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Re: Green's Function for Fresnel Reflection?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg101115] Re: Green's Function for Fresnel Reflection?
*From*: alexhavy <alexhavy at comcast.net>
*Date*: Wed, 24 Jun 2009 06:34:10 -0400 (EDT)
*References*: <h1en3o$i4i$1@smc.vnet.net>
On Jun 18, 8:45 pm, AES <siegman at stanford.edu> wrote:
> Can anyone point to a two-dimensional Green's Function for the classical
> Fresnel reflection problem of a linearly polarized infinite plane wave
> striking the planar interface between two unbounded half-spaces with
> different refractive indices?
>
> [This is admittedly a math-and-physics rather than Mathematica question;
> but there's an incredible level of mathematical smarts on this group,
> and I'd like to implement the answer in some extended Mathematica
> calculations.]
>
> Specific case of interest can be described as a y-polarized E field with
> k vector in the x,z plane; interface in the x,y (or y,z) plane; with
> wave going from higher to lower index half space so that TIR occurs
> above a critical angle, as treated using plane-wave analysis in every
> elementary optics or e-m theory text.
>
> A closed-form or special-function Green's function solution for a point
> source instead of an infinite plane wave would be extremely useful; and
> digging into Helmholtz/Weyl decomposition to derive something like this
> looks like very heavy going.
>
> Extension to point source located in higher-index planar slab between
> two lower-index half spaces (aka a planar waveguide) would be truly
> great.
I do not think the closed-form formula for the Green's tensor exist in
half-plane geometry, except for long range asymptotics. The integral
expression via plane wave expansion can be found in Maradudin and
Mills, PRB 11(4) 1392 (1975). Very recently G. Panasyuk et al obtained
a short distance expansion for G(r,r'), works like a charm: J. Phys.
A: Math. Theor. 42 No 27 (10 July 2009) 275203. This is suitable for a
point source within a surface proximity. The Fortran code can be found
at http://whale.seas.upenn.edu/CODES
This code can also find G numerically via direct integration for,
virtually, any distances. Hope this will be helpful.
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