by single-mode guides with tunable phase shifters. Lengths of the MMI regions and phases in the interconnect regions
are optimized. Sensitivity to length, phase shift, wavelength, and width of the optimized structures is also presented.

These optimized parameters are compared to those predicted using two different approximations. In the first approx-
imation, the modes are assumed to have zero intensity at the boundaries of the guide and the length of the MMI regions
is determined by the difference between the propagation two lowest order modes. As will be shown, in this first approx-
imation only the first order correction term in the Taylor series is retained in the expression for the propagation con-
stants. A second approximation will be shown, in which the length of the MMI section is determined by the difference
between the two lowest order propagation constants when calculated exactly. Switching properties of these devices
designed with both of these approximations will be compared to those designed using the optimization of the exact
modal analysis.

THEORY
Using the effective index method, a three dimensional rib waveguide device is modeled by a two dimensional infi-
nite slab structure. A typical NxN MMI structure with i = 1, 2, ..., N single-mode inputs, m = 0, 1, ... modes in the guid-
ing region, j = 1, 2, ..., N single-mode outputs, and a step index difference from effective index, n, in the guiding region
to no outside is shown in Fig. 1. The width of the MMI region is W, and the length is zo. The electric field in each 73
region is given by

 

 

 

 

 

 

 

 

 

 

Bei i{=1,2,..,.N Input Region
Eee m=0,1,... MMI Region
Eouj J=1L2,...N Output Region (a)
No
a
Ein, N N 1 Eout, 1
4
—’ eae
Eo En j Eu, j
——_—— Z —_
WwW
Ein, 1 1 N out, N
:
Input oS MMI dig Output
Region ¢ ue Region “a Region
0 Zo

Fic. 1. TWO DIMENSIONAL VIEW OF N X N GUIDING STRUCTURE SHOWING SINGLE-MODE INPUT AND OUTPUT GUIDE POSITIONS.

In the simplest approximation, the electric field modes in the MMI region are assumed to be zero at x = +W/2. A
correction to this approximation can be made by defining a mode-dependent effective width calculated using the Goos-
Hanschen shift, but for high-contrast guides the correction is quite small and often neglected. Using the actual width,
the dispersion relation may be written