Self-trapped leaky waves in lattices: discrete and Bragg soleakons

We propose lattice soleakons: self-trapped waves that self-consistently populate leaky modes of their self-induced defects in periodic potentials. Two types, discrete and Bragg, lattice soleakons are predicted. Discrete soleakons that are supported by combination of self-focusing and self-defocusing nonlinearities propagate robustly for long propagation distances. They eventually abruptly disintegrate because they emit power to infinity at an increasing pace. In contrast, Bragg soleakons self-trap by only self-focusing, and they do not disintegrate because they emit power at a decreasing rate.

investigated: lattice solitons and lattice breathers. During evolution, the shape of lattice solitons is preserved while it oscillates in lattice breathers. Still, the wave-packets of both lattice solitons and lattice breathers exhibit exponential decay in the trapped directions, resulting from the balance between the nonlinearity and lattice dispersion/diffraction.
Another division of self-trapped lattice states is according to the location of their eigenvalues (eigen-energies or propagation constants) in the band structure. The linear modes of lattices are Floquet-Bloch waves, with their spectra divided into bands that are separated by gaps in which propagating modes do not exist [16]. The eigenvalues of a self-trapped lattice state can reside in the semi-infinite gap, in which case it is often termed discrete soliton [2,4,5,7,15] or discrete breather [1,3,14,15], or in a gap between two bands, hence termed gap soliton [6,9] or Bragg soliton [8]. Notably, discrete and gap solitons often exhibit different properties because discrete solitons are trapped through total internal reflections where gap solitons are localized by Bragg reflections [11]. A prime example for a system in which self-localized lattice waves have been investigated experimentally is optical nonlinear waveguide arrays [17][18][19][20][21][22]. Discrete solitons [17][18][19][20][21]23], discrete breathers [24], gap solitons [22,23] and gap breathers [10], as well as more complicated structures such as vector lattice solitons [25,26] and incoherent lattice solitons [27,28], have been explored in one and two dimensional arrays of waveguides.
Lattice solitons and lattice breathers have their counterparts in nonlinear homogeneous media. In homogeneous media, however, a different type of self-confined states, which thus far was not considered in lattices, was recently proposed: self-trapped leaky mode -a soleakon [29]. A soleakon induces a waveguide through the nonlinearity and populates its leaky mode self-consistently. As shown in Ref. [29], soleakons exhibit very different properties from solitons and breathers. By their nature, soleakons emit some power to infinity during propagation and therefore decay. However, if a double-barrier W structure waveguide is induced, a waveguide structure that can give rise to long-lived leaky modes [30], then the decay rate can be very small. In such cases, soleakons exhibit stable propagation, largely maintaining their intensity profiles, for very long propagation distances (orders of magnitude larger than their diffraction lengths). In order to selfinduce the desired W-shape waveguide, Ref. [29] proposed using media with nonlocal self-defocusing and local self-focusing nonlinearities. This case can be realized for example in glass, polymers, etc. (which exhibit both a negative nonlocal thermal selfdefocusing and the optical Kerr self-focusing). Beyond optics, Bose Einstein condensate can also display simultaneous nonlocal nonlinearity through dipole-dipole interaction and local self-focusing by van der Waals interaction [31]. Still, the requirement for a proper superposition of wide negative and narrow positive nonlinearities is a restricting factor in the obtainability and impact of soleakons.
Here, we propose and demonstrate numerically soleakons that propagate in arrays of slab wave-guides. Two types of lattice soleakons are predicted: discrete soleakons and Bragg soleakons. Discrete soleakons are supported by combination of nonlocal defocusing and local focusing nonlinearities that jointly induce a ring-barrier wave-guide structure. This waveguide gives rise to long-lived leaky modes that reside within the first band of the lattice transmission spectra. The decay rate of discrete soleakons increases during propagation. Consequently, they eventually disintegrate abruptly, emitting all their power to delocalized radiation. The predicted Bragg soleakons are supported by self-focusing nonlinearity only. Interestingly, the decay rates of Bragg soleakons decrease during the propagation, hence, Bragg soleakons continue to propagate without disintegration. Lattice soleakons of both types were found numerically. We studied their dynamics using semianalytical model and verified our theoretical predictions by numerical simulations of beam propagation.
Soleakons are nonlinear entities associated with linear leaky modes of their self-induced waveguide. Let us discuss leaky modes first. Leaky modes are solutions of the propagation equation when applying outgoing boundary conditions [32]. A leaky mode is a superposition of radiation modes (continuum states), forming a wave-packet that is highly localized at the vicinity of the structure, but oscillatory outside the waveguide and diverges exponentially far away from it. The propagation constant of a leaky mode is a complex quantity, with the imaginary part associated with unidirectional power flow from the localized section to the radiative part. However, the decay rate can be made extremely small, yielding long-lived localized modes. Interestingly, the real part of the propagation constant resides within a band of non-localized propagating modes. As such, the spatial spectrum of a leaky mode belongs entirely to radiation modes. In order to excite a leaky mode, one has to excite properly its localized section, which resembles a bound state.
Because a leaky mode is not a true eigen-mode, the radiation modes comprising it dephase, hence radiation is constantly emitted away at a distinct angle.
Lattice soleakons are universal entities that can be excited in many nonlinear lattices.
However, for concreteness we analyze here optical lattice soleakons in waveguide arrays and use the corresponding terminology. Specifically, we assume a bulk media with linear refractive index change in the form of array of slab sinusoidal wavegudes:

(dash brown curves) bands versus
x k for the modes with different k y . For a constant k y , the transmission spectra of the waveguide array is divided into bands that are separated by gaps in which propagating modes do not exist. Such a gap for modes with k y =0 is shown by the brown region in Fig. 1(b).
However, as shown in Fig. 1(b), these gaps are full with propagating modes with other k y 's. In other words, the transmission spectrum of the 1D lattice potential does not include gaps. Instead, it consists of a semi-infinite band continuously filled with delocalized propagating modes and a semi-infinite gap above it.
Next we consider propagation of a beam in a nonlinear array of slab wave-guides. Such a nonlinear array of slab waveguides can, for example, be optically induced in photorefractives [15,16] or by periodic voltage biasing in liquid crystals [17]. The complex amplitude of a paraxial beam that propagates in this medium is described by the (2+1)D Nonlinear Schrödinger equation: where  is propagation constant. The solution of Eq. 3 was found numerically using the self-consistency method with modifications for finding soleakons [27].
We explored two different types of soleakons: discrete and Bragg soleakons. Like discrete solitons, discrete soleakons also bifurcate from the upper edge of the first band.
But, in contrast to discrete solitons, that reside in the semi-infinite gap, propagation constant of discrete soleakons must be "shifted" downward into the first band. This can be realized by self-defocusing nonlinearity. However, as proposed in ref. [27], the combination of nonlocal self-defocusing and localized self-focusing leads to long-lived soleakons. In particular, we assume saturable self-focusing and nonlocal self-defocusing nonlinearities: The slope of these lines, which is given by changes from 0 [point A in Fig. 2(a and c)] to infinity [point B in Fig. 2(a and c)].
Therefore the normals to these curves cover  2 angle. The directions of these normals correspond to the directions of the power radiation in real space (direction of maximum localization in k-space corresponds to the direction of maximum delocalization in real space). Thus, our discrete soleakons radiate power to all directions. Finally, Fig. 2d shows the induced waveguide structure that exhibits a negative ring structure which is the two dimensional version of the one-dimensional double-barrier waveguide which is known to support long-lived leaky modes.
The decay rate of the soleakon given by the imaginary part of propagation constant versus its localized power is shown in Fig. 3(a). Such monotonically decreasing dependence is explained in Fig. 3 To verify our theoretical predictions we followed Ref. [29] and developed a semi-  Fig. 3(a)]. Figure 3(c) shows P(z) from the model against P(z) from direct numerical simulations of beam propagation (the fine matching was obtained when the input beam in the beam propagation method corresponded to 1.0125 times the calculated wave-function from the self-consistency method). The agreement is good only until z~115 cm because at small power levels, the self-consistency method did not converge After z~115 cm, the soleakon disintegrates abruptly loosing all its power to delocolized radiation. Figure 3c also shows the power of a linear leaky mode that propagates in the fixed wave-guide that was induced at z=0. The power of the linear leaky mode decays exponentially because the decay rate is constant. This comparison shows that the soleakon indeed decays at the increasing rate. The intensity profiles of the beam found by the self-consistency method for several values of its power corresponding to z=0, z=107cm and z=115cm [denoted by circles in Fig. 3(c)] are shown in Fig. 3(d, e) and Fig. 2(c) respectively. These plots show that during propagation, the soleakon localized section indeed becomes wider and weaker while the radiation part gets stronger.
The discrete soleakons in the array of slab wave-guides presented above are similar to the soleakons in homogeneous media [29] in that they both require a combination of nonlocal defocusing with local focusing nonlinearities and decay at increasing rate during and radiation modes (Eq. 4). In Bragg soleakons, the slope of these lines, given by Eq.
(5), is finite, hence the normals to these curves cover the specific angles in the upper and lower half planes of the Fourier space as shown by black dashed lines in Fig. 4(c). The directions of these normals correspond to the directions of the power radiation in real space. Therefore Bragg soleakons radiate power into the specific angles which is reflected by the characteristic bow-tie shape of the radiation part of the soleakon [ Fig.   4(b)].
The decay-rate of Bragg soleakons monotonically decrease with decreasing power of the localized section (Fig. 5a). This dependence, which is opposite to the dependence of discrete soleakons (Fig. 3a), is related to the fact that in Bragg soleakons, the spatial widths increase and the bandwidth decrease as a result of the decrease in soleakon localized power (Figs. 5b and 5c). On the other hand, the plane wave-numbers of the "inresonace" radiation modes yR k lie outside the band given by Eq. 4. Therefore the power leakage results in the reduction of the spectral overlap between the soleakon and radiation modes and hence in weaker radiation and smaller decay rate of the soleakon.