High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission

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Abstract
This is Paper I of a series on high-frequency scattering of a scalar plane wave by a transparent sphere (square potential well or barrier). It is assumed that (ka)⅓≫1,|N−1|½(ka)⅓≫1, where k is the wave-number, a is the radius of the sphere, and N is the refractive index. By applying the modified Watson transformation, previously employed for an impenetrable sphere, the asymptotic behavior of the exact scattering amplitude in any direction is obtained, including several angular regions not treated before. The distribution of Regge poles is determined and their physical interpretation is given. The results are helpful in explaining the reason for the difference in the analytic properties of scattering amplitudes for cutoff potentials and potentials with tails. Following Debye, the scattering amplitude is expanded in a series, corresponding to a description in terms of multiple internal reflections. In Paper I, the first term of the Debye expansion, associated with direct reflection from the surface, and the second term, associated with direct transmission (without any internal reflection), are treated, both for N > 1 and for N < 1. The asymptotic expansions are carried out up to (not including) correction terms of order (ka)−2. For N > 1, the behavior of the first term is similar to that found for an impenetrable sphere, with a forward diffraction peak, a lit (geometrical reflection) region, and a transition region where the amplitude is reduced to generalized Fock functions. For N < 1, there is an additional shadow boundary, associated with total reflection, and a new type of surface waves is found. They are related to Schmidt head waves, but their sense of propagation disagrees with the geometrical theory of diffraction. The physical interpretation of this result is given. The second term of the Debye expansion again gives rise to a lit region, a shadow region, and a Fock-type transition region, both for N > 1 and for N < 1. In the former case, surface waves make shortcuts across the sphere, by critical refraction. In the latter one, they excite new surface waves by internal diffraction.