Analytical formulas for far-field radiated energy and angular momentum of metallic thin films

Abstract: We investigate far-field radiation of energy, linear momentum, and angular momentum from two-dimensional electron systems, focusing on metallic thin films described by the Drude conductivity. Using the Keldysh formalism within the non-equilibrium Green's function framework, we derive analytical expressions for radiative power, force, and torque. To enable angular momentum radiation, an out-of-plane magnetic field is applied to break reciprocity, resulting in gyrotropic terms in the permittivity tensor. By approximating the emitter as a thin film, the photon Green's functions can be solved analytically. Expressions for the Poynting vector and Maxwell's stress tensor can subsequently be extracted from the lesser Green's function, which governs the field correlations. The final radiation formulas can be expressed in terms of Fresnel coefficients, revealing an insightful connection to energy conservation via Kirchhoff's law. Using the Wigner transform, the analytical expression for the radiative torque can also be related to the generalized Fresnel coefficients. Numerical calculations based on the optical conductivity of bismuth are presented to corroborate the analytical results. These results provide a unified framework for energy, momentum, and angular momentum radiation in gyrotropic thin films.