Exact decomposition of a Gaussian-averaged nonlinear function.

In nonlinear optics we are frequently measuring the functional dependence of a physical parameter on light intensity. This immediately presents a problem, since by their nature real optical beams have a varying transverse intensity profile. Any nonlinear measurement will inherently be different at the center of the beam than it is at the edges. A spatially integrated whole-beam measurement must be decomposed to yield the true functional form of the intensity dependence. One approach is to create a uniform rectangular transverse intensity profile by passing a Gaussian beam through a narrow spatial aperture. This never works well, since the center of a Gaussian is not perfectly uniform and the sharp edges of the aperture produce near-field diffraction rings. Another approach is to assume some nonlinear functional form and to fit it to the spatially integrated experimental data. This reverse procedure does not necessarily result in a unique answer. In this Letter we present an exact decomposition equation for Gaussian beams that retrieves the true intensity dependence of a measured parameter. Let the Gaussian beam have the transverse intensity profile