VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17DECEMBER 2001

Discrete Diffraction Managed Spatial Solitons

Mark J. Ablowitz and Ziad H. Musslimani Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, Colorado 80309-0526 (Received 25 June 2001; published 30 November 2001) Motivated by recent experimental observations of anomalous diffraction in linear waveguide array, we propose a novel model governing the propagation of an optical beam in a diffraction managed nonlinear waveguide array. This model supports discrete spatial solitons whose beamwidth and peak amplitude evolve periodically. A nonlocal integral equation governing the slow evolution of the soliton amplitude is derived and its stationary soliton solutions are obtained.

DOI: 10.1103/PhysRevLett.87.254102 PACS numbers: 05.45.Yv

Discrete spatial solitons in nonlinear media have at- Kerr media in the presence of both normal and anomalous tracted considerable attention in the scientific community diffraction. Importantly, even though optical diffraction for many years [1]. They have been demonstrated to exist and chromatic dispersion originate from different physics in a wide range of physical systems such as atomic chains (the former being geometric and the latter being media de- [2,3], molecular crystals [4], biophysical systems [5], elec- pendent), nevertheless, discrete diffraction spatial solitons trical lattices [6], and recently in arrays of coupled nonlin- and dispersion managed solitons share many properties ear optical waveguides [7,8]. Such discrete excitations can which highlight the universality and diversity of solitons. form when discrete diffraction is balanced by nonlinearity. A nonlocal integral equation governing the slow evolution In an optical waveguide array this can be achieved by using of the beam amplitude is derived, and its stationary soli- an intense laser beam which locally changes the nonlinear ton solutions are obtained. Such averaged beams lose peak refractive index of the waveguides via the Kerr effect and power during propagation, hence, since the overall power in turn decouples them from the remaining waveguides. is preserved, they become wider; at the end point of the After almost a decade since their first theoretical pre- diffraction map, a full recovery of the initial beam peak diction [9], discrete solitons in an optical waveguide array power occurs. The competition between varying diffrac- were experimentally observed [7]. When a low intensity tion and self-focusing nonlinearity offers many new excit- beam is injected into one waveguide, the propagating field ing physical possibilities. spreads over the adjacent waveguides, hence, experiencing We begin our analysis by considering an infinite array of discrete diffraction. However, at sufficiently high power, weakly coupled optical waveguides with equal separation the beam self-trap (to form a localized state) in the center d. The equation which governs the evolution of the elec- waveguides. Soon thereafter, many fascinating properties tric field En, according to nonlinear coupled mode theory of discrete solitons were reported: for example, the ex- [7,9,16], is given by perimental observation of linear and nonlinear Bloch os- ≠En 2 cillations in AlGaAs waveguides [10]; temperature tuned ෇ iC͑En11 1 En21͒ 1 ikwEn 1 ikjEnj En , (1) polymer waveguides [11], and in an array of curved op- ≠z tical waveguides [12]. Moreover, the dynamical behavior where C is the coupling constant between adjacent wave- of discrete solitons was also experimentally realized [8]. guides which is given by an overlap integral of the two Discrete solitons have unique properties that are absent in modes of such waveguides; k is a constant describing the their bulk counterparts [13]. The most noticeable one is nonlinear index change, z is the propagation distance, and the possibility of designing the diffraction properties of a kw is the propagation constant of the waveguides. To fa- linear waveguide array to produce anomalous diffraction cilitate understanding, we first recall basic properties of [14]. As a result, self-focusing and defocusing processes discrete diffraction of a linear array. When a cw mode of can be achieved in the same medium (structure) and wave- the form En͑z͒ ෇ A exp͓i͑kzz 2 nkx d͔͒ is inserted into length that leads to the possibility of observing dark soli- the linearized version of Eq. (1) it yields tons in self-focusing Kerr media [15]. k ෇ k 1 2C cos͑k d͒ . (2) In this Letter, inspired by the recent experimental ob- z w x servations of anomalous diffraction in a linear waveguide In close analogy to the definition of dispersion, discrete 00 ෇ 2 ͑ ͒ array, we propose a novel model governing the propaga- diffraction is given by kz 22Cd cos kx d . As first tion of an optical beam in a diffraction managed nonlin- pointed out in [14], an important feature of the last 00 ear waveguide array. Using a cascade of tilted segments relation is that kz can change sign depending on kx . of an array of waveguides with positive average diffrac- Indeed, the diffraction becomes positive in the range tion, we develop a nonlinear discrete model that describes p͞2 , jkxdj #p; hence, a light beam can experience the evolution of an optical field in self-focusing nonlinear anomalous diffraction. In practice, the sign and value of

254102-1 0031-9007͞01͞87(25)͞254102(4)$15.00 © 2001 The American Physical Society 254102-1 VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17DECEMBER 2001 the diffraction can be controlled and manipulated by (a) launching light at a particular angle or equivalently by ∆(ζ) (b) ∆ tilting the waveguide array. This in turn allows the 1 possibility of achieving a “self-defocusing” (with positive Kerr coefficient) regime which leads to the formation 1−θ 2 of discrete dark solitons [15]. Similar to [14], we use 2L θ 2 ζ a cascade of different segments of waveguide, each ∆ piece being tilted by an angle zero, and g, respectively. 2 The actual physical angle g (the waveguide tilt angle) is related to the wave number k by the relation [14] FIG. 1. Schematic presentation of the waveguide array (a) and x of the diffraction map (b). sing ෇ kx͞k, where k ෇ 2pn0͞l0 (l0 ෇ 1.53 mm is the central wavelength in vacuum and n0 ෇ 3.3 is the ͑ ͑0͒͒ ෇ ͑ ͑1͒͒ ෇ linear refractive index). In this way, we generate a wave- J fn 0, J fn 2Fn , (6) guide array with alternatingp diffraction. To model such a i͑k 12C͒z 0 where ,ءf e w and z ෇ z͞z ءsystem, we define E ෇ P n n ≠A D͑z͒ n ء ء with P and z being the characteristic power and nonlin- J ͑An͒ϵi 1 ͑An11 1 An21 2 2An͒ , ear length scale, respectively. Substituting these quantities ≠z 2h2 into Eq. (1) yields the following (dropping the prime) ≠f͑0͒ n da ͑0͒ ͑0͒ ͑0͒ diffraction-managed discrete nonlinear Schrödinger Fn ෇ i 1 ͑fn1 1fn2 2 2f ͒ ≠Z 2h2 1 1 n (DM-DNLS) equation: 1 jf͑0͒j2f͑0͒. df D͑z͞z ͒ n n n ෇ w ͑ ͒ j j2 i 2 fn11 1fn21 2 2fn 1 i fn fn , dz 2h To solve at order 1͞zw [see Eq. (6)], we introduce the (3) discrete Fourier transform, X1` 2 ˆ ͑ ͒ ෇ ͑0͒͑ ͒ 2iqnh , C cos͑kx d͒ ෇ D͑z͞zw͒͑͞2h ͒, f0 q, z, Z fn z, Z eءand z ͒ء෇ 1͑͞kP ءwith z where D͑z͞z ͒ is a piecewise constant periodic func- n෇2` w Z (7) tion that measures the local value of diffraction. Here, h p͞h f͑0͒͑z Z͒ ෇ fˆ ͑q z Z͒eiqnh dq . , , n , 0 ء ͞ ϵ zw 2L z , with L being the actual length of each 2p 2p͞h waveguide segment [see Fig. 1(a) for schematic represen- tation]. Equation (3) describes the dynamical evolution of The solution is therefore given in the Fourier representation a laser beam in a Kerr medium with varying diffraction. by When the “effective” nonlinearity balances the average fˆ 0͑q, z, Z͒ ෇ cˆ ͑Z, q͒ exp͓2iV͑q͒C͑z͔͒ , (8) diffraction, then bright discrete solitons can form. R The coupling constants that correspond to the experi- ͑ ͒ ෇ ͓ ͑ ͔͒͞ 2 ͑ ͒ ෇ z ͑ 0͒ 0 with V q 1 2 cos qh h and C z 0 D z dz . mental data reported in [14] (for 2.5 mm waveguide sepa- The amplitude cˆ ͑Z, q͒ is an arbitrary function whose ෇ 21 ration and width) are found to be C 2.27 mm [17], dynamical evolution will be determined by a secularity ෇ 21 21 ഠ W and condition associated with Eq. (6). In other words, the 300 ءk 3.6 m W . For typical power P ഠ 1 mm and ءwaveguide length L ഠ 100 mm, we find z condition of the orthogonality of Fn to all eigenfunctions zw ഠ 0.2. Hence, it is natural to construct an asymptotic Fn of the adjoint linear problem which, when written in theory based on small zw. It is in this parameter regime the Fourier domain, takes the form that diffraction managed spatial solitons can be experimen- Z 1 tally observed. To this end, we consider the case in which dz Fˆ ͑q, z, Z͒Fˆ ͑q, z, Z͒ ෇ 0, (9) the diffraction takes the form 0 where Fˆ , Fˆ are the Fourier transform of Fn, Fn, respec- 1 y y D͑z͞zw͒ ෇ da 1 D͑z͞zw͒ , (4) tively. Here, Jˆ Fˆ ෇ 0 with Jˆ being the adjoint operator z w to Jˆ ϵ id͞dz2D͑z ͒V͑q͒. Substituting (8) into Fˆ and where da is the average diffraction (taken to be positive) performing the integration in condition (9) yields the fol- and D͑z͞zw͒ is a periodic function. Since in this case lowing nonlinear evolution equation for cˆ ͑Z, q͒: Eq. (3) contains both slowly and rapidly varying terms, ͑ ͒ ෇ ͞ dcˆ Z, q we introduce new fast and slow scales as z z zw and i ෇ daV͑q͒cˆ ͑Z, q͒ 2 R͓cˆ ͑Z, q͔͒ , Z ෇ z, respectively, and expand fn in powers of zw: dZ Z f ෇ f͑0͒͑z, Z͒ 1 z f͑1͒͑z, Z͒ 1 O͑z2 ͒ . (5) n n w n w R ϵ dq K͑q, q1, q2͒cˆ ͑q1͒cˆ ͑q2͒ (10)

ء Substituting Eqs. (5) and (4) into Eq. (3), we find that 3 cˆ ͑q1 1 q2 2 q͒ , the leading order in 1͞zw and the order 1 equations are, respectively, given by where dq ϵ dq1dq2 and the kernel K is defined by 254102-2 254102-2 VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17DECEMBER 2001

Z h2 1 1.2 1 K͑ ͒ ෇ ͓ ͑ ͒ ͑ ͔͒ (a) (b) q, q1, q2 2 dz exp iC z x q, q1, q2 , 2 2

4p 0 | | n n φ φ

µ ∂ µ ∂ | | Y2 ͑ ͒ h͑q 2 q͒ 0.6 0.5 ෇ 4 h q1 1 q2 j x 2 cos sin . h 2 j෇1 2 Equation (10) governs the evolution in Fourier space of 0 −0 − an optical beam in a coupled nonlinear waveguide array −10 0 n 10 10 5 0 n 5 10 in the regime of strong diffraction. In the special case of FIG. 2. Mode profile in physical space obtained from Eq. (11) the two-step diffraction map shown in Fig. 1(b), i.e., when (solid line) and from the average method (dashed line). Parame- two waveguide segments are tilted by angle zero and g ters are as follows: (a) vs ෇ 1, h ෇ 1, s ෇ 0.17, D1 ෇ 2D2 ෇ alternatively, we have D͑z ͒ ෇ D1 for 0 # jzj ,u͞2 and 0.681, da ෇ 1.135, and zw ෇ 0.2;(b)vs ෇ 1, h ෇ 0.5, s ෇ 1, ෇ ෇ ෇ ෇ D2 in the region u͞2 , jzj , 1͞2, where u is the fraction D1 2D2 4, da 1, and zw 0.2. K of the map with diffraction D1. In this case, the kernel 0 0 2 2 ͑0͒ ෇ ͑0͒ ͑ ͒ 00 takes the simple form K ෇ h sin͑sx͒͑͞4p sx͒ with s ෇ fn jfn qj exp iQn which has power E0 . Then 00 ͓uD1 2 ͑1 2u͒D2͔͞4. Importantly, these parameters can ͑1͒ ෇ ͑0͒ ͞ 00 fn fn E0 E0 is the new guess and, in general, be related to the experiments reported in [14]. To achieve the mth iteration takes the form a waveguide configuration with alternate diffraction, we q X` ෇ ͞ ͑ ෇ ͒ ͑ ͒ ͑ ͒00 ͑ ͒00 use two values of kxd 0 and 2p 3 h 1 which cor- m11 ෇ E ͞E 00 m E 00 ෇ j m j2 ± fn 0 m fn , m fn . responds to waveguide tilt angles g ෇ 0 and 3.43 .In n෇2` this case we find, for u ෇ 0.5, D1 ෇ 2D2 ෇ 0.681, da ෇ (12) 1.135, and s ෇ 0.17. Different sets of parameters with a smaller angle g are also realizable. Next, we look for a In Fig. 2 the mode profiles associated with a stationary so- stationary solution for Eq. (10) (for the particular kernel lution are depicted for two typical parameter values. The given above) in the form cˆ ͑Z, q͒ ෇ cˆ s͑q͒ exp͑ivsZ͒. In- profiles are obtained by using both the integral equation ap- serting this ansatz into (10) leads to proach as well as the averaged method. The evolution of 1 these discrete diffraction managed solitons are illustrated cˆ ͑q͒ ෇ R͓cˆ ͑q͔͒ ϵ M͓cˆ ͑q͔͒ , (11) s d V͑q͒ 1v s s in Figs. 3 and 4 for the same two sets of parameter values a s as in Fig. 2, respectively. We note that initially the beam ˆ ͑ ͒ which implies that the mode cs q is a fixed point of the has zero chirp [23]. During propagation, a chirp develops nonlinear functional M. To numerically find the fixed and the peak amplitude of the beam begins to decrease and, point, we employ a modified Neumann iteration scheme as a result, the beam becomes wider (due to conservation [18–20] and write√ Eq. (11)! in the form of power). A full recovery of the soliton’s initial ampli- 3͞2 a͑cˆ ͑m͒͒ tude and width is achieved at the end of the map period. cˆ ͑m11͒͑q͒ ෇ s M͓cˆ ͑m͒͑q͔͒, m $ 0, s ͑ ˆ ͑m͒͒ s This breathing behavior is shown in Figs. 3 and 4 for both b cs Z strongly and moderately confined beams, respectively. ͑ ͒ To establish the relation between the two approaches and a ෇ jcˆ m ͑q͒j2 dq ; s to highlight the periodic nature of these new solitons, we Z calculate the nonlinear chirp by both the integral equation ෇ ˆ ͑m͒͑ ͒ ͓ ˆ ͑m͒͑ ͔͒ b cs q M cs q dq . approach and the averaging method. It is clear that for small values of the map period, zw, the asymptotic analy- The factors a and b are introduced to stabilize an sis is in good agreement with the averaging method, as otherwise divergent Neumann iteration scheme. This shown in Fig. 5. We also mention briefly that the method method is used to find stationary soliton solutions to of analysis associated with Eq. (3) can be modified to ac- the integral equation (10) which in turn provides an count for situations where the average diffraction is small, asymptotic description of the diffraction-managed DNLS Eq. (3). It should be also noted that we can obtain periodic diffraction-compensated soliton solutions directly from Eq. (3). The technique is similar to that originally pro- posed for finding periodic dispersion-managed solitons in communications problems [21,22]. The averaging proce- dure does not require that the map period, zw , be small. To implement the method, initially we startP with a guess, for ͑0͒ ෇ ͑ ͒ ෇ ` 2͑ ͒ instance fn sech nh with E0 n෇2` sech nh . ͑0͒0 Over one period, this initial ansatz will evolve to fn FIG. 3. Beam propagation over one period using Eq. (8) as which in general will have a chirp [23]. We then de- the initial condition obtained by a direct numerical simulation ͑ ͒00 ͑ ͒ ͑ ͒0 0 ෇ ͑ 0 0 2iQn ͒͞ fine an average: fn fn 1fn e 2, where of Eq. (3). Parameters are the same as used in Fig. 2(a). 254102-3 254102-3 VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17DECEMBER 2001

M. J. A. is partially supported by the Air Force Office of Scientific research, under Grant No. F49620-00-1-0031, and NSF under Grant No. DMS-0070792. Z. H. M. was supported by the Rothschild Foundation.

FIG. 4. Beam propagation over one period (a) and station- [1] D. Henning and G. P. Tsironis, Phys. Rep. 307, 333 (1999); ary evolution (b) obtained by a direct numerical simulation of O. M. Braun and Y.S. Kivshar, Phys. Rep. 306, 1 (1998); Eq. (3) evaluated at end map period. Parameters are the same S. Flach and C. R. Willis, Phys. Rep. 295, 181 (1998); as used in Fig. 2(b). F. Lederer and J. S. Aitchison, Les Houches Workshop on Optical Solitons, edited by V. E. Zakharov and S. Wabnitz (Springer-Verlag, Berlin, 1999). ෇ i.e., da ø 1. In such a situation we writep da eDa, [2] A. C. Scott and L. Macneil, Phys. Lett. 98A, 87 (1983). D ෇ eDa 1D͑z͒, z ෇ j͞e, and fn ෇ eCn. Then it [3] A. J. Sievers and S. Takeno, Phys. Rev. Lett. 61, 970 is found that Cn satisfies (1988). [4] W. P. Su, J. R. Schieffer, and A. J. Heeger, Phys. Rev. Lett. dC D ͑j͞e͒ 42, 1698 (1979). n ෇ i ͑C 1C 2 C ͒ [5] A. S. Davydov, J. Theor. Biol. 38, 559 (1973). 2 n11 n21 2 n dj 2h [6] P. Marquii, J. M. Bilbaut, and M. Remoissenet, Phys. Rev. 2 1 ijCnj Cn , (13) E 51, 6127 (1995). [7] H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, Phys. Rev. Lett. 81, 3383 (1998). ͑ ͞ ͒ ෇ 1 ͑ ͞ ͒ [8] R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and where D j e Da 1 e D j e . The model (13) is valid in parameter regimes which applies to a physical situ- Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999). ation where the average diffraction is small. [9] D. N. Christodoulides and R. J. Joseph, Opt. Lett. 13, 794 (1988). In conclusion, we have developed a new discrete equa- [10] R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and tion governing the evolution of an optical beam in a wave- Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999). guide array with varying diffraction. This equation has [11] T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and a novel type of discrete spatial soliton solution which F. Lederer, Phys. Rev. Lett. 83, 4752 (1999). breathes under propagation and, as a result, gains a non- [12] G. Lenz, I. Talanina, and C. Martijn de Sterke, Phys. Rev. linear chirp. A full recovery of the soliton initial power is Lett. 83, 963 (1999). achieved at the end of each diffraction map. This opens [13] A. B. Aceves, C. De Angelis, S. Trillo, and S. Wabnitz, the possibility of fabricating a customized waveguide ar- Opt. Lett. 19, 332 (1994). ray which admits specialized diffraction-managed spatially [14] H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchi- confined solitary waves. A nonlocal integral equation gov- son, Phys. Rev. Lett. 85, 1863 (2000). erning the slow evolution of the soliton amplitude is de- [15] R. Morandotti, H. Eisenberg, Y. Silberberg, M. Sorel, and J. Aitchison, Phys. Rev. Lett. 86, 3296 (2001). rived and its stationary solutions are obtained. [16] S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, Appl. Phys. Lett. 22, 46 (1973). [17] A. Yariv, Optical Electronics (Saunders, Philadelphia, 1991). (a) (b) 0.1 0.1 [18] V.I. Petviashvili, Sov. J. Plasma Phys. 2, 257 (1976). [19] M. J. Ablowitz and G. Biondini, Opt. Lett. 23, 1668 (1998). [20] M. J. Ablowitz, Z. H. Musslimani, and G. Biondini (to be 0 0 published). chirp

chirp [21] I. R. Gabitov and S. K. Turitsyn, Opt. Lett. 21, 327 (1996);

−0.1 −0.1 S. K. Turitsyn, Sov. Phys. JETP Lett. 65, 845 (1997). [22] J. H. B. Nijhof, W. Forysiak, and N. J. Doran, IEEE J. Sel. Top. Quantum Electron. 6, 330 (2000). −0.2 −0.2 [23] In close analogy to the definition of a chirp in disper- 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 peak amplitude peak amplitude sion managed solitons, here, the chirp of a discrete beam fn͑z͒ ෇ jfn͑z͒j exp͓iQn͑z͔͒ is defined to be the coeffi- FIG. 5. Periodic evolution of the beam chirp versus maximum 2 ͑ ͒ peak power. Solid line represents the leading order approxi- cient of n in the expansion of Qn z in a “Taylor se- ෇ ͑ ͒ഠ ͑ ͒ mation, i.e., Eq. (8); dashed line represents numerical solution ries” around n 0. In other words, if Qn z c0 z 1 ͑ ͒ ͑ ͒ 2 ͑ ͒ of the DM-DNLS, and dotted line depicts the evolution of the c1 z n 1 c2 z n 1 … , then the chirp is given by c2 z . leading order solution under Eq. (3). Parameters are the same This definition is valid for moderately localized solitons as used in Fig. 2(b) with zw ෇ 0.2 (a) and 0.1 (b). and breaks down for strongly localized beams.

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