Insight into a new geometrical approach to beam fanning in BaTiO3

64 J. Opt. Soc. Am. B/Vol. 18, No. 1 /January 2001 Mailhan et al.

Insight into a new geometrical approachto beam fanning in BaTiO3

C. Mailhan

Equipe de Recherche en Photonique et Optoelectronique, Supelec, 2 rue Edouard Belin, 57070 Metz, France, andLaboratoire Materiaux Optiques a Proprietes Specifiques, Centre Lorrain d’Optique et d’Electronique des

Solides, Universite de Metz et Supelec, 2 rue Edouard Belin, 57070 Metz, France

M. Goetz

Equipe de Recherche en Photonique et Optoelectronique, Supelec, 2 rue Edouard Belin, 57070 Metz, France, andRegion Champagne Ardenne, Service Enseignement Superieur et Recherche, 5 rue de Jericho,

57037 Chalons en Champagne, France

N. Fressengeas

Equipe de Recherche en Photonique et Optoelectronique, Supelec, 2 rue Edouard Belin, 57070 Metz, France

G. Kugel

Equipe de Recherche en Photonique et Optoelectronique, Supelec, 2 rue Edouard Belin, 57070 Metz, France, andLaboratoire Materiaux Optiques a Proprietes Specifiques, Centre Lorrain d’Optique et d’Electronique des

Solides, Universite de Metz et Supelec, 2 rue Edouard Belin, 57070 Metz, France

Received March 9, 2000

The fanning behavior of the two interacting laser beams in a double phase-conjugate mirror (DPCM) obviouslyinfluences the characteristics of the mirror. We present a two-dimensional experimental study and model ofthe beam-fanning patterns obtained in nominally undoped and cobalt-doped (20 parts in 106) BaTiO3 samples.We propose a numerical method based on beam coupling theory that allows us to determine, for each crystal,which directions will develop significant fanning. We thus interpret the beam-fanning patterns by focusing onthe influence of the incident beam’s width and incidence angle. The time evolution of the intensity in each ofthese directions is also studied and modeled. This model has been applied to DPCM to enable both incidentbeams to generate fanning in a common direction, with good results in terms of response time, efficiency, andstability. © 2001 Optical Society of America

OCIS codes: 190.5330, 190.7070.

1. INTRODUCTIONBeam fanning (BF) is known to occur in various photore-fractive materials when they are illuminated by a laserbeam. This phenomenon consists of an asymmetric defo-calization of light1 and plays a basic role in self-pumpedphase-conjugate2 and double phase-conjugate mirrors(DPCM’s),3,4 particularly in BaTiO3 crystals. The pro-cess of BF has been a subject of interest for approximately20 years.

Banerjee and Misra5 interpreted BF as a deterministicphenomenon (deterministic BF) by considering an asym-metric refractive-index profile created by symmetric illu-mination, which in fact consists of a light-induced wave-guide. Iteration of the process leads to progressivedistortion of the beam. An alternative interpretation(Ref. 6 and references therein) is based on the fact thatBF results from amplification by two-wave mixing ofwaves scattered by inhomogeneities at the entrance faceof or inside the crystal, (denoted random BF by Banerjeeand Misra5). Several models to describe beam fanninghave been developed; most of them are bidimensional.7–12

0740-3224/2001/010064-11$15.00 ©

Explanations of the whole fanning pattern, including thediffraction ring and lobes, have been presented.13,14

Even if most papers have dealt with the steady-state ef-fect, the time dependence of fanning has also beeninvestigated.6,15 One study of BF, performed to improvethe efficiency of a DPCM in a fiberlike Bi12TiO20 sample,was made by Kamshilin et al.16 The approach of Kam-shilin et al. is in many aspects similar to ours. Indeed, inthe present paper we intend, as Kamshilin did, to find theparticular direction in which fanning intensity reaches itsmaximum so we can optimize the overlap of the two coun-terpropagating fanning lobes used in the (DPCM).4 Ourexperiment uses bulk samples of BaTiO3 instead of fiber-like Bi12TiO20. Moreover, we are interested in the tem-poral evolution of the fanning as well as in the influenceof the width of the beam; the former was not consideredby Kamshilin et al.

Therefore we propose here a new experimental and the-oretical approach to energy distribution of fanning inBaTiO3. Inasmuch as the basic motivation for this re-search is to use our results to improve the characteristics

2001 Optical Society of America

Mailhan et al. Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. B 65

of a DPCM in terms of stability, efficiency, and responsetime, we consider two beams entering the two oppositecrystal faces orthogonally to the ferroelectric c axis. (Forconvenience we call the faces that correspond to beamstraveling approximately in the direction of the ferroelec-tric axis and in the opposite direction the 1c and 2c crys-tal faces, respectively.) Experimental conditions havebeen adapted to reduce the problem to two dimensions (ina plane including the ferroelectric axis) and to eradicateconical diffraction17 in the DPCM.

In what follows, first we present the experimentalsetup with which to record typical BF patterns obtainedfor each sample, depending on geometrical conditions andbeam width. To interpret easily the preliminary resultsobtained, we propose an extension to the classic beam-coupling theory, aiming at determining numerically thedirection in which fanning develops with the highest in-tensity as a function of the incident angle. We do this byconsidering the characteristics of the two crystals and forthe 1c or 2c face illumination. The influence of thebeam waist is studied. Being aware of the complexity ofthe physical system considered, we decided to adopt aphenomenological approach, first explaining the most in-tense fanning directions observed experimentally. Pro-gressive improvements are then added to the model, per-mitting the interpretation of the whole fanning patternand of its time dependence.

The method that we have developed enables us to pre-dict the directions in which significant fanning is devel-oped by iterating a simple graphic process: Whereasother methods are focused on a thorough description ofthe fanning pattern, we focus only on those particularangle values. This method is dictated by our ultimategoal, namely, improving the DPCM significantly.

2. EXPERIMENTA. Experimental SetupOur study was conducted with two different samplesof BaTiO3. The first one (called an undoped crystal)is nominally undoped BaTiO3 (4.38 mm 3 3.02 mm3 4.39 mmic) that was grown at the Chinese Academy ofSciences in Beijing. The second sample (called a dopedcrystal) is made from BaTiO3 doped with 20 parts in 106

of Co (5 mm 3 2.6 mm 3 5 mmic) and was grown atthe Forschungsinstitut fur Mineralische und MetallischeWerkstoffe Edelsteine/Edelmetalle GmBH, Idar Ober-stein, Germany. The experimental setup, shown in Fig.1, consists in fact of a DPCM configuration. We use anAr1 ion laser (l 5 514 nm) with a maximum outputpower of 75 mW and a 1/e2 Gaussian beam waist of 0.5mm. The beam goes through an optical isolator to pre-vent the laser from being perturbed by reflected waves.The light entering the crystal is extraordinarily polarizedby a half-wave plate. The original beam is split into twoparts, and a shutter is placed in each one to selectwhether the incoming beam will enter the 1c or the 2cface. Each beam is focused in a horizontal plane at theentrance of the crystal by means of a cylindrical lens toprevent conical diffraction.18 The beam waist in theother direction at the entrance of the crystal is controlledby means of a second cylindrical lens whose axis is per-

pendicular to that of the first lens. Three different 1/e2

waist values (w0 5 48 mm, w0 5 24 mm, and w05 12 mm) were tested. The sample lies upon automatedrotation and translation stages to permit accurate controlof the lateral position and the angle of incidence of thebeam (with a precision of 1 mm and 1/1000°, respectively).For determination of the BF pattern of both the 1c andthe 2c entrance faces, each optical arm of the bench isused independently. For each case we measure the in-tensity of light transmitted in every direction of the fan-ning pattern by moving a photodiode along a rail. Theparameters used in the experimental study as well as inthe theoretical investigation are the following:

u i is the angle inside the crystal between the directionof the ferroelectric axis and the linearly transmittedbeam, which we call the incident beam in what follows;

u f is the fanning angle, i.e., the angle inside the crystalbetween the direction of the ferroelectric axis and the di-rection of maximum fanned intensity that is due to pho-torefractive fanning;

w0 is the waist of the incident beam in the horizontalplane outside the crystal (see Fig. 2);

l' is the width of the beam inside the crystal that cor-responds to the 1/e2 waist w0 outside the crystal andwhose value we modify by rotating the crystal (Fig. 2):

l'~u i! 52w0 cos~u i!

cos$arcsin@n sin~u i!#%. (1)

We must point out that the beams that we considerhere must be focused at the entrance of the crystal so the

Fig. 1. Experimental setup used for study of beam fanning. (a)Description of the experimental apparatus used with either (b)shutter 1 or (c) shutter 2 opened, for propagation of light in thedirection of the c axis or in the opposite direction.


region in which they can be assimilated into plane waves(the Rayleigh region) is longer than the crystal. With theparameters in our experiment, the smallest possiblelength of the Rayleigh region is 4.8 mm for a crystal withan index of n 5 2.4. Therefore we can neglect the dif-fraction of the beams inside the crystal and consider the1/e2 width nearly constant.

B. Preliminary Experiments and Description of TypicalFanning PatternsA typical fanning pattern for a beam entering the 1c faceof a crystal is represented in Fig. 3 for various incidentlaser powers. The pattern consists of a transmitted (re-fracted but not fanned) spot in the incident direction (u i)and of fanned light whose absolute maximum intensityappears in the direction u f . The incident beam consistsindeed of one main beam and additional less-intense com-ponents that we call satellites of the main beam. Thesesatellites result from partial reflection and transmissionof the beam through the parallel faces of the beam split-ters in the experimental setup. The main beam and itssatellites are consequently parallel before they are fo-cused by the cylindrical lenses. They enter the crystal atslightly different incident angles. Many secondary localmaxima appear (Fig. 3); they are detailed in Subsection2.B below. We shall explain in particular why we choseonly some of them as local maxima associated with themain beam and thus labeled u f

(1,2,3...) in Fig. 3.If the beam enters the 1c face (2c face), then u i < u f

(u f < u i). In both cases the fanning pattern containsmany peaks of intensity. Shifting the crystal or chang-ing the intensity of the incident beam (from 20 to 300 mWin the undoped crystal) does not affect the general shapeof the pattern, even if that effect is not shown here.

We studied the effect of the increase in u i on the inten-sity distribution by rotating the crystal about its verticalaxis. We observed that the ratio of global fanned inten-sity to linearly transmitted intensity increased with u i ,as did the ratio of the maximum intensity values. Thiswas true for both entrance faces considered.

The qualitative behavior was the same for the othersample. We can deduce from our results (in both crystalsand for the 1c and 2c faces) that the fanned light splitsinto different directions and tends to go in the 1c direc-tion; as Xie et al.10 theoretically demonstrated, angle u fstrongly depends on u i and on the beam width.

Fig. 2. Evolution of the width of the beam inside the crystal asa function of incident angle.

C. Fanning-Angle Variation versus Incident Angle andInfluence of the Width of the Incident Beam forTwo SamplesAs we already mentioned, we recorded several fanningpatterns with different values of beam waists in our twosamples. For a beam in the 1c direction, widening thebeam significantly increased the fanning effect, thus di-minishing the value of fanning angle u f for a given valueof u i (see Fig. 4).

The typical evolution of u f as a function of u i is illus-trated in Fig. 5 for the undoped crystal with the 1c en-trance face and for three values of the waist. It clearlyappears that the width of the beam has no influence onthe curve for small incident angles. A limit on the inci-dent angle seems to exist for each value of the waist abovewhich the fanning behavior is clearly modified. Equation(1) enables us to evaluate the width l' of the beam insidethe crystal that corresponds to each critical angle. Belowthe critical point, the fanning pattern does not depend onthe width of the beam. That is why the next experimentswere conducted with narrow beams; the distinction be-tween narrow and broad beams is explained in Section 5below.

Figure 6 shows the experimental (filled circles) depen-dence of fanning angle u f as a function of incidence angle

Fig. 3. Fanning distribution in an undoped BaTiO3 crystal foran incident angle u i 5 23° on entrance face 1c. Fanning for200- and 100-mW intensities are shown. The curve referred toas ‘‘2*100 mW’’ represents the pattern obtained with 100 mWmagnified by a factor of 2. The angles are taken inside the crys-tal.

Fig. 4. Fanning patterns obtained in a doped BaTiO3 crystalwith u i 5 18.6° and 100-mW incident intensity, depending on thebeam width.


u i for two different crystals and the 1c and 2c entrancefaces. The behavior of the two faces is not symmetrical.Furthermore, the doped sample, that is to say, the crystalwith the higher effective number of empty traps NA , isresponsible for the stronger deviation in light that resultsfrom the fanning phenomenon. The solid and dashedcurves in Fig. 6 correspond to the theoretical model of theprocess developed in the case of the narrow-beam ap-proximation and illustrated in Subsection 2.A.

We also performed an experiment to determine thetime dependence of the fanning pattern. The correspond-ing results are presented and discussed in Section 5 be-low.

3. THEORETICAL BACKGROUNDThe interpretation that we propose for the fanning pat-terns observed is based on a deliberately simplified two-wave mixing approach and study of the optimization ofthe beam-coupling gain.

Fig. 5. Dependence of fanning angle on incident angle for sev-eral beam waists.

Fig. 6. Comparison of measured (filled circles) and theoreticallydetermined (curves) fanning angles as a function of incidentangle for Co-doped and undoped BaTiO3 samples. Upper andlower dashed curves, theoretical maxima of gain for NA 5 0.43 1016 cm23; solid curves, NA 5 0.15 3 1016 cm23. Dark filledcircles, experimental points of the doped crystal; pale filledcircles, experimental points of the undoped crystal. Curves andfilled circles located over the first bisecting line correspond to 2corientation; others, to 1c orientation.

Consider two coherent waves with wave vectors k1 andk2 interfering in a photorefractive crystal. The buildingof an index grating of wave vector K 5 k2 2 k1 leads,through wave mixing19 and under the assumption ofslowly varying envelopes, to a transmitted signal beamintensity that can be expressed as

Is~z !

Is~0 !5

Is~0 ! 1 Ip~0 !

Ip~0 ! 1 Is~0 !exp~g z !exp@~g 2 a!z#. (2)

In the undepleted-pump approximation for two beams(pump and signal) that have been interacting at length z,Eq. (2) gives

Is~z ! 5 Is~0 !exp@~g 2 a!z#, (3)

where Is and Ip are the signal and the pump beams, re-spectively, z is the interaction length, and a and g are theabsorption coefficient and the coupling gain.

In the case studied here, we assume that the two inter-fering beams originate from diffusion in the inhomogene-ities of the crystal, where the diffused signal is progres-sively amplified by the two-wave mixing process. Suchan assumption can be made for BaTiO3 crystals. Fur-thermore, in what follows we have to remember that r42 isthe highest electro-optic coefficient in BaTiO3. Accordingto the research reported in Refs. 3 and 20, the beam-coupling gain g for direction u f amplified by pump direc-tion u i for both extraordinarily polarized waves can be ex-pressed as

g~u i , u f! 5~2p!2n3kBT

cos~u i 2 u f /2!el

ne

nor42 cosS u i 1 u f

2 D3 sin~u i 1 u f!

L

L2 1 LD2 , (4)

where n is the effective refractive index, kB is the Boltz-mann constant, T is the temperature, e is the electricalcharge of the electron, l is the wavelength, ne is the ex-traordinary index, and no is the ordinary index. L andLD are, respectively, the grating spacing and the Debyelength. They can be written as follows:

L 5l

2 3 sin~u i 2 u f /2!, (5)

LD 52p@«a«0 cos2~u i 1 u f /2! 1 «c«0 sin2~u i 1 u f /2!#1/2

eA NA

kBT

,

(6)where «0 is the vacuum dielectric permittivity, «a and «care the eigenvalues of the dielectric tensor related to the aand c axes, respectively, and NA is the total number ofempty traps. For doped and undoped BaTiO3 crystalsand l 5 514 nm, «a . 4200 and «c . 150.

The values of the refractive indices and the experimen-tal angles that can be reached inside the crystal implythat 0° < u f 1 u i/2 < 27° and consequently that cos2(ui1 uf /2) @ sin2(ui 1 uf /2). Therefore

LD .2pA«a«0 cos~u i 1 u f /2!

e~NA /kBT !1/2 . (7)

The beam-coupling gain can be then expressed by


One can see that this gain depends on experimental pa-rameters (u i , l, T) and on crystal parameters(n, ne , no , «a , NA) as well as on u f , which is considered avariable. This means that for a given crystal, and if land T are fixed by the experimental conditions, g can beconsidered a function of only u i and u f .

4. MODEL FOR THEORETICALDETERMINATION OF THE FANNINGANGLESA. Graphic Determination of the Fanning Angle andInfluence of the Parameters NA and TIn our theoretical model we assume that fanning origi-nates from two-wave mixing between an incident pumpbeam and a signal that results from diffusion in the inho-mogeneities of the crystal. Therefore the amplified direc-tion of the diffused beam should correspond to the maxi-mization of the beam-coupling gain times the effectiveinteraction length of the incident and fanning directions.This means that, for any given u i , two-wave mixingshould amplify mostly the direction u f that maximizesthis product. We shall, however, first consider that theinteraction length is nearly constant. This assumptionenables us to determine the possible fanning directions,as we shall show below. The value of the effective inter-action length obviously depends on the incidence angleand is taken into account in a second step in which weevaluate the intensity distribution in these directions.Moreover, we show in what follows that the main fanningdirection is amplified not only by the incident beam butalso by other, less-intense fanning directions, which ex-plains that deriving the product gleff would not be a goodsolution here. As we shall see, this iterative methodgives results that are in good agreement with the experi-ment.

Fanning angle u f associated with u i can thus be foundby solution of the equation

]g~u i , u f!

]u f5 0. (9)

This solution cannot be found easily in an analytical way,but mathematical software tools permit us to solve theproblem numerically by implicitly plotting u f as a func-tion of u i . We performed such a calculation, using pa-rameters with the values defined in Section 2. For agiven doping rate (NA given), the solution consists of twocurves: The lower one (upper one) corresponds to an in-cident beam entering the 1c(2c) face, as is shown to-gether with experimental points, in Fig. 6 for a tempera-ture T of 300 K and for two samples that exhibit differenteffective numbers of empty traps NA . As the determina-tion of u f is obtained by derivation of g, it is obvious that

the only two parameters that are to be considered hereare temperature and the effective number of empty traps.

The direct influence of temperature on the value of u ffor a given u i has been shown to be negligible comparedwith the influence of the doping rate. Nevertheless, tem-perature obviously has a huge indirect influence on thevalue of coupling gain g through the gain’s being propor-tional to r42 . Indeed, this electro-optic coefficient isknown to be strongly temperature dependent for BaTiO3(Ref. 19) near the ambient temperature owing to the vi-cinity of the structural phase transition. In our determi-nation of the fanning angle, we took T 5 300 K andr42 5 1640 pm/V.

We experimentally observed that the higher NA is, themore the fanned beam is attracted by the c axis, as weshow in our model, in which NA has been considered anadjustable parameter. The result of the numerical fit-ting between the theoretical curves and the experimentaldata from in Fig. 6 yields NA 5 0.15 3 1016 cm23 for theundoped sample and NA 5 0.4 3 1016 cm23 for the dopedsample for «a 5 4200. In both cases, excellent agree-ment between theory and experiment is obtained in thewhole incident angle range tested (u i < 24.5°). More-over, if the value of NA is taken to fit the experimentalpoints with the theoretical curve for the 1c entrance face,the experimental points obtained with the 2c entranceface are on the upper theoretical curve.

Notice that the process used here yields a simple wayto determine the value of NA in a photorefractive mate-rial, provided that «a and T are known with sufficient pre-cision. We intend to carry out further, independent, two-wave mixing experiments to verify the values of NA .

B. Determination of the Successive Fanning DirectionsWe have determined the dependence on incident angle ofthe maximum intensity fanning angle. In this Sectionand to interpret with more detail the typical beam-fanning pattern observed (Fig. 3), we now focus on the ex-istence of various peaks inside the fanning pattern itself.

The main idea is that the former process (that is, am-plification of a diffused beam through two-wave mixing)can be iterated: We assume that the incident beam is re-sponsible for significant amplification in a direction u f

(1),which itself can be considered a pump for two-wave mix-ing with a new fanning direction, u f

(2). The values ofu f

(1) and u f(2) can be iteratively found by use of the im-

plicit plot of Fig. 6. We can generalize this process bynaming the nth fanning direction u f

(n), with n increasingas we move away from the incident angle. The nth fan-ning direction acts as a pump, following the formerly ex-posed model, and amplifies mostly the diffused intensityin what we call the n 1 1st direction.

The process explained above is shown in Fig. 7, wherethe various angles u f

(n) that correspond to maxima ob-

g~u i , u f! 5~2p!2n3kBT

e

ne

nor42

cos~u i 1 u f /2!sin~u i 1 u f!

sin~u i 2 u f!H F l

2 sin~u i 1 u f /2!G2

1 F2pAkBT~«a«0 /NA!1/2 cos~u i 1 u f /2!

qG 2J . (8)


served in the fanning pattern of the undoped sample (Fig.3) are clearly identified: The incident u i 5 23° is respon-sible for amplification in the u f

(1) 5 20.7° direction, whichamplifies u f

(2) 5 18.4° and leads to u f(3) 5 16.2°.

Our experimental results are that u f(1) 5 21.1°, u f

5 u f(2) 5 18.8°, and u f

(3) 5 17°, in good agreement withour predictions (Fig. 7, inset).

A careful observation of the experimental BF patternindicates the presence of two additional effects:

• Together with the main maxima that are due to theiterative process already explained, other peaks appear.In the fanning pattern, the secondary maxima observedcan be interpreted within our model as successive fanningbeams that are due to the satellites of the incident beam(Fig. 7). Some of them can be quite intense, dependingon the value of the corresponding amplification coeffi-cient. The photodiode that we used for our experimentsis 1 cm wide; therefore the measurements for the small(high) values of u i in the case of the 1c (2c) entranceface are averaged, and some peaks thus cannot appear inthe fanning pattern (Fig. 3).

• As can be particularly well observed for the 48-mmbeam waist in the undoped sample (Fig. 5), for incidentangles greater than ;18.6° the experimental data becomeincompatible with the model used here. Indeed, themain BF angle u f is observed at 15.6°, whereas the theorypredicts that u f

(1) 5 17.7° and u f(2) 5 15.5°; the last

value is rather similar to the measured value. The sameobservation can be made for the other incident angles.This deviation from the model, which always occurs forbeams that can be qualified as broad, are in fact due to athreshold of the l' value and consequently to its influenceon the effective interaction length between the two mixedbeams. So far this result has not been explained by ourtheoretical model (considering that the effective interac-tion length is constant) and is studied in Section 5.

Fig. 7. Theoretical determination of the successive fanningangles in the undoped crystal for u i 5 23°. Long-dashed line,the first bisecting line; solid curves, the couples (u i , u f) that cor-respond to ]g/]u f 5 0 for 1c and 2c propagation. Short-dashed lines, successive fanning angles that can be determinedfor a 1 c propagation. Inset, corresponding BF pattern experi-mentally observed.

5. INTENSITY DISTRIBUTION OF THEFANNED LIGHT AND ITS TIMEDEPENDENCEA. Modeling of Amplification in a Fanning Directionand Influence of the Beam WidthAs we mentioned above, many directions can develop sig-nificant amplification as a result of the iteration describedabove, apparently depending on the width of the incidentbeam. In this section we determine from our modelwhich direction is most important in terms of intensity bytaking into account the function beam width l' . As theformer model permitted the determination of the fanningangles with little error, we chose here to consider thevariation of the effective interaction length that affectsnot the values of the successive fanning angles but thedistribution of intensity among these directions. Indeed,Eq. (3) shows that amplification of a fanning direction de-pends on gleff . To take this dependence into account, weconsider the amplification for each couple of interactingdirections. We make a distinction between the interac-tion of two narrow and two broad beams by estimatingthe effective interaction length leff between the incidentand the fanned beams.

From now on we shall identify as narrow beams a pairof incident and fanned beams that do not overlap on theoutput face (see Fig. 8). As shown in Fig. 8, the interac-tion length leff

NB between these narrow beams is not lim-ited by the crystal length and can be arbitrarily chosenequal to the length of the line that bisects the angleformed by the direction of the incident beam and that ofthe fanned beam:

leffNB 5

l'

cos2~u i 1 u f /2!~tan u i 2 tan u f!. (10)

For broad beams, the effective interaction length leffBB is

the same as before but is limited by the length of the crys-tal. Therefore it depends only on the angles and thelength of the crystal, l0 , in the direction of the c axis:

leffBB 5

l0

cos~u i 1 u f /2!. (11)

We have already shown that the actual behavior of u fas a function of u i follows the theoretical curve for narrow

Fig. 8. Definition of the effective interaction length leff of twobeams inside the crystal, showing the difference in the influenceon leff of narrow and broad beams.


beams (see Fig. 6). Thresholds experimentally observedin Fig. 5 are related to the widening of the beam.

A threshold appears experimentally when, for a givenwaist w0 , u i is increased continuously. For any given u i ,however, a threshold waist can be theoretically calculatedbecause it corresponds to the limit between broad andnarrow beams; its value is found by solution of

leffBB 5 leff

NB, (12)

which is equivalent to

l'

cos2~u i 1 u f /2!~tan u i 2 tan u f!5

l0

cos~u i 1 u f /2!,

(13)

where l' is given by Eq. (1). The experimental results,shown in Fig. 5 for a waist w0 5 24 mm (w0 5 48 mm) ex-hibit a threshold for an incident angle of u i 5 24° (u i5 19°). The theoretical results given by Eq. (12) for u i5 24° (u i 5 19°) are w0 5 21 mm (w0 5 53 mm). Theresults obtained are in good agreement with the experi-mental data.

Moreover, for broad beams the effective interactionlength of two beams increases with the values of theangles, whereas, for narrow beams, the higher the fan-ning angle is, the smaller leff is. It is possible here to un-derstand qualitatively why high-order fanning directionscan be the most amplified ones when the incident beam(and subsequently the fanned beam) is broad.

Trying to understand our experimental data with theseproperties, we have become aware that the intensity dif-fracted in a direction u f

(n) is due not only to the contribu-tion of the incident intensity but also to the action of in-termediate fanning beams. We consider that theundepleted-pump approximation is still true for amplifi-cation by any of the successive fanning directions. UsingEq. (3), and assuming that each of the beams taken as apump amplifies the intensity diffused in the fanning di-rection considered, we can write an approximate expres-sion for the intensity in the nth fanning direction:

If @u f~n !# 5 I0@u f

~n !#)i51

n21

exp $g@u f~i !, u f

~n !#leff @u f~i !, u f

~n !#%

3 exp $g@u i , u f~n !#leff @u i , u f

~n !#%. (14)

This expression of the intensity of light fanned in the u f(n)

direction includes the contributions to amplification of allprevious fanning directions. In our experiment, we usedBaTiO3 samples; we consider therefore that absorption(near 1 cm21 for the undoped and 2 cm21 for the dopedsamples if l 5 514 nm, according to Garrett et al.21) canbe neglected compared with amplification. If @u f

(n)# de-pends on the quantity of light initially (z 5 0) diffused bythe material I0 . Assuming that the diffused intensity isgreater in directions that are near the incident direction,that is, that I0@u f

(n)# < I0@u f(n21)# < I0(u i), Eq. (14) does

describe the influence of the width of the beam on thesteady-state fanning pattern obtained with the undopedcrystal (in which the high-order fanning angles can carrya large amount of intensity). However, it does not ac-count for the fanning pattern in the doped crystal forbroad beams. In that case, even when one is considering

a homogeneous distribution of intensity $i.e., I0@u f(n)#

5 I0@u f(n21)# 5 I0(u i), which is the limit case of the in-

equality and should favor high-order fanning angles%, themodel predicts [according to Eq. (14)] that the first fan-ning direction will always carry the maximum intensity.

B. Temporal Evolution of Light Intensity in the MainFanning DirectionsTo improve our understanding of the phenomenon dis-cussed above, we decided to investigate experimentallyand interpret the temporal evolution of a fanning pattern.Therefore we recorded the temporal evolution of the in-tensity until steady state in the main directions of fan-ning. The results of measurements performed on the twosamples are rather similar. Those obtained in the un-doped sample are represented in Fig. 9 for three values ofincident power and an incident angle of 23°. In the re-sults shown and to determine the successive transfers ofintensity from one direction to the next one, we were in-terested in broad beams that allowed at least the first di-rection @u f

(1) 5 20.7°# nearly to disappear at steady state.We can observe that the intensity in the direction u f

(1)

does appear first, but it decreases as soon as the next di-rection @u f

(2) 5 18.4°# grows, which then gives way to thenext direction. The process continues until one direction

Fig. 9. Temporal evolution of the energy distribution in the fan-ning pattern for various incident intensities recorded in an un-doped BaTiO3 sample. The beam width is the same, and the in-cident angle is 23°.


gets to a stable state. The time needed for stabilizing thefanning pattern depends on the power of the incidentbeam.

Because Eq. (14) is a valid model only for the steadystate, to compare experimental observation and theory wemust bring in the grating temporal evolution for eachcouple of interacting beams. According to Yeh,22 gaing(u i , u f , t) reaches the steady-state value g(u i , u f) givenin Eq. (8), with the following time evolution:g(u i , u f , t) 5 g(u i , u f)$1 2 exp@2(t/t)#%. For each coupleof considered directions (each grating), we determined asteady-state value g as well as a specific time constant t.With these values and the physical parameters of the un-doped crystal, we calculated the temporal evolution of in-tensity in the main fanning directions. The results,which are represented in Fig. 10(a), show interestingagreement in the growing phase of the fanned intensitybut clearly fail to fit the general shape of the evolution.Evidence is given that the experimental steady state doesnot result simply from successive amplifications but alsodepends on energy exchanges and that consequently theundepleted-pump approximation does not apply here inthe way in which we had used it previously.

C. Proposal of a Model for Fanned IntensityThe model used so far leads to two conclusions: Compar-ing the theoretical and experimental results in Figs. 9and 10(a), we can see that the time during which eachbeam grows is rather well described by the model. More-over, high-order fanning directions reach steady state ingood agreement with the theory. Nevertheless, the mainproblem with the model is that the intensity in the inter-mediate fanning directions do not lose energy after thebeams have grown. Therefore we propose to modify Eq.(14) by considering a new approximation based on the fol-lowing idea: The experimental results show that the in-tensity of light is lost by a given fanning direction whenthis intensity is used to build up new gratings that arededicated to amplification of the next directions. Thismeans, within the energy-conservation principle (exceptfor absorption), that each beam acts as a depleted pumpto amplify the next direction. Thus the amplified inten-

Fig. 10. (a) Theoretical temporal evolution of the energy distri-bution in the fanning pattern with the first model and (b) resultsof the calculation performed for the improved model with deple-tion of the pumps taken into account. Both models are for anundoped BaTiO3 sample. The parameters were chosen to fit asclosely as possible the steady-state experimental ratio of the in-tensities in the various fanning directions.

sity in a given direction originates from the nearest direc-tion that pumps it. The expression of intensity in a di-rection u f

(n) can be written as follows:

If@u f~n !#

5 I0@u f~n !#)

i51

n21

3 expH g~u f~i !, u f

~n !!F1 2 expS 2tt D G leff @u f

~i !, u f~n !#J

3 expH g@u i , u f~n !#F2 expS 2

tt D G leff @u i , u f

~n !#

2al0

cos u f~n !J 2 (

j5n11

`

Io@u f~ j !#expH g@u f

~n !, u f~ j !#

3 F1 2 expS 2tt D G leff @u f

~n !, u f~ j !#J . (15)

The calculation performed with Eq. (15) yields a theo-retical behavior that is in better agreement with the gen-eral experimental calculation, especially in the case of theundoped sample [see Fig. 10(b)]. The model correspond-ing to Eq. (15) is still not sufficient to compare with thedata from the doped sample. With the characteristic pa-rameters of the doped sample, and under the assumptionof homogeneous diffused intensity $i.e., that I0@u f

(n)#5 I0@u f

(n21)# 5 I0(u i)%, we took a width l' 5 1 mm forthe incident beam and simulated the temporal evolution.These conditions (which should favor high-order fanningangles) and Eq. (15) lead to a calculated maximum inten-sity in the first fanning direction that can be written asu f 5 u f

(1), which is obviously not in agreement with theexperimental evidence. Indeed, with a broad beam weexperimentally obtained the second- or even third-orderfanning direction as the direction of the maximum fan-ning intensity. These considerations made us include anew parameter in the model.

D. Necessity to Use a Model with Deep and ShallowTrapsOur aim is to perform an accurate calculation, which canbe applied to various samples, of the time evolution of BFintensities in determined directions. To do so, in accor-dance with Garrett et al.21 we introduce beam-couplinggain g that includes the intensity-dependent factor h(I0)because BaTiO3 and Co-doped BaTiO3 can be consideredtwo impurity-level systems (deep and shallow traps21).This implies that the Debye screening wave vector kD isexpressed by

kD2 5 kOD

2 1 kOT2, (16)

where kOD and kOT are the Debye screening wave vectorsfor deep and shallow traps, respectively.

In that model, the quantity h(I0) satisfies the followinginequality:

0 , h~Is! 51

kD2 S kOD

2 1kOT

2

1 1 b/sT Is~0 !D < 1,

(17)

where sT is the shallow-trap excitation cross section andb is the thermal ionization rate. To simplify, we considerthat h(Is) depends only on the intensity.


We also ought to take the absorptive gain into account,but this gain is negligible in our experimental conditions.The grating wave vectors that are part of the fanning pro-cess are smaller than 5.2 mm21, and, according to Garrettet al.,21 in undoped and in 20-parts-in-106 Co-dopedBaTiO3 the absorptive gains that correspond to such val-ues are less than 0.1 cm21, whereas with the extraordi-nary polarization states that are chosen here the beam-coupling gain reaches 10 cm21. These values allow us toneglect absorptive gain compared with beam-couplinggain. The gain can now be expressed as

g 5~2p!2n3kBT

e

ne

nor42

cos~u i 1 u f /2!sin~u i 1 u f!

sin~u i 2 u f!H F l

2 sin~u i 1 u f /2!G2

1 F2pAkBT~«a«0 /NA!1/2 cos~u i 1 u f /2!

qG 2J h~I !. (18)

Fig. 11. Influence of h on the theoretical temporal evolution ofthe intensity in four expected fanning directions with parametersof a nominally undoped crystal. The parameters used here arethose used in the experiment represented in Fig. 9.

We assume here that h is independent of incident angleu i , so this quantity does not affect the determination ofthe fanning angles.

The systematic application of the final model based onboth Eqs. (15) and Eq. (18) allows us to determine theo-retically the BF pattern’s time evolution with the follow-ing main results:

• If the beam is narrow enough, only the first fanningdirection will be significantly amplified.

• A broad beam gives birth to fanning in the second-or third-order fanning direction or even farther, as in Fig.4. The intensity can then be parted between the main di-rections in a way that depends on the initial diffusion inthe crystal and on the geometry and parameters of thesample.

• We can modify parameter h(I0) by changing thevalue of the incident intensity. According to inequality(17), that parameter grows when the intensity increases.The influence of h(I0) on the steady-state energy distri-bution is shown in Fig. 11. We can see that increasingh(I0) makes a lower-order fanning direction become moreintense. If we increase intensity (Fig. 9), the same thingis observed, as the maximum intensity that was in thethird direction goes to the second fanning direction.

6. FIRST DOUBLE PHASE-CONJUGATEMIRROR EXPERIMENTAL RESULTSThe experimental setup used for the DPCM is nearly thesame as shown in Fig. 1(a). Two additional beam split-ters enable us to measure both incident and conjugatewaves for each beam with four photodiodes. The crystaltested here is the undoped sample. The beam enteringthe 1c face has an incidence angle equal to the Brewsterangle to minimize the instabilities that are due to reflec-tions of the conjugate wave on this face. We chose thesecond incident angle such that both incident beams gen-erate fanning in the same direction. The cylindricallenses focus the beams to select the first fanning directionfor each with the same width (;60 mm) inside the crystal.Both incident intensities are 100 mW.

We define the reflectivity of a conjugate wave as thevalue of the conjugate intensity that emerges from oneface divided by the intensity entering the opposite face,with both values of intensities taken inside the crystal,which means taking Fresnel transmission coefficientsinto account.


These conditions allowed us to optimize the DPCM bysimply translating the incident beams horizontally andvertically to yield the best possible overlap of the twofanned beams in the sample. The phase-conjugate wavesthat we obtained build up in less than 10 s, with the high-est reflectivity equal to 35%. As shown in Fig. 12, thephenomenon is highly stable (for 10,000 s). We are nowstudying the behavior of DPCM’s for wider beams.

7. CONCLUSIONWe have studied both experimentally and theoreticallythe beam-fanning pattern that occurs in doped and un-doped BaTiO3 crystals. The pattern has been shown todepend strongly on the incident angle of light, on thewaist of the beam and its direction of propagation, and onthe doping ratio of the samples. The fanning process hasbeen interpreted with an iterative method on the basis ofthe two-wave mixing model, which allows the fanning di-rections to be determined accurately. We presented anextension of the model to evaluate the temporal evolutionof fanning that validates the fanning-angle determinationprocess presented above.

We believe that the temporal evolution of intensity inthe various fanning directions combined with the influ-ence of the incident beam width allows us to explain someof the instabilities that have been reported for cat configu-ration (self-pumped phase-conjugate mirror using inter-nal reflections in the crystal2) phase conjugation23–25 andfor the DPCM.26 The time constants for establishing suc-cessive directions of fanning are comparable with the pe-riods of the instabilities that we observed in the crystalsthat we studied. The instabilities may be due to the in-stallation of successive directions of fanning. The associ-ated gratings are then used for four-wave mixing andphase conjugation. If any of the gratings disappears, be-cause of a higher-order fanning beam coming from the op-posite pump beam (in the case of a DPCM) or the reflected

Fig. 12. Temporal evolution of the two efficiency coefficients of aDPCM in the undoped crystal. Darker curve, reflectivity of theconjugate beam coming from the 1c face; lighter curve, for the2c face. Inset, enlargement representing the first 300 s of thesame curves.

beam (in the case of the cat), the corresponding phase-conjugate intensity will also vanish. The erasure can bethought to occur in the way described in Ref. 7 for a beamthat does not fulfill the Bragg condition with the gratingdescribed. We used the results presented here to achievea fast establishing, stable, and efficient DPCM.

ACKNOWLEDGMENTSThe authors thank D. Rytz, of the Forschunginstitut furMineralische und Metallische Werkstoffe Edelsteine/Edelmetalle (Idar-Oberstein, Germany) for theBaTiO3:Co crystal on which part of our experiments wereconducted. This research was supported in part by theRegion Lorraine.

C. Mailhan’s e-mail address: [email protected].

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Insight into a new geometrical approach to beam fanning in BaTiO3