## Abstract

We study localized surface waves at the interface between linear dielectric and biased centrosymmetric photorefractive (CP) crystals. If the propagation constant *b* is fixed, the energy of localized surface waves increases with the order of localized surface waves. For low *b* values, the considerable part of the energy of localized surface waves is concentrated in the linear dielectric and decreases with an increase in *b*. For high *b* values, the part of the energy of localized surface waves concentrated in the nonlinear CP crystals is always higher than that in the linear dielectric and increases with *b*. The stability properties of these localized surface waves are also discussed in detail.

© 2013 OSA

## 1. Introduction

Optical spatial solitons in photorefractive media have attracted much attention due to the possibility of their observation at very low laser power levels and great potentials for applications in optical switching and routing. At present, many branches of photorefractive solitons are known: screening solitons [1, 2], photovoltaic solitons [3, 4], screening-photovoltaic solitons [5, 6], and solitons in centrosymmetric photorefractive (CP) crystals [7–10], all of which form in bulk photorefractive materials. On the other hand, the interface of linear and photorefractive materials can support surface waves [11–22]. Delocalized photorefractive surface waves at the interface between linear dielectric and biased photorefractive crystals [18] or unbiased photorefractive crystals [19], which were predicted and observed [20], have infinite energy due to the presence of the long slowly decaying oscillating tails going into the volume of the photorefractive crystals [20]. Of particular interest are localized photorefractive surface waves, which have not oscillating tails in the volume of the photorefractive crystals. Such localized surface waves at the interface between linear dielectric and biased photorefractive crystals have been predicted [21, 22]. However, it is not clear whether localized surface waves at the interface between linear dielectric and biased CP crystals are also possible.

In this paper, we analyze localized surface waves at the interface between linear dielectric and biased CP crystals. The lowest energy of localized surface waves appears when the propagation constant *b* is equal to a certain threshold value. When *b* is less than such a certain threshold value, the considerable part of the energy of localized surface waves is concentrated in the linear dielectric and decreases with an increase in *b*. When *b* is greater than such a certain threshold value, the part of the energy of localized surface waves concentrated in the nonlinear CP crystals is always higher than that in the linear dielectric and increases with *b*. The stability properties of these localized surface waves have been investigated numerically and we have found that they are stable.

## 2. Theoretical model

Let us consider an optical beam that propagates near the interface between linear dielectric occupying the half-space *x* ≥ 0 and centrosymmetric photorefractive crystals occupying the half-space *x* < 0 along the *z*-axis and is allowed to diffract only along the *x* direction. Moreover, let us assume that the beam is linearly polarized along the *x*-axis and that the external bias electric field is applied in the same direction. For illustration purposes, let the centrosymmetric photorefractive crystal be potassium lithium tantalate niobate (KLTN) with its optical *c*-axis oriented along the *x*-axis. In this case, the propagation of the optical beam is described by the standard shortened wave equations for the complex amplitude *A*(*x*, *z*) of the light field:

*n*is the unperturbed refractive index of the CP crystal,

*g*is the effective quadratic electro-optic coefficient,

_{eff}*ε*

_{0}and

*ε*, respectively, are the vacuum and relative dielectric constants,

_{r}*E*is the space-charge field inside the CP crystal,

_{sc}*k*= 2

*πn/λ*is the wave number in the area of the CP crystal,

*λ*is the free-space wavelength of the lightwave used,

*k*

_{0}= 2

*πn*

_{0}/

*λ*is the wave number in the area of the linear dielectric,

*n*

_{0}is the dielectric refractive index. For relatively broad beam configurations, the space-charge field can be obtained from the Kukhtarev-Vinetskii model and is approximately given by [2]

*I*= |

*A*|

^{2}is the intensity of the light beam,

*K*is the Boltzmann constant,

_{B}*T*is the absolute temperature,

*e*is the electron charge, and

*I*is the dark irradiance of the crystal,

_{d}*E*

_{0}is the value of the space-charge field at

*x*→ ±∞. If the spatial extent of the optical wave is much less than the

*x*-width

*l*of the CP crystal, then under a constant voltage bias

*V*,

*E*

_{0}is approximately given by ±

*V/l*. Moreover, for simplicity, let us adopt the following dimensionless coordinates, i.e., let $\xi =z/{k}_{0}{x}_{0}^{2}$ and

*s*=

*x/x*

_{0}, where

*x*

_{0}is an arbitrary transverse scale. By employing these latter transformations and by substituting Eq. (2) into Eq. (1b), the complex amplitude of the light field is found to satisfy:

The first term in the right-hand side of Eq. (3b) describes the diffraction spreading of the beam, the second term characterizes the interaction of the beam with the interface between linear dielectric and CP crystals, the third term describes the beam self-focusing caused by the local drift component of nonlinear response, and the the fourth and fifth term describe the beam self-bending of first and second orders caused by the diffusion component of the nonlinear response of the CP crystal, respectively. In this study, we consider the following examples: Let *E*_{0} = 2×10^{5}V/m, *λ* = 0.5μm, and *x*_{0} = 9μm. The KLTN parameters [9] are taken to be *n* = 2.2, *T* = 21°C, and *g _{eff}* = 0.12m4C

^{−2}. For this set of values,

*β*= 3.6,

*μ*

_{1}= 0.1, and

*μ*

_{2}= 0.00071.

## 3. Numerical results

For a solitary beam, the wave field amplitude *A* can be expressed as
$A=\sqrt{{I}_{d}}u\left(s\right)\text{exp}\left(ib\xi \right)$, where *b* is the real propagation constant and the envelope *u*(*s*) is the real function. Substitution of this form of *A* into Eqs. (3a) and (3b) yields

*u*and

*du/ds*should match the continuity conditions at the boundary point

*s*= 0. In the area of nonlinear CP medium, Eq. (4b) cannot be solved analytically and should be integrated numerically, for example, by the shooting method that reduces a two-point boundary problem to the Cauchy problem. The initial conditions are chosen by using the fact that in the area of linear dielectric the solutions of Eq. (4a) can be readily obtained and are given by

*u*=

*m*exp[−(2

*b*)

^{1/2}

*s*], where

*m*is the free parameter describing the strength of the nonlinear effects. By varying parameters

*b*and

*m*, and integrating Eq. (4b), we obtained various profiles of surface waves at the interface between linear dielectric and biased CP crystals.

Now let us classify all possible types of solutions of Eq. (4b) by using quite general treatment based on the analogy of Eq. (4b) for the envelope of the surface wave with the equation describing the motion of a mechanical particle in a potential well with nonlinear dissipation, where the wave envelope *u* is equivalent to the particle position and the transverse coordinate *s* is equivalent to time. In the area of the nonlinear CP crystal (*s* < 0) Eq. (4b) can be readily written in the following form:

*U*= (

*p*−

*b*)

*u*

^{2}+

*β*(1 +

*u*

^{2})

^{−1}−

*β*and

*T*= (1/2)(

*du/ds*)

^{2}are the potential and kinetic energies of the particle with unit mass, respectively, and the right-hand side of Eq. (5) shows the nonlinear friction force. Figure 1 depicts typical profiles of the potential well

*U*for different values of

*p*−

*b*when

*β*= 3.6. The potential well is symmetric with respect to the point

*u*= 0. This figure shows only the right part of the potential well corresponding to positive values of

*u*.

For *p*−*b* > *β*, the potential well has a single stable stationary point *u* = 0 (see dashed curve in Fig. 1). In this case, a mechanical particle with a nonzero initial energy *U* + *T* performs damped oscillations (as *s* varies from 0 to −∞), passing from the right wing of the potential well to the left wing and losing its energy because of the influence of nonlinear friction. When *s* → −∞, this particle asymptotically approaches the stable equilibrium position *u* = 0. This type of particle motion corresponds to the well-known delocalized surface waves [21], which have long oscillating tails in the volume of the CP crystal. Figure 2 shows the profiles of such waves for two different values of *m* when *β* = 3.6, *p* = 6, *b* = 1, *μ*_{1} = 0.1, and *μ*_{2} = 0.00071.

For 0 < *p* − *b* < *β*, the potential well has two stable (
$u=\pm \sqrt{\sqrt{\beta /\left(p-b\right)}-1}$) and one unstable (*u* = 0) stationary points, as shown in Fig. 1. A particle with a nonzero initial energy *U* + *T* will be periodically transferred from the right wing of the potential well (corresponding to positive *u*) into the left wing of the well (corresponding to negative *u*) until it stops at one of the two stable stationary points
$u=\pm \sqrt{\sqrt{\beta /\left(p-b\right)}-1}$ or at the unstable stationary point *u* = 0 because of the energy loss. The former case corresponds to the shock surface waves with an infinite energy [21], which have a nonzero asymptotic form at *s* → −∞. Figure 3 depicts typical profiles of the shock surface waves of the first three orders. Note that the order of a wave is defined by the number of intersections of its envelope with the *s*-axis, including the point *s* = 0. When *s* → −∞, shock surface waves represent damped oscillations superimposed on the constant background where height is given by
$\pm \sqrt{\sqrt{\beta /\left(p-b\right)}-1}$, as shown in Fig. 3. On the other hand, shock surface waves in the diffusion medium are highly unstable since they have zero harmonic in the spatial spectrum and are affected by modulation instability.

Of particular interest is the case when a particle describing a surface wave stops at the unstable stationary point *u* = 0. This situation corresponds to the formation of localized surface waves without oscillating tails in the volume of the CP crystal. For low *b* values, Fig. 4(a) depicts the profiles of localized surface waves of the first three orders, whereas Fig. 4(b) depicts the profiles of the second order localized surface waves for three different values of *b*. The energy
$W={\int}_{-\infty}^{\infty}{u}^{2}\left(s\right)ds$ conveyed by localized surface waves of first three orders versus the propagation constant is shown in Fig. 5. Figure 5 demonstrates that the lowest energy of localized surface waves appears when *b* is equal to a certain threshold value and the energy of localized surface waves increases with the order of localized surface waves when *b* is fixed. Moreover, Figs. 4(a) and 4(b) indicate that the considerable part of the energy of localized surface waves can be concentrated in the linear dielectric and decreases with an increase in *b*. For high *b* values, the profiles of localized surface waves of the first three orders are shown in Fig. 6(a), whereas the profiles of the third order localized surface waves for three different values of *b* are shown in Fig. 6(b). It is clearly seen from Figs. 6(a) and 6(b) that the part of the energy of localized surface waves concentrated in the nonlinear CP crystal is always higher than that in the linear dielectric and increases with *b*.

The stability properties of these localized surface waves will be discussed. In particular, their stability is investigated here using a beam propagation method. Localized surface waves given by Eqs. (4a) and (4b) have been used as the input beam profiles. As an example, consider the second order localized surface modes for *b* = 0.01 and 0.03 in Fig. 4(b). Figures 7(a) and 7(b) depict the evolution of the second order localized surface modes for *b* = 0.01 and 0.03 in Fig. 4(b) when their input amplitude have been perturbed by 20%. Moreover, consider the third order localized surface modes for *b* = 0.06 and 1 in Fig. 6(b). Figures 7(c) and 7(d) show the evolution of the third order localized surface modes for *b* = 0.06 and 1 in Fig. 6(b) when their input amplitude have been perturbed by 20%. The solitary behavior of these localized surface modes is evident in these figures since they propagate unchanged. Similarly, a stability study of other localized surface modes in Figs. 4(b) and 6(b) shows that they are also stable.

Finally, for *p* − *b* < 0, the potential well has a single unstable stationary point *u* = 0 (see dash-dot curve in Fig. 1). In this case, the finite motion of a particle with a nonzero energy *U* + *T* is not possible and it is useless to speak about surface waves.

## 4. Conclusion

In conclusion, we have studied localized surface waves at the interface between linear dielectric and biased CP crystals. We have shown that when *b* is fixed, the energy of localized surface waves increases with the order of localized surface waves. We have found that for low *b* values the considerable part of the energy of localized surface waves is concentrated in the dielectric and decreases with an increase in *b* and that for high *b* values the part of the energy of localized surface waves concentrated in the nonlinear CP crystals is always higher than that in the linear dielectric and increases with *b*. The stability of these localized surface waves have been investigated numerically and it has been found that they are stable.

## Acknowledgment

The work was supported by the Natural Science Foundation of Chinese Tianjin (No. 13JCY-BJC16400) and by the National Natural Science Foundation of China (Nos. 10674176 and 10474136).

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