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

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.

Near/far-side angular decompositions of Legendre polynomials using the amplitude-phase method

“…The amplitude-phase representation of the Legendre polynomial turns out to be different from the typically used propagating angular waves as defined by Nussenzveig and Fuller [17][18][19][20]. The differences are however not manifested, as shown in this study, under semiclassical conditions where scattering involves significant contributions from large angular momentum quantum numbers.…”

Debye series for light scattering by a nonspherical particle

“…(9) differ by a minus sign to what is standardly written down in Mie theory for A mn and B mn [40,41]. This new convention will have all of our final Debye series expansion of A mn and B mn differ from what is in the literature [15,16] by a minus sign.…”

“…The Debye series was first proposed by Debye in his studies on light scattering by a circular cylinder with normal plane-wave incidence [13]. The Debye series for the scattering amplitudes for plane-wave incidence on a sphere was first worked out by van der Pol and Bremmer [14] and was greatly popularized by Nussenzveig [15]. As a complement, the Debye series for the interior amplitudes for a sphere was derived in Ref.…”

Scattering of a tightly focused beam by an optically trapped particle

“…24,25 An incoming partial wave l is in part diffracted by the sphere (½), it is in part externally reflected from the sphere surface ͑ϪR l external ͞2͒, and it is in part transmitted through the sphere…”