Holographic scattering and its applications

Mostafa A. Ellabban1 Martin Fally

Romano A. Rupp

Institut fur Experimentalphysik, Universitat Wien,

Strudlhofgasse 4, A-1090 Wien, Austria

Theo Woike

Institut fur Mineralogie und Geochemie, Universitat zu Koln,

Zulpicherstraße 49b, D-50674 Koln, Germany

Mirco Imlau

Fachbereich Physik, Universitat Osnabruck,

Barbarastraße 7, D-49069 Osnabruck, Germany

28th March 2001Running title: Holographic scattering

1on leave from: Tanta University, Egypt

ABSTRACT

Propagation of laser radiation through aphotorefractive crystal is accompanied by agradual increase of unintentional photoin-duced light scattering. The process emergesfrom an initial scattering of the incidentbeam (pump beam) at the air-crystal bound-ary or from inhomogeneities within the crys-tal causing a slight deviation from the lightdistribution of the pump beam. By thephotorefractive effect small refractive-indexchanges are recorded which, subsequently,diffract the pump beam. If the origi-nally scattered wave and the wave diffractedfrom the pump beam are in phase, scat-tering grows continuously by amplification.The photoinduced refractive-index inhomo-geneities can be regarded as parasitic holo-grams, the process as holographic scatter-ing (HS). Those spurious structures, whichare recorded at the same time as the de-sired holograms, are among the main draw-backs limiting the applicability of thick holo-graphic recording media in data storage andimage processing applications.

On the other hand, HS can advanta-geously be exploited in applications for op-tical communication systems, optical limit-

ing, motion detection, and plays an impor-tant role in the generation of self-pumpedphase conjugators and the realisation of dou-ble phase conjugate mirrors. Further, HSis a common phenomenon that reflects theproperties of various material parameters,thus being a valuable tool for material inves-tigations. Its major advantage as comparedto two-wave mixing is that experiments aremuch simpler to perform (only one beam, nosensitivity to vibrations).

This review outlines the origin of HS,the parameters influencing it, presents tech-niques how to avoid HS and applications inwhich HS plays an essential role. More-over, we give a short overview on materi-als exhibiting that phenomenon, focussingon recent results for three model sub-stances with differing mechanism of pho-torefractive response: the acentric crys-tals SrxBa1−xNb2O6 (SBN) and LiNbO3,as well as the centrosymmetric crystalNa2[Fe(CN)5NO] · 2H2O (sodium nitroprus-side, NaNP).

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1. INTRODUCTION

The history of holographic scattering (HS)is as long as the history of the photorefrac-tive effect. In 1966, Ashkin et al. recognisedthe phenomenon of optical damage to resultfrom a light-induced refractive-index changeand suggested its electro-optic origin [1]. Inthe same publication we find the first pho-tograph of the HS phenomenon (Fig. 1a inRef. [1]). Rather soon the photorefractiveeffect was utilised for the storage of volumephase holograms [2, 3] thus opening a newgrandiose field for nonlinear optics.

The generic observation of HS is the fol-lowing: If a laser beam propagates througha photorefractive crystal there is a gradualincrease of light scattering with time, andthus a considerable decrease of the transmis-sion. This scattering shares two character-istic properties with volume holograms: itbuilds up with a time constant of the sameorder of magnitude characteristic for holo-graphic recording and disappears immedi-ately upon a slight rotation of the crystal ora change of the wavelength, just in the sameway as the reconstructed wave-fields of vol-ume holograms do upon violation of Bragg’scondition.

Because of this volume holographic na-ture it is quite common to call this phe-nomenon holographic scattering [4], a termwhich we adopt in this article. Other namesare in use as well, and in the next paragraphwe will give and explain some of them. Ofcourse each has its well-founded justificationby pointing out specific aspects or character-istics of HS. The drawback is that it does notjust facilitate literature search. It even mayimpede understanding of common physicalprinciples behind and interrelations betweenthe many observations reported so far.

In the context of optical devices - in par-ticular for information processing and stor-age - HS is part of the optical noise [5]

generated in the systems, namely the onecaused by optical nonlinearity of the mate-rials or by their holographic recording ca-pabilities. This is stressed by the designa-tions photoinduced [6] or light-induced scat-tering of light [7]. The nonlinearities in-volved have much similarities to stimulatedBrillouin and Raman scattering which sug-gests to call them stimulated photorefrac-tive scattering [8]. In some photorefractivemedia the incident beam spreads out in abroad fan which is called beam fanning [9].An accompanying effect often confused withHS is the well-known self-focussing or self-defocussing phenomenon of Gaussian beams.Both phenomena give rise to and consti-tute the optical damage [10] of a medium.Self-(de)focussing may be regarded as (usu-ally) large-scale optical damage originatingfrom the deviation of the real beam pro-file from the ideal, i.e., infinitely extendedplane wave of constant intensity, while HSis a micro-scale optical damage [11] origi-nating from the deviation of optical prop-erties from the ones of an ideally homoge-neous medium. The first effect can readilybe calculated once the beam profile is givenand causes some kind of “deterministic beamfanning”, mostly into small angles. In con-trast, HS emerges from noise and leads toa more wide-angle type of scattering, occa-sionally also called random beam fanning [12,13]. HS and self-(de)focussing have in com-mon that they are based on the optical non-linearity of the recording medium and on thenonlinearity of the recording-reading feed-back. In addition, self-defocussing and HSare asymmetric in the presence of a non-localrecording mechanism [9, 11]. Yet, while thefirst occurs also with incoherent light, coher-ence is an absolutely necessary requirementfor HS (cf. Fig. 3 in Ref. [11]). The funda-mental difference is namely the one betweenrefraction and diffraction.

The most elementary process of HS can

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be understood by considering the combinedaction of four plane wave components: twoof them writing a “parasitic” hologram, andone being reconstructed upon a fourth inci-dent plane wave component. If the first pairand the second pair of waves coincide, therather broad distributions (beam fanning)of free HS appear. In the other case phasematching conditions apply and scattering isoften confined to narrow cones (conical scat-tering [14]). Provided that the combined ac-tion above is reciprocal, i.e., is a true four-wave mixing process, it is called parametricscattering [15] because of its analogy withother parametric scattering processes in non-linear optical media.

There are two necessary conditions to ob-tain light-induced scattering: an initiatingsource of scattering (defects, scratches, inho-mogeneities, fluctuations [16]), the so calledseed scattering, and a sufficiently effectiveamplification mechanism for that scatteredradiation. The latter implies that the effectcan only be generated by coherent radiation.

It is known from standard holographythat steady state amplification of one of thebeams in two-wave mixing requires a non-local recording mechanism, i.e., the inter-ference pattern of the incident beams needsto be phase shifted with respect to therefractive-index pattern [17, 18]. The sameapplies to the multi-wave mixing of HS andmeans that HS is necessarily associated witha special nature of the photorefractive effect.For example in electro-optic crystals wherethe refractive-index change is caused by anelectric field which is generated by the exci-tation of electrons or ions during illumina-tion, the decisive point is the charge trans-port mechanism (photovoltaic effect, diffu-sion or drift).

We would like to emphasise the general-ity of HS: it may occur in virtually all pho-torefractive media (defined here in the mostgeneral sense of the term, namely media in

which the refractive-index is changed underirradiation with light). Experimental resultswere interpreted by two-wave [19–24], three-wave [21, 25–27], and four wave interactionschemes [28, 29]. The steady state scat-tering patterns which are observed in dif-ferent photorefractive materials range fromtwo rings in photopolymers [30–32], ”dots”,ring patterns, or lobes of scattered light inLiNbO3 [1, 6, 19, 21, 26, 33–38], a more com-plex lobe structure, “dots”, rings and lines inBaTiO3 [21, 26, 27, 39–47], diffracted lobesin LiTaO3 [25, 29, 48], to a homogeneousisotropic scattering distribution [49] and asharp scattering ring caused by flip scatter-ing [50] in NaNP.

It is beyond the scope of this review todeal with all peculiarities of the host of ma-terials which have been reported to exhibitHS so far. Instead we exemplarily discussthree model cases in more detail: SBN as theelectro-optic prototype for a nonlocal (diffu-sion [51]), LiNbO3 as the electro-optic pro-totype for prevailing local (photovoltaic [52])recording mechanism, and NaNP where am-plification by beam coupling is forbidden asthe crystal is centrosymmetric [49].

2. ORIGIN OF HS

Light-induced scattering may roughly beexplained by the standard model of self-amplification of parasitic holograms [19, 21,53]:

Irregularities at the air-crystal boundaryor inhomogeneities within the crystal causeweak initial scattering which acts as the seedfor the subsequent holographic amplificationprocess. By the interference pattern of theincident beam and its scattered field a holo-gram of the scattered wave is recorded. Inphotorefractive materials with linear electro-optic effect for example, the refractive in-dex can be modulated by the light-inducedbuild up of a space-charge field. This is

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achieved by the excitation of charge carriersupon the non-uniform illumination of the in-terference pattern, their migration from thebright regions and subsequent trapping atnew sites in the dark regions. The recordedhologram is simultaneously read out by thesame incident wave. In linear recording me-dia the wave field reconstructed from thehologram differs from the seed wave onlyby its intensity and a possible overall phaseshift. Both quantities exclusively depend onthe recording mechanism (Changes of thephase of the incident wave equally affect allwave fields involved, so they do not haveany consequences). Constructive interfer-ence of the reconstructed field can occur onlyin two cases: If an absorption hologram wasrecorded, the recording mechanism must belocal and there must be gain, i.e., negativeabsorption. If a refractive-index hologram isrecorded, a non-local recording mechanismis needed, since the reconstructed wave isshifted by ±π/2 with respect to the seedwave. Hence, in first approximation thereis no change of the amplitude of the sumof both fields. However, non-local record-ing mechanisms provide an additional phaseshift of ±π/2. Then reconstructed and seedfield may sum up to a (weakened or) am-plified scattered wave. In the case of am-plification, this intensified wave field writesa stronger hologram with the incident wavewhich has again the correct readout phasefor amplification. This positive feed-backcontinues to amplify the scattered field un-til limitations like pump-wave depletion leadto a stationary state. The build-up processof the parasitic hologram is determined bya highly non-linear kinetics which cannot bedescribed by a simple theoretical model.

3. CLASSIFICATION OF HS

HS can be classified according to two maincategories:

I. Non-flip/flip scatteringDepending on whether the state of lightpolarisation is conserved in the scatter-ing process or flipped into the orthog-onal polarisation state, we call it - inanalogy to spin flip scattering of neu-trons - non-flip or flip scattering, respec-tively. In the field of holography non-flipscattering is usually called isotropic scat-tering [37, 38] whereas flip scattering iscalled anisotropic scattering [19, 21, 54].To avoid any confusion we strongly recom-mend to discontinue in the use of the latterterms since in many other fields of physicsisotropic/anisotropic scattering means scat-tering with an isotropic/anisotropic direc-tional characteristics (scattering indicatrix).

In the context of polarisation we alwaysrefer to the electric component of the elec-tromagnetic field and use the following ab-breviations below: o,e polarisation for or-dinary and extraordinary polarisation (opti-cally uniaxial crystals); a, b, c for polarisa-tion of light along crystallographic a, b, or caxes (optically biaxial crystals.)

Flip scattering requires non-diagonal ele-ments of the light-induced change δεij of thedielectric tensor at optical frequencies. Forexample, in linear electro-optic media flipscattering is possible if at least one of thenon-diagonal elements of the tensor rijnKn

is non-zero [19, 21]. Here rijn denotes a com-ponent of the electro-optic tensor, Kn of thegrating vector, and we adopt Einstein’s sumconvention.II. Free/matched scattering

In the simplest case parasitic holograms havethe features of two-beam interactions, i.e.,they may be regarded to result from couplingof plane wave components in pairs whichare exactly the same ones which interact inread-out. Bragg conditions are thereby au-tomatically fulfilled for all scattered waves.As no further restrictions apply, the dis-tribution of free scattering is rather broad

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Figure 1: Classification scheme of HS.

and smooth though mostly anisotropic. Pro-vided that we may assume that seed scat-tering is in most cases to a good approxima-tion isotropic, the anisotropy is in the firstplace a consequence of the anisotropy of theholographic gain. Therefore it reflects theanisotropy of material properties. This scat-tering, not confined to further phase match-ing conditions, is called free scattering.

Scattered light can also be amplified asa result of mixing of more than two planewave components imposing more stringentrestrictions by phase matching conditions inwhich case we call it matched scattering. Theangular distribution of scattering observedin the far field then exhibits rather sharp,well-defined structures. There are two spe-cial cases of matched scattering which aremost frequently investigated: conical scatter-ing and parametric scattering. Conical scat-tering refers to a situation where the scat-tered wave vectors form cones and the termparametric scattering was coined in analogyto the notion used for degenerate four-wavemixing [55, 56] and refers to true mixing ofmore than two waves. Parametric scatteringcan be either conical or directional (“dots”).Conical scattering can be parametric or freeHS. A classification scheme is sketched inFig. 1.

Figure 2: Far field distribution of free HS in a 1 mmthick sample of SBN61 doped with 0.7 mol% Ce:Non-flip type of scattering is induced by e polarisedpump light of wavelength λp = 633 nm and exhibitsan asymmetric distribution with preference towardsthe negative end of the c-axis.

3.1. Free holographic scattering

3.1.1. Non-flip scattering:

Non-flip scattering is the most frequentlyobserved type of HS [1, 9, 11, 21, 33, 43,57–61]. The shape of the scattering pat-tern is mainly determined by the directionsof maximum gain. For electro-optic crys-tals these depend in the first place on thesymmetry of the electro-optic tensor [22,62, 63]. Asymmetric scattering patterns asshown in Figs. 2 and 3 are the character-istic fingerprints of amplification due to apure non-local recording mechanism. Whilethere is only one scattering lobe for SBN(Fig. 2), a more complicated distribution oc-curs in BaTiO3 (Fig. 3). In order to explainthe two additional side lobes deviating fromthe c-axis direction, elastic and piezoelec-tric properties have to be accounted for [43].In the stationary state one would expectfor LiNbO3 completely asymmetric scatter-ing with respect to the c-axis, too: for +camplification, for −c even damping. Thisis approximately observed only for ratherthick LiNbO3 crystals illuminated by an ex-traordinarily (e) polarised beam [21, 37] be-cause of the large interaction length (Fig. 4).However, the scattering pattern is symmet-ric for thin samples [64]. Interestingly, an

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Figure 3: Far-field distribution of free non-flip scat-tering in BaTiO3 [43]. The +c-axis points to theright.

asymmetric far-field distribution is found forBi12SiO20 under an externally applied elec-tric field (Fig. 5, [65]). It has been suggestedto originate from frequency detuning by aquasi-elastic (=slightly inelastic) scatteringprocess [65].

Only recently has HS been discovered inthe centrosymmetric NaNP (Fig. 6). Themechanism leading to the strong free scat-tering with an isotropic far-field distributionis still not clear (see Sec. 5.6.).

Flip scattering:

In most textbooks of optics it is statedthat mutual orthogonally polarised electro-magnetic waves cannot interfere, and in-deed, the interference pattern has constantintensity. Therefore the suggestions that flipscattering might occur in a free holographicscattering process sounds like a sacrilege.Even more than that: the necessary holo-graphic amplification mechanism for free flipscattering is not only required to permitholographic recording by orthogonally po-larised waves but in addition to be a non-local writing mechanism! Nonetheless, suchan odd process was discovered to occur inLiNbO3 [66] and BaTiO3 [67]. In princi-ple the phenomenon can take place in me-

Figure 4: Nearly symmetric far field pattern of freenon-flip HS in LiNbO3 induced by an e polarisedpump beam. The −c-axis points to the right.

dia with bulk photovoltaic effect. The phe-nomenological description of its photocur-rent density is given by

jl = βlmneme∗n (1)

where em denotes the mth component ofthe electric amplitude of the electromagneticfield and the asterisk the complex conjugate.As can be seen, mixing of mutual orthogo-nally polarised wave fields results in a pho-tocurrent provided that there exists a non-zero element βlmn of the photovoltaic tensorwith m 6= n. From the general symmetryrelation βlmn = β∗

lnm it is evident that such atensor element may be split up into its realsymmetric part βL

lmn = βLlnm and an imag-

inary antisymmetric part βClmn = −βC

lnm,where the superscripts L and C remind onthe fact that the first contribution to thephotovoltaic current density can be gener-ated by linearly and the second by circularlypolarised light. By defining a second ranktensor bC

lk via the relationship βClmn = bC

lkεkmn

with εkmn being the totally antisymmetricunit tensor we have

jl = βLlmneme∗n + ibC

lk (~e× ~e ∗)k . (2)

The second term on the right hand sideis imaginary, i.e., it describes the non-localcontribution to the current density responsi-ble for holographic amplification of free flipscattering [55].

Investigating HS with incident radiationpolarised at angles of 0, 45 and 90 with

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Figure 5: Far field free non-flip scattering inBi12SiO20 generated under an applied electric field(+c-axis pointing to the right [65]).

respect to the crystallographic c-axis, an effi-cient pumping process from e polarised lighttowards o polarised light was shown to oc-cur in LiNbO3:Fe [68]. This free flip scat-tering is a quasi steady state phenomenon,especially pronounced for o polarised pumpbeams. It arises after a transient incubationperiod of strong non-flip HS by virtue of non-local photovoltaic contributions by the ten-sor bC

mn [19]. HS is mirror symmetric with re-spect to a plane containing the c-axis and thepump beam direction (Fig. 7). As bC

12 < 0 inLiNbO3:Fe, the e polarised mode is amplifiedat the expense of the o polarised mode andvice versa for LiTaO3:Cu, because bC

12 > 0holds in the latter case [25].

3.2. Matched Holographic Scattering

Those cases where more than two plane wavecomponents of free HS are involved in thegeneration of HS can easily be recognised,since phase matching conditions apply inthick holographic media. They impose strin-gent restrictions on the scattering indicatrixof those processes and cause strikingly nar-row structures in the scattering patterns ascompared to the broad distributions charac-teristic for free HS.

As a prototype example we consider herethe interaction of four plane wave compo-

Figure 6: Free non-flip HS in NaNP (c-cut, b-polarised pump beam).

nents: pump waves α and β with fixed wavevectors ~kα and ~kβ and scattered waves γ and

δ. The tips of the vectors ~kγ or ~kδ of scat-tered waves with the same photon energy lieon ellipsoidal surfaces centred at the origin~k = ~0 of the reciprocal space, called herethe γ and δ ellipsoids. For simplicity onlythe δ waves are assumed here to write therefractive index structure with wave α, whilethe waves γ are generated by diffraction ofthe read-out wave β. In linear recordingmedia strongest response is to be expectedfor Fourier components with grating vector+ ~K = ~kδ − ~kα (image) and − ~K = ~kα − ~kδ

(conjugate image). As ~kα is fixed, the set

of all grating vectors ~K is the δ ellipsoidwith its origin shifted by −~kα. By virtueof the inversion symmetry of the δ ellipsoid,all grating vectors − ~K end on a shifted δellipsoid with its centre positioned at +~kα.These two ellipsoids are called the ellipsoidsof the structure factor or the structure ellip-soids in short. Now, consider a δ wave prop-agating collinearly with ~kα. If both have thesame polarisation state, we have nα = nδ forthe pertinent refractive indices, otherwise wemay have nα < nδ or nα > nδ resulting in thethree possibilities of the structure ellipsoidssketched in Fig. 8. The γ ellipsoid is thelocus of all waves that fulfill energy conser-vation when generated from wave β by solidstate excitations of frequency Ω. However,only the subset of these can really be gener-ated efficiently for which a solid state exci-

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Figure 7: Free flip scattering in LiNbO3:Fe in a planeperpendicular to the optic c-axis [19].

tation with spatial frequency ~K ′ exists suchthat the quasi-momentum, ~kγ = ~kβ + ~K ′, is

conserved. Because of ~K ′ = ~kγ − ~kβ this

is the case on a γ ellipsoid shifted by −~kβ

in the reciprocal space and is called Ewald’sellipsoid. Being interested only in the quasi-static excitations induced by α and δ, theconsiderations come down to demand fulfill-ment either of the Bragg condition

~kγ − ~kβ = ~kδ − ~kα or

~kγ − ~kβ = ~kα − ~kδ

Sturman et al. call the first possibilityA process, the second one B process [69],we prefer the designations S and S∗ pro-cess to remind the fundamental differencein the physics: the first corresponds to re-construction of the signal and the second ofthe (pseudoscopic) conjugate signal in theholographic read-out [70]. The Bragg condi-tions are fulfilled on the lines of intersectionof Ewald’s ellipsoid with the structure ellip-soids. As the intersection line of two ellip-soidal surfaces is always an ellipse, the scat-tered waves emerge on cones as sketched inFig. 9 for some rather general case. This iswhy such type of scattering is called coni-cal scattering. Note that with the above as-

(a)

(b)

(c)

Figure 8: Sketch of the structure ellipsoids (full line)

recorded by a wave with ~kα and all scattered waves

with ~kδ (tips ending on the dashed δ ellipsoids) for thecases (a) nα < nδ, (b) nα = nδ (c) nα > nδ. Left andright side: Image and conjugate image, respectively.

sumptions in general two cones are gener-ated, one with the cone axis along ~kα + ~kβ

(S process) and another with the cone axis

along ~kα − ~kβ (S∗ process). There remainsone exception where conical scattering can-not occur: the degenerate case with ~kβ = ~kα.This limiting case is free HS and has beendiscussed in Sec. 3.1..

To distinguish the different variants ofconical scattering, we introduce the notation〈γ|δα|β〉 for S processes and 〈γ|δα|β〉∗ forS∗ processes, where α, δ are the waves gen-erating the Fourier component + ~K and − ~K,respectively, and β is the read-out wave bywhich the scattered wave γ is produced.

So far we discussed processes where onepair of waves interacts by a writing pro-cess and the second by a reading processon the same grating. If the reciprocal pro-cess occurs, too, i.e., if both pairs are cou-pled by writing and reading, we call it para-

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Figure 9: Ewald’s construction explaining conicalscattering (Here for case (c) of Fig. 8). Bragg condi-tion is fulfilled at the intersection of Ewald’s ellipsoid(dashed) with the structure ellipsoid S (open squares)or with its conjugate S∗ (open circles). The scatteredγ waves (dotted) emerge on cones with axis of ro-

tation parallel ~kα + ~kβ (S process) or with axis of

rotation parallel ~kα − ~kβ (S∗ process).

metric scattering. For example, the case〈γ|αδ|β〉+〈δ|βγ|α〉 would denote a paramet-ric scattering process, and it goes withoutany saying that all scattering by parametricfour-wave processes are also conical scatter-ing processes. In many cases it is quite dif-ficult and always cumbersome to judge fromexperiments whether a true parametric scat-tering process is present. That is why theterm ”parametric scattering” is often usedsynonymously for ”conical scattering”.

Many examples of parametric HS were re-ported to appear as ring structures, occa-sionally as unclosed lines, on a screen placedbehind the sample [21, 26, 27, 6, 39, 40,71, 72]. Two or more conical scatteringsets may interact and thus single out dis-tinct spots (”dots”) [6] where either the in-tensity sums up (incoherent superposition)or constructive mixing of the amplitudes ofmore than four waves takes place. This di-rectional ”beam” scattering is a special fea-ture of multi-wave parametric scattering (inthe above generalised sense). As an excel-lent review article is available on parametric

scattering [69], we consider here only someselected cases.

3.2.1. Non-flip scattering:

The “trivial case” of conical scattering is theone observed by simply detuning the angleof incidence of the pump beam after freeHS had developed in a thick holographicrecording medium [30, 31]. Thereby theBragg condition is violated for all but thescattered wave vectors on the intersectionsof the Ewald ellipsoid with both structureellipsoids and two ring-shaped bright pat-terns contrast on a screen behind the sampleagainst the dark background. Sometimes thegain indicatrix or aperture limitations per-mit to see only one of these rings (Fig. 27).

Angular and wavelength dependence [30,31, 73] as well as the diffraction fine struc-ture [32] of the “trivial case” of conical scat-tering have been investigated. Trivial con-ical scattering is quite practical to analysefree HS or to unambiguously decide whetherfree HS had occurred in a sample. For exam-ple, in all materials with a quadratic electro-optic effect, HS can be avoided in lowest or-der if holographic writing is performed with-out application of an external electric field,because the writing process for the first spa-tial harmonics is latent. After latent writ-ing the hologram can be read out by apply-ing an external electric field (read-out field)without interference from HS. In contrast,free HS is generated by writing under ex-ternally applied field as can be proven bythe appearance of conical HS upon devia-tion from the original direction of the pumpbeam (Fig. 10, [74]) .

3.2.2. Flip scattering:

The first case of parametric flip-scatteringwas discovered in LiTaO3 [25] and is ofthe type 〈e|eo|o〉. The phenomenon is inso far peculiar, as it is caused by a local

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Figure 10: Far field pattern of conical HS in PLZT,(read out beam tilted against the original directionof the pump beam [74]).

recording process involving the symmetricphotovoltaic tensor element βL

131 and has aquadratic dependence on the sample thick-ness. Holographic flip scattering of the type〈o|ee|e〉∗ has been discovered in BaTiO3 andLiNbO3 crystals [21, 26, 27, 54].

As LiTaO3 is positively birefringent theanalogous phenomenon is of the type〈e|oo|o〉∗ [48]. As for most types of para-metric flip scattering, this angle is sensitiveto the birefringence of the photorefractivecrystal [21]. As sketched in Fig. 11, un-der strict phase matching condition an e po-larised pump beam can be diffracted in thesecrystals to an o polarised beam resulting in acone of well-defined aperture angle (Fig. 12).

HS in the centrosymmetric NaNP does notonly occur as free non-flip HS (see above)but also as parametric and free flip scatter-ing [50] (for details see 5.6.).

4. PARAMETERS AFFECTING HSAND THEORETICAL MODELING

Most of the applications (see sec.6.2.) whichare based on beam fanning require both, fastresponse and high gain, to exploit the beamfanning efficiently. There are few materialsthat may meet those criteria (see Sec.5.) andmuch research work is under way to enhance

Figure 11: Sketch of the geometry for wave-vectors allowing for flip scattering in the reciprocal

space:~kp. . . incident wave-vector of the pump beam,~ks(e, o). . . wave-vectors of the e and o scattered waves,~K. . . grating vector.

beam fanning and to increase the speed. Inother applications, beam fanning must besuppressed as effective as possible or eveneliminated like for optical interconnections,image amplification, etc.

In general, HS is influenced by the ma-terial itself (type of material and doping),sample preparation (thickness of the sample,surface quality, number of defects), the in-tensity and the geometry of illumination, thewavelength of the radiation, the beam profile,and a host of other parameters, see e.g. [11].From the fact that interference is a basic re-quirement for HS (see Sec. 2.) it is evidentthat the coherence of the light source andde-coherence processes in the sample play anessential role.

It sounds plausible that the main contri-bution to the seed scattering is generatedat the entrance face (because of the largerefractive-index change at the air-crystalboundary) or at least close to it (where thepump wave is hardly depleted by absorptionor scattering). Sun et al. [75] advocate infavour of this view, because they did not finda difference of HS for a beam focussed on athin and a thick sample, respectively. How-ever, the argument is not that convincing,because they did not account for the effectof self-focussing and wave-guiding.

Computer simulations seem even to con-

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Figure 12: Conical flip scattering of the type 〈o|ee|e〉in BaTiO3 in a plane perpendicular to the c-axis [21].

tradict a model of surface seed scatter-ing [76]. Gu and Yeh investigate there-fore seed scattering by randomly distributed,screened charges in the volume of the crys-tal [23]. Seed scattering by static spa-tial fluctuations of the refractive-index andof the photovoltaic tensor was extensivelytreated by Obukhovskiι et al. [4, 77] who ex-tended the theory to include also static spa-tial fluctuations of the dark conductivity andthe tensor of photoconductivity [78].

Diffraction from holograms of a planepump wave and of seed waves emerging frompoint scatterers in the volume is consideredby Forshaw [32]. Thereby the interactionwith the scattering of the read-out wavefrom the same point scatterers as well as theeffect of deviation from the Bragg angle wereconsidered, but not the nonlinear recording

Figure 13: Parametric (ring) and free (central part)flip scattering in NaNP (c-cut).

dynamics in non-latent photorefractive me-dia.

The non-linear dynamics of free HS wastaken into account for surface as well as vol-ume seed scattering in Ref. [24, 60]. Eventoday the work of Marotz et al. [60] maybe regarded as the most complete theoryof free HS. Retaining the transverse diffrac-tion term in the wave equation, the origin ofphase conjugate scattering was analysed inRef. [79]. In the final stage of HS intensityzeros at lines where the wave field changessign might be important [8] thus suggestinga link to optical vortices [80].

For static volume seed scattering, calcu-lations by Obukhovskiι and Stoyanov con-firm that for local recording processes thereis only transient amplification for HS, i.e.,in the steady state no HS can persist [4].Hence, amplification mechanisms for HSproposed for local recording processes relyon Rayleigh noise or other sources of quasi-elastic scattering [79, 81].

The nonlinear dynamics of parametricscattering resulting from two incident pumpwaves was modeled in Ref. [35, 39, 41, 46]and [72] (most general theory). Novikov etal. developed a theory for generation of sub-harmonic HS [40].

In many cases the general behaviour of HScan be fitted by the following approximativeformulae:

Is(t)

Is(0)=

exp (Γeffd)− 1

Γeffd(3)

for scattered beams with intensity Is [82] andby

Ip(t)

Ip(0)=

1 + m0

1 + m0 exp (Γeffd)(4)

for the transmitted pump beam intensityIp [21, 83]. Both formulae were derived fromeffective two-beam approximations [21, 84],where Γeff = Γ [1− exp (−t/τ)] denotes theeffective amplification, t the time, τ the ef-fective relaxation time, and m0 the initial

11

scattering ratio.For the particular case of sillenites and

fiber-like crystals with small electro-optic co-efficients as compared to BaTiO3 or SBNthe diffusion-type photorefractive nonlinear-ity is small. The application of an alter-nating external electric field, in particular ofsquare pulse form, may be used to achievestrong beam coupling and thus enhancementof HS [82].

It has been often overlooked that non-local recording does not necessarily meanrecording by a diffusion field ED only. Thecomplete expression for the non-local contri-bution ENLto the space charge field is givenby [85]:

ENL = ED(1 + ED/Eq) + E2

0/EqED

(1 + ED/Eq)2 + E2

0/E2q

(5)

where Eq is the maximum space-charge fieldattainable for a given trap or donor den-sity [86] and E0 = Eext+Eph. As the formulasuggests, a constant photovoltaic field Eph aswell as a constant externally applied electricfield Eext enhance HS. The latter effect wasnot only experimentally demonstrated by di-rect application of an external field but bymore subtly exploiting pyroelectric and de-polarisation fields generated by thermal andirradiation treatments [87].

As pointed out by Montemezzani et al.,besides the properties usually considered(dielectric, conductive and electro-optic),in some cases piezoelectric and mechanicalproperties play a decisive role for amplifica-tion of HS [43].

Moreover, the importance of the wave-length of the recording beam, the angle of in-cidence [42], the crystal temperature [83], ad-ditional incoherent illumination [84], and thepolarisation of the recording beam [38, 88,89] for the presence and strength of HS havebeen underlined by various authors. Angu-lar and polarisation dependence discussed byBelendez et al. [90] might not originate from

noise gratings but be caused by optically in-duced birefringence of the medium.

While no dependence of HS was found forbeam diameters of 7 mm and 1 mm [58],Zhang et al. [61] report a decrease of HS forbeam diameters below 0.9 mm, particularlystriking for diameters below 15 µm wherescattering vanishes. This was attributedto the decrease of the effective interactionlength [22] leff = L [1− exp (−D/L sin Θ)]for holographic amplification with decreas-ing beam diameter D at a given scatteringangle Θ and sample length L.

5. MATERIALS EXHIBITING HS

To give an impression on the generality ofthe phenomenon we pick up some completelydifferent materials which exhibit HS: acen-tric electro-optic crystals (SBN, BaTiO3 andLiNbO3), fiber-like materials (Bi12SiO20), acentrosymmetric crystal (NaNP) and pho-topolymers (PMMA).

5.1. SBN

SBN is an important representative for diffu-sion driven holographic light scattering [91].The large linear electro-optic coefficients [92,93] guarantee a strong amplification of thescattered light, provided that a wavelengthis chosen at which a significant chargetransport can be excited (Fig. 2). E.g.,strong light scattering can be observed incerium doped (0.7 mol%) Sr0.61Ba0.39Nb2O6(SBN61) with a wavelength λp=633 nm ofan e polarised pump wave in the rangeof the relaxor type phase transition at T= 325 K. The fact that scattering is di-rected antiparallelly to the polar c-axis ofthe single crystal proves that the chargecarrier redistributed by diffusion are elec-trons. Charge transport based on a hop-ping model and photo-induced conductivityare discussed in Ref. [94, 95]. The inten-

12

sity distribution of the scattered light alongthe c-axis (Fig. 14) results from the depen-dence of the holographic gain on the anglebetween pump beam and scattered waves.The gain is proportional to the effective

Figure 14: Angular dependence of the scattered in-tensity in the a − c-plane for SBN61 doped with 0.7mol% Ce and a thickness of d=1 mm. Pump beam:e polarisation, λp = 633 nm. The −c-axis points tothe right.

linear electro-optic coefficient reff(T ) andthe space charge field Esc(Θ(T )), which de-pends on the angle Θ(T ) due to the tem-perature dependent screening length [96]:Γ ∝ reff(T )Esc(Θ(T )). As a consequence,the apex angle of the maximum of thescattered light and its amplitude dependstrongly on temperature and doping con-centration. On the other hand the over-all behaviour of linear electro-optic coeffi-cients and space charge fields can thereforebe studied by HS as a function of tempera-ture near phase transitions since the depen-dence is particularly pronounced.

Pure SBN61 exhibits a ferroelectricrelaxor-kind phase transition from thetetragonal polar point symmetry group 4mmto the 4/mmm high temperature phase atabout T = 360 K [97]. Doping with cerium,strongly lowers the phase transition tem-perature [98], whereby Ce3+ occupies Sr2+

sites in off-centre positions [99, 100]. This

special kind of phase transition was anal-ysed by HS for the first time in Ref. [101].Far field patterns of scattered radiation areshown in Fig. 15 at temperatures belowand above the relaxor phase transition. Inthe same measure as the total cross sec-tion of HS increases when approaching thephase transition, transmission of the pumpbeam decreases (Fig. 16). The broad min-

Figure 15: Intensity distribution of non-flip scatter-ing for SBN61 doped with 0.7 mol% Ce for differenttemperatures induced by an e polarised pump beamat λp = 633 nm.

imum without any critical behaviour is ex-plained by a random field Ising model [102,103], whereby the random fields act as pin-ning centres for the ferroelectric polarisa-tion, which consequently decays over a largetemperature range [104]. Further, the factthat the minima shown in Fig. 16 occur atdifferent temperatures for different Ce con-centrations demonstrates the dependence ofthe phase transition temperature on the dop-ing concentration. Applying external volt-age on the crystal along the c-axis, flip scat-tering can be observed as a ring in oppositedirection to the intensity of free HS [102].

13

Figure 16: Temperature dependence of the transmit-ted intensity for SBN61 with two different dopingconcentrations.

Measuring the cone angle of this parametricscattering process, the birefringence and es-pecially the order parameter (the ferroelec-tric polarisation) can be traced as a func-tion of temperature. Again a relaxor kindbehaviour is found. The examples discussedso far show that special features of all kindsof phase transitions can simply be analysedby HS.

5.2. Barium titanate

BaTiO3 is maybe the most popular photore-fractive crystal, especially for laboratory ex-periments based on beam fanning such asphotorefractive passive phase conjugate mir-rors, self-pumped phase conjugate mirrors ingeneral [53, 5, 105, 106], and optical nov-elty filters [107, 108]. At room tempera-ture BaTiO3 is a ferroelectric crystal witha phase transition at about 5C. This re-sults not only in an extremely large dielec-tric constant ε11 = 4400 [109], but also ina large electro-optic coefficient of r51 = 1300pm/V [109], and hence a high gain factor forthe diffusion recording mechanism. Further-more, the response time is still acceptablyfast.

One source of seed scattering may beseen in a light-induced increase of Rayleigh

scattering. The effect is particularly pro-nounced at the ferroelectric phase transitiontemperature and was attributed to photo-fluctuations [110].

A single pump beam incident upon thecrystal may produce stimulated scatteringin the near-forward direction [9], the back-ward direction [111], in a ring scattering pat-tern [21, 26], or in a variety of complicatedscattering patterns [63]. Different scatteringrings and lines are observed when two wavespenetrate the crystal [39, 41, 44, 47, 71, 67,112–117]. Most of the applications of beamfanning presented in Sec. 6.2. were set upemploying BaTiO3.

5.3. Lithium niobate

Lithium niobate is a fine model crystal forthe study of effects connected with nonlin-ear optical properties. Like BaTiO3 it is fer-roelectric at ambient conditions, thus acen-tric with non-vanishing linear electro-opticcoefficients. Lithium niobate is the mostwidely studied photorefractive crystal, andhas been doped with various ions (preferablyiron), sometimes even with two species [118,119, 7, 120]. In Fe-doped crystals the photo-voltaic effect is by far the dominant chargetransport mechanism, hence local responseprevails. At the first glance this wouldexclude any amplification mechanism andhence the existence of HS. However, the ef-fect of optical damage was even discoveredin LiNbO3 [1] and it turned out to be verystrong. Now, not understanding but ratheraccepting the presence of HS we would guessthat due to a small contribution of a nonlo-cal charge transport mechanism (e.g. diffu-sion) the scattering pattern would be deter-mined by the symmetry of the electro-optictensor. This would require amplification to-wards the +c-direction of LiNbO3 (posi-tive electro-optic coefficient) and depletionalong the −c-direction (negative value) like,

14

e.g., in SBN. However, again LiNbO3 doesnot fulfill our imagination as from the ex-periment we see that amplification occurs inboth directions albeit without full symme-try, see Fig. 4! Several models to explainthis phenomenon were proposed [77, 78, 81].The ideas are based on either spatial fluctu-ations of the photovoltaic tensor or tempo-ral fluctuations of the refractive-index andof the intensity of the pump wave. In oneof those papers the importance of the in-teraction between the scattered waves waspointed out, a fact which was always ignoredwhen analysing the scattering process.

Light-induced scattering has been exten-sively studied from various points of viewin LiNbO3 [1, 19, 21, 33–36, 66, 7, 121]In this chapter we restrict the topic to re-cent work performed on the angular, spectraland polarisation dependence of the transmit-ted intensity during reconstruction of para-sitic holograms for the non-flip type of freeHS. A phenomenological model based onthe Ewald sphere construction [122, 123]explains the major features of the recon-struction curves [37]. Finally some applica-tions [38, 124] are discussed. The investiga-tions shed some light on the question of theoptimal conditions at which the effect of HScan be minimised, or at which advantageoususe can be made for material characterisa-tion.

Genesis and evolution of parasitic holo-grams can best be studied with a single beamsetup. By choosing a geometry with max-imised effect, i.e. by irradiating the sam-ple with an e polarised pump beam of wave-length λp = 488 nm impinging perpendic-ularly onto the sample surface. In Fig. 17the time dependence of the transmitted in-tensity is shown, for an oxidised LiNbO3 :crystal doped with 0.1 wt.% Fe2O3 in themelt and a thickness of about 2.6 mm. Thesolid line represents a fitting curve which de-scribes the time dependence of the recording

behaviour (Equ. (4)) [21] and thus gives in-formation about some characteristic param-eters of the crystal, e.g., the photoconduc-tivity [21]. After reaching the steady state

Figure 17: Time dependence of the transmitted in-tensity in LiNbO3 under illumination with a singlelaser beam.

rotations were performed either around anaxis parallel (ω-rotation) or perpendicular(φ-rotation) to the crystallographic c-axis,reading out the parasitic holograms at dif-ferent wavelengths and polarisation states.In Fig. 18 the transmitted intensity as afunction of the rotation angle 1 ω is shown.Those reconstruction curves include a lotof features that characterise the crystal, itsphotorefractive properties, and the recordedparasitic holograms. For the reconstructionconditions shown in Fig. 18 (ω-rotation, epolarisation [37]), the transmission curveshave the following characteristic features:

• For λr < λp two minima appear atω 6= 0 being symmetric with respect tothe origin ω = 0, and exhibiting a maxi-mum there. The rotation angle at whichthe minima occur increases as the differ-ence between the reading and recordingwavelengths increase.

1In all graphs the angles are given within the crystal, i.e.they are corrected for the refractive index.

15

Figure 18: Transmitted intensity as a function of therotation angle ω for various reading wavelengths λr

and e polarisation.

• For λr ≥ λp only one minimum existsat ω = 0 and the transmission valueincreases as the λr increases.

In Fig. 19 the transmission curve togetherwith the corresponding scattering patterns isshown for the case of a rotation perpendicu-lar to the c-axis and for e polarisation. Oneimportant difference between the two typesof rotation which can easily be seen is that atλr = λp the half-widths differ tremendously.The FWHM for the φ-rotation is nearly anorder of magnitude smaller than that for anω-rotation.

Inspecting Fig. 19 different characteristicscan be observed [38]:

• All transmission curves for λr 6= λp ex-hibit two minima.

• An asymmetry in the depth of theminima is observed which reflects theasymmetry in the scattering patternalong the c-axis behind the crystal (seeFig. 4). We want to call the reader’sattention to a surprising point concern-ing the relative depth of the two min-ima: It depends on the sense of therotation and the ratio of wavelengthsλr/λp which of the minima is shalloweror deeper, respectively.

Figure 19: Transmitted intensity and three of the cor-responding scattering patterns as a function of therotation angle φ, for various reading wavelengths λr

and e polarisation.

The results for the same experiment butusing an o polarised beam for reconstruc-tion [38] are shown in Figs. 20 and 21. Ingeneral, the results resemble the previouslydiscussed case when reconstructing with epolarised light. However, there are two maindifferences:the transmission values are much higher inthe case of reading with o polarised light(note the scale when comparing Fig. 18 and20!) and the absolute minimum value oftransmission does not occur at λr = λp butat λr < λp. The main features of the ex-perimental observations can be explained bya simple phenomenological model based onEwald’s construction [37]: Assume that in-terference of the pump beam and of thescattered wave field lead only to a smallmodulation, and hence a linear response ofthe refractive-index. Nonzero values of therefractive-index change δn ~(K) will only beobtained if the energy is conserved, i.e. theBragg condition is fulfilled. This confines the(reciprocal) space of allowed wave-vectors

for nonzero δn ~(K) to the surface of twoellipsoids, the primary and conjugate im-ages (Grey area in Fig. 22) representing the”structure factor” of the parasitic gratings.

16

Figure 20: Angular dependence (ω) of the transmit-ted intensity for reconstruction with o polarised light.

Inspecting again the scattering pattern ofLiNbO3(Fig. 4) we recognise that the inten-sity is more or less exclusively distributed inthe kx−kz-plane. This two-dimensionality ofthe problem introduces new particular fea-tures, and moreover simplifies the model.It is evident that intensity for the scatter-ing process can only be provided from theregion of the intersection of the kx − kz-plane with the two ellipsoids, called ”record-ing ellipses”. When reconstructing the para-sitic hologram the totally scattered intensityis given by the integral value of the inter-section region containing intensity (record-ing ellipse) and simultaneously satisfying theBragg condition geometrically representedby the Ewald sphere (reading ellipsoid orsphere). Fig. 22 schematically shows therecording ellipses in addition to the readingellipses or circles at different geometries atφ = ω = 0 for different wavelengths in re-ciprocal space. The grey part represents therecording ellipses with a strongly exagger-ated width and ellipticity for a better repre-sentation.

Despite of its extreme simplification thismodel explains the main features of theabove presented results. It correctly pre-dicts the occurrence of minima and double-minima under rotation around a crystal axisand the behaviour when changing the wave-

Figure 21: Angular dependence (φ) of the transmit-ted intensity for reconstruction with o polarised light.

length and/or polarisation. The experimen-tal results can qualitatively be understoodonly by inspecting Fig. 22 and consideringthe corresponding reading ellipses or circles.

Reconstruction and recording of parasiticholograms can be used to extract the follow-ing information:

1. From the recording curve the dielec-tric relaxation time and the photocon-ductivity can be determined, as wellas an estimation of the gain and therefractive-index changes are possible.

2. From the asymmetry of the scatteringpattern the direction of the positive+c-axis direction of the crystal can befound, knowing the sign of charge carri-ers and the sign of the effective electro-optic coefficient.

3. From reconstruction with o polarisedlight the optical character can be deter-mined, i.e., whether the crystal is neg-ative or positive birefringent. The oc-currence of the lowest minimum valueat λr < λp or at λr > λp is indicative,respectively.

4. From reconstruction at the Bragg angleusing o and e polarised light we can es-timate the value of birefringence for thecrystal, or the ratio n3

or13/n3er33.17

Figure 22: Geometrical representation of the record-ing ellipses and either reading ellipses or circles atthe recording (Bragg) condition for different readingwavelengths in the reciprocal space. The grey partrepresents the recording ellipses. The reading ellipse(solid line) corresponds to the case of e–e polarisationand λp = λr. The reading circles correspond to thecase of e–o polarisation at λp = λr (bold solid line),λp < λr (dashed line) and λp > λr (dotted line).The short-dashed line is the reading ellipse for e–epolarisation, λp = λr and a rotation around an axisperpendicular to the c−axis.

To minimise the effect of parasitic hologramsit is therefore advisable to use o polarisationand wavelengths higher than the recordingwavelength for reconstruction.

5.4. Sillenites

Under ambient conditions the electro-opticcoefficients are too small for fanning ef-fects in low-gain materials such as sillenites,Bi12SiO20, Bi12TiO20 and Bi12GeO20. How-ever, the application of a proper exter-nal electric field results in the formationof a refractive-index grating which is phaseshifted by π/2 with respect to the inter-ference pattern. Therefore a strong energyexchange between the pump beam and thescattered light occurs. It was shown that thehighest gain in sillenites is obtained by ap-plying the electric field parallel to the 〈111〉

crystallographic axis [82]. The fanning pat-tern depends strongly on the polarisation ofthe pump beam [125]. A two-dimensionalcoupled wave theory [65, 126, 127] is usedto explain the angular distribution of fan-ning light in sillenite crystals under an ap-plied external electric field. Fast response ofbeam fanning can be obtained in fiber-likephotorefractive samples [128].

5.5. PLZT

Krumins et al. investigated HS in thetransparent ceramics of compositionPb0.9La0.1(Zr0.65Ti0.35)0.975O3 (PLZT-10/65/35) as a function of light intensity,electric field and thickness [59, 74, 129].Essentially, steady-state HS is caused by anon-local recording mechanism and exhibitsan asymmetric distribution of scatteredradiation depending on the direction ofthe applied external field. It has, however,been notified that there is also HS for dom-inating local recording mechanism havinga strikingly different, namely symmetric,distribution of HS [59]. In two-wave mixingexperiments HS reduces the two-beam mod-ulation, strongly deteriorates the regularity(and thus the contrast) of the two-beaminterference pattern with increasing expo-sure and externally applied field strength,and even may be responsible for a differenceof the diffraction efficiency measured witha read-out beam from both writing beamdirections [85] (scattering analogue of theBorrmann effect [54]).

PLZT-10/65/35 is a material with a quasi-quadratic electro-optic effect. Without ex-ternal electric field, i.e., for recording bydiffusive charge transport, refractive-indexgratings with spatial frequency K can-not be generated, and self-diffraction andholographic amplification is effectively sup-pressed. For such a latent recording processHS cannot build up, a property which those

18

ceramics share with all photorefractive ma-terials exhibiting a quadratic electro-opticeffect.

5.6. Sodium nitroprusside

5.6.1. Metastable electronic states

Single crystals of NaNP are grown from anaqueous solution. NaNP belongs to theorthorhombic centrosymmetric space groupPnnm [130]. The photorefractive behaviourresults from light-induced excitation of twometastable electronic states, SI and SII,thus being fundamentally different fromthe one in electro-optic materials. Themetastable electronic states are localisedat the nitrosylprusside-anion, [Fe(CN)5NO],and can be excited by light in the blue-green spectral range. Their lifetime is ex-tremely large at low temperatures(e.g., 108

s at T = 138K). The population of themetastable states depends on exposure Q =It, where I is the laser intensity and t theillumination time [131]. De-excitation is re-alised by illumination in the red and in-frared spectral range or by thermal heatingabove T=150 K (SII) and T=200 K (SI),respectively. Further details on the natureof the metastable electronic states are givenin Ref. [131]. Light-induced phase transi-tions from the centrosymmetric space groupPnnm into one of the acentric subgroupsPmn21, P21212, P2, P21, and P1, exceptPnn2, can be excluded, because neutron andX-ray diffraction measurements with two-dimensional detectors [132, 133] show thatthe diffraction conditions for the (hkl) reflec-tions in the space group Pnnm are conservedfor the ground state and the metastablestates [134], i.e., new reflections are notfound. Further, Mossbauer spectroscopyshows identical temperature dependence ofthe quadrupol splitting and isomer shift forthe ground state and metastable state SI ofNaNP, respectively [135].

5.6.2. Photorefractive properties

Excitation of metastable electronic statesis accompanied by a strong change of theabsorption coefficient [136] and of the re-fractive index [137]. Elementary holo-graphic gratings can be written in theblue-green spectral range by excitation andin the red-and infrared spectral range byde-excitation of the metastable states, re-spectively. In general these are mixedabsorption-/refractive-index gratings. Therefractive-index change ∆n is the main con-tribution responsible for the high diffractionefficiency of NaNP and attains its maximumwith 1.4× 10−2 for λ=514 nm. The remark-ably large refractive index changes are a pos-sible key for the explanation of the observedholographic scattering to be discussed be-low [137].

Interestingly, the refractive-index ampli-tude is still pretty high in the red (∆n ≈10−3 at λ= 633 nm) and near infrared(∆n ≈ 10−4 at λ = 1047 nm) spectral re-gion. Build-up of holographic gratings is inagreement with the facts known for the ki-netics of the population and de-excitation ofthe metastable electronic states. The record-ing mechanism becomes increasingly non-linear upon overexposure. As a consequence,the first spatial harmonics of the refractive-index amplitude possess a maximum and thediffraction efficiency vanishes asymptoticallyfor long exposure (Fig. 23).

With regard to HS a test shows that holo-graphic phase gratings can be written evenfor small modulation of the incident light in-terference pattern using a two-beam interfer-ometer. Fig. 23 shows the dependence of thediffraction efficiency η = IRd/(IRd + IRt) =sin2(π∆nd)/(λ cos θ) on exposure for an in-tensity ratio of the incoming writing beamsof 500:1. Here the indices d and t refer tothe diffracted and transmitted beam, respec-tively (see experimental scheme in the insetof Fig. 23). For a b-cut sample of a thick-

19

ness of 150 µm a maximum diffraction ef-ficiency of 2% is reached at an exposure of140 J/cm2 for λp = 514 nm,a-polarisation,and an angle of incidence of 20 (thus avoid-ing interference with HS). As mentioned,the diffraction efficiency vanishes in steadystate. Using two-beam coupling analy-

Figure 23: Diffraction efficiency η as a function ofexposure Q .

sis the written phase gratings are found tobe in phase with the interference pattern,excluding amplification by energy transfervia two-beam coupling. Hence, holographicscattering was believed to be forbidden inthis photorefractive system. Beside writingholograms with parallelly polarised waves,recording with mutually orthogonally po-larised waves is possible with a pronounceddiffraction efficiency of up to 95% (corre-sponding to ∆n = 10−3), which does not de-crease to zero for large exposure (in contrastto holographic gratings written with paral-lelly polarised waves).

5.6.3. Non-flip scattering

HS is usually investigated with an unex-panded beam of an argon-ion laser strikingthe sample perpendicularly to the entranceface.

A broad isotropic intensity distribution in-

dicating free HS appears at temperaturesbelow T = 200 K in a-, b- and c-cuts pro-vided that polarisation is in the a − b-plane(Fig. 24.). In contrast, there is neither a

(a) (b) (c)

Figure 24: Intensity distributions observed in trans-

mission for an exposure of Q = 50[J/cm2]. (a) a-cut,b-polarisation (b) b-cut, a-polarisation (c) c-cut, a-polarisation. For c-cut and b-polarisation see Fig. 6.

change of transmission nor HS for polarisa-tion parallel to the c-axis (Fig. 25), but HSreappears immediately, if a read-out polar-isation in the a − b-plane is chosen. These

(a) (b)

Figure 25: Intensity distribution in transmission forc-polarisation. (a) a-cut (b) b-cut

findings do not depend on the direction ofthe pump beam. They agree with the po-larisation dependence observed for writingof elementary holographic phase gratings inNaNP [137] and strongly suggest a photore-fractive origin. Measurements at differentwavelengths demonstrate, that the broaddistribution of free HS arises in the blue-green spectral range between 350 < λ < 580nm, i.e., in a range where metastable elec-tronic states can be excited. In the redand infrared spectral range free HS can onlybe generated after pre-exposure in the blue-

20

green spectral range, thus being caused byde-excitation of metastable states in anal-ogy to the results of two-wave mixing ex-periments. Variation of the crystal thick-ness between 0.1 and 2 mm has no influ-ence on the isotropy of the scattered inten-sity distribution. Fig. 26 shows the de-

Figure 26: Intensity of the transmitted beam to-gether with the intensity scattered at an angle δ =5 as a function of the exposure Q. The experimentalarrangement is illustrated schematically in the inset.

pendence of the transmitted and scatteredintensity, respectively, as a function of ex-posure, for a b-cut crystal with thicknessd= 150 µm and a-polarisation. The inten-sity of the transmitted beam first decreasesand that of the corona increases with expo-sure until extrema are reached for Q ≈ 50J/cm2 and Q ≈ 80 J/cm2, respectively. Fur-ther exposure increases the intensity of thetransmitted laser beam and decreases thescattered intensity. Saturation is obtainedfor Q ≈2000 J/cm2, in accordance withthe maximum population of the metastablestate SI for the same polarisation [131]. Insaturation both intensities differ from theirinitial value. This means that the writ-ten gratings do not completely disappearas it is the case for elementary holographicphase gratings written in a two-beam con-figuration [137]. The major argument thatwe are really dealing with HS is the exis-tence of a scattering cone. In the presenceof HS deviations in the read-out angle and

in wavelength lead to characteristic diffrac-tion cones, as the Bragg condition is vio-lated for nearly all parasitic gratings exceptthe ones on the cones. Forshaw [31] ex-plains this phenomenon by making use ofan Ewald construction, in which the pri-mary and conjugated spheres of the wave-vectors for recorded parasitic gratings in-tersect with the Ewald sphere for read-out.Diffraction of the read-out light occurs atgratings that lie on the rim of the circularcuts, so diffracted waves propagate on thesurface of two characteristic cones (primaryand conjugate spheres). The isotropic distri-bution changes into a diffraction cone, whichoriginates in the exposed region when thecrystal is rotated about an axis within thecrystal plate and gives a characteristic brightring in projection, as shown in Fig. 27. Thevertical line belongs to the diffraction coneof the primary image, which has a large apexangle. The b-cut was pre-illuminated up to

Figure 27: Holographic non-flip scattering pattern forNaNP. The sample was rotated with respect to therecording.

an exposure of Q = 50 [J/cm2] with light

of wavelength λp = 514 nm, ~E||a axis, andread out under an angle of α = 15. Theappearance of this phenomenon shows thesame dependence on the polarisation statesof the recording and read-out beams and onthe wavelengths as free HS. The apex angle

21

of the diffraction cone is given by:

ξ = 2 arctan

[sin α

cos α ± λp/λr

]. (6)

The positive sign belongs to the conjugatesphere and the negative sign to the sphereof the image [31, 138]. The apex angle de-pends on the read-out angle α and the wave-length λr with respect to the pump beamconfiguration (λp, α = 0) and reduces toξ = α if λp = λr for the conjugate sphere.As the apex angle of the diffraction cone forthe primary sphere starts with π and de-creases to zero with increasing read-out an-gle, measurements are not possible withinthe experimentally given maximum apex an-gle of ± 20. The apex angle ξ of the diffrac-tion cone for the conjugate sphere was mea-sured as a function of the read-out angle αfor λr ≡ λp = 514.5 nm (squares in Fig. 28)and 632.8 nm = λr > λp = 514.5 nm (cir-cles in Fig. 28). All experimental resultsare in good agreement with the theoreticallyexpected behaviour derived from the Ewaldconstruction. This is illustrated by the solidlines in Fig. 28.

Figure 28: Apex angle as a function of read-out anglefor two different read-out wavelengths (cf. text).

5.6.4. Holographic flip scattering

In addition to build-up of the already identi-fied holographic non-flip scattering a diffrac-

tion cone with an apex angle of θ514 =(52 ± 1) shows up for illumination withan unexpanded beam of an argon ion laserat wavelength λ = 514.5 nm propagatingalong the c-axis (c-cut, thickness d=300µm,b-polarisation). It can be observed as a ring(width of 2) encircling the broad distribu-tion of free HS (Fig. 13). The light of thiscone is polarised orthogonally to the polar-isation of the pump beam, i.e., parallelly tothe crystallographic a-axis. The cone is alsoobserved in backward reflection. The non-flipped and flipped part of the light scat-tered in forward direction can be separatedby a polariser which is inserted in the trans-mission pattern after the scattering has oc-curred. Returning to sec. 2., the resultingscattering patterns for a polarised (Fig. 6)and b polarised (Fig. 13) light are shown,respectively. Note, that in addition to theflip scattering cone flip scattering exists alsofor free HS (Fig. 13). Both, non-flip and flipscattering can be generated with light in theblue-green spectral range in which the exci-tation of metastable electronic states takesplace. The apex angle of the flip scatteringcone is about 0.5 degree smaller if generatedat 457.9 nm. The intensity scattered intothe cone nearly vanishes, if the crystal is de-tuned by about 1 with respect to the pumpbeam. Both observations imply that a Braggcondition has to be obeyed. This demon-strates, that the cone originates from diffrac-tion from holographically recorded parasiticgratings. With an a-polarised pump beam inthe same experimental geometry, the conecannot be observed within the maximumpossible observation angle of θ=65 . Free aswell as conical HS disappear with the decayof the metastable electronic state at about200 K.

22

5.7. Photopolymers

Photopolymers are supercooled liquids andtherefore optically isotropic. When Moranand Kaminow studied the behaviour andcharacteristics of holographic diffractiongratings using PMMA sensitised for UVlight (λp = 325 nm) by peroxidation [30],they observed a curious two-ring scatteringpattern behind their sample. The patternconsists of an inner and an external ringboth passing through the bright spot pro-duced by the direct laser beam. The in-ternal angles of the horizontal beam for theinner and external rings were found to de-pend on the ratio of reading to recordingwavelength λr/λp. The ring pattern can-not be accounted for by simple scattering.Later the appearance of diffraction rings wasexplained to be caused by the interferencebetween the original laser beam and lightscattered from internal or surface inhomo-geneities in unexposed PMMA [31]. The ef-fect is closely related to the Venetian blindphenomenon [122] which has been explainedusing the Ewald sphere concept. One ofthe diffraction rings is produced by diffrac-tion of light from the primary image termof the holographic pattern, while the otherring originates from the conjugate image. Aquantitative comparison for VIS sensitisedPMMA with theory is published in Ref. [73].HS similar to PMMA was observed in manyother photopolymers [32, 139–141].

6. VIRTUE AND DRAWBACKS OF HSIN APPLICATIONS ANDMATERIAL RESEARCH

One of the major applications of pho-torefractive materials is image amplifica-tion. Weak optical signals experience gainif feeded with pump light due to the non-linearity of the medium. Unfortunately, thespontaneous optical noise is amplified, too,

and strongly limits the usefulness of such de-vices.

Another important application of pho-torefractives is data storage by means oftwo-wave mixing techniques. Here againHS turns out to be a serious obstacle forrecording data with a low error-bit-rate. Insuch applications HS is disadvantageous andtherefore a crucial task to avoid or at leastto suppress it.

On the other hand HS may be quite usefulfor certain purposes, too: e.g., it can initi-ate the self pumped phase conjugation pro-cess [63, 142].

In what follows we summarise numerousmethods published in the literature on howto efficiently suppress HS or how to make useof it.

6.1. Suppression of HS

In general, the most important limitations ofthe practical application of thick holographicrecording media are connected with the ori-gin of strong light-induced scattering duringthe process of optical recording. The pres-ence of parasitic gratings (noise gratings)transfers almost all the light from the initialbeam to the scattering directions and greatlycorrupts the quality of any optical system.Fan-out light, i.e. light which is scatteredout of the pump beam, is amplified, and notonly adds strong noise to the reconstructedimages, but also depletes the energy of thepump as well as the signal beam, leadingto a reduction of the net gain. The largerthe two-beam coupling gain, the stronger thelight-induced scattering. Therefore, beamfanning is a major problem in applicationslike optical data storage. Thus suppressingthat effect becomes a main goal in opticalimage processing by two-wave coupling inphotorefractive materials. Many techniqueswere proposed and demonstrated to elimi-nate or suppress the light-induced scattering

23

(beam fanning) while maintaining high gainfor the two-wave mixing image amplification.

There are three different attempts how tosuppress the light-induced scattering. Twoof them are effective in suppressing therise of parasitic gratings during the record-ing process. The first technique utilisesthe fact that the growth- and the erasure-rate differ in time for the parasitic grat-ings and the signal gratings, respectively.The second method tackles the problemby optimising the crystal itself by dopingwith damage-resistant (anti-photorefractive)dopants, which strongly prevent from ampli-fying scattered radiation. The third methodaims at minimising the effect of already ex-isting parasitic gratings during the recon-struction of the image.

6.1.1. Suppressing the buildup of HS

Rajbenbach et al. were the first to proposeand experimentally demonstrate suppressionof HS during optical amplification [143]. Byslowly rotating the recording medium, aBaTiO3 or a Bi12SiO20 crystal, they achievea low-noise signal but a large beam ampli-fication. The idea is that rotations pro-vide different local velocities for the differ-ent scattering sources in the crystal andhence blur the relative phase between themin the crystal volume. The signal beam to beamplified, however, records a moving signalgrating with constant modulus of the grat-ing vector. As the photorefractive responsetime for the signal grating is much smallerthan for parasitic gratings a rotation veloc-ity in between those two time scales givesthe possibility to suppress the buildup of HSand at the same time to amplify the signal.

Another technique which is applicable fornarrow bandpass photorefractive amplifiers,i.e., amplifiers where the gain has a sharpresonance around a certain coupling angle,is based on the fact that one main reason

for HS originates from oscillations betweenthe crystal faces [144]. To avoid the lat-ter and to maintain a high signal amplifica-tion, the entrance face of the crystal is tiltedwith respect to the pump beam while set-ting the coupling angle to its optimal value.In such a geometry the directions in whichparasitic recording would occur are dampedout, whereas the signal exhibits high gain.

A pulsed read-out scheme to suppress thegrowth of noise gratings is suggested in [145].The authors exploit the fact that HS doesnot occur below a certain intensity thre-shold, and that the temporal response of HSis slow. Thus initially they use an inten-sity below the threshold to record a grating,followed by a short pulse of the high inten-sity pump beam. Hence, the energy trans-fer from the signal to the pump beam takesplace efficiently and HS does not occur be-cause of its slow response time.

Rabinovich et al. demonstrated that beamfanning can be suppressed using an achro-matic grating arrangement [146, 147]. Theyuse a multicoloured beam which is diffractedfrom an auxiliary grating. The zeroth andfirst diffraction orders are used as pump andsignal beams which record a grating with thesame spacing for all colours. If the ratio ofintensities is chosen properly, beam fanningcan be suppressed whereas the signal of eachcolour can be amplified via two-wave mixingas the signal gratings have the same spac-ing and phase. A quite similar approach hasbeen chosen in [148, 149]. A standard two-wave mixing setup is combined with a thirdbeam of different wavelength. The presenceof the latter hinders the formation of HSwhile only slightly reducing the signal grat-ing, again employing the fact that the timeresponse of HS is slower than that of thetwo-wave mixing process. A slightly modi-fied version where the recording and the era-sure due to a third beam are performed peri-odically has been suggested and experimen-

24

tally verified by Zhang et al. [150]. The au-thors use the fact that the buildup time ofthe grating is much less than that of the HSresponse time and with the erasure time itis vice versa.

In electro-optic crystals high gain usuallycan be obtained for e polarised light, andtherefore beam fanning is strong, too. Us-ing a pump beam with an adjustable po-larisation direction enabled He and Yeh [84]to optimise the signal to noise ratio, as theordinary polarised component is a type ofincoherent erasure. The advantage of thistechnique is that the magnitude of the era-sure can be easily controlled and that theerasing beam is automatically aligned withthe pump beam. A further development ofthat technique for low pump-to-signal ratiosshows that it is advantageous having both,the signal and the pump beam, in mixed po-larisation states [151].

A new scheme for high-gain, noise-free amplification by two-beam coupling inKNbO3 crystals using the beam fanning ef-fect was suggested and demonstrated by Xuet al. [152]. The idea is as follows: The two-wave coupling coefficient depends amongother things on the angles of the beams withrespect to the c-axis of the crystal. A signalbeam traveling along the positive c-axis willbe amplified not only by the pump beam butalso by the major part, i.e. within a cer-tain angular range, of scattered light beamswhich are initiated by the pump beam.

Most of the above described techniquesare useful mainly if amplification is needed.A method to suppress HS and to record holo-grams has been successfully demonstratedby Sun et al. [153]. In addition to therecording-beams (He-Ne-laser) impinging onthe crystal asymmetrically with respect tothe c-axis they use the light of an Ar+-laserto reduce the HS. The efficiency of that tech-nique depends on the intensity of the latter.

A setup with an active feedback sys-

tem adjusting the phase between the light-interference and the refractive index pattern(self-stabilised setup) for hologram record-ing not only provides higher diffraction effi-ciencies but also leads to a strongly reducedlevel of light scattering in LiNbO3 [154]. Thereduction in HS may be due to the movingrecording pattern of light being unable to ef-ficiently record the parasitic holograms thatare responsible for scattering.

A further possibility to significantly re-duce HS is to raise the crystal tempera-ture [83]. As the net space-charge fieldat high temperatures is weakened by thecompensation of the photoinduced electronicfield with the thermally activated ions, theS/N ratio increases.

Only recently multidomain periodicallypoled samples of LiNbO3:Y:Fe were used forrecording a phase grating of high diffrac-tion efficiency. At the same time, the multi-domain structure inhibit the nonlinear re-sponse at low spatial frequencies, i.e. unde-sired HS is suppressed [10].

6.1.2. Optimising the crystal composition

Various authors made attempts to opti-mise the crystal composition itself, suchthat HS is minimised. Usually they use socalled damage-resistant dopants, i.e. anti-photorefractive additives.

The properties of LiNbO3 crystals dopedwith Mg, Zn, or In, respectively, were inves-tigated by Volk et al. [121]. The photore-fraction decreases due to an increase in thephotoconductivity, keeping the dark conduc-tivity constant. Therefore, those dopantsmake LiNbO3 damage-resistant [155]. Theincrease in the photoconductivity is directlyrelated to the reduction of the intrinsic de-fects of NbLi because of their substitutionby damage-resistant dopants. Moreover, theexistence of a threshold impurity concentra-tion of the dopants is reported which mani-

25

fests itself in abrupt changes of some physi-cal properties.

Zhang et al. theoretically proposed athreshold effect of the incident light inten-sity for HS in double doped LiNbO3-crystals(:Fe,Mg or Zn or In) [118]. The thresholdshould be strongly depending on the darkconductivity, the intensity of HS on the con-centration of the iron ion at the lithium site.On the basis of this model the competitionbetween beam fanning and signal amplifica-tion is studied [156]. As a consequence ofthat competition an optimum photovoltaicfield and pump intensity for signal ampli-fication are predicted. Experimental stud-ies of this effect were performed by Kam-ber et al. [119, 7, 120]. They found thatthe two-beam coupling gain has a maximumvalue at a certain incident light intensitywhereas the effect of HS is still negligible.The results show that doping with damage-resistant dopants is a good technique tosuppress HS while maintaining the desiredphotorefractive properties, e.g. gain (seeFig. 29). The S/N ratio and the resolution

(a) (b)

Figure 29: Image amplification (a) without and (b)with suppression of beam fanning [157].

of the holographic reconstructed patterns inthese kinds of double-doped crystals are veryhigh and the threshold light intensity abovewhich HS occurs increases. Moreover, dou-bly doped LiNbO3 (Fe,Mn) not only sup-presses HS but considerably increases thelifetime of stored holograms [158].

6.1.3. Post-suppression

Several techniques are aiming at minimisingthe influence of parasitic gratings during thereconstruction of holograms.

For photographic emulsions (silver halide)HS and the reconstruction of parasitic holo-grams was studied extensively [88, 123, 159,160]. It has been shown experimentally thatat the optimum replay wavelength the para-sitic gratings dominate and the characteris-tics of the planar gratings can be regainedonly by considerably changing the replaywavelength [160]. The importance of the rel-ative orientation of the polarisation of therecording and the reconstructing beams insilver halides was emphasised in Ref. [88].In general, the effect of already recordedparasitic gratings on the reconstruction ofthe hologram can be minimised by choosingthe replay parameters as different as possi-ble from the recording conditions. Changingthe wavelength [161], the polarisation [88]or the angle of incidence significantly dimin-ishes the influence of parasitic gratings. Adetailed study of the angular, spectral andpolarisation dependence for LiNbO3:Fe hasbeen performed by Fally et al. [37, 38].

6.2. Applications of HS

One of the most exciting applications ofbeam fanning is to initiate optical phase con-jugation. A lot of work has been done on thattopic, however, after giving the basic princi-ple and the role of HS, we will restrict our-selves to other applications. Material char-acterisation by employing a simple set-up isanother useful possibility for HS, as well asincoherent-to-coherent conversion or the usefor optical novelty filters or optical limiters.

The major advantage of beam fanning foruse in phase conjugation lies in the fact thatthe whole device is extremely simple, i.e.it merely exists of a photorefractive crys-tal. Degenerate four-wave mixing is gener-

26

ated by one incident beam, the second nec-essary pump beam rises as a consequenceof HS. Those beams are internally reflectedand retro-reflected near a corner of the crys-tal, fulfilling the conditions for phase con-jugation. In addition the whole device isself-aligning. In Fig. 30 images using sucha self-pumped phase conjugator and a mir-ror in the presence of a phase distorter areshown, respectively. Already in the 80ies

(a) (b)

Figure 30: Image using (a) a self-pumped conjugatorand (b) a mirror in the presence of a phase distorter,respectively [63].

possible applications of beam fanning, liketo construct a one-way optical device (opti-cal diode) or a bistable optical device withan erasable memory, were discussed in anexcellent review by Feinberg [9] and later inRef. [162].

If a single pump beam passes through aphotorefractive crystal exhibiting high gainHS will occur in certain directions. Thescattered light is distributed over a wideregion (cf. section 5.) depending on thesymmetry properties of relevant parameters(e.g. electro-optic coefficient). That scat-tered light trivially fulfills Bragg’s conditionbecause it acted as the signal beam duringrecording of the parasitic gratings. In the re-ciprocal space using the terms of the Ewaldconcept this means that the Ewald-sphere iscontinuously filled. In other words: there ex-ist a host of different grating vectors which

allow for efficient scattering. Up to now wewere dealing with non-flip scattering. Onthe other hand the possibility of flip scatter-ing to occur is highly restricted. This is thecase because Bragg’s condition is only ful-filled for a particular grating vector whichconnects the isotropic Ewald-sphere (o po-larisation state) with the anisotropic Ewald-ellipsoid (e polarisation state). The resultof that restriction is that a sharp scatter-ing ring appears whose polarisation is po-larised perpendicularly with respect to thepump beam. Employing this fact, Rupp andDrees established a simple method to mea-sure the birefringence from the cone angleof the flip scattering [21]. The techniquewas employed to determine the Li/Nb ratioof LiNbO3 crystals [163]. The compositionof LiNbO3 influences the e refractive-indexand thereby the birefringence. This methodturned out to be quite complementary tomeasuring the optical absorption edge anddetermining the Li/Nb ratio as for the lat-ter very pure crystals are necessary becausethe presence of impurities shifts the absorp-tion edge [164, 165]. Employing the methodof holographic flip scattering the photore-fractive effect requiring impurity centres isutilised.

In addition to the value of birefringenceits sign can be deduced from the polarisa-tion of the incident light [42]. Rupp et al.studied the temperature dependence of lightscattering phenomena, mainly the flip scat-tering and hence again the birefringence, inSBN:Ce up to the phase transition temper-ature [101].

The time dependent measurements of thetransmitted or scattered radiation offer asimple and advantageous way to estimatesome physical parameters related to holo-graphic experiments, e.g. the photoconduc-tivity [21].

For holographic storage media which haveto be processed chemically after recording

27

usually a shrinkage or swelling of the holo-gram is unavoidable. The swelling (thick-ness change) is not known but can be cal-culated from the reconstruction of parasiticgratings at different wavelengths and an-gles [166]. In general, if there is no change inthickness a single dip appears in the trans-mission curve at the same recording anglewhen reconstructing the parasitic hologramusing the same wavelength. Changes in thethickness show up in two dips as the record-ing and replay Ewald-spheres are not thesame. Reconstructing parasitic hologramsin general gives information on any changesbetween recording and replay, in particularon the photochemical processing like bleach-ing [167, 168]. Therefore, it is possible toanalyse and optimise processes which are im-portant in the production of holograms orhigh quality holographic optical elements.

Lee et al. utilised the effect of HS toperform colour conversion or incoherent-to-coherent conversion [169]. The basic idea isquite simple: At first they generate as in-tense as possible parasitic gratings in thephotorefractive crystal (BaTiO3) using a co-herent (red) laser beam. Next another(green) laser beam carrying an input patternimpinges on the crystal, thus spatially selec-tive erasing the parasitic gratings. In theregions where the erasure took place the redlaser beam is transmitted without fanning,and hence projects the image in red, i.e. thecolour has been converted from green (in-put) to red (output). This set-up behaves asa transmission spatial light modulator, withthe output patterns being positive replicasof the input patterns. An amended methodof incoherent-to-coherent conversion with re-spect to the resolution was introduced byZhang et al. [170, 171]. The authors illumi-nate a photorefractive crystal with the co-herent light of a laser beam which results inthe generation of parasitic holograms, henceconsiderably decreasing the transmitted in-

tensity. An incoherent image bearing beamspatially modulates the transmission of thecoherent beam as it erases parts of the par-asitic holograms. This results in a coherentpositive replica of the incoherent image witha resolution of up to 90 line pairs/mm. Thelatter depends on the nature of the fanningprocess and the quality of the imaging sys-tem.

While the applications described up tonow pay attention to amplification, the factthat the pump beam is strongly de-amplifiedcan be quite interesting, too. Cronin-Golomb and Yariv describe the use of HSfor the design of optical limiters [172]. Alaser beam passing through a photorefrac-tive crystal exhibiting HS loses much of itspower into a broad fan of light. Thus thecrystal can be thought as an optical limiterwhich dumps excess power away from the in-cident direction into the fanning directions.Some of the features associated with fanninglimiters are as follows:

• The damage thresholds for the crystalsby overheating are quite high becausethe decrease of radiation due to fanningis mostly deflected, not absorbed.

• Choosing an appropriate crystal theymay be effective over the entire visiblespectrum.

• The crystals can be refreshed as often asneeded by erasing the fanning gratingswith bright, even incoherent, light.

• The effect is highly angular and wave-length dependent (violation of Bragg’slaw)

A quite interesting application using a beamfanning optical limiter was demonstrated byBialkowski [173]. Here the effect is used todetect low photothermal (IR) spectroscopysignals. The sample (a gas) is excited by

28

the IR-beam which is in line with a visi-ble probe beam passing through the opticallimiter. The portion of visible light passingthrough the latter is indicative for refractive-index changes of the sample due to the IR-radiation. Thus the limiter is used as anadaptive spatial filter, reminding of noveltyfilters. An optical novelty filter is display-ing the changing parts of an image whilesuppressing the unchanged parts. Thus, itcan be regarded as a time-differentiator fortwo-dimensional images. It is extremely use-ful for data reduction and it can dramati-cally reduce the image processing time ow-ing to its ability to perform parallel opticalprocessing in comparison to electronic meth-ods. The application of novelty filters fortransient phase detection has been realisedin different ways [174]. The simplest typeof a photorefractive novelty filter is basedon beam fanning, has been suggested byCronin-Golomb et al. [175], and was realisedexperimentally by Ford et al. [107] for thefirst time. Thereby an input signal beam isde-amplified by HS. As the input varies, theparasitic hologram written in the crystal ad-justs to the change according to the photore-fractive response time. Thus the changingcomponents of the image become brighteras the de-amplification is disturbed, in otherwords: the violation of Bragg’s law leadsto a higher transmitted intensity. Becauseof the latter the system is sensitive to anychange of the input field, including its am-plitude, phase, polarisation, or wavelength.An amended method of a beam fanning nov-elty filter makes use of an additional externalphase shift in one part of the signal wave,thus enhancing the phase resolution [176].As a strong photorefractive effect is requiredto obtain sufficient HS for the novelty fil-tering an optimisation study on that topichas been carried out for BaTiO3 only re-cently [177].

A similar idea is used for monitoring the

vibration of a rough object [178]. Thetechnique utilises the fact that the trans-mitted intensity through the photorefractiveBi12SiO20 crystal exhibiting HS is a param-eter to detect time variations, in particularthe variations of laser speckles. and hencevibrations of a rough object. The set-up issimple and turns out to be as sensitive asinterferometric methods. It effectively usesmost of the light power scattered by the ob-ject and provides a linear response on thedisplacement amplitude.

An important application of HS is to im-prove the characteristics of thin photorefrac-tive crystals (waveguides, fibers). Kamshilinet al. studied theoretically and experimen-tally the propagation of light beams in a thinphotorefractive crystal with diffusion mech-anism [128, 179, 180]. The strong beam fan-ning renders possible that the light energyis self-channeled near the surface. Applyingan ac electric field in addition results in ashorter photorefractive response time whichis highly interesting for, e.g., fast phase con-jugators.

Thermal fixing of holographic gratings isan important task for possible applicationsas data storage media (LiNbO3) [181, 182].The hologram is recorded at ambient tem-peratures leading to a spatial distributionof electrons and via the electro-optic effectto a space charge field. When increasingthe temperature, ions become mobile, com-pensate for the electron distribution, andhence the space charge field vanishes. Re-turning to room temperature and illuminat-ing the sample with incoherent light redis-tributes the electrons, thus leading to anionic grating which gives rise to a very sta-ble hologram. A crucial parameter for thistechnologically relevant process is the acti-vation energy for thermal fixing. Ellabban etal. proposed a simple method to determinethe activation energy for thermal fixing pro-cesses in photorefractive materials making

29

use of HS [124]. After recording a parasitichologram at room temperature, the time de-pendence of the transmitted intensity at thefixing temperature is measured, thereby de-termining the compensation time constant.The temperature dependence of the compen-sation time constants permits the evaluationof the thermal activation energy. A compari-son with the standard two-wave mixing tech-nique shows that the values of thermal acti-vation energy agree excellently, avoiding anycomplications in the set-up (vibration insen-sitive) or problems with too high refractive-index modulations.

In a recent idea an all-optical computingdevice to perform incoherent digital opera-tions including the NAND, NOR, and XORgates was proposed [183]. The advantage ofthe proposed set up is that it is able to workwith mutually incoherent inputs as well asmutually coherent inputs. It makes use ofself-generating parasitic gratings and theirerasure by another beam in two photorefrac-tive self-pumped phase conjugators. In theself-pumped phase conjugator’s output con-trolling process, a train of e polarised wavesis usually used to generate the phase conju-gate waves, and a train of o polarised wavesis used to suppress the self-pumped phaseconjugation of the e polarised wave. There-fore, turning on or off the interfering o po-larised wave stops or revives the phase con-jugate e polarised wave, i.e. a self-pumpedphase conjugate optical switch is obtained.

7. SUMMARY

In this review definitions for basic notionsand concepts of HS are given and a stan-dardised nomenclature is proposed. The ori-gin of HS and its appearance in a varietyof materials have been discussed with par-ticular regard to those media with highestrefractive-index change. Some recent resultsobtained for SBN, LiNbO3, and NaNP are

summarised. As HS is a serious nuisancein many applications where photorefractivematerials play a role, an overview on the sta-tus of the proposals is given by which HS canbe sufficiently suppressed. To account forthe increasing popularity of methods in ma-terials research based on HS, a summary ofthe possibilities and achievements is given.

Acknowledgement

The authors are grateful to E. Tillmanns forcontinuously supporting the group. M. A.Ellabban acknowledges a scholarship of theOAD. This review has been supported bymeans of the project WTZ A2-1998 of theAustrian academic exchange service.

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