  Motivated by the physics literature on "photonic doping" of scatterers made
from "epsilon-near-zero" (ENZ) matrials, we consider how the scattering of
time-harmonic TM electromagnetic waves by a cylindrical ENZ region $\Omega
\times \mathbb{R}$ is affected by the presence of a "dopant" $D \subset \Omega$
in which the dielectric permittivity is not near zero. Mathematically, this
reduces to analysis of a 2D Helmholtz equation $\mathrm{div}\, (a(x)\nabla u) +
k^2 u = f$ with a piecewise-constant, complex valued coefficient $a$ that is
nearly infinite (say $a = \frac{1}{\delta}$ with $\delta \approx 0$) in $\Omega
\setminus \bar{D}.$ We show (under suitable hypotheses) that the solution $u$
depends analytically on $\delta$ near $0$, and we give a simple PDE
characterization of the terms in its Taylor expansion. For the application to
photonic doping, it is the leading-order corrections in $\delta$ that are most
interesting: they explain why photonic doping is only mildly affected by the
presence of losses, and why it is seen even at frequencies where dielectric
permittivity is merely small. Equally important: our results include a PDE
characterization of the leading-order electric field in the ENZ region as
$\delta \to 0$, whereas the existing literature on photonic doping provides
only the leading-order magnetic field.
