Re: Modeling of fields in waveguides

Rolf Wester wrote:
> Thank you very much for your helpfull reply. I still have a question 
> concerning the filed components. If I have axial and transversal 
> symmetry I get uncoupled wave equations for the field components. 

Axial and cylindrical symmetry, or axial symmetry and a stack of planar 
layers (e.g. a slab guide) will lead to a Laplacian that separates. 
Rectangular symmetry, e.g. a rectangular guide, doesn't let you separate 
the Laplacian outside the rectangle.  It can't, partly on account of the 
jumps in the perpendicular E and tangential D--the jumps occur only at 
the rectangle's edges, so the field
outside the rectangle can't be the product of two one-variable functions 
in x and y.

> If I 
> solve the wave equation for the Ex component how do I get the other 
> field components Ey, Ez, Hx, Hy and Hz? I'm not sure whether the 
> procedure I outlined above is correct. Are there any field components 
> that are 0 everywhere (transversal electric -> Ez = 0)? My problem is 
> how can I be sure to get all the nonvanishing field components correctly 
> when only solving the wave equation for a single component (or may by Ex 
> and Ey).

Dielectric guides are intrinsically more complicated than metal guides. 
 For example, a circular metal cylinder has two families of modes, TE 
and TM, and the axial component of the (non-transverse) field can be 
used as a potential function to derive all the other components.  In 
step-index optical fibres (which are about the simplest dielectric 
guides going), there are not two but four families of modes (EH, HE, TE, 
and TM).  Things get uglier in a hurry for more complicated guides.

The usual method is to use a mode solver to get the eigenmodes of the 
waveguide, and then construct a suitable analytic approximation after 
having a look at the fields, if you can find one.  Plugging in a 
functional form *a priori* is liable to lead to poor results, unless 
you're sure that it approximates the true fields well.  Sines and 
cosines are non-starters.  There's a very good book, "Introduction to 
Optical Waveguide Analysis: Solving Maxwell's Equation and the 
Schrdinger Equation" by Kenji Kawano & Tsutomo Kitoh, that I'm working 
through slowly myself.  Professor Siegman put me on to it.

> Regarding your last paragraph what numerical scheme would you recommend, 
> FE, FD, ..?

It depends.  Mode solvers aren't hard to write, but I haven't done one. 
My simulation expertise mainly centres round optimizing FDTD simulators 
for design synthesis, so I don't have much advice to give here.  FDTD is 
great for relatively small things (roughly, things that will fit inside 
a 100-wavelength cube), is easy to code, runs pretty fast for a 3-D full 
EM solver, and so on, but uses a boatload of memory and is much slower 
than BPM.

Cheers,

Phil Hobbs