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

TL;DR

We develop a non-equilibrium Green's-function framework to quantify far-field radiative energy, linear momentum, and angular momentum from gyrotropic two-dimensional metallic thin films. By solving the Dyson equation in a thin-film geometry and expressing the photon Green's functions in terms of generalized Fresnel coefficients, we obtain closed-form formulas for radiative fluxes that connect to Kirchhoff's law in zero magnetic field and enable angular-momentum radiation under an out-of-plane field. The approach yields unified expressions for emitted power, force, and torque through spectral integrals involving the Bose distribution $N(\omega)$ and Fresnel coefficients, with the Wigner transform facilitating the angular-momentum flux calculation. Numerical study on Bi confirms the framework, showing modest power enhancement by a magnetic field and a non-monotonic dependence of radiated torque on $B$, highlighting magneto-optical control of angular momentum transfer in nanoscale photonic systems.

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.

Figures (4)

• Figure 1: Schematic diagram of the system, where gray represents a 2D thin-film material applied with an out-of-plane magnetic field $\bm{B}$. The radiation quantities of interest are the power $(\langle I \rangle)$, force $(\langle N_z \rangle)$, and torque $(\langle M_z\rangle)$.
• Figure 2: Plots for (a) Power spectrum ($S_I$) and (b) torque spectrum ($S_M$) using the analytical formulas, Eqs. \ref{['eqn:define_si']} and \ref{['eqn:define_sm']}, respectively. Magnetic field used ranges from 01. Drude parameters used are shown in Fig. \ref{['tab:frequencies']}. Room temperature (300K) is used in all simulations.
• Figure 3: Heatmap to visualize how the angular momentum spectrum varies with the applied magnetic field. Each horizontal slice shows $S_M(\omega)$ (defined in Eq. \ref{['eqn:define_sm']}) at a particular level of $B$.
• Figure 4: Heatmap to visualize the dependence of the total radiative torque ($\langle M_z \rangle$) on the applied magnetic field ($B$) and temperature ($T$).