Abstract

Fundamental limits for path lengths of light in isotropic absorbers are calculated. The method of calculation is based on accounting for occupied states in optical phase space. Light trapping techniques, such as scattering or diffraction, are represented by the way how the available states are occupied. One finding of the presented investigation is that the path length limit is independent of the light trapping mechanism and only depends on the conditions for light incidence to, and escape from the absorber. A further finding is that the maximum path length is obtained for every light trapping mechanisms which results in a complete filling of the available states in phase space. For stationary solar cells, the Yablonovitch limit of 4dn2, with n the refractive index of the absorber, is a very good approximation of this limit.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Goetzberger, “Optical confinement in thin Si-solar cells by diffuse back reflectors” 15th IEEE Photovoltaic Specialists Conference, 867–870 (1981).
  2. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. El. Dev. 29(2), 300–305 (1982).
    [CrossRef]
  3. M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
    [CrossRef]
  4. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982).
    [CrossRef]
  5. J. C. Miñano, “Optical confinement in photovoltaics,” in Physical Limitations to the Photovoltaic Solar Energy Conversion, A. Luque A., Araújo G. L. (Ed.), Adam Hilger, Bristol, UK. (1990).
  6. T. Markvart, “Solar cell as a heat engine: energy-entropy analysis of photovoltaic conversion,” Phys. Status Solidi 205(12), 2752–2756 (2008).
    [CrossRef]
  7. T. Kirchartz, in Physics of Nanostructured Solar Cells, V. Badescu (Ed.) (Nova Science Publishers, 2009), pp. 1–40.
  8. Z. Yu and S. Fan, “Angular constraint on light-trapping absorption enhancement in solar cells,” Appl. Phys. Lett. 98(1), 011106 (2011).
    [CrossRef]
  9. The original article was published by J. Liouville in the Journale de Mathematique 3, (1838), 349. Better sources are textbooks on statistical mechanics or quantum systems. One example is W. Blaschke, Vorlesungen ueber Differential-Geometrie I, Springer Verlag Berlin, 68 (1924).
  10. M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 1999), pp. 724 – 726.
  11. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd edition, Volume 8 (Butterworth-Heinemann, 1984), pp. 257 – 264.
  12. N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
    [CrossRef]
  13. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
    [CrossRef] [PubMed]
  14. A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
    [CrossRef]
  15. J. Gjessing, A. S. Sudbo, and E. S. Marstein, “A novel back-side light trapping structure for thin silicon solar cells,” J. Euro. Opt. Soc. 6, 11020 (2011).
  16. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010).
    [CrossRef] [PubMed]
  17. 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(3), 2790–2797 (2012).
    [CrossRef] [PubMed]
  18. M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
    [CrossRef] [PubMed]
  19. M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
    [CrossRef]
  20. M. Peters, J. C. Goldschmidt, and B. Bläsi, “Angular confinement and concentration in photovoltaic converters,” Sol. Energy Mater. Sol. Cells 94(8), 1393–1398 (2010).
    [CrossRef]
  21. C. Ulbrich, M. Peters, B. Bläsi, T. Kirchartz, A. Gerber, and U. Rau, “Enhanced light trapping in thin-film solar cells by a directionally selective filter,” Opt. Express 18(S2), A133–A138 (2010).
    [CrossRef] [PubMed]
  22. B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
    [CrossRef]
  23. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010).
    [CrossRef] [PubMed]
  24. C.-J. Winter, R. L. Sizmann, and L. L. Vant-Hull, eds., Solar Power Plants, Springer-Verlag, 2008), pp. 297–300.
  25. Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Phys. Rev. Lett. 109(17), 173901 (2012).
    [CrossRef] [PubMed]
  26. P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
    [CrossRef]
  27. P. Panek, M. Lipinski, and J. Dutkiewicz, “Texturization of multicrystalline silicon by wet chemical etching for silicon solar cells,” J. Mater. Sci. 40(6), 1459–1463 (2005).
    [CrossRef]
  28. S. C. Baker-Finch, K. R. McIntosh, and M. L. Terry, “Isotextured silicon solar cell analysis and modeling 1: optics,” IEEE J. Photovolt. 2(4), 457–464 (2012).
    [CrossRef]

2014 (1)

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

2012 (4)

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Phys. Rev. Lett. 109(17), 173901 (2012).
[CrossRef] [PubMed]

S. C. Baker-Finch, K. R. McIntosh, and M. L. Terry, “Isotextured silicon solar cell analysis and modeling 1: optics,” IEEE J. Photovolt. 2(4), 457–464 (2012).
[CrossRef]

M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
[CrossRef] [PubMed]

2011 (3)

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

J. Gjessing, A. S. Sudbo, and E. S. Marstein, “A novel back-side light trapping structure for thin silicon solar cells,” J. Euro. Opt. Soc. 6, 11020 (2011).

Z. Yu and S. Fan, “Angular constraint on light-trapping absorption enhancement in solar cells,” Appl. Phys. Lett. 98(1), 011106 (2011).
[CrossRef]

2010 (5)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010).
[CrossRef] [PubMed]

M. Peters, J. C. Goldschmidt, and B. Bläsi, “Angular confinement and concentration in photovoltaic converters,” Sol. Energy Mater. Sol. Cells 94(8), 1393–1398 (2010).
[CrossRef]

C. Ulbrich, M. Peters, B. Bläsi, T. Kirchartz, A. Gerber, and U. Rau, “Enhanced light trapping in thin-film solar cells by a directionally selective filter,” Opt. Express 18(S2), A133–A138 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010).
[CrossRef] [PubMed]

2008 (1)

T. Markvart, “Solar cell as a heat engine: energy-entropy analysis of photovoltaic conversion,” Phys. Status Solidi 205(12), 2752–2756 (2008).
[CrossRef]

2005 (1)

P. Panek, M. Lipinski, and J. Dutkiewicz, “Texturization of multicrystalline silicon by wet chemical etching for silicon solar cells,” J. Mater. Sci. 40(6), 1459–1463 (2005).
[CrossRef]

2002 (2)

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
[CrossRef]

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
[CrossRef]

1987 (1)

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

1982 (2)

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. El. Dev. 29(2), 300–305 (1982).
[CrossRef]

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982).
[CrossRef]

1975 (1)

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
[CrossRef]

Aberle, A. G.

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
[CrossRef] [PubMed]

Alexander, D. T. L.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Baker-Finch, S. C.

S. C. Baker-Finch, K. R. McIntosh, and M. L. Terry, “Isotextured silicon solar cell analysis and modeling 1: optics,” IEEE J. Photovolt. 2(4), 457–464 (2012).
[CrossRef]

Ballif, C.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Battaglia, C.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
[CrossRef] [PubMed]

Bläsi, B.

Boccard, M.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Campbell, P.

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

Cantoni, M.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Charrière, M.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Cody, G. D.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. El. Dev. 29(2), 300–305 (1982).
[CrossRef]

Cui, Y.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Despeisse, M.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Dutkiewicz, J.

P. Panek, M. Lipinski, and J. Dutkiewicz, “Texturization of multicrystalline silicon by wet chemical etching for silicon solar cells,” J. Mater. Sci. 40(6), 1459–1463 (2005).
[CrossRef]

Escarré, J.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Fan, S.

Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Phys. Rev. Lett. 109(17), 173901 (2012).
[CrossRef] [PubMed]

Z. Yu and S. Fan, “Angular constraint on light-trapping absorption enhancement in solar cells,” Appl. Phys. Lett. 98(1), 011106 (2011).
[CrossRef]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010).
[CrossRef] [PubMed]

Forberich, K.

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
[CrossRef] [PubMed]

Gerber, A.

Gjessing, J.

J. Gjessing, A. S. Sudbo, and E. S. Marstein, “A novel back-side light trapping structure for thin silicon solar cells,” J. Euro. Opt. Soc. 6, 11020 (2011).

Goetzberger, A.

A. Goetzberger, “Optical confinement in thin Si-solar cells by diffuse back reflectors” 15th IEEE Photovoltaic Specialists Conference, 867–870 (1981).

Goldschmidt, J. C.

M. Peters, J. C. Goldschmidt, and B. Bläsi, “Angular confinement and concentration in photovoltaic converters,” Sol. Energy Mater. Sol. Cells 94(8), 1393–1398 (2010).
[CrossRef]

Green, M. A.

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
[CrossRef]

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
[CrossRef]

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

Haug, F. J.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Hsu, C. M.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Kirchartz, T.

Lipinski, M.

P. Panek, M. Lipinski, and J. Dutkiewicz, “Texturization of multicrystalline silicon by wet chemical etching for silicon solar cells,” J. Mater. Sci. 40(6), 1459–1463 (2005).
[CrossRef]

Luque, A.

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

Markvart, T.

T. Markvart, “Solar cell as a heat engine: energy-entropy analysis of photovoltaic conversion,” Phys. Status Solidi 205(12), 2752–2756 (2008).
[CrossRef]

Marstein, E. S.

J. Gjessing, A. S. Sudbo, and E. S. Marstein, “A novel back-side light trapping structure for thin silicon solar cells,” J. Euro. Opt. Soc. 6, 11020 (2011).

Marti, A.

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

McIntosh, K. R.

S. C. Baker-Finch, K. R. McIntosh, and M. L. Terry, “Isotextured silicon solar cell analysis and modeling 1: optics,” IEEE J. Photovolt. 2(4), 457–464 (2012).
[CrossRef]

Mellor, A.

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

Mendes, M. J.

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

Panek, P.

P. Panek, M. Lipinski, and J. Dutkiewicz, “Texturization of multicrystalline silicon by wet chemical etching for silicon solar cells,” J. Mater. Sci. 40(6), 1459–1463 (2005).
[CrossRef]

Peters, M.

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
[CrossRef] [PubMed]

M. Peters, J. C. Goldschmidt, and B. Bläsi, “Angular confinement and concentration in photovoltaic converters,” Sol. Energy Mater. Sol. Cells 94(8), 1393–1398 (2010).
[CrossRef]

C. Ulbrich, M. Peters, B. Bläsi, T. Kirchartz, A. Gerber, and U. Rau, “Enhanced light trapping in thin-film solar cells by a directionally selective filter,” Opt. Express 18(S2), A133–A138 (2010).
[CrossRef] [PubMed]

Phong, B. T.

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
[CrossRef]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Raman, A.

Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Phys. Rev. Lett. 109(17), 173901 (2012).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010).
[CrossRef] [PubMed]

Rau, U.

Sahraei, N.

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

M. Peters, C. Battaglia, K. Forberich, B. Bläsi, N. Sahraei, and A. G. Aberle, “Comparison between periodic and stochastic parabolic light trapping structures for thin-film microcrystalline Silicon solar cells,” Opt. Express 20(28), 29488–29499 (2012).
[CrossRef] [PubMed]

Söderström, K.

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Sudbo, A. S.

J. Gjessing, A. S. Sudbo, and E. S. Marstein, “A novel back-side light trapping structure for thin silicon solar cells,” J. Euro. Opt. Soc. 6, 11020 (2011).

Terry, M. L.

S. C. Baker-Finch, K. R. McIntosh, and M. L. Terry, “Isotextured silicon solar cell analysis and modeling 1: optics,” IEEE J. Photovolt. 2(4), 457–464 (2012).
[CrossRef]

Tobias, I.

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

Ulbrich, C.

Venkatara, S.

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982).
[CrossRef]

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. El. Dev. 29(2), 300–305 (1982).
[CrossRef]

Yu, Z.

Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Phys. Rev. Lett. 109(17), 173901 (2012).
[CrossRef] [PubMed]

Z. Yu and S. Fan, “Angular constraint on light-trapping absorption enhancement in solar cells,” Appl. Phys. Lett. 98(1), 011106 (2011).
[CrossRef]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010).
[CrossRef] [PubMed]

ACS Nano (1)

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(3), 2790–2797 (2012).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

Z. Yu and S. Fan, “Angular constraint on light-trapping absorption enhancement in solar cells,” Appl. Phys. Lett. 98(1), 011106 (2011).
[CrossRef]

Commun. ACM (1)

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
[CrossRef]

IEEE J. Photovolt. (1)

S. C. Baker-Finch, K. R. McIntosh, and M. L. Terry, “Isotextured silicon solar cell analysis and modeling 1: optics,” IEEE J. Photovolt. 2(4), 457–464 (2012).
[CrossRef]

IEEE Trans. El. Dev. (1)

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. El. Dev. 29(2), 300–305 (1982).
[CrossRef]

J. Appl. Phys. (1)

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62(1), 243–249 (1987).
[CrossRef]

J. Euro. Opt. Soc. (1)

J. Gjessing, A. S. Sudbo, and E. S. Marstein, “A novel back-side light trapping structure for thin silicon solar cells,” J. Euro. Opt. Soc. 6, 11020 (2011).

J. Mater. Sci. (1)

P. Panek, M. Lipinski, and J. Dutkiewicz, “Texturization of multicrystalline silicon by wet chemical etching for silicon solar cells,” J. Mater. Sci. 40(6), 1459–1463 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

Nat. Mater. (1)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Opt. Exp. (1)

N. Sahraei, K. Forberich, S. Venkatara, A. G. Aberle, and M. Peters, “Analytical solution for haze values of aluminium-induced texture (AIT) glass superstrates for a-Si:H solar cells,” Opt. Exp. 22(S1), A53–A67 (2014).
[CrossRef]

Opt. Express (3)

Phys. Rev. Lett. (1)

Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Phys. Rev. Lett. 109(17), 173901 (2012).
[CrossRef] [PubMed]

Phys. Status Solidi (1)

T. Markvart, “Solar cell as a heat engine: energy-entropy analysis of photovoltaic conversion,” Phys. Status Solidi 205(12), 2752–2756 (2008).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010).
[CrossRef] [PubMed]

Prog. Photovolt. Res. Appl. (3)

A. Mellor, I. Tobias, A. Marti, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt. Res. Appl. 19(6), 676–687 (2011).
[CrossRef]

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
[CrossRef]

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt. Res. Appl. 10(4), 235–241 (2002).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

M. Peters, J. C. Goldschmidt, and B. Bläsi, “Angular confinement and concentration in photovoltaic converters,” Sol. Energy Mater. Sol. Cells 94(8), 1393–1398 (2010).
[CrossRef]

Other (7)

T. Kirchartz, in Physics of Nanostructured Solar Cells, V. Badescu (Ed.) (Nova Science Publishers, 2009), pp. 1–40.

J. C. Miñano, “Optical confinement in photovoltaics,” in Physical Limitations to the Photovoltaic Solar Energy Conversion, A. Luque A., Araújo G. L. (Ed.), Adam Hilger, Bristol, UK. (1990).

The original article was published by J. Liouville in the Journale de Mathematique 3, (1838), 349. Better sources are textbooks on statistical mechanics or quantum systems. One example is W. Blaschke, Vorlesungen ueber Differential-Geometrie I, Springer Verlag Berlin, 68 (1924).

M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 1999), pp. 724 – 726.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd edition, Volume 8 (Butterworth-Heinemann, 1984), pp. 257 – 264.

C.-J. Winter, R. L. Sizmann, and L. L. Vant-Hull, eds., Solar Power Plants, Springer-Verlag, 2008), pp. 297–300.

A. Goetzberger, “Optical confinement in thin Si-solar cells by diffuse back reflectors” 15th IEEE Photovoltaic Specialists Conference, 867–870 (1981).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic sketch of the three situations for light incidence and escape investigate in this work.

Fig. 2
Fig. 2

Sketch of Lambertian scattering. The incident light cone (green) is scattered at the scattering surface. Lambertian scattering implies that the available phase space (blue) is completely occupied after a single scattering event. Light that is scattered into the escape cone leaves the system after reaching the front surface again. Note that the hemisphere is only a graphical representation of phase space and does not represent the density of states which is given in Eq. (2).

Fig. 3
Fig. 3

Path length for light in an absorber scattered with Phong characteristics for different Phong exponents p and different refractive indices of the absorber n.

Fig. 4
Fig. 4

Sketch of the 90 degree conical tilt. The incident cone (green) is tilted by 90 degree by the scattering surface and, for geometrical reasons, halved. The available phase space (blue) is not completely filled as the occupied phase space (red) is conserved. b) shows why scattering on a geometrical scatterer results in a “V” shaped path. The red path is the one observed here; the grey paths are its geometrical counterparts. Symmetry requires that red and grey paths are interchangeable; reciprocity requires that the direction of light paths can be reversed. Considering the light intensity in each path it follows, that the solid line is the effective path for the calculation of the path length.

Fig. 5
Fig. 5

Sketch of the 90 degree spherical tilt. The incident light cone (green) is scattered at the scattering surface. Phase space is conserved so that only a fraction of the available phase space (blue) is occupied (red). The spherical tilt results in a ring shaped occupation of available states in phase space.

Fig. 6
Fig. 6

Sketch of the successive filling of phase space concept. a) shows the way how successive filling is treated in the calculation; the path length for non-overlapping rings with equal area on the available phase space is calculated and summed up. When the escape cone is reached, light is exiting the system and the summation stops. The sequence of the summation is of no consequence and the approach can therefore be generalised to all techniques that result in a complete filling of phase space. b) shows a different sketch of how the successive filling works. Blue dots mark scattering events. For light with different angles of incidence αi, also the scattering angle αs is different. As phase space is conserved, a unique relation exists between αi and αs. As an example the path for one specific angle of incidence is shown.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

Ω= 1 2π dt Ω sun (t)
DOS( θ )=cosθsinθ
N air N abs = 0 arcsin 1 n cosθsinθd θ 0 π 2 cosθsinθd θ = 1 2 sin ( arcsin 1 n ) 2 1 2 = 1 n 2
L=2d( π 2 π 2 dθ 1 sinθ cosθsinθ ) k=0 ( 1 Ω n 2 ) k =4 n 2 Ω d
L=4 n 2 d
L=4d k=0 ( 1 sin θ s 2 n 2 ) k =4 n 2 sin θ s 2 d1.8 10 5 n 2 d
ζ = 0 arc sin 1 n cos θ p sin θ d θ 0 π 2 cos θ p sin θ d θ = 1 ( 1 1 n 2 ) 1 + p 2
L = 4 d k = 0 ( ( 1 1 n 2 ) 1 + p 2 ) k = 4 1 1 ( 1 1 n 2 ) 1 + p 2 d
S= Ω air n 2 = Ω abs cosθsinθd θ 0 π 2 cosθsinθd θ
L=2d Ω abs sinθd θ 0 π 2 cosθsinθd θ
S= Ω n 2 = 0 θ m sinθarccos θ θ m d θ 0 π 2 cosθsinθd θ
L=2d 0 θ m sinθarccos θ θ m d θ 0 θ m cosθsinθd θ = 4d θ m sin θ m 2
L=4dn 1 Ω ( 1 π + O 3 [ n 2 Ω ] )2.26 n 2 Ω
S = Ω n 2 = π 2 θ d π 2 cos θ s i n θ d θ 0 π 2 cos θ sin θ d θ = sin θ d 2
L = 2 d π 2 θ d π 2 1 cos θ cos θ sin θ d θ π 2 θ d π 2 cos θ sin θ d θ = 2 d sin θ d 1 2 sin θ d 2 = 4 d n 2 Ω
L = 4 d n = 14 d | n = 3.5
L=4d n 2 sin θ s 851dn=2980d | n=3.5
Ω n 2 = π 2 θ dn π 2 θ dn+1 cosθsinθdθ 0 π 2 cosθsinθd θ θ dn+1 = 1 2 arccos[ cos2 θ dn 2 Ω n 2 ]
L=2d[ π 2 θ d1 π 2 sinθdθ π 2 θ d1 π 2 cosθsinθdθ +...+ π 2 θ dn+1 π 2 θ dn sinθdθ π 2 θ dn+1 π 2 θ dn cosθsinθdθ +... ] =8d[ sin θ d1 cos2 θ d1 +...+ sin θ dn+1 sin θ dn cos2 θ dn+1 cos2 θ dn +... ] =4d n 2 Ω[ sin θ d1 +...+sin θ dn+1 sin θ dn +... ] =4d n 2 Ω
L=4d n 2 Ω [ sin θ d1 +...+sin θ dn+1 sin θ dn +...sin θ c ] =4d n 2 Ω [ sinπsin θ c ] =4d n 2 Ω [ 1 1 n ]=4d n(n1) Ω

Metrics