompanets IN, Kotova SP, Morozov VN. Associative access to data in holographic memories with controlled transparancies. Sov J Quantum Electron 1978;8:740–3. [21] Vasil’ev AA, Kotova SP, Morozov VN. Pseudorandom signals as key words in an associative holographic memory. Sov J Quantum Electron 1979;9:1440–1. [22] Park SK, Miller KW. Random number generators: good ones are hard to nd. J Assoc Comput Mach 1988;31:1192–201.

Adil Lahrichi received his B.S.E.E. and M.S.E.E. in electrical engineering and his M.S. in applied mathematics from the University of Southern California in 1986 and 1989, respectively. He received his Ph.D. in electrical engineering from the University of Colorado Boulder in 1995. At the University of Southern California, his research focused on optical disk storage and ber optics. At the University of Colorado Boulder his research work included optical interconnects, volume holographic storage, system optimization and analysis, and photorefractive crystals with numerous contributions in these areas. He also worked extensively in the areas of optical correlators, birefringent lters, and liquid crystals. In 1996, he joined the guided wave optics laboratory at the University of Colorado Boulder, to work on optical waveguides, phase conjugation, and transistors oscillators. He has authored several papers and presented his work at numerous conferences. He also pioneered and invented a reversible process to x holograms in photorefractive materials using incoherent ultraviolet radiation. Presently, he is a senior research and development scientist at Fotonix, where he continues his research endeavors in the eld of photonics and develops systems and products for optical communications, storage, and medical diagnostics. Further, he continues to consult actively in the development of photonic based products.