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 dealingwith 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. The following original results are presented in the context of both of these models:• an analytic expression for the low-frequency temperature is found for a hyperbolic tangent background profile, valid in the entire parameter space; it is well-known that the spectrum is approximately thermal at low frequencies, but a universally valid expression for the corresponding temperature is an original development;• an analytic expression for the spectrum, valid over almost the entire frequencyrange, when the velocity profile parameters lie in the regime where the low-frequencytemperature is given by the Hawking prediction; previous work has focused on thelow-frequency thermal spectrum and the characterization of the deviations fromthermality, rather than a single analytic expression; and• a new unexplored regime where no group-velocity horizon exists is examined; theHawking spectra are found to be non-zero here, but also highly non-thermal, andare found, in the limit of small deviations, to vary with the square of the maximumdeviation; the analytic expression for the case with a horizon is found to carry overto this new regime, with appropriate modifications.Furthermore, the thesis examines the results of a classical frequency-shifting experimentin the context of fibre-optical horizons. The theory of this process is presented for both a constant-velocity and a constantly-decelerating pulse, the latter case taking account of the Raman effect. The resulting spectra are at least qualititively explained, but there is a discrepancy between theory and experiment that has not yet been accounted for.
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