## Abstract

Studies have shown that 1^{st} order coupling coefficient dispersion can cause significant effects on the propagation of short pulses in a twin-core coupler. In this paper, we have extended the study to the case of nonidentical cores and investigated the effect of 2^{nd} order dispersive coupling coefficient on switching dynamics. A pair of new coupled nonlinear equations have been presented and analyzed.

© 2003 Optical Society of America

## 1. Introduction

Nonlinear directional couplers made of two cores have been studied extensively owing to their potential applications especially in ultra-rapid all-optical switching processing [1–3]. Most of previous research work concentrated on two identical cores which are difficult to produce in practice, so it’s necessary to understand the switching dynamics of two-nonidentical-core couplers. An important idea in nonlinear switching is to use soliton pulse, which maintains their shape in the presence of weak perturbations, while they are supported by the balance between dispersive and nonlinear effects.

It has been shown that 1^{st} order dispersive coupling coefficient (i.e., intermodal disperson) can give rise to significant effects on pulse propagation in a twin-core directional coupler. The purpose of this paper is to analyze the influence of 2^{nd} order dispersive coupling coefficient on two-nonidentical-core coupler. To include the effects arising from the dispersion properties of the coupling coefficient, new coupled-mode equations including 2^{nd} order coupling coefficient dispersion have been presented.

## 2. Analysis

It has been shown that intermodal dispersion usually has greater effect and can break up pulses launched into twin-core coupler [4]. Higher-order coupling coefficient dispersion has often been ignored in most cases and only intermodal dispersion has been considered. However, it’s important to observe the switching dynamics when higher-order coupling coefficient dispersion is considered in two-nonidentical-core couplers.

We present the coupled-mode equations that contain the 2^{nd} order dispersive coupling coefficient and govern the pulse dynamics in two-nonidentical-core couplers with Kerr nonlinearity. The two parallel nonidentical cores embedded in a common cladding, are different in their radii denoted by ρ_{1} and ρ_{2}, but have the same refractive index.

The new coupled equations that contain the second-order dispersive coupling coefficient are shown as:

where subscripts 1 and 2 denoting the large and small core respectively. a_{1}(Z,T) and a_{2}(Z,T) represent the normalized varying envelopes of the pulses carried by the fundamental modes of the two cores in isolation. *k*″_{1(2)} denotes the group velocity dispersion of the large (small) core individually and ν_{g1(2)} describes the group velocity.

*C*
_{12} and *C*
_{21} are the coupling coefficients. *C*′_{12} and *C*′_{21} account for the intermodal dispersion, and the terms that contain *C*″_{12} and *C*″2l describe the 2^{nd} order dispersive coupling coefficients. To simplify the two equations, a normalized coordinate system that moves with the waves at the group velocity of small core is adopted, i.e.,
$Z=-\frac{{k}_{2}^{"}}{{t}_{0}^{2}}z,T=\frac{1}{{t}_{0}}\left(t-\frac{z}{{v}_{g2}}\right),$
where t_{0} is width of the input pulse. Then the coupled-mode nonlinear equations for two-nonidentical-core couplers which contain 2^{nd} order dispersive coupling coefficient are obtained as follows:

Due to the radii difference between the two cores, all the parameters of each core are different, which leads to the asymmetry of the equations and increases the complexity of solving the equations. A numerical technique, Fourier series analysis technique [5], is used to solve the Eqs. (3) and (4). First *a*
_{1} and *a*
_{2} are expanded in a Fourier series and substituted into (3) and (4), then a set of first-order partial differential equations are obtained. The set of equations obtained can be solved by Runge-Kutta method in the frequency domain. Employing the inverse Fourier transform, we can get the time domain solution.

We analyze a typical two-nonidentical-core fiber, which has a refractive index difference of 0.0055 between the cores and the cladding, corresponding to weakly guiding case, and is operated at the wavelength of 1.55µm. Other parameters include core-to-core separation d=9.75µm, pulse width t_{0}=10fs, ρ_{1}=3.25µm and ρ_{2} / ρ_{1} =0.9..

In an ideal soliton-based communication system, input pulses launched into fiber should be unchirped. In order to investigate the effect of 2^{nd} order coupling coefficient dispersion, we launch a pulse into the large core. The initial conditions are given by: *a*
_{1}(0,*T*) = *A*sec*h*(*T*) and *a*
_{2}(0,*T*) = 0, where A is the amplitude of input pulse. We vary the amplitude of input pulse and study the shift of threshold power point without/ with the presence of 2^{nd} order coupling coefficient dispersion.

First we analyze the case while ignoring the effect of 2^{nd} order coupling coefficient dispersion, i.e., setting *C*″_{12} =*C*″_{21}=0. It is well known that a nonlinear fiber coupler can function as an all-optical switch, and the pulse can couple back and forth between the two cores under the threshold value. Above it the pulse will essentially trap in the input core and cannot couple to the other core.

We vary the amplitude of the input pulse in a practical range while all other initial parameters are kept the same. In the simulation, we narrow down the range of A from 4.0 to 4.5. The propagation characteristics are shown in Fig. 1 (a) and Fig. 1 (b) for A = 4.0 and 4.5, respectively. In the figures., the parameters U and V are defined as: U(Z,T)=|a_{1}(Z,T)|_{2} and V(Z,T)=|a_{2}(Z,T)|^{2}. At A = 4.0, the pulse can still couple back and forth between the two cores with little distortion, while at A = 4.5, the pulse stays mainly in one core. These phenomena form the basis of an intensity-dependent optical switch. Our simulation results show that there is a great power transfer from large core to small core for A = 4.0 to 4.3. At A = 4.4, the pulse is almost trapped in the input core. Thus, we can conclude that the amplitude threshold is between 4.3 and 4.4.

Then we consider the effects of 2^{nd} order coupling coefficient dispersion on pulse propagation. Similarly we vary the amplitude of the input pulse and the propagation characteristics are shown in Fig. 2 (a) and (b) for amplitude A = 4.5 and 4.9 respectively. At A =4.5, the pulse can still couple between the two waveguides. For A = 4.9, the pulse stays mainly in the input core.

The difference is that the threshold amplitude value is larger than that without the presence of *C*″_{12}, and *C*″_{21}. The results show that a greater coupling of power from core 1 to core 2 for A between 4.5 and 4.7. For A=4.8, less power can be coupled. At A = 4.9, little power can be transferred from core 1 to core 2. Therefore the amplitude threshold level is between 4.8 and 4.9. Compared with Fig. 1, the 2^{nd} order dispersive coupling coefficient has the effect of increasing the threshold amplitude above which the input pulse cannot couple to the other core and only stay in the input core.

## 3. Conclusion

By employing the Fourier series analysis technique, we have solved numerically the new coupled equations describing pulse switching in two-nonidentical-core couplers that contain the 2^{nd} order dispersive coupling coefficient. In particular, we have studied the effects of the 2^{nd} order coupling coefficient on soliton switching dynamics in a two-nonidentical-core coupler. We have found that 2^{nd} coupling-coefficient dispersion can increase the threshold amplitude and they should be an important factor of consideration in the design of two-nonidentical-core nonlinear directional couplers.

## References

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