Security optical data storage in Fourier holograms

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1 Securiy opical daa sorage in Fourier holograms Wei-Chia Su, Yu-Wen Chen, 2, * Yu-Jen Chen, Shiuan-Huei Lin, 3 and Li-Karn Wang 2 Graduae Insiue of Phoonics, Naional Changhua Universiy of Educaion, Changhua 500, Taiwan 2 Insiue of Phoonics Technologies, Naional Tsing Hua Universiy, Hsinchu 3003, Taiwan 3 Deparmen of elecrophysics, Naional Chiao Tung Universiy, Hsinchu 30050, Taiwan *Corresponding auhor: d9280@oz.nhu.edu.w Received 7 November 20; revised 9 January 202; acceped 9 January 202; posed 7 January 202 (Doc. ID 57738); published 5 March 202 We have proposed and demonsraed a holographic securiy sorage sysem ha is implemened wih a shif muliplexing echnique. The securiy funcion of his sorage sysem is achieved by using a microdiffuser (MD) for random phase encoding of he reference beams. The apparaus of random phase encoding in his sysem offers an addiional and flexible funcion during he recording processes. The sysem can generae holographic securiy memory or nonsecuriy holographic memory via using he MD or no. The sorage capaciy and he average signal-o-noise value of he securiy sorage sysem are 6 bis μm 2 and 3.5, respecively. Laeral shifing seleciviy in his holographic securiy sorage sysem is heoreically analyzed and experimenally invesigaed. 202 Opical Sociey of America OCIS codes: , , Inroducion Volume-holographic sorage has received increasing aenion owing o is poenial high sorage capaciy and fas access rae [ 3]. A novel echnique for holographic sorage called collinear holography has been invesigaed in recen years [4]. This apparaus is implemened wih a coaxial opical srucure o record holograms. The signal and reference beams in his sysem can be generaed by using a spaial ligh modulaor (SLM), and hey are usually disribued on he inner and ouer surface region of he SLM. These wo beams propagae along he opical axis of he sysem and pass hrough a Fourier lens, and hen finally hey inerfere a he focal area of he Fourier lens. A recording maerial is locaed a he focal area of he Fourier lens for he holographic recording. I has been proven ha he collinear holographic sorage sysem performs high poenial in miniaurizaion of he opical archiecure and could be compaible wih he exising disc sorage sysems. Shimura e al. have also shown ha an addiional X/2/ $5.00/0 202 Opical Sociey of America random phase encoding in he reference pixels improves he imaging performance in a collinear holographic daa sorage sysem [5]. Meanwhile, random phase encoding in holographic sorage is one aracive and imporan issue for securiy sorage due o he growing demand for proecion of informaion [6 3]. Random phase encoding generaed by using a microdiffuser (MD) has shown grea advanages for holographic securiy daa sorage owing o heir difficuly in duplicaion [4]. The Fourier-archiecure-based holographic sorage sysem wih random phase encoding has been sudied for a long ime [5 8]. In his paper, he radiional Fourier-archiecure-based holographic sorage sysem is modified o become a coaxial opical srucure. Accordingly, a holographic securiy sorage sysem based on random phase encoding is presened, and he sysem is implemened wih a shifing muliplexing echnique. As shown in Fig., he MD was placed in he reference arm in order o generae a random phase wavefron for encrypion of he holographic sorage. The sored image can be rerieved from he medium if he original MD is placed in he original spaial posiion o generae he same random phase wavefron for 20 March 202 / Vol. 5, No. 9 / APPLIED OPTICS 297

Fig.. (Color online) Securiy holographic sorage implemened wih a MD. holographic reconsrucion. However, he sored image can no be rerieved if users lack he original MD.

2 Fig.. (Color online) Securiy holographic sorage implemened wih a MD. holographic reconsrucion. However, he sored image can no be rerieved if users lack he original MD. Accordingly, securiy sorage in his sysem is implemened by his MD. If he MD is moved away from he reference arm, he shif muliplexing mechanism for holographic sorage can sill be performed in his sysem, and he whole sysem becomes very similar o a sysem presened by Seckman e al. [9]. Consequenly, he characerisic of he MD in his sysem offers an addiional funcion during he recording process. We can generae a holographic securiy or nonsecuriy memory opionally by moving he MD ino he reference arm or moving i away in each holographic recording. As a resul, a holographic securiy and nonsecuriy memory boh can be generaed individually by using our proposed sysem. In our previous sudy, we developed similar echniques o implemen encrypion-selecable holographic sorage sysems [20,2]. However, he sudy in his manuscrip is differen from our previous publicaions. In comparison wih he sysem proposed in [20], he holographic securiy sorage proposed in his paper can be accessed wihou using phaseconjugaed readou algorihms. In [20], we have shown ha he fideliy of he phase-conjugaed reference beam is an imporan facor ha significanly affecs he signal-o-noise raio (SNR) of he reconsruced holograms. In addiion, a scheme of encrypion-selecable holographic sorage in LiNbO 3 using angular muliplexing based on 90 geomery is described in [2]. Currenly, we find ha he sudy of encrypion-selecable holographic sorage based on Fourier holograms is sill less discussed. Therefore, no only is he holographic securiy sorage sysem demonsraed in his paper, bu he encrypion-selecable funcion of he proposed sysem is also discussed. The effec of he MD on shifing seleciviy in his holographic securiy sorage sysem is analyzed heoreically and experimenally. The experimenal resuls show ha sorage capaciy wih 6 bis μm 2 in he securiy sorage sysem can be obained and ha he average SNR of he rerieved daa image is Securiy Holographic Sorage Sysem The experimenal apparaus of he holographic securiy sorage sysem is shown in Fig. 2. We used a diode-pumped, solid-sae laser a 532 nm as he ligh source. The laser beam was collimaed and spli ino wo pars, one being he signal beam and he oher being he reference beam. The signal beam was inciden upon he inpu image and was hen passed hrough he beam splier (BS) direcly. The reference beam was also inciden on he BS, bu i refleced from he BS. These wo beams propagaed along he opical axis of he Fourier lens L2 and hen passed hrough i. The focal lengh of lens L2 was 50 mm. In he reference arm, a MD was placed a 5 mm in fron of he recording medium in order o generae random phase wavefrons, and he MD was fixed a he same posiion for all he following encrypion processes. In his sudy, handmade ground glass was used as he required MD. The ground glass in he experimen was made by grounding a fla glass wih Al 2 O 3 powders. Therefore, a random phase disribuion on he surface of he ground glass diffuser was obained. The deailed specificaions of he ground glass can be found in our previous work [6]. Based on he coaxial geomery, he signal beam would inerfere wih he random phase wavefrons of reference beam in he focal area of he lens L2. A PQ-PMMA maerial [22] wih dimensions of 20 mm 20 mm 2 mm was used as he holographic recording medium, and i was locaed a he focal area of he lens L2. The recording maerial was mouned upon a wo-dimensional ranslaion sage (piezo-moor driven linear sage, Newpor) for implemening shif muliplexing of holographic sorage. Based on he shif muliplexing algorihm, holograms were recorded rack by rack in he horizonal and verical direcions of PQ-PMMA during he recording processes. These holograms beween adjacen racks should be sored by shifing he recording maerial wih a disance larger han or a leas equal o he shifing seleciviy of his sysem. The analysis of he shifing seleciviy will be presened in he nex secion. In he reading processes, a Fourier lens L3 locaed beween recording medium and CCD plane performed an inverse Fourier ransform of L SF M HP PBS M3 HP2 M2 BS Inpu image L2 MD H L3 CCD Fig. 2. Experimen seup for a securiy collinear holographic sysem: SF, spaial filer; L, lens; M, mirror; PBS, polarizaion beam splier; HP, half-wave plae; MD, microdiffuser; H, hologram. 298 APPLIED OPTICS / Vol. 5, No. 9 / 20 March 202

$he diffracion beam.$

3 he diffracion beam. When he signal beam was blocked, he original reference beam was used as he reading beam and he medium was shifed o he corresponding recording posiion; he sored image could hen be perfecly reconsruced on he CCD plane. 3. Laeral Seleciviy Shifing seleciviy is an imporan parameer for affecing sorage capaciy. The laeral shifing seleciviy can be analyzed heoreically by using he VOHIL model (he volume hologram being an inegraor of he lighs emied from elemenary ligh sources) [23]. As shown in Fig. 3, a convergen spherical wave was inciden on he MD o generae he random phase wavefron. Thus, we can wrie he composie wavefron on he hologram plane as R w x 3 ; y 3 d 2 d 2 A w expjϕx ; y expjkr dx dy ; () where d is he dimension of he illuminaion region of he MD, A w is he ampliude of each spherical wave, ϕx ; y is he iniial random phase of each poin source of MD, and r x 3 x 2 y 3 y 2 z 3 z 2 2 is he disance beween he MD and he hologram. The objec beam was a plane wave inciden upon he inpu image, which was placed a Δx o disance from opical axis o he image cener and hen passed hrough he Fourier lens L. Therefore, he inerference fringes of he hologram records can be wrien as Hx 3 ; y 3 jr w x 3 ; y 3 j 2 jifsx o Δx o ; y o gj 2 R wx 3 ; y 3 IfSx o Δx o ; y o g R w x 3 ; y 3 I fsx o Δx o ; y o g; (2) where If g represens he Fourier ransform funcion and Sx o ; y o is he inpu image funcion. During he reading process, we used a random phase wavefron o read he hologram. The reading wave on he hologram plane can be expressed as R r x 3 ; y 3 d 2 d 2 A r expjϕx 2 ; y 2 expjkr 2 dx 2 dy 2 ; (3) where A r is he ampliude of reading wave, ϕx 2 ; y 2 is he iniial random phase of each poin source of MD used for encoding he reading wave, and r 2 is he disance beween he MD and he hologram, which can be expressed as r 2 x 3 x 2 2 y 3 y 2 2 z 3 z For he paraxial condiion, r z 0 and r 2 z 0, where z 0 is he disance beween MD and he hologram. When he recording maerial is only laerally shifed and causes a relaive displacemen of speckle wave wih a disance of Δx x 2 x, and Δy y 2 y, we can express he diffracion as D T 2 T R r H exp jk 2 r dr T 2 T 2 ly 2 lx 2 dy 2 dx 2 T 2 l y 2 l x 2 d y 2 d x 2 IfSx o Δx o ; y o gja r jja w j exp j k x 2z 3 x Δx 2 y 3 y Δy 2 0 x 3 x 2 y 3 y 2 T exp jk 2 r dx dy dx 3 dy 3 dr; (4) where r z 3 cos α, T cos α, α is he propagaion angle of he signal beam, and is he hickness of hologram. Therefore, he diffracion inensiy can be expressed as x o x x 3 Wriing Plane wave y MD z H d T α r z 3 y o x 2 y 3 Plane wave x o y2 Reading z 2 Fig. 3. Inpu image L (Color online) Theoreical model for analysis of shifing seleciviy in securiy collinear holograms. z 0 20 March 202 / Vol. 5, No. 9 / APPLIED OPTICS 299

I jdj 2 lx 2 l x 2 w 2 w 2 e j2π x 0 λf Δx x 3 dx 3 dx 2 sinc 2 dx Δx d x 2 Sx 0 Δx 0 ; y 0 dx 0 dy 0 sinc 2 dy Δy e j2π Δx x dx ly 2 sinc 2 2 cos α 2 cos α λ cos 2 α l y 2 cos α ejk S 2 cos α e j2π

$paern. Here we can define he laeral seleciviy as he shif deviaion for he firs zero diffracion obained from he sinc funcions in Eq. (5). The heoreical simulaion and experimenal resuls are shown in Fig.$

4 I jdj 2 lx 2 l x 2 w 2 w 2 e j2π x 0 λf Δx x 3 dx 3 dx 2 sinc 2 dx Δx d x 2 Sx 0 Δx 0 ; y 0 dx 0 dy 0 sinc 2 dy Δy e j2π Δx x dx ly 2 sinc 2 2 cos α 2 cos α λ cos 2 α l y 2 cos α ejk S 2 cos α e j2π y 0 λf Δy x 3 dy 3 dy 2 e j2π λ cos α Δx 0 f Δx ; f Δy z 0 z 0 d y 2 z 3 2 dz 3 e j2π Δy λz y 0 dy 2 ; (5) where f is he focal lengh of lens L, andw is he dimension of he illuminaion region of he paern. Here we can define he laeral seleciviy as he shif deviaion for he firs zero diffracion obained from he sinc funcions in Eq. (5). The heoreical simulaion and experimenal resuls are shown in Fig. 4. The parameers used in he heoreical simulaion are λ 532 nm, 0 5 mm, d x 2.5 mm, d y 2 mm, 2 mm, and f 50 mm. As shown in Fig. 5, a chessboard paern wih pixels was used as he inpu paern. From our experimenal measuremen resuls, he laeral seleciviy of collinear holographic sorage by shifing a single hologram was abou only.3 and.5 μm in he horizonal (x) and verical (y) direcions, respecively. 4. Opical Implemenaion In his secion, a pracical holographic securiy sorage sysem using he shif muliplexing echnique was demonsraed. In order o keep a higher SNR of he reconsruced daa image, he shif disance for soring nex adjacen hologram in his pracical demonsraion sysem was se as 5 μm in boh laeral direcions [24]. According o he [24], we chose more han he hird Bragg null in order o make sure ha he reconsruced daa image could keep a higher SNR. In his demonsraion, 00 image pages were recorded wihin his maerial wih 0 rows. Each Normalized diffracion inensiy simulaion resuls(x direcion) simulaion resuls(y direcion) measuremen resuls(x direcion) measuremen resuls(y direcion) row conained 0 holograms, and hese 0 holograms were parially spaially overlapped, bu he shif separaion of each adjacen hologram was 5 μm. Afer he recording in he firs row was compleed, he maerial was horizonally shifed o is original posiion and addiionally shifed in a verical direcion wih 5 μm. We repeaed he row recording process described above unil he whole 00 holograms were sored wihin hese 0 rows. The dimension of each pixel of he chessboard paern wih pixels was 500 μm 500 μm. Therefore, he DC erm dimension of he signal beam locaed a he focal plane was around 00 μm 00 μm. We can find hese 00 holograms were parially spaially overlapped. The spaial disribuions of hese 00 holograms are illusraed in Fig. 6. Figure 7(a) o (e) are he s, 45h, 50h, 55h, and 00h rerieved images obained from his sysem, respecively. Figure 7(f) shows he rerieved images of encrypion sorage wihou using he correc MD. The sorage capaciy of our pracical sysem can be expressed as N M Δx Δy ; (6) where M is he pixel number of he daa pages. According o Eq. (6), he securiy sorage capaciy of our experimenal sysem will reach6 bis μm 2. The SNR of each rerieve image ha was encryped by MD based on he coaxial geomery was also calculaed. In our experimenal seup, he chessboard paern wih pixels (500 μm in size) was used as a binary paern. The magnificaion of 0 mm 0 mm Laeral displacemen (micromeer) Fig. 4. (Color online) Theoreical and experimenal resuls of laeral seleciviy for securiy collinear holograms. Fig. 5. Inpu image: chessboard paern wih pixels. 300 APPLIED OPTICS / Vol. 5, No. 9 / 20 March 202

(a) (b) (c) Fig. 6. (Color online) Spaial disribuion of 00 sored holograms wihin he recording maerial. There are a oal of 00 holograms in 0 racks. Each rack conains 0 shif-muliplexed holograms.

Therefore, he magnificaion of he rerieved image on he CCD plane was modulaed such ha one pixel on he rerieved image corresponded o one superpixel on he CCD image sensor.

Alhough 20 20 camera pixels on he rerieved image represen bi, he edges were lef ou, so only he cenral 0 0 pixels effecively wihin a superpixel was used for calculaion of SNR.

8, he average SNR value of 00 pages daa of rerieved images was abou 3.5. 5.

5 (a) (b) (c) Fig. 6. (Color online) Spaial disribuion of 00 sored holograms wihin he recording maerial. There are a oal of 00 holograms in 0 racks. Each rack conains 0 shif-muliplexed holograms. our rerieved image sysem was abou 0.9. Therefore, he magnificaion of he rerieved image on he CCD plane was modulaed such ha one pixel on he rerieved image corresponded o one superpixel on he CCD image sensor. One superpixel was composed of camera pixels. For each rerieved image, here were superpixels used for SNR calculaion. Alhough camera pixels on he rerieved image represen bi, he edges were lef ou, so only he cenral 0 0 pixels effecively wihin a superpixel was used for calculaion of SNR. The SNR formula can be expressed as SNR μ μ 0 σ 2 ; (7) σ2 0 where μ, μ 0 and σ, σ 0 are he mean value and deviaion of he deeced energy for one and zero bis, respecively [25]. As shown in Fig. 8, he average SNR value of 00 pages daa of rerieved images was abou Securiy-Selecable Sorage In his paper, we exend he concep menioned above o configure a securiy-selecable holographic sorage (d) (e) (f) Fig. 7. (Color online) Rerieved images of he sored paerns in his collinear holographic sorage sysem. (a) (e) Rerieved images by using he same MD locaing a he correc posiion. (f) Diffracion resuls of encrypion sorage wihou using he correc MD. sysem. We can generae a holographic securiy or nonsecuriy memory opionally by using he same sysem. A holographic securiy memory means ha he whole sored images in he memory are encryped during he recording processes, and a holographic nonsecuriy memory means ha whole sored images in he memory are no encryped during he recording processes. If users choose nonencrypion recording for all he sored images in his sysem, he MD should be moved away from he reference arm and he muliplexing sorage mechanism in our sorage sysem becomes similar o he convenional shif-muliplexed holographic sorage sysem [9]. If users choose encrypion recording for all he sored images in his sysem, we sill perform shif muliplexing for recording, bu he MD is moved ino he reference arm for each exposure. Thus, a holographic securiy memory described in Secion 4 can be generaed. In our sysem design, each securiy-selecable sorage sysem of volume holography has a unique MD for encrypion and decrypion. The MD in each sysem only can be shifed in horizonal direcion bu i Fig. 8. (Color online) SNR of rerieved images for he securiy collinear holographic sorage sysem. 20 March 202 / Vol. 5, No. 9 / APPLIED OPTICS 30

Sored daa in holographic nonsecuriy memory can be easily rerieved in anoher sysem because only spherical reference wave is required for he readou.

6 (a) Fig. 9. (Color online) (a), (b) Rerieved images of nonencrypion sorage implemened wih he same sysem. canno be removed from he sysem wihou desrucion. Holographic nonsecuriy memory generaed in an arbirarily recording sysem can be used as a removable memory. Sored daa in holographic nonsecuriy memory can be easily rerieved in anoher sysem because only spherical reference wave is required for he readou. Conrary o nonsecuriy memory, he readou of holographic securiy memory generaed by an arbirary recording sysem in anoher sysem is impossible because lack of he original encrypion key o decryp he securiy informaion. Encryped holographic memory only can be accessed in he original recording sysem. Accordingly, each sysem can generae is own encryped memory and herefore offers high securiy for conen proecion. In our demonsraion of nonsecuriy sorage, he shif seleciviy of shif-muliplexed holographic sorage wihou using a MD in his sysem is 6 μm and 90 μminx and y direcions, respecively. However, we have noiced he shifing seleciviy in horizonal and verical direcions for nonsecuriy sorage are quie differen for his convenional shif-muliplexed holograms owing o he Bragg degeneraion in he verical direcion [6,2,26,27]. Accordingly, 5 image pages were recorded wihin his maerial wih 3 rows. Each row conained 5 holograms, and hese 5 holograms were parially spaially overlapped, bu he shif separaion of each adjacen hologram was 6 μm. In addiion, he shif separaion of each adjacen row was 90 μm. Afer he recording in he firs row was compleed, we repeaed he row recording process described above unil all 5 holograms were sored wihin hese 3 rows. Figures 9(a) and (b) are he rerieved images of he nonencrypion sorage. The average SNR value of 5 pages daa of rerieved images was abou Conclusion In his paper, we have proposed a holographic securiy sorage sysem by using a shif muliplexing echnique. A MD played an imporan role in encrypion and decrypion of holographic sorage and reconsrucion. Shif seleciviy of he proposed securiy holographic sorage sysem was analyzed. The experimens demonsrae a securiy holographic sorage of 00 holograms wihin his sysem, and he sorage capaciy and he average SNR value of he presened securiy collinear holographic sorage sysem were 6 bis μm 2 and 3.5, respecively. Selecive (b) encrypion sorage also can be achieved by moving he MD ino he reference arm or moving i away during he recording process. Nonsecuriy holographic sorage using he same sysem was also presened. This work was suppored by he Naional Science Council of Taiwan under conrac no. NSC E MY3 Reference. H. J. Caulfield, D. Psalis, and G. Sincerbox, Holographic Daa Sorage (Springer-Verlag, 2000). 2. S. S. Orlov, W. Phillips, E. Bjornson, Y. Takashima, P. Sundaram, L. Hesselink, R. Okas, D. Kwan, and R. Snyder, High-ransfer-rae high-capaciy holographic disk daasorage sysem, Appl. Op. 43, (2004). 3. G. W. Burr, C. M. Jefferson, H. Coufal, M. Jurich, J. A. Hoffnagle, R. M. Macfarlane, and R. M. Shelby, Volume holographic daa sorage a an areal densiy of 250 gigapixels in: 2, Op. Le. 26, (200). 4. H. Horimai, X. Tan, and J. Li, Collinear holography, Appl. Op. 44, (2005). 5. T. Shimura, S. Ichimura, R. Fujimura, K. Kuroda, X. D. Tan, and H. Horimai, Analysis of a collinear holographic sorage sysem: inroducion of pixel spread funcion, Op. Le. 3, (2006). 6. C. C. Sun and W. C. Su, Three-dimensional shifing selecively of random phase encoding in volume holograms, Appl. Op. 40, (200). 7. C. Denz, K.-O. Muller, F. Visinka, and T. Tschudi, Digial volume holographic daa sorage using phase-coded holographic memory sysem, Proc. SPIE 3802, (999). 8. J. F. Heanue, M. C. Bashaw, and L. Hesselink, Encryped holographic daa sorage based on orhogonal-phase-code muliplexing, Appl. Op. 34, (995). 9. W. C. Su and C. H. Lin, Three-dimensional shifing seleciviy of decrypion phase mask in double random phase encoding holographic memory, Op. Commun. 24, 29 4 (2004). 0. O. Maoba and B. Javidi, Encryped opical memory sysem using hree-dimenional keys in he Fresnel domain, Op. Le. 24, (999).. O. Maoba and B. Javidi, Encryped opical sorage wih angular muliplexing, Appl. Op. 38, (999). 2. G. Unnikrishnan, J. Joseph, and K. Singh, Opical encrypion sysem ha uses phase conjugaion in a phoorefracive crysal, Appl. Op. 37, (998). 3. O. Maoba, Y. Yokohama, M. Miura, K. Nia, and T. Yoshimura, Reflecion-ype holographic disk memory wih random phase shif muliplexing, Appl. Op. 45, (2006). 4. W. C. Su, C. C. Sun, Y. C. Chen, and Y. Ouyang, Duplicaion of phase key for random-phase-encryped volume holograms, Appl. Op. 43, (2004). 5. Y. Takeda, Y. Oshida, and Y. Miyamura, Random phase shifers for Fourier ransformed holograms, Appl. Op., (972). 6. C. B. Burckhard, Use of a random phase mask for he recording of Fourier ransform holograms of daa masks, Appl. Op. 9, (970). 7. O. Maoba and B. Javidi, Secure holographic memory by double-random polarizaion encrypion, Appl. Op. 43, (2004). 8. M. Yamazaki and J. Ohsubo, Opimizaion of encryped holograms in opical securiy sysems, Op. Eng. 40, (200). 9. G. J. Seckman, A. Pu, and D. Psalis, Sorage densiy of shifmuliplexed holographic memory, Appl. Op. 40, (200). 20. C. C. Sun, W. C. Su, B. Wang, and A. E. T. Chiou, Laeral shifing sensiiviy of a ground glass for holographic encrypion 302 APPLIED OPTICS / Vol. 5, No. 9 / 20 March 202

7 and muliplexing using phase conjugae readou algorihm, Op. Commun. 9, (200). 2. W. C. Su and C. C. Sun, Encrypion-selecable holographic sorage in LiNbO 3 wih angle muliplexing, Microw. Op. Technol. Le. 42, (2004). 22. S. H. Lin, K. Y. Hsu, W.. Chen, and W. T. Whang, Experimenal characerizaion of phenanhrenequinone-doped poly (mehyl mehacrylae) phoopolymer for volume holographic sorage, Op. Eng. 42, (2003). 23. C. C. Sun, A simplified model for diffracion analysis of volume holograms, Op. Eng. 42, (2003). 24. G. Barbasahis, M. Levene, and D. Psalis, Shif muliplexing wih spherical reference waves, Appl. Op. 35, (996). 25. C. Moser and D. Psalis, Holographic memory wih localized recording, Appl. Op. 40, (200). 26. C. C. Sun, M. S. Tsaur, W. C. Su, B. Wang, and E. T. Chiou, Two-dimensional shifing olerance of a volume-holographic correlaor, Appl. Op. 38, (999). 27. C. C. Sun, W. C. Su, B. Wang, and Y. OuYang, Diffracion seleciviy of holograms wih random phase encoding, Op. Commun. 75, (2000). 20 March 202 / Vol. 5, No. 9 / APPLIED OPTICS 303

Announcement. Allowed

Announcement. Allowed 9//05 nnouncemen Firs es: Sep. 8, Chap. -4 llowed wriing insrumen poce calculaor ruler One 8.5" " paper conaining consans, formulas, and any oher informaion ha you migh find useful (NOT any inds of soluions).