Transmission of Electromagnetic Waves through a Conducting Slab. II. The Peak at Cyclotron Resonance

Abstract

This is the second in a series of three papers concerned with the propagation of electromagnetic waves through a metallic slab of finite thickness. In the situation studied here, the relation between the current in the slab and the electric field which drives it is highly nonlocal because of the finite electron mean free path. We calculate the transmission amplitude under these conditions, assuming that there is a steady magnetic field normal to the faces, and assuming also that electrons within the slab scatter diffusely at the surfaces. A transmission peak is predicted at cyclotron resonance which, in the limit of slab width much larger than an electron mean free path, has a line shape which approaches the square root of a Lorentzian and a phase which shifts by $\ensuremath{\pi}$ across the line. All of this is superposed on the Gantmakher-Kaner oscillations which are also present. This resonant behavior, which is absent when the electrons are assumed to be scattered specularly from the surface of the slab, can be understood in terms of electron transport from Fresnel zones on the Fermi sphere which expand at resonance so that the first zone covers the entire hemisphere. The absence of this resonance when the electrons are scattered specularly is tentatively ascribed to a fortuitous cancellation which arises because the equatorial electrons, whose response gives rise to currents which shield the interior of the metal, themselves undergo a resonance which excludes the field from the metal. In carrying out these calculations using the two-sided Wiener-Hopf technique for the nonlocal wave equation, we found that the technique itself can be simplified considerably if the quantity of interest is the transmission amplitude (the field at the emergent face of the slab) rather than the full-field amplitude in the slab. We show that this simplification arises when certain short-range parts of the field are eliminated.