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Efficient computation of thermal radiation from biperiodic layered metasurfaces using the T-matrix method

Martin Gabbert1, Markus Nyman2, Lukas Rebholz1, Carsten Rockstuhl1,2, and Ivan Fernandez-Corbaton2

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Phys. Rev. B 113, 195306 – Published 22 May, 2026

DOI: https://doi.org/10.1103/xwt4-mf93

Abstract

Metasurfaces are becoming important tools for the control of thermal radiation. Understanding their functional possibilities on computational grounds requires evaluating the response of the biperiodic layered system for many degrees of freedom, including several radiation directions and polarizations, while varying lattice spacing, thicknesses, and/or materials of homogeneous layers, over a range of frequencies. The diverse set of cases that need to be considered in simulations prompts for efficient numerical tools to handle them. To respond to this need, we present a method for computing the thermal radiation from metasurfaces that combines the directional Kirchhoff's law with efficient T-matrix-based calculations. We show that such a method can accurately reproduce experimental data from a metasurface made of platinum square plates. Additionally, we predict highly circularly polarized emissivity from a chiral metasurface. When comparing CPU-times, the method outperforms other approaches such as rigorous coupled wave analysis already at the modest number of 61 cases per frequency.

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References (70)

  1. J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, Coherent emission of light by thermal sources, Nature (London) 416, 61 (2002).
  2. M. H. Eriksen, J. R. Deop-Ruano, J. D. Cox, and A. Manjavacas, Chiral light–matter interactions with thermal magnetoplasmons in graphene nanodisks, Nano Lett. 25, 313 (2025).
  3. D. Costantini, A. Lefebvre, A.-L. Coutrot, I. Moldovan-Doyen, J.-P. Hugonin, S. Boutami, F. Marquier, H. Benisty, and J.-J. Greffet, Plasmonic metasurface for directional and frequency-selective thermal emission, Phys. Rev. Appl. 4, 014023 (2015).
  4. C. Blanchard, L. Wojszvzyk, C. Jamois, J.-L. Leclercq, C. Chevalier, L. Ferrier, P. Viktorovitch, I. Moldovan-Doyen, F. Marquier, J.-J. Greffet, and X. Letartre, Metallo-dielectric metasurfaces for thermal emission with controlled spectral bandwidth and angular aperture, Opt. Mater. Express 12, 1 (2022).
  5. V. Fernández-Hurtado, F. J. García-Vidal, S. Fan, and J. C. Cuevas, Enhancing near-field radiative heat transfer with Si-based metasurfaces, Phys. Rev. Lett. 118, 203901 (2017).
  6. A. C. Overvig, S. A. Mann, and A. Alù, Thermal metasurfaces: Complete emission control by combining local and nonlocal light-matter interactions, Phys. Rev. X 11, 021050 (2021).
  7. G. Perrakis, A. C. Tasolamprou, G. Kenanakis, E. N. Economou, S. Tzortzakis, and M. Kafesaki, Passive radiative cooling and other photonic approaches for the temperature control of photovoltaics: A comparative study for crystalline silicon-based architectures, Opt. Express 28, 18548 (2020).
  8. B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, Near-field radiative thermal transport: From theory to experiment, AIP Adv. 5, 053503 (2015).
  9. J. C. Cuevas and F. J. García-Vidal, Radiative heat transfer, ACS Photonics 5, 3896 (2018).
  10. M. F. Picardi, K. N. Nimje, and G. T. Papadakis, Dynamic modulation of thermal emission—A tutorial, J. Appl. Phys. 133, 111101 (2023).
  11. M. M. R. Elsawy, S. Lanteri, R. Duvigneau, J. A. Fan, and P. Genevet, Numerical optimization methods for metasurfaces, Laser & Photonics Reviews 14, 1900445 (2020).
  12. S. A. Schulz, R. F. Oulton, M. Kenney, A. Alù, I. Staude, A. Bashiri, Z. Fedorova, R. Kolkowski, A. F. Koenderink, X. Xiao, J. Yang, W. J. Peveler, A. W. Clark, G. Perrakis, A. C. Tasolamprou, M. Kafesaki, A. Zaleska, W. Dickson, D. Richards, A. Zayats, et al., Roadmap on photonic metasurfaces, Appl. Phys. Lett. 124, 260701 (2024).
  13. B. Gallinet, J. Butet, and O. J. F. Martin, Numerical methods for nanophotonics: Standard problems and future challenges, Laser & Photonics Reviews 9, 577 (2015).
  14. P. Monk, Finite Element Methods for Maxwell's Equations (Oxford University Press, Oxford, 2003).
  15. L. Li, New formulation of the Fourier modal method for crossed surface-relief gratings, J. Opt. Soc. Am. 14, 2758 (1997).
  16. M. G. Moharam, T. K. Gaylord, E. B. Grann, and D. A. Pommet, Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings, J. Opt. Soc. Am. 12, 1068 (1995).
  17. P. Waterman, Matrix formulation of electromagnetic scattering, Proc. IEEE 53, 805 (1965).
  18. D. Beutel, I. Fernandez-Corbaton, and C. Rockstuhl, treams – a T-matrix-based scattering code for nanophotonics, Comput. Phys. Commun. 297, 109076 (2024).
  19. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, Scattering-matrix approach to multilayer diffraction, J. Opt. Soc. Am. 12, 1097 (1995).
  20. G. Kirchhoff, Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht, Ann. Phys. 185, 275 (1860).
  21. H. Baltes, I on the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment, in Progress in Optics, edited by E. Wolf (Elsevier, North Holland, 1976), pp. 1–25.
  22. A. Monti, A. Alu, A. Toscano, and F. Bilotti, Surface impedance modeling of all-dielectric metasurfaces, IEEE Trans. Antennas Propag. 68, 1799 (2019).
  23. A. I. Dimitriadis, D. L. Sounas, N. V. Kantartzis, C. Caloz, and T. D. Tsiboukis, Surface susceptibility bianisotropic matrix model for periodic metasurfaces of uniaxially mono-anisotropic scatterers under oblique TE-wave incidence, IEEE Trans. Antennas Propag. 60, 5753 (2012).
  24. J. G. Smith, I. R. Hooper, N. Clow, A. P. Hibbins, and S. A. R. Horsley, Method for extracting the surface impedance of a generic reflective metasurface, Sci. Rep. 14, 27141 (2024).
  25. U. Hohenester, N. Reichelt, and G. Unger, Nanophotonic resonance modes with the nanobem toolbox, Comput. Phys. Commun. 276, 108337 (2022).
  26. A. Doicu and T. Wriedt, Calculation of the T matrix in the null-field method with discrete sources, J. Opt. Soc. Am. 16, 2539 (1999).
  27. D. W. Mackowski, Discrete dipole moment method for calculation of the T-matrix for nonspherical particles, J. Opt. Soc. Am. 19, 881 (2002).
  28. I. Fernandez-Corbaton, D. Beutel, C. Rockstuhl, A. Pausch, and W. Klopper, Computation of electromagnetic properties of molecular ensembles, ChemPhysChem 21, 878 (2020).
  29. B. Zerulla, M. Krstić, D. Beutel, C. Holzer, C. Wöll, C. Rockstuhl, and I. Fernandez‐Corbaton, A multi‐scale approach for modeling the optical response of molecular materials inside cavities, Adv. Mater. 34, 2200350 (2022).
  30. J. D. Mazo-Vásquez, M. Nyman, M. Krstić, L. Rebholz, C. Rockstuhl, and I. Fernandez-Corbaton, Studying thermal radiation with T-matrices, Phys. Rev. B 112, 054307 (2025).
  31. Do Van Lam, D. T. Dung, U. N. T. Nguyen, H. S. Kang, B. Bae, H. Kim, M. Lim, D. Kim, J. Kim, and S. Lee, Metal‐organic frameworks as a thermal emitter for high‐performance passive radiative cooling, Small Methods 9, 2401141 (2025).
  32. D. Theobald, A. Egel, G. Gomard, and U. Lemmer, Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles, Phys. Rev. A 96, 033822 (2017).
  33. A. Egel, L. Pattelli, G. Mazzamuto, D. S. Wiersma, and U. Lemmer, CELES: CUDA-accelerated simulation of electromagnetic scattering by large ensembles of spheres, J. Quant. Spectrosc. Radiat. Transfer 199, 103 (2017).
  34. T. Martin, T-matrix method for closely adjacent obstacles, J. Quant. Spectrosc. Radiat. Transfer 234, 40 (2019).
  35. D. Schebarchov, E. C. L. Ru, J. Grand, and B. Auguié, Mind the gap: Testing the Rayleigh hypothesis in T-matrix calculations with adjacent spheroids, Opt. Express 27, 35750 (2019).
  36. T. Rother and S. C. Hawkins, Notes on Rayleigh's hypothesis and the extended boundary condition method, J. Acoust. Soc. Am. 149, 2179 (2021).
  37. J. Barkhan, M. Ganesh, and S. Hawkins, Approximation of radiating waves in the near-field: Error estimates and application to a class of inverse problems, J. Comput. Appl. Math. 401, 113769 (2022).
  38. A. G. Lamprianidis, C. Rockstuhl, and I. Fernandez-Corbaton, Transcending the Rayleigh hypothesis with multipolar sources distributed across the topological skeleton of a scatterer, J. Quant. Spectrosc. Radiat. Transfer 296, 108455 (2023).
  39. I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, Objects of maximum electromagnetic chirality, Phys. Rev. X 6, 031013 (2016).
  40. T. Wriedt and J. Hellmers, New scattering information portal for the light-scattering community, J. Quant. Spectrosc. Radiat. Transfer 109, 1536 (2008).
  41. J. Hellmers and T. Wriedt, New approaches for a light scattering Internet information portal and categorization schemes for light scattering software, J. Quant. Spectrosc. Radiat. Transfer 110, 1511 (2009).
  42. G. Gouesbet, T-matrix methods for electromagnetic structured beams: A commented reference database for the period 2019–2023, J. Quant. Spectrosc. Radiat. Transfer 322, 109015 (2024).
  43. M. I. Mishchenko, Comprehensive thematic T-matrix reference database: a 2017–2019 update, J. Quant. Spectrosc. Radiat. Transfer 242, 106692 (2020).
  44. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, T-matrix computations of light scattering by nonspherical particles: A review, J. Quant. Spectrosc. Radiat. Transfer 55, 535 (1996).
  45. M. Fruhnert, I. Fernandez-Corbaton, V. Yannopapas, and C. Rockstuhl, Computing the T-matrix of a scattering object with multiple plane wave illuminations, Beilstein J. Nanotechnol. 8, 614 (2017).
  46. R. M. A. Ekeroth, A. García-Martín, and J. C. Cuevas, Thermal discrete dipole approximation for the description of thermal emission and radiative heat transfer of magneto-optical systems, Phys. Rev. B 95, 235428 (2017).
  47. C. Guo, B. Zhao, and S. Fan, Adjoint Kirchhoff's law and general symmetry implications for all thermal emitters, Phys. Rev. X 12, 021023 (2022).
  48. J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, Adaptive finite element method for simulation of optical nano structures, physica status solidi (b) 244, 3419 (2007).
  49. JCMsuite6.4 documentation: Parameter reference, Multipole Expansion, JCMwave GmbH, Berlin (2024).
  50. See Supplemental Material at http://link.aps.org/supplemental/10.1103/xwt4-mf93 for a discussion on the required multipolar order and the convergence behavior of the T-matrix method and RCWA, which includes Refs. [3, 15, 43, 48, 52, 54, 66, 67, 68, 69].
  51. N. Asadova, K. Achouri, K. Arjas, B. Auguié, R. Aydin, A. Baron, D. Beutel, B. Bodermann, K. Boussaoud, S. Burger, M. Choi, K. M. Czajkowski, A. B. Evlyukhin, A. Fazel-Najafabadi, I. Fernandez-Corbaton, P. Garg, D. Globosits, U. Hohenester, H. Kim, S. Kim, et al., T-matrix representation of optical scattering response: Suggestion for a data format, J. Quant. Spectrosc. Radiat. Transfer 333, 109310 (2025).
  52. B. Zerulla, R. Venkitakrishnan, D. Beutel, M. Krstić, C. Holzer, C. Rockstuhl, and I. Fernandez‐Corbaton, A T‐Matrix based approach to homogenize artificial materials, Adv. Opt. Mater. 11, 2201564 (2023).
  53. D. Beutel, I. Fernandez-Corbaton, and C. Rockstuhl, Unified lattice sums accommodating multiple sublattices for solutions of the Helmholtz equation in two and three dimensions, Phys. Rev. A 107, 013508 (2023).
  54. A. Egel, Y. Eremin, T. Wriedt, D. Theobald, U. Lemmer, and G. Gomard, Extending the applicability of the T-matrix method to light scattering by flat particles on a substrate via truncation of Sommerfeld integrals, J. Quant. Spectrosc. Radiat. Transfer 202, 279 (2017).
  55. D. Beutel, A. Groner, C. Rockstuhl, and I. Fernandez-Corbaton, Efficient simulation of biperiodic, layered structures based on the T-matrix method, J. Opt. Soc. Am. B 38, 1782 (2021).
  56. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, Optical properties of metallic films for vertical-cavity optoelectronic devices, Appl. Opt. 37, 5271 (1998).
  57. M. F. Schubert and A. M. Hammond, Fourier modal method for inverse design of metasurface-enhanced micro-LEDs, Opt. Express 31, 42945 (2023).
  58. V. Liu and S. Fan, S4 : A free electromagnetic solver for layered periodic structures, Comput. Phys. Commun. 183, 2233 (2012).
  59. X. Garcia-Santiago, M. Hammerschmidt, J. Sachs, S. Burger, H. Kwon, M. Knöller, T. Arens, P. Fischer, I. Fernandez-Corbaton, and C. Rockstuhl, Toward maximally electromagnetically chiral scatterers at optical frequencies, ACS Photonics 9, 1954 (2022).
  60. E. C. L. Ru, W. R. C. Somerville, and B. Auguié, Radiative correction in approximate treatments of electromagnetic scattering by point and body scatterers, Phys. Rev. A 87, 012504 (2013).
  61. J. D. Fischbach, F. Betz, N. Asadova, P. Tassan, D. Urbonas, T. Stöferle, R. F. Mahrt, S. Burger, C. Rockstuhl, F. Binkowski, and T. J. Sturges, A framework to compute resonances arising from multiple scattering, Adv. Theor. Simul. 8, 2400989 (2025).
  62. P. Garg, J. D. Fischbach, A. G. Lamprianidis, X. Wang, M. S. Mirmoosa, V. S. Asadchy, C. Rockstuhl, and T. J. Sturges, Inverse‐Designed dispersive Time‐Varying nanostructures, Adv. Opt. Mater. 13, 2402444 (2025).
  63. A. Zhan, R. Gibson, J. Whitehead, E. Smith, J. R. Hendrickson, and A. Majumdar, Controlling three-dimensional optical fields via inverse Mie scattering, Sci. Adv. 5, eaax4769 (2019).
  64. L. Kuhn, T. Repän, and C. Rockstuhl, Inverse design of core-shell particles with discrete material classes using neural networks, Sci. Rep. 12, 19019 (2022).
  65. N. Asadova, K. Boussaoud, J. Meyer, F. Tristram, and C. Rockstuhl, A T-matrix database to promote information-driven research in nanophotonics, arXiv:2602.02101.
  66. A. Egel, D. Theobald, Y. Donie, U. Lemmer, and G. Gomard, Light scattering by oblate particles near planar interfaces: On the validity of the T-matrix approach, Opt. Express 24, 25154 (2016).
  67. J. W. Gibbs, Fourier's series, Nature (London) 59, 200 (1898).
  68. E. Popov and M. Nevière, Maxwell equations in Fourier space: Fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media, J. Opt. Soc. Am. 18, 2886 (2001).
  69. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings, J. Opt. Soc. Am. 24, 3781 (2007).
  70. M. Gabbert, treams_metasurface add-on to treams (2025), https://github.com/tfp-photonics/treams_metasurface.git.

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Phys. Rev. B 113, 195306– Published 22 May, 2026
  • Received 24 August 2025
  • Revised 13 February 2026
  • Accepted 6 April 2026
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