The Hankel transform in n-dimensions and its applications in optical propagation and imaging

Wave propagation is considered in multidimensional reciprocal space. For the first Rayleigh-Sommerfeld diffraction integral, the propagating field can be represented by homogeneous and inhomogeneous components. These add up to give a propagating component on a hemispherical surface in reciprocal space, and an evanescent component that lies totally outside the corresponding sphere. If evanescent waves can be neglected, the 3D angular spectrum method, entailing inverse Fourier transformation of the weighted hemisphere, can be used to calculate efficiently the propagated field. This basic concept is applied in spaces of different dimensionality. For functions displaying hyperspherical symmetry in nD space, the corresponding Hankel transformation leads to Hankel-transform pairs. Tables of functions relevant in wave propagation, diffraction, and information optics are presented. The two-dimensional (2D) case is particularly important as it can be applied to propagation in planar wave guides, surface plasmonics, and cross sections of propagationally invariant fields, as well as to fringe analysis and image processing in two dimensions.