CALTECH EE 243A - Discrete self-focusing in nonlinear

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794 OPTICS LETTERS / Vol. 13, No. 9 / September 1988Discrete self-focusing in nonlinear arrays of coupled waveguidesD. N. ChristodoulidesDepartment of Computer Science and Electrical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015R. I. JosephDepartment of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, Maryland 21218Received April 7, 1988; accepted May 31, 1988We show that a nonlinear array of coupled waveguides can exhibit discrete self-focusing that in the continuumapproximation obeys the so-called nonlinear Schrodinger equation. This process has much in common with thebiophysical model of Davydov.Over the past few years, the dynamic behavior of non-linear atomic chains has been the subject of consider-able research.' Prototype models employed in thesestudies are, for example, the Fermi-Pasta-Ulam an-harmonic lattice2and the integrable Toda lattice' withexponential atomic interactions. Unlike atomicchains, however, the collective behavior of optical non-linear arrays has so far received little attention. Suchone-dimensional arrays of coupled optical waveguidesare feasible today by exploiting recent advances inmicrofabrication technology. Moreover, these struc-tures can exhibit interesting nonlinear properties.3The well-known directional coupler,4which involvesonly two coupled waveguides, is clearly a special caseof this sort of system. The nonlinear version of thiscoupler was first discussed by Jensen5and is a poten-tial candidate for all optical-switching applications.3'6In a rather different context, recent theoretical stud-ies7,8have considered the transverse propagation(along the x direction of our Fig. 1) of optical waves innonlinear superlattices. For this case, Chen andMills7predict gap solitons with possible bistable prop-erties, and Sipe and Winful8 show that nonlinearSchr6dinger-type solitons can evolve in these periodicstructures.In this Letter we show that a coherent optical fieldpropagating along a nonlinear array (along the z axis ofFig. 1) obeys a nonlinear difference-differential equa-tion that resembles that of Davydov in biophysics.9-"The soliton model in a-spiral protein molecules pro-posed by Davydov more than a decade ago attemptedto explain some of the fundamental questions in bio-energetics, such as transfer, storage, and movement ofvibrational energy in polypeptides.10Our model,which can be considered the optical counterpart ofthat of Davydov, exhibits similar features, such asdiscrete self-focusing. As will be shown, this discreteprocess can be described in the continuum approxima-tion by a nonlinear Schr6dinger equation. The powerthreshold required for self-focusing to occur and theconditions necessary for spatial instabilities to takeplace are given in terms of relevant parameters.Consider an array of coupled optical waveguides asshown in Fig. 1. We assume that this array is losslessand infinite (big enough) and that it is comprised ofidentical, regularly spaced waveguides. The distancebetween successive waveguides is D. Furthermore, letus assume that the array is made from a nonlinearmaterial whose refractive index increases linearly withthe intensity of the optical field. By using the formal-ism of coupled-mode theory and by considering onlynearest-neighbor interactions, it can be shown that theelectric field propagating in the nth waveguide obeysthe following nonlinear difference-differential equa-tion3-5:idEn + #En+c(En+1 + En-l) + XIEnl2Endz+ ,(lEn+l 12+ IEn-112En = 0, (1)where ,3 is the field propagation constant of each wave-guide, c is the coupling coefficient, and X and 1A arepositive constants. The nonlinear term XIEnI2En de-scribes the self-phase modulation that takes place inthe nth waveguide, and the last term of Eq. (1) arisesfrom the nonlinear overlap of the adjacent modes.5As pointed out in previous studies,3'5the self-phase-modulation term dominates the nonlinear process,e.g., X >> gA, and hence from this point we set ,4 = 0 inEq. (1). Having done that, Eq. (1) reduces to theequation previously examined in connection with Da-vydov's model.',0"The nonlinear spatial dispersive properties of theFig. 1. A nonlinear array of coupled waveguides.0146-9592/88/090794-03$2.00/0 © 1988, Optical Society of America- -September 1988 / Vol. 13, No. 9 / OPTICS LETTERS 795array can be obtained from Eq. (1) by assigning to Enthe formE, = A exp[i(kzz - k~x,)], where x, = nD. Inthis case, the nonlinear dispersion equation readilyfollows and is given bykf = i + 2c cos(kD) + XIA12.(2)ci)I-zIt is interesting to note that the linear part of Eq. (2)resembles the E-k dispersion equation in the bandtheory of solids.'2The dispersive character of thisarray structure becomes evident when only one wave-guide is initially excited, say, for example, Eo = AO andE+n = 0 for n # 0 at z = 0. With these initial condi-tions, the linear part of Eq. (1) has the solution4En(z)= Ao(i)n exp(i/z)Jn(2cz), where Jn(x) is a Bessel func-tion of order n. This latter result shows that asymp-totically (z - c) the field tunnels away from the cen-tral waveguide and eventually equipartition of powertakes place. This diffusion or tunneling can bethought of as a sort of discrete one-dimensional dif-fraction.To study Eq. (1) (with g = 0) we write En(z) as En(z)= bn(z)exp[i(2c + #)zJ. Substituting this form intoEq. (1) we obtaini d n + c(QPn+l + ;n-l - 2cbn) + XIcIbnl42In = 0. (3)1.0 F-l0.8k0.6p0.4k-7 -6 -5 -4 -3 -200x/D(a)U.b! -I-zI-We can further simplify the problem by employing thetransformation45n(Z) = (pc/X)'1/2fn(z),(4)where fn(z) is a normalized dimensionless function,i.e., ,n IfnI2 = 1, and p is a dimensionless parameter ofthe problem. It follows directly from Eq. (3) that thesum En 4)nj12 = p(c/X) is conserved during propagation(power-conservation law), and therefore the quantityp(c/X) is associated with total power flowing down inthis array. From Eqs. (3) and (4) we find thati dfn.dz + c(fn~l + fA.1 - 2/n) + pcifAI2fn = 0. (5)0.6 [-0.4k0.21-IIIIIIIIIIIIIIIIIIIIII I1,-, a0-4 -3 -2 -I 0x/D(b)3 4 5 6 71 2 3 4Fig. 2. Intensity patterns of the self-focusing formation.The solid bars represent the quantities IfJI2 (discrete model),and the dashed curves represent the soliton envelopesech2(x/x0) as obtained from the continuum approximationfor the same value of p. (a) Self-focusing formation when p1.92; (b)


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