# Analytical expression for electro-field around rectangle waveguide?

1. Sep 23, 2004

### Cao Yu

i am now trying to figure out the electro-field around a rectangle waveguide. the refractive index of the waveguide is n1, which is larger than that of the cladding, namely n2, outside. we may regard the waveguide embeded in the cladding material.

as following shown, the waveguide crossection is the region I. It is surrounded by region II, III, IV, V and four corner regions.

i i
i III i
i i
---------------------------------------------------
i i
V i I i IV
i i
---------------------------------------------------
i i
i II i
i i

it is easy to get the electrical field expression in II, III, IV and V, by using EIM or purtabation method. Now my question is how to get the analytical expression in the four corner region, when the mode in the waveguide is near cutoff?

when the mode is far from cutoff, we will separate the mode into uncoupled X(x) and Y(y) solutions. in this way, we can tract the results for Ex and Ey separately. but if the mode is near cutoff, the x and y dependent solutions will be strongly coupled through the boundary conditions in the corner regions.

people normally neglect the field in the corner regions as their approximation. but the case i am studying now can not take this approximation. so i need to know all the electrical field distributuion around the waveguide.

anyboday has good idea? thx a lot.

2. Sep 23, 2004

### Cao Yu

oh, the picture gets into a mass. hope someboday can understand it.

III and II are the regions just above and below the waveguide core, which is noted as I.

IV and V are the regions on the right and left of the core I.