The following calculation method permits the evaluation of the
transverse distribution of the field amplitude Ψ(x,y,z) propagating in a
waveguide. This method is based on a standard method for propagation in
free space, which is using the Fast Fourier Transform (FFT). It is very
simple in practice, and it does not require a mode decomposition for the
waveguide propagation. It uses the properties of periodicity and symmetry of
the FFT in the transverse plane in order to take into account the influence of
the waveguide. The advantage of this calculation method is the full
compatibility with the case of a guided wave (waveguide propagation), an
unguided wave (free space propagation) and any intermediate regime where
the guiding occurs softly or only in a limited part of the waveguide; for
example when an input wave is not guided at the entrance of the
waveguide, and becomes guided at the end. In practice, these conditions
are observed experimentally for example in the infrared “Free Electron
Lasers”, which are broadband tunable lasers, and some of them are using a
waveguide, in the center of the optical cavity, coupled with free space areas.
The calculation method presented here works in this special configuration.
On the other hand, although not using a mode decomposition, this method
allows also a rapid decomposition in eigenmodes TEpq and TMpq. Using only
one symmetry operator and one FFT, it gives the whole set of mode
amplitudes.