  We present measurements of the complete spatio-temporal Fourier spectrum of
Faraday waves. The Faraday waves are generated at the interface of two
immiscible index matched liquids of different density. By use of a new
absorption technique we are able to determine the bifurcation scenario from the
flat surface to the patterned state for each complex spatial and temporal
Fourier component separately. The measured surface spectra at onset are in good
agreement with the predictions from a linear stability analysis. For the
nonlinear state our measurements show in a direct manner how energy is
transferred from lower to higher harmonics and we quantify the nonlinear
coupling coefficients. Furthermore we find that the nonlinear coupling
generates constant components in the spatial Fourier spectrum leading to a
contribution of a non oscillating permanent sinusoidal deformed surface state.
A comparison of a hexagonal and a rectangular pattern reveals that spatial
resonances only can give rise to a spectrum that violates the temporal
resonance conditions given by the weakly nonlinear theory.
