Complex analytic dependence on the dielectric permittivity in ENZ materials: The photonic doping example

Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon-near-zero” (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region $\Omega \times R$ 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 $\mathrm{div}(a\left(x\right)\nabla u)+k^{2}u=f$ with a piecewise-constant, complex valued coefficient a that is nearly infinite (say $a=\frac{1}{\delta }$ with $\delta \approx 0$) in $\Omega \setminus \overline{D}$. We show (under suitable hypotheses) that the solution u depends analytically on δ 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 δ 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 the 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.