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Discrete Diffraction Managed Spatial Solitons

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Discrete Diffraction Managed Spatial Solitons 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 ͒ .
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