Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Direct observation of the violation of Kirchhoff’s law of thermal radiation


Thermal emission—the process through which all objects with a finite temperature radiate electromagnetic energy—has generally been thought to obey reciprocity, where the absorbed and emitted radiation from a body are equal for a given wavelength and angular channel. This equality, formalized by Gustav Kirchhoff in 1860, is known as Kirchhoff’s law of thermal radiation and has long guided designs to control the emitted radiation. Removing the constraint of Kirchhoff’s law unlocks a multitude of applications and designs for thermal emitters. Decoupling the absorptivity and emissivity relationship can be leveraged to achieve novel functions, ranging from reducing re-emission losses to the Sun in the context of solar energy harvesting systems to radiative camouflage. Here we report the direct measurements of an inequality between the spectral directional emissivity and absorptivity for a photonic design that supports a guided-mode resonance coupled to a magneto-optic material. This inequality occurs under the application of an in-plane magnetic field that modifies the normally diagonal permittivity tensor to a non-diagonal tensor in magneto-optic InAs, resulting in an antisymmetric relationship where the magnetic tuning of enhanced emissivity for a given angle correlates with decreased absorptivity for the same angle.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic of magnetically tunable emission and absorption of GMR structure coupled to n-InAs and comparison of zero-field absorptivity and emissivity.
Fig. 2: Violation of Kirchhoff’s law in absorptivity and emissivity measurements.
Fig. 3: Onsager–Casimir relations for emissivity and absorptivity in a magnetic field.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


  1. Planck, M. The Theory of Heat Radiation (Forgotten Books, 2013).

  2. Greffet, J.-J. et al. Coherent emission of light by thermal sources. Nature 416, 61–64 (2002).

    Article  ADS  Google Scholar 

  3. Lu, G. et al. Engineering the spectral and spatial dispersion of thermal emission via polariton-phonon strong coupling. Nano Lett. 21, 1831–1838 (2021).

    Article  ADS  Google Scholar 

  4. Caldwell, J. D. et al. Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators. Nano Lett. 13, 3690–3697 (2013).

    Article  ADS  Google Scholar 

  5. Jun, Y. C., Lik, T. S. & Ellis, A. R. Doping-tunable thermal emission from plasmon polaritons in semiconductor epsilon-near-zero thin films. Appl. Phys. Lett. 105, 131109 (2014).

    Article  ADS  Google Scholar 

  6. Xu, J., Mandal, J. & Raman, A. P. Broadband directional control of thermal emission. Science 372, 393–397 (2021).

    Article  ADS  Google Scholar 

  7. Park, J. et al. Dynamic thermal emission control with InAs-based plasmonic metasurfaces. Sci. Adv. 4, eaat3163 (2018).

    Article  ADS  Google Scholar 

  8. Liberal, I. & Engheta, N. Near-zero refractive index photonics. Nat. Photon. 11, 149–158 (2017).

    Article  ADS  MATH  Google Scholar 

  9. Luo, C., Narayanaswamy, A., Chen, G. & Joannopoulos, J. D. Thermal radiation from photonic crystals: a direct calculation. Phys. Rev. Lett. 93, 2113905 (2004).

    Article  Google Scholar 

  10. De Zoysa, M. et al. Conversion of broadband to narrowband thermal emission through energy recycling. Nat. Photon. 6, 535–539 (2012).

    Article  ADS  Google Scholar 

  11. Baranov, D. G. et al. Nanophotonic engineering of far-field thermal emitters. Nat. Mater. 18, 920–930 (2019).

    Article  ADS  Google Scholar 

  12. Laroche, M., Carminati, R. & Greffet, J.-J. Coherent thermal antenna using a photonic crystal slab. Phys. Rev. Lett. 96, 123903 (2006).

    Article  ADS  Google Scholar 

  13. Kirchhoff, G. On the relationship between the emissivity and absorptivity of a body for heat and light. Ann. Phys. 109, 275–301 (1860).

    Article  Google Scholar 

  14. Lorentz, H. A. The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light. Verl. K. Akad. W Amsterdam 4, 176 (1896).

    MATH  Google Scholar 

  15. Miller, D. A. B., Zhu, L. & Fan, S. Universal modal radiation laws for all thermal emitters. Proc. Natl Acad. Sci. USA 114, 4336–4341 (2017).

    Article  ADS  Google Scholar 

  16. Khandekar, C., Khosravi, F., Li, Z. & Jacob, Z. New spin-resolved thermal radiation laws for nonreciprocal bianisotropic media. New J. Phys. 22, 123005 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  17. Guo, C., Zhao, B. & Fan, S. Adjoint Kirchhoff’s law and general symmetry implications for all thermal emitters. Phys. Rev. X 12, 021023 (2022).

    Google Scholar 

  18. Green, M. A. Time-asymmetric photovoltaics. Nano Lett. 12, 5985–5988 (2012).

    Article  ADS  Google Scholar 

  19. Fan, S. Thermal photonics and energy applications. Joule 1, 264–273 (2017).

    Article  Google Scholar 

  20. Buddhiraju, S., Santhanam, P. & Fan, S. Thermodynamic limits of energy harvesting from outgoing thermal radiation. Proc. Natl Acad. Sci. USA 115, 3609–3615 (2018).

    Article  ADS  Google Scholar 

  21. Huang, R., Miranowicz, A., Liao, J.-Q., Nori, F. & Jing, H. Nonreciprocal photon blockade. Phys. Rev. Lett. 121, 153601 (2018).

    Article  ADS  Google Scholar 

  22. Sounas, D. L. & Alù, A. Non-reciprocal photonics based on time modulation. Nat. Photon. 11, 774–783 (2017).

    Article  ADS  Google Scholar 

  23. Zhu, L. & Fan, S. Near-complete violation of detailed balance in thermal radiation. Phys. Rev. B 90, 220301 (2014).

    Article  ADS  Google Scholar 

  24. Zhao, B., Guo, C., Garcia, C. A., Narang, P. & Fan, S. Axion-field-enabled nonreciprocal thermal radiation in Weyl semimetals. Nano Lett. 20, 1923–1927 (2020).

    Article  ADS  Google Scholar 

  25. Pajovic, S., Tsurimaki, Y., Qian, X. & Chen, G. Intrinsic nonreciprocal reflection and violation of Kirchhoff’s law of thermal radiation in planar type-I magnetic Weyl semimetal surfaces. Phys. Rev. B 102, 165417 (2020).

    Article  ADS  Google Scholar 

  26. Wang, S. S., Magnusson, R., Bagby, J. S. & Moharam, M. G. Guided-mode resonances in planar dielectric-layer diffraction gratings. J. Opt. Soc. Am. 7, 1470–1474 (1990).

    Article  ADS  Google Scholar 

  27. Fattal, D., Li, J., Peng, Z., Fiorentino, M. & Beausoleil, R. G. Flat dielectric grating reflectors with focusing abilites. Nat. Photon. 4, 466–470 (2010).

    Article  ADS  Google Scholar 

  28. Kamboj, A. et al. All-epitaxial guided-mode resonance mid-wave infrared detectors. Appl. Phys. Lett. 118, 201102 (2021).

    Article  ADS  Google Scholar 

  29. Khaleque, T. & Magnusson, R. Light management through guided-mode resonances in thin-film silicon solar cells. J. Nanophoton. 8, 083995 (2014).

    Article  ADS  Google Scholar 

  30. Paskov, P. P. Refractive indices of InSb, InAs, GaSb, InAsxSb1–x, and In1–xGaxSb: effects of free carriers. J. Appl. Phys. 91, 1890 (1997).

    Article  ADS  Google Scholar 

  31. Heyman, J. N. et al. Terahertz emission from GaAs and InAs in a magnetic field. Phys. Rev. B 64, 085202 (2001).

    Article  ADS  Google Scholar 

  32. Liu, M. et al. Nonreciprocal thermal radiation in ultrathin magnetized epsilon-near-zero semiconductors. Preprint at (2022).

  33. Shayegan, K. J., Zhao, B., Kim, Y., Fan, S. & Atwater, H. A. Nonreciprocal infrared absorption via resonant magneto-optical coupling to InAs. Sci. Adv. 8, eabm4308 (2022).

    Article  Google Scholar 

  34. Reddy, H. et al. Temperature-dependent optical properties of plasmonic titanium nitride thin films. ACS Photonics 4, 1413–1420 (2017).

    Article  Google Scholar 

  35. Boyd, R. W. Radiometry and The Detection of Optical Radiation (Wiley, 1983).

  36. Lalanne, P. et al. Quasinormal mode solvers for resonators with dispersive materials. J. Opt. Soc. Am. A 36, 686–704 (2019).

    Article  ADS  Google Scholar 

  37. Li, J., Li, Z. & Shen, S. Degenerate quasi-normal mode theory for near-field radiation between plasmonic structures. Opt. Express 28, 34123–34136 (2020).

    Article  ADS  Google Scholar 

  38. Casimir, H. B. G. On Onsager’s principle of microscopic reversibility. Rev. Mod. Phys. 17, 350 (1945).

    Article  ADS  Google Scholar 

  39. Zhao, B. et al. Near-complete violation of Kirchhoff’s law of thermal radiation with a 0.3 T magnetic field. Opt. Lett. 44, 4204 (2019).

    Article  ADS  Google Scholar 

  40. Xiao, Y., Sheldon, M. & Khats, M. A. Super-Planckian emission cannot really be ‘thermal’. Nat. Photon. 16, 397–401 (2022).

    Article  ADS  Google Scholar 

  41. Taliercio, T., Guilengui, V. N., Cerutti, L., Tournié, E. & Greffet, J.-J. Brewster ‘mode’ in highly doped semiconductor layers: an all-optical technique to monitor doping concentration. Opt. Express 22, 24294–24303 (2014).

    Article  ADS  Google Scholar 

  42. Fan, S. & Li, W. Photonics and thermodynamics concepts in radiative cooling. Nat. Photon. 12, 182–190 (2022).

    Article  ADS  Google Scholar 

  43. Park, Y., Zhao, B. & Fan, S. Reaching the ultimate efficiency of solar energy harvesting with a nonreciprocal multijunction solar cell. Nano Lett. 22, 448–452 (2022).

    Article  ADS  Google Scholar 

  44. Zhao, B. et al. Nonreciprocal thermal emitters using metasurfaces with multiple diffraction channels. Phys. Rev. Applied 16, 064001 (2021).

    Article  ADS  Google Scholar 

  45. Hwang, J. S., Xu, J. & Raman, A. Simultaneous control of spectral and directional emissivity with gradient epsilon-near-zero InAs photonic structures. Preprint at (2022).

  46. Zhang, Z. & Zhu, L. Broadband nonreciprocal thermal radiation. Phys. Rev. Applied 19, 014013 (2023).

    Article  ADS  Google Scholar 

  47. Liu, E. et al. Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal. Nat. Phys. 14, 1125–1131 (2018).

    Article  Google Scholar 

  48. Hadad, Y., Soric, J. C. & Alù, A. Breaking temporal symmetries for emission and absorption. Proc. Natl Acad. Sci. USA 113, 3471–3475 (2016).

    Article  ADS  Google Scholar 

  49. Gangaraj, S. A. H. & Monticone, F. Drifting electrons: nonreciprocal plasmonics and thermal photonics. ACS Photonics 9, 806–819 (2022).

    Article  Google Scholar 

  50. Bergmann, J., Heusinger, M., Andrä, G. & Falk, F. Temperature dependent optical properties of amorphous silicon for diode laser crystallization. Opt. Express 20, A856–A863 (2012).

    Article  ADS  Google Scholar 

  51. Zhong, F. et al. Angle-resolved thermal emission spectroscopy characterization of non-Hermitian metacrystals. Phys. Rev. Applied 13, 014071 (2020).

    Article  ADS  Google Scholar 

  52. Adambukulam, C. et al. An ultra-stable 1.5 tesla permanent magnet assembly for qubit experiments at cryogenic temperatures. Rev. Sci. Instrum. 92, 085106 (2021).

    Article  ADS  Google Scholar 

Download references


We acknowledge discussions with A. Laucht and D. H. Drew. This work has been supported by DARPA NLM (grant no. HR00111820046). K.J.S. acknowledges support from the NSF for a graduate research fellowship.

Author information

Authors and Affiliations



H.A.A. and S.F. conceived the project. K.J.S. fabricated the samples and carried out the measurements, supported by S.B. K.J.S. led the data analysis and modelling, with support from S.B. and B.Z. H.A.A. supervised the project. K.J.S., S.B. and H.A.A. wrote the manuscript with input from all other authors.

Corresponding author

Correspondence to Harry A. Atwater.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Photonics thanks Denis Baranov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Refractive indices of InAs and a-Si and heated emissivity measurements.

(a) Refractive index values (n and k, solid and dot-dashed lines respectively) of the InAs wafer for various temperatures. We observe shifts in n and k that correspond to a redshifting of the ENZ-wavelength of the wafer. (b) Temperature-dependent refractive index values of the deposited a-Si. The refractive index grows as a function of temperature. The extinction coefficient is negligible in the wavelength range. (ce) Emissivity measurements taken at 50, 100, and 150 °C, respectively. The blue, green, and red curves are obtained by solving E E.D. 1 using the average measured refractive index of a-Si at each temperature. Both (a) and (b) were measured and fitted using a J.A. Woollam IR-VASE and the WVASE software.

Extended Data Fig. 2 Fabrication process of sample.

(a) We begin with a bare InAs wafer for which we have already determined the temperature-dependent optical properties. (b) a 2-μm-thick a-Si layer is deposited on top of the InAs wafer via PECVD. (c) A 750-nm-thick layer of resist, ZEP520A, is spun onto the wafer and the grating pattern is written into the resist via EBPG. (d) The pattern is then developed, and remaining resist is baked. The resist left on the sample surface acts as a mask for the ICP-RIE process. (e) The resist mask is removed overnight and the a-Si GMR structure is cleaned and imaged with an optical microscope and SEM.

Extended Data Fig. 3 SEM images of a-Si layer and grating structure.

(a) Side-view SEM image of the 2-μm-thick a-Si layer on top of the InAs wafer. This sample was not used for the final grating structure, but rather for the optical characterization of the a-Si layer. (b) Top-view SEM of the fabricated grating structure.

Extended Data Fig. 4 Schematics of the emissivity and absorptivity measurement setups.

(a) For emissivity measurements, the sample is mounted on a heater at the rotation axis of a goniometer, centered between the pole pieces of an electromagnet. The goniometer allows us to rotate the sample around the z-axis and probe the outgoing radiation at an angle θ. The sample emission is collected through a ZnSe lens and sent through an external port into an FTIR. We place a polarizer in front of the detector to resolve the TM and TE emission. (b) The absorptivity measurement system uses an FTIR source and detector mounted on a 2-θ rotation stage. Instead of an electromagnet, we use a Halbach array of permanent magnets to apply the magnetic field. The sample is still heated to the same temperatures as in the emissivity measurements so that we have a direct comparison of the absorptivity and emissivity as defined by the Kirchhoff’s Law.

Extended Data Fig. 5 Normalization of the emissivity data for θ = 65°.

Raw traces of the collected emission from a Carbon black reference (black) and the sample (red) at three different temperatures. The blue trace is the emission of Aluminum at 24 °C and represents the background emission contributions from the system. Traces taken at different temperatures are offset for clarity.

Extended Data Fig. 6 Fitting results from the fine magnetic field sweep at θ = 65°.

(a) Experimental data and fits for three different magnetic fields. (b) The extracted InAs contribution to the emissivity (amplitude) for varying magnetic field, modeled as a sigmoid. (c) Central wavelength of the InAs contribution (that is the sigmoid) for varying magnetic field. We observe a blueshift of the emission edge with increasing magnetic field. (d) The resonant emissivity contribution (amplitude) of the GMR structure is modified, but the spectral shift is dictated by the InAs magnetic field response (b and c). (e) Change in the resonant amplitude for all magnetic field values.

Extended Data Fig. 7 Experimental and simulated emissivity change for non-zero magnetic field exhibiting saturation effect as resonant sample emissivity approaches Blackbody limit.

(a) Experimental data (black points) with quadratic fit (solid red) for the full magnetic field range and linear fit (dot-dash blue) for the positive magnetic field range. (b) The same plot as in (a) for our simulated photonic structure. The magnitude of the change in emissivity as a function of magnetic field is larger, however we observe the same saturation behavior. Note that our resonant peak emissivity at θ = 65° is at a slightly longer wavelength (λ = 12.68 μm).

Extended Data Fig. 8 Experimental data comparing different regimes of the spectral and amplitude tuning of the peak emissivity for three angles.

(a) Emissivity for θ = −25° for three magnetic field values. At small angles, the spectral overlap of the InAs emissivity edge (13 μm) and the +1st order GMR (12.1 μm) is minimal, resulting in no tuning of the spectral position or magnitude of the emissivity peak of the overall structure. (b) At θ = 55°, the spectral position of the emissivity peak is tuned by the direction of the InAs emissivity edge shift (blueshift for +1.0 T and redshift for −1.0 T) relative to the +1st order GMR. We term this angular range where the InAs emissivity edge begins to change the spectral position of the emissivity peak as the ‘critical angle’ θC. (c) For large angles, we get strong emissivity amplitude tuning with little shift in the spectral position of the resonant emissivity.

Supplementary information

Supplementary Information

Supplementary Figs. 1–7, discussion and refs. 1–4.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shayegan, K.J., Biswas, S., Zhao, B. et al. Direct observation of the violation of Kirchhoff’s law of thermal radiation. Nat. Photon. 17, 891–896 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing