A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrodinger equation with the addition of a bulk diffraction term and an external  optical trap  is derived from first principles, i.e., Maxwell equations. When the nonlinearity is of the defocusing type, a family of unstaggered localized modes are numerically constructed. It is shown that the equation with an induced potential is well-posed and gives rise to localized dynamically stable nonlinear modes. The derived model is of the Gross-Pitaevskii type, a nonlinear Schrodinger equation with a linear optical potential, which also models Bose-Einstein condensates in a magnetic trap

Ablowitz, Mark J.

Julien, Keith

Musslimani, Ziad H.

Weinstein, Michael I.

English

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University of Central Florida STARS Faculty Bibliography 2000s Faculty Bibliography 1-1-2005 Wave dynamics in optically modulated waveguide arrays Mark J. Ablowitz Keith Julien Ziad H. Musslimani University of Central Florida Michael I. Weinstein Find similar works at: https://stars.library.ucf.edu/facultybib2000 University of Central Florida Libraries http://library.ucf.edu This Article is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for inclusion in Faculty Bibliography 2000s by an authorized administrator of STARS. For more information, please contact STARS@ucf.edu. Recommended Citation Ablowitz, Mark J.; Julien, Keith; Musslimani, Ziad H.; and Weinstein, Michael I., "Wave dynamics in optically modulated waveguide arrays" (2005). Faculty Bibliography 2000s. 4936. https://stars.library.ucf.edu/facultybib2000/4936 Wave dynamics in optically modulated waveguide arraysMark J. Ablowitz,1 Keith Julien,1 Ziad H. Musslimani,2 and Michael I. Weinstein31Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, Colorado 80309-0526, USA2Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA3Department of Applied Physics and Applied Mathematics, Columbia University, 200 S.W. Mudd - MC4701 New York,New York 10027, USAsReceived 20 January 2005; published 11 May 2005dA model describing wave propagation in optically modulated waveguide arrays is proposed. In the weaklyguided regime, a two-dimensional semidiscrete nonlinear Schrödinger equation with the addition of a bulkdiffraction term and an external “optical trap” is derived from first principles, i.e., Maxwell equations. Whenthe nonlinearity is of the defocusing type, a family of unstaggered localized modes are numerically con-structed. It is shown that the equation with an induced potential is well-posed and gives rise to localizeddynamically stable nonlinear modes. The derived model is of the Gross-Pitaevskii type, a nonlinearSchrödinger equation with a linear optical potential, which also models Bose-Einstein condensates in a mag-netic trap.DOI: 10.1103/PhysRevE.71.055602 PACS numberssd: 42.65.WiWave propagation in nonlinear periodic structures dis-plays unique phenomena that are absent in homogeneousmedia. The interplay between periodicity and nonlinearitycan lead to the formation of discrete or lattice solitons, whichwere predicted theoretically in the context of optical wave-guide arrays f1g and then experimentally observed in f2g.Until recently, discrete solitons were considered experimen-tally in one-dimensional geometry f2g. However, by makinguse of the photorefractive screening nonlinearity one can“write” either one- or higher-dimensional optical waveguidearrays by interfering pairs of plane waves f3g. Indeed, suchlocalized structures were experimentally observed in two-dimensional geometries f4g.In this paper we study wave propagation in opticallymodulated waveguide arrays, starting from the full time-harmonic three-dimensional Maxwell’s equations. For thecase where the periodic modulation along the y direction ismuch larger than the periodic modulation along the x direc-tion we derive, using multiscale asymptotic analysis, a semi-discrete nonlinear Schrödinger equation with the addition ofbulk diffraction term and an external “optical trap.” Whenthe nonlinearity is of the defocusing type swhere in the ab-sence of modulation no finite energy solitons are knowndunstaggered localized modes are numerically constructed.The fundamental properties such as the well-posedness ofthe equation, existence, and the dynamical stability associ-ated with a special class of localized wave solutions, i.e.,stationary wave, or ground state, are discussed. The semidis-crete model is derived from the scalar nonlinear Helmholtzequation. Below we briefly outline the justification for ne-glecting vectorial effects under certain physical assumptions.A more general and detailed study of scalar and vector semi-discrete nonlinear Schrödinger sNLSd type models will begiven elsewhere.We begin by considering the three-dimensional Maxwellequations governing time-harmonic solutions of frequencyv0„2E − = s= · Ed + k02sE + Pd = 0, k0 =v0c. s1dHere, ==]xiˆ+]yjˆ+]zkˆ , E=Esx ;v0d denotes the complex en-velope of the electric field, P=PsEsxd ;v0d denotes the po-larization field, containing both linear and nonlinear re-sponses; we further assume the nonlinear polarization to beof Kerr type f5g where the second component of the polar-ization is given byP2 = xE2 + dssuE1u2 + s1 + gduE2u2 + uE3u2dE2 + gsE12 + E32dE2*d;s2dwhere g is a constant, d is proportional to the nonlinear indexchange of refraction, and x is a function of x and y; the otherpolarization components are found by cyclically changingthe indices s1→2→3→1d. We consider propagation in the zdirection through a photonic structure sinvariant in zd havingnontrivial spatial variations in the sx ,yd plane due to x. Aschematic of the kind of transverse structure we consider isgiven in Fig. 1. This structure has a rapid periodic variationin x and a slow modulation in y. In nondimensional terms,this corresponds to the assumed form x=xsx ,«1/2yd, where «is a small dimensionless parameter. The period in x is oforder 1 whereas a typical distance in y is of order «−1/2. Wefurther assume that the nondimensional nonlinear index ofrefraction is small in size fOs«dg. Then, analysis ofMaxwell’s equations s1d shows that E3 /E2=Os«d andE1 /E2=Os«3/2d and to leading order]zE3 = − ]yE2. s3dThen the second component of Maxwell’s equations s1dleads to the following nonlinear Helmholtz equation„2C + f2sx,ydC + «huCu2C = 0, s4dwhere C is the envelope wave function, which is propor-tional to the optical field E2 s„2=]x2+]y2+]z2d. f2sx ,yd=1+xis the linear refractive index of the waveguide structure, h isproportional to d and sgnh= +1 and sgnh=−1 correspond to,respectively, the cases of self-focusing and self-defocusingPHYSICAL REVIEW E 71, 055602sRd s2005dRAPID COMMUNICATIONS1539-3755/2005/71s5d/055602s4d/$23.00 ©2005 The American Physical Society055602-1nonlinearity. We assume that the linear refractive index ap-pearing in s4d takes the formf2sx,yd = F2sxd + « aFp2sx,«1/2yd , s5dwhere a= ±1. F2sxd is the refractive index of a grating struc-ture in the x longitudinal coordinate, which may be viewedas a superposition of spatial translates of a basic waveguidewith index profile F02sxd, which we assume to be singlemoded. Thus, F2sxd,omFm2 sxd, Fmsxd=F0sx−mDd. Here,«Fp2sx ,«1/2yd is a weak modulation of refractive index sslowin y and fast in xd. One can create a photonic structure of thistype by illuminating a photorefractive crystal with a pair ofinterfering one-dimensional plane waves weakly modulatedalong the y direction with a wavelength larger than the wave-length along the x direction. We now analyze wave propaga-tion in a nonlinear optical two-dimensional array and discussthe physical phenomena that result.We exploit the weak nonlinearity ssmall «d in s4d and s5dto construct a multiple scale expansion of the envelope,C. We seek C as a superposition of x− translates of theisolated single mode wave function with slowly varying am-plitudes f6gC < om=−‘+‘AmsZ,Ydcmsxdeimz, s6dwhere Z=«z and Y =«1/2y are slow propagation and modula-tion scales, respectively. Here, csxd is the single waveguidemode and m its corresponding eigenvalued2cdx2+ sF02sxd − m2dc = 0, s7dand cmsxd=csx−mDd. Substituting the expansion s6d intothe Helmholtz equation s4d and making use of s7d, we findthat the corrections to s6d are small provided the projectionsofom=−‘+‘ F2i«mcm]Am]Z + cmF2sxd − Fm2 sxdAm + «cm]2Am]Y2+ a«Fp2sx,YdcmAm + «h om8,m9=−‘+‘cmcm8cm9* AmAm8Am9* G= 0, s8donto all c j are of order «d ,d.1. This yields ssee also f6gdi]An]Z+ CsAn+1 + An−1d + g]2An]Y2+ aVnsYdAn + kuAnu2An = 0,s9dwhere «C=1/c0e (F2sxd−Fn±12 sxd)cn±1cn*dx, VnsYd=1/c0eFp2sx ,Yducnu2dx, k=h /c0e ucnu4dx, and c0=2me ucnu2dx; g=1/ s2md. Note that we are considering theregime where only nearest neighbor waveguides contributeto order «. Equation s9d governs the slow evolution of An ina weakly modulated optically induced waveguide array. Nextwe examine linear propagation and then highlight somephysical nonlinear phenomena that are predicted by themodel s9d. For the ideal one-dimensional waveguide arrayfg=k=VnsYd;0g, the propagating field experiences discretediffraction due to optical tunneling to adjacent sites and ex-hibits a typical discrete diffraction pattern with the intensitymainly concentrated in the outer lobes f7g. However, in thepresence of modulation sgÞ0 and VnÞ0d, and in the quasi-two-dimensional configuration swhen modulating along the ydirectiond, the waveguide action prevents the beam from dif-fracting. It should also be noted that a similar derivation inthe case when two fields are initially present, i.e., nontrivialE1 ,E2 leads to a vector system whose first componentsatisfies,i]Ans1d]Z+ CsAn+1s1d + An−1s1d d + g]2Ans1d]Y2+ aVnsYdAns1d+ kfs1 + gduAns1du2 + uAns2du2gAns1d + gAns1d2Ans2d*= 0,s10dand the second equation is obtained by cyclically changingthe indices s1→2→1d. In a future publication we will givethe derivation in detail. We now discuss the results obtainedfor the model s9d which are depicted in Figs. 2–4. First, inboth self-focusing and self-defocusing cases, propagatingbeams of any finite power do not collapse or filament. This isin contrast to the continuum analog, the two-dimensionalcubic-focusing NLS equation, whose solutions with suffi-cient initial power are well known to develop singularities ata finite distance into a bulk Kerr propagation medium f8g.That the semidiscrete character inhibits collapse, ssee Fig. 4d,can be understood by an argument based on the conservedintegrals of s9dFIG. 1. A typical modulated waveguide array.ABLOWITZ et al. PHYSICAL REVIEW E 71, 055602sRd s2005dRAPID COMMUNICATIONS055602-2N = onE uAnu2dYH = ConE uAn+1 − Anu2 + u]YAnu2 − aVnuAnu2 − k2 uAnu4dY .That no singularity can form as the wave form propagatesthrough increasing Z is a direct consequence of the Z− inde-pendence of N and H and the semidiscrete Sobelev-Gagliardo-Nirenberg inequality sdSNGd for functions fnsYddefined on the integer lattice times the real continuumf9–11g: Let fW= (fnsYd), tfW= (fn+1sYd) and ifWipp=one ufnupdY.Then, there is a universal constant C*.0 such that for all fWifWi44 ł C*i]Y fWi2ifW − tfWi2ifWi22. s11dIn particular, the Z independence of N and H and the in-equality s11d together imply upper bounds on i]Y fWsZdi2 andufWsZ ,Ydu in terms of H and N swhich are independent of Zand Yd. These bounds break down when passing to thecontinuum limit. It should be remarked that in the de-focusing case, gk,0, when VnsYd;0, there are no localizednonlinear modes. In this case, a finite energy concentrationdiffractively spreads and attenuates in amplitude. However,the optical trapping potential, VnsYd, introduces the possibil-ity of stable bright solitonlike states. Indeed, the existenceand importance of such nonlinear defect modes has beenstudied in the context of a plasma model f12g and in thetrapping of nonlinear pulses in fiber gratings with localizeddefects f13g. Such nonlinear defect modes are solutions toEq. s9d of the form An=Bne−inZ and satisfynBn + CsBn+1 + Bn−1d + g]2Bn]Y2− VnsYdBn + kuBnu2Bn = 0,s12dwhere n.0 is the propagation constant. In the above, wetook a=−1 and h=−1. The existence and stability of thesestates follows from their variational characterization as localminima of the energy functional H subject to fixed totalpower, N. Here too, the inequality s11d plays a role in that itimplies the boundedness from below of the constrained en-ergy. For simplicity, in our numerics we consider a perturbedindex, which is locally parabolic swhich can be induced by aweak sinusoidal refractive indexd, Fp2sx ,Yd= a˜x2+b˜Y2 lead-ing to the induced potentialVnsYd = an2 + bY2. s13dTo numerically construct the bound states solutions to Eq.s12d we first define the Fourier transform F and its inverseF−1Bˆ sk,qd = FfBnsYdg = on=−‘+‘ E−‘+‘BnsYde−isqn+kYddY , s14dBnsYd = F−1fBˆ sk,qdg =1s2pd2E−‘+‘ E−p+pBˆ sk,qde+isqn+kYddkdq .s15dThe idea of the method ssee also Ref. f14gd is to make thechange of variables: BnsYd=uQnsYd, Bˆ sk ,qd=uQˆ sk ,qd,where uÞ0 is a constant to be determined from a consis-tency condition. Taking the Fourier transform on Eq. s12dand using the above change of variables we getFIG. 3. Localized semidiscrete, two-dimensional soliton solu-tion to Eq. s12d in the presence of a one-dimensional “discrete trap”sa=1/2; b=0.1d for the defocusing nonlinearity. The parametersare C=2, n=4, g=1, and k=−1.FIG. 4. Localized semidiscrete, two-dimensional soliton solu-tion to Eq. s12d without a trap fVnsYd;0g for the focusing nonlin-earity. The parameters are C=2, n=1, g=1, and k= +1.FIG. 2. Localized semidiscrete, two-dimensional soliton solu-tion to Eq. s12d in the presence of a semidiscrete two-dimensionaltrap sa=b=1/2d for the defocusing nonlinearity. The parametersare C=2, n=4, g=1, and k=−1.WAVE DYNAMICS IN OPTICALLY MODULATED… PHYSICAL REVIEW E 71, 055602sRd s2005dRAPID COMMUNICATIONS055602-3Vsk,qdQˆ sk,qd − FfVnsYdQnsYdg = − u2FfkuQnu2Qng ,s16dwhere Vsk ,qd=n+2C cossqd−gk2. Multiplying Eq. s16d byQˆ *sk ,qd and integrating over the sk ,qd space, we findu2 =−E Qˆ *sk,qdhVsk,qdQˆ sk,qd − FfVnsYdQnsYdgjdkdqE Qˆ *sk,qdFfkuQnu2Qngdkdq.s17dSince Vsk ,qd vanishes for n=−2C cossqd+gk2, we add andsubtract sr+2CdQˆ sk ,qd in Eq. s16d where r is an arbitrarypositive number. Then the iteration will take the followingform:Qˆ sm+1d = r + n + 2Cr + 2Cf1 − cossqdg + gk2Qˆ smdsk,qd−FfVnsYdQnsmdsYdg − susmdd2FfkuQnsmdu2Qnsmdgr + 2Cf1 − cossqdg + gk2,s18dwhere usmd is defined by the right-hand side of s17d with Qset equal to Qsmd. Typical examples of self-localized beamsare shown in Figs. 2–4. In Fig. 2 we have a trap in both nand Y and the mode is localized equally in both directions. InFig. 3, we depict a trap with a=1/2 and b=0.1, which ismuch longer in the Y direction than the discrete n. We alsonote that when the trap is only a function of n sb=0d, thecorresponding mode is only localized in the n direction;similarly it turns out that when the trap is localized in the ydirection se.g., a=0d then the mode is only localized in the Ydirection. Finally we find that when the trap is “turned off”Vn=0—then we find a new localized mode in the focusingnonlinear case when gk=1 ssee Fig. 4d.In conclusion, a model describing wave propagation inoptically modulated waveguide arrays is derived from Max-well’s equations. In the weakly guided regime, a discretenonlinear Schrödinger equation with the addition of a bulkdiffraction term and an external “optical trapping potential”is derived. In the defocusing regime, where in the absence ofmodulation no finite energy solitons are known, the inducedoptical trap prevents the beam from defocusing, resulting ina stable unstaggered localized mode. These results also es-tablish a connection to the modeling of Bose-Einstein con-densation where discrete optical lattices with a potential in-duced by a magnetic trap have been studied scf. f15gd.M.J.A. was partially supported by the Air Force Office ofScientific Research under Grant No. F-49620-03-1-0250 andby the NSF under Grant No. DMS-0303756 M.I.W. was par-tially supported by the NSF under Grant No. DMS-0412305.K.J. acknowledges partial support from the University ofColorado.f1g D. N. Christodoulides and R. J. Joseph, Opt. Lett. 13, 794s1988d.f2g H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J.Aitchison, Phys. Rev. Lett. 81, 3383 s1998d.f3g N. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, andM. Segev, Phys. Rev. E 66, 046602 s2002d.f4g J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoul-ides, Nature sLondond 422, 147 s2003d.f5g R. W. Boyd, Nonlinear Optics, 2nd ed. sAcademic Press,SanDiego, CA, 2003d.f6g M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276s2003d.f7g A. Yariv, Optical Electronics in Modern Communications sOx-ford University Press, Oxford, 1997d.f8g P. L. Kelley, Phys. Rev. Lett. 15, 1005 s1965d.f9g M. I. Weinstein and B. Yeary, Phys. Lett. A 222, 157 s1996d.f10g M. I. Weinstein, Nonlinearity 12, 673 s1999d.f11g B. Yeary, Ph.D. thesis, University of Michigan, 1996 sunpub-lishedd.f12g H. A. Rose and M. I. Weinstein, Physica D 30, 207 s1988d.f13g R. H. Goodman, R. E. Slusher, and M. I. Weinstein, J. Opt.Soc. Am. B 19, 1635 s2002d.f14g M. J. Ablowitz and Z. H. Musslimani, Opt. Lett.sto be pub-lishedd.f15g P. Meystre, Atom Optics sSpringer, Berlin, 2001d.ABLOWITZ et al. PHYSICAL REVIEW E 71, 055602sRd s2005dRAPID COMMUNICATIONS055602-4

Wave dynamics in optically modulated waveguide arrays

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Wave dynamics in optically modulated waveguide arrays

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