## Abstract

We report on the existence and stability of solitons in a defect embedded in a square optical lattice based on a photorefractive crystal with focusing saturable nonlinearity. These solitons exist in different bandgaps due to the change of defect intensity. For a positive defect, the solitons only exist in the semi-infinite gap and can be stable in the low power region but not the high power region. For a negative defect, the solitons can exist not only in the semi-infinite gap, but also in the first gap. With increasing the defect depth, these solitons are stable within a moderate power region in the first gap while unstable in the entire semi-infinite gap.

© 2010 OSA

## 1. Introduction

Solitary waves have been researched widely in nonlinear media. Recently, the existence of solitons in optical lattices with nonlinearity has attracted vast attention [1–5]. In a periodic optical medium, linear light propagation exhibits Bloch bands and forbidden bandgaps. Gap solitons form by the nonlinear coupling between forward- and backward-propagating waves when both experience Bragg scattering from the periodic structures. The interplay between nonlinearity and periodicity results in many new effects in nonlinear waves. For instance, solitons in optical lattices can exist not only under focusing nonlinearity, but also under defocusing nonlinearity [6–8]. Gap solitons can exist in different bandgaps [5,9]. Optical lattices support different types of solitons such as ground state solitons, twisted solitons, complex soliton trains, and vortex solitons [10–13]. In experiment, periodic structures can be created by various methods such as sophisticated fabrication techniques, laser writing, and optical induction techniques [1]. These methods provide an advantage for us to research spatial solitons. For instance, two-dimensional (2D) lattice solitons have been observed in optical lattices with various geometry structures [14–16].

2D linear defect modes can be moved from lower bandgaps to higher ones by changing the defect strength [17]. Linear defect modes at different wavelengths were observed in 2D optical lattices with a defect [18]. Moreover, 2D defect surface solitons were observed experimentally in a hexagonal waveguide array [19]. Therefore, it is interesting to know whether 2D optical lattices with a defect can support localized nonlinear modes. We report on that defect solitons (DSs) in 2D square optical lattices with a defect can exist in different bandgaps when the defect strength (or defect intensity) is changed. The stability of the 2D defect solitons is also studied analytically and numerically.

## 2. The model

Here we consider that a probe beam is launched at the defect site embedded in 2D square optical lattice which is optically induced in a based on photorefractive crystal with a focusing saturable nonlinearity. The probe beam is an extraordinary beam which experiences a large electro-optic coefficient. Meanwhile, the lattice beam, which is an ordinary light, is launched through an amplitude mask to form the square optical lattice with a defect. Therefore, the propagation of the probe beam through the defect-embedded 2D optical lattice with focusing saturable nonlinear media is described by the nonlinear Schrödinger equation [1,10]

*U*is the slowly varying amplitude of the probe beam and

*I*is the transverse intensity distribution of the lattice beam with a defect as described by

_{L}*I*is the lattice peak intensity,

_{0}*z*is the propagation distance (in units of $2{k}_{1}{D}^{2}/{\pi}^{2}),$ ${k}_{1}={k}_{0}{n}_{e}$,

*n*is the unperturbed refractive index along the extraordinary axis, ${k}_{0}=2\pi /{\lambda}_{0}$ is the wave number (${\lambda}_{0}$ is the wavelength in vacuum),

_{e}*D*is the lattice spacing,

*x*and

*y*are the transverse distances (in units of $D/\pi $),

*E*is the applied dc field voltage [in units of ${\pi}^{2}/({k}_{0}^{2}{n}_{e}^{4}{D}^{2}{\gamma}_{33})$], ${\gamma}_{33}$is the electro-optic coefficient of the crystal for the extraordinary polarization, and

_{0}*ε*is the modulation parameter for the defect intensity. For a positive defect

*ε*>0, the lattice intensity

*I*at the defect site is higher than that in the surrounding regions. For a negative defect

_{L}*ε*<0, the lattice intensity

*I*at the defect site is lower than that in the surrounding regions. The intensity distributions of optical lattices with a negative defect (

_{L}*ε*= −0.5) and a positive defect (

*ε*= 0.5) are displayed in Figs. 1(b) and 1(c), respectively. In this paper, we choose the physical parameters as$D=30\mu m$,${\lambda}_{0}=0.5\mu m$,${n}_{e}=2.3$,${\gamma}_{33}=280pm/V$. Thus, one

*x*or

*y*unit corresponds to 6.4

*μm*, one

*z*unit corresponds to 2.3

*mm*, and one

*E*unit corresponds to 20

_{0}*V/mm*. We take

*I*= 3 and

_{0}*E*= 6, which are typical experimental conditions [11].

_{0}In order to obtain the soliton solutions in the bandgaps, we search for the Floquet-Bloch spectrum by substituting $U(x,y,z)=u(x,y)\mathrm{exp}[i({k}_{x}x+{k}_{y}y)-i\mu z)]$ into the linear version of Eq. (1) and obtain eigenvalue equation as follow

*u*(

*x*,

*y*) possesses the same periodicity as the lattices,

*μ*is a real propagation constant,

*k*and

_{x}*k*are wave numbers in the first Brillouin zone. We calculate this equation by the plane wave expansion method to obtain the bandgap diagram as shown in Fig. 1(a). Using above given parameters, we can obtain the boundaries of the allowed bands and provide the range of the semi-infinite gap and the first gap as $\mu <3.58$ and $4.41<\mu <5.55$, respectively.

_{y}The stationary soliton solutions in the model Eq. (1) are sought in the form$U(x,y)=u(x,y)\mathrm{exp}(-i\mu z)$, where *u*(*x*, *y*) is a real-valued function and satisfies the nonlinear equation

*u*(

*x*,

*y*) can be calculated by the modified squared-operator iteration method [20]. The power of solitons is defined by $P={{\displaystyle {\int}_{-\infty}^{\infty}{\displaystyle {\int}_{-\infty}^{\infty}\left|u\right|}}}^{2}dxdy$.

To indicate the stability of defect solitons, we perturb them as $U(x,y,z)=\{u(x,y)+[v(x,y)-w(x,y)]\mathrm{exp}(\delta z)+{[v(x,y)+w(x,y)]}^{*}\mathrm{exp}({\delta}^{*}z)\}\mathrm{exp}(-i\mu z)$, where the superscript “*” represents the complex conjugation, and $v,w<<1$. Substituting this perturbation into Eq. (1) and linearizing it, we can get

We can obtain the growth rate Re(δ) by a numerical method [21]. If Re Re(δ)>0, the DSs are linearly unstable, otherwise they are linearly stable.

## 3. Numerical results

To further study the DSs’ robustness, we add a noise to the inputted DSs by multiplying them with$[1+\rho (x)]$ and let them propagate along the *z* direction by 200 units distance, where $\rho (x)$ is a Gaussian random function with $<\rho >=0$ and $<{\rho}^{2}>={\sigma}^{2}$ (The adopted *σ* is equal to 10% of the input soliton amplitude).

First, we choose *ε* = −0.5 as a typical case for the negative defect lattice. The DSs exist not only in the semi-infinite gap, but also in the first gap. Figure 2(a)
shows the power of DSs versus propagation constant *μ*. In the semi-infinite gap, the DSs can stably exist in the range of $2.42\le \mu \le 3.33$. As an example, Fig. 2(c) shows the profile of DS for *μ* = 2.8 [point A in Fig. 2(a)]. The DSs are trapped at the defect site and maintain their profiles at z = 100 [Fig. 2(d)] and z = 200 [Fig. 2(e)]. The VK criterion tells us that the solitons are stable if dp/dμ<0 and unstable if dp/dμ>0 [5]. Apparently, in the range of $2.42\le \mu \le 3.33$, where dp/dμ<0, the DSs are stable according to VK criterion. However, we can see in Fig. 2(a) that the power of DSs does not monotonically decrease with propagation constant when $\mu \ge 3.34$, which is similar to that of on-site lattice solitons near the first band [22]. The DSs cannot stably exist in the region close to the first band since dp/dμ>0, which accords with the VK criterion. Figure 2(f) shows the profile of DS for *μ* = 3.4 [point B in Fig. 2(a)]. Figures 2(g) and 2(h) show the profiles of DS at z = 100 and z = 200, respectively, where the DS cannot maintain its profile upon propagation. When $\mu \le 2.41$, the power of DSs exponentially grows with decreasing propagation constant. Apparently, the DSs are exponentially instable in the high power region and such instability is different from the usual VK instability caused by a sign change in the slope of the power diagram [5]. Such instability property is inherited from that of defect solitons in one-dimension (1D) optical lattices which also possesses an exponential instability [5]. The physical explanation for such phenomenon is not yet clear and needs further investigation. As an example, we give the profile of DS for *μ* = 2.4 [point C in Fig. 2(a)] as shown in Fig. 3(a)
. The DS cannot be trapped at the defect site at z = 100 [Fig. 3(b)] and z = 200 [Fig. 3(c)]. To further verify solitons’ instability, we numerically calculate Eq. (5) to obtain the perturbation growth rate Re(δ) as shown in Fig. 2(b). In the first gap the DSs can stably exist. The power of DSs is gradually decreasing with propagation constant, where dp/dμ<0 as shown in Fig. 2(a), which implies that the DSs are stable according to VK criterion. Taking *μ* = 4.5 [point D in Fig. 2(a)], Figs. 3(d), 3(e), and 3(f) show the profiles of DS at z = 0, z = 100, and z = 200, respectively. When *μ* = 4.7 [point E in Fig. 2(a)], we also give the profiles of DS at z = 0, z = 100, and z = 200 as shown in Figs. 3(g), 3(h), and 3(i), respectively. In the first gap these solitons can maintain their shapes with most of their energies concentrated on the defect site along propagation.

Next, with increasing defect depth, we find that the DSs are not always stable in the first gap. We take *ε* = −1 as an example. Figure 4(a)
shows the power of the solitons versus propagation constant. The DSs can stably exist only in $4.77\le \mu \le 5.06$, according to the diagram of growth rate Re(δ) as shown in Fig. 4(b). Figure 4(c) shows the profile of DS for *μ* = 5.0. Figures 4(d) and 4(e) are the profiles of DS at z = 100 and z = 200, respectively. In the first gap, near the first band ($\mu \le 4.76$) the DSs are unstable in the high power region although the slope of the power diagram satisfies the VK stability criterion. As an example, we give the profile of DS for *μ* = 4.68 [Fig. 4(f)] and its profiles at z = 100 [Fig. 4(g)] and z = 200 [Fig. 4(h)], respectively. The DSs finally break up. In the lower power region we find that the power of DSs isn’t monotonically decreasing with the propagation constant. When $\mu \ge 5.07$, the DSs cannot stably exist. As an example, taking *μ* = 5.08, Figs. 5(a)
, 5(b), and 5(c) show the profiles of DS at z = 0, z = 100, and z = 200, respectively. We can see that the DS completely breaks at z = 200. In the semi-infinite gap, although we can obtain the soliton solutions, they are unstable. As an example, we also give the profile of DS for *μ* = 3.0 [Fig. 5(d)] and its profiles at z = 100 [Fig. 5(e)] and z = 200 [Fig. 5(f)], respectively. The DSs cannot be trapped at defect site along propagation.

Finally, we choose *ε* = 0.5 as a typical case for the positive defect lattice. Figure 6(a)
shows the power of DSs versus propagation constant *μ*. The DSs only exist in the semi-infinite gap. Moreover, the DSs are stable in the low power region but unstable in the high power region [Fig. 6(a)], which is similar to that of defect solitons in 1D optical lattices [5]. We give the profiles of DSs for *μ* = 3.0 [point B in Fig. 6(a)] and *μ* = 1.4 [point A in Fig. 6(a)] as shown in Figs. 6(c) and (f), respectively. For *μ* = 3.0, the DS can keep its profiles at z = 100 [Fig. 6(d)] and z = 200 [Fig. 6(e)], but for *μ* = 1.4 the DS cannot keep its original shape at z = 100 [Fig. 6(g)] and z = 200 [Fig. 6(h)]. The perturbation growth rate Re(δ) is not zero that means the DSs are unstable in the case $\mu \le 1.79$ but stable in the case $\mu >1.79$ in the semi-infinite gap [Fig. 6(b)]. We notice that several 2D defect soliton solutions in positive defect lattices have been shown in Ref [23] for routing and control light propagation. Provided no band-gap diagram and instability analysis in the reference, we can only judge from the profiles of those defect solitons that they might belong to those shown in Figs. 6(e) and 6(f) in this paper.

By the way, we also investigated the stability of the DSs for various lattice strengths. We found that higher lattice strength usually possessed a narrowed stable region. Therefore, a change of lattice strength would not affect the generality of our discussion.

## 4. Summary

To summarize, we have revealed the existence of DSs at the defect site embedded in the square optical lattices with focusing saturable nonlinearity. For a negative defect, the DSs can exist in both of the semi-infinite gap and the first gap. When the defect depth is moderate, in the semi-infinite gap the DSs are only stable within a moderate power region, while they are stable in the entire first gap. However, with increasing defect depth, the DSs can only stably exist in the moderate power region in the first gap. For a positive defect, the DSs exist only in the semi-infinite gap and are stable in the low power region but not in the high power region.

## Acknowledgments

We are very grateful to Dr. X. S. Wang for his helpful discussion. This work is supported by the National Natural Science Foundation of China (10874047). W. H. Chen and X. Zhu equally contribute to this work.

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