## Abstract

We proposed a new all-optical switch by using the phase modulation of spatial solitons. The proposed structure is composed of the nonlinear Mach-Zehnder interferometer (MZI) with the straight control waveguide, the uniform nonlinear medium and the nonlinear output waveguides. The local nonlinear MZI functions like a phase shifter. The light-induced index changes in the local nonlinear MZI make the output signal beam routing in the uniform nonlinear medium. The all-optical switching scheme employs angular deflection of spatial solitons controlled by phase modulation created in the local nonlinear MZI. By properly launching the control power and increasing the length of the uniform nonlinear medium, this device can be generalized to a 1×N all-optical switch. It would be a potential key component in the applications of ultra-high-speed optical communications and optical data processing system.

© 2007 Optical Society of America

## 1. Introduction

Demands for ultra-high-speed optical signal processing and computing are increasing dramatically. All-optical ultrafast switching devices and logic gates based on the optical Kerr effect in a nonlinear waveguide have been of particular interest for high-bit rate optical communication and optical computing systems. The interest in nonlinear waveguide device, which has been growing steadily in recent years, stems from their potential use for ultrafast all optical signal processing and optical computing systems. The confinement of optical beam in the small core area and its diffractionless propagation over long distance increase the efficiency of the nonlinear interaction and permit the use of relatively weak nonlinearities. It has been shown that these nonlinear waveguides possess a variety of novel and exciting features such as power-dependent propagation constants and field profiles leading to novel feasibilities for all-optical signal processing and optical computing.

Several all-optical switching and logic devices using optical nonlinearity have ever been proposed and implemented [1–12]. Most of the conventional all-optical devices are based on uniformly nonlinear structure. Therefore, the whole of the waveguide has optical nonlinearity uniformly. In optical waveguide structure of uniform nonlinearity, several interesting optical properties have been shown, however, there are still more attractive propagation characteristics in waveguide structures combined a nonlinear material with a linear one. Some theoretical studies about the optical waveguide structures made from linear and nonlinear materials have been proposed, for example, a waveguide structure composed of linear films bounded by nonlinear media [13–19] or a nonlinear film sandwitched between linear media [20–24].

Recently, there has been great interesting in the Mach-Zehnder waveguide interferometer device [25–28]. The Mach-Zehnder waveguide interferometer device has been developed for use of modulating, switching, and logic gates, etc. Most of them are operated by the principles of electric-optic effect where the change of refractive index in the arms of Mach-Zehnder interferometer is produced by applying the external electric field. The all-optical Mach-Zehnder waveguide interferometer device has been presented previously [29–30]. In this paper, we propose a new 1×N all-optical switching device by using the local nonlinear MZI with a straight control waveguide based on the phase modulation of spatial solitons. The light-induced index changes in the MZI break the symmetry of the structure and make the output signal beam routing in the uniform nonlinear medium. By properly launching the control power and increasing the length of the uniform nonlinear medium, the numerical results show that this device can be generalized to a 1×N all-optical switch.

## 2. Analysis process

The proposed structure is composed of the local nonlinear MZI with the straight control waveguide, and the uniform nonlinear medium, as shown in Fig. 1. The scheme employs angular deflection of spatial solitons controlled by the phase modulation created in the local nonlinear MZI which functions like a phase shifter. The presence of the evanescent tail of the optical beam in the straight control waveguide will introduce the phase difference between the two arms of the local nonlinear MZI, and will lead to a final position shift of the signal beam at the end of the proposed structure. The nonlinear MZI with two local nonlinear waveguides is used to propagate the signal beam. Since the control beam is useless in the following process, a lossy medium at the end of a straight control waveguide is used to attenuate it. For the local nonlinear MZI, the branching angle is *θ*, w_{1} the width of the control waveguide, w_{2} the width of the Mach-Zehnder waveguide, *L*
_{1} the length of local nonlinear waveguides, *L*
_{2} the length of Mach-Zehnder waveguide, *L*
_{3} the length of the uniform nonlinear medium, and *l* the distance between the local control waveguide and the nearest signal waveguide.

In this section, we use the modal theory to derive the general formulas that can be used to analyze the multilayer optical waveguide structure with nonlinear guiding films, as shown in Fig. 2. The multilayer optical waveguide structure is composed of nonlinear guiding films ($\left(\frac{m-1}{2}\phantom{\rule{.2em}{0ex}}\mathrm{layers}\right)$ layers), interaction layers ($\left(\frac{m-3}{2}\phantom{\rule{.2em}{0ex}}\mathrm{layers}\right)$ layers), cladding, and substrate. The total number of layers is m (*m*=3,5,7,…). The cladding and substrate layers are assumed to extend to infinity in the +x and −x direction, respectively. The major significance of this assumption is that there are no reflections in the x direction to be concerned with, except for those occurring at interfaces.

For simplicity, we consider the transverse electric polarized waves propagating along the z direction. The wave equation can be reduced to

with solutions of the form

where *ω* is the angular frequency, *k*
_{0} is the wave number in the free space, and β is the effective refractive index. For a Kerr-type nonlinear medium [54–56], the square of the refractive index of the guiding film can be expressed as[31–33]:

where *n*
_{0i} and *α _{i}* are the linear refractive index and the nonlinear coefficient of the

*i*-th layer nonlinear guiding film, respectively. The transverse electric field in each layer can be expressed as:

$$i=3,5,\dots ,m-2\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\mathrm{in}\phantom{\rule{.2em}{0ex}}\mathrm{the}\phantom{\rule{.2em}{0ex}}\mathrm{interaction}\phantom{\rule{.2em}{0ex}}\mathrm{layers}$$

$$i=2,4,\dots ,m-1\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\mathrm{in}\phantom{\rule{.2em}{0ex}}\mathrm{the}\phantom{\rule{.2em}{0ex}}\mathrm{guiding}\phantom{\rule{.2em}{0ex}}\mathrm{film},\phantom{\rule{.2em}{0ex}}\mathrm{for}\phantom{\rule{.2em}{0ex}}\beta <{n}_{i}$$

$$i=2,4,\dots ,m-1\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\mathrm{in}\phantom{\rule{.2em}{0ex}}\mathrm{the}\phantom{\rule{.2em}{0ex}}\mathrm{guiding}\phantom{\rule{.2em}{0ex}}\mathrm{film},\phantom{\rule{.2em}{0ex}}\mathrm{for}\phantom{\rule{.2em}{0ex}}\beta <{n}_{i}$$

where cn is a Jacobian elliptic function, and the constants *d* and *w* are the widths of the guiding film and the interaction layer, respectively.

The constants *p _{i}*,

*b*,

_{i}*A*,

_{i}*l*,

_{i}*b′*,

_{i}*A′*, and

_{i}*l′*can be expressed as

_{i}${p}_{i}={k}_{0}\sqrt{{\beta}^{2}-{n}_{i}^{2}},$ ${b}_{i}^{2}=\frac{\sqrt{{q}_{i}^{4}+2{\alpha}_{i}{k}_{0}^{2}{K}_{i}-{q}_{i}^{2}}}{{\alpha}_{i}{k}_{0}^{2}},$ ${A}_{i}={\left[\left({a}_{i}^{2}+{b}_{i}^{2}\right)\left(\frac{{\alpha}_{i}{k}_{0}^{2}}{2}\right)\right]}^{1\u20442},$ ${l}_{i}=\frac{{b}_{i}^{2}}{\left({a}_{i}^{2}+{b}_{i}^{2}\right)},$ ${b\prime}_{i}^{2}=\frac{\sqrt{{Q}_{i}^{4}+2{\alpha}_{i}{k}_{0}^{2}{K}_{i}}+{Q}_{i}^{2}}{{\alpha}_{i}{k}_{0}^{2}},$ ${A}_{i}\prime ={\left[\left({a\prime}_{i}^{2}+{b\prime}_{i}^{2}\right)\left(\frac{{\alpha}_{i}{k}_{0}^{2}}{2}\right)\right]}^{1\u20442},$ ${l}_{i}\prime =\frac{{b\prime}_{i}^{2}}{\left({a\prime}_{i}^{2}+{b\prime}_{i}^{2}\right)},$

where the constants *a _{i}*,

*a′*,

_{i}*q*,

_{i}*Q*,

_{i}*K*,

_{i}*x*

_{0}

*and*

_{i}*x′*

_{0}

*are all constants which can be determined by a numerical method on a computer.*

_{i}## 3. Numerical results and discussions

A general method for analyzing the multilayer planar optical waveguide with a localized arbitrary nonlinear guiding film has been proposed in the previous section. The wave Eq. (1) can be solved numerically by using the beam propagation method [34] with 4096 transverse sampling points and a longitudinal step length Δz=0.05*µm*. The numerical data have been calculated with the values: *θ*=0.53°, w_{1}=1*µm*, w_{2}=2*µm*, *L*
_{1}=4000*µm*, *L*
_{2}=9000*µm*, *L*
_{3}=3000*µm*, *l*=3.5*µm*, *n _{f}*

_{0}=

*n*

_{u}_{0}=1.55,

*n*=

_{c}*n*

_{c}_{0}=1.545, the free space wavelength

*λ*=1550

*nm*,

*α*=6.3786×10

^{-12}

*m*

^{2}/

*V*

^{2}(for MBBA liquid crystal)[31–33], the optimum input signal power is fixed at P

_{0}=23

*W*/

*m*, and the weak control power is varied from 0 to 0.5P

_{0}. The symbol Δd is used to denote the position shift of the output signal beam propagating throughout the uniform nonlinear medium and P

_{c}/P

_{0}the normalized control power. The position shift Δd as a function of the normalized control power P

_{c}/P

_{0}is shown in Fig. 3. The numerical results show the possibility of controlling the switching of a beam in the local nonlinear MZI by means of a relatively weak control beam in a parallel linear waveguide. The presence of the evanescent tail of the control beam will introduce an asymmetry in the local nonlinear MZI, and will lead to a final position shift of the signal beam at the end of the propose structure. For further understanding the results shown above, we show some numerical examples of the waves propagating along the structure, as shown in Figs. 4 (a)~(e). When there is no control beam, the evolution of output signal beam propagating straight through the uniform nonlinear medium is shown in Fig. 4 (a). When the control beam is on, the output signal beam will route in the uniform nonlinear medium as shown in Figs. 4 (b)~(e). For example, when the control power reaches P

_{c}=0.0573P

_{0}, the output signal beam will be switched to the left side of uniform nonlinear medium with the position shift Δd=-12

*µm*, as shown in Fig. 4 (b), when the control power reaches P

_{c}=0.0528P

_{0}, the output signal beam will be switched to the left side of uniform nonlinear medium with the position shift Δd=-6

*µm*, as shown in Fig. 4 (c), when the control power reaches P

_{c}=0.0246P

_{0}, the output signal beam will be switched to the right side of uniform nonlinear medium with the position shift Δd=6

*µm*, as shown in Fig. 4 (d), and when the control power reaches P

_{c}=0.021P

_{0}, the output signal beam will be switched to the right side of uniform nonlinear medium with the position shift Δd=12

*µm*, as shown in Fig. 4 (e). As the results shown above, we can design a 1×N all-optical switching device by using the position shift of the output signal beam.

The proposed 1×N all-optical switching device is composed of the local nonlinear MZI with a straight control waveguide, a uniform nonlinear medium, and N nonlinear output waveguides, as shown in Fig. 5. *L*
_{4} is denoted the length of the nonlinear output waveguides. The beam propagation method is used to simulate the evolution of optical waves propagating along the proposed structure. For the simulation, we propose a 1×7 all-optical switching device with the local nonlinear MZI, as shown in Fig. 6. The numerical data have been calculated with the values: *θ*=0.53°, w_{1}=1*µm*, w_{2}=2*µm*, *L*
_{1}=4000*µm*, *L*
_{2}=9000*µm*, *L*
_{3}=3000*µm*, *L*
_{4}=2000*µm*, *l*=3.5*µm*, *λ*=1.55*µm*, *n _{f}*

_{0}=

*n*

_{u}_{0}=1.55,

*n*=

_{c}*n*

_{c}_{0}=1.545,

*α*=6.3786×10

^{-12}

*m*

^{2}/

*V*

^{2}, P

_{0}=23

*W*/

*m*, and the weak control power is varied from P

_{c}=0.01P

_{0}to P

_{c}=0.063P

_{0}. When there is no control beam, the output signal beam will propagate straight through the central output waveguide A, as shown in Fig. 7 (a). When the control beam is on, the output signal beam will be switched from one output waveguide to another, as shown in Figs. 7 (b)~(g). When the control power reaches P

_{c}=0.061P

_{0}, the output signal beam will be switched to the left output waveguide B with the position shift Δd=-30

*µm*, as shown in Fig. 7 (b), when the control power reaches P

_{c}=0.0578P

_{0}, the output signal beam will be switched to the left output waveguide C with the position shift Δd=-20

*µm*, as shown in Fig. 7 (c), when the control power reaches P

_{c}=0.0529P

_{0}, the output signal beam will be switched to the left output waveguide D with the position shift Δd=-10

*µm*, as shown in Fig. 7 (d), when the control power reaches P

_{c}=0.025P

_{0}, the output signal beam will be switched to the right output waveguide E with the position shift Δd=10

*µm*, as shown in Fig. 7 (e), when the control power reaches P

_{c}=0.021P

_{0}, the output signal beam will be switched to the right output waveguide F with the position shift Δd=20

*µm*, as shown in Fig. 7 (f), and when the control power reaches P

_{c}=0.0118P

_{0}, the output signal beam will be switched to the right output waveguide G with the position shift Δd=30

*µm*, as shown in Fig. 7 (g). For the numerical results shown above, we confirm that the proposed device could function as a new 1×N all-optical switching device by properly launching the control power and increasing the length of the uniform nonlinear medium.

## 4. Conclusions

We proposed a new all-optical switch by using the position shift of the spatial solitons controlled by phase modulation created in the local nonlinear MZI with a straight control waveguide. The local nonlinear MZI functions like a phase shifter. The presence of the evanescent tail of the optical beam in the straight control waveguide will introduce the phase difference between the two arms of the local nonlinear MZI, and will make the output signal beam switching from one output waveguide to another. By properly launching the control power and increasing the length of the uniform nonlinear medium, this device can be generalized to a 1×N all-optical switch. It would be a potential key component in the applications of ultra-high-speed optical communications and optical data processing systems.

## Acknowledgment

This work was partly supported by National Science Council R. O. C. and Ministry of Education R. O. C. under Grant No. 95C9031 and 95TSFC9031.

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