Tuesday, August 26, 2008

Analytical Formulation of Normal Modes in Symmetrical Directional-coupler

The analysis of the optical power transfer in the linear step index directional-coupler based on the couple-mode theory is inaccurate for a small gap. This problem has been previously overcome by using the normal-modes approximation. Commonly,

this approximation has been solved by numerical methods such as Fourier transform or finite difference. In this paper, the Helmholtz equation is, instead, analytically solved by using a characteristic matrix of multiplayer waveguides in order to find the electric field and its propagation constant of the normal-modes. The importance of these analytical formulas, is that a phase shift of the normal modes along the propagation can be easily analyzed.

1. Introduction

In integrated optics areas, the directional-couplers are the major interest with potential applications to optical communications, i.e, used to fabricate low-loss optical switches[1], high speed modulators[2], polarization splitter[3] and wavelength demultiplexer/multiplexer[4]. Due to coupling effect, optical power can be transferred from one waveguide to another adjacent waveguide as a result of the overlap in the evanescent fields of the two guides. The amount of power transferred between the waveguides depends upon the waveguide parameters, i.e, the guided wavelength, the confinement of the individual waveguides, the separation between them, the length over which they interact, and the phase mismatch between the individual waveguides[5].

The power transfer of two waveguides in the directional-couplers has been treated extensively utilizing the coupled-mode method, but as shown in [6] this method becomes less accurate when the waveguides get too close. An alternative choice is the normal-modes approximation. This approximation taken full account of the entire structure and solves for modal indices and guided fields of the supermodes. In the normal-modes approach, the characteristic of the directional-couplers are then represented by interferences between the guided fields of the supermodes[7], i.e symmetrical and asymmetrical modes. In practical, the directional-couplers are made in 3-D structure, consist of waveguides with finite lateral dimensions. In order to obtain the exact solutions of normal modes, the 3-D is usually reduced to 2-D guides structure[7],[8]. Hence in the 2-D guides, the two parallel waveguides with their surrounding medium can be considered as a single structure, so that the normal-modes of the such structure can be solved by method of multilayer waveguides. In this paper we use the multilayer waveguides to formulate the optical electric fields in the symmetrical directional-couplers. The expression of such the guided fields derived by method of multilayer waveguides given by Kogelnik[9], and Rohedi[10].

For Detail Visit http://rohedi.com

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