Calculation of forward-scattering amplitudes from a focused beam: exact values for the gaussian parameters

Abstract

The equation for radiative transfer is solved in the case of forward scattering of a laser beam that may be focused into a medium to attain its minimum in vacuo width beneath the surface. The moments of up to second order in the angular distribution of the radiance are calculated for an arbitrary point within the scattering medium and used to obtain a Gaussian fit to the radiance distribution at that point in order to reduce the mass of required computation to a small set of fundamental parameters essential for considerations of design and safety. The Gaussian parameters—irradiance, tilt, and angular breadths of the radiance distribution — are Fourier transformed back to configuration space, where they appear as one-dimensional Hankel transforms. For calculations of the off-beam scattering amplitudes in the focal plane of the unscattered beam, a \\ldwasp-waist\\rd approximation is introduced, in which the radial profile of the beam shrinks to a δ-function in the focal plane. A method for the improved convergence of the Hankel transforms used to evaluate the Gaussian parameters in this approximation is derived. It is then shown that these single-ray solutions, which are mainly independent of the details of the (narrow-angle) beam, can be used as Green's functions, enabling the exact scattering amplitudes to be recovered by convolving them with the normalized profile of the unscattered beam in any plane of interest. Thus even for broad beam profiles and points near the beam axis, the computation of the exact Gaussian parameters requires only a single further integration of the relevant Green's functions over everywhere positive, i.e., nonoscillatory, functions. The asymptotic bahavior of the Green's function for the irradiance is studied at small radial distances r, and an approximative formula for this function is derived.