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  • Open Access

Upper Bounds on Broadband Absorption and Application to Solar Cells

Stéphane Collin*

Maxime Giteau

  • *Contact author: stephane.collin@cnrs.fr
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PRX Energy 5, 023006 – Published 26 May, 2026

DOI: https://doi.org/10.1103/1tzg-hgqx

Abstract

We address the question of the optimal broadband absorption of waves in an open dissipative system and its key implications for solar cells. We develop a general framework for absorption induced by multiple overlapping resonances, based on quasinormal modes and radiative and nonradiative decay rates. We demonstrate that the upper bounds on broadband absorption in a slab of thickness d take the simple form A=1−exp(−Fαd), where α is the absorption coefficient and F is the path enhancement factor. We apply these results to sunlight absorption in photovoltaics and answer the long-standing debate on the best light-trapping strategy in solar cells. For angle-independent absorption enhanced by scattering, we derive the isotropic upper bound F=4n2 (where n is the refractive index), extending the well-known Lambertian (or Yablonovitch) limit beyond the ray-optics and weak-absorption regimes. For angle-restricted illumination, we show that F can be further increased up to 8πn2/3 using multiresonant absorption induced by periodical patterning. These results have a general scope in the field of wave physics and open new opportunities to maximize absorption, detection, and attenuation of electromagnetic or mechanical waves in ultrathin devices.

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Popular Summary

How thin can a solar cell be while still absorbing most of the sunlight? This question is central to reducing material use, cost, and environmental impact in energy technologies. Usually, efficient absorption requires relatively thick layers, with a well-known enhancement limit set by random light scattering: the Lambertian (or Yablonovitch) limit.

Here, the authors present a universal framework to calculate the maximum possible absorption of light (or any wave) in lossy materials. By leveraging multiple overlapping resonances, they derive a simple formula for the absorption upper bound that relies on the material’s absorption coefficient and an effective path length. The effective path length can be maximized with multiresonant absorption induced by periodic patterns, exceeding conventional limits and answering the long-standing debate on the best light-trapping strategy in solar cells.

These results identify clear pathways toward highly efficient, ultrathin solar cells and, more broadly, compact absorbers for waves ranging from light to sound and radio frequencies.

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References (65)

  1. S. Molesky, W. Jin, P. S. Venkataram, and A. W. Rodriguez, T operator bounds on angle-integrated absorption and thermal radiation for arbitrary objects, Phys. Rev. Lett. 123, 257401 (2019).
  2. M. Gustafsson, K. Schab, L. Jelinek, and M. Capek, Upper bounds on absorption and scattering, New J. Phys. 22, 073013 (2020).
  3. P. Chao, B. Strekha, R. Kuate Defo, S. Molesky, and A. W. Rodriguez, Physical limits in electromagnetism, Nat. Rev. Phys. 4, 543 (2022).
  4. K. N. Rozanov, Ultimate thickness to bandwidth ratio of radar absorbers, IEEE Trans. Antennas Propag. 48, 1230 (2000).
  5. M. Benzaouia, G. Tokić, O. D. Miller, D. K. P. Yue, and S. G. Johnson, From solar cells to ocean buoys: Wide-bandwidth limits to absorption by metaparticle arrays, Phys. Rev. Appl. 11, 034033 (2019).
  6. O. Acher, J. M. L. Bernard, P. Maréchal, A. Bardaine, and F. Levassort, Fundamental constraints on the performance of broadband ultrasonic matching structures and absorbers, J. Acoust. Soc. Am. 125, 1995 (2009).
  7. M. Yang, S. Chen, C. Fu, and P. Sheng, Optimal sound-absorbing structures, Mater. Horiz. 4, 673 (2017).
  8. S. Qu and P. Sheng, Microwave and acoustic absorption metamaterials, Phys. Rev. Appl. 17, 047001 (2022).
  9. W. J. Padilla, Y. Deng, O. Khatib, and V. Tarokh, Fundamental absorption bandwidth to thickness limit for transparent homogeneous layers, Nanophotonics 13, 1623 (2024).
  10. S. Scheel, Sum rule for local densities of states in absorbing dielectrics, Phys. Rev. A 78, 013841 (2008).
  11. P. Chao, R. K. Defo, S. Molesky, and A. Rodriguez, Maximum electromagnetic local density of states via material structuring, Nanophotonics 12, 549 (2023).
  12. O. D. Miller, A. G. Polimeridis, M. T. Homer Reid, C. W. Hsu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, Fundamental limits to optical response in absorptive systems, Opt. Express 24, 3329 (2016).
  13. E. Yablonovitch, Statistical ray optics, J. Opt. Soc. Am. 72, 899 (1982).
  14. M. A. Green, Lambertian light trapping in textured solar cells and light-emitting diodes: Analytical solutions, Prog. Photovoltaics: Res. Appl. 10, 235 (2002).
  15. J. C. Miñano, in Physical Limitations to Photovoltaic Energy Conversion (Adam Hilger, Bristol, 1990), pp. 50–83.
  16. U. Rau, U. W. Paetzold, and T. Kirchartz, Thermodynamics of light management in photovoltaic devices, Phys. Rev. B 90, 035211 (2014).
  17. I. Massiot, A. Cattoni, and S. Collin, Progress and prospects for ultrathin solar cells, Nat. Energy 5, 959 (2020).
  18. J.-F. Guillemoles, T. Kirchartz, D. Cahen, and U. Rau, Guide for the perplexed to the Shockley-Queisser model for solar cells, Nat. Photonics 13, 501 (2019).
  19. W. Shockley and H. J. Queisser, Detailed balance limit of efficiency of p-n junction solar cells, J. Appl. Phys. 32, 510 (1961).
  20. J. Wong, S. T. Omelchenko, and H. A. Atwater, Impact of semiconductor band tails and band filling on photovoltaic efficiency limits, ACS Energy Lett. 6, 52 (2021).
  21. See the Supplemental Material http://link.aps.org/supplemental/10.1103/1tzg-hgqx for the full derivation of the theoretical framework, additional discussions on the historical perspective on light trapping, critical comments on previous works, and additional results of absorption models applied to silicon and GaAs slabs.
  22. P. Campbell and M. A. Green, The limiting efficiency of silicon solar cells under concentrated sunlight, IEEE Trans. Electron Devices 33, 234 (1986).
  23. S. Eyderman, S. John, and A. Deinega, Solar light trapping in slanted conical-pore photonic crystals: Beyond statistical ray trapping, J. Appl. Phys. 113, 154315 (2013).
  24. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  25. H. Haus, Waves and Fields in Optoelectronics, Solid State Physical Electronics Series (Prentice-Hall, Englewood Cliffs, NJ, USA, 1984).
  26. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, Coherent perfect absorbers: Time-reversed lasers, Phys. Rev. Lett. 105, 053901 (2010).
  27. W. R. Sweeney, C. W. Hsu, and A. D. Stone, Theory of reflectionless scattering modes, Phys. Rev. A 102, 063511 (2020).
  28. A. D. Stone, W. R. Sweeney, C. W. Hsu, K. Wisal, and Z. Wang, Reflectionless excitation of arbitrary photonic structures: A general theory, Nanophotonics 10, 343 (2021).
  29. C. Guo and S. Fan, Reciprocity constraints on reflection, Phys. Rev. Lett. 128, 256101 (2022).
  30. P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, Light interaction with photonic and plasmonic resonances, Laser Photonics Rev. 12, 1700113 (2018).
  31. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, Quasinormal-mode expansion for waves in open systems, Rev. Mod. Phys. 70, 1545 (1998).
  32. S. Fan, W. Suh, and J. D. Joannopoulos, Temporal coupled-mode theory for the Fano resonance in optical resonators, J. Opt. Soc. Am. A 20, 569 (2003).
  33. S. Collin, Nanostructure arrays in free-space: Optical properties and applications, Rep. Prog. Phys. 77, 126402 (2014).
  34. C. Battaglia, C.-M. Hsu, K. Söderström, J. Escarré, F.-J. Haug, M. Charrière, M. Boccard, M. Despeisse, D. T. L. Alexander, M. Cantoni, Y. Cui, and C. Ballif, Light trapping in solar cells: Can periodic beat random? ACS Nano 6, 2790 (2012).
  35. S. Eyderman, S. John, M. Hafez, S. S. Al-Ameer, T. S. Al-Harby, Y. Al-Hadeethi, and D. M. Bouwes, Light-trapping optimization in wet-etched silicon photonic crystal solar cells, J. Appl. Phys. 118, 023103 (2015).
  36. Z. Yu, A. Raman, and S. Fan, Fundamental limit of nanophotonic light trapping in solar cells, Proc. Natl. Acad. Sci. 107, 17491 (2010).
  37. V. Grigoriev, A. Tahri, S. Varault, B. Rolly, B. Stout, J. Wenger, and N. Bonod, Optimization of resonant effects in nanostructures via Weierstrass factorization, Phys. Rev. A 88, 011803(R) (2013).
  38. A. Krasnok, D. Baranov, H. Li, M.-A. Miri, F. Monticone, and A. Alú, Anomalies in light scattering, Adv. Opt. Photonics 11, 892 (2019).
  39. A. Mellor, I. Tobías, A. Martí, M. J. Mendes, and A. Luque, Upper limits to absorption enhancement in thick solar cells using diffraction gratings, Prog. Photovoltaics: Res. Appl. 19, 676 (2011).
  40. Z. Yu, A. Raman, and S. Fan, Fundamental limit of light trapping in grating structures, Opt. Express 18, A366 (2010).
  41. H.-L. Chen, A. Cattoni, R. De Lépinau, A. W. Walker, O. Höhn, D. Lackner, G. Siefer, M. Faustini, N. Vandamme, J. Goffard, B. Behaghel, C. Dupuis, N. Bardou, F. Dimroth, and S. Collin, A 19.9%-efficient ultrathin solar cell based on a 205-nm-thick GaAs absorber and a silver nanostructured back mirror, Nat. Energy 4, 761 (2019).
  42. S. Divitt and L. Novotny, Spatial coherence of sunlight and its implications for light management in photovoltaics, Optica 2, 95 (2015).
  43. C. Menzel, C. Rockstuhl, and F. Lederer, Advanced Jones calculus for the classification of periodic metamaterials, Phys. Rev. A 82, 053811 (2010).
  44. Z. Yu, A. Raman, and S. Fan, Thermodynamic upper bound on broadband light coupling with photonic structures, Phys. Rev. Lett. 109, 173901 (2012).
  45. Z. Liu, S. E. Sofia, H. S. Laine, M. Woodhouse, S. Wieghold, I. M. Peters, and T. Buonassisi, Revisiting thin silicon for photovoltaics: A technoeconomic perspective, Energy Environ. Sci. 13, 12 (2020).
  46. N. M. Haegel et al., Photovoltaics at multi-terawatt scale: Waiting is not an option, Science 380, 39 (2023).
  47. M. A. Green, Improved silicon optical parameters at 25∘C, 295 K and 300 K including temperature coefficients, Prog. Photovoltaics: Res. Appl. 30, 164 (2022).
  48. T. Niewelt, B. Steinhauser, A. Richter, B. Veith-Wolf, A. Fell, B. Hammann, N. E. Grant, L. Black, J. Tan, A. Youssef, J. D. Murphy, J. Schmidt, M. C. Schubert, and S. W. Glunz, Reassessment of the intrinsic bulk recombination in crystalline silicon, Solar Energy Mater. Solar Cells 235, 111467 (2022).
  49. M.-C. van Lare and A. Polman, Optimized scattering power spectral density of photovoltaic light-trapping patterns, ACS Photonics 2, 822 (2015).
  50. M. Burresi, F. Pratesi, F. Riboli, and D. S. Wiersma, Complex photonic structures for light harvesting, Adv. Opt. Mater. 3, 722 (2015).
  51. R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional problem with smooth boundary surface, Ann. Phys. 60, 401 (1970).
  52. D. M. Callahan, J. N. Munday, and H. A. Atwater, Solar cell light trapping beyond the ray optic limit, Nano Lett. 12, 214 (2012).
  53. H. A. Atwater and A. Polman, Plasmonics for improved photovoltaic devices, Nat. Mater. 9, 205 (2010).
  54. I. Massiot, N. Vandamme, N. Bardou, C. Dupuis, A. Lemaître, J.-F. Guillemoles, and S. Collin, Metal nanogrid for broadband multiresonant light-harvesting in ultrathin GaAs layers, ACS Photonics 1, 878 (2014).
  55. S. Bhattacharya, I. Baydoun, M. Lin, and S. John, Towards 30% power conversion efficiency in thin-silicon photonic-crystal solar cells, Phys. Rev. Appl. 11, 014005 (2019).
  56. K. Vynck, R. Pierrat, R. Carminati, L. S. Froufe-Pérez, F. Scheffold, R. Sapienza, S. Vignolini, and J. J. Sáenz, Light in correlated disordered media, Rev. Mod. Phys. 95, 045003 (2023).
  57. F. Bigourdan, R. Pierrat, and R. Carminati, Enhanced absorption of waves in stealth hyperuniform disordered media, Opt. Express 27, 8666 (2019).
  58. J. Buencuerpo, T. E. Saenz, M. Steger, M. Young, E. L. Warren, J. F. Geisz, M. A. Steiner, and A. C. Tamboli, Efficient light-trapping in ultrathin GaAs solar cells using quasi-random photonic crystals, Nano Energy 96, 107080 (2022).
  59. E. Camarillo Abad, H. J. Joyce, and L. C. Hirst, Transparent quasi-random structures for multimodal light trapping in ultrathin solar cells with broad engineering tolerance, ACS Photonics 9, 2724 (2022).
  60. N. Tavakoli, R. Spalding, A. Lambertz, P. Koppejan, G. Gkantzounis, C. Wan, R. Röhrich, E. Kontoleta, A. F. Koenderink, R. Sapienza, M. Florescu, and E. Alarcon-Llado, Over 65% sunlight absorption in a 1μm Si slab with hyperuniform texture, ACS Photonics 9, 1206 (2022).
  61. M. Li, S. Xiao, Y.-Y. Bai, and B.-Z. Wang, An ultrathin and broadband radar absorber using resistive FSS, IEEE Antennas Wirel. Propag. Lett. 11, 748 (2012).
  62. V. Leroy, A. Strybulevych, M. Lanoy, F. Lemoult, A. Tourin, and J. H. Page, Superabsorption of acoustic waves with bubble metascreens, Phys. Rev. B 91, 020301(R) (2015).
  63. S. Huang, Z. Zhou, D. Li, T. Liu, X. Wang, J. Zhu, and Y. Li, Compact broadband acoustic sink with coherently coupled weak resonances, Sci. Bull. 65, 373 (2020).
  64. S. Brûlé, E. H. Javelaud, S. Enoch, and S. Guenneau, Experiments on seismic metamaterials: Molding surface waves, Phys. Rev. Lett. 112, 133901 (2014).
  65. S. Brûlé, S. Enoch, and S. Guenneau, Emergence of seismic metamaterials: Current state and future perspectives, Phys. Lett. A 384, 126034 (2020).

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PRX Energy 5, 023006– Published 26 May, 2026
  • Received 23 December 2025
  • Revised 2 April 2026
  • Accepted 28 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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