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“Rainbows” in Homogeneous and Radially Inhomogeneous Spheres: Connections with Ray, Wave, and Potential Scattering Theory

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Advances in Interdisciplinary Mathematical Research

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 37))

Abstract

This chapter represents an attempt to summarize some of the direct and indirect connections that exist between ray theory, wave theory, and potential scattering theory. Such connections have been noted in the past and have been exploited to some degree, but in the opinion of this author, there is much more yet to be pursued in this regard. This article provides the framework for more detailed analysis in the future. In order to gain a better appreciation for a topic, it is frequently of value to examine it from as many complementary levels of description as possible, and that is the objective here. Drawing in part on the work of Nussenzveig, Lock, Debye, and others, the mathematical nature of the rainbow is discussed from several perspectives. The primary bow is the lowest-order bow that can occur by scattering from a spherical drop with constant refractive index n, but zero-order (or direct transmission) bows can exist when the sphere is radially inhomogeneous. The refractive index profile automatically defines a scattering potential but with a significant difference compared to the standard quantum mechanical form: the potential is k-dependent. A consequence of this is that there are no bound states for this system. The correspondences between the resonant modes in scattering by a potential of the “well-barrier” type and the behavior of electromagnetic “rays” in a transparent (or dielectric) sphere are discussed. The poles and saddle points of the associated scattering matrix have quite profound connections to electromagnetic tunneling, resonances, and “rainbows” arising within and from the sphere. The links between the various mathematical and physical viewpoints are most easily appreciated in the case of constant n, thus providing insight into possible extensions to these descriptions for bows of arbitrary order in radially inhomogeneous spheres (and cylinders).

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Acknowledgements

I have been heavily influenced by the work of Professors H. M. Nussenzveig and J. A. Lock in the preparation of this chapter. I would particularly like to thank Professor Lock for his generous advice, detailed and constructive suggestions on this material (also pointing out an error in Appendix 5), and permission to use the quotation from his paper [3]. The comments of an anonymous reviewer also contributed significantly to the improvement of this chapter and are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Old Dominion University, Norfolk, VA, 23508, USA

    John A. Adam

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  1. John A. Adam
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Correspondence to John A. Adam .

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  1. , Dept of Mathematics & Computer Sciences, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Bourama Toni

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Adam, J.A. (2013). “Rainbows” in Homogeneous and Radially Inhomogeneous Spheres: Connections with Ray, Wave, and Potential Scattering Theory. In: Toni, B. (eds) Advances in Interdisciplinary Mathematical Research. Springer Proceedings in Mathematics & Statistics, vol 37. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6345-0_3

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