Solitons in the model of nonlinear photonic crystals with the transverse structure based on two-dimensional (2D) quadratic- or rhombic-shaped Kronig-Penney (KP) lattices are studied by means of numerical methods. The model can also applies to a Bose-Einstein condensate (BEC) trapped in a superposition of linear and nonlinear 2D periodic potentials. The analysis is chiefly presented for the self-repulsive nonlinearity, which gives rise to several species of stable fundamental gap solitons, dipoles, four-peak complexes, and vortices in two finite bandgaps of the underlying spectrum. Stable solitons with complex shapes are found, in particular, in the second bandgap of the KP lattice with the rhombic structure. The stability of the localized modes is analyzed in terms of eigenvalues of small perturbations, and tested in direct simulations. Depending on the value of the KP’s duty cycle (DC, i.e., the ratio of the void’s width to the lattice period), an internal stability boundary for the solitons and vortices may exist inside of the first bandgap. Otherwise, the families of the localized modes are entirely stable or unstable in the bandgaps. With the self-attractive nonlinearity, only unstable solitons and vortices are found in the semi-infinite gap.
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1. Introduction and the model
In this work, we consider the basic model of periodically patterned photonic media with the two-dimensional (2D) transverse structure, which are uniform in the longitudinal direction, z. These media represent quasi-2D photonic crystals [1, 2], which are tantamount, as a matter of fact, to photonic-crystal fibers (PCFs) [3–11]. The transverse structures that we deal with can be fabricated in the form of a host medium perforated by a system of parallel square-shaped voids, which are filled with a different material, featuring a nearly matched refractive index, n 0, simultaneously with a strong contrast in the Kerr coefficient, n 2. Two different types of such 2D structures, that we will call square-shaped and rhombic ones, respectively, are displayed in Figs. 1 and 2. Both types may be considered as 2D versions of the Kronig-Penney (KP) lattice, diverse forms of which were investigated in photonics in the 1D geometry, in terms of the nonlinear transmission [14–17] and spatial solitons alike [18–24]. 1D linear and nonlinear potentials of the KP type where also investigated in models of Bose-Einstein condensates (BECs) [25, 26]. The 2D KP structures may be realized in all-solid PCFs, cf. Refs. [12, 27], or as PCFs with the voids infiltrated by liquids with specially chosen optical properties [13, 28–31]. The latter setting admits an additional possibility to control optical properties of the medium by means of electric fields [28, 29].
Such composite optical media have drawn considerable attention as realizations of nonlinear lattices, i.e., media with periodically modulated effective nonlinear potentials, which support specific species of solitons, see recent review . Other possibilities for the creation of non-linear lattices may be based on the use of doping techniques in solid-state waveguides , and structures induced in liquid crystals be means of a properly patterned external electric field .
On the other hand, in the more traditional situation, with only n 0 subject to the spatially periodic modulation, while n 2 is constant, the PCF is replaced by the nonlinearly uniform medium equipped with a linear lattice potential ; in the limit of a very deep lattice potential, this medium becomes effectively discrete [36–39]. In particular, the 2D linear potential lattice of the KP type gives rise to stable nonlinear Bloch waves , as well as solitons and solitary vortices , including the case when the uniform nonlinearity includes a “supercritical” self-focusing quintic term . It is also relevant to mention that 2D variants of the KP potential for the electron gas were experimentally created in solid-state physics, by means of superlattice heterostructures .
The model of the nonlinear photonic crystal outlined above, which includes the periodic modulation of both n 0 and n 2 in the transverse plane, is based on the 2D nonlinear Schrödinger (NLS) equation for the spatial evolution of the complex amplitude of the electromagnetic field, Ψ(x,z):Fig. 1, describes the periodic lattice of the nonlinear guiding channels, each of width D and depth U > 0, which are separated by linear buffer stripes of width L – D, i.e., D/L ≡ DC may be defined as the “duty cycle” of the guiding set: Eq. (1) correspond, respectively, to the self-focusing and self-defocusing nonlinearity. In this study, we fix the normalizations in Eq. (2) by setting L = 2π, and consider a value of the modulation depth, U = 10, which makes it possible to present generic results, while the DC of the modulation pattern [Eq. (2)] will be given two characteristic values, 0.75 and 0.5.
The juxtaposition of two square patterns [Eq. (2)] with DC ≤ 0.5, shifted relative to each other in the diagonal direction by L/ 2, gives rise to the rhombic structure. In particular, it takes the form displayed in Fig. 2 in the case of DC = 0.5. Only this case will be considered below as a representative of the 2D KP lattices of the rhombic type.
A straightforward estimate demonstrates that, with periodicity L = 10λ of the microstructure guiding the light with wavelength λ, the depth U = 10 corresponds to the refractive-index contrast δn 0 ≃ 0.02 between the channels and the host medium, in the all-solid setting or the one infiltrated with a liquid. The intensity of the input light beams, which create various solitons reported below, must be high enough, to generate the characteristic nonlinearity length on the order of few millimeters, assuming λ ∼ 1 μm.
The “checkerboard”-like pattern adopted in Eqs. (1) and (2) may also be created in a virtual form, rather than as the solid structure in the material medium, by means of electromagnetically-induced transparency, as recently proposed in Ref. . Further, the same model applies, in a completely different physical setting, to the description of effectively 2D patterns in BECs (Bose-Einstein condensates). The linear potential of the KP type, which appears in Eqs. (1) and (2), can be created by dint of magnetic lattices  (the well-known technique based on optical lattices [46,47] usually gives rise to effective sinusoidal potentials in the BEC), while the nonlinearity pattern may be induced by a system of focused laser beams, which modify the local scattering length via the optically-controlled Feshbach resonance [48, 49]. In the latter case, Eq. (1), with evolutional variable z replaced by time t, plays the role of the corresponding Gross-Pitaevskii equation.
The objective of this work is to find fundamental and vortical 2D solitons in Eq. (1), supported by the combination of the linear and nonlinear potentials determined by the KP modulation function [Eq. (2)], for both signs of the nonlinearity, self-focusing and defocusing. In the former case, solitons reside in the semi-infinite gap of the respective linear spectrum (where they turn out to be unstable, for the parameter values considered below), while in the latter case gap solitons will be found in two finite bandgaps. In previous works, solitons in 2D PCF media were found in other settings. Gap solitons and soliton trains in periodic and quasi-periodic 2D arrays of cylindrical voids were reported in Ref. . The symmetry classification of fundamental and vortical solitons in hexagonal and more general periodic transverse structures were developed in detail for fundamental solitons in Refs. [6, 7], and for vortices—in Ref. .
The rest of the paper is structured as follows. In Section II methods of the numerical analysis used in this work are outlined (we rely on numerical methods, as the present 2D model is too difficult for analytical considerations). In Section III, the results are collected for gap solitons, their bound states (dipoles and four-peak complexes), and vortices of two types (off- and on-site-centered ones), all found in two finite bandgaps of the lattice’s linear spectrum, in the case of the quadratic transverse structure (see Fig. 1) and self-defocusing nonlinearity. The stability of the various solitary modes is investigated through the calculation of eigenvalues for small perturbations, and is verified by dint of direct simulations. In Section IV, the results are reported for the localized modes in the model based on the rhombic KP lattice, see Fig. 2. In that case, fundamental solitons, including those with complex shapes (found in the second finite bandgap) may be stable, but vortices are not. In Section IV, we also briefly summarize results obtained for fundamental solitons and vortices in the semi-infinite gap of the model with the self-attractive nonlinearity, all of which turn out to be unstable. The paper is concluded by Section V.
2. The framework of the analysis
2.1. Stationary solutions and the bandgap spectrum
Stationary soliton solutions to Eq. (1) are sought for as Ψ = eiβzΦ(x,y), where β is the propagation constant, and real wave function Φ(x,y) obeys the equationEq. (3) were constructed by means if the modified Newton’s method developed in Ref.  (see also book ). Families of soliton solutions are characterized by the total power (or norm, in terms of the BEC model), Eq. (1), together with the Hamiltonian of this equation.
Before proceeding to the analysis of solitons and vortices in the photonic-crystal model, we aim to construct the linear bandgap spectrum of the 2D KP modulation pattern [Eq. (2)]. Pursuant to the Floquet-Bloch theory, solutions to the linearized version of Eq. (3) can be sought for as Φ(x,y) = e ikxx+ikyy ϕ (x,y,kx, ky), where β = β (kx, ky) is dispersion relation, wavenumbers kx, ky belong to the first Brillouin zone, while ϕ (x,y,kx, ky) is a periodic function of x and y with the periodicity of the underlying KP lattice [Eq. (2)]. The dispersion relation for the 2D Bloch wave functions is found by solving the ensuing linear eigenvalue problem:Figs. 1(b), 1(c), and 1(d) and 2(b) display the numerically generated dispersion relation and the corresponding bandgaps of Eq. (5) along the edge of the irreducible Brillouin zone (points Γ → X → M → Γ ) for, respectively, the square lattice [Eq. (2)] with DC = 0.75, 0.5 and 0.25, and for the rhombic lattice with DC = 0.5. Note that finite bandgaps are absent in Fig. 1 when DC and/or U are too small—e.g., in the case shown in Fig. 1(d) for DC = 0.25. The bandgaps also disappear when the depth of the KP lattice falls to U ≈ 2.5 for DC = 0.75.
2.2. The linear-stability analysis of solitons
To develop the stability analysis for 2D solitons, the solution is perturbed as follows:Eq. (1) and linearizing, we arrive at the following linear-stability eigenvalue problem: 51].
3. Photonic crystals with the square transverse structure
3.1. Fundamental solitons and their bound states
In the model with the square-lattice modulation function defined as per Eq. (2) and self-repulsive (defocusing) nonlinearity, which corresponds to σ = −1 in Eq. (1), 2D fundamental gap solitons have been found in the two finite bandgaps. Also found were diagonally oriented dipoles, i.e., bound states of two solitons with opposite signs, and in-phase four-soliton complexes of the square and rhombus types (alias off-site- and on-site-centered ones, respectively, i.e., rings formed by four peaks, without or with an empty site at the center—see, e.g., Ref. ). Here, we consider the complexes that carry no phase structure (i.e., they are not quadrupoles). For U = 10 and DC = 0.75, these families of the gap solitons are characterized by dependences of the integral power on the propagation constant, P(β) [see Eq. (4)], which are shown in Fig. 3(a) for the two finite bandgaps. Actually, the second bandgap is very narrow in this case,Fig 1(b). Other panels in Fig. 3 display typical examples of stable and unstable fundamental and compound solitons found in the first bandgap. Examples of stable solitons of the same types, but found in the second bandgap, are displayed in Fig. 4. As concerns other types of gap-soliton complexes, such as straight dipoles (ones oriented along bonds of the KP lattice, rather than diagonally) and quadrupoles, they were found too, but these species of the localized patterns turn out to be completely unstable at DC = 0.75.
The stability of the solitons, for which the corresponding analysis reveals no eigenvalues with nonzero real parts, has been corroborated by direct simulations (not shown here). The simulations of the evolution of the solitons with small real parts, which typically account for the instability in the present case (see Fig. 3), performed by means of the standard split-step Fourier-transform method, with absorbers installed at edges of the integration domain, demonstrate a fairly weak instability, in the form of a very slow decay of perturbed solitons, as shown in Fig. 5.
The situation for the gap solitons and their bound complexes is different at DC = 0.5. In this case, the second finite bandgap does not exist, while the families of the fundamental solitons and all their bound states, viz., straight and diagonal dipoles, square- and rhombus-shaped complexes, as well as square-shaped quadrupoles, are all entirely stable in the first bandgap (recall that the straight dipoles and quadrupoles were completely unstable at DC = 0.75). The corresponding P(β) dependences, as well as generic examples of the stable solitons and their bound states, are displayed in Fig. 6.
Similarly to the four-soliton complexes without the phase structure, that were considered above, solitary vortices with topological charge 1 can be built, in the first and second finite bandgaps, as squares and rhombuses, alias off-site- and on-site-centered four-peak rings, which carry the phase circulation of 2π (see Refs.  and  for a review of vortex solitons in linear lattices). For DC = 0.75, the P(β) curves for the vortices of both types and their typical examples are shown in Fig. 7(a) (rhombic vortices were not found at β < 4.4). Simulations of perturbed unstable vortices (not shown here in detail) demonstrate the evolution which is quite similar to that shown for the weakly unstable gap solitons and complexes in Fig. 5, i.e., a slow decay (with the loss of the vorticity).
Finally, for DC = 0.5 (recall that there is the single finite bandgap in this case) solutions for square-shaped vortices were not found, while rhombic vortices exist and are stable in the entire bandgap in this case, as shown in Fig. 9. The fact that vortical rhombuses are more robust, in the sense of the existence and stability, than their square-shaped counterparts, is a general feature which was also found in other models .
4. Photonic crystals with the rhombic transverse structure
4.1. Simple and complex gap solitons
In the model based on the rhombic structure with DC = 0.5, which is shown in Fig. 2, only fundamental gap solitons were found in the first finite bandgap, but not their bound states in the form of dipoles, squares, or rhombuses. The P(β) curve for the solitons in two finite bandgaps of this photonic crystal are shown in Fig. 10(a). The (in)stability of the gap solitons is illustrated in Fig. 10(b) by plotting the largest instability growth rate versus propagation constant β, the zero value of the latter implying the stability. In fact, stability intervals in the first bandgap are quite narrow: there are two of them on the top branch of the respective curve P(β),Figs. 10(c) and (d), respectively.
The comparison of Fig. 10 to Fig. 3 demonstrates that the instability of the gap solitons (of those of them which are unstable) is much stronger in the present case than in the model of the photonic crystal with the square transverse structure. Accordingly, direct simulations, displayed in Figs. 11(a) and 11(b), demonstrate a much faster decay of the unstable solitons, cf. Fig. 5.
The second bandgap hosts complex localized modes, both stable and unstable ones, which seem as bound states of fundamental solitons, see generic examples in Fig. 12. However, the simplest among these species, at least—the rhombus which is shown in the top row of Fig. 12(a)—is actually a fundamental gap soliton by itself, in the sense that there are no single-peak solitons that would build this four-peak pattern. As for the stable eight-peak ring shown in the bottom row of Fig. 12(a), and the complex unstable pattern displayed in Fig. 12(b), the relation between the shapes and powers of the three modes presented in Fig. 12, which pertain to the common value of the propagation constant, suggests that, in principle, the eight-peak ring may be realized as a superposition of two four-peak fundamental rhombuses, and the unstable pattern shown in Fig. 12(b) may be built as a complex of five eight-peak rings [the evolution of the latter unstable pattern is displayed in Fig. 11(c)]. We stress that all these three species of the complex modes carry no phase structure.
Localized vortex patterns, with rather complex shapes, were also found in the two finite bandgaps of this model, but they all turn out to be unstable. Typical example of the shapes and instability spectra are displayed in Fig. 13. Direct simulations of the evolution of the unstable vortices demonstrate a quick transition to chaotic dynamics (not shown here).
4.3. The self-focusing nonlinearity
In the same model with the self-attractive nonlinearity, which corresponds to σ = +1 in Eq. (1), ordinary fundamental solitons supported by the lattice structure  can be found in the semi-infinite gap, but they are unstable. The P(β) dependence for this family, along with a typical example of the shape and instability of the fundamental soliton, are displayed in Fig. 14. Rhombic vortices have also been found in the semi-infinite gap, but they are completely unstable too, suffering a fast decay (not shown here). Square-shaped vortices could not be constructed in the model with the self-focusing nonlinearity.
This work reports results of the systematic analysis of 2D solitons, their bound states of various types, and solitary vortices in two fundamental models of nonlinear optical crystals and/or optical-crystal fibers, with the KP (Kronig-Penney) square or rhombic lattice of voids filled by a linear material. The same model can be realized in terms of the BEC (Bose-Einstein condensate) loaded into a combination of linear and nonlinear 2D lattice potentials. The solitons and solitary vortices can be created experimentally in the physical settings of both types.
The analysis was mainly focused on the models with the self-repulsive nonlinearity, which gives rise to many species of stable solitons and vortices in the two lowest finite bandgaps of the respective linear spectrum. In particular, stable fundamental gap solitons feature intricate shapes in the second finite bandgap of the photonic crystal with the rhombic transverse structure. The stability of the various modes was investigated through the calculation of the corresponding eigenvalues, and verified via direct simulations of the perturbed evolution. Depending on the value of the lattice’s DC (duty cycle, i.e., the ratio of the void’s width to the pitch of the KP lattice), an intrinsic stability border for the gap solitons, complexes built of them, and solitary vortices can be found inside the first finite bandgap. In other cases, the families of solitons and vortices are completely stable or unstable inside the respective finite bandgaps. The case of the self-attraction was briefly considered too, but only unstable solitary modes were found in that case in the semi-infinite gap.
The work may be extended in other directions, to consider additional physically relevant issues. In particular, a challenging problem is mobility of the solitons in the present setting (it is known that 2D gap solitons may be mobile in linear lattices [54, 55]). Another natural extension may aim to cover a broader range of parameters of the square-shaped and rhombic KP lattices, such as DC and the depth of the lattice, U, and to generalize the model for an anisotropic version of the lattice, cf. Refs.  and , where experimental and theoretical results were reported for solitons in anisotropic 2D linear-lattice potentials.
The work of T.M. was supported by the Thailand Research Fund through grant RMU5380005.
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