On the Wave Speed of Thermal Radiation Inside and Near the Boundary of an Absorbing Material

Planck's law describes thermal radiation into vacuum from a black body in thermal equilibrium. This law can be easily adapted to describe radiation into a transparent medium with a constant refractive index, and it admits a less trivial extension to radiation into a transparent medium with a nonconstant refractive index. However, this law cannot be straightforwardly generalized to describe thermal radiation into absorbing media and, in particular, to describe thermally exited electromagnetic fields inside the radiating body itself. We first analyze Planck's law and show why it cannot be straightforwardly extended to radiation into an absorbing medium. The derivation of this law relies on the assumption that a radiated field admits decomposition into normal modes, which cannot exist in absorbing media that are characterized by a complex-valued refractive index $n=n′+in″$⁠, whose imaginary part describes the rate of energy dissipation. Correspondingly, the speed of electromagnetic waves in absorbing media $c=c0/n$⁠, where c₀ is the speed of light in vacuum, is also complex-valued, which suggests that the conventional concept of a complex valued wave speed is not suitable for modeling thermal radiation. We demonstrate that complex-valued wave speeds adequately describe waves that carry signals, such as radio waves and laser beams. Such waves decay because they pass some of their energy to the medium. The energy absorbed by the medium is eventually reradiated, but in studies focused on the transmission of signals, the reradiated fields are ignored as noise. In order to study thermal radiation in an absorbing material, one must treat the material and the radiation together as a closed system. The energy in such a system is conserved, and its distribution between the material and radiation does not change in time. This radiation admits decomposition into normal modes, which makes it possible to extend Planck's law to radiation into absorbing materials. This paper proposes a model of thermal radiation in an absorbing medium as a closed, energy conserving system. The radiation field has normal modes that correspond to an effective speed of wave propagation. Assuming that an absorbing material and the radiation in it are in thermal equilibrium, we show that deep inside the material, the average speed of photons is given by a frequency and temperature-dependent expression $c∗=c0/(1+e−ℏω/κT)$⁠. While this result is independent of the material, we further show that close to the boundary of the medium, the speed of thermal radiation depends in a complex way on the refractive index and the extinction coefficient of the material, as well as the direction of propagation and the distance from the material's surface.