The features of geometrical phase of modulated wave
propagated in plane waveguide
Yu.N. Zayko
Volga Region Academy of State Service, Saratov, Russia
The results of this paper are concerned with the revealing
of geometrical phase for modulated waves, especially
pulses, propagated in dispersive media, for instance in plane
metallic waveguide. It is shown here that distortions of a
signal 
(z -
coordinate, t - time, 
- carrying frequency) according to
dispersion may be represented as a rotation of a phase vector 
=
(a,b) in plane space which is leaning against basic vectors
and 
. This rotation leads to
transformation of vector with the help of definite matrix W,
which is an analog of wellknown Jones matrix in polarisation
optics [1]. Matrix W depends on dispersive character of
medium. This geometric picture can be supplied with the mapping of
phase space, mentioned above to a hemisphere, which radius is
equal to maximum of amplitude of a signal.
Two parameters of a
mapping are the polar angles of vector 
. The main result
consists of that closed paths on this hemisphere drawn up by the
end of which are embracing the polar axis are absent. There
exist only such a paths along which change of geometrical phase
is nonmonothonic. These oscillations of
geometrical phase or momentary frequency were found in [2].
The specific feature of this effect is that the geometric phase
oscillates with the period of carrying frequency. It's whole
deviation during the period is equal to zero what is in
agreement with the zero spatial angle which is drawn up by the
end of wave vector 
on unit sphere in space of wavevectors.
Notice, that wave vector is that of one of
two plane waves constructing the ordinary 
wave in waveguide:
their directions are at the angle 
to the longitudinal axis of waveguide,
is a cutoff frequency [3].