A metamaterial slab is built

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Optics & Photonics · Wave propagation and light in media

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

Problem

A metamaterial slab is built, as in the wire-and-split-ring construction of section 7.3, so that at some fixed frequency its real, lossless permittivity and permeability are both negative:

\epsilon=-2.4

and

\mu=-1.6

. A wave polarised along

\hat{\mathbf x}

and travelling inside the slab is written

\mathbf E=E_0\hat{\mathbf x}e^{i(kz-\omega t)}

, with

\mathbf H=\dfrac{k}{\omega\mu_0\mu}\,\hat{\mathbf z}\times\mathbf E

from Faraday's law, as used in the text. ((a)) Show that

k^2=\epsilon\mu k_0^2

is positive for this slab, so

k

is real and the wave truly propagates, and explain briefly why this would fail if only one of

\epsilon,\mu

were negative instead. ((b)) Compute the time-averaged Poynting vector

\mathbf S=\tfrac12\mathrm{Re}(\mathbf E\times\mathbf H^*)

in terms of

k

, and use the requirement that energy flows away from whatever launched the wave (taken as

+\hat{\mathbf z}

) to fix the sign of

k

. Hence give the refractive index

n=k/k_0

of the slab, magnitude and sign. ((c)) A student argues: ‘since

n

is negative, the energy in this wave must be travelling backwards, against

+\hat{\mathbf z}

.’ Say whether this is correct, and if not, identify which velocity, phase or energy, actually reverses.

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