April 15, 1995 / Vol. 20, No. 8 / OPTICS LETTERS 837

Nonstationary self-focusing in photorefractive media

A. A. Zozulya and D. Z. Anderson

Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80309-0440

Received November 21, 1994

We consider nonstationary self-focusing of an optical beam propagating in a photorefractive medium in theframework of a two-dimensional model.

PACS numbers: 42.65.Hw, 42.50.Ne.

Self-focusing of light is one of the most basic mani-festations of nonlinearity in any optical medium.Recently there has been considerable interest inobserving this effect in photorefractive nonlinearmedia1,2 and also in its theoretical description.3 Be-low we theoretically investigate self-focusing inphotorefractive media in the framework of a two-dimensional model. We show that the self-focusingin photorefractive media is essentially nonstation-ary in nature. The magnitude of the process in thetransient regime is determined by the values of theexternal electric field or the photogalvanic nonlin-earity. The magnitude of the self-focusing at the re-laxation stage of the transient regime and in steadystate depends critically on the value of dark conduc-tivity. There is no steady-state self-focusing whenthis parameter goes to zero.

The material response of a photorefractive mediumis governed by the following set of equations4:

≠N1D y≠t ­ sb 1 sIemd sND 2 N1

D d 2 jneN1D , (1a)

= ? se0eEd ­ r , (1b)

≠ry≠t 1 = ? J ­ 0 , (1c)

r ­ esN1D 2 NA 2 ned , (1d)

J ­ emneE 1 mkBT=ne 1 bphsND 2 N1D dcIem . (1e)

Here ND , N1D , NA, and ne are the density of donors,

ionized donors, acceptors, and conducting electrons,respectively; b and s are the thermal and photoex-citation coefficients; Iem ­ jBsr, tdj2 is the intensityof electromagnetic radiation; j is the recombinationconstant; E is the amplitude of the static electricfield; e is the elementary charge; e0 is the electricpermeability of vacuum; e is the static dielectric ten-sor; r and J are the charge and the electric cur-rent densities, respectively; m is the electron mobility;kB is Boltzmann’s constant; and T is the tempera-ture. The photogalvanic tensor is assumed to havethe largest component bph generating current alongthe direction of the axis of spontaneous polarizationof the medium (axis c), and other components areneglected.

The amplitude Bsr, td of an electromagnetic beampropagating in a photorefractive medium obeys the

0146-9592/95/080837-03$6.00/0

equations

f≠y≠l 2 siy2kd='2 2 ingBsr, td ­ 0 , (2)

nsr, td ­ 2sky2dn2ep ? srel ? Ed ? ep , (3)

where l is the coordinate in the direction of propaga-tion of the beam, ='

2 is a transverse Laplace operator,k is the wave number of light in the medium, rel isthe electro-optic tensor, and ep is the unity vector inthe direction of polarization of the beam.

We consider the following two-dimensional geome-try of interaction: the beam has unlimited extentalong the axis y, is polarized along the axis z, propa-gates along the axis x, enters at the face x ­ 0, andexits at x ­ Lx. The characteristic transverse size ofthe beam is much smaller than the size of the crys-tal. The photogalvanic current is generated alongthe z axis (c is parallel to the z axis); the externalelectric field (if any) is also applied along the z axis.The nonlinear refractive index is due to the rzzz sr33dcomponent of the electro-optic tensor. As a result,we are left with the longitudinal direction of propa-gation x and the transverse coordinate z. The quasi-static electric field E has only a z component andchanges along the z axis.

In Eqs. (1) the characteristic value of the electricfield is determined by E ­ kBTkDye ; eNAye0eckD ,where kD ­ se2NAykBTe0ecd1/2 is the Debye wave num-ber and ec is the component of the static dielec-tric tensor along the c axis. We also introduce thedimensionless intensity of the electromagnetic radi-ation, which includes a contribution from the thermalexcitation of carriers: I ­ sIem 1 bysdyI0 ; Iem 1 Id,where I0 is some characteristic intensity (e.g., in thecenter of the beam). The term Id will be referredto as the normalized dark intensity. In typical con-ditions Id ,, 1, so this last term is often neglected;for the problem that we are considering, its propertreatment is of extreme importance.

For an externally applied electric field the sidefaces of the crystal perpendicular to the z axis arekept at the potential difference V. In the absence ofthe beam this potential difference corresponds to thefield Eext ­ 2VyLz everywhere in the crystal, whereLz is the size of the crystal along z. Generation ofcharge carriers by light changes the value of the elec-tric field inside the beam in the illuminated region,

1995 Optical Society of America

838 OPTICS LETTERS / Vol. 20, No. 8 / April 15, 1995

but its asymptotic value outside remains the sameif the transverse size of the beam d is much lessthan the transverse size of the crystal Lz. Hencethe asymptotic boundary conditions for the electricfield should read Esx, z ! 6`, td ! Eext. Note alsothat Iemsx, z ! 6`, td ! 0, so I sx, z ! 6`, td ! Id.With these boundary conditions and under the as-sumptions of ND .. NA .. n0 and meye0ecj ,, 1, wesee that Eqs. (1) may be reduced to

I 21 ≠

≠tE 1 E

µ1 1 kD

21 ≠

≠zEE

∂21

2

µ1 1 kD

21 ≠

≠zEE

∂ 22

kD22 ≠2

≠z2E

­ 2kD21E

µ1 1 kD

21 ≠

≠zEE

∂21

3≠

≠zln I 1

Id

IsEext 1 Ephd 2 Eph , (4)

where Eph ­ bphjNAyems is the characteristic photo-voltaic field and t ­ tyt0 st0 ­ e0ecjNAyemsI0ND d.

For a Gaussian beam Iem ­ exps28z2yd2d,assuming that kDd .. 1 and Id ,, 1, thesteady-state distribution of the electric field near itscenter, according to Eq. (4), is given by the relation

E ­ fIdEext 2 s1 2 IddEphg 116EkDd

zd

1 8IdsEext 1 Ephdz2

d21 . . . . (5)

The physics described by Eqs. (4) and (5) is as fol-lows: immediately after the beam is turned on theamplitude of the electric field E in the medium isequal to that of the externally applied field Eext.The beam generates charge carriers that start to re-distribute themselves by drift and diffusion, therebychanging the electric field in the illuminated regionof the crystal and resulting in the initial buildup ofthe induced nonlinear lens. This lens brings abouttransient self-focusing or defocusing of the beam, de-pending on the sign of the product r33sEext 1 Ephd.The redistribution of charges also creates a contribu-tion to the refractive index that varies linearly acrossthe beam (the term proportional to ≠Iy≠z), result-ing in the bending of the beam toward the opticalaxis.5,6 At times exceeding several characteristic re-laxation times, the induced lens starts to relax to itssteady-state value. The physics of the relaxation isdetermined by the redistribution of charges, whichtends to screen the externally applied electric fieldand counterbalance the photogalvanic driving term.The screening becomes more and more pronouncedat larger times. Its steady-state magnitude is deter-mined by the ratio of conductivities in the illuminatedand dark regions of the crystal or, equivalently, bythe value of the dark intensity Id. If Id ­ 0, the illu-minated region is ideally conducting compared withthe rest of the crystal, and after the charges have hadtime to redistribute themselves the screening is abso-lute. The external field does not penetrate the beam,

and the static electric field E inside exactly coun-terbalances the photogalvanic driving term. In thislimit the induced nonlinear lens decays to zero andthe effect of self-focusing or defocusing completelydisappears. For nonzero values of Id the lens relaxesto a nonzero steady-state value determined by thevalue of Id; thus the magnitude of the self-focusing ordefocusing is directly related to the magnitude of Id.

Figure 1 illustrates results of numerical solu-tions of Eqs. (2) and (4) for a Gaussian beamBin ­ exps24z2yd2d. Figure 1(a) shows the tempo-ral evolution of the output diameter of the beamfout ­ dsx ­ Lxdyd for several values of the darkintensity. The initial output value of the diameteris determined by diffraction and for the parametersof the calculations equals fout ­ 1.86. As the non-linearity starts to build up, the magnitude of therefractive index grows, resulting in the self-focusingof the beam. At later times the screening comesinto play, quenching the self-focusing. The steady-state magnitude of the effect is proportional to thevalue of Id. Figure 1(b) shows the spatial evolutionof the diameter inside the medium for t ­ 5 (dot-ted curves) and t ­ 50 (solid curves), demonstratingthat the initial stage of evolution of the beam isweakly dependent on the value of Id, but later evo-

Fig. 1. (a) Temporal evolution of the output normal-ized diameter of an input Gaussian beam, (b) spatialevolution of the diameter in the medium for t ­ 5and t ­ 50 with d ­ 30 mm, Lx ­ 4 mm, n ­ 2.3,kDyk ­ 0.35, l ­ 0.633 mm, kn2r33ELx ­ 32, andsEext 1 EphdyE ­ 20.2.

April 15, 1995 / Vol. 20, No. 8 / OPTICS LETTERS 839

Fig. 2. Same as Fig. 1 but for sEext 1 EphdyE ­ 20.6.

lution is directly determined by it. The diametersfor this and Fig. 2 are calculated according to theformula d2sxd ­ 16P 21 s sz 2 z0d2Iemsx, zddz, wherez0 ­ P 21 s zIemsx, zddz is the z coordinate of thecenter of the beam and P ­ s Iemsx, zddz is itstotal power.

Figure 2(a) shows the output diameter of the beamas a function of time for EextyE ­ 20.6, with all otherparameters the same as in Fig. 1; Fig. 2(b) shows thespatial evolution of the diameter inside the mediumfor t ­ 5 (dotted curves) and t ­ 50 (solid curves).The nonlinearity in this case turns out to be too large,so the beam overfocuses and its minimum diameterlies inside the medium (for still larger values of thenonlinearity the beam diameter may have severalmaxima and minima inside the medium).

To summarize, we have developed a theory of self-focusing in photorefractive media in the frameworkto a two-dimensional model. We have shown that itis an essentially nonstationary process. The magni-

tude of a nonlinear lens, induced by the beam andresponsible for its self-focusing or defocusing, is afunction of time. At zero time there is no lens. Atsome intermediate times, when a sufficiently largedensity of charge carriers has been generated buttheir redistribution is still incomplete, the strength ofthe induced lens reaches its maximum value. At stilllater times the continuing redistribution of chargesstarts to screen the external field and the photogal-vanic term, and the magnitude of the lens relaxesto its steady-state value. This value is directly de-pendent on the value of the dark intensity Id. Webelieve that our results agree qualitatively with theexperimental observations of Refs. 1 and 2. Ourmodel is in disagreement with the two-dimensionalmodel of self-focusing developed in Ref. 3. Thesteady-state theory of Ref. 3, which predicts self-focusing, does not include the dark intensity sId ­ 0d.According to our results, in steady-state this darkintensity is necessary. The reason for the discrep-ancy between our results and those of Ref. 3 is asfollows. In Ref. 3 a particular steady-state solutionof linearized two-dimensional versions of Eqs. (1)was used with the amplitude of a light beam consist-ing of two plane waves. A general solution for anarbitrary transverse distribution of the beam was as-sumed to be a linear superposition of such solutionswith plane waves being all possible pairs from theFourier decomposition of the beam amplitude. Thisassumption is incorrect, as indicated by the resultsof this Letter. For example, the effect of screeningwas lost.

This research was supported by National ScienceFoundation grant PHY90-12244.

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