Delocalization of the Information Stored in a Holographic Memory

( ? y . L . P .

Delocalization of the Information Stored in

a Holographic Memory* *

*

^

B y itlu m in a tin g th e d a ta m ask of a holographic m em ory ^y an a rra y of c o h e ren t p o in t sources, th e in fo rtn a tio n sto red in a ho lo g ram can be delocalized, i.e. th e re exists a th ree dim en sio n al reg io n a ro u n d th e focal p lan e over w hich th e in fo rm atio n is d is trib u te d uniform ly. In consequence of th e m eth o d described sto rag e re d u n d a n c y can be achieved, an d som e p ro p e rtie s of th e F o u rie r sp ectru m can be, m oreover, k e p t in th e v ic in ity of th e F o u rie r plane. T he calculations h av e been verified b y a m odel experim ent.

1. Introduction

1 §. In fo rm a tio n stored in holographic m em ory has th e form of th e F o u rie r sp ectru m of a p rim a ry d a ta m ask, c o n tain in g tra n s p a re n t an d opaque squares, corresponding to th e b it c o n ten t. W hen th e d a ta m ask is illum inated by a plane or h om o centric wave, th e n each sq u are gives th e sam e am p litu d e d istrib u tio n in th e F o u rie r p lan e, b u t th e phase facto r is p ro p o rtio n al to th e lo catio n of th e bit-square. The F o u rie r sp ectrum of th e whole d a ta m ask is produced b y th e in terferen ce of th e in div i­ d ual b it spectra. Such a k in d of sp ectru m will be called a Mwi/orw sp e ctral d istrib u tio n .

F o r reco n stru ctio n of a b it-sq u are from its holographic record only th e c e n tra l p a rt of th e d iffractio n p a tte rn is needed, i.e. th e region defined by resolutio n req u irem en ts. F ro m th e uniform ity it follows consequently th a t for th e reco n stru ctio n of th e whole d a ta m ask th e sam e region is needed as for an ind iv idu al b it-sq u are. T hus, th e system of a holographic m em ory can be th e follow ing [1 ]: A block of d a ta is realized in th e fo rm of a d a ta m ask, th e m ask is th e n reco rd ed in a hologram of fin ite e x te n t, called su bhologram . The whole storage a re a consists of side by side subholo­ gram s.

D ifficulties arise, how ever in th e realizatio n of th e storage red u n d a n c y . A m ere increase in

* I n s titu te for P hysics, B u d a p est.

** This w ork h a s been sponsored by th e S ta te Office of T echnical D evelopm ent.

th e size of th e subhologram s allows to record only sp atial frequencies t h a t a re u n im p o rta n t for th e re c o n stru c tio n ; th e reliab ility of th e device does n o t increase w hile its cap ac ity dim inishes. A fu rth e r pro b lem is t h a t th e F o u rie r spectru m a p p e ars only in th e im age plane of th e lig h t source. H ence, th e sm allest m isalignm ent of th e recording p lan e in th e direction of th e lig h t p ro p ag a tio n leads to th e recording of th e F resn el in ste a d of th e F rau n lio f- fer diffraction. I n th e F resn el diffractio n p a t ­ te rn th e field s c a tte re d b y th e b it sq uares is n o t d istrib u ted u n ifo rm ly an y m ore, th e p a tte rn s of th e in d ivid u al b its being locally shifted. A dam age of th e h ologram p la te m ay lea d to th e loss of some b its in th e re c o n stru c te d im age.

The en h ancem ent of th e region over w hich th e in form atio n is d istrib u te d uniform ly is c a l­ led delocalization. D elocalization can be ach ie­ ved in several w ays. T he sim plest m eth o d is th e illum ination of th e d a ta m ask th ro u g h a gro u n d glass p late . T hen, how ever, th e speckle noise of th e re c o n stru c te d field will be v e ry high, like in th e case of rough surface objects [2]. A m ore a d v a n tag eo u s p ro ced u re is th e illu m in a ­ tio n of th e m ask th ro u g h a d iffractio n g ra tin g [3], since due to v a rio u s orders of th e d iffra c te d lig h t F ra u n h o ffe r p a tte rn s a re sh ifted a n d consequently, th e sp e ctru m is m ultip led.

The m ethod in v e stig a te d b y us [ I ] can be reg ard ed as a generalizatio n of th e d iffractio n g ratin g m ethod. E a c h b it sq u a re is illu m in ated b y an in d iv id u al p o in t source (Fig. 1). The F o u rie r sp ectrum of a b it, illu m in a te d in th is w ay, is th e co n v o lu tio n of th e sp e ctru m of

w here a single h it a n d th a t of th e illu m in a tin g beam .

This re su lta n t sp ectru m consists of an infinite n u m b er of elem en tary hologram s given by a plan e wave source. Thus, th e r e s u lta n t

spec-Fig. 1. P o in t m a trix illu m in atio n

f — poin t sources, J f — d a ta m a s k : FT/ — F o u rie r tra n sfo rm lo ts, 77 — hologram plane, / — focal len g th

of th e F o u rie r tran sfo rm lens

tru m is g ettin g b lu rred regularly. The coherent p o in t sources can be produced by using a fly's eye optics, th e distance betw een th e le n sle ts being eq u al to th e side of th e b it squares. Such m eth od w as used for re d u n d a n t recording of p ic tu re ­ like inform ation [5]. The fly 's eye illu m inatio n has also been described in [6] w ith o u t d etailed analysis.

2. Theory

2 §. A t firs t we shall discuss th e m ost p o p u lar F o u rie r (or quasi-F o u rier) m eth o d of tra n s p a ­ ren cy tra n sfo rm atio n . A plane w ave falls on th e d a ta m ask (Fig. 2) a n d th e hologram is

F ig. 2. P lan e w ave illu m in a tio n 717 — d a ta m ask, 7<'7i — F o u rie r tra n sfo rm lens, 7, —

reference beam

reg istered in th e focal plane of a lens. L et th e tra n sp a re n c y of th e b inary m ask co n tain in g A" b its be V V ?(3ku, ,'/u) =

y y

- ( X X )

l/i; a re th e co ordinates on th e b in ary m ask ; 771, _{is th e size of a b i t ; A ^y is th e distance betw een} th e b it-c en tre s; a n d = 0 or 1 depending on th e con tent. E lectric field in th e hologram

/by) is [7]: -^*(''7/? /by) f ( F r ( 4 r + ?///) = — e x p — — ----V ) / / a ) X

x

w here 7 denotes th e w av elen g th , A', th e a m p litu ­ de of th e incident lig h t w ave a n d / is th e focal len g th of th e F o u rie r lens. T ak ing in to account (1) we get for th e field (in th e following we neglect th e p ro p o rtio n a lity facto rs):

1! where -^Jiy/2 ^(^yy; /fyy) = f -Uiy/2 - bn/2 < 2 Try x e x p l ---X a n d /by) V .V = A_a(Arx + b /;,) j. (5)

^(37/'//yy) fu n ctio n will be called th e M specb'dw /;/ucbow, because it describes th e sp ectru m of a single bit. The F (tr^ , /by) fu n ctio n is called th e cu/dc/d .s-prcb'/rw /Mucbo/q since it describes th e sp e ctru m of a m ask on w hich th e bits a re rep resen ted b y d fu nctions. As an exam ple Fig. 3 shows th e co n te n t sp ectrum

F ig. 3. A re s u lt of th e oue dim ensional c o m p u te r sim u la ­ tio n of th e c o n te n t sp e c tru m

fu n ctio n of 20 ran d o m ly selected b its of a one­ dim ensional m ask. I t can be seen th a t th e co n te n t sp ectru m spreads over th e whole holo­ gram plane, so th e sp ectru m to be recorded on th e hologram for th e reliab le storage is d eterm ined by th e b it sp e ctru m function.

F o r p lan e w ave illu m in a tio n th e b it spectru m fu n ctio n is

Kbbu- 1/n ) - D ^ s in c ^ " j b n c ^

(6) w here sinc(;r) = simr/.r.

T he sp ectru m has app reciable values only in th e v icin ity of th e op tical axis (.r^ = 0, 2/n = 0).

The spectru m described in (3), (1), (3), (6), is recorded on th e F o u rie r h ologram of th e d a ta m ask.

In th e reco n stru ctio n process th e in te n sity d istrib u tio n a t th e d e te c to r gq,, y ^ plane is

.Vo)

const} j* J

w here /* is th e focal len g th of th e re a d out F o u rie r lens, a n d a re th e coordinates of th e cen tre of th e sp ectral b a n d used for th e reco n stru ctio n an d A denotes th e w id th of th e rec o n stru c te d F o u rie r sp ectru m . The d efo rm a­ tio n of a bit of th e re c o n stru c te d signal is shown in Fig. I as a fu n ctio n of th e re c o n stru c te d sp ectral rang e (i.e. /;) for th e case = 0,

=

b-Fig. 4. D eform ation of a b it of the. re c o n stru c te d signal as a fun ctio n of th e re c o n stru c te d sp e c tra l range

(calculation)

I t can be seen t h a t th e m ain p a r t of th e inform ation is co n tain ed w ithin th e spectral ran ge determ ined by A r'^ = y ^

= 0 ; th e re is no reaso n to ta k e larg e r areas for reconstructio n as th e o u tp u t signal increases only slightly.

M oreover, a n y la te ra l shift of th e cen tre of th e reco n stru cted a re a (from th e origin of coordi­ nates = 0, y", = 0) will lead to h ig her los­ ses (Fig. 3). Therefore, from p ra c tic a l p o in t

/ 7 / \ ---/ \ + -/ / \ /' \ /' \ / \ / \ / \ w

Fig. 5. In te n sity d istrib u tio n in th e d e te c to r plane for several values

7t = oo for th e co n tin u o u s curve, 7t = 1A //D ^ for th e o th e r curves (calculation)

of view th e re c o n stru c tio n region of th e holo­ gram can be defined by *

iyn! < ( S )

Therefore th ere is also no reaso n to record larger subhologram s, such subhologram s are not r e ­ d u n d a n t, an d losses in th e hologram yield inform ation losses in th e re c o n stru c te d im age.

Fig. 6. L ens m a trix illu m in a tio n

F J7 — lens m a trix , J f — d a ta m ask, F F — F o u rie r tran sfo rm lens, 7? — h o logram plane, F — focal plan e

3 §. L et now a m a trix of lenses be placed in fro n t of th e d a ta m ask (see F ig. 6) so, th a t th e axis of each lens in te rse c t th e d a ta m ask a t th e centre of th e corresponding b it square. The b its

* The lim it defined in (8) is tw ice th a t req u ested by th e R ayleigh criterion.

a re illu m in ated by spherical waves

F .e x p ^ + '

w here

r — denotes th e distance betw een th e focal p lan e of th e lens m atrix a n d th e d a ta m ask. T he field a t a distance / ' m easu red from the F o u rie r lens is th en .1/ f _exp [* m /Ar ) in 1 1 — %?i,} exp ¿y t (9) (F or sim plicity we discuss only a one-dim ensi­ onal case, th e relations for tw o dim ensions are obvious.) T he firs t fac to r in th e in te g ran d in (9) describes th e spherical w ave, th e second is th e tra n sp a re n c y of th e m ask, th e th ir d describes th e focusing action of th e F o u rie r tra n sfo rm lens, a n d th e fo u rth is th e p ro p ag a tio n term . A fter some bo rin g calculations for th e field d iffracted by th e &-th sq u are we get

= e x p { - ^ [ ( R d ^ ( a + y ) - a T r ( l + p ) ) 2 +

+ y)]j x

is th e F resnel in teg ral, y is th e defocusing facto r ch aracterizin g th e m isalignm ent of th e hologram p late, defined as

a n d a sta n d s for

y = ¿ - . f f

a (1 1)

The fo rm u la (10) is h ard ly d isputable. W e shall con cen trate th e discussion on th e role of th e p a ra m e te r a containin g both th e defo­ cusing facto r a n d th e spherical wave illum ina­ tio n of th e bit. If tt^ -0 an assim p to tic a p p ro ­ x im atio n is to be used [8] a n d [10] tu rn s into th e F o u rie r tra n sfo rm of th e square(6). The c a ­ se, a = 0 b u t y ^ 0 an d r =/= oo, corresponds to th e realizatio n of th e F o u rie r tra n sfo rm atio n , w hen th e F o u rie r plane is th e im age p lane of th e source [7]. The case r ^ o o an d y-sO has been discussed in § 2.

A m ore im p o rta n t case is when a is a large nu m b er, i.e. th e distance y is m uch less th a n th e focal len g th / , b u t th e m isalignm ent is n o t too high (y 1). T he d istrib u tio n rem ains n early uniform , if for th e tw o (in dep end ent of th e co­ o rd in ates a?,,) te rm s (11) of th e F resn el in teg ral a rg u m e n t th e following cond ition holds :

-- —-- ^

---/ a I 2

i.e. w hen th e defocusing fac to r

a M

2A; A j; y 2AA^

F o r = 1.5, % = 50 a n d y/y = 30 th e lim it of th e rela tiv e m isalign m ent is

lyl 4 1/15 ^ 6 % , w hich can be easily achieved.

I n th e absence of th e lens m a trix (y = oo) th e te rm d e p e n d en t on th e locatio n of th e sq u are will be

A'/li; l y (see (10) a n d (11)), in ste a d of

th u s th e accuracy of th e lo n g itu d in a l alig n m en t is fa c ilita te d by th e fa c to r y/y.

The field d istrib u tio n m u st be calculated fro m F resn el in teg rals. I n Fig. 7 we show some c a lc u la te d d istrib u tio n s for some v alu es of

y/y. As it can be ex p ected , th e w itd h of th e in te n sity d istrib u tio n c u rv e increases w ith th e

y/y ratio . ( I t is clear, t h a t th e case, w hen th e ap p ro x im atio n of th e g eo m etrical optics is v alid, th e illu m in atio n of th e F o u rie r p lan e by a b it betw een th e shadow m arg ines is c o n sta n t.)

F ig. 7. D ependence of th e in te n s ity d istrib u tio n of th e b it sp e ctru m on th e focal le n g th of th e fly 's eye lenses ( D ^ = 0.19 m m , / = 300 m m , A = 6328 A) (ca lcu la­

tion)

W hen th e whole data, m ask is illu m in ated by a lens m atrix , th e field d istrib u tio n in th e holo­ g ram p lan e is

-E(ah?) = A

w here -E^aq?) is given b y (10).

The rec o n stru c te d in te n s ity d istrib u tio n in th e d e te c to r plan e is

I(a?^) = const] < -E(.r^)e ' da?g] . -A+a^

(12) The re c o n stru c te d in te n s ity d istrib u tio n (12) fo r th e case / / r = 100, y = 0 w as calculated

F ig. 8. C alculated in te n s ity d is trib u tio n a t th e d etecto r p la n e in th e case of fly 's eye lens illu m in a tio n for se­

v e ra l it v alu es; = 0

num erically. Fig. 8 p rese n ts th e c a lc u la te d d istrib u tio n in th e case of = 0 for several v a lu ­ es of A. C om paring Figs 4 a n d 8 we can see, t h a t a satisfacto ry re c o n stru c tio n w ith p lan e w ave an d fly's eye lens illu m in a tio n m eth o d s req u ires th e sam e sp e ctral b a n d (A = 1 ) . T he e n la rg e ­ m en t of th e re c o n s tru c te d a re a (A > 1) gives an enlarged signal in th e d e te c to r p ro p o rtio n a l to th e en largem ent of th e re c o n stru c te d area, i.e. th e storage red u n d a n c y is p ro p o rtio n a l to th e enlarg em en t of th e h o lo gram size. I n F ig. 9 th e rec o n stru c te d in te n sity d istrib u tio n is p lo t­ te d for b a n d w id th A = /A /7 l„ b u t th e cen tre shifted up to = 3.75 The g ra p h shows no significant d isto rtio n com pared w ith th e centred b a n d (cu rv e a in F ig . 9), in con­ tra d ic tio n to th e case of th e p lan e w ave il­ lum ination (Fig. 5) w hich pro ves th e equivalence of different p a rts of th e sp e ctru m .

Fig. 9. C alculated in te n s ity d istrib u tio n a t th e d e ­ te c to r plane for several values (It = 0 .7 5 A //D ^; -D3; = 0.19 m m ; / = 300 m m ; A = 6328 A ; / / r = 100)

3. Experimental

§ 4. F o r th e sake of sim plicity in th e e x ­ p erim en ts we h av e used a pinhole in ste a d of th e fly 's eye lens a rra y . T he a rra n g e m e n t of th e pinholes on an opaqu e m ask w as id en tic a l to th a t of th e b its in th e b in a ry m ask. P lacing th e pinhole a rra y in ste a d of th e lens a rra y th e b its were illu m in a te d b y th e beam s d iffra c te d from th e pinholes. T he d istan ce betw een th e pinhole a rra y a n d th e b in a ry m ask was chosen so t h a t each b it be illu m in a te d b y lig h t com ing only from th e pinhole. I t is obvious th a t p in ­

hole a rra y can n ot replace th e fly 's eye lens a rr a y in every respect because of th e loss in lig h t power.

The d a ta m ask used for te st is shown in

F ig. 10, while Fig. I t d em o n strates its im age w hen centred stops of v ario u s d iam e te r were p u t in th e focal plane of th e F o u rie r tra n sfo rm

lens. F o r p ictu res a-d p arallel wave an d for pictures aa-d d m a trix illum inatio n were used. A ccording to § .1, no significant d ev iatio n for sy m m etrical sp e ctra is observed a t both kinds of illu m in ation . The size /; = is sa tis­ fac to ry for m aking fair records.

Fig. 12 is a side-band p ic tu re using stop of diam eter 2Â//_D^ shifted by 3/4A //7t^. Fig. 12a was tak e n w ith p a ra lle l beam , Fig. 12b with a m atrix . The difference betw een th e two re ­ cords is obvious. The fu rth e r a d v a n ta g e of th e m atrix illu m in atio n is d e m o n stra ted in Fig. 13, showing th e result of th e off focal plane m isalig­ nm ent. The im age of a plane wave illum inated m ask is cut off a t th e corners a n d strongly d isto rte d in th e sides, in d icatin g th e absence of th e u n ifo rm ity , while th e p ictu re take]] w ith th e m a trix shows no local losses in th e co nten t, th e u n ifo rm ity being observed.

* * * * *

* w w # # # # .

# w # # w # w #

w # #

# #

* w w w w w w *

w w w w w w w w

W W W

w w

F ig. 11. I n te n s ity d istrib u tio n a t th e d e te c to r p la n e : a —d: p a ra lle l w ave illu m in a tio n , a a - d d : ma t r i x il­ lu m in atio n , a , aa : 7;. = 1 (A//D jt;). b , b b : A = 0 . 8 3 ( 4 / / c,ee, : A = 0.7 (A//Z),M), d ,d d : A = 0 .0 ( 4 //D j/) , ,r ^ = 0

a b

b ig. 12. E x p e rim e n ta l com parison of th e p la n e w ave (a) and m a trix (b) ¡d o m in atio n m eth o d (7t = 0.75 = 0.75 (A //D ^)).

a b

Fig. 13. E x p e rim e n ta l com parison of th e p lan e w ave (a) and m a trix (b) illu m in a tio n m eth o d (7t = 1 (A //D ^), ". //r = 1"". У = "-I-'')

4. Conclusions

C alculation a n d ex p erim en ts sliow th a t a sa tisfa c to ry sto rag e red u n d a n c y can be achie­ ved w hen each b it of th e b in ary m ask is illu m in ated se p ara te ly b y an iden tical d iv e r­ g en t beam . The use of d iv erg en t illum in atio n does n o t increase th e m inim al size of th e hologram , so th is m eth o d does n o t decrease th e m ax im ally a tta in a b le sto rage d ensity. T he req u ired sto rag e re d u n d a n c y can be achie­ ved by a p ro p o rtio n a l en larg em en t of th e holo­ g ram size. U sing o u r m eth o d th e spectrum rem ains unifo rm for th e hologram s of out-of F o u rie r planes ju s t as well. C onsequently, F o u rie r tra n sfo rm lenses w ith larg e a n g u lar a p e rtu re can be used, a n d th e recording m aterial need n o t be m a tc h e d to th e c u rv e d focal plane, because th e p ro p erties of th e F o u rier tra n sfo rm are conserved in d ep th .

*

* *

T h an k s are due to Mr. F . K irhly for his active assistance in th e ex p erim en ts.

Le d ép lacem en t des in fo r m a tio n s m a g a sin é e s dans u n e m é m o ir e h o lo g ra p h iq u e

A u m oyen de l'éc laira g e de la m a sq u e c o n te n a n t des données de la m ém oire h o lo g rap h iq u e p a r un systèm e des sources de lum ière pon ctu elles cohérentes il est possible de déplacer les in fo rm atio n s stockées dans un hologram m e ; cela v e u t dire q u 'a u to u r du plan focal il ex iste une zone de tro is dim ensions, su r laquelle les in fo rm atio n s so n t disposées u n ifo rm é ­ m ent. E n se se rv a n t de c e tte m éth o d e on p e u t o b te n ir u n surplus de m ém oire; en plus, ce rta in e s p ro p rié té s d u spectre de F o u rie r peuvent, être prése rv é es d an s l'en to u ra g e du p la n de F o u rie r. L es calculs o n t été vérifiés p ra tiq u e m e n t su r u n modèle.

Перемещение информации, хранимой в голографической памяти Путем освещения маски с данными голографическом памяти системой когерентных точечных источников можно перемещать информацию, хранимую в голограмме. Иначе говоря: существует трехмерная зона вокруг фокальной плоскости, на которой информация одинаково размещена. Применением описанного метода можно получить избы­ ток памяти, а некоторые свойства спектров Фурье могут, сверх того, сохраниться вблизи плоскости Фурье. Расчеты проверены эмпирически на модели. ОрттсА ArPLiCATA V , 3 -4 , 1975 4 1

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