Approximate scaling relationships of light diffraction by apertures with complicated shapes.

Numerical and analytical simulations of Fraunhofer diffraction by oddly shaped apertures and exponential light amplitudes show that within an accuracy of 20-40%, the dependencies of the normalized encircled energy epsilon on ?, the fraction of the energy transmitted by the aperture within a given polar angle ? are similar in shape when ? is normalized by lambda/rho. Here rho is the size parameter of the aperture geometry (its shape and size) and of the light-amplitude profile. For a small ?, rho = rho(l,eff) = (Sigma(eff)/pi)((1/2)) depends on the effective aperture area Sigma(eff), which is calculated through the axial light intensity. For a large ?, rho = rho(2,eff) = 2Sigma(eff)/Rho(eff) depends on Sigma(eff) and the corresponding perimeter Rho(eff). In the case of a uniform light-amplitude distribution the normalization of rho(1) and rho(2) corresponds to the well-known expansion of epsilon(?) for small and large ?.