1. Introduction

Since discrete solitons were first predicted in nonlinear waveguide array lattices in 1988 [1] and then successfully observed a decade later [2], which have attracted extensive attention in diverse branches of science such as biological systems, nonlinear optics, and solid state physics. That optical wave propagates in periodic optical lattices, for example, an array of optical waveguides and optically-induced lattice, exhibits many new intriguing phenomena [3, 4]. The periodic photonic structures can provide an unique way for controlling light propagation, and the latest development have demonstrated that soliton switching and filtering can be completed by use of the property of discrete soliton [5, 6, 7]. Periodic modulation of the refractive index modifies the diffraction properties of light and strongly affects nonlinear propagation and localization of light [8, 9]. In these periodic structures, wave dynamics is governed by the interaction between optical discrete diffraction/coupling effects and material nonlinearity. When these two effects reach an balance, the waves form self-localized modes which known as lattice solitons[10]. To date, the arrays made from materials having intensity-dependent Kerr [2], quadratic [11], as well as photorefractive nonlinearities [12] have been shown to support such discrete localized states-discrete solitons.

A standard theoretical approach to study discrete optical solitons is based on the tight-binding approximation and the effective discrete nonlinear model – that is the discrete nonlinear Schrödinger (DNLS) equation and their continuous analog employed for analyzing stationary localized solutions [13, 14]. This approach allows the prediction of many types of nonlinear modes and the research on the properties of discrete models and continuous models of nonlinear periodic optical lattices [14, 15, 16], and the latter was used in our research work.

In this paper, we analyze numerically the self-trapping behavior of the beam propagation in optically induced waveguide lattice which is created via the plane wave interference in the photorefractive nonlinear medium, and also give the relation between the trapping properties and the biased field. Applied appropriately oriented external field on the optical induced waveguide, the localization modes of light waves, namely lattice solitons, can be achieved. When the voltages of the biased field are changed in quantity, the lattice solitons exhibit different localization properties. The wave localization is also associated with the lattice period and the input position of light pulse. And the ratio of the waist width of input beam to lattice period is related with the number of channels in which the discrete solitons are excited. However, the increase of lattice intensity almost does not affect the propagation behavior of soliton.

2. Theory expression

The optically induced waveguide lattices were established in a photorefractive crystal by periodic diffraction-free intensity patterns that result from plane-wave superposition [17]. Figure 1 shows model of photorefractive crystal with applied field and the lattice intensity. As shown in Fig. 1(a), c axis of the crystal is parallel to x axis. When the plane waves linearly polarized along the ordinary y axis are input in the incident face, the transverse periodic refractive index modulation is achieved for creating the optical lattices. They are highly nonlinear for extraordinary light that is polarized along x axis [15]. In our analysis, Intensity of one-dimensional periodic optical lattices [Fig 1(b)] is described by the following:

where I ₀ is the maximum lattice intensity, and D is the lattice spacing.

We consider the propagation of optical beam in the model mentioned above. With the beam propagation along the longitudinal z and the periodical refractive index modulation along the transverse x, the evolution of the dimensionless field envelope amplitude u is governed by the following one-dimensional continuous NLS [15]:

where k=k ₀ n_e and n_e is the refractive index for extraordinary polarized beam, r ₃₃ represents the electro-optical coefficient and E ₀ stands for the applied field intensity. E_sc is the steady state space charge-field given by

where K_B is the Boltzmann constant, T is the temperature, and e is the electron charge. In Eq. (3), the first term describes dominant screening nonlinearity of the photorefractive crystal, and the second term arises in weak diffusion effects. However, it does not affect the formation of discrete solitons so that it is ignored here. In photorefractive optical lattices, I(x) = I_g(x)+|u|² is total normalized light intensity.

 

Fig. 1. (a). Photorefractive crystal with applied field. (b) Optical induced periodic waveguide lattice via plane waves interference.

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To simplify our analysis, the transformations of ξ = z/(k ₀ n_ex ₀ ²) and η = x/x ₀ are used, where x ₀ is the characteristic beam width, and the following normalized equation is obtained from Eq. (2) and Eq. (3):

where V ₀ is proportional to the biased field, exactly, V ₀ = k ₀ ² n_e ⁴ r ₃₃ E ₀ x ₀ ²/2. And for convenience, we set d = D/x ₀, then Eq. (1) is transformed to the following:

In our analysis, the profile of input light is selected in the form of sech-like beam:

where η ₀ and Ω characterize the center position and the waist width of the input pulse, respectively.

3. Results and discussions

The evolution of the beam with varying biased field is reviewed by numerical simulation under the fixed conditions of the input pulse center at η ₀=0, the maximum lattice intensity I ₀=1, the lattice spacing d=2 and waist width Ω=1. In all of simulation, the propagation distance is considered from ξ =0 to ξ =20. Firstly, we consider the cases of low biased field at V ₀=1. The simulation results are shown in Fig. 2. In the presence of lattice, the beam reaches a steady localized state after a short distance with weak diffraction, the discreteness cannot be seen even the existence of the lattice potential as shown in Fig. 2(a). When we ignore the lattice, which means the beam propagation only in a photorefractive crystal without the periodic refractive index modulation of transverse scalar, the profile of amplitude always keep a steady state that can be seen in Fig. 2(b), namely soliton state.

 

Fig. 2. Evolution of the beam at V ₀ =1 (a) with optical lattice (d =2) and (b) without the lattices.

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Secondly, when V ₀ is increased to 30, as shown in Fig. 3, both the case with the lattice and without the lattice are very different from that of V ₀ =1. Discrete behavior of the beam was observed distinctly in optical lattice. After light has propagated a short distance, due to the interaction between the lattice potential and probe beam u, the beam trapped quickly in the four center lattice sites and the energy of the soliton in the two inner sites are much greater than that in the two outer sites [as shown in Fig. 3(a) and Fig. 3(c)]. However, if without the lattice, the compression of beam and the slight increase of the beam peak are observed due to the increase of nonlinearity [as shown in Fig. 3(b) and Fig. 3(d)].

Thirdly, the case of V ₀ = 80 is checked for comparing with that of above, and the results are similar to those of V ₀ =30. The beams are all localized in the four center lattice sites, however, comparing with that of V ₀ =30, there is a more power concentrated in the two outer lattices after propagation a short distance as shown in Fig. 4(a) and Fig. 4(c). Therefore, solitons are excited in more channels with the increase of applied voltage as long as the incident sech-type beam covers all of these lattices. That is the discreteness of soliton is more obviously than that of the weak nonlinearity under low applied voltage. For that of without the lattice, the evolution of the beam is shown in Fig. 4(b) and Fig. 4(d). It can be see from Fig. 4(b) and Fig. 4(d) that the nonlinear focusing effect is so strengthening that the beam is compressed more due to the high nonlinearity at the high biased field.

 

Fig. 3. Propagation of the beam at three different sites along ξ direction at V ₀ =30 (a) with optical lattice and (b) without the lattice in the photorefractive material; (c) corresponding to the case of (a) and (d) the case of (b).

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Fig. 4. Evolution of the amplitude of beams at three different sites along ξ direction at V ₀ =80. (a) with optical lattice and (b) without the lattice in the photorefractive material.; (c) corresponding to the case of (a) and (d) the case of (b).

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Lastly, Fig. 5 gave the evolution of the beam amplitude in the center site (η = 0) at three sites (ξ = 1,8,16) with biased field V ₀. Figure 5(a) shows that of light beam propagation in the optical waveguide lattice with d =2. Figure 5(b) shows that of without the lattice. It can be seen clearly that the amplitude of the beam at the site η =0 varies with the external biased field. When V ₀ is increased, the energy of the beam trapped into the lattices and formed discrete solitons in optical lattice, but it is compressed for the case of without lattice. The results are consistent with the facts that were shown in Fig. 3 and Fig. 4.

 

Fig. 5. The amplitude |u| at the center site (η =0) versus biased field V₀ for (a) with lattice spacing d =2 and (b) without the lattice.

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Furthermore, it is also interesting to observe how the parameters of lattice spacing d affect the formation and propagation of discrete solitons in optical waveguide lattices. We considered the cases of d = 5 at V ₀ =30 and V ₀ =80, respectively. The hyperbolic secant beam is input at η = 0 and the results are shown in Fig. 6. It is found that the beam trapped in the center lattice and the two nearest lattices dependence on the V ₀, which does not like the cases of d = 2. At the small lattice spacing (d = 2), the beam easily trapped in the neighboring lattices (left and right). As shown in Fig. 6(a), the beam trapped strongly in the centric lattice and relatively weaker in the adjacent two lattices. However, with the increase of V ₀, then the beam mostly trapped into the two adjacent lattices [as shown in Fig. 6(b)].

 

Fig. 6. Propagation of the beam in optical waveguide lattice with d =5 at η =0 for (a) V ₀ =30, and (b) V ₀ =80.

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Then we observed the formation of discrete solitons is closely dependence on the position of input pulse in optical lattices, which is reflected by the parameter η ₀. By fixed d = 2, and the incident beam is input at the first left lattice where η ₀ = -1, the evolution of the beam at V ₀ =30 and V ₀ =80 is shown in Fig. 7. Compared Fig. 7 with Fig. 3(a) and Fig. 4(a), it can be found that their discrete behavior is different, but the symmetry of patterns is same in all of these cases. When the beam is input in one lattice, the propagation of energy is mainly as discrete solitons by trapped in the lattice and the adjacent lattices (see Fig. 7). When the beam is input in the position between two lattices, then the beam trapped in the left and right adjacent lattices and thus formed the discrete solitons [Fig. 3(a) and Fig. 4(a)].

 

Fig. 7. Propagation of the beam which is input at the first left lattice in optical waveguide lattice with d =2 for (a) V ₀ =30,

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To further investigate the effects of beam width and lattice intensity on the propagation of discrete soliton, Eq. (4) is numerically analyzed with different values of Ω and I ₀. Since one of the meaningful factors that determine physically the propagation regime of laser beam in periodic structure is not the width of beam itself but rather the ratio of the beam width to lattice period [19], lattice period is fixed by setting d =2 and the waist width of input soliton is varied in our numerical simulations. We found that the solitons are excited in more channels, which is illustrated in Fig.8(a) and Fig. 8(b), with the increase of the waist width of the soliton by decreasing the value of Ω. However, when only the lattice intensity is increased by increasing the value of I ₀, there is hardly changing for the propagation behavior of discrete soliton (as shown in Fig. 9) except for the distance they trapped in lattices [compared Fig. 4(a) with Fig. 9(a) and Fig. 9(b)]).

 

Fig. 8. Evolution of the beam at V ₀ =80 for (a) Ω =0.4, I ₀ =1; (b) Ω =0.2, I ₀ =1. Other parameters are the same as in Fig. 3.

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Fig. 9. Evolution of the beam at V ₀ =80 for (a) Ω =1, I ₀ =10. (b) Ω =1, I ₀ =20. Other parameters are the same as in Fig.3.

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4. Conclusions

In conclusion, the properties of discrete solitons in optical induced waveguide lattice depend on many parameters such as biased field V ₀, lattice period and the position of the input pulse. With appropriate V ₀, the generation of the discrete solitons is observed in optical waveguide lattice and the discrete localized states are also varies with V ₀. If without the lattice in the photorefractive crystal, the propagation of the beam exhibits different phenomena with that of the lattice for variant V ₀. The lattice spacing and the input position of the light pulse also affect the discreteness of the beam, but they cannot determine whether the discrete solitons can be formed. Though it is found that the discrete behavior are different at different input position of original pulse, the symmetry of patterns is remained. In addition, the soliton propagation in the waveguide lattice is also affected by the ratio of the waist width of beam to lattice period, for example, the wider waist of input beam, the more lattices localized in. However, the lattice intensity has hardly affection on the propagation of discrete soliton except that the generation of localization happens quickly or slowly. Therefore, these results are useful reference for observing discrete solitons experimentally in optical induced waveguide lattice. It is also significant to control propagation of light in the optical induced waveguide lattices for practical application.