  Hawking radiation, despite its presence in theoretical physics for over
thirty years, remains elusive and undetected. It also suffers, in its original
context of gravitational black holes, from conceptual difficulties. Of
particular note is the trans-Planckian problem, which is concerned with the
apparent origin of the radiation in absurdly high frequencies. In order to gain
better theoretical understanding and, it is hoped, experimental verification of
Hawking radiation, much study is being devoted to systems which model the
spacetime geometry of black holes, and which, by analogy, are also thought to
emit Hawking radiation. These analogue systems typically exhibit dispersion,
which regularizes the wave behaviour at the horizon but does not lend itself
well to analytic treatment, thus rendering Hawking's prediction less secure. A
general analytic method for dealing with Hawking radiation in dispersive
systems has proved difficult to find.
  This thesis presents new numerical and analytic results for Hawking emission
spectra in dispersive systems. It examines two black-hole analogue systems: it
begins by introducing the well-known acoustic model, presenting some original
results in that context; then, through analogy with the acoustic model, goes on
to develop the lesser-known fibre-optical model.
