<p>Neste trabalho investigamos o padrão de difração no regime de Fresnel (campo próximo) e de Fraunhofer (campo distante) para um conjunto de <italic>n</italic>fendas não simétricas e de larguras diferentes. Partindo da fórmula de difração de Fresnel-Kirchhoff conseguimos obter uma expressão para a intensidade da onda difratada por um conjunto de <italic>n</italic> fendas de tamanhos arbitrários onde é possível observar a transição do regime de Fresnel para o regime de Fraunhofer.</p>

We describe electromagnetic propagation in a relativistic electron gas at finite temperatures and carrier densities. Using quantum electrodynamics at finite temperatures, we obtain electric and magnetic responses and general constitutive relations. Rewriting the propagator for the electromagnetic field in terms of the electric and magnetic responses, we identify the modes that propagate in the gas. As expected, we obtain the usual collective excitations, i.e., a longitudinal electric and two transverse magnetic plasmonic modes. In addition, we find a purely photonic mode that satisfies the wave equation in vacuum, for which the electron gas is transparent. We present dispersion relations for the plasmon modes at zero and finite temperatures and identify the intervals of frequency and wavelength where both electric and magnetic responses are simultaneously negative, a behavior previously thought not to occur in natural systems.

We compute the real and imaginary parts of the electric permittivities and magnetic permeabilities for relativistic electrons from quantum electrodynamics at finite temperature and density. A semiclassical approximation establishes the conditions for neglecting nonlinear terms in the external electromagnetic fields as well as electron-electron interactions. We obtain both the electric and magnetic responses in a unified manner and relate them to longitudinal and transverse collective plasma oscillations. We demonstrate that such collective modes are thresholds for a metamaterial regime of the electron plasma which exhibits simultaneously negative longitudinal permittities and permeabilities. For nonzero temperatures, we obtain electromagnetic responses given by one-dimensional integrals to be numerically calculated, whereas for zero temperature we find analytic expressions for both their real/dispersive and imaginary/absorptive parts.

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We show that a gas of relativistic electrons is a left-handed material at low frequencies by computing the effective electric permittivity and effective magnetic permeability that appear in Maxwell’s equations in terms of the responses appearing in the constitutive relations, and showing that the former are both negative below the same frequency, which coincides with the zero-momentum frequency of longitudinal plasmons. We also show, by explicit computation, that the photonic mode of the electromagnetic radiation does not dissipate energy, confirming that it propagates in the gas with the speed of light in vacuum, and that the medium is transparent to it. We then combine those results to show that the gas has a negative effective index of refraction n e f f = − 1 . We illustrate the consequences of this fact for Snell’s law, and for the reflection and transmission coefficients of the gas.