Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law

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Abstract
Contrary to common belief, neither reciprocity of the bidirectional reflectance distribution function (BRDF) nor the directional form of Kirchhoff’s electromagnetic radiation law can be demonstrated on the basis of energy conservation. The BRDF is generally considered reciprocal as an extension of Helmholtz reciprocity, but Helmholtz reciprocity does not always hold. We describe the flaw in a thermodynamic demonstration of reciprocity that uses an enclosure calculation. Some conclusions can be drawn from the enclosure calculation, but reciprocity requires more restrictive conditions. We conclude that, although they can be violated, reciprocity and the directional form of Kirchhoff’s law generally hold because of the quantum-mechanical principle of time-reversal invariance, which applies to most materials.