Hi,

I'm trying to understand the basics of waveguide propagation. As far as
I understood so far the EM fields in waveguides (constant properties in
propagation direction) cannot in general, as in the case of metallic
waveguides, be seperated in transversal electric and transversal
magnetic fields. So if I would have solved the wave equations for the Ex
and Ey components by use of BPM the Ez component would follow from:

dEz 1 d n^2 Ex d n^2 Ey
--- = - --- ( -------- + -------- )
dz n^2 dx dy

This is an initial value problem (what initial value do I have to choose?)
Knowing all the electric field components the Hx, Hy and Hz components
follow from:

rot E = -i w mu0 H


Similar procedures hold in the case that I would first have solved the
wave equations for Hx and Hy respectively.

An alternative procedure would be to solve the wave equations for Ex, Ey
and Hx, Hy simultaneously. In that case I would have to know the initial
values not only of Ex, Ey but also the approriate initial values of Hx,
Hy fitting the electric field components.
(What alternative is the better one?)

If the transverse components are uncoupled I can employ the higher order
Pade approximant BPM schemes for solving the wave equations while in the
case that the transverse components are coupled there are only numerical
schemes for solving the usual parabolic BPM.

That's my understanding of waveguide modeling so far. If I'm wrong in
any point I would be very appreciative for corrections.

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.

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).

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

Rolf Wester wrote:

Hi,

I'm trying to understand the basics of waveguide propagation. As far
as I understood so far the EM fields in waveguides (constant
properties in propagation direction) cannot in general, as in the case
of metallic waveguides, be seperated in transversal electric and
transversal magnetic fields. So if I would have solved the wave
equations for the Ex and Ey components by use of BPM the Ez component
would follow from:

dEz 1 d n^2 Ex d n^2 Ey
--- = - --- ( -------- + -------- )
dz n^2 dx dy

This is an initial value problem (what initial value do I have to
choose?)
Knowing all the electric field components the Hx, Hy and Hz components
follow from:

rot E = -i w mu0 H


Similar procedures hold in the case that I would first have solved the
wave equations for Hx and Hy respectively.

An alternative procedure would be to solve the wave equations for Ex,
Ey and Hx, Hy simultaneously. In that case I would have to know the
initial values not only of Ex, Ey but also the approriate initial
values of Hx, Hy fitting the electric field components.
(What alternative is the better one?)

If the transverse components are uncoupled I can employ the higher
order Pade approximant BPM schemes for solving the wave equations
while in the case that the transverse components are coupled there are
only numerical schemes for solving the usual parabolic BPM.

That's my understanding of waveguide modeling so far. If I'm wrong in
any point I would be very appreciative for corrections.


It depends on the symmetry of the waveguide. You're assuming
translational invariance down the axis, so the axial dependence will
always separate out. If it's radially symmetric, you can separate the
Laplacian in cylindrical coordinates as usual.

When the guide is rectangular, the Laplacian approximately
separates--doing this gets the fields wrong in the wedges extending off
each corner, but the fields are generally small there. This is where
the metal guide case is much easier.

Asymmetric guides, e.g. ridge guides, don't allow separation of
variables in the transverse direction.

The other thing that may cause you some grief is the continuity
conditions. At a material interface, tangential E and perpendicular D
are continouous. That means that there's an abrupt jump in tangential D
and perpendicular E, which makes smooth approximations very error-prone,
especially as the index contrast increases.

Cheers,

Phil Hobbs

Thank you very much for your helpfull reply. I still have a question

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.

I'm going to bye 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