We derive four laws relating the absorptivity and emissivity of thermal emitters. Unlike the original Kirchhoff radiation law derivations, these derivations include diffraction, and so are valid also for small objects, and can also cover nonreciprocal objects. The proofs exploit two recent approaches. First, we express all fields in terms of the mode-converter basis sets of beams; these sets, which can be uniquely established for any linear optical object, give orthogonal input beams that are coupled one-by-one to orthogonal output beams. Second, we consider thought experiments using universal linear optical machines, which allow us to couple appropriate beams and black bodies. Two of these laws can be regarded as rigorous extensions of previously known laws: One gives a modal version of a radiation law for reciprocal objects-the absorptivity of any input beam equals the emissivity into the "backward" (i.e., phase-conjugated) version of that beam; another gives the overall equality of the sums of the emissivities and the absorptivities for any object, including nonreciprocal ones. The other two laws, valid for reciprocal and nonreciprocal objects, are quite different from previous relations. One shows universal equivalence of the absorptivity of each mode-converter input beam and the emissivity into its corresponding scattered output beam. The other gives unexpected equivalences of absorptivity and emissivity for broad classes of beams. Additionally, we prove these orthogonal mode-converter sets of input and output beams are the ones that maximize absorptivities and emissivities, respectively, giving these beams surprising additional physical meaning.Kirchhoff radiation laws | thermal radiation | optical modes | solar energy conversion | mode conversion R adiation laws relating the absorptivity and emissivity of an object are at the core of the thermal physics of radiation and are particularly interesting for understanding limits to efficiency of solar energy conversion (e.g., refs. 1 and 2), for example. The core relation is Kirchhoff's radiation law (3-6), which equates the absorptivity and emissivity of a surface. This law is often extended to a "directional" version, which equates the absorptivity of a given direction of input beam and the emissivity into the opposite or "reversed" direction. Typical textbook approaches (5, 6) to the directional law trace back to Planck's rederivation (4) of Kirchhoff's approach (3). Both Kirchhoff's and Planck's treatments explicitly make two assumptions: (i) The optical properties of the object are reciprocal (e.g., excluding Faraday rotation); (ii) diffraction is neglected, presuming objects much larger than a wavelength and using ray rather than wave optics. It is, however, now known that such a directional radiation law is not correct for nonreciprocal systems (1, 7), and nanophotonic structures for the control of thermal radiation (8-22) can have feature sizes much smaller than the wavelength. Given the fundamental thermodynamic importance and the technological relevan...