Dynamic of total internal reflection (2+1)D bright beams on photorefractive SBN61:Ce crystal

E. Alvarado-Méndez; M. Trejo-Durán; J. M. Estudillo-Ayala; J. A. Andrade-Lucio; E. Vargas-Rodríguez; M. de G. García-Hernández

doi:10.1364/OE.20.000754

1. Introduction

The study of the interaction between radiation and mater at nonlinear interfaces opens the possibility of implementing several applications such as optical switching if a reflected beam is controlled varying the power of the input beam without changing the incident angle [1–7]. Switching devices based on the interaction of spatial and temporal solitons have been suggested to exploit the effects of beam steering and mutual soliton trapping. The one dimensional case of total internal reflection has been studied experimentally and theoretically. Some physical characteristic effects observed in all cases are filamentation, optical bistability, surface waves, and Goos-Hänchen shift [8–14]. The refraction of black and gray solitons was studied theoretically as well as the existence of a unique total nonrefraction angle for gray solitons Sánchez-Curto et al. [15].

Recently, the physical phenomena of soliton reflection have been studied experimentally at the boundary of quadratic nonlinear media [16,17]; in these works, the beam incident upon the interface of periodically poled LiNbO₃, and linear beam transmission or soliton reflection was observed. Effects of remote boundaries on soliton dynamics in nonlinear media with a large range of nonlocality can lead to soliton steering and oscillation in predetermined trajectories [18].

Other important physical phenomena in nonlinear interfaces are the arrays of spatial solitons. Scheuer et al. [19] investigated the dynamics of spatial soliton arrays, they found that the repulsive forces between the solitons are counteracted by the potential produced by nonlinear interfaces of the waveguide and stabilize array. Reflection and refraction of spatial solitons at the nonlinear interface formed by nematic liquid crystals were studied by Peccianti et al. [20], one interesting result was the Goos-Hänchen shift effect.

The dynamics of propagation of (2 + 1)D beams is rather complicated. A question for (2 + 1)D beams was if they are stable; Saffman et al. [21] showed theoretically and numerically that the circular soliton do not exist in photorefractive media due to the direction of propagation. However, Shih and associates [22] have observed experimentally solitons in photorefractive crystals with circular symmetry. Low order filamentation in anisotropic self-focusing media was reported by Zozulya et al. [23] and they obtained large optical nonlinearities using low-power cw lasers in photorefractive materials. A detailed theory considering the propagation of (2+1)D beams in photorefractive media is presented by Saffman et al. [24]. They showed that single solitons are elliptical beams and include asymmetric filamentation into several beam lets in parallel direction of the external electric field. Other interesting results on (2+1)D propagation were reported by Carrasco et al. [25] in quadratic nonlinear media, with long distance of propagation (20 lengths diffraction) which consist of ring formation and splitting of the beam by a nonlinear interface formed by a photorefractive SBN61:Ce crystal and air.

In this paper, we report experimental results that show evidence of the formation of several beams stimulated by nonlinear interface in SBN61:Ce-air under application of an external electric field.

2. Mathematical model for the nonlinear interface

We assume that the light beam is launched within the nonlinear medium at an angle towards the lateral interface, as it is shown in Fig. 1 .

Fig. 1 Beam filamentation falling upon an interface between nonlinear and linear media.

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The mathematical model used is an extension to that for the (1+1)D problem [8] (the nonlinear medium is characterized by a refractive index of saturable form) and it is represented by the next (2 + 1)D nonlinear Schrödinger equation (NLSE),

i \frac{\partial A}{\partial Z} = \frac{1}{2} (\frac{\partial^{2} A}{\partial X^{2}} + \frac{\partial^{2} A}{\partial Y^{2}}) + η f (X, Y) [Δ + \frac{μ | A |^{2}}{1 + μ | A |^{2}}] A

where

Δ = (n_{01} - n_{02}) / n_{2} | A_{0} |^{2}

represent the normalized refractive index difference;

n_{01}

and

n_{02}

are the linear refractive index of different media with

n_{01} > n_{02}

;

η = n_{01} k_{0} x_{0}^{2} y_{0}^{2} / (1 / k_{0} δ n_{0})

is the nonlinear coefficient medium,

x_{0}

and

y_{0}

are the width in x-axis and y-axis respectively,

δ n_{0} = (1 / 2) r n^{3} V_{0} / L

is the nonlinear contribution to the refractive index,

r

is the effective electro-optic coefficient,

V_{0}

is the external applied voltage, and

L

is the transverse width of the crystal;

μ = I_{s a t} / a_{0}^{2}

is the saturation parameter,

a_{0}^{2} = I_{\max} (0)

is the intensity of the uniform background illumination.

Z = z / L_{D}

is the normalized propagation distance to the diffraction length,

L_{D} = k_{0} n_{01} x_{0}^{2}

,

X = x / x_{0}

and

Y = y / y_{0}

are the transversal coordinate normalized to the initial beam width in

x

and

y

direction, respectively. The complex amplitude of the beam is normalized to the initial amplitude peak value; the interface lies in the

y - z

plane at

X = 0

;

f (X, Y) = [1 - \tanh (k Y)]

describes the spatial behavior of the interface, and

k

represents the steepness of the interface. If the interface is abrupt, we considered a step function for

f (X, Y)

;

f (X, Y) = 0

for

X \leq 0

and

f (X, Y) = 1

for

X > 0

. In our numerical simulations

A (x, y, z = 0) = A_{0} \exp {1 / 2 [(x - x_{0}^{2}) + (y - y_{0}^{2})]} \exp [- i V (y - y_{0})]

is the initial condition, parameter V characterizes the incidence angle in normalized coordinate. Numerically we found that angles of incidence

V \leq 0.5

are the condition for total internal reflection (TIR); to V > 0.5 the beam is split into reflected and transmitted and its energy is not conserved [26].

If the beam approaches to the interface and is reflected, its peak intensity and width remains practically the same, proving that the V parameter is small enough. In consequence, the energy is approximately conserved during the reflection at the nonlinear interface [8]. Only this case is studied in this paper.

In Fig. 2(a) the beam intensity in the input face is showed, the units are normalized, the interface is represented by a dotted line; Fig. 2(b) shows the reflected beam by the interface with $V = 0.5$ , $x_{0} = 5$ , $y_{0} = 5$ , $μ = 1$ and $η = 7$ parameters. The beam is self-focused perpendicular to propagation direction due to anisotropy of the photorefractive crystal (PRC) exhibited [27]. If the nonlinearity is increased ( $η = 7$ ), modulation instability effect produces oscillations in the beam when is launched to the interface (Fig. 3 ); the beam is broken up in two new beams stimulated by nonlinear interface. For a best understanding of the interface we study the behavior of the beam in the interface for the stationary case.

Fig. 2 Numerical simulations of total internal reflection of the (2+1)D beam. a) input face; b) output face without voltage. The parameters are $V = 0.5$ , $y_{0} = 5$ , $x_{0} = 5$ , $η = 7$ .

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Fig. 3 Numerical simulations of the broken up (2+1)D beam. The parameters are $V = 0.5$ ; $y_{0} = 5$ ; $x_{0} = 5$ ; $η = 10$ .

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The 2D modulational instability has been studied by Saffman et al. in photorefractive media [24]. However, the dynamics with nonlinear interface is different.

Two cases are presented. First, the effect on the beam when is launched to the nonlinear interface (Fig. 4(a) ); the graphic at the top shows propagation and reflection in the space; the graphic at the bottom is an orthogonal view. The high nonlinearity ( $η = 10$ ) and the interface brake the beam in two new beams. Second, when the beam is propagated without interface, oscillations in intensity are presented; nevertheless, the beam is not broken, Fig. 4(b).

Fig. 4 The beam propagation when: a) it is reflected by nonlinear interface, b) the beam is propagated in nonlinear medium without interface. In both cases the numerical values are $x_{0} = 5$ , $V = 0.5$ , $η = 10$ .

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For best clarity about the beam behavior when it interacts with the interface, we show the transversal profiles of the beam in x-y plane to different propagation distance (Z), only for one particular case ( $x_{0} = 5$ , $η = 10$ ). We considered the stationary case; thus, there is dependence only on x-coordinate according to NLSE.

Figure 5(a) shows the profile of the initial beam Z = 0. The sequence Figs. 5(b)-5(e) show the beam behavior when it is closed to the nonlinear interface, which is pointed out by the arrow, for Z = 5, 10, 15 and 18 respectively. The beam presents oscillations due to high nonlinearity; however, it is not broken. The next sequence Figs. 5(f)-5(i) show the beam behavior when it is reflected by the nonlinear interface, for Z = 20, 25, 30 and 35 respectively. The broken up of the beam is stimulated by the nonlinear interface (Fig. 5(f)), and the phase accumulated due to nonlinearity depends on the propagation length. The incident energy is split by the interface and is trapped in two new soliton-like beams. The characteristic curve between width and intensity determine whether the beams reflected are or not soliton [8].

Fig. 5 Gaussian beam in x-y plane to different values of propagation distance. (a) initial beam profile for Z = 0; (b) Z = 5; (c) Z = 10; (d) Z = 15; (e) Z = 18; sequence of the beam when is reflected by nonlinear interface: f) Z = 20; g) Z = 25; h) Z = 30; i) Z = 35; the beam is broken up in two new beams stimulated by nonlinear interface;

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Figure 6 shows the comparison of the propagation of (2 + 1)D beam with and without interface; the total distance propagation is constant, $Z = 12$ , and with different nonlinearity values. Figures 6(a), 6(c), and 6(e) show the intensity profile of the beam without interface; Figs. 6(b), 6(d), and 6(f) show the reflected beam by the interface, with different values of nonlinearity ( $η = 7, 10, 13$ , respectively). From Fig. 6, we observe symmetrical filamentation in the case without interface. However, if interface is considered, the asymmetrical filamentation of the beam takes place. These numerical results agree with the fission of N-soliton in planar waveguide [28] and with theoretical research made by Ye et al. [29].

Fig. 6 The pictures (a), (c), and (e) correspond to the beam propagation within nonlinear interface. The pictures (b), (d) and (f) refer to the beam reflection with nonlinear interface, at different values of nonlinearity.

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3. Experimental setup

In the experimental setup used, we study the total internal reflection of (2+1)D beam. This is similar to that used in our previous investigations of (1+1)D case [8] and it is shown in Fig. 7 . A He-Ne CW laser a (632.8 nm, 10 mW) was used to illuminate uniformly the photorefractive crystal (PRC) and another laser b (632.8 nm 20 mW) was used to generate a circular Gaussian beam of 200 μm of diameter. We use SBN61:Ce PRC with electrooptic coefficient ( $r_{33} = 224$ pm/V) and refractive index $n = 2.33$ with $0.1 %$ wt of Ce, and dimensions 5 × 5 × 5 mm³. The focusing beam was propagated along 10 mm of crystal, perpendicular to the c-axis with a parallel polarization to it. A high voltage power supply was used to apply an external electric field. The input and output intensity distributions were captured with a CCD camera and a frame grabber system.

Fig. 7 Experimental setup for studying the total internal reflection of (2+1)D beam in SBN61:Ce photorefractive crystal.

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Two mirrors were used to control the angle of incident Gaussian beam (2+1)D at the input face of the PRC. We study the special condition of total internal reflection of (2+1)D beams, that was obtained when the angle between the incident and normal of the input face was 2.57°, very close to the nonlinear interface. The beam was reflected totally by the face (001) without external electric field applied. In the experimental results, the output intensity distribution obtained for different external applied voltages are shown in Figs. 8 -10 . In each case the effect of an external applied voltage was studied as function of time, taking image each 10 seconds; 50s after, the beams were stable; the Figs. 8(a), 9(a) , and 10(a) show the beam at the output face without voltage. 600 V/cm were applied in PRC and the temporal evolution of the reflected beam is shown in Fig. 8. After 50 seconds the beam acquires an elliptic form, reaching the steady state (Fig. 8(f)).

Fig. 8 Dynamics of the beam for 600 V/cm each 10 s. (f) the self-focusing beam is observed at 50 s.

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Fig. 10 Dynamics of the beam for 1000 V/cm each 10 s. (f) the beam is broken up in four new beams at 50 s.

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Fig. 9 Evolution of the beam for 800 V/cm each 10 s. (f) the beam is broken up in two new beams.

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The output distribution presents an intense central elliptic beam surrounded by a low intensity modulated elliptic ring. The behavior obtained by the central lobe was very similar to that, reported by Zozulya et al. [23] in the experimental confirmation of its theory presented in [24]; an initial circular symmetric Gaussian beam evolved to an elliptical beam.

When the applied external electric field was 800 V/cm (Fig. 9), the dynamic is very interesting and it starts after 20 seconds of applying external field, Fig. 9(c), the output distribution is very similar to that obtained in previous case at 50 seconds; a main lobe with high intensity surrounded by a low intensity ring. Nonetheless, the intensity of the central lobe is not kept due to part of it is at the ring (Fig. 9(d)). Practically after 40 s the central lobe disappeared and the surrounding ring seems to disperse in many spots. At 50 seconds (Fig. 9(f)) two circular bright spots could be observed; one with higher intensity than the other (closer to the interface) and we have obtained good agreement between experimental and numerical results (Figs. 3 and 6(d)).

In the case of 1000 V/cm after 20 seconds (Fig. 10(c)) the output distribution is very similar to the obtained at 20 s with 800 V/cm (Fig. 9(c)) and at 50 s with 600 V/cm (Fig. 8(f)). However, 40 s later (Fig. 10(e)), the main lobe reduced its intensity and gave rise to low intensity circular spots; it was possible to identify three circular spots, the main, with higher intensity aligned perpendicular to the interface and some irregular spots above of it.

Finally, for a better comprehension, numerical and experimental results are compared and shown in Fig. 11 . We analyze the steady state of the beams at 50 s, regarding Fig. 10(f) pictures (8), (9) and (10). We found numerically three important regions of the nonlinearity (η) and accumulate phase by distance of propagation (Z) as function of nonlinearity, where the experimental parameters were, initial width of the beam 200 μm, linear refractive index of 2.33, electro-optic coefficient, r = 224 pm/V, length of the crystal L = 1 cm, and λ = 632.8 nm. First region, for $3 \leq η \leq 8$ , the beam is only self-focused after of TIR and is stable for 50 s (see Fig. 11(a) to $η = 7.81$ ); second region, for $8 < η \leq 11.5$ , the beam is launched to the nonlinear interface and it is broken after the reflection in two beams (see Fig. 11(b)) for $η = 10 .42$ ); third region, for $η > 11.5$ (see Fig. 11(c)) for $η = 13 .1$ ) the beam is broken in multiple beams stimulated by the nonlinear interface, and the accumulating phase by the nonlinearity which depends on the propagation length. Values of the experimental nonlinearity have a constant difference among them of 2.6, approximately.

Fig. 11 Numerical and experimental comparison: a) first case, transversal profile of intensity of the beam for 600 V/cm (Fig. 8(f)); b) numerical transversal profile intensity for η = 7.81; c) second case, 800 v/cm (Fig. 9(f)); d) numerical profile intensity for η = 10.42; e) third case, 1000 V/cm (Fig. 10(f)); f) numerical profile intensity for η = 13.1.

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3. Conclusion

In conclusion, we have presented numerical an experimental results of the dynamics of total internal reflection of (2+1)D beams in nonlinear interface formed by SBN61:Ce PRC and air. When external electric field is applied, multiple beams appear in the parallel direction of application of the external electric field. The dependence of the intensity of the beam with electric field opens new applications for optical switching devices. Both results: experimental and numerical are consistent with each other.

Acknowledgments

This work was financially supported by Guanajuato-University-DINPO grant 30/11, 5/11 and SEP/PROMEP 103.5/09/7407. This article is dedicated in memory to Dr. G. E. Torres-Cisneros.

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Dynamic of total internal reflection (2+1)D bright beams on photorefractive SBN61:Ce crystal

Abstract

1. Introduction

2. Mathematical model for the nonlinear interface

3. Experimental setup

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (1)

Optics Express