Surface vortex solitons

We discover the existence of vortex solitons supported by the surface between two optical lattices imprinted in Kerr-type nonlinear media. Such solitons can feature strongly noncanonical profiles, and we found that their properties are dictated by the location of the vortex core relative to the surface. The refractive index modulation forming the lattices at both sides of the interface results in complete stability of the vortex solitons in wide domains of their existence, thus introducing the first known example of stable topological solitons supported by a surface.

A new important possibility, never addressed to date, is the existence of vortex solitons at the surface between two different materials. Surfaces can support waves confined at the very interface, and nonlinear surface waves on various interfaces were studied in solid-state physics, in nonlinear and near-field optics (see, e.g., [15][16][17]).
Nonlinearity drastically alters refraction scenario for optical beams and results in bistability and transition between regimes of total internal reflection and complete transmission [18]. Optical surface waves have been observed in photorefractive materials with diffusion nonlinearity [19] and at the interface of uniform and layered media, including photonic crystals [20]. Arrays of waveguides made with currently available technologies allow formation of solitons at the interface between optical lattices, a landmark concept put forward recently in [21].
In this Letter we discover the existence of vortex solitons at an interface of two periodic lattices imprinted in Kerr-type focusing nonlinear media. To the best of our knowledge, the existence of surface vortices has not been reported so far, not even for interfaces of uniform materials. We found that surface vortex solitons feature strongly asymmetric shapes and noncanonical phase distributions. Importantly, we found that the lattices forming the interface results in complete stability of the vortex solitons in wide domains of their existence. Also, we reveal nontrivial relation between interface parameters and existence domains of surface vortex solitons.
We consider propagation of laser radiation at the interface of two periodic lattices imprinted in the focusing media with Kerr-type saturable nonlinearity, described by the nonlinear Schrödinger equation for dimensionless complex amplitude of the light field q : Here the transverse η and longitudinal ξ coordinates are scaled in terms of beam width and diffraction length, respectively, and S is the saturation parameter. The function stands for total refractive index profile, where is the depth of periodic part of the lattice, Ω is its frequency,  (1) yields the system Upon searching for various vortex soliton profiles we solved system (2)  we term such vortices "off-site" since phase singularity is located between local lattice maxima. Four main intensity lobes (whose separation is minimal for off-site case) are clearly resolvable in vortex profile, but two of them located at are smaller than those located at η , so that vortex become strongly asymmetric, especially for big refractive index steps δ . The phase distributions for asymmetric surface vortices are noncanonical [22], i.e. on a ring of radius r whose center coincides with phase singularity, phase does not grow linearly, but rather possesses alternating regions of slow (in the vicinity of local intensity maxima) and fast growth. While increasing δ results in stronger asymmetry, the growth of the periodic modulation depth leads to stronger localization of vortex energy near local lattice maxima.
We found that at fixed δ and there exist lower b and upper b cutoffs for existence of off-site surface vortices. In contrast to canonical vortices in uniform is not constant and varies with b , , and δ . We found that close to both lower and upper cutoffs one of the quantities U abruptly tends to zero. The domain of existence for surface vortices is presented in Fig. 2(a). The width of existence domain in b is maximal (though still limited) for δ , and quickly shrinks with increase of δ , so that above certain critical value of step in the constant part of refractive index off-site surface vortices do not exist at the lattice interface. For a fixed δ the width of existence domain gets broader with decrease of the lattice depth . Though dependencies are nonmonotonic close to cutoffs, the total energy flow U still is a monotonically increasing function of propagation constant ( Fig. 2(b)). The maximal energy flow carried by the vortex soliton residing at the interface quickly decreases with increase of . We found that close to lower cutoff surface vortices are typically less localized than their counterparts near upper cutoff. This is especially pronounced at small and moderate values of , when low-energy vortices expand over several neighboring lattice maximums, moreover, the expansion in the region can be much more pronounced than that for η . Vortices with strongly asymmetric shapes at δ are typically well localized in both cutoffs. The striking feature of surface vortices is that critical value of refractive index step decreases with increase of the lattice depth (Fig. 2(c)), i.e. stronger asymmetries can only be achieved at interfaces formed by shallower lattices.
One of the central results of this Letter is that strongly asymmetric vortices at interface of two lattices can be made completely stable in the substantial part of their existence domain. To analyze stability of surface vortex solitons, we searched for the perturbed solutions of Eq. (1) in the form q w , where and u are real and imaginary parts of perturbation that can grow with complex rate upon propagation. Linearization of Eq. (1) around w ,w yields the system of equations where we suppose that u . We solved this system numerically in order to find the perturbation profiles and associated growth rates δ . The outcome of stability analysis for off-site surface vortices performed for various sets of parameters δ and is summarized in Fig. 3. It was found that such strongly asymmetric vortices are stable in the most part of their existence domain at moderate and high depths of periodic modulation. There are two instability domains located near the lower and upper cutoffs (Fig. 3(c)). The instability domain near upper cutoff is too narrow and 4(a)), while unstable representatives of off-site vortex families decay via progressively growing oscillations of their intensity lobes ( Fig. 4(b)).
In addition, it is worth mentioning that besides simplest off-site surface vortices we found a variety of other asymmetric vortex solitons families.
Most representative examples of on-site vortices that reside mainly at the left or at the right of the interface are shown in Figs. 1(d) -1(f) (notice, that upon searching for such vortices we translated the lattice by π in the vertical direction for convenience). We found that on-site vortices are much more sensitive to variation in the height of refractive index step δ and typically require δ so that the existence domain substantially departs from that for their off-site counterparts.
Nevertheless, on-site vortices also feature strongly asymmetric shapes (see, e.g. Fig.   1(e)), and can be made completely stable in certain domains of their existence.
Finally, we would like to remark that we also found that lattices with defocusing nonlinearity also can support asymmetric vortex solitons. /