Self-focusing and soliton formation in media with anisotropic nonlocal material response

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1 EUROPHYSICS LETTERS 20 November 1996 Europhys. Lett., 36 (6), pp (1996) Self-focusing an soliton formation in meia with anisotropic nonlocal material response A. A. Zoulya 1, D. Z. Anerson 1, A. V. Mamaev 2 ( ) an M. Saffman 2 1 JILA, University of Colorao - Bouler, CO , USA 2 Department of Optics an Flui Dynamics, Risø National Laboratory Postbox 49, DK-4000 Roskile, Denmark (receive 28 June 1996; accepte in final form 4 October 1996) PACS Tg Optical solitons; nonlinear guie waves. PACS Jx Beam trapping, self-focusing, an thermal blooming. PACS Hw Phase conjugation, optical mixing, an photorefractive effect. Abstract. We investigate self-focusing in bulk meia with anisotropic nonlocal photorefractive response. Analytical results emonstrate the possibility of the existence of anisotropic soliton solutions. Self-focusing of Gaussian beams an their convergence to elliptically shape soliton solutions is investigate theoretically an emonstrate experimentally. In typical nonlinear optical meia the material respons to the presence of the optical fiel B(r) by a nonlinear change in its refractive inex δn that is an algebraic (local) function of the light intensity. This local response in its simplest case δn B 2 results in the canonical nonlinear Schröinger equation for the amplitue of light propagating in the meium [1]. Higher-orer nonlinearities result in various forms of local saturable response [2]. In photorefractive meia the change in the refractive inex is proportional to the amplitue of the static electric fiel inuce by the optical beam. Fining the material response therefore requires solving globally an elliptic-type equation for an anisotropic electrostatic potential with a source term ue to light-inuce generation of mobile carriers [3] that has no irect analogs in nonlinear optics. The closest structurally similar counterparts probably are Davey-Stewartson equations [4] escribing nonlinear ispersive waves in flui ynamics or Zakharov equations for parametrically couple electromagnetic an Langmuir waves in plasmas [5]. Meia with a photorefractive nonlinearity turn out to be convenient for stuying complex spatial ynamics. These inclue formation of patterns [6], self-focusing [7], [8], an transverse moulation instability of solitary stripe beams [9] with formation of optical vortices [10]. In this work we present the theory of steay-state two-transverse-imensional solitons in bulk meia with nonlocal anisotropic photorefractive response an analye their properties in etail. Evolution of arbitrary Gaussian beams, their convergence to elliptically shape ( ) Permanent aress: Institute for Problems in Mechanics, Russian Acaemy of Sciences, Prospekt Vernaskogo 101, Moscow, Russia. c Les Eitions e Physique

2 420 EUROPHYSICS LETTERS soliton asymptotic states, the rate of this convergence, an its epenence on the parameters of the problem are investigate both numerically an experimentally. Propagation of an optical beam B(r) in a photorefractive meium is governe by the set of equations [3], [9], [10] [ x i ] 2 2 B(r) = i ϕ B(r), (1a) 2 ϕ + ln(1 + B 2 ) ϕ = ln(1 + B 2 ). (1b) Here = ŷ( / y) + ẑ( / ) an ϕ is the imensionless electrostatic potential inuce by the beam with the bounary conitions ϕ(r ) 0. The imensionless coorinates (x, y, ) are relate to the physical coorinates (x, y, ) by the expressions x = αx an (y, ) = kα(y, ), where α = (1/2)kn 2 r eff E ext. Here k is the wave number of light in the meium, n is the inex of refraction, r eff is the effective element of the electro-optic tensor, an E ext is the amplitue of the external fiel irecte along the -axis far from the beam. The normalie intensity B(r) 2 is measure in units of saturation intensity I, so that the physical beam intensity is given by B(r) 2 I. Equations (1) are vali for relatively wie beams an neglect the part of the nonlinearity responsible for asymmetric stimulate scattering since it is not important in the range of parameters iscusse here (for etails see [3]). For typical spatial structures of light in the conitions when this secon part ominates see [11]. The strong anisotropy of eqs. (1) oes not allow raially symmetric solutions. In the simplest approximation the beam shoul be treate as elliptical an characterie by two iameters y an along the y an axes, respectively. Its amplitue in the parabolic approximation can be expresse in the form B = I m exp [ 4y2 2 y i y2 2 y + i 2 ] + iθ, (2) y 2 where I m = I in y (0) (0)/ y an θ(x) is the nonlinear phase change. The primes enote ifferentiation with respect to x. In the unsaturate regime I m 1 eq. (1b) reuces to 2 ϕ = B(r) 2 /. Its solution in cylinrical coorinates y = r cos ψ, = r sin ψ takes the form ϕ(r, ψ) = I m 16 2 r 2k+1 F k (r 2 ) sin(2k + 1)ψ, (3a) F k = r 2 k=0 τ ττ 2k 2 τ τ k+1 Ak (τ ) exp[ aτ ], 0 A k = ( 1) k+1 [δ k,0 + (1 δ k,0 )I k (bτ) + I k+1 (bτ)]. Here a = 4( 2 y + 2 ), b = 4( 2 y 2 ), an I k are moifie Bessel functions. The refractive inex in the central region of the beam accoring to eqs. (3) is given by the expression (3b) (3c) ν(y, ) = 8I m 2 y 2 + ( 2 y + 2 y ) 2 2 ( y + ) 2. (4) Substituting expressions (4) an (2) into eq. (1a), one arrives at the set of equations [12] y = 16 3 [ y 4 κf 3 (F + 1) 2], = 16 3 [ 4 κ(f + 2)(F + 1) 2 ], (5) where κ = I in y (0) (0) an F = y (x)/ (x) is the beam iameter ratio. Equations (5) emonstrate a nontrivial nonseparable way in which both iameters of the beam contribute

3 a. a. oulya et al.: self-focusing an soliton formation in meia with etc y y / DIAMETERS MAXIMUM INTENSITY PROPAGATION DISTANCE Fig. 1. Fig. 2. Fig. 1. Diameters of the soliton solution of eq. (1) an their ratio vs. its maximum intensity. The insert shows the soliton intensity cross-sections for I m = 5. Fig. 2. Evolution of iameters of an initially roun Gaussian beam for I in = 0.5 (soli) an 6 (ashe curves). to its evolution. In particular they show the possibility of a self-channele propagation y, = const that correspons to the value of the iameter ratio F etermine by the relation F 3 = F + 2 or F = F The self-channele beam is narrower in the irection of the applie fiel an wier in the perpenicular irection. Stability analysis reveals that the evolution of perturbations aroun this solution is of an oscillatory character in the framework of an aberrationless approximation. The analytical results inicate that eqs. (1) may have soliton solutions B(x, y, ) = b(y, ) exp[iλx], where λ is a real propagation constant. These solutions have been foun numerically by iteratively solving eqs. (1) using the renormalisation proceure ue to Petviashvili [13]. Note that this approach is exact an oes not imply any approximations (e.g., a Gaussian ansat) mae previously in the theoretical treatment. The lowest-orer soliton solution of eqs. (1) turns out to be a smooth one-humpe beam that is narrower along the coorinate an wier in the perpenicular irection. Figure 1 shows the iameters y an of the soliton solution an their ratio y / as functions of the soliton maximum intensity I m = b(0, 0) 2. The iameters have been calculate at the 1/2 level of the maximum intensity. The insert presents the cross-sections of the soliton intensity for I m = 5 along the (narrower curve) an the y (wier curve) axes. The values of the iameters are inversely proportional to the square root of the soliton maximum intensity y, 1/ I m for I m 0, logarithmically proportional to it ( y, ln I m ) in the opposite limit I m, an pass through shallow minima in between. Notice that the iameter ratio in fig. 1 is strikingly close to that given by the analytical formulas, eq. (5). The absolute values of the iameters for I m 1 are about two times off. Numerical analysis of the evolution of arbitrary initial beams inicates that the soliton solutions are attractors of the set of eqs. (1). The rate of convergence to these solutions epens primarily on the normalie intensity I m. We have foun that evolution of an input beam in general is characterise by oscillations of both its iameters, in qualitative agreement with the results of the analytical approach. In the unsaturate limit I m 1 these oscillations are strongly ampe an their spatial perio scales as 1/I m. The initially roun beam becomes

4 422 EUROPHYSICS LETTERS elliptical an converges to the soliton solution. Increasing I m up to about unity ecreases the perio of the oscillations, while still keeping them reasonably heavily ampe, thereby ecreasing the length of the spatial transient. Further increase in I m up to several units still ecreases the spatial perio of oscillations but also sharply ecreases the relaxation rate an increases the spatial transient length. In the very high-saturation regime (roughly I m > 50) the oscillation perio starts to grow an the relaxation rate remains small, so the spatial transient length remains large. The characteristic perio of oscillations in this case may excee the length of the meium. By ajusting input parameters the output profiles of the beam in the high-saturation transient regime can be mae to have a broa range of shapes starting from beams that are more elongate along the y-axis to those elongate along the -axis an incluing roun output beams in between. Figure 2 shows spatial evolution of the input Gaussian beam B in = I in exp[ 2 ln 2(r/) 2 ] for I in = 0.5 (soli) an 6 (ashe curves). Insie the meium the beam unergoes self-focusing an its intensity I m becomes 1.1 an 7 27, respectively. Note the rapi relaxation to the soliton solution for I in = 0.5 an a very long transient in the high-saturation regime. Note also that the longituinal span of fig. 2 consierably excees the characteristic lengths of photorefractive meia, concrete estimates will be given below. Figure 2 emonstrates that the best parameter region to observe solitons correspons to a moerate-saturation regime when the intensity of the beam is about the saturation intensity. In the unsaturate limit I m 1 the oscillations of an input beam are ampe but the overall evolution scale goes up as 1/I m. In the high-saturation regime I m 1 the amping rate is too small an the spatial transients last a very long istance. The experimental arrangement was similar to that use in ref. [9], [10]. A 10 mw He-Ne laser beam (λ = 0.63 mm) was passe through a variable beam splitter an a system of lenses controlling the sie of the beam waist. The beam was irecte into a photorefractive crystal of SBN:60 ope with 0.002% by weight Ce. The beam propagate perpenicular to the crystal ĉ-axis, an was polarie along it to take avantage of the largest component of the electro-optic tensor of SBN r 33. The crystal measure 10 mm along the irection of propagation, an was 9 mm wie along the ĉ-axis. A variable c voltage was applie along the ĉ-axis to control the value of nonlinear coupling. Images of the beam at the output face of the crystal were recore with a CCD camera. The iameters were measure from scans of the output images at the 1/2 level of the maximum intensity. The effective saturation intensity was varie by illuminating the crystal from above with incoherent white light. The epenence of the photorefractive coupling constant on saturation intensity I in the absence of an applie fiel takes the form [14] γ = γ 0 I beam /(I beam + I ). By measuring the two-beam coupling gain with the white-light source off (I = 0) an on (I 0) the ratio I m = I beam /I has been etermine. Experiments in the high-saturation regime I m 1 showe that the shape of the input roun beam at the output from the nonlinear meium changes as a function of the voltage, the input iameter of the beam, an the focusing conitions. At high voltages the center of the output beam was also isplace along the ĉ-axis by up to a couple of beam iameters ue to self-bening [3]. We have observe ellipticities y / ranging from about 0.5 to 5, incluing roun output beams. This is illustrate in fig. 3 which shows the output iameters of the roun input 26 mm iameter beam as functions of the applie voltage in the high-saturation regime. The inserts show the beam profile at 0.5 an 0.9 kv. Note how the beam iameter ratio value y / flips over between 0.4 an 0.7 kv an then reverts to values larger than unity. Recent experimental observations of photorefractive self-focusing in two transverse imensions have claime that solitons have cylinrical symmetry [8]. Our results contraict this

5 a. a. oulya et al.: self-focusing an soliton formation in meia with etc DIAMETERS [µm] y DIAMETERS [µm] y APPLIED VOLTAGE [kv] Fig. 3. Fig APPLIED VOLTAGE [kv] Fig. 3. Ouput iameters vs. applie voltage in the high-saturation regime. The power of the input beam was 50 mw. Fig. 4. Ouput iameters vs. applie voltage in the moerate-saturation regime. The power of the input beam was 4.2 mw. claim an we believe that the results in ref. [8] shoul be ientifie as a transient stage of convergence to a soliton. As a proof of their claim the authors of ref. [8] presente a sie-view image of the beam passing through the crystal an several beam profiles at istances up to 0.7 mm apart, that, accoring to the authors, showe the same beam iameters. The technique of imaging through the nonlinear meium has been characterie in ref. [7] as not vali ue to high uncontrollable istortions. As concerns the profiles, the corresponing FWHM iameters measure from the ata in the last of ref. [8] turn out to vary by about 25% (from 10.7 mm to 13.3 mm). The experiments were conucte in the high-saturation regime where the evolution of a light beam is accompanie by long transient oscillations. Accoring to our estimates the spatial perio of these oscillations was slightly larger than the sie of the crystal. By an appropriate choice of parameters (see fig. 3) the output beam iameter ratio can be mae to have many values (in particular, be equal to unity). The eviations from a roun shape at istances of about 0.7 mm shoul be about plus/minus ten percent, which is in goo agreement with the experimental ata of ref. [8]. Figure 4 shows the output iameters vs. the applie voltage for the same beam as in fig. 3 but with twelve times smaller power. It emonstrates rapi convergence of the output beam profile to an elliptical shape elongate along the y-axis characteristic for the soliton solution. The insert is the output intensity istribution for 1.0 kv external voltage. The measure normalie input beam intensity for fig. 4 was I m 0.6 with about 30% uncertainty. In fig. 3 the laser power was set 12 times higher than in fig. 4, so the value of I m in fig. 3 is known to be 12 times higher than in fig. 4 with high accuracy. The measure value of the electro-optic coefficient for our crystal is r 33 = 360 pm/v. Soli curves in fig. 3 an 4 are theoretical comparisons obtaine by solving eqs. (1) with a 26 mm iameter collimate input Gaussian beam an I m = 6 an 0.5, respectively. Note that the only parameter in the theoretical calculations with a relatively large experimental uncertainty was I m in fig. 4, which was chosen to give the best agreement with the ata. The resulting agreement for both figures, consiering the complexity of the system being moelle, is strikingly goo. Quantitative ifferences in fig. 3 are probably ue to the non-gaussian shape of the beam an some uncertainty in the position of its waist with respect to the crystal face.

6 424 EUROPHYSICS LETTERS For the above parameters an 1.0 kv voltage the 26 mm input iameter translates to 5.8 in units of fig. 2 an the one-centimeter crystal length to the propagation istance x Soli an ashe curves in fig. 2 in the range 0 < x < 23 show the evolution of the beam iameters insie the crystal for the 1.0 kv/cm points in fig. 3 an 4, respectively. As is seen from fig. 2 the spatial transients at the output from the crystal for I in = 0.5 are largely gone an the beam is close to its asymptotic soliton shape. This allows us to conclue that ata in fig. 4 present experimental observation of convergence to the soliton solutions foun in this paper. *** AAZ an DZA acknowlege the support of NSF grant PHY an the Optoelectronics Computing Center, an NSF Engineering Research Center. The work at Risø was supporte by the Danish Natural Science Research Council. REFERENCES [1] Chiao R. Y., Garmire E. an Townes C. H., Phys. Rev. Lett., 13 (1964) 479. [2] Marburger J. H., Prog. Quantum Electron., 4 (1975) 35; Karlsson M., Phys. Rev. A, 46 (1992) 2726; Suter D. an Blasberg T., Phys. Rev. A, 48 (1993) 4583; Dowell M. et al., Phys. Rev. A, 52 (1995) [3] Zoulya A. A. an Anerson D. Z., Phys. Rev. A, 51 (1995) [4] Ablowit M. J. an Segur H., Solitons an the Inverse Scattering Transform (SIAM, Philaelphia) [5] Zakharov V. E., Zh. Ėksp. Teor. Fi., 62 (1972) 1745 (Sov. Phys. JETP, 35 (1972) 908). [6] Hona T., Opt. Lett., 18 (1993) 598; Mamaev A. V. an Saffman M., Europhys. Lett., 34 (1996) 669. [7] Iturbe-Castillo M. D. et al., Appl. Phys. Lett., 64 (1994) 408; Iturbe-Castillo M. D. et al., Opt. Commun., 118 (1995) 515. [8] Duree G. et al., Phys. Rev. Lett., 71 (1993) 533; Shih M.-F. et al., Electron. Lett., 31 (1995) 826; Shih M.-F. et al., Opt. Lett., 21 (1996) 324. [9] Mamaev A. V., Saffman M. an Zoulya A. A., Europhys. Lett., 35 (1996) 25; Mamaev A. V., Saffman M., Anerson D. Z. an Zoulya A. A., Phys. Rev. A, 54 (1996) 870. [10] Mamaev A. V., Saffman M. an Zoulya A. A., Phys. Rev. Lett., 76 (1996) [11] Zoulya A. A., Saffman M. an Anerson D. Z., Phys. Rev. Lett., 73 (1994) 818. [12] Zoulya A. A. an Anerson D. Z., in Proceeings of the International Conference on Photorefractive Materials, Effects an Devices, eite by D. Z. Anerson an J. Feinberg (Estes Park) 1995, p [13] Petviashvili V. I., Fi. Plamy, 2 (1976) 469 (Sov. J. Plasma Phys., 2 (1976) 257). [14] Kukhtarev N. V. et al., Ferroel., 22 (1979) 949.

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