Improved Analysis of the Coupling of Optical Waves into Multimode Waveguides Using Overlap Integrals

The optical multimode interconnection technology has become important due to increasing data rates in modern multi-processor systems. In this context, the coupling of optical waves into multimode waveguides has to be analyzed. Ray optical methods are preferred, but are not applicable for all geometries. As full-wave analysis using the mode matching technique is very extensive, we present a simple method based on overlap integrals, to calculate the coupling efficiency of impinging optical waves. Our approach neglects reflected waves. The corresponding error is corrected by applying a transmission factor, which is that of a plane wave irradiating a dielectric half space. We verify our approach at a planar slab waveguide and a cylindrical fiber, as their mode spectra are well known. The mode matching technique is applied as reference method and the impinging wave is a Gaussian beam with varying angles of incidence and lateral displacements. Introduction Future computing hardware will operate with on-chip clock rates up to 10 GHz. To provide proportionate data transfer rates in multi-processor systems, an optical interconnection technology based on board-integrated optical channel waveguides is needed. These waveguides have to be highly multimodal to be adaptable to the existing PCB mass production [1]. The system design of optical interconnects requires an effective set of design tools with each basic algorithm being efficient with respect to time needs and memory requirements. The aim of this paper is to present an effective approach to describe the coupling of optical waves into a multimodal step index waveguide based on overlap integrals. Investigation of these coupling processes by full-wave analysis using the mode matching technique is very extensive, as a lot of reflected and transmitted modes have to be regarded. The total number of modes needed by this method might exceed a few thousand. Thus a very large linear system of equations has to be solved. Therefore, this method is very time and memory consuming. Another disadvantage is that only closed structures can be analyzed. An often used approximation is to neglect the reflected waves. In this case, evaluating the boundary conditions for the transverse field components by means of the mode orthogonality leads to explicit expressions for the amplitudes of the transmitted modes involving overlap integrals. The error caused by neglecting reflected waves can be corrected by applying a mode-independent transmission factor. The waveguide design we want to analyze consists of waveguides with rectangular cross section. As no analytical expressions for the modes of these waveguides are known, we look at two basic waveguides with known mode spectra to verify our approach. The first one is a planar slab waveguide, and the second is a cylindrical fiber. In our calculations, the incident wave is a Gaussian beam, which irradiates the waveguide interface with varying angles of incidence and lateral displacements. Since we use the mode matching method as the reference method, we first give a short introduction to this method. The Mode Matching Method In order to apply the mode matching method discrete mode spectra must be provided. Therefore, a closed structure with e.g. perfect electric conducting (PEC) walls, like in Fig. 1, has to be considered. Let the transverse components of the electric and magnetic fields in the discontinuity plane z = 0 be given by { E (s) t H (s) t } = ∑ k C (s) k { e (s) k h (s) k } , with s ∈ {i, r, t} . (1) The upper indices i, r, and t denote incoming, reflected, and transmitted waves, respectively. Due to the lossless materials the orthogonality relation 1 2 ∫ a ( e (s) k × { h m }∗) z da = Q (s) k δkm (2) Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 465