Abstract

In this paper we introduce the three-dimensional formulation of the OPTOS formalism, a matrix-based method that allows for the efficient simulation of non-coherent light propagation and absorption in thick textured sheets. As application examples, we calculate the absorptance of solar cells featuring textures on front and rear side with different feature sizes operating in different optical regimes. A discretization of polar and azimuth angle enables a three-dimensional description of systems with arbitrary surface textures. We present redistribution matrices for 3D surface textures, including pyramidal textures, binary crossed gratings and a Lambertian scatterer. The results of the OPTOS simulations for silicon sheets with different combinations of these surfaces are in accordance with both optical measurements and results based on established simulation methods like ray tracing. Using OPTOS, we show that the integration of a diffractive grating at the rear side of a silicon solar cell featuring a pyramidal front side results in absorption close to the Yablonovitch Limit enhancing the photocurrent density by 0.6 mA/cm2 for a 200 µm thick cell.

© 2015 Optical Society of America

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2015 (3)

J. Eisenlohr, N. Tucher, O. Höhn, H. Hauser, M. Peters, P. Kiefel, J. C. Goldschmidt, and B. Bläsi, “Matrix formalism for light propagation and absorption in thick textured optical sheets,” Opt. Express 23(11), A502–A518 (2015).
[Crossref] [PubMed]

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (4)

M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells—optimum parameters and loss mechanisms,” Prog. Photovolt. Res. Appl. 20(7), 862–873 (2012).
[Crossref]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

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]

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

2011 (4)

A. Mellor, I. Tobias, A. Marti, and A. Luque, “A numerical study of bi-periodic binary diffraction gratings for solar cell applications,” Sol. Energy Mater. Sol. Cells 95(12), 3527–3535 (2011).
[Crossref]

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]

S. C. Baker-Finch and K. R. McIntosh, “Reflection of normally incident light from silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 19(4), 406–416 (2011).
[Crossref]

S. Baker-Finch and K. R. McIntosh, “One-dimensional photogeneration profiles in silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 20, 1–11 (2011).

2009 (1)

A. K. Chu, J. S. Wang, Z. Y. Tsai, and C. K. Lee, “A simple and cost-effective approach for fabricating pyramids on crystalline silicon wafers,” Sol. Energy Mater. Sol. Cells 93(8), 1276–1280 (2009).
[Crossref]

1998 (1)

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

1997 (1)

1995 (2)

1990 (1)

P. A. Basore, “Numerical modeling of textured silicon solar cells using PC-1D,” IEEE Trans. Electron. Dev. 37(2), 337–343 (1990).
[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]

1983 (1)

P. Sheng, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579 (1983).
[Crossref]

1982 (1)

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
[Crossref]

Baker-Finch, S.

S. Baker-Finch and K. R. McIntosh, “One-dimensional photogeneration profiles in silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 20, 1–11 (2011).

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]

S. C. Baker-Finch and K. R. McIntosh, “Reflection of normally incident light from silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 19(4), 406–416 (2011).
[Crossref]

Basore, P. A.

P. A. Basore, “Numerical modeling of textured silicon solar cells using PC-1D,” IEEE Trans. Electron. Dev. 37(2), 337–343 (1990).
[Crossref]

Benick, J.

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

A. Mellor, H. Hauser, C. Wellens, J. Benick, J. Eisenlohr, M. Peters, A. Guttowski, I. Tobías, A. Martí, A. Luque, and B. Bläsi, “Nanoimprinted diffraction gratings for crystalline silicon solar cells: implementation, characterization and simulation,” Opt. Express 21(S2Suppl 2), A295–A304 (2013).
[Crossref] [PubMed]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

Bläsi, B.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

J. Eisenlohr, N. Tucher, O. Höhn, H. Hauser, M. Peters, P. Kiefel, J. C. Goldschmidt, and B. Bläsi, “Matrix formalism for light propagation and absorption in thick textured optical sheets,” Opt. Express 23(11), A502–A518 (2015).
[Crossref] [PubMed]

A. Mellor, H. Hauser, C. Wellens, J. Benick, J. Eisenlohr, M. Peters, A. Guttowski, I. Tobías, A. Martí, A. Luque, and B. Bläsi, “Nanoimprinted diffraction gratings for crystalline silicon solar cells: implementation, characterization and simulation,” Opt. Express 21(S2Suppl 2), A295–A304 (2013).
[Crossref] [PubMed]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells—optimum parameters and loss mechanisms,” Prog. Photovolt. Res. Appl. 20(7), 862–873 (2012).
[Crossref]

Brendel, R.

R. Brendel, “Sunrays. A versatile ray tracing program for the photovoltaic community,” in Proceedings of the 12th European Photovoltaic Solar Energy Conference1994, pp. 1339–1342.

Campbell, P.

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

Chu, A. K.

A. K. Chu, J. S. Wang, Z. Y. Tsai, and C. K. Lee, “A simple and cost-effective approach for fabricating pyramids on crystalline silicon wafers,” Sol. Energy Mater. Sol. Cells 93(8), 1276–1280 (2009).
[Crossref]

Drießen, M.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

Eisenlohr, J.

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

J. Eisenlohr, N. Tucher, O. Höhn, H. Hauser, M. Peters, P. Kiefel, J. C. Goldschmidt, and B. Bläsi, “Matrix formalism for light propagation and absorption in thick textured optical sheets,” Opt. Express 23(11), A502–A518 (2015).
[Crossref] [PubMed]

A. Mellor, H. Hauser, C. Wellens, J. Benick, J. Eisenlohr, M. Peters, A. Guttowski, I. Tobías, A. Martí, A. Luque, and B. Bläsi, “Nanoimprinted diffraction gratings for crystalline silicon solar cells: implementation, characterization and simulation,” Opt. Express 21(S2Suppl 2), A295–A304 (2013).
[Crossref] [PubMed]

Feldmann, F.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

Fromherz, T.

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Gaylord, T. K.

Goetzberger, A.

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

Goldschmidt, J. C.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

J. Eisenlohr, N. Tucher, O. Höhn, H. Hauser, M. Peters, P. Kiefel, J. C. Goldschmidt, and B. Bläsi, “Matrix formalism for light propagation and absorption in thick textured optical sheets,” Opt. Express 23(11), A502–A518 (2015).
[Crossref] [PubMed]

Graf, M.

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

Grann, E. B.

Green, M. A.

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

Guttowski, A.

Hartmann, P.

Hauser, H.

J. Eisenlohr, N. Tucher, O. Höhn, H. Hauser, M. Peters, P. Kiefel, J. C. Goldschmidt, and B. Bläsi, “Matrix formalism for light propagation and absorption in thick textured optical sheets,” Opt. Express 23(11), A502–A518 (2015).
[Crossref] [PubMed]

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

A. Mellor, H. Hauser, C. Wellens, J. Benick, J. Eisenlohr, M. Peters, A. Guttowski, I. Tobías, A. Martí, A. Luque, and B. Bläsi, “Nanoimprinted diffraction gratings for crystalline silicon solar cells: implementation, characterization and simulation,” Opt. Express 21(S2Suppl 2), A295–A304 (2013).
[Crossref] [PubMed]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells—optimum parameters and loss mechanisms,” Prog. Photovolt. Res. Appl. 20(7), 862–873 (2012).
[Crossref]

Heine, C.

Hermle, M.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells—optimum parameters and loss mechanisms,” Prog. Photovolt. Res. Appl. 20(7), 862–873 (2012).
[Crossref]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

Hingerl, K.

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Hohenester, U.

Höhn, O.

Jantsch, W.

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Jurek, M. P.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

Kiefel, P.

Lalanne, P.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

Lee, B. G.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

Lee, C. K.

A. K. Chu, J. S. Wang, Z. Y. Tsai, and C. K. Lee, “A simple and cost-effective approach for fabricating pyramids on crystalline silicon wafers,” Sol. Energy Mater. Sol. Cells 93(8), 1276–1280 (2009).
[Crossref]

Leiner, C.

Li, L.

Luque, A.

A. Mellor, H. Hauser, C. Wellens, J. Benick, J. Eisenlohr, M. Peters, A. Guttowski, I. Tobías, A. Martí, A. Luque, and B. Bläsi, “Nanoimprinted diffraction gratings for crystalline silicon solar cells: implementation, characterization and simulation,” Opt. Express 21(S2Suppl 2), A295–A304 (2013).
[Crossref] [PubMed]

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]

A. Mellor, I. Tobias, A. Marti, and A. Luque, “A numerical study of bi-periodic binary diffraction gratings for solar cell applications,” Sol. Energy Mater. Sol. Cells 95(12), 3527–3535 (2011).
[Crossref]

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]

A. Mellor, I. Tobias, A. Marti, and A. Luque, “A numerical study of bi-periodic binary diffraction gratings for solar cell applications,” Sol. Energy Mater. Sol. Cells 95(12), 3527–3535 (2011).
[Crossref]

Martí, A.

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]

S. C. Baker-Finch and K. R. McIntosh, “Reflection of normally incident light from silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 19(4), 406–416 (2011).
[Crossref]

S. Baker-Finch and K. R. McIntosh, “One-dimensional photogeneration profiles in silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 20, 1–11 (2011).

Meinhardt, G.

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Mellor, A.

A. Mellor, H. Hauser, C. Wellens, J. Benick, J. Eisenlohr, M. Peters, A. Guttowski, I. Tobías, A. Martí, A. Luque, and B. Bläsi, “Nanoimprinted diffraction gratings for crystalline silicon solar cells: implementation, characterization and simulation,” Opt. Express 21(S2Suppl 2), A295–A304 (2013).
[Crossref] [PubMed]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

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]

A. Mellor, I. Tobias, A. Marti, and A. Luque, “A numerical study of bi-periodic binary diffraction gratings for solar cell applications,” Sol. Energy Mater. Sol. Cells 95(12), 3527–3535 (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]

Milenkovic, N.

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

Moharam, M. G.

Morf, R. H.

Müller, C.

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

Nemitz, W.

Peters, M.

Pommet, D. A.

Rothemund, R.

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Rüdiger, M.

M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells—optimum parameters and loss mechanisms,” Prog. Photovolt. Res. Appl. 20(7), 862–873 (2012).
[Crossref]

Schweitzer, S.

Sheng, P.

P. Sheng, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579 (1983).
[Crossref]

Sommer, C.

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, and A. Luque, “A numerical study of bi-periodic binary diffraction gratings for solar cell applications,” Sol. Energy Mater. Sol. Cells 95(12), 3527–3535 (2011).
[Crossref]

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]

Tobías, I.

Tsai, Z. Y.

A. K. Chu, J. S. Wang, Z. Y. Tsai, and C. K. Lee, “A simple and cost-effective approach for fabricating pyramids on crystalline silicon wafers,” Sol. Energy Mater. Sol. Cells 93(8), 1276–1280 (2009).
[Crossref]

Tucher, N.

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[Crossref]

J. Eisenlohr, N. Tucher, O. Höhn, H. Hauser, M. Peters, P. Kiefel, J. C. Goldschmidt, and B. Bläsi, “Matrix formalism for light propagation and absorption in thick textured optical sheets,” Opt. Express 23(11), A502–A518 (2015).
[Crossref] [PubMed]

Umundum, T.

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Wang, J. S.

A. K. Chu, J. S. Wang, Z. Y. Tsai, and C. K. Lee, “A simple and cost-effective approach for fabricating pyramids on crystalline silicon wafers,” Sol. Energy Mater. Sol. Cells 93(8), 1276–1280 (2009).
[Crossref]

Wellens, C.

Wenzl, F. P.

Yablonovitch, E.

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. Sheng, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579 (1983).
[Crossref]

Energy Procedia (2)

H. Hauser, A. Mellor, A. Guttowski, C. Wellens, J. Benick, C. Müller, M. Hermle, and B. Bläsi, “Diffractive backside structures via nanoimprint lithography,” Energy Procedia 27, 337–342 (2012).
[Crossref]

N. Tucher, J. Eisenlohr, H. Hauser, J. Benick, M. Graf, C. Müller, M. Hermle, J. C. Goldschmidt, and B. Bläsi, “Crystalline silicon solar cells with enhanced light trapping via rear side diffraction grating,” Energy Procedia 77, 253–262 (2015).
[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. Antenn. Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
[Crossref]

IEEE Trans. Electron. Dev. (1)

P. A. Basore, “Numerical modeling of textured silicon solar cells using PC-1D,” IEEE Trans. Electron. Dev. 37(2), 337–343 (1990).
[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]

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P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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Prog. Photovolt. Res. Appl. (5)

S. Baker-Finch and K. R. McIntosh, “One-dimensional photogeneration profiles in silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 20, 1–11 (2011).

M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells—optimum parameters and loss mechanisms,” Prog. Photovolt. Res. Appl. 20(7), 862–873 (2012).
[Crossref]

S. C. Baker-Finch and K. R. McIntosh, “Reflection of normally incident light from silicon solar cells with pyramidal texture,” Prog. Photovolt. Res. Appl. 19(4), 406–416 (2011).
[Crossref]

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]

R. Rothemund, T. Umundum, G. Meinhardt, K. Hingerl, T. Fromherz, and W. Jantsch, “Light trapping in pyramidally textured crystalline silicon solar cells using back-side diffractive gratings,” Prog. Photovolt. Res. Appl. 21, 747–753 (2012).

Sol. Energy Mater. Sol. Cells (3)

A. K. Chu, J. S. Wang, Z. Y. Tsai, and C. K. Lee, “A simple and cost-effective approach for fabricating pyramids on crystalline silicon wafers,” Sol. Energy Mater. Sol. Cells 93(8), 1276–1280 (2009).
[Crossref]

A. Mellor, I. Tobias, A. Marti, and A. Luque, “A numerical study of bi-periodic binary diffraction gratings for solar cell applications,” Sol. Energy Mater. Sol. Cells 95(12), 3527–3535 (2011).
[Crossref]

J. Eisenlohr, B. G. Lee, J. Benick, F. Feldmann, M. Drießen, N. Milenkovic, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Rear side sphere gratings for improved light trapping in crystalline silicon single junction and silicon-based tandem solar cells,” Sol. Energy Mater. Sol. Cells 142, 60–65 (2015).
[Crossref]

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R. Branke and A. Heimsath, “Raytrace3D power tower - a novel optical model for central receiver systems,” in SolarPACES 2010,16th Solar Power And Chemical Energy Systems International Symposium (2010).

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“PV-Lighthouse,” www.pvlighthouse.com.au .

A. Taflove and S. C. Hagness, Computational electrodynamics. The Finite-Difference Time-Domain Method, 3rd (Artech House, 2005).

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

R. Brendel, “Sunrays. A versatile ray tracing program for the photovoltaic community,” in Proceedings of the 12th European Photovoltaic Solar Energy Conference1994, pp. 1339–1342.

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Figures (11)

Fig. 1
Fig. 1 Angle discretization for three-dimensional OPTOS calculations. Figure a) shows the discretized surface of the three-dimensional half sphere and figure b) its projection onto the surface plane. The channels vary in polar and azimuth angle. The discretization is chosen in a way to exhibit angle channels with equal area and symmetry elements as highlighted in red and depicted in figure b). These symmetry elements can be used to reduce the number of calculations.
Fig. 2
Fig. 2 Results of the azimuth angle convergence analysis. The total absorptance of a system with planar front and diffractive grating at the rear side was calculated for different numbers of azimuth angle channels and compared with an exact discretization.
Fig. 3
Fig. 3 Redistribution matrix for a regular inverted pyramid front side at wavelength of 1100 nm calculated via ray tracing. (a) shows a sketch of light interaction with the silicon-air surface (b) shows a simplified version of the matrix where all entries of different φ-values have been summarized only into the dependency of θ.
Fig. 4
Fig. 4 Redistribution matrix for a Lambertian scatterer calculated analytically. (a) schematically shows the interaction of light at a Lambertian surface. (b) shows the complete redistribution matrix with all polar and azimuth angles. (c) shows the simplified version where all entries of different φ-values have been summed up for each given angle θ.
Fig. 5
Fig. 5 Redistribution matrix for a binary crossed grating structure on the rear side with a period of 1 µm at a wavelength of 1100 nm. (a) shows a sketch of light interaction with the silicon-air surface (b) shows a simplified version of the matrix where all entries of different φ-values have been summarized only into the dependency of θ.
Fig. 6
Fig. 6 Absorptance of a silicon wafer with a planar front side and a binary crossed grating rear side. Since the measurement was carried out with an incoming angle of 8°, we used this angle for the simulation as well.
Fig. 7
Fig. 7 Simulated and measured absorptance of a silicon wafer with a pyramidal front side texture (inverted (inv), upright (up), random (rand) or regular (reg) pyramids) and a planar rear side.
Fig. 8
Fig. 8 Simulated (closed symbols) and measured (open symbols) absorptance for a silicon sheet with pyramidal front side and planar reflector (orange) or binary crossed diffraction grating (green) at the rear side. The peak of the absorptance gain (blue, right scale) due to the rear side grating occurs at a similar wavelength in the simulation as in the experiment, whereas the total gain is larger for the simulated system.
Fig. 9
Fig. 9 Simulated generation profile for a silicon sheet with pyramidal front side and planar rear side. In OPTOS the front surface is considered as effective two-dimensional surface that redistributes light. This leads to the profiles G(z). By geometrical considerations the data can be transformed to the profiles G(ζ), where ζ denotes the distance to the pyramid surface. Data of G(ζ) obtained with OPTOS is in good agreement with data obtained with the Sentaurus Device ray tracing tool.
Fig. 10
Fig. 10 Absorptance calculated with OPTOS of a 200 µm thick silicon solar cell with inverted pyramidal front with double layer ARC and different rear surface structures combined with a perfect reflector. The diffraction grating (orange) and Lambertian (dark green) rear lead to an absorptance gain (blue, right scale) due to increased light trapping compared to the planar rear (light green). Both systems approach the Yablonovitch limit for a system with zero reflectance.
Fig. 11
Fig. 11 Scheme of the power redistribution and the nomenclature for the matrices and the power distribution vector.

Equations (11)

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N azimuth = c azimuth r polar
B,C=( ( θ 1 , φ 1 )( θ 1 , φ 1 ) ( θ 1 , φ 2 )( θ 1 , φ 1 ) ( θ n , φ m )( θ 1 , φ 1 ) ( θ 1 , φ 1 )( θ 1 , φ 2 ) ( θ 1 , φ 2 )( θ 1 , φ 2 ) ( θ n , φ m )( θ 1 , φ 2 ) ( θ 1 , φ 1 )( θ n , φ m ) ( θ 1 , φ 2 )( θ n , φ m ) ( θ n , φ m )( θ n , φ m ) )
v=( p( θ 1 , φ 1 ) p( θ 1 , φ 2 ) p( θ 1 , φ m ) p( θ 2 , φ 1 ) p( θ 2 , φ 2 ) p( θ n , φ m ) );D=( e αd/cos θ 1 0 0 e αd/cos θ n )
v 2 =DCDv ' 0 ;v ' 2i = (BDCD) i v ' 0 ; v 2i+1 = (DBDC) i Dv ' 0 v ' 2i+1 = (CDBD) i CDv ' 0 ; v 2i = (DCDB) i1 v 2 P ' i = j (v ' i ) j ; P i = j ( v i ) j
Abs=Ab s down +Ab s up =( i=0 i max (P ' 2i P 2i+1 ) )+( i=1 i max (P ' 2i1 P 2i ) )
Abs= P 0 P 1 +P ' 1 P 2 +P ' 2 P 3 +P ' 3 P 4 +... = j ( i=0 (BDCD) i v 0 i=0 (DBDC) i D v 0 + i=0 (CDBD) i CD v 0 i=0 (DCDB) i DCD v 0 ) j = j ( (IBDCD) 1 v 0 (IDBDC) 1 D v 0 + (ICDBD) 1 CD v 0 (IDCDB) 1 DCD v 0 ) j
D(z)=( e αz/cos θ 1 0 0 e αz/cos θ n )
A dn (z)=( 1 e αz/cos θ 1 0 0 1 e αz/cos θ n ); A up (z)=( 1 e α(dz)/cos θ 1 0 0 1 e α(dz)/cos θ n )
Ab s dn (z)= j ( i=0 i max A dn (z)v ' 2i ) j = j ( A dn (z) i=0 i max v ' 2i ) j
Ab s up (z)= j ( i=0 i max A up (z)v ' 2i+1 ) j = j ( A up (z) i=0 i max v ' 2i+1 ) j
Abs(z)=Ab s dn (z)+Ab s up (0)Ab s up (z)

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