1. Introduction

Light propagation in periodic photonic structures exhibits many intriguing phenomena. Examples in the linear regime include transmission spectrum band-gaps, Bloch modes, and diffraction management. In the nonlinear regime, a typical example is the self-localized wave packets (discrete or gap solitons), which have been a subject of intense investigation [1]. Much of the work has been done in optically induced photonic lattices in photorefractive crystals [2–6]. Since the photorefractive nonlinearity is intrinsically anisotropic and nonlocal [7], a nature question arises: how would the anisotropy and nonlocality influence the self-localized states in induced photonic lattices, or in nonlinear discrete systems in general? In a continuum photorefractive system, both theoretical and experimental studies have shown that coherent and incoherent elliptical solitons can arise from either anisotropic nonlinearity or anisotropic coherence (correlation function) [8–10]. However, such issues have received little attention in a discrete system [11–16]. Recently, theory has shown that anisotropic linear diffraction and moving elliptical discrete solitons exist in two-dimensional waveguide lattices induced in photorefractive media based on an isotropic model [12, 13]. Although an anisotropic model has been utilized to analyze the formation of discrete solitons [14–16], the elliptical discrete solitons possessing obvious ellipticity were never observed in experiment. In this paper, we predict in theory and demonstrate in experiment a novel class of elliptical discrete solitons. We show by employing an anisotropic photorefractive model that the soliton ellipticity is originated from enhanced photorefractive anisotropy and nonlocality under a nonconventional bias condition.

2. The theoretical model

We first propose a model, considering both the photorefractive anisotropy and different orientation configurations, to describe the propagation of a probe beam in an optically induced photonic lattice under a nonconventional bias field. A coordinates system is constructed with placing the x axis along one of the principal axes of the lattice beam as shown in Fig. 1. The angles of the external bias field and the crystalline c-axis with respect to the x axis are denoted by θ_e and θ_c, respectively. The probe and lattice beam propagate collinearly along the z axis. Here, the polarization directions of the probe and lattice beam are always kept to be parallel and perpendicular to the c-axis, respectively. Then the dimensionless equations governing the steady state propagation of the probe beam in the induced lattice can be written as [2, 4–5, 7, 9, 17]

 

Fig. 1. Geometry of the relative orientation of the crystalline c-axis, bias field, and lattice beam.

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where ∇=x̂(∂/∂x)+ŷ(∂/∂y), B(r⃗) is the amplitude of the probe beam, φ is the light-induced electrostatic potential with a boundary condition of φ(r⃗+Λ)=φ(r⃗), Λ is the period of the lattice beam, E ₀ is the external bias field, V(x, y)=cos²(πx/Λ)cos²(πy/Λ) is the intensity of the lattice beam, and I is normalized by the dark irradiance of the crystal (including background illumination). The dimensionless coordinates (x, y, z) are related to the physical coordinates (x′, y′, z′) by the expressions (x, y)=(kl)^1/2(x′, y′) and z = lz′, where l = 0.5kn _e ² γ ₃₃ E ₀. Here, k is the wave number of light in the crystal, n _e is the extraordinary refractive index, and γ ₃₃ is the electro-optic coefficient.

The solitary solutions of Eqs. (1) can be found in the form B(x, y, z)=b(x, y)exp(iβz), where β is the propagation constant, and the real envelope b(x, y) satisfies the following equation

where φ is also determined by Eqs. (1b) and (1c).

3. Variation of induced lattice structures

Figure 2 depicts the normalized index distributions of the induced lattices $\left(\Delta n=\frac{\partial \phi }{\partial x}\cos {\theta }_c+\frac{\partial \phi }{\partial y}\sin {\theta }_c\right)$ , where θ̣_c corresponding to the three rows (form top to bottom) are 0, π/8, and π/4 (see the green arrows), respectively, and θ_e for (a)–(e) are equal to θ_c, θ_c+π/4, θ_c+π/2, θ_c+3π/4, and θ_c+π (see the purple arrows), respectively. The number given in the upper left corner of each figure corresponds to the index modulation depth of the lattice structure at E ₀ = 1. From Fig. 2, one can see that an identical lattice beam can induce different photonic lattice structures under various bias conditions (as shown in each row), and likewise for an identical bias condition, the lattice structures created by lattice beams with different orientations are distinct (as shown in each column). It should be noticed that under the same bias condition, the index modulation depth of the induced photonic lattice depends strongly on the orientations of the lattice beam. The bottom panels of Figs. 2(a) and 2(e) correspond to self-focusing lattices and self-defocusing backbone lattices under conventional bias conditions, as used in previous experiments of 2D discrete and gap solitons [3, 5, 6]. The top and bottom panels in Fig. 2(a) represent similar lattices used in Ref. [15]. From the index profiles illustrated in Fig. 2(c), one can see that the shape and location of the index maxima and minima are dramatically different from that of the lattice-beam intensity, indicating enhanced photorefractive anisotropy and nonlocality by a bias field perpendicular to the c-axis. The induced photonic lattice structures under such a bias condition could lead to novel discrete self-localized states, as we shall demonstrate below.

 

Fig. 2. Refractive index distributions of optically induced photonic lattices under different conditions, where in each figure the green and purple arrow indicate the directions of the crystalline c-axis and bias field, respectively, and the number in the upper left corner corresponds to the index modulation depth at E ₀ = 1.

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4. Numerical results of discrete diffraction and self-trapping

Now we study the linear and nonlinear propagation dynamics in the induced lattices in the presence of enhanced nonlocality and anisotropy by use of both beam propagation method and the iteration method. Let $B(x,y)=\sqrt{0.6}\exp \left[\frac{-\left(x^2+y^2\right)}{64}\right]$ , E ₀ = 1, Λ = 8, and z = 110, the output intensity pattern of the probe beam (for on-site excitation) after linear propagation is depicted in Fig. 3(a), where the panels from left to right correspond to the cases of θ_e = θ_c = π/4; θ_e = 3π/4, θ_c = π/4; θ_e = 5π/8, θ_c = π/8; and θ_e = π/2, θ_c = 0, respectively. Notice that, for comparison, we include one conventional bias case, i.e., θ_e = θ_c = π/4. In the presence of nonlinearity, the input Gaussian beams can evolve into self-trapped states as shown in Figs. 3(b). It is clear that different conditions will lead to different patterns of linear diffractions and nonlinear self-trapped states. The soliton solutions with various peak intensities obtained by Petviashvili iteration method [18] for each case are displayed in Figs. 3(c) and 3(d), which shows that the solitons with higher peak intensities exhibit stronger localization. By comparing Figs. 3(b) and 3(c), we can see that the soliton profiles at lower peak intensities obtained by the nonlinear beam propagation and iteration method are very similar. Note that in the presence of photorefractive anisotropy, even under conventional bias condition, the soliton profile is somewhat elliptical (see the left column in Fig. 3). With enhanced anisotropy and nonlocality under E ₀⊥c, the ellipticity of the solitons will be dramatically increased, and the orientations of elliptical solitons can be altered by changing the relative orientations of lattice beam, bias direction, and the c-axis. For the case with θ_e = π/2 and θ_c = 0, the equivalent lattice spacing is reduced while the index modulation is relatively low [see Fig. 2(c), top], thus the coupling between waveguide channels is much stronger, and both linear diffraction and nonlinear self-trapped state resembles that in the continuum limit (not well discretized).

 

Fig. 3. Numerical results for discrete diffraction and self-trapped states. (a)–(b) are the output patterns of the probe beam under linear (a) and nonlinear (b) propagation obtained by beam propagation method. Animations show the corresponding evolution dynamics. (c)–(d) show soliton solutions at different peak intensities obtained with iteration method. The columns from left to right are for the cases of θ_e = θ_c = π/4; θ_e = 3π/4, θ_c = π/4; θ_e = 5π/8, θ_c = π/8; and θ_e = π/2, θ_c = 0, respectively. [Media 1][Media 2]

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5. Experimental results of discrete diffraction and self-trapping

To perform experimental demonstrations, we use the experimental setup similar to that used in Ref. [5]. The photonic lattices are created by sending a partially coherent beam through an amplitude optical mask, and a focused Gaussian laser beam as a probe beam propagates collinearly with the lattice beam. The probe beam and the lattice beam have the same wavelength of 532nm, and their polarization directions are kept to be parallel and perpendicular to the c-axis, respectively. A photorefractive SBN:60 crystal (with dimensions of 6.5mm×6.5mm×6.7(c)mm and 0.025% by weight chromium dopant) is used. The intensity ratio between the probe and lattice beam is adjusted to be about 3:1. In our experiment, the lattice spacing is about 23µm. We first block off the probe beam until the lattice structure arrive steady-state, after that the linear and nonlinear transport of the probe beam is monitored simply by taking its instantaneous (before nonlinear self-action) and steady-state (after self-action) output patterns after propagating through the lattice. Because the induced lattice potential in our experiment is relatively shallow due to the weak absorption of the crystal sample at 532nm, thus a simulation using the experimental parameters is also performed for comparison.

Typical experimental results are shown in Fig. 4 (with all intensities normalized to their maxima), where (a)–(c) are corresponding to the cases of θ_e = θ_c = π/4; θ_e = 3π/4, θ_c = π/4; and θ_e = 5π/8, θ_c = π/8, respectively. The first and second rows in Figs. 4(a)–4(c) are for linear and nonlinear output beam patterns, respectively. The left column is the three-dimensional display of the middle column. The right column shows the simulation results obtained using our experimental parameters. The bias voltage for observations of Fig. 4(a) is 2kV and that for Figs. 4(b) and 4(c) is 3kV. It is obvious that under linear condition, the input beam undergoes discrete diffraction. While in the presence of nonlinearity, the probe beam can evolve into an elliptical self-trapped state. The observed patterns of discrete diffraction and elliptical solitons are in good agreement with those theoretical results obtained from the anisotropic photorefractive model. We also emphasize that, compared with the moving elliptical discrete solitons previous studied with an isotropic model [12, 13], the elliptical discrete solitons presented here all undergo on-axis propagation, and they do not arise from the anisotropic diffraction.

6. Summary

In summary, we have demonstrated a novel class of elliptical discrete solitons in a nonconventionally biased photorefractive crystal. The ellipticity of the discrete solitons is originated from an enhanced photorefractive anisotropy and nonlocality. It is revealed that both the bias direction and the orientation of the lattice beam can dramatically influence the structure and modulation depth of the lattices. Our experimental observations are in good agreement with the numerical simulations. These results open the door for investigating how anisotropy and nonlocality influence the formations of lattice structure and lattice solitons, such as elliptical gap solitons and vortex solitons. Since the lattice structure can be tuned at ease by altering the relative orientations among the c-axis, bias field, and lattice beam, we expect this can lead to a novel approach for band-gap engineering as well as diffraction and refraction management.

 

Fig. 4. Experimental observations of elliptical discrete solitons. Results in (a)–(c) correspond to the cases of θ_e = θ_c = π/4; θ_e = 3π/4, θ_c = π/4; and θ_e = 5π/8, θ_c = π/8, respectively, where the left (3D intensity plots) and middle (2D transverse patterns) columns are experimental results, and the right column is from numerical simulation for comparison. The top and bottom rows in (a)–(c) show the linear and nonlinear output beam patterns, respectively.

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