## Abstract

We propose new all-optical logic gates containing a local nonlinear Mach-Zehnder interferometer waveguide structure. The light-induced index changes in the Mach-Zehnder waveguide structure make the output signal beam propagate through different nonlinear output waveguides. Based on the output signal beam propagating property, various all-optical logic gates by using the local nonlinear Mach-Zehnder waveguide interferometer structure with two straight control waveguides have been proposed to perform XOR/NXOR, AND/NAND, and OR/NOR logic functions.

©2008 Optical Society of America

## 1. Introduction

All-optical ultrafast photonic devices using nonlinear optical effect for applications to optical communications and optical signal processing systems have been interested popularly. There has been great interest in the possibility of using nonlinear optical waveguide devices as ultrafast all-optical switching and logic devices for optical signal processing and optical communication systems. Several all-optical switching and logic devices using optical nonlinearity have ever been proposed and implemented [1–8]. 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 one or two nonlinear media [9–14] or a nonlinear film sandwitched between linear media [15–18] or arbitrary layers with nonlinear media [19].

Recently, there has been great interesting in the Mach-Zehnder waveguide interferometer device [20–23]. The Mach-Zehnder waveguide interferometer device has been developed for the 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 [24–25]. In this paper, based on the principle of optical couplers, a local nonlinear Mach-Zehnder waveguide interferometer structure will be proposed. The nonlinear Mach-Zehnder interferometer with two straight control waveguides will be used to design various all-optical logic gates with XOR/NXOR, AND/NAND, and OR/NOR functions. The light-induced index changes in the Mach-Zehnder waveguide break the symmetry of structure and make the output signal beam propagate through different nonlinear output waveguides. These characteristics are investigated by using the beam propagation method (BPM) [26]. By properly fixing the input signal power and launching the control beams, the numerical results show that the proposed waveguide structure could function as all-optical XOR/NXOR, AND/NAND, and OR/NOR logic gates.

## 2. Analysis

The proposed waveguide structures of all-optical logic gates are composed of the local nonlinear Mach-Zehnder waveguide interferometer with two straight control waveguides and two output waveguides as shown in Fig. 1 (a)–(c). 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. Here we consider three cases of the different local nonlinear distribution in two arms of the nonlinear Mach-Zehnder interferometer. For the first case, the local nonlinear waveguides are located on two arms of the nonlinear Mach-Zehnder interferometer, as shown in Fig. 1(a). For the second case, the local nonlinear waveguide is only located on the left arm of the nonlinear Mach-Zehnder interferometer, as shown in Fig. 1(b). For the third case, the local nonlinear waveguide is only located on the right arm of the nonlinear Mach-Zehnder interferometer, as shown in Fig. 1(c). The nonlinear Mach-Zehnder interferometer with two local nonlinear waveguides functions like a phase shifter and is used to propagate the signal beam. Since the control beams are useless in the following process, a lossy medium at the end of two straight control waveguides is used to attenuate it. For the nonlinear Mach-Zehnder interferometer, the branching angle is *θ _{1}*,

*w*the width of control waveguide,

_{1}*w*the width of Mach-Zehnder waveguide,

_{2}*L*the length of Mach-Zehnder waveguide,

_{1}*L*the length of nonlinear output waveguides, and

_{2}*L*the length of local nonlinear waveguides. The distance between the local control waveguide and the nearest signal waveguide is

_{3}*l*and the branching angle between the output guides is

*θ*.

_{2}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 (
$\frac{m-1}{2}$
layers), interaction layers (
$\frac{m-3}{2}$
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, the square of the refractive index of the guiding film can be expressed as[27–28]:

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:

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

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

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

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*′

_{i}, and

*l*′

_{i}can be expressed as

where the constants *a _{i}*,

*a*′

_{i},

*q*,

_{i}*Q*,

_{i}*K*,

_{i}*x*

_{0i}, and

*x*′

_{0i}are all constants which can be determined by a numerical method on a computer.

## 3. Numerical Results

In this paper, the wave Eq. (1) was solved numerically by using the BPM with 4096 transverse sampling points and a longitudinal step length Δz=0.05*µm*. The numerical data have been calculated with the value: the total propagation distance *z*=12500*µm*, *θ _{1}*=0.5°,

*θ*=0.25°,

_{2}*l*=3.5

*µm*,

*w*=1

_{1}*µm*,

*w*=2

_{2}*µm*,

*L*=10000

_{1}*µm*,

*L*=2500

_{2}*µm*,

*L*=4100

_{3}*µm*, α=6.3786µm

^{2}/V

^{2},

*n*=1.55,

_{f0}*n*=1.545, the free space wavelength

_{c0}*λ*=1.55

*µm*. We first examine the XOR/NXOR logic functions, as shown in Fig. 3(a)–(d). Figure 3 shows the typical evolutions of the input light waves propagating along the structure with the input control power P

_{c}=23.7 W/m and the input signal power P

_{s}=79 W/m. When there is no control straight through the central output guide C, as shown in Fig. 3(a). When only the right control guide B is excited, the output signal beam will propagate through the right output guide D, as shown in Fig. 3(b). When only the left control guide A is excited, the output signal beam will propagate through the right output guide D, as shown in Fig. 3(c). When both of the control guides A and B are excited simultaneously, the output signal beam will propagate straight through the central output guide C, as shown in Fig. 3(d). As the results shown above, the output port C functions as an NXOR Gate and the output port D functions as an XOR gate. All logic states of the XOR and NXOR gates are shown in Table 1.

We next examine the AND/NAND logic functions, as shown in Fig. 4(a)–(d). Figure 4 shows the typical evolutions of the input light waves propagating along the structure with the input control power P_{c}=30 W/m and the input signal power P_{s}=60 W/m. When there is no control beam, the output signal beam will propagate through the right output guide D, as shown in Fig. 4(a). When only the right control guide B is excited, the output signal beam will propagate through the right output guide D, as shown in Fig. 4(b). When only the left control guide A is excited, the output signal beam will propagate through the right output guide D, as shown in Fig. 4(c). When both of the control guides A and B are excited simultaneously, the output signal beam will propagate straight through the central output guide C, as shown in Fig. 4(d). As the results shown above, the output port C functions as an AND gate and the output port D functions as a NAND gate. All logic states of the AND and NAND gates are shown in Table2.

Finally, we examine the OR/NOR logic functions, as shown in Fig. 5(a)–(d). Figure 5 shows the typical evolutions of the input light waves propagating along the structure with the input control power P_{c}=31.2 W/m and the input signal power P_{s}=78 W/m. When there is no control beam, the output signal beam will propagate through the right output guide D, as shown in Fig. 5(a). When only the right control guide B is excited, the output signal beam will propagate straight through the central output guide C, as shown in Fig. 5(b). When only the left control guide A is excited, the output signal beam will propagate straight through the central output guide C, as shown in Fig. 5(c). When both of the control guides A and B are excited simultaneously, the output signal beam will propagate straight through the central output guide C, as shown in Fig. 5(d). As the results shown above, the output port C functions as an OR gate and the output port D functions as a XOR gate. All logic states of the OR and NOR gates are shown in Table3.

For the results shown above, we confirm that the proposed waveguide structure could really function as all-optical XOR/NXOR, AND/NAND, and OR/NOR logic gates.

## 4. Conclusions

In conclusion, new all-optical logic devices have been proposed by using the property of repulsion and attraction between optical light waves of two arms in the nonlinear Mach-Zehnder interferometer. When the control beam is on or off, the output signal beam will propagate through different output waveguides. The numerical results show that the proposed devices could really function as all-optical XOR/NXOR, AND/NAND, OR/NOR logic gates. It could be a potential key component in the application of optical signal processing and optical computing. In principle, the logic elements can be implemented by combining the nonlinear material with the planar lightwave circuit fabrication technique. Moreover, several new approaches have been proposed and demonstrated which can realize this device farther.

## Acknowledgement

The authors would like to thank M. L. Whang and R. Z. Tasy for their help. This work was partly supported by National Science Council of Taiwan under Grants NSC 96-2221-E-151-025 and NSC 96-2221-E-151-024-MY3 and Ministry of Education of Taiwan under Grants 95C9031, 95TSFC9031, and 95A6077.

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