## Abstract

We propose an all-optical switching device based on the interaction property between optical spatial solitons. By launching the nonlinear symmetric modes for the relative phase relation Π into the uniform nonlinear medium, the repulsive property between spatial solitons will be observed. Based on the repulsive property, a new all-optical switching device will be proposed.

© 2006 Optical Society of America

## 1. Introduction

All-optical switching devices based on the optical Kerr effect in a nonlinear waveguide have been object of great interest for high-bit rate optical communication systems and ultra-fast information processing systems. In the past, a number of all-optical switching devices have been proposed by using a nonlinear interferometer [1–2], a nonlinear directional coupler [3], and a nonlinear waveguide junction [4–6]. Further, a number of theoretical studies about wave propagating along the guided-wave systems made from linear and nonlinear material have also been reported [7–18].

Recently, the application of the spatial soliton in the all-optical device [19–28] has been discussed ardently. Spatial solitons are the result of balance between diffraction effect and self-focusing effect. The light beams propagating through a bulk medium tend to broaden due to diffraction. However, the spatial broadening due to diffraction can be compensated by using the features of nonlinear optics which make the evolutions of light beams sharpen due to self-focusing. Once the diffraction effect and the self-focusing effect balance for each other, the spatial solitons can be formed [29–30]. This is a similar situation that exists in the time domain for the optical pulses. The optical pulses propagating along a fiber tend to broaden due to group velocity dispersion. Once the group velocity dispersion effect and the self-phase modulation effect balance for each other, the temporal optical solitons can be formed.

The interaction between spatial solitons has attracted much attention because they resemble real particles in the interaction properties [31–32]. When one fundamental soliton is launched parallel to another, they will attract or repel each other, depending on the relative phase relation between them. The interaction is attractive with the relative phase of 0 in phase, whereas the interaction is repulsive with the relative phase of *π* out of phase. The strength of interaction also depends upon the initial separation distance between spatial solitons.

Based on the repulsion between spatial solitons, a new all-optical switch will be proposed. By launching the nonlinear symmetric modes with the relative phase of *π* into the uniform nonlinear medium, the repulsive property between spatial solitons is observed. These characteristics are investigated by using the beam propagation method (BPM) [33]. By fixing the input signal power and changing the control power, the numerical results show that this device could really function as an all-optical switch.

## 2. Analysis

The structure of the proposed all-optical switching device is shown in Fig. 1. It is divided into three sections: the input section, the uniform nonlinear medium section and the output section. The lengths of the three sections are *L*
_{1}, *L*
_{2}, and *L*
_{3}, respectively. In the input section, the straight signal guide is in the center and the two outward guides are control guides. The two control guides are parallel to the signal guide and thus the control beam and the signal beam can repel each other in the nonlinear medium. The separated distance between the control guide and the signal guide is denoted S, and the width of each guiding film is denoted W. In the nonlinear medium section, the spatial solitons are repelling each other. In the output section, the nonlinear waveguides are used to couple out the spatial solitons excited by the input signal beams. The separation between each output guide is sufficiently large to prevent the mutual coupling. Since the control beam is useless in the following process, we may use a lossy medium to attenuate it. For simplicity, we consider the case of *TE* waves propagating along the structure as

where *k*
_{0} is the wave number in the free space and β is the effective refractive index, and we have taken the field to be homogeneous in the y direction. Taking into account the slowly varying envelope approximation, we obtain the following equation for E(x, z):

where the subscripts f, c, and u are used to denote the guiding film, the cladding, and the uniform nonlinear medium, respectively. For a Kerr-type nonlinear medium, the square of the refractive index ${n}_{i}^{2}$ can be expressed as:

where *n*_{io}
is the linear refractive index of the nonlinear medium and α is the nonlinear coefficient (α=0, for the linear medium). All the refractive indices in the proposed structure are *n*_{i}
= *n*_{co}
in the cladding of the input and output sections, ${n}_{i}=\sqrt{{n}_{\mathit{fo}}^{2}+\alpha {\mid E\mid}^{2}}$ in the guiding film of the input section, ${n}_{i}=\sqrt{{n}_{\mathit{fo}}^{2}+\alpha {\mid E\mid}^{2}}$ in the guiding film of the output section, and
${n}_{i}=\sqrt{{n}_{\mathit{uo}}^{2}+\alpha {\mid E\mid}^{2}}$ in the uniform nonlinear medium section.

## 3. Results and discussions

We use the BPM to simulate *TE* waves propagating along the structure. For the calculations, we choose the following numerical data: the transverse sampling points N=1280, a longitudinal step length Δ*z* = 0.05 μm, the width of each guiding film W=1.5μm, the free space wavelength *λ* =1.55μm, L_{1}=50μm, L_{2}=220μm, L_{3}=50μm, *n*
_{f0} = 1.53, *n*
_{c0} = *n*
_{u0} = 1.5, *α* =6.3786*μm*
^{2} /*V*
^{2}, *S*=2.5μm, the optimum input signal power is fixed at *P*
_{0} =0.07*W*/*mm*, and the input control power is varied from 0.82*P*
_{0} to 1.16*P*
_{0} The symbol Δd is used to denote the position shift of the output signal beam propagating throughout the uniform nonlinear medium and the symbol P_{c}/P_{0} is used to denote the normalized control power. The position shift Δd as a function of the normalized control power P_{c}/P_{0} with the left control beam is shown in Fig. 2. Symmetry guarantees similar results when the right control beam is on. The results shown above can be used to design an all-optical 1×N switching device by using the position shift of the output signal beam. For example, we proposed a 1×9 all-optical switching device, as show in Fig. 3.

When there is no control beam, the output signal beam will propagate straight through the output waveguide E, as shown in Fig. 4. When the left control beam is on, the output signal beam will swing in the uniform nonlinear medium by using the repulsive property between optical spatial solitons. When the control power reaches P_{c} = 0.832P_{0}, the output signal beam will be switched to the right output waveguide A with the position shift Δd = 2*μm*, as shown in Fig. 5, when the control power reaches P_{c} = 0.864P_{0}, the output signal beam will be switched to the right output waveguide B with the position shift Δd = 4*μm*, as shown in Fig. 6, when the control power reaches P_{c} = 0.91P_{0}, the output signal beam will be switched to the right output waveguide C with the position shift Δd = 6*μm*, as shown in Fig. 7, and, last, when the control power reaches P_{c} = 1.02P_{0}, the output signal beam will be switched to the right output waveguide D with the position shift Δd = *μm*, as shown in Fig. 8. Symmetry guarantees similar results when the right control beam is on. The output signal beams will be switched to the left output guides from one waveguide to another. These results shown above can be used to design an all-optical switching device by using the position shift of the output signal beam.

In order to further understand the influence of the separated distance S and the width of each guiding film W of the position shift, more detail results are shown in Figs. 9–10. Figure 9 shows that when we fix the width of each guiding film W=1.5μm and Po=0.07W/mm, and increase the separated distance S, the position shift will also decrease consequently. Because when S increases, the repulsive force between the spatial solitons will decrease. Figure 10 shows that when we fix S=2.5 μm and Po=0.07W/mm, and increase the width of each guiding film W, the position shift will increase at the beginning and then decrease consequently. Because when the normalized control power P_{c}/P_{o} is less than 0.85, as W increases, the repulsive force between the spatial solitons will also increase; but when P_{c}/P_{o} is large than 0.85, as W increases, the repulsive force between the spatial solitons will decrease.

## 4. Conclusions

In the paper, we have designed a new 1×N all-optical switch by using the spatial soliton repulsion. The proposed structure is an all-optical switch controlled by two control beams. If there is no control beam, the signal beam will be coupled out in the central output guide. When the control beam is on, the output signal beam will swing in the uniform nonlinear medium. The numerical results show that the proposed waveguide structure could really function as a all-optical switching device. It is a potential key component in the application of optical signal processing and optical computing systems.

## Acknowledgments

This work was partly supported by National Science Council R. O. C. under Grant No. 94-2215-E-151-001.

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