Book contents
- Frontmatter
- Contents
- Preface
- 1 Critical effects in semiclassical scattering
- 2 Diffraction and Coronae
- 3 The rainbow
- 4 The glory
- 5 Mie solution and resonances
- 6 Complex angular momentum
- 7 Scattering by an impenetrable sphere
- 8 Diffraction as tunneling
- 9 The Debye expansion
- 10 Theory of the rainbow
- 11 Theory of the glory
- 12 Near-critical scattering
- 13 Average cross sections
- 14 Orbiting and resonances
- 15 Macroscopic applications
- 16 Applications to atomic, nuclear and particle physics
- References
- Index
5 - Mie solution and resonances
Published online by Cambridge University Press: 02 December 2009
- Frontmatter
- Contents
- Preface
- 1 Critical effects in semiclassical scattering
- 2 Diffraction and Coronae
- 3 The rainbow
- 4 The glory
- 5 Mie solution and resonances
- 6 Complex angular momentum
- 7 Scattering by an impenetrable sphere
- 8 Diffraction as tunneling
- 9 The Debye expansion
- 10 Theory of the rainbow
- 11 Theory of the glory
- 12 Near-critical scattering
- 13 Average cross sections
- 14 Orbiting and resonances
- 15 Macroscopic applications
- 16 Applications to atomic, nuclear and particle physics
- References
- Index
Summary
It is nice to know that the computer understands the problem, but I would like to understand it too.
(Attributed to E. P. Wigner)The Mie series solution for the scattering of an electromagnetic plane wave by a homogeneous sphere is introduced. Though ‘exact’, ‘a mathematical difficulty develops which quite generally is a drawback of this “method of series development”: for fairly large particles … the series converge so slowly that they become practically useless’ (Sommerfeld 1954). What is still worse, numerical studies reveal the occurrence in Mie cross sections of very rapid and complicated fluctuations, known as the ‘ripple’, that are extremely sensitive to small changes in the input parameters. They are related to orbiting and resonances, which may also be detected through their contribution to nonlinear optical effects, such as lasing, that are observed in liquid droplets.
The Mie solution
We consider a monochromatic linearly polarized plane electromagnetic wave with wave number k incident on a homogeneous sphere of radius a and a complex refractive index (relative to the surrounding medium) N = n + iκ, where n is the real refractive index and κ is the extinction coefficient (Jackson 1975). The time factor exp(–ωt) where ω = ck is the circular frequency, is omitted.
Taking the origin at the center of the sphere, the z axis along the direction of propagation of the incident wave and the x axis along its direction of polarization, the incident electric field is E0 = exp(ikz) ◯. In spherical coordinates, the components of the scattered electric field at large distances are of the form (Bohren & Huffman 1983)
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- Diffraction Effects in Semiclassical Scattering , pp. 37 - 44Publisher: Cambridge University PressPrint publication year: 1992
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