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November 15, 1994 / Vol. 19, No. 22 / OPTICS LETTERS 1825

Polarization spatial chaos in second-harmonic generationStefano Trillo

Fondazione Ugo Bordoni, Via B. Castiglione 59, 00142 Rome, ItalyGaetano AssantoDipartimento di Ingegneria Elettronica, III University of Rome, Via Eudossiana 18, 00184 Rome, Italy

Received June 6, 1994We investigate second-harmonic generation beyond the scalar approximation. We show that the tensorial natureof the second-order susceptibility may lead to the onset of spatial chaos in the polarization evolution of theinteracting fields. We discuss the effect of chaos on situations of interest.

eakness of third-order nonlinearities is an ob-of all-optical deviceslevels. This problem has mo-interest in quadratic media, infrequency (FE) field may experi-r phase change as a result of its inter-second-harmonic (SH).'4two-wave mixing has been investigated un-the hypotheses that only a FF and a SH scalarinteract through an effective nonlinear coeffi-In this case the field evolutions are describedby the closed-form solutions to the coupled-4 or by phase-plane methods. 5' 6 Con-in this Letter we demonstrate that theof the coupled-mode approach to theappearance of spatiallybehaviors of the fields. The importance ofprediction is twofold. First, the onset of spa-chaos poses a fundamental limitation on theof SH generation as well as on the mode

of a SH seed.57 Second, vectorial interac-generalize the cascading effect based on theapproach, for which the generation of the SHphase shift of theIn the vectorial case a FF beam may ac-a cascaded process that impliesto the SH field as an intermediate step.

3'8 it is important tothe range of parameters for which the fields remain regular and predictable.Here we consider the interaction between two lin-polarized cw components of an FF field (withvarying amplitudes E0j, j = 1, 2) and a scalarof an SH field (say E 2 ). The coupled-equations governing the parametric interaction

'i'i =XyjjEEog*exp(iAkjz)+ X12E2E0,3-i* expi Ak + Ak2Z

dE2 2 2dz =Xi1 Eo,l ex~p(- Ak 1z) Y22Eo,2 exp(- iAk 2 Z)+ 2X1Eo,1Eo,2 exp(- Ak 1 2 ) (1)

where Akj = k(2wo) kj(co) = 2co(n2a - nj)/cis the wave-vector mismatch, Xij = w[21(6oc3n2ni ni )]" 2 x(2), with i, j = 1, 2, where X(2) X(2)(c;2w,-w) =X12)(2o); w, w) is the corresponding el-ement of the susceptibility tensor (measured inmeters per volt), and the field amplitudes E 2 andEo,j are such that their square moduli directlygive the intensity (in watts per square meter).Equations (1) describe satisfactorily the vectorialprocess under the hypotheses that the orthogonalSH component is highly mismatched and hencedoes not couple to the other fields. Henceforthwe consider a purely parametric process with realx(. In this case Eqs. (1) admit of the conservationof the total intensity, I = 1E,112 + 1E0,212 + E212,and may be cast in the usual Hamiltonian form,5-i dAj/dz = aH/aA*, in terms of the new variablesAj = T2EEo,/7 and A3 = E2/jrI. The field Hamil-tonian H =H(Al,A2 , A3,Al*,A2*, A3*), which repre-sents another conserved quantity of Eqs. (1), may bedecomposed as H = H1 + H2 + H12, where

A2A 3 *H. = Xii 2 exp(-iAkjz) + c.c. (j = 1,2),(2a)

H 1 2 = Xl2AlA 2 A3 * exp(Ak 1 Ak 2 z) + c.c. (2b)represent the Hamiltonian of the (two) type I andthe type II interactions, respectively. The numberof effective degrees of freedom of Eqs. (1) may be re-duced by changes of variables (i.e., canonical transfor-mations). From Hamiltonian equations (2) note thatthe phases q5j = arg(Aj) appear only through thecombination 01,2 Akl, 2 + 0 3 - 201,2. The variablesconjugated to Oj(j = 1, 2) are the corresponding FFintensity (or power) fractions 7-q A2/2 =IEo 2 /I.From Eqs. (1) we obtain that the variables qj and Ojobey

dqj aHd; aOl' dOj _d; aHa7 j (j == ,2), (3)where v- Ak2z, and the reduced Hamiltonian Hreads as

0146-9592/94/221825-03$6.00/0 1994 Optical Society of America

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1826 OPTICS LETTERS / Vol. 19, No. 22 / November 15, 1994

U)0C.).'-

0

1

00 20 40 60 80 100 120distance

(a)

00L.a..4-

000 20 40 60 80 100 120distance

(b)V)0.a).6-.1)10

1

00 10 20 3distance C

(c)Fig. 1. Power fractions of the SH (77,and the FF (n,, solid curve and 272,fields versus distance ' for an angle /771(0) = 0-9997, 772(0) = 0.00031. Here i

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November 15, 1994 / Vol. 9, No. 22 / OPTICS LETTERS 182710 1

S

a) 10

0 10,.-1

1-1-2-3

l0 -4 101 10 2 103distance 10 4 10 52. Maximal Lyapunov exponent Amversus distancefor different values of the input amplitude p. Here

772(0) = 0.01, XI = 0.5, and K = 0.

10C.).a-4-

0co;_1

4-)

01

00 5 10input amplitude p

3. Generated SH power fraction versus normalizedK = 0, = 0.1, and (a) 6max Ak2 L = 2,Vmax= 20.

9 "' defined as the limit for g X of= (1/')ln[llt(')1J/11t(' = 0)11]. Here t is thevector, which evolves according to the equa-= _ 1(afi/axj)tj, where x (01,0I, 772, 02) andi = 1, . .. ,4.

show the estimated Lyapunov exponentfor 71,(0)= 0.99, 772(0) = 0.01. The monotoni-Amobtained for p = 0.3 correspondsregular (periodic or quasi-periodic) trajectory.threshold for chaos the exponent converges toWe obtained similar results in a wide range

Because in a real experiment access to the lon-evolution is generally impossible, the on-chaos may be demonstrated by monitoringSH as a function of the input am-Although in the chaotic regimeof power result in completely differ-

ent dynamic evolutions, this becomes evident onlyfor sufficiently long (with respect to the rate of di-vergence Am,of nearby trajectories) sample lengths.In fact, as Fig. 3(a) shows for ; = 2, the SH powerfraction seems a regular function of p, whereas itturns out to be clearly irregular for ; = 20 [Fig. 3(b)].Whenever pulsed light is employed, because similaroscillations occur in the transmission of the FF com-ponents the FF pulse will emerge completely depolar-ized in the nonlinear interaction. For given valuesof p and Ak2L the corresponding peak intensity inphysical units is I = (pAk2L/X11L). For instance,pAk2L = 10 gives, in a crystal L = 1 cm long at A=2'ic/w = 1 Am, a peak intensity of I = 3.4 GW/cmfor X(2)/[n2. (n1)2 ]"2 = 1 pm/V.In summary, the inclusion of the polarization prop-erties in the description of SH generation in a trans-parent medium leads us to predict, for the first timeto our knowledge, the onset of spatial chaos. Thechaotic regime appears as the stochastic evolution ofboth the FF polarizations and the generated SH. Wepoint out that, here, spatial chaos has the features ofa conservative (Hamiltonian) system, as opposed tothe temporal chaos predicted for intracavity SH gen-eration described by a dissipative (i.e., with lossesand pumping) system.'2

The authors are grateful to G. I. Stegeman for en-lightening discussions. The work of S. Trillo wascarried out in the framework of an agreement be-tween the Fondazione Ugo Bordoni and the IstitutoSuperiore Poste e Telecomunicazioni. G. Assantoacknowledges support from 1993 MURST 40%.References1. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I.Stegeman, E. W. Van Stryland, and H. Vanherzeele,Opt. Lett. 17, 28 (1992).2. G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland,and G. Assanto, Opt. Lett. 18,13 (1993).3. D. C. Hutchings, J. S. Aitchison, and C. N. Ironside,Opt. Lett. 18, 793 (1993).4. J. A. Armstrong, N. Bloembergen, J. Ducuing, andP. S. Pershan, Phys. Rev. 6, 1918 (1962).5. S. Trillo, S. Wabnitz, R. Chisari, and G. Cappellini,Opt. Lett. 17, 637 (1992).6. S. A. Akhmanov and R. V. Khoklov, Problems of Non-

linear Optics (Gordon & Breach, New York, 1972),pp. 155-165.7. D. J. Hagan, M. Sheik-Bahae, Z. Wang, G. I.Stegeman, E. W. Van Stryland, and G. Assanto, Opt.Lett. 19, 1305 (1994).8. G. Assanto, I. Torelli, and S. Trillo, Electron. Lett. 30,733 (1994).9. A. J. Lichtenberg and M. A. Lieberman, Regular andStochastic Motion (Springer-Verlag, New York, 1983),pp. 151-211, 260-285.10. S. Wabnitz, Phys. Rev. Lett. 58, 1415 (1987).11. G. Benettin, L. Galgani, and J. M. Strelcyn, Phys. Rev.A 14, 2338 (1976).12. C. M. Savage and D. F. Walls, Opt. Acta 30, 557(1983).