Elsevier

Optics Communications

Volume 300, 15 July 2013, Pages 249-256
Optics Communications

Defect modes and solitons supported by optically induced lattices in highly saturable nonlinear media with quadratic electro-optic effect

https://doi.org/10.1016/j.optcom.2013.03.008Get rights and content

Abstract

We theoretically investigate the defect modes and defect solitons supported by optically induced lattices in highly saturable nonlinear media with quadratic electro-optic effect. It is shown that positive and negative defect modes bifurcate from the right and left edge of each band into the bandgaps, respectively, and the distances between the eigenvalues of defect mode and band edge increase with the defect strength. Defect solitons bifurcate from every infinitesimal linear defect mode. When the defect is attractive, lower power defect soliton can propagate stably, but the soliton with high intensity are unable. In repulsive defects, defect soliton are stable in entire regions of their existence in lower bandgaps but only the higher power solitons are stable in higher bandgaps. Especially, when the repulsive defect is strong, both the defect mode and soliton power diagram branches terminate on the right edge of this gap and then appear in higher band gaps, and these solitons can propagate stably in whole soliton existing regime.

Introduction

Spatial optical solitons are robust self-localized optical beams that evolve with a stationary intensity profile when the medium nonlinearity exactly balances diffractive spreading. Several mathematical models are commonly accepted for description of soliton evolution in nonlinear optical materials [1]. Among these optical solitons, photorefractive optical solitons forms a specific branch, thanks to their feature of formation at low laser power level, stability in more than one transverse dimension due to the saturable nonlinearity and a variety of potential important applications in all optical switching and signal processing [2], [3], [4], [5], [6]. These solitons arise from the index change produced by the photorefractive effect to a photoinduced internal field, and the perturbed index nonlinear change Δn for incident beam slowly varying envelope U is scales with [γ1/(1+|U|2)2/(m+1)] with γ=0 or 1, where m is material parameter. For the case γ=1, m=1, the above model reduces to that in photovoltaic photorefractive (PR) crystals [4]; γ=0, m=1 and γ=0, m=0 correspond to the noncentrosymmetric and centrosymmetric PR crystals due to Kerr electro-optics effects (in fact, 90% of the point group classes lacking inversion symmetry can provide pure Kerr electro-optics effects in appropriate optical configurations) [6], [8]; for photorefractive polymers [7], γ=0 and m can ranges from less than 1 to greater than 3. Especially, Model with highly saturable nonlinearities are the most typical ones in nonlinear optics. A prime example of the typical centrosymmetric photorefractive crystals is paraelectric Potassium lithium tantalite niobate (KLTN) which exhibits enhanced photorefractive properties under external electric field, for combing the advantages of high diffraction efficiency, fast electro-optic response time and very large Kerr coefficients, and has been considered as the best material for electroholographic applications [9], [10]. One and two dimensional spatial solitons in this kind of PR crystals have been systematically investigated by DelRe et al. [11], [12]. these solitons resulted from quadratic electro-optic effect (dc Kerr effect) require a smaller nonlinearity than that in noncentrosymmetric PR crystals with linear electro-optic effect (Pockels effect).

A new direction in soliton physics is to predict novel soliton existence, stability, and dynamics in periodic photonic lattices [13]. One of effective physical methods to create various optical lattices is optical induction technology [14], [15]. This method has led to the observation of several types of solitons under photorefractive nonlinearity. Such as surface solitons [16], vortices solitons [17], random-phase lattice solitons [18], Bessel lattices solitons [19], [20], and photovoltaic lattice solitons [21]. It is well known that the main feature of the wave propagation in periodic systems is the existence of a set of multiple forbidden gaps where linear waves undergo Bragg reflection from the periodic structure. Moreover, when a local defect is introduced into these periodic structures, light can be trapped in the defect site in the form of linear defect modes with their propagation constants lying inside the gaps of these periodic structures. In the past years, one dimensional and two dimensional defect modes and solitons have been investigated in biased photorefractive crystals with linear electro-optic effect or photovoltaic effect [22], [23], [24], [25], [26]. However, defect modes and solitons supported by optically induced lattices in highly saturable nonlinear media with quadratic electro-optic effect have not been analyzed.

In this paper, we focus our attention on light propagation in one-dimensional optical induced lattices with a defect site in centrosymmeric photorefractive crystals. For localized defects, we show numerically that both the positive (attractive) and negative (repulsive) defect can support defect modes and defect solitons in these lattices. Positive and negative defect modes bifurcate from the right and left edge of each band into the bandgaps, respectively, and the distances between the eigenvalues of defect mode and band edge increase with the defect strength. As the negative defect strength increase, the negative defect modes move from lower band gaps to higher ones. The same phenomena are observed when the applied dc field is strong enough for the fixed repulsive defect lattices. The confinement of the positive defect modes increases with the defect strength, however, the most localized negative defect modes do not appear when the lattices have strongest negative defect. Another optical phenomenon, defect solitons closely related the defect modes and their stability properties in one-dimensional optical induced lattices with a defect site in centrosymmeric photorefractive crystals are also studied. The power curves of these defect solitons bifurcate from the corresponding defect modes and move to edges of their respective band gaps. Furthermore, as the negative strength increases, similar to the bifurcation of negative defect mode, the defect soliton power diagram branches move to edges of higher band gap and then appear in higher band gaps. The linear stability analysis by using perturbation method shows that positive defect solitons are linearly stable at low power region but unstable in higher power regime. For a negative defect, the defect solitons are stable in entire regions of their existence in the lower bandgaps but only the solitons at low power region are stable in higher bandgaps. These results are found to be in excellent agreement with the numerical simulation.

Section snippets

Theory model

We employ a similar arrangement that has proven to be a very effective physical method for the creation of various periodic photonic lattices in noncentrosymmetric PR crystal to induce one-dimensional photonic lattice with a local defect in a KLTN crystal. The crystal principal axes are aligned with the x, y and z directions of the system, and an external biased electrical field is applied to the sample along x direction. Both the lattice beam with a single-site defect and low intensity probe

Defect modes in one-dimensional optically induced lattices with quadratic electro-optic effect

It is instructive to analyze the linear propagation of light beams in the optical lattice. When the energy flows of soliton beams are small, the optical lattices induce a bandgap structure in the linear Schrödinger spectral problem:μu=2us2E0u[1+IL(s)]2

Bounded solutions of this linear equation are called Bloch modes, and the corresponding frequencies μ form the Bloch bands. Based on the Floquet–Bloch theory, we can obtain the Bloch spectrum by solving the above equation with u(x)=w(s+T)exp(iks

Defect solitons and their stability supported by optically induced lattices with quadratic electro-optic effect

The above results are closely related the defect solitons in one-dimensional optical lattices with a defect. We can show that each linear defect mode generates a family of defect solitons with the propagation constant μ above the linear defect mode eigenvalues, and these stationary states of Eq. (1) can be further solved numerically. To elucidate the linear stability of the solitons, we searched for perturbed solutions in the form [28]q(s,ξ)={q(s,ξ)+[v1(ξ)v2(ξ)]eiδξ+[v1(⁎)(ξ)+v2(⁎)(ξ)]eiδ(⁎)ξ}

Conclusions

In conclusion, we have thoroughly studied defect modes and defect solitons supported by optically induced lattices in highly saturable nonlinear media with quadratic electro-optic effect. It is found that positive and negative defect modes bifurcate from the right and left edge of each band into the bandgaps, respectively, and the distances between the eigenvalues of defect mode and band edge increase with the defect strength. As the negative defect strength increase, the branch of negative

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant no. 11247259) and Shandong Provincial Natural Science Foundation, China (Grant no. ZR2012AQ005).

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      Obviously, the drift direction points at the medium with stronger nonlinearity [26]. It is well known that the defect modes supported optically induced photonic lattices in semi-infinite bandgap are symmetric and the peak appears at the defect site, however, those defect modes in the first bandgap are antisymmetic and the peak appears in the neighboring lattice channel [22,27]. As a result, soliton profile laterally drifts toward the optical lattice and uniform medium in the semi-infinite and first bandgap, respectively.

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