Alternative to Mie Theory for Analysis of Homogenous Sphere

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Mie theory is typically applied to analyze the interaction of light with a homogeneous sphere in a uniform dielectric medium. When considering a sphere surrounded by two different media, such as one half in a substrate and the other half in air, the complexity increases. An analytic solution may not be straightforward, but additional boundary conditions could be introduced to adapt the existing equations. Alternatively, deriving solutions from Maxwell's equations could provide the necessary framework for this scenario. Addressing these complexities is essential for accurate modeling of light absorption and scattering in such configurations.
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I know that mie theory is used to analyze the absorption/scattering/extinction of a homogeneous sphere within a homogeneous dielectric medium. However, if I wanted to perform the same analysis on a sphere enclosed by two different media, is there an analytic solution to this?
 
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How by two different media? Do the media form shells? Or the lower half in one medium, the upper half in another?
 
One half of the sphere is embedded in a substrate, and the upper half is in air/vaccum.
 
maybe if you add additional boundary conditions to the equations in 3.5.2 (link below) you can get a solution suited to your needs... otherwise derive from maxwell...

http://evan.ucsd.edu/class/sioc251/Lecture_Notes_files/mie_detailed.pdf
 
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