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Spherical Vector Wave Functions and Surface Green's Functions

1. The problem statement, all variables and given/known data

I'm currently doing research to try and find a way of calculating simultaneously both the internal and external fields of elliptic shaped dielectric and metallic nanoparticles with a depression on one of its sides so as it would look like a bean. We use spherical vector wave functions to expand the propagating wave this much I know however I dont actually know how to explain what spherical vector wave functions are so any simplified explanation would be helpful. Also we are going to use a surface Green's function to give an exact relation between the internal and scattered fields and the incident fields however again I've covered calculating Green's functions in other classes but not sure what a Layman's explanation of them would be so any information you could help me with would be very much appreciated.
Thanks :)

2. Relevant equations

3. The attempt at a solution

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