Analytical expression for electro-field around rectangle waveguide?

[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? thanks a lot.

 

[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.

 

[Graeme Robinson]

The analytical expression for the electro-field around a rectangle waveguide can be obtained by solving the Maxwell's equations for the electric field in each region of the waveguide. The electric field in region I, where the waveguide is located, can be written as a combination of the x and y components, Ex and Ey, respectively. The electric field in the other regions, II, III, IV, and V, can be obtained using the boundary conditions at the interfaces between the regions.

To obtain the analytical expression for the electric field in the corner regions, where the mode is near cutoff, we can use the method of separation of variables. This involves separating the mode into uncoupled X(x) and Y(y) solutions and then combining them to get the total electric field in the corner regions. However, in this case, the x and y dependent solutions will be strongly coupled due to the boundary conditions in the corner regions.

In order to accurately obtain the electric field distribution in the corner regions, it is important to consider the full coupling between the x and y components of the electric field. Neglecting the field in the corner regions may lead to an inaccurate solution, especially when the mode is near cutoff.

One approach to obtaining the analytical expression for the electric field in the corner regions could be to use numerical methods such as finite element or finite difference methods. These methods can handle the strong coupling between the x and y components of the electric field and provide accurate results for the electric field distribution around the waveguide.

Another approach could be to use perturbation methods, where the solution is obtained by making small changes to a known solution. In this case, the known solution could be the electric field distribution in the other regions of the waveguide. However, this method may be more challenging as the perturbation equations can become complex when dealing with strong coupling between the x and y components of the electric field.

In conclusion, obtaining the analytical expression for the electric field distribution around a rectangle waveguide, particularly in the corner regions near cutoff, can be a challenging task. Careful consideration and accurate methods, such as numerical or perturbation methods, must be used to obtain an accurate solution.