A spherical water droplet of refractive index scatters light of vacuum wavelength (a…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Wave propagation and light in media

Conditions of Validity, Ch Integral equations, kernels and the Fredholm alternative for the last time

Problem

A spherical water droplet of refractive index

n=1.33

scatters light of vacuum wavelength

\lambda=632.8\,\mathrm{nm}

(a He-Ne laser), modelled as a scalar wave with potential strength

U=k^2(n^2-1)

k=2\pi/\lambda

. For a uniform sphere of radius

a

the symmetrised Hilbert-Schmidt norm of the Born kernel is

\|K\|_{\rm HS}=|U|a^2/2

, and the Born series is guaranteed to converge when

\|K\|_{\rm HS}<1

. ((a)) Compute

U

\mathrm{m^{-2}}

. ((b)) Find the largest radius

a

for which convergence of the Born series is guaranteed. ((c)) For a droplet of radius

a=100\,\mathrm{nm}

, compute

\|K\|_{\rm HS}

and state whether convergence is guaranteed.