Abstract
A set of functions, cn(x), is presented that minimize beam dispersion for a fixed-size exit aperture. The root-mean-square deviation of the set of functions and their Fourier transforms are calculated and compared with Gaussian Fourier transform pairs. It is shown for Cn(k), the Fourier transforms of the cn(x), that as n increases, the energy in the central lobe of Cn(k) increases, given that the cn(x) are defined such that each is smoother than the previous one. It is also shown that the cn(x) become narrower and the Cn(k) become broader as n increases. When the root mean square is used as the measure of the width of the functions, for n → ∞ the product of the variances of cn(x) and Cn(k) approaches 1/2. Additionally, it is shown that the cn(x) span the [−1, 1] Hilbert space.
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