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How does a transparent sphere's shadow fade as $n\to 1$?

Consider radiation of wavenumber $k$ incident on a transparent sphere of radius $a$ and index of refraction $n$ in the limit $ka \gg 1$. (Equivalently, consider a quantum-mechanical particle of momentum $k$ incident on a spherical square-well potential taking value $V_0 > 0$ inside radius $a$ and vanishing outside.) It is known that when $ka (n-1)^{3/2} \gg 1$, there is a diffractive peak in the far forward direction which destructively interferes with the unmolested ($n=1$ wave), effectively producing the sphere's shadow:

$$S_0 (\theta) \approx ka \sqrt{\frac{\theta}{\sin \theta}} \frac{J_1 (ka \theta)}{\theta}, \quad \theta \ll (ka/2)^{1/3}$$

Here $S_0$ is the S-function, i.e., the scattering amplitude, and $J$ is the Bessel function. See Eq.(9.16) / p.96 of "Diffraction Effects in Semiclassical Scattering" by Nussenzveig (2006), or Eq.(5.115) / p.183 of "Scattering of Waves from Large Spheres" by Grandy (2005). The restriction $ka (n-1)^{3/2} \gg 1$ is stated explicitly at the beginning of Nussenzveig, J. Math. Phys. 10, 82 (1969).

However, this formulas clearly breaks down for $ka (n-1)^{3/2} \lesssim 1$ since it remains finite as the sphere vanishes, $n \to 1$.

Where is this transition discussed?

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  • $\begingroup$ Note that this question is a more specific version of this vague one whose answer cannot be used. $\endgroup$ – Jess Riedel May 23 '16 at 16:35
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    $\begingroup$ Hi Jess. I'm afraid I haven't looked into this before and would probably simply begin with Mie scattering equations and try to get them to tell me something useful from first principles. Did you know there has been quite a bit of work done on this kind of thing for studying the behavior of photonic crystals in the last 15 years. $\endgroup$ – WetSavannaAnimal May 25 '16 at 1:16

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