Lattice surface solitons in diffusive nonlinear media driven by the quadratic electro-optic effect

Kaiyun Zhan; Chunfeng Hou

doi:10.1364/OE.22.011646

1. Introduction

The surface soliton propagation dynamic at the interface of nonlinear media with periodic refractive-index has become a considerate topic in nonlinear optics for their potential important applications in optical sensing, switching and exploration of intrinsic and extrinsic surface characteristics [1]. There are known two simplified approaches to achieve stable surface solitons at the interface of nonlinear media. One of the physical mechanisms that support the existence of surface solitons is the nonlocal nonlinearity. In a highly nonlocal nonlinear medium, the refractive index is determined by the intensity distribution over the entire transverse plane. In this case, both the sample geometry and bounded conditions strongly affect the trajectory and stability of surface solitons. Surface solitons with weak or strong nonlocality have been investigated systematically in diffusive Kerr-type materials [2, 3], nonlocal thermal materials [4–7] and liquid crystals [8, 9]. Because some periodic structures with or without defects induced by nonlinearity can confine a light beam propagating along the surface, many fantastic optical surface solitons supported by photonic and optical lattices have been a subject of intense study in Kerr nonlinear media [10, 11] and photorefractive media with drift nonlinearity, since the depth, period, and structure of these lattices can be easily modified in experiment [12–17].

It is well known that the photorefractive nonlinearity is classified as drift, photovoltaic and diffusion, among which drift and photovoltaic are local nonlinearity inducing beam focusing or defocusing, and diffusion is nonlocal nonlinearity inducing beam bending [18–21]. Xu reveals that the domains of existence and the stability of gap solitons supported by optical lattices in photorefractive crystals are strongly depend on the diffusive effect. Both the diffusion and lattice strength modify the gap soliton mobility. Under the action of diffusive response, the soliton center jumps into a neighboring lattice channel, with the number of jumps depending on the amplitude of the input beam [22].

In fact, the pure nonlocal diffusive nonlinearity can support stable spatial soliton states. Christodoulides [23] and Crosignani [24] have predicted one-dimensional soliton states can propagate undistorted by means of exploiting the diffusion photorefractive nonlinearity in unbiased noncentrosymmetric and centrosymmetric photorefractive uniform media, respectively. Recently, Kartashov et al. [25] have just shown that surface solitons supported by optical lattices can be formed in unbiased photorefractive media with asymmetrical diffusion nonlinearity due to linear electro-optic effect. In contrast to the case of interface of uniform diffusive medium, such solitons exist when diffusion nonlinearity would cause light bending not only toward the lattice edge but also when it bends light against it.

In this Letter we reveal that nonlocal diffusion nonlinearity can result in surface soliton formation at the interface produced by one-dimensional optically lattices in unbiased centrosymmetric photorefractive media, where the change in refractive index is governed by quadratic electro-optic effect [26]. Two types of surface lattice solitons were found in this paper, self-focusing surface solitons, residing on the semi-infinite gap as well as self-defocusing surface solitons with eigenvalues in the first gap. We also demonstrate that in addition to surface gap solitons which bifurcate from infinitesimal linear Bloch modes, there is also a family of twisted surface solitons in the first gap with a self-defocusing nonlinearity, which can be considered combination of several odd solitons. The stabilities of these surface solitons are also investigated.

2. Theory model

We consider the propagation of an optical beam at an interface of a semi-infinite lattice imprinted in a highly diffusive nonlinear medium with saturable nonlinearity driven by the quadratic electro-optic effect. In centrosymmetric media, the change in refractive index can be expressed as $Δ n = Δ n_{L} + Δ n_{P R}$ [27], $Δ n_{P R} = - n_{e}^{3} g_{eff} ε_{0}^{2} {(ε_{r} - 1)}^{2} E_{sc}^{2} / 2$ is the photoinduced nonlinearity perturbation, and the lattice pattern is described by $Δ n_{L}$ , where $n_{e}$ is the unperturbed index of refraction, g_eff is the effective quadratic electro-optic coefficient, ε₀ and ε_r are the vacuum and relative dielectric constants, respectively. E_sc is the space charge field in the material resulting from the external bias electric field and the displacement of charge. Under unbiased condition, E_sc depends on only the nonlocal diffusive nonlinearity [21]. The propagation dynamic of the beam is governed by the following one-dimensional nonlinear Schrödinger (NLS) equation for the dimensionless complex amplitude of the light field q:

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

where the transverse s and longitudinal ξ coordinates are scaled to the input beam width and diffraction length, respectively;

β = \pm 1

with the upper (lower) sign corresponding to a self-defocusing (self-focusing) nonlinearity; the parameter p characterizes the scaled lattice depth; the function

R (s) = 0

at

s < - π / 2

and

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

at

s \geq - π / 2

describe the transverse periodic lattice modulation pattern. We stress that this type of nonlinearity put forward here is appropriate in unbiased centrosymmetric photorefractive crystals, a prime example of the typical centrosymmetric photorefractive crystals is paraelectric Potassium lithium tantalite niobate (KLTN), where the change in refractive index is governed by quadratic electro-optic effect, and such above lattice can be technologically fabricated, for example, by the method of using focused light beams coupled into the waveguide arrays [28], or induced optically in this type of crystal whose photosensitivity strongly depends on wavelength and a sharp transition between the uniform region lattices might be created by erasing part of the imprinted lattice [11]. In this model, the saturable parameter

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

, where

k_{0} = 2 π / λ_{0}

,

λ_{0}

is the free-space wavelength. One character of operating a centrosymmetric photorefractive material in paraelectric phase is that the dielectric constant is strong temperature-dependent. In the mean-field approximation region, the dependence on the temperature follows the Curie-Weiss law with a dependence of

{(T - T_{c})}^{- 1}

, where T_c is the Curie temperature [29, 30]. On the basis of this principle, we can adjust the crystal temperature to change surface soliton properties. Equation (1) conserves the power or energy flow

P = \int_{- \infty}^{\infty} {| q |}^{2} d s

.

We search the stationary soliton solutions of Eq. (1) in the form $q (s, ξ) = u (s) \exp (i μ ξ)$ that can be characterized by the propagation constant $μ$ , which should fall into the gaps of the lattice spectrum, as shown in Fig. 1(a). All possible propagation constant values lay in gaps (white regions) where Eq. (1) admits Bloch wave solutions, while in the bands (gray regions) periodic waves do not exist. These gaps from top to bottom are called sequentially as the semi-infinite gap, first gap, second gap, and so on in this paper. Without loss of generality, we choose the particular value p = 5 for the lattice normalized parameter in all the calculations. In this case, we find that the region of semi-infinite gap is $μ \geq 3.577$ 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 were solved numerically by a relaxation method. To elucidate the stability of surface lattice solitons driven by the quadratic electro-optic effect, we search for perturbed solutions of Eq. (1) in the form [31]

q (s, ξ) = {u (s) + [v_{1} (ξ) - v_{2} (ξ)] e^{i δ ξ} + [v_{1}^{*} (ξ) + v_{2}^{*} (ξ)] e^{- i δ^{*} ξ}} e^{i μ ξ},

where v₁ and v₂ are small perturbations with the complex growth rate

δ = δ_{r} + i δ_{i}

and the asterisk means complex conjugation. The standard linearization procedure around the localized solution u(s) for Eq. (1) yields the following linear eigenvalue problem

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

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

for growth rate δ and perturbation components v₁, v₂, which were solved numerically by a finite-difference method. For

δ_{i} = 0

, the solitons are linearly stable; otherwise, they are linearly unstable.

Fig. 1 (a) Gap structure of a uniform lattice for different lattice depth p. Gray regions show bands and white regions correspond to gaps. (b) Energy flow versus propagation constant for β = −1at S = 1, 1.5 and 5. (c) Existence domain of surface solitons versus S.

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3. Surface lattice solitons supported by optical lattices with a self-focusing nonlinearity

First, we consider the properties of localized waves supported by an interface of a lattice imprinted in a diffusive nonlinear media with a self-focusing nonlinearity ( $β = - 1$ ). We find that surface solitons can exist in the semi-infinite gap, and bifurcate from the lower edge of Bloch band into the corresponding gap, as illustrated in Fig. 1(b), The dramatic increase of power with propagation constant at higher saturable parameter is attributed to the strong coupling of self-focusing nonlinearity at the surface of lattice, thus creating the enhanced diffraction, which in turn requires stronger nonlinearity to balance it. Figure 1(b) also shows that the energy flow P is a monotonic function of the propagation constant, and it increases with μ in the existence domain. Based on Vakhitov-Kolokolov (VK) stability criterion, this family of surface solitons is VK stable in their existence region, which is similar to other fundamental solitons [32]. Moreover, Fig. 1(b) also shows that the soliton existence domain becomes much narrower with the increase of S. There exists both lower and upper cutoffs on the propagation constant for surface solitons as depicted in Fig. 1(c), and the existence domain shrinks with the growth of saturation parameter. The upper cutoff decreases monotonically and closes to the lower one with increasing S. When saturation parameter exceeds a critical value $S_{c r} \approx 9$ , surface solitons cannot be found in the semi-infinite gap.

Representative profiles of three solitons belonging to the semi-infinite gap are shown in Figs. 2(a)–2(c), which show that these soliton energy distributions center in the nearest-to interface lattice and become more localized with increasing of power flow. It is shown that, for a fixed μ, both the power and amplitude of surface soliton decrease as the S. This is because the diffusive nonlinearity governed by the soliton light beam without sufficiently high intensity cannot compensate the lattice with this soliton, stronger nonlinearity (higher S) is necessary to support surface soliton with higher peak intensity. The instability criterion defined by the VK stability criterion can be confirmed by a rigorous linear stability analysis. Based on the study of the imaginary part of the growth rate of the perturbation δ_i versus the propagation constant µ, we found that there was no complex or purely imaginary in their existence region, that is to say, this family of surface solitons is completely stable in the entire existence domain. Figures 2(d)–2(f) show the linearization eigenvalue spectrums of surface solitons corresponding to Figs. 2(a)–2(c). To confirm the stability analysis results, we numerically integrate Eq. (1) with a standard beam propagation method code, using the stationary solutions under 10% random initial perturbations as the initial inputs. Their evolutions are shown in Figs. 2(g)–2(i) corresponding to profiles displayed by Figs. 2(a)–2(c), respectively. We see that these solitons are robust and exhibit stable evolution under perturbation. We have also tried several other random-noise perturbations, and the evolution results are qualitatively the same.

Fig. 2 Examples of surface soliton profiles at (a) S = 1.5, µ = 3.6; (b) S = 1.5, µ = 4.2; (c) S = 0.5, µ = 4.2. (d)– (e) are complex planes of the respective stability eigenvalues corresponding to (a)–(c). (g)– (i) are the stable evolutions of surface solitons under 10% random initial perturbation corresponding to (a)–(c), respectively.

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4. Surface lattice solitons supported by optical lattices with a self-defocusing nonlinearity

In the model with the self-defocusing nonlinearity, i.e., $β = 1$ , surface gap solitons can be readily found in first finite gap. It is shown that such surface lattice solitons do not occupy the whole gap in the higher saturable parameter domain, shown in Fig. 3(a). There also exists both lower and upper cutoffs on the propagation constant for surface gap solitons as depicted in Fig. 3(c). The upper cutoff remains basically the same and the lower one increases with increasing S. That is to say, the domain of soliton existence becomes narrower and shrinks completely at S_cr≈10. A similar picture takes place for surface solitons at the self-focusing interface, with the only difference being that the upper cutoff on the surface gap solitons with self-focusing nonlinearity decreases and lower cutoff remains fairly constant as S rises.

Fig. 3 (a) Energy flow versus propagation constant for β = 1at S = 0.5, 1.5 and 3. (b) Energy flow curves in low power domain. (c) Existence domain of surface solitons versus S at β = 1.

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Generic examples of surface gap solitons in first finite gap are displayed in Figs. 4(a) and 4(b). Figures 4(c) and 4(d) show the complex planes of the respective stability eigenvalues corresponding to Figs. 4(a) and 4(b). It is shown that surface solitons in the first gap exhibit exponential instability in an extremely narrow region near the edge of Bloch band, where the slope of power curve changes to a positive one ( $d P / d μ > 0$ ), see Fig. 3(b). Linear stability analysis also revealed the presence of very weak oscillatory instability in the vicinity of the edge of Bloch band. This behavior is attributed to the resonant energy redistribution between the gaps which is similar to the gap solitons in other periodic optical systems [31]. The linear stability of surface gap solitons discussed above is further corroborated by direct numerical simulations of these solitons under perturbations. The conclusion of the stability analysis is that except for a very narrow region close to Bloch band almost all surface solitons are stable. To demonstrate, we again choose the surface solitons at µ = 2.5 and 3.534, and perturb them by 10% random-noise perturbation. Figure 4(e) illustrates the stable propagation of the perturbed surface soliton at µ = 2.5 which retains its input structure during the distance. When µ = 3.534, development of instability causes pronounced amplitude oscillation and transformation of the internal structure of unstable soliton, shown in Fig. 4(f), and the intensity distribution of such soliton profile is found drifting within lattices after a distance.

Fig. 4 Examples of surface soliton profiles at (a) S = 1.5, µ = 2.5; (b) S = 1.5, µ = 3.534. (c)– (d) are complex planes of the respective stability eigenvalues corresponding to (a)–(b). (e)– (f) are the evolution of surface solitons under 10% random initial perturbation corresponding to (a)–(b), respectively.

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Besides the above surface gap solitons associated with the first gap, we have found several other types of surface soliton for the self-defocusing nonlinearity which have more complex profiles in general. An example of a field profile of a twisted mode is depicted in Fig. 5(c). The envelope of a twisted solution is in phase within the first channel, while among neighboring elements of the lattice a phase shift of π appears. Figure 5(a) shows that the energy flow P for both twisted and above mentioned surface gap solitons are a monotonically decaying function of the propagation constant µ, and the power of twisted soliton is higher than that of surface gap soliton at the same propagation constant. This behavior is similar to the bifurcation of gap solitons in periodic potential with local nonlinearity. Linear stability analysis reveals that twisted solitons are unstable in the entire domain of their existence, similar to the even solitons in local [33] and Kerr-type nonlocal media [34]. The perturbation growth rate for twisted solitons is drastically reduced with a decrease of S [Fig. 5(b)]. Figure 5(d) shows the unstable evolution of surface twisted soliton corresponding to profile displayed by Fig. 5(c) at µ = 2.5 in the first gap, and always exhibits chain-type evolution due to the oscillator instability, which is different from that in Kerr-type nonlocal optical lattices where the twisted solitons move across the lattice and are accompanied by radiation losses that eventually lead to soliton capture in one of the lattice channels due to the exponential and oscillatory instabilities [34].

Fig. 5 (a) Energy flow of twisted (solid line) and surface gap (dashed line) solitons versus propagation constant for β = 1. (b) Imaginary part of perturbation growth rate versus propagation constant. (c) Profile of twisted surface soliton at μ = 2.5. (d) Evolution of surface soliton under 10% random initial perturbation corresponding to (c).

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Finally, it is interesting to compare the above results with those previously obtained in noncentrosymmetric photorefractive crystals where the change in refractive index is governed by linear electro-optic effect. First of all, Fig. 3(c) shows that surface lattice solitons supported by optical lattices with a self-defocusing nonlinearity can occupy the whole first finite gap in the lower saturable parameter domain. This behavior is different from that in noncentrosymmetric photorefractive crystals where the surface solitons do not occupy the whole gap [25]. Physically, the difference is attributed to the fact that spatial solitons resulted from quadratic electro-optic effect require a smaller nonlinearity than that in noncentrosymmetric photorefractive crystals with linear electro-optic effect [26]. Second, in the self-focusing case, our results show that the intensity profiles of surface solitons in semi-finite gap are entirely positive and localized at the input channel. However, surface solitons from first gap driven by the linear electro-optic effect exhibit multi-humped profiles, become less localized with decreasing propagation constant and feature an almost linearly decreasing tail inside the uniform medium [25]. Furthermore, we present what is to our knowledge the first prediction of twisted surface solitons in diffusive nonlinear media driven by the quadratic electro-optic effect, these results can be directly extended to the case of saturable diffusive nonlinear media with linear electro-optic effect. Finally, as pointed out above, the saturable parameter S is one of the key parameters for control of soliton properties. It is well known that saturable parameter S is strongly dependent on the crystal temperature through three physical factors, i.e., the dielectric constant, diffusion process and the dark irradiance, which is different from that in noncentrosymmetric photorefractive crystals where only the diffusion process and the dark irradiance are considered. This suggests that our results can provide other new way of optical control scheme for optical solitons.

5. Conclusions

In this article, we have thoroughly studied surface lattice solitons and their stability properties in diffusive nonlinear media driven by the quadratic electro-optic effect with self-focusing and self-defocusing saturable nonlinearity. This type of lattice and nonlinearity is appropriate in unbiased centrosymmetric photorefractive crystals. Both in the self-focusing and self-defocusing cases, powers of surface solitons dramatic vary with propagation constant at higher saturable parameter. Under a self-focusing nonlinearity, it was shown that surface gap solitons can exist in the semi-infinite gap, and bifurcate from the lower edge of Bloch band into the corresponding gap. These surface solitons belonging to the semi-infinite gap can propagate stably in whole existence domain. Surface gap solitons for the case of self-defocusing nonlinearity can realize in first gap but are absent in the semi-infinite gap, and these surface solitons can propagate stably in whole existence domain except for an extremely narrow region close to the Bloch band, where such solitons suffer from a exponential instability, and energy flows are destroyed and tend to drift into the lattice after a distance. Lastly, we demonstrate that in first gap, in addition to surface gap solitons which bifurcate from infinitesimal linear Bloch modes, the lattice in a defocusing medium can support specific solitons with more complex profile that are termed twisted solitons. The power of twisted solitons is higher than that of surface gap soliton at the same propagation constant. Moreover, it is shown that these twisted solitons are unstable in the entire domain of their existence, and exhibit chain-type evolution under random initial perturbation.

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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Lattice surface solitons in diffusive nonlinear media driven by the quadratic electro-optic effect

Abstract

1. Introduction

2. Theory model

3. Surface lattice solitons supported by optical lattices with a self-focusing nonlinearity

4. Surface lattice solitons supported by optical lattices with a self-defocusing nonlinearity

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (4)

Optics Express