Gaussian scaling laws for diffraction: top-hat irradiance and Gaussian beam propagation through a paraxial optical train

摘要

Abstract: 1415@36@A one-to-one correspondence between the parameter spaces of propagated top-hat (plane wave, flat top) irradiance profiles and Gaussian beams has been proposed. The diffractive 'size' of the propagated top hat is estimated by calculating that of an equivalent Gaussian beam with the same Fresnel number. Given the propagation Fresnel number, one also knows the detailed intensity and phase profiles of the propagated top hat within the Gaussian envelope; a library of plots of beam profiles is provided for both rectangular and circular top hats propagated over a large spectrum of Fresnel numbers. For Fresnel numbers less than approximately 10, the Gaussian envelope is shown to enclose roughly 90 percent of the top hat's total power. The formalism thus allows one to perform the simple matrix manipulations of Gaussian beam propagation to determine the propagated top-hat beam envelope and then use the look-up tables of beam profiles to determine the detailed intensity and phase of the plane as it propagates through a paraxial optical train. The equivalent Gaussian method allows one to include lowest order diffraction effects when designing an optical system instead of relying solely on geometrical optics. For beams that are not significantly different from top hats, one can approximate their propagated profiles by using this method with a modified wavelength, lengthened to account for non-ideal beam spreading. Examples include propagation through a focus and a one-to-one imaging system, both encountered in ring resonator designs, and design and implementation of a multiwavelength imaging laser diagnostic optical train. A final example is extension to non-orthogonal optical systems using the analysis of J. A. Arnaud to propagate the Gaussian beam.!