# Universal modal radiation laws for all thermal emitters

## Abstract

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 outputmore »

- Authors:

- Stanford Univ., CA (United States). Ginzton Lab.

- Publication Date:

- Research Org.:
- Stanford Univ., CA (United States); Energy Frontier Research Centers (EFRC) (United States). Light-Material Interactions in Energy Conversion (LMI)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); US Air Force Office of Scientific Research (AFOSR)

- OSTI Identifier:
- 1351031

- Alternate Identifier(s):
- OSTI ID: 1465693

- Grant/Contract Number:
- SC0001293; FA9550-15-1-0335

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Proceedings of the National Academy of Sciences of the United States of America

- Additional Journal Information:
- Journal Volume: 114; Journal Issue: 17; Journal ID: ISSN 0027-8424

- Publisher:
- National Academy of Sciences, Washington, DC (United States)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Kirchhoff radiation laws; thermal radiation; optical modes; solar energy conversion; mode conversion

### Citation Formats

```
Miller, David A. B., Zhu, Linxiao, and Fan, Shanhui.
```*Universal modal radiation laws for all thermal emitters*. United States: N. p., 2017.
Web. doi:10.1073/pnas.1701606114.

```
Miller, David A. B., Zhu, Linxiao, & Fan, Shanhui.
```*Universal modal radiation laws for all thermal emitters*. United States. doi:10.1073/pnas.1701606114.

```
Miller, David A. B., Zhu, Linxiao, and Fan, Shanhui. Mon .
"Universal modal radiation laws for all thermal emitters". United States.
doi:10.1073/pnas.1701606114.
```

```
@article{osti_1351031,
```

title = {Universal modal radiation laws for all thermal emitters},

author = {Miller, David A. B. and Zhu, Linxiao and Fan, Shanhui},

abstractNote = {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.},

doi = {10.1073/pnas.1701606114},

journal = {Proceedings of the National Academy of Sciences of the United States of America},

number = 17,

volume = 114,

place = {United States},

year = {Mon Apr 10 00:00:00 EDT 2017},

month = {Mon Apr 10 00:00:00 EDT 2017}

}

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Works referenced in this record:

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