  The Bogoliubov dispersion relation for the elementary excitations of the
weakly-interacting Bose gas is shown to hold for the case of the
weakly-interacting photon gas (the ``photon fluid'') in a nonlinear Fabry-Perot
cavity. This relation, which is also derived by means of a linearized
fluctuation analysis in classical nonlinear optics, implies the possibility of
a new superfluid state of light. The theory underlying an experiment in
progress to measure this dispersion relation is described, and another
experiment to test the prediction of the superfluidity of this state of light
is proposed.
