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The Kirchhoff law of thermal radiation, relating emissivity {\epsilon} andabsorptance {\alpha}, has been originally formulated for opaque bodies inthermodynamic equilibrium with the environment. However, in many systems ofpractical importance, both assumptions are often not satisfied. In this work,we revisit the century-old law and examine the limits of its applicability inan example of Er:YAG and Er:YLF dielectric crystals, potential radiationconverters for thermophotovoltaic applications. In our experiments, the (80at.%) Er:YAG crystal was opaque between 1.45 {\mu}m and 1.64 {\mu}m. In thisspectral range, its absorptance {\alpha}({\lambda}) is spectrally flat anddifferentiates from unity only by a small amount of reflection. The shape ofthe emissivity spectrum {\epsilon}({\lambda}) closely matches that ofabsorptance {\alpha}({\lambda}), suggesting that the Kirchhoff law canadequately describe thermal radiation of opaque bodies, even if the requirementof thermodynamic equilibrium is not satisfied. The(20 at.%) Er:YLF crystal hadsmaller size, lower concentration of Er ions, and it was not opaque.Nevertheless, its spectrum of emissivity {\epsilon}({\lambda}) had almost thesame shape (between 1.45 {\mu}m and 1.62 {\mu}m) as the spectrum of absorptance{\alpha}({\lambda}) derived from the transmission measurements. Thisobservation is in line with our prediction that the spectra of emissivity andabsorptance should have identical shapes in optically thin slabs. We, thus,show that the Kirchhoff law of thermal radiation can be extended (with caution)even to not-opaque bodies away from the thermodynamic equilibrium.

Thermal radiation of Er doped dielectric crystals: Probing the range of applicability of the Kirchhoff law