A metal with real

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Condensed Matter, Materials & Soft Matter · Wave propagation and light in media

What an Instrument Can See, Ch Media that are designed: plasmons, metamaterials and photonic crystals

Problem

A metal with real, lossless permittivity ϵm=−14.2\epsilon_m=-14.2 fills z<0z<0; a transparent dielectric of permittivity ϵd=2.1\epsilon_d=2.1 fills z>0z>0. Look for a bound, p-polarised wave running along xx as ei(kxx−ωt)e^{i(k_xx-\omega t)}, with the only nonzero magnetic component Hy={H0 eikxx e−κdz,z>0,H0 eikxx eκmz,z<0,H_y=\begin{cases}H_0\,e^{ik_xx}\,e^{-\kappa_dz}, & z>0,\\ H_0\,e^{ik_xx}\,e^{\kappa_mz}, & z<0,\end{cases} where κd>0\kappa_d>0 and κm>0\kappa_m>0 so the field decays away from the interface on each side. For this polarisation, Ampère's law gives Ex=1iωϵ0ϵj∂Hy∂zE_x=\dfrac{1}{i\omega\epsilon_0\epsilon_j}\dfrac{\partial H_y}{\partial z} in each medium jj. ((a)) Compute ∂Hy/∂z\partial H_y/\partial z on each side, and hence show that matching ExE_x at z=0z=0 (with HyH_y already continuous there) gives ϵmκd+ϵdκm=0\epsilon_m\kappa_d+\epsilon_d\kappa_m=0. ((b)) Combine this with κd2=kx2−ϵdk02\kappa_d^2=k_x^2-\epsilon_dk_0^2 and κm2=kx2−ϵmk02\kappa_m^2=k_x^2-\epsilon_mk_0^2 to eliminate κd\kappa_d and κm\kappa_m, and show that kx=k0ϵmϵd/(ϵm+ϵd)k_x=k_0\sqrt{\epsilon_m\epsilon_d/(\epsilon_m+\epsilon_d)}. ((c)) State, in terms of ϵm\epsilon_m and ϵd\epsilon_d, the condition under which κd\kappa_d and κm\kappa_m come out real and positive, confirm it holds for the values given, and evaluate kx/k0k_x/k_0.

Browse the problem library