Surface lattice solitons in diffusive nonlinear media with spatially modulated nonlinearity

Kaiyun Zhan; Zhiyong Jiao; Xi Li; Yulei Jia; Hui Zhang

doi:10.1364/OE.23.024048

1. Introduction

The optical propagation dynamics in nonlinear media with periodic refractive-index has become a considerate topic in nonlinear optics, because of their intriguing physical properties and their potential applications [1]. A large amount of research on spatial optical solitons has been focused on Kerr [2] and saturable nonlinear media with photorefractive effect for easily realization of linear and nonlinear optical lattices [3]. At presents, several types of lattice solitons [4–6 ] have been observed and the research scopes on spatial solitons have been expanded from low dimensional geometries to high dimensional ones, from fundamental soliton states to multipole soliton states, from simple nonlinear models to complex (hybrid) ones [7], from long wavelength to subwavelength confinement of the optical field [8]. Diffusive nonlinearity is a universal nonlocal effect that usually appears during space-charge field displacement, carrier diffusion and heat transfer process [9]. It is well known that a biased photorefractive(PR) crystal exhibits drift and diffusion components of the nonlinear response, among which drift is local nonlinearity inducing beam focusing or defocusing, and diffusion is nonlocal nonlinearity inducing beam bending in the presence of drift nonlinear response [10–12 ]. It has been shown that the domains of existence and the stability of fundamental and higher-order solitons supported by optical lattices in biased PR crystals are strongly depend on the diffusive effect [2,13 ]. Under the action of diffusive response, the soliton center shifts toward the neighboring lattice channel, with the number of jumps depending on the strength of diffusion nonlinearity. Nevertheless, the pure nonlocal diffusive nonlinearity can support self-trapping of optical beams propagating in unbiased PR uniform crystals [14,15 ]. Spatial solitons in diffusive nonlinear media not only possess the merit of nonlocal solitons but also possess the special dynamic behavior. Recently, surface solitons supported by optical lattices have been investigated theoretically in unbiased PR media with diffusion nonlinearity due to linear and quadratic electro-optic effect [16,17 ].

In most previous studies, nonlinearity was spatially uniform inside the material. In actual practice, a number of tools have been developed to control and manipulate the spatial profile of the nonlinearity. For example, one may tune the nonlinearity profile by changing the concentration of dopants [18], or change the strength of the nonlinearity by tuning the interatomic interactions [19], and for some PR crystals grown through the top-seeded solution method, the nonlinearity profile can be controlled by adjusting the growth temperature, pulling rate, rotating rate and other growth parameters [20,21 ]. For this reason, the use of spatially modulated nonlinearity coefficient is a unique possibility to strongly affect and control the beam dynamics in nonlinear media. The objective of this work is to study surface lattice solitons in diffusive nonlinear media with spatially modulated nonlinearity in the one-dimensional (1D) case.

In this paper, we address the existence and stability of surface lattice solitons at the interface of optical lattices in diffusive nonlinear media with spatially modulated nonlinearity. For both self-defocusing and self-focusing diffusive spatially modulated nonlinearity, two families of gap and twisted surface solitons exist. It is shown that the modulated nonlinearity strength strongly affects the properties of such surface lattice solitons.

2. Theory model

The nonlinear light propagation at an interface of a semi-infinite lattice imprinted in a high diffusive nonlinear medium with spatially modulated nonlinearity obeys the following 1D nonlinear Schrödinger (NLS) equation [17,22 ],

i \frac{\partial q}{\partial z} + \frac{1}{2} \frac{\partial^{2} q}{\partial x^{2}} + p R (x) q - β χ (x) \frac{q}{{(1 + S {| q |}^{2})}^{2}} {(\frac{\partial {| q |}^{2}}{\partial x})}^{2} = 0,

q (x, z)

is the dimensionless complex envelope of the optical field, which depends on the transverse coordinate x and the longitudinal propagation distance z; here x and z are scaled to the beam width x ₀ and the diffraction length

k_{0} n_{0} x_{0}^{2}

, respectively, where

k_{0} = 2 π / λ_{0}

,

λ_{0}

is the free-space wavelength, n ₀ is the background refractive index. The parameter p characterizes the scaled depth of the optical lattice, defined as

R (x) = 0

at

x < - π / 2

and

R (x) = \cos^{2} (x)

at

x \geq - π / 2

.

β = \pm 1

defines the nonlinearity sign, namely self-defocusing

(β = + 1)

or self-focusing

(β = - 1)

. It is stressed that this model is appropriate in a unbiased centrosymmetric PR crystal with spatial modulated nonlinearity parameter

χ (x)

. A prime example of the typical centrosymmetric PR crystals is paraelectric Potassium lithium tantalite niobate (KLTN) [22], where the change in refractive is governed by quadratic electro-optic effect. S is the saturable coefficient depending on the nonlinear material parameters, In this model, the saturable parameter is given by

S = {[k_{0}^{2} n_{0}^{4} g_{eff} {(K_{B} T ε_{0})}^{2} {(ε_{r} - 1)}^{2} / (2 e^{2})]}^{- 1 / 2}

, where g _eff is the effective quadratic electro-optic coefficient, ε ₀ and ε _r are the vacuum and relative dielectric constants, respectively, K _B is the Boltzmann constant, T is the absolute temperature, e is the electron charge. We can adjust the crystal temperature to change saturable parameter, here we set S = 1.5. The x-dependent nonlinearity coefficient is taken in the special form,

χ (x) = 1 + σ \exp (- x^{2} / 128)

with

σ

being the modulated strength [6]. In the case

σ = 0

, the NLS Eq. (1) is simplified to standard 1D NLS equation with a periodic lattice, whose soliton solutions, as well as their stability, have been discussed [17]. Recall that Eq. (1) conserves the power or energy flow

U = \int_{- \infty}^{\infty} {| q |}^{2} d x

.

We search the stationary surface soliton solutions of Eq. (1) in the form $q (x, z) = w (x) \exp (i μ z)$ that can be characterized by the propagation constant μ. Such states only exist for μ values falling into the gaps of the lattice Floquet-Bloch spectrum, as shown in Fig. 1(a) . We choose p = 5 throughout this letter, and the region of semi-infinite gap is found to be $μ \geq 3.577$ and the first, second finite gap are $1.252 \leq μ \leq 3.538$ and $- 0.3065 \leq μ \leq 0.7538$ , respectively. The stationary solutions are solved numerically by the Newton’s iteration method and developed squared-operator method [23]. Furthermore, in order to test the stability of surface lattice solitons, we employ a linear stability analysis for perturbed solutions as [24]

q (x, z) = {w (x) + [v_{1} (x) - v_{2} (x)] e^{i δ z} + [v_{1}^{*} (x) + v_{2}^{*} (x)] e^{- i δ^{*} z}} e^{i μ z},

where v ₁ and v ₂ are small perturbations that can grow with the complex rate

δ = δ_{r} + i δ_{i}

upon propagation and the asterisk means complex conjugation. Inserting the above ansatz into Eq. (1) and linearization around w(x) for Eq. (1) leads to the following linear eigenvalue problem:

δ v_{1} = - \frac{1}{2} \frac{d^{2} v_{2}}{d x^{2}} + [\frac{4 β χ w^{2}}{{(1 + S w^{2})}^{2}} {(\frac{d w}{d x})}^{2} + μ - p R] v_{2},

δ v_{2} = - \frac{1}{2} \frac{d^{2} v_{1}}{d x^{2}} + [\frac{4 β χ w^{2} (3 - S w^{2})}{{(1 + S w^{2})}^{3}} {(\frac{d w}{d x})}^{2} + μ - p R] v_{1} + \frac{8 β χ w^{3}}{{(1 + S w^{2})}^{2}} \frac{d w}{d x} \frac{d v_{1}}{d x},

which we solve numerically to find the growth rate. Obviously, Solutions with

δ_{i} > 0

indicate the soliton instability, as the perturbations grow exponentially, otherwise, solitons are stable.

Fig. 1 (a) Bandgap lattice spectrum for different lattice depth p. Gray regions are Bloch bands. (b) Existence and stability regions for gap solitons in the plane of (σ, μ) of the self-focusing nonlinear system. (c) Energy flow versus μ for σ = −0.5 and 5 . (d) Energy flow versus σ for μ = 4 and 4.6.

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3. Surface solitons and their stability for self-focusing diffusive nonlinearity

To begin with, in this section, we investigate surface solitons and their stability supported by optical lattice with spatially modulated self-focusing nonlinearity $(β = - 1)$ . Numerically, we find two families of surface solitons bifurcating from band edge in the semi-infinite gap.

3.1. Surface gap solitons

The families of surface gap solitons at different modulated strength σ, bifurcate from the lower edge of Bloch band into the corresponding gap, as shown in Fig. 1 (c), and their localization depends on μ value, in terms of which there exits both lower and upper cutoff on μ at different modulated strength σ, see Fig. 1(b). The upper cutoff increases monotonically and expands the asymptotic maximum to μ = 4.8 at large σ. This is accompanied by a decrease of energy flow U, as shown in Fig. 1(d). The existence region of these solitons also expands when the propagation constant is close to the edge of Bloch band, and the minimum value of modulated strength σ approaches −1. This is attributed to the necessary requirement of self-focusing nonlinearity $(β = - 1)$ in the semi-infinite gap for supporting such solitons, and the nonlinear refractive index sign would be changed in the region surrounding the center (x = 0) when σ is less than −1. Figure 1(c) shows that U is a monotonic function of μ, and increases with μ in the existence domain. Representative profiles of two surface solitons belonging to the semi-infinite bandgap are shown in Figs. 2(a) and 2(b) corresponding to the circled points a (μ = 4, σ = −0.1) and b (μ = 4.6, σ = 5) of Fig. 1 (d). They show that these solitons are entirely positive with different propagation constant and their intensity localize at the input lattice channel. Based on Vakhitov-Kolokolov (VK) stability criterion [25], this family of surface fundamental solitons is VK stable in their existence region. Linear stability analysis also reveals that there is no complex or purely imaginary δ, as shown in the insets of Fig. 2(a) and 2(b). That is to say, these solitons are stable in the whole domain of their existence, These results are confirmed by direct simulations of propagation of perturbed soliton solutions in Eq. (1). Figure 2(c) and 2(d) display examples of the evolution of stable surface solitons corresponding to Fig. 2(a) and 2(b), which are robust and propagate without any noticeable deformations under 10% random initial perturbation.

Fig. 2 Profiles and perturbation eigenvalue spectrums (inset) of gap solitons at μ = 4, σ = −0.1 (a) and μ = 4.6, σ = 5 (b) marked in Fig. 1(d). (c) and (d) are stable evolutions of gap solitons under 10% random noise perturbations corresponding to (a) and (b), respectively.

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3.2. Surface twisted solitons

We also find that a family of higher-order surface twisted soliton supported by optical lattice with spatially modulated self-focusing nonlinearity can exist in the semi-infinite gap. They can be viewed as combinations of several odd solitons and is in phase within the first channel, while among neighboring elements of the lattice a phase shift of π appears. Figure 3(a) shows the existence and stability domains of such solitons on modulated strength σ. Apparently, for the existence curves of twisted solitons, a similar trend is observed to surface gap solitons, but the faster saturation of the present case expands to the asymptotic maximum at μ = 4.7, which is lower than the gap soliton case. Figure 3 (b) and 3(c) show the soliton power diagrams on propagation constant μ and modulated strength σ, respectively. Similar to gap solitons mentioned above, at fixed σ, the soliton energy flow monotonically increases with increase of propagation constant μ, and decreases with growing σ for the same μ. It is well known that the instability of multipole solitons cannot be predicted by the VK stability criterion. Thus, a comprehensive linear instability analysis based on Eqs. (3) and (4) should be performed. Our result shows that such twisted solitons are stable in their existence region for the small values of modulated strength (σ<-0.23), however, there is a very narrow unstable region near the upper cutoff, as shown in Fig. 3(a), where the imaginary part of the perturbation growth rate δ is nonzero and the exponential instability develops for the strong modulated strength, as shown in Fig. 3(d). To demonstrate, we do direct numerical simulations of Eq. (1) perturbed by input noise on amplitude. We choose the twisted solitons at σ = 0.9 and 3 with μ = 4.4 corresponding to the circled points a and b of Fig. 3(c), and the corresponding linearization eigenvalue spectrums of these solitons are shown in the insets in Figs. 4(a) and 4(b) . Figure 4(c) illustrates the unstable propagation of perturbed surface soliton at σ = 0.9 and μ = 4.4, and breaks up due to the oscillator instability. When σ = 3 and μ = 4.4, this corresponding twisted mode is robust and exhibits stable evolution under the noise perturbation, as shown in Fig. 4(d).

Fig. 3 (a) Existence and stability regions for twisted solitons in self-focusing nonlinear media. (b) Energy flow versus μ for σ = −0.5 and 5. (c) Energy flow versus σ for μ = 3.8 and 4.4.Circles correspond to profiles shown in Fig. 4. In (b) and (c), stable and unstable branches are plotted by solid and dashed curves, respectively. (d) Perturbation growth rate versus σ at μ = 4.4.

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Fig. 4 Profiles and perturbation eigenvalue spectrums (inset) of twisted solitons at μ = 4, σ = 0.9 (a) and σ = 3 (b) corresponding to points marks as circles in Fig. 3(c). Unstable (c) and stable (d) propagations of twisted soliton shown in (a) and (b), respectively.

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4. Surface solitons and their stability for self-defocusing diffusive nonlinearity

In this section, we constrain our studies on surface solitons and their stability supported by optical lattice with spatially modulated self-defocusing nonlinearity ( $β = 1$ ). Gap and twisted surface solitons belonging to the first finite gap of the linear spectrum are found.

4.1. Surface gap solitons

In the model with the self-defocusing nonlinearity, i.e., $β = 1$ , surface gap solitons can be readily found in first finite gap. The existence area of the such solitons on the (σ, μ) plane is presented in Figs. 5(a) and 5(b) , which show that there also exists both lower and upper cutoffs on the propagation constant for surface gap solitons for different σ, they do not occupy the whole gap and shrinks with the decrease of σ in the low modulated strength domain, as shown in Fig. 5(a). Linear stability analysis indicates that a very narrow instability band emerges when the propagation constant approaches to the upper cutoff, presented in Fig. 5(b), where the slope of power curve changes to a positive one ( $d P / d μ > 0$ ), see Fig. 5(c). This behavior is attributed to the resonant energy redistribution between the gaps. Figure 5(d) shows the energy flow is also a monotonically decreasing function of the modulation strength σ, the stable and unstable intervals of the present solitons are intertwined, and unstable solitons have relatively high powers. Two typical examples of gap surface solitons marked in Fig. 5(d) are displayed in Figs. 6(a) and 6(b) . It is shown that these solitons exhibit multihumped profiles and become less localized when the propagation constant approaches to the upper cutoff. In high modulated strength domain, the perturbed surface soliton propagates stably and retains its input structure during the distance, as shown in Fig. 6 (d). The unstable propagation of gap solton at μ = 4.4 and σ = 0.9 is illustrated in Fig. 6 (c), which shows the intensity distribution of such soliton profile drifts within lattices after a distance.

Fig. 5 (a), (b) Areas of existence and stability of gap solitons for self-defocusing diffusive nonlinearity on the (σ, μ) plane. (c) Energy flow curves versus μ for σ = 5. The inset shows the case in low power domain. (d) Energy flow versus σ at μ = 3.535, solid and dash lines indicate their linear stability and instability, respectively. Imaginary part of perturbation growth rate versus σ at μ = 3.535 are shown in the inset of this figure.

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Fig. 6 Profiles of two gap solitons at μ = 4, σ = 0.9 (a) and σ = 3(b) marked in Fig. 5(d), the corresponding perturbation eigenvalue spectrums are shown in the insets. Evolution of unstable (c) and stable (d) gap solitons corresponding to (a) and (b), respectively.

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4.2. Surface twisted solitons

In this self-defocusing nonlinearity case, the twisted surface solitons belonging to the first finite gap are also found. Similar to gap surface solitons in this model, the present soliton existence domain shrinks with decreasing σ at σ<-0.65 and can occupy the whole gap in the high modulated strength region, see Figs. 7(a) and 7(b) . Unlike the case in the semi-infinite gap with spatially modulated self-focusing nonlinearity, shown in Fig. 3, these twisted solutions are predicted to be unstable in their entire existence area by a comprehensive stability analysis based on Eqs. (3) and (4) . The perturbation growth rates on σ and the corresponding power diagram with same parameters are plotted in Fig. 7(c), which shows that the imaginary component of the complex rate δ is nonzero in whole existence domain. It is also shown that the perturbation growth rate falls with σ when the propagation constant approaches to the upper cutoff, while far from the cutoff maximum peak of imaginary part of δ is monotonically increased. Let μ _cr be the propagation constant at which the imaginary part of δ has its peak for a given modulated strength σ, dependence of μ _cr on σ is shown in Fig. 7(d). Regarding the evolution of these solitons under perturbations, two examples with μ = 2, σ = 0.2 and μ = 3.4, σ = 2, marked by circles in Fig. 7(c), are presented in Fig. 8 together with the soliton profiles, linear stability spectrums and dynamical evolutions, which shows that they always exhibit chain-type evolutions due to the oscillator instability.

Fig. 7 (a) Existence and instability regions for twisted solitons in the plane of (σ, μ) of the self-defocusing nonlinear system. (b) Energy flow versus μ for σ = −0.3 and 3. (c) Energy flow versus σ for μ = 2 and 3.4. δ _i versus σ are shown in the inset of this figure. (d) Dependence of μ _cr on σ. Dash lines indicate existence region.

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Fig. 8 Profiles and perturbation eigenvalue spectrums (inset) of twisted solitons at μ = 2, σ = 0.2 (a) and μ = 3.4, σ = 2 (b) marked in Fig. 7(c). (c) and (d) are unstable evolutions of twisted solitons corresponding to (a) and (b), respectively.

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5. Summary and discussion

To conclude, surface lattice solitons and their stability in diffusive nonlinear media with spatially modulated nonlinearity have been analyzed. Two families of lattice solitons, i.e., gap and twisted surface solitons can be formed at the interface of between a one-dimensional photonic lattice and a uniform medium with self-defocusing or self-focusing nonlinearity. It is shown that the properties of these surface solitons are sensitive to the spatially modulated strength, solitons existence and stabilities can be effectively controlled by adjusting the modulated strength. These surface solitons can realize in high modulated strength domain (σ>20) where these solitons have very low powers, and the minimum value of modulated strength σ approaches −1 in order to guarantee the signs of the nonlinear refractive index . Both gap and twisted surface solitons for the case of self-focusing nonlinearity can be founded in the semi-infinite gap, and bifurcate from the lower edge of Bloch band into the corresponding gap. Their existence curves expands to the asymptotic maximum with the growth of modulated strength. Gap surface solitons belonging to the semi-infinite gap can propagate stably in whole existence domain, twisted surface soliton are also linearly stable in lower modulated strength region, but, there is a very narrow unstable region near the upper cutoff in high modulated strength region, where the unstable propagation of perturbed surface soliton breaks up due to the oscillator instability. Under a self-defocusing nonlinearity, surface gap and twisted solitons can exist in first gap, for high modulated strength, these soliton can realize in the whole gap, but do not occupy the whole gap and shrinks with the decrease of σ in the lower modulated strength domain. Stability analysis shows that gap surface solitons can propagate stably in whole existence domain except for an extremely narrow region close to the Bloch band, and energy flows of unstable soliton are destroyed and tend to drift into the lattice after a distance. Twisted solitons belonging to this gap are unstable in the entire domain of their existence, and exhibit chain-type evolution under random initial perturbation. These results can be extended to other physical diffusive settings, including one-dimensional complex spatially modulated potentials and two-dimensional geometries, and may provide several novel ways for beam controlling in micro structure.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant No. 11247259), Shandong Provincial Natural Science Foundation, China (Grant No. ZR2012AQ005), and the Fundamental Research Funds for the Central Universities (Grant No. 14CX02156A).

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Surface lattice solitons in diffusive nonlinear media with spatially modulated nonlinearity

Abstract

1. Introduction

2. Theory model

3. Surface solitons and their stability for self-focusing diffusive nonlinearity

3.1. Surface gap solitons

3.2. Surface twisted solitons

4. Surface solitons and their stability for self-defocusing diffusive nonlinearity

4.1. Surface gap solitons

4.2. Surface twisted solitons

5. Summary and discussion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (4)

Optics Express