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A brief introduction to GLMT

Lorenz-Mie Theory provides a rigorous and semi-analytical solution to the scattering of a monochromatic plane wave by a sphere. The sphere is assumed to be homonenous and isotropic and its interaction with light is elastic (no light frequency modification) and linear (optical properties do not depend on the lightning power). It is noteworthy that this theory has been developped independently in the framework of ether theory by Ludvig Lorenz [1], [2] and in the maxwell electromagnetic framework by Gustav Mie [3] and Peter Debye [4]. This theory is known as ’Mie Theory’ , ’Lorenz-Mie Theory’ or ’Lorenz-Mie-Debye Theory’ depending on whether one is German, French or careful to not offend anyone. For my part, I conform to the Rouen school where i come from. Beyond the names, two different approaches can be distinguished. The first one, the most popular in the anglosphere is based on Vectorial Potential (Herz-Debye potential). The second, is based on scalar potentials as introduced by Bromwich [5] and further developped by Borgnis [6]. In the common case of a sphere, both methods are equivalent [7]. Following the french tradition i use the Bromwich potential approach.

From G. MIe Paper (1908)

A family of methods has been developped starting from LMT. These methods known as Separable Variable Methods (SVM) provide a rigorous description of light scattering by objects of various shape, but all regular, as cylinder or ellipsoid for exemple. Some of these methods has been extended to a non-uniform ligthning (focus beams). They are called Generalized Lorenz Mie Theory or Theories (GLMT). Obviously, light scattering by particle can be described using other methods, approximation or model, analytical or numerical, but SVM (and GLMT) have several advantage.
At first, they provide an exacte solution to Maxwell equations for light scattering by linear isotropic and homogenous particles. In particular, there’s no assumption on particle size or refractive index and the scattering particle is not described through a discret mesh. Being semi-analytical, SVM allow fast computation of integral quantities, such as scattering cross section or albedo. Also, light scattering in a small part of the space can be considered alone without computing ligth propagation in the whole volume in and around the scattering object.
However, SVM show several limitations. At first, Semi-analytical means that Special functions (Bessel finction, Lengendre polynomials, ...) involving in LMT (and other SVM) have to be computed numerically. As a consequence some limitations arise in practice on particle size, especially for non-spherical particles. Beyond these numerical constraints, common to all approaches, the main limitations are probably the regular shape imposed to the particle, and the inability of these methods to consider non linear and non elastic interaction such as fluorescence, raman scattering, Kerr effect, which is increasingly restricting with the development of powerfull laser sources.

[1L. Lorenz. Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge. Vidensk. Selk.Skr., 6 :1–62, 1890.

[2L. Lorenz. Sur la lumière réfléchie et réfractée par une sphère transparente. 1898. Lib.Lehmann et Stage, œuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner.

[3G. Mie. Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen. Ann. der Phys., 25 :377–452, 1908.

[4P. Debye. Der Lichtdruck auf Kugeln von beliebigem Material. Ann. der Phys., 4(30) :57–136, 1909.

[5T.J. Bromwich. Electromagnetic waves. Phil. Mag., 38 :143–164, 1919.

[6F.E. Borgnis. Elektromagnetische Eigenschwingungen dielektrischer Raüme. Ann. der Phys., 35 :359–384, 1939.

[7J.A. Lock. Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle. J. Opt. Soc. Amer. A, 10(4) :693–706, 1993.