Analysis and applications of the speckle patterns registered in aphotorefractive BTO crystal

Myrian Tebaldi, Alberto Lencina, N�eestor Bolognini*

a Centro de Investigaciones �OOpticas, CIOp (CONICET, CIC) and OPTIMO (Dpto.de Fisicomatem�aatica,Facultad de Ingenier�ııa, UNLP), P.O. Box 124, 1900 La Plata, Argentina

Received 1 September 2001; received in revised form 2 November 2001; accepted 5 December 2001

Abstract

The features of the volume speckle pattern recording in a sillenite BTO crystal are investigated. The modulation of

the speckle grains appears when the image of a random diffuser is formed onto the crystal by an optical system whose

pupil consists of two identical apertures. In fact, the volume nature of the grains generates speckle overlapping regions

where the modulation takes place. The 3-D modulated speckle pattern stored in the crystal leads to a space-charge field

by drift of photocarriers, resulting in a refractive index modulation. By observing the diffracted read-out light is es-

tablished that the index modulation in the speckle overlapping regions lead to a remarkable angular selectivity. The

statistical nature behavior of the speckle distributions is considered in the theoretical analysis. A comparison between

the theoretical predictions and the experimental results is done. The analysis leads to the adequate selection of the write-

in parameters as it is illustrated in the optical processing applications presented. � 2002 Published by Elsevier Science

B.V.

PACS: 42.30.Ms; 42.30.Va; 42.40.Ht

Keywords: Speckle; Photorefractive BTO crystal

1. Introduction

When an optically rough surface is illuminatedby coherent light, speckles appear in front of thesurface. The speckle patterns have important ap-plications in the field of metrology. In particular,

speckle techniques have been extensively used forthe measurements of small displacements, tilts,deformations, surface roughness and contouring[1–3]. In these techniques, photographic plates orfilm are used as recording medium requiring wetprocedures thereby avoiding real-time measure-ments. To overcome this drawback the photore-fractive materials have been used in speckletechniques [4–7]. For example, ferroelectric mate-rials such as LiNbO3 offer good storage capacity,high read-out efficiency, high resolution and re-versibility but have a low sensitivity. By contrast,

15 February 2002

Optics Communications 202 (2002) 257–270

www.elsevier.com/locate/optcom

*Corresponding author. Tel.: +54-221-4840280/2957; fax:

+54-221-4712771.

E-mail address: myrianc@odin.ciop.unlp.edu.ar (M. Te-

baldi).

0030-4018/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0030-4018 (02 )01084-2

the BaTiO3 is very sensitive but is slow concerningits response time. Also, the crystals of the sillenitefamily have a high photosensitivity and high car-rier mobility which permit achievements of fastresponse time that make them adequate for real-time holography, optical phase conjugation, am-plification of weak light signals, image processingand speckle applications [8].The general formula for undoped sillenites

crystals is Bi12XO20 or BXO where the symbol Xstands for Ge, Si or Ti. The photorefractiveproperties of these materials depend, directly orindirectly, on the nature of X. BSO and BGO arephotosensitive roughly in the blue-green region ofthe spectrum, whereas in BTO the edge of theoptical absorption band is shifted toward the redregion of the spectrum [9]. As a consequence BTOcrystals are suitable for recording in the red spec-tral region. In general, the optical activity hasthe effect of reducing the effective thickness of thecrystal and thus the diffraction efficiency of thestored holograms. The BTO optical activity at0:63 lm is considerably lower (rotary powerq ¼ 6:3 deg/mm [10,11]) than BSO ðq ¼ 25 deg/mm) and BGO (q ¼ 24 deg/mm), making it ap-propriate for holographic interferometry applica-tions as well as reversible recording medium inoptical memories [12,13]. Besides, BTO crystals arepromising because of their high photoconductivityand electro-optic coefficient.The BSO crystal was employed for real-time

holographic interferometry and speckle arrange-ments to determine displacements and rotations ofdiffusing objects [14]. In previous papers, the use ofa thick BSO crystal as a speckle recording mediumwas proposed [15–17].In this paper, the storage in a BTO crystal of

modulated speckle patterns is analyzed. It is basedon the modulation of the 3-D speckle grains thatappears when the image of a coherentlyilluminated random diffuser is formed onto thecrystal by an optical system whose pupil consistsof two identical apertures. In the write-in process,the intensity speckle pattern (wavelengthk ¼ 633 nm) is imaged and stored onto the crystalby applying an external electric field. This patternis encoded as a spatial distribution of the space-charge field, which induces, through the linear

electro-optic effect, a spatial variation of the re-fractive index.The main features of the diffracted read-out

light are discussed. The purpose is to identify thediffraction efficiency properties required in speckleapplications concerning optical signal processingand metrology. That is, to control the diffractionefficiency according to the experiment proposed.For instance, if a multiple image storage is done,then it would be convenient to have a high angularselectivity to avoid the filtering procedure whenreconstructing a determined image. In this case, tooptimize the angular selectivity the distance be-tween the pupil apertures should be increased. Onthe contrary, in some applications a low angularselectivity could be convenient. In case that thiscondition is fulfilled, the use of different pupilapertures between exposures in a multiple expo-sure routine allow to display simultaneously sev-eral relative in-plane displacements [18,19]. Thisexperiment would imply to narrow the aperturepupil separation to reduce the angular selectivityand to obtain an equal diffraction efficiency asso-ciated to each aperture pair.In Section 2 a theoretical analysis of speckle

registering in a BTO crystal is done. In the read-outstep, the diffraction efficiency is characterized interms of different write-in parameters (pupil aper-tures: separation, diameter and orientation) al-lowing then to select the appropriate workingconditions. The external applied field turns out tobe a key parameter in the diffraction efficiency de-pends analysis. For instance, it is demonstrated thatthe diffraction efficiency depends strongly on theangle formed between the index grating-vector andthe applied field. On the other hand, it is establishedthat under the experimental conditions analyzedthe diffusion field is negligible. Besides, by observ-ing the diffracted read-out light is established thatthe index modulation in the speckle overlappingregions lead to a remarkable angular selectivity.In Section 3, a comparison between the theo-

retical predictions and the experimental results isdone. In the theoretical calculation of the diffrac-tion efficiency two approaches are considered. Inone case, it is obtained by using a mean modula-tion value. In the another approach, the statisticalnature of the speckle intensity distribution is taken

258 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270

into account. The previous analysis leads to theadequate selection of the write-in parameters inthe applications presented.

2. Features of speckle pattern encoding in the BTOcrystal

As mentioned above, in BTO crystals the edgeof the absorption band is shifted toward the redregion of the spectrum in comparison with BSOand BGO crystals. Let us consider the experi-mental set-up of Fig. 1. In the write-in process adiffuser is illuminated by a collimated He–Ne laser(kw ¼ 633 nm) beam. An image of this input isformed in the crystal, by using a lens L1 located atthe u–v plane. The distance between the lens andthe crystal is ZC and the distance between the inputand the lens is Z0. Besides, to image the inputdiffuser, a pupil mask is placed immediately infront of the lens. The pupil mask has two circularapertures (ai and aj) of diameter D symmetricallylocated with respect to the axis of the imagingsystem. The apertures ai and aj are centered atpoints ðui; viÞ and ðuj; vjÞ, respectively. The dis-tance between the respective centers is

dij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuj � uiÞ2 þ ðvj � viÞ2

qand the line that joins

them forms an angle aij ¼ tan�1 ½ðvj � viÞ=ðuj � uiÞ�with the u-axis.When a double aperture pupil mask is placed in

front of the lens, each point in the image planereceives two contributions, one from each aper-ture. Then, a speckled image of the input diffuser isproduced through each aperture. The complexamplitude of waves going through different aper-tures are statistically independent from each othersince different components of the angular spectrumof scattered light are accepted by the apertures.Thus, the speckle image distribution formedthrough one aperture is uncorrelated with thedistribution obtained through another aperture.Moreover, the resulting speckle pattern in theBTO crystal appears as the interference of thementioned distributions because they are coherent.Let us consider a pair of spatially coincidentimage speckle grains. Their phases are constantand they are coherent. Then, these grains arefringe modulated and the fringes are orthogonal tothe line joining the aperture centers defined by thepoints ðui; viÞ and ðuj; vjÞ and form an anglep=2� aij with the X-axis. If the aperture pair isvertical, the fringes run horizontally. Note that as

Fig. 1. Experimental set-up: R: diffuser; L1;L2 and L3: lenses; M1 and M2: masks; BS: beam splitter; CS: collimation system, P:Fourier plane, P�: Imaging plane.

M. Tebaldi et al. / Optics Communications 202 (2002) 257–270 259

a consequence of the statistical nature of thespeckle phenomenon, the fringes spatial periodassociated to the registered speckle grains shouldbe treated as an average value. This average spatialperiod is pij ¼ ZCkw=dij where dij represents themean separation between due apertures.At this point it is important to emphasize the

volume nature of the speckles by regarding thatthe registering procedure itself takes place in avolume medium. Let us consider an object pointthat generates an image speckle obtained by dif-fraction through the pupil apertures As it is wellknown, the average diameter and length of theimage speckle grains can be expressed asSX ðkW ZCÞ=D and SZ kW ðZC=DÞ2, respec-tively. Then, the speckle dimensions decrease inproportion as the aperture diameter increases.Note that the SX and SZ values in our write-inexperimental conditions implies that a pair ofspeckle grains have a depth overlapping regiongiven by

Lspeckle ¼SZ if h0ij < arctan

SxSz

� �;

SXsinðh0ijÞ

otherwise

8<: ð1Þ

and h0ij ¼ dij=ð2ZCÞ is the write-in-angle.It should be noted that the average intensity at

the crystal is proportional to the pupil aperturearea, because the apertures limit the energy pass-ing through the pupil in each case.The intensity speckle pattern distribution

IðX ; Y Þ imaged onto the crystal is

IðX ; Y Þ ¼ jAðX ; Y Þj2

¼ C1

Z Z Z ZA0ðx0; y0ÞP ðu; vÞ

� exp �2pi

kW

x0Z0

�þ XZC

�u

þ y0Z0

þ YZC

�m

� dx0 dy0 dudm

2

ð2Þ

C1 is a constant factor, A0ðx0; y0Þ is the complexamplitude distribution at the input plane andP ðu; vÞ is the pupil function of the system.In the experimental set-up a BTO crystal which

is cut in the transverse electro-optic configurationis employed. The intensity distribution received by

the crystal creates photocharges that drift due tothe external electric field which is applied along[�11 1 0] direction of the crystal. These photochargesdrift from the highly illuminated into the less il-luminated regions where they are trapped. Thegeneration rate of photocharges is proportional tothe light pattern received by the crystal. Thesecharges develop a space-charge field that partiallycompensates the external field. Besides, by re-garding the experimental conditions the diffusioncharge transport mechanism is negligible. A re-sulting internal field is obtained at each point andthe system arrives at a steady-state situation.Thus, the speckle pattern intensity distributionreceived by the crystal is encoded as the spatialdistribution of the resulting electric field strengthat each point, which induces, through the linearelectro-optic effect the crystal exhibits, the corre-sponding spatial variation of the refractive indexDn.In summary, the resulting speckle grains that

are fringe modulated develop a photorefractiveindex grating in the whole volume speckle grain.Let us define the index grating-vector ~XXij so thatj~XXijj ¼ ð2pÞ=pij ¼ ð2p dijÞ=ðZC kWÞ.In the read-out process, a plane wave from an

Ar laser (wavelength kR ¼ 514 nm) is employed(see Fig. 1). The observation is done at the Fourierplane (U–V). Taking into account that the in-equality Dn n always holds (n is the non-per-turbed refractive index) it can be demonstratedthat the amplitude transmittance of the crystal isproportional to the input intensity pattern. Theencoded input is Fourier transformed in a con-ventional way. Then, the read-out intensity dis-tribution at the U–V plane results:

IfðU ; V Þ ¼ C2

Z ZIðX ; Y Þ

� exp

�� i 2p

kRfðXU þ YV Þ

dX dY

2

;

ð3Þwhere f is the focal length of lens L2 and C2 is aconstant. The spectrum IfðU ; V Þ contains thespatial frequencies that are the information carri-ers for the input signal. Then, the intensity in theFourier plane consists of three circular diffraction

260 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270

spots having all the same spectral width, which isdetermined by the aperture diameters D of thepupil. The fringes that modulate the refractiveindex pattern give rise to the side diffraction spotssymmetrically located to the zero-order spot. Asfar as D increases, higher input frequencycomponents are admitted by the system. Besides,the position of the side spots relative to the cen-tral spot is determined by the parameters dij, f , aij,kR.The BTO is a volume recording medium. As

mentioned above, the speckle grains are modu-lated by fringes that develop a photorefractiveindex grating of spatial period pij in the wholevolume speckle grain. The volume nature of theregister becomes apparent when the diffraction ofthe read-out beam is observed at the Fourierplane. The coupled wave theory [20] predicts anon-zero diffraction efficiency when a plane waveprobe is incident at volume gratings at an angleslightly different from that determined by theBragg condition, In accordance with this modelthe diffraction efficiency of the index grating pro-duced by a pair of interfering speckle grains in thetransmission geometry should behave as

g ¼ j2

j2 þ ðn=2Þ2sin2 L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 þ n

2

� 2s0@

1A; ð4Þ

where

n ¼ jXijjcosðhijÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 � sin2 ðhijÞq

0B@

1CADhij

is the off-Bragg parameter, jXijj is the modulus ofthe index grating-vector, hij is the external Braggangle in the read-out step (hij ¼ ðkR=kWÞh0ij, whereh0ij ¼ dij=ð2ZCÞ is the write-in angle), Dhij is the off-Bragg read-out angle,

j ¼ pnDn

kRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 � sin2 ðhijÞ

qis the coupling coefficient, n is the refractive indexat kR and Dn is the index grating modulationdepth. Also, Dn depends on the external appliedfield and its explicit expression is [8]

Dn ¼ n3r41m2

ð~EEa �XXijÞ2 þ E2D1þ ðED=EqÞ� �2 þ ðð~EEa �XXijÞ2=E2qÞ

" #1=2

ð5Þ

where r41 is the electro-optic coefficient,

m ¼ 2ffiffiffiffiffiffiffiI1I2

p

I1 þ I2

is the modulation of the interfering speckles (I1and I2 are the intensity of the write-in interferingspeckles), ED is the diffusion field, Eq is the maxi-mum space-charge field, Ea is the external appliedfield and �XXij is the versor index grating. Note thatin our experimental, conditions the diffusioncharge transport mechanism is negligible and EDcould be omitted.As mentioned above, the volume nature of the

speckle grains introduces a new parameter that isthe overlapping region of the speckle grain. Thedepth of the overlapping region L is given by

L ¼ Lspeckle if Lspeckle < LZ ;LZ otherwise:

�ð6Þ

Eq. (4) predicts a sharp peak at the Bragg angle.Moreover, the angular half-width of the Braggpeak strongly depends on the writing angle h0ij andthe depth L. Then, it is possible to choose ade-quately the write-in parameters to reduce or toincrease the angular selectivity according to theapplication to be proposed.As it is well known, the speckle patterns are

statistical in nature [21]. In our case, two speckledistributions are obtained in the image plane, onefrom each aperture. Each speckle field is charac-terized by a randomly varying intensity I and aphase U. The intensity distribution of the specklepattern generated by one aperture is assumed to bethe gathering of ideal spots grðx� xrÞ, whoseshape and position xr are random. Note that theintensity value of each ideal spot is different. Thespread of the function gr is related to the averageradius of the speckles. Also, the intensity distri-bution of the speckle pattern generated by the re-maining aperture is assumed to be the gathering ofideal spots frðx� xrÞ where both frðx� xrÞ andgrðx� xrÞ have similar features. Then, the strengthof the modulation depth depends on the relative

M. Tebaldi et al. / Optics Communications 202 (2002) 257–270 261

intensity of the speckle grains that interfere. Thus,for a vector position xr a random modulation mr isobtained. Indeed, statistical distribution of mod-ulation appears in the crystal. It should be notedthat a statistical treatment should be applied to themodulation m in the index grating modulationdepth given by expression (5). In our case, thistreatment is considered by using a mean modula-tion value and taking into account the statisticalnature of the speckle intensity distribution.

3. Experimental results

In the following analysis different experimentalresults are presented which support the predictionsdescribed by Eq. (4).In the experiments a BTO crystal which is cut in

the transverse electro-optic configuration is em-ployed. The directions h�1110i, h001i and h110i ofthe crystal coincidewith theXYZ axes and the lineardimensions are LX ¼ LY ¼ LZ ¼ 8 mm, respec-tively. Besides, ZC ¼ 490 mm and Z0 ¼ 130 mm.Owing toZC remains fixed along the experiments,

the parameter D governs the speckle dimensions.Let us take two examples: if D ¼ 2:25 mm, thenSX 0:137 mm and SZ 30:02 mm whereas ifD ¼ 4:7 mm, then SX 0:066 mm and SZ 6:88mm.It is experimentally observed that to optimize

the reconstruction it is necessary to satisfy theBragg condition that corresponds to an anglehij ðkR=kWÞðdij=ð2 ZCÞÞ. This result can beconfirmed by observing Fig. 2. The results of

Fig. 2(a) and (b) correspond to off and on Braggread-out, respectively. In Fig. 2(a) the read-outstep is done by a beam that impinges normallyonto the crystal and we referred to it as a fixedgeometry. This read-out configuration is usualwhen a two-dimensional recording medium isemployed. Note that the off Bragg angle in thefixed geometry Dhij ¼ hij ðkR=kWÞðdij=ð2 ZCÞÞ isthe same for both diffraction orders. It is apparentthat the off-Bragg read-out reduces the intensity ofthe diffraction spots in comparison with the brightsingle diffraction spot resulting from the Bragg-matched read-out.In Fig. 3 different values of the apertures sepa-

ration dij are employed. These results correspondto the fixed reconstruction geometry. Note that theindex grating modulation depth Dn is formed viathe linear electro-optic effect and is proportional tothe induced space-charge field. In our experimentalconditions, the fringes have spatial frequencieslower than 350 lines/mm. As it is well known, forthese frequencies the diffusion transport ofphotocharges is negligible and the drift carriermechanism predominates in the index grating de-velopment. Then, in the drift dominant regime, theindex modulation remains unchanged irrespectivethe variation that dij undergoes, provided that afixed value of the applied field ~EEa is maintained.Note that different values of dij imply differentvalues of Dhij and L in the read-out process. Arelative low value of dij implies a low value of theoff-Bragg parameter n in Eq. (4) and therefore thesystem behaves approximately Bragg matchedwith a large effective thickness. On the contrary, a

Fig. 2. (a) Off-Bragg reconstruction and (b) on-Bragg reconstruction, for D ¼ 4 mm, dij ¼ 16 mm and Ea ¼ 8:75 kV=cm.

262 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270

high value of n resulting from a higher dij reducesthe diffraction efficiency because the effectivethickness of the crystal is shortened. In summary,the diffraction efficiency of each speckle grain

volume grating behaves as a transmission volumehologram in agreement with Eq. (4) providing thatthe adequate effective overlapping depth L is usedas established by Eq. (6). From Fig. 3 it must be

Fig. 3. Diffraction patterns and intensity profiles in terms of the aperture separation dij corresponding to an off-Bragg reconstructionfor D ¼ 4 mm and Ea ¼ 8:75 kV=cm.

M. Tebaldi et al. / Optics Communications 202 (2002) 257–270 263

pointed out that a good tolerance in the off-Braggreconstruction is achieved by using a relativelysmall apertures separation. However, the tolerancecould be increased if the apertures diameter D isincreased.Note that, the first-order statistical momenta

are useful position parameters to describe the ex-perimental conditions because the average dif-fraction efficiency is the observable to bemeasured. This approach is considered in the fol-lowing analysis.The agreement between the theoretical analysis

of Section 2 and the experimental results of Fig. 3can be confirmed by observing Fig. 4. In Fig. 4 thediffraction efficiency in terms of the aperture sep-aration dij is depicted. The parameters utilized are:D ¼ 4 mm, Ea ¼ 8:75 kV=cm. The solid line inFig. 4 corresponds to the theoretical calculation ofthe diffraction efficiency by employing a first-orderstatistical momentum through the introduction ofthe mean modulation value hmi. This value hmi isobtained by using the average intensity hI1i andhI2i detected at the crystal plane for both write-inspeckle distributions

hmi

¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihI1ihI2i

phI1i þ hI2i

!:

Then, the diffraction efficiency (see Eq. (4)) is cal-culated by replacing in Eq. (5) m by hmi.The dotted line in Fig. 4 is obtained by con-

sidering the statistical nature of the speckle inten-sity distribution. In this case, the expressionutilized is

g ¼Z Z

j2ðI1; I2Þj2ðI1; I2Þ þ ðn=2Þ2

� sin2 L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2ðI1; I2Þ þ

n2

� 2s0@

1A 1

hI1ihI2i

� exp�� I1

hI1i

�þ I2hI2i

dI1 dI2: ð7Þ

It should be pointed out that there is not a re-markable deviation between the theoretical curvesmentioned above (solid and dotted lines of Fig. 4).These curves are obtained by using a different Lvalue for each aperture separation dij in accor-dance with Eq. (6). Besides, the triangular dotsrepresented in Fig. 4 are obtained from the peakvalues of the experimental intensity profiles of Fig.3. The triangular experimental dots are in agree-ment with the theoretical calculations.On the other hand the dashed curve depicts the

theoretical calculation when L ¼ LZ is considered.However for dij ¼ 8:5 mm, Lspeckle 9 mm and fordij ¼ 16 mm, Lspeckle 4:7 mm. These values giveus an estimation of the average overlapping depth.From the range of dij employed is clear that thereplacement of the overlapping depth by thecrystal depth, that is the approximation L ¼ LZ

brings a bad evaluation of the diffraction efficiencyas can be confirmed by comparing the dashedcurve with the experimental dots.Fig. 5 shows the diffraction spots and the cor-

responding intensity profile in terms of the exter-nal applied field for a given pupil apertureseparation. In these cases the read-out beam sat-isfy the Bragg condition. A strong dependence onthe applied field is observed. These results confirmthat the drift carrier contribution that develops theindex grating dominates.

Fig. 4. Diffraction efficiency in terms of the aperture separation

dij for D ¼ 4 mm, Ea ¼ 8:75 kV. The solid and dotted linescorrespond to the theoretical calculation of the diffraction effi-

ciency by considering a mean modulation value and the sta-

tistical nature of the speckle intensity distribution, respectively.

The triangular dots are obtained from the peak values of the

experimental intensity profiles of Fig. 3. The dashed curve

corresponds to the theoretical calculation when L ¼ LZ ¼ 8 mmis considered.

264 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270

In Fig. 6 the diffraction efficiency in terms of theexternal applied field is depicted. The parametersutilized are: D ¼ 2:25 mm and dij ¼ 6 mm. Thesolid and dotted lines in Fig. 6 correspond to the

theoretical calculation of the diffraction efficiencyby considering a mean modulation value and thestatistical nature of the speckle intensity distribu-tion (Eq. (7)), respectively. The similarity between

Fig. 5. Diffraction patterns and the corresponding intensity profiles in terms of the external applied field corresponding to an on-Bragg

reconstruction for dij ¼ 6 mm and D ¼ 2:25 mm.

M. Tebaldi et al. / Optics Communications 202 (2002) 257–270 265

both curves is apparent. Note that in this case,L ¼ LZ ¼ 8 mm is utilized because Lspeckle 22:5mm > LZ . The triangular dots are obtained fromthe peak values of the experimental intensity pro-file of Fig. 5. The same recording time is employedfor all the experimental registers. The disparity inthe low value range of the applied field is a con-sequence of that the saturation was not reached.The behavior of the diffraction efficiency cor-

responding to different orientations of the pupilapertures for a fixed value dij is observed in Fig. 7.The read out is done in the fixed geometry. Theparameters are: dij ¼ 8 mm, D ¼ 4:7 mm andEa ¼ 7:25 kV=cm. The different orientations of thepupil are described by the angle aij. For the ex-perimental frequencies the diffusion transport ofphotocharges is negligible and the drift carriermechanism predominates. Under this conditionthe field ~EEa introduces an anisotropic behaviorwhen the grating is built up. That is, the projectionof the field ~EEa onto the direction of the grating-vector ~XXij establishes the strength of the drift car-rier contribution. Moreover, the index gratingmodulation depth Dn is proportional to the scalarproduct ~EEa �XXij (see Eq. (5)). Thus, as far as theangle aij between the external applied field ~EEa andthe grating-vector ~XXij increases, the diffractionefficiency decreases accordingly. In both vectors

are perpendicular each other, the drift mechanismdoes not contribute to the index grating formationand the diffraction efficiency tends to a negligiblevalue.In Fig. 8 the diffraction efficiency in terms of the

pupil aperture orientation is depicted. The pa-rameters utilized are: dij ¼ 8 mm, D ¼ 4:7 mm andEa ¼ 7:25 kV=cm. The solid and dotted linescorrespond to the theoretical calculation of thediffraction efficiency by considering a mean mod-ulation value and the statistical nature of thespeckle intensity distribution (Eq. (7)), respec-tively. The triangular dots are obtained from thepeak values of Fig. 8. The agreement between thetheoretical curves and the experimental values isapparent.The response time is an important feature when

considering a multiple register of speckle patterns.As it is well known, the time required to produce aphotorefractive grating depends on the efficiencyof the charge generation and transport process.The diffraction efficiency evolution of the modu-lated speckle recording in a BTO crystal is shownin Fig. 9. The read-out procedure is destructive. Toavoid this degradation a mechanical chopper isemployed. In this case the average incident inten-sities are hI1i� 6 lW and hI2i� 32 lW atk ¼ 633 nm, Ea ¼ 8:75 kV=cm and dij ¼ 8 mm.Under our experimental conditions the steady-state diffraction efficiency reaches 31%.Note that the average intensity at the crystal is

proportional to the pupil aperture area. The av-erage exposure should be kept constant. That is ifthe diameter aperture is reduced the recording timeshould be increased accordingly.The preceding results are useful to predict the

features of the encoded speckle distributions whenusing a multiple-aperture pupil. In fact, whenreconstructing the encoded speckle distribution,the refractive index modulation corresponding toeach elemental index grating acts as a carrier fre-quency. To explain the distribution of diffractedspots when using a multiple aperture, the com-bined effect of all possible individual fringe systemsmust be considered. It can be inferred that thedistance dij between the aperture centres, thepupils diameter D, the angle aij between ~EEa and ~XXij

and the off Bragg read-out angle Dhij, determine

Fig. 6. Diffraction efficiency in terms of the external applied

field for: dij ¼ 6 mm and D ¼ 2:25 mm. The solid and dottedlines correspond to the theoretical calculation by considering a

mean modulation value and the statistical nature of the speckle

intensity distribution, respectively. The triangular dots are ob-

tained from the peak values of the experimental intensity profile

of Fig. 5.

266 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270

Fig. 7. Diffraction pattern corresponding to an off-Bragg reconstruction for dij ¼ 8 mm, D ¼ 4:7 mm and different pupil apertureorientations (aij ¼ 0�; 26�; 47�; 60�).

Fig. 8. Diffraction efficiency in terms of the pupil, aperture

orientation for dij ¼ 8 mm, D ¼ 4:7 mm. The solid and dottedlines correspond to the theoretical calculation by considering a

mean modulation value and the statistical nature of the speckle

intensity distribution, respectively. The triangular dots are ob-

tained from the peak values of the intensity profiles in Fig. 7.

Fig. 9. Diffraction efficiency evolution for the BTO crystal

corresponding to average incident intensities hI1i 6 lWand hI2i 32 lW at k ¼ 633 nm, Ea ¼ 8:75 kV=cm and dij ¼8 mm.

M. Tebaldi et al. / Optics Communications 202 (2002) 257–270 267

the diffraction efficiency of each spot. This be-haviour should be taken into account to optimizethe diffraction efficiency when designing a multi-ple-aperture pupil arrangement. In Fig. 10(a) ascheme of the multiple-aperture pupil employedand in Fig. 10(b) its corresponding diffractionpattern in the fixed read-out geometry are de-picted. This arrangement gives almost the sameefficiency in all the spots. Note that d12 > d13,a12 ¼ 0�, a13 ¼ 40� and a23 ¼ �40�. Let us observein Eq. (3) the dependence of the diffraction effi-ciency in terms of aij and dij.It is apparent that a higher d12 in comparison

with d13 which decreases the efficiency is compen-sated by a higher contribution corresponding toa12 ¼ 0� in comparison with a13 ¼ 40�.To illustrate the diffraction efficiency analysis

two applications are presented. In the first case, atransparency is placed in the diffuser plane. If twoimages are sequentially encoded into the crystal byusing the same aperture pupil, then the encodedspeckle distributions are modulated by identicalindex gratings. In the read-out process, the spec-tral components of both images are overlapped.The pupil employed has circular apertures of di-ameter D ¼ 2:25 mm, which are separated by adistance dij ¼ 6 mm. As input signal a triangularshaped opaque mask attached to the diffuser,whose position is modified between exposures isemployed. In the first column of Fig. 11 the inputsignal corresponding to the first exposure and thephotorefractive read-out reconstructed image

corresponding to this exposure are displayed. Inthe second column of Fig. 11 the input signalcorresponding to the second exposure and thecorresponding photorefractive read-out recon-structed image are displayed. In the third columnthe digital OR operation between the input signalsand the output reconstruction image when adouble exposure recording is done in the crystalare shown. Note that the retrieved overlapping ofthe two input images in a single frame suggests aconnection between this image operation andthe union of two sets (the input apertures) in thecontext of the ensemble theory and also with theOR logical operation. By associating each re-trieved input image as an ensemble of brightpoints, the image depicted in the third columncorresponds to the union operation between theinputs. Besides, a connection with the OR opera-tion can be established because in the third columnof Fig. 11 the bright points of the first and secondcolumn are retrieved.Concerning the experimental parameters se-

lected and from the previous analysis, a gooddiffraction efficiency in the fixed read-out geometryis obtained. In this case, a filtering procedure isrequired in the reconstruction step.The retrieval of the image can be done easily by

filtering out one of the corresponding side-orderdiffraction spots in the Fourier plane, by means ofa mask with a circular hole. It is clear that this stepcould be simply avoided by selecting another pa-rameters set. Then, the angular selectivity involved

Fig. 10. (a) Scheme of the multiple-aperture pupil employed and (b) the corresponding diffraction pattern in the fixed read-out

geometry.

268 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270

could be used for filtering procedures. In doing so,the read-out beam should be matched to the write-in Bragg angle of a certain encoded input, toproduce the isolated reconstruction of the diffrac-tion spot associated to the input.In Fig. 12 a metrological application is pre-

sented. In this case, two images of the diffuser aresequentially registered in the crystal. Between the

exposures, the diffuser undergoes an in-plane dis-placement. In the read-out step, an on-Bragg dif-fracted order modulated by fringes whose periodand orientation depend on the displacement of thediffuser is shown.

4. Conclusions

In this work, the features of modulated specklepatterns encoded as a photorefractive index grat-ing in a BTO crystal is analyzed. The Bragg lawfulfillment clearly demonstrates the volume natureof the induced grating that modulates each specklegrain.It is confirmed that the diffraction efficiency of

the encoded speckle distribution behaves as pre-dicted in the coupled-wave theory for transmissiongeometry. In particular, the diffraction efficiencyby considering a mean modulation value and thestatistical nature of the speckle intensity distribu-tion is theoretically analyzed. These theoreticalcalculations do not exhibit a noticeable deviation

Fig. 11. Input signals and the corresponding single exposure photorefractive read-out reconstructed image. The digital OR operation

between the input signals and the output reconstruction image corresponding to the double exposure recording (D ¼ 2:25 mm,dij ¼ 6 mm, Ea ¼ 7:25 kV=cm).

Fig. 12. On Bragg reconstruction of a double exposure dis-

playing an in-plane displacement.

M. Tebaldi et al. / Optics Communications 202 (2002) 257–270 269

between them and are in good agreement with theexperimental data.As predicted by the coupled-wave theory, the

efficiency could be optimized by Bragg matchingthe read-out angle. However, the experimentalconditions (pupil aperture separations, pupil aper-ture diameter, crystal thickness, applied voltage) inthe fixed read-out geometry (as usual when usingplane recording media) to obtain a good toleranceconcerning off-Bragg reconstruction is investigated.This analysis demonstrates the good perfor-

mance of BTO crystals to be employed in real-timeimage processing and metrological applications.For instance, further investigation could be doneregarding multiple exposure by using differentmultiple apertures in each exposure. Besides, theirsensitivity in the red spectral region enables to usediode lasers concerning different applications.Also, this analysis suggests further investiga-

tions of modulated speckle techniques by usingphotorefractives, to take advantage of reusabilityand high storage capacity of these volume media.Furthermore, other well-known developments ofphotorefractive holography, for instance fixingprocedures, image amplification, selective erasure,wavelength multiplexing, etc., could be imple-mented on the basis of a photorefractive specklerecording.

Acknowledgements

This research was performed under the auspi-cious of CONICET and Faculty of Engineering ofthe National University of La Plata (Argentina).A. Lencina acknowledges to CIC (Argentina).

References

[1] M. Franc�on, Laser Speckle and Applications in Optics,Academic Press, New York, 1979.

[2] R.K. Erf, Speckle Metrology, Academic Press, New York,

1978.

[3] J.C. Dainty (Ed.), Laser Speckle and Speckle Phenomena,

Springer-Verlag, Berlin, 1975.

[4] H.J. Tiziani, K. Leonhardt, J. Klenk, Opt. Commun. 34

(1980) 327.

[5] H.J. Tiziani, J. Klenk, Appl. Opt. 20 (1981) 1467.

[6] R. Tripathi, G.S. Pati, A. Kumar, K. Singh, Opt. Com-

mun. 149 (1998) 355.

[7] K.S. Dharmasaktu, A. Kumar, K. Singh, Optik 112 (2001)

49.

[8] P. Huignard, P. G€uunter (Eds.), Photorefractive Material

and their Applications II, Springer-Verlag, Berlin, 1989,

p. 205.

[9] L. Arizmendi, J.M. Cabrera, F. Agullo-Lopez, Int. J.

Optoelectron. 7 (1992) 149.

[10] A. Feldman, W.S. Brower, D. Horowitz, Appl. Phys. Lett.

16 (1970) 201.

[11] E. Shamonina, V.P. Kamenov, K.H. Ringhofer, G. Cedil-

nik, A. Kiebling, R. Kowarschik, D.J. Webb, Opt. Com-

mun. 146 (1998) 62.

[12] D. Tonchev, S. Zhivkova, M. Miteva, Appl. Opt. 29 (1990)

4753.

[13] A.A. Kamshilin, M.P. Petrov, Opt. Commun. 53 (1985)

23.

[14] H.J. Tiziani, Opt. Acta 29 (1982) 463.

[15] M. Tebaldi, L. Angel Toro, M.C. Lasprilla, N. Bolognini,

Opt. Commun. 155 (1998) 342.

[16] L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, Opt.

Commun. 168 (1999) 55.

[17] M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, Opt. Eng. 39

(2000) 3232.

[18] L. Angel Toro, M. Tebaldi, N. Bolognini, M. Trivi, J. Opt.

Soc. Am. A 17 (2000) 107.

[19] M. Tebaldi, L. Angel Toro, M. Trivi, N. Bolognini, Opt.

Commun. 182 (2000) 95.

[20] H. Kogelnik, Bell Syst. Tech. J. 48 (1969) 2909.

[21] J.C. Dainty, E. Wolf (Eds.), Progress in Optics, vol. XIV,

Elsevier Science, Amsterdam, 1977.

270 M. Tebaldi et al. / Optics Communications 202 (2002) 257–270