On Fri, 16 Jun 2006, Cheng Cosine wrote:

Given a Helmholtz eqn with constant coefficients, assuming solution in
negative z-dir is neglegible, i.e., no reflecting wave in negative z-dir,
one can then use 2d FFT in xy-plane to solve the eqn. But when applying
the same argument to x- and y-dir, one can argue that wave still
propogate
in positive and negative x- and y-dir since no simplifications are made
in
those directions. That is, there are totally 5 propagation directions:
positive z-dir and neg/pos directions of x- and y-dirs. Then how can one
say that: it is a "plane wave in positive z-direction"?

Use separation of variables in Cartesian coordinates to find a general
solution to the Helmholtz equation. You end up with 2 arbitrary parameters
k_x and k_y which can vary from -infinity to +infinity for every
individual solution. Looking at the solutions, you can see that these
parameters are the x and y components of the wavevector.

Since you know the wavenumber, you can find k_z from

k^2 = k_x^2 + k_y^2 + k_z^2.

This gives you 2 distinct families of solutions: those where you take the
positive square root, and those where you take the negative square root.
These are plane waves propagating (at least partly) in the +z and -z
directions (except for k_z=0). Only when k_x=k_y=0 do the plane waves
propagate purely in the +z or -z directions.

The general solution is then an integral over some amplitude density
multiplied by these solutions, integrated over all k_x, k_y.

If for some given z=A, if all sources have z<A, then you only have
non-zero contributions from modes with k_z>0 for z>A.

(That's assuming exp(ik.r-iwt) convention.)

Does this mean that one can neglect negative z-direction and then use
FFT
to march along positive z-direction to solve ANY given wave sources
without
errors?

No. A technical "no" because FFT isn't exact - you need the continuous
2D Fourier transformation in general.

A more general "no" because it isn't enough to know the sources - you need
to know the fields over an entire z=constant plane such that all of the
sources are on the -ve side of the plane. For the very restricted case of
a plane wave with wavevector in the +z direction incident on a screen with
some apertures, it's a very useful method. If you have a collection of
sources spread out through a 3D volume, you'll first need to find the
fields in a plane. In your post on sci.physics.electromag, you ask about
point sources. For point sources you can do this easily enough by using
the appropriate Green function. But then you don't need to bother with
Fourier transforms at all, since you can find the fields anywhere.