Abstract

This paper addresses a number of important and underlying theoretical and practical questions regarding the recently developed wave optics light-trapping theory for solar cells. We provide a rigorous and complete justification of its mathematical validity, using the second law of thermodynamics as well as fundamental properties of resonant scattering as described by the temporal coupled-mode theory. For maximal absorption, all optical modes supported by a solar cell structure must couple and only couple to the incident channel of sunlight radiation. For the first time to the best of our knowledge, we derive the ultimate limit of light trapping, which depends positively on the number of optical modes and hence on the periodicity of the structure. The ultimate limit is reached when every mode in the structure can couple only to the incident channel. Furthermore, we predict the theoretical optimal operating regimes of nanophotonic solar cells under practical scenarios. Our work reveals significant gaps between state-of-the-art nanophotonic solar cell designs and the ultimate limit, pointing to important future opportunities for nanophotonic light management for efficiency enhancement and cost reduction of solar cells.

© 2019 Optical Society of America

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References

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2017 (2)

S. Buddhiraju and S. Fan, “Theory of solar cell light trapping through a nonequilibrium Green’s function formulation of Maxwell’s equations,” Phys. Rev. B 96, 035304 (2017).
[Crossref]

G. Y. Abdel-Latif, M. F. O. Hameed, M. Hussein, M. A. Razzak, and S. S. Obayya, “Electrical characteristics of funnel-shaped silicon nanowire solar cells,” J. Photonics Energy 7, 047501 (2017).
[Crossref]

2016 (2)

Z. Wang, Z. Wang, and Z. Yu, “Photon management with index-near-zero materials,” Appl. Phys. Lett. 109, 051101 (2016).
[Crossref]

P. Yu, J. Wu, S. Liu, J. Xiong, C. Jagadish, and Z. M. Wang, “Design and fabrication of silicon nanowires towards efficient solar cells,” Nano Today 11, 704–737 (2016).
[Crossref]

2015 (1)

E. D. Kosten, B. K. Newman, J. V. Lloyd, A. Polman, and H. A. Atwater, “Limiting light escape angle in silicon photovoltaics: ideal and realistic cells,” IEEE J. Photovoltaics 5, 61–69 (2015).

2014 (6)

K. X. Wang, Z. Yu, V. Liu, M. L. Brongersma, T. F. Jaramillo, and S. Fan, “Nearly total solar absorption in ultrathin nanostructured iron oxide for efficient photoelectrochemical water splitting,” ACS Photon. 1, 235–240 (2014).
[Crossref]

Z. Yu, S. Sandhu, and S. Fan, “Efficiency above the Shockley-Queisser limit by using nanophotonic effects to create multiple effective bandgaps with a single semiconductor,” Nano Lett. 14, 66–70 (2014).
[Crossref]

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4, 175 (2014).
[Crossref]

M. L. Brongersma, Y. Cui, and S. Fan, “Light management for photovoltaics using high-index nanostructures,” Nat. Mater. 13, 451–460 (2014).
[Crossref]

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, A. Raman, Y. Cui, and S. Fan, “Light trapping in photonic crystals,” Energy Environ. Sci. 7, 2725–2738 (2014).
[Crossref]

2013 (9)

V. K. Narasimhan and Y. Cui, “Nanostructures for photon management in solar cells,” Nanophotonics 2, 187–210 (2013).
[Crossref]

C. Wang, S. Yu, W. Chen, and C. Sun, “Highly efficient light-trapping structure design inspired by natural evolution,” Sci. Rep. 3, 1025 (2013).
[Crossref]

M. Zeman, O. Isabella, S. Solntsev, and K. Jäger, “Modelling of thin-film silicon solar cells,” Sol. Energ. Mat. Sol. Cells 119, 94–111 (2013).
[Crossref]

R. A. Pala, J. S. Q. Liu, E. S. Barnard, D. Askarov, E. C. Garnett, S. Fan, and M. L. Brongersma, “Optimization of non-periodic plasmonic light-trapping layers for thin-film solar cells,” Nat. Commun. 4, 2095 (2013).
[Crossref]

G. Gomard, R. Peretti, E. Drouard, X. Meng, and C. Seassal, “Photonic crystals and optical mode engineering for thin film photovoltaics,” Opt. Express 21, A515–A527 (2013).
[Crossref]

S. Wang, B. D. Weil, Y. Li, K. X. Wang, E. Garnett, S. Fan, and Y. Cui, “Large-area free-standing ultrathin single-crystal silicon as processable materials,” Nano Lett. 13, 4393–4398 (2013).
[Crossref]

A. Naqavi, F.-J. Haug, C. Battaglia, H. P. Herzig, and C. Ballif, “Light trapping in solar cells at the extreme coupling limit,” J. Opt. Soc. Am. B 30, 13–20 (2013).
[Crossref]

S. A. Mann and E. C. Garnett, “Extreme light absorption in thin semiconductor films wrapped around metal nanowires,” Nano Lett. 13, 3173–3178 (2013).
[Crossref]

E. D. Kosten, J. H. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

2012 (13)

M. A. Green, “Time-asymmetric photovoltaics,” Nano Lett. 12, 5985–5988 (2012).
[Crossref]

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, 2790–2797 (2012).
[Crossref]

S. Mokkapati and K. R. Catchpole, “Nanophotonic light trapping in solar cells,” J. Appl. Phys. 112, 101101 (2012).
[Crossref]

R. Peretti, G. Gomard, C. Seassal, X. Letartre, and E. Drouard, “Modal approach for tailoring the absorption in a photonic crystal membrane,” J. Appl. Phys. 111, 123114 (2012).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, Y. Cui, and S. Fan, “Absorption enhancement in ultrathin crystalline silicon solar cells with antireflection and light-trapping nanocone gratings,” Nano Lett. 12, 1616–1619 (2012).
[Crossref]

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

A. Bozzola, M. Liscidini, and L. C. Andreani, “Photonic light-trapping versus Lambertian limits in thin film silicon solar cells with 1D and 2D periodic patterns,” Opt. Express 20, A224–A244 (2012).
[Crossref]

H. Tan, R. Santbergen, A. H. Smets, and M. Zeman, “Plasmonic light trapping in thin-film silicon solar cells with improved self-assembled silver nanoparticles,” Nano Lett. 12, 4070–4076 (2012).
[Crossref]

A. Niv, M. Gharghi, C. Gladden, O. D. Miller, and X. Zhang, “Near-field electromagnetic theory for thin solar cells,” Phys. Rev. Lett. 109, 138701 (2012).
[Crossref]

K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. 11, 1017–1022 (2012).
[Crossref]

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11, 174–177 (2012).
[Crossref]

D. M. Callahan, J. N. Munday, and H. A. Atwater, “Solar cell light trapping beyond the ray optic limit,” Nano Lett. 12, 214–218 (2012).
[Crossref]

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

2011 (3)

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

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

M. A. Green, “Enhanced evanescent mode light trapping in organic solar cells and other low index optoelectronic devices,” Prog. Photovolt. 19, 473–477 (2011).
[Crossref]

2010 (2)

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

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

2008 (3)

J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, and D. W. Prather, “Thin film silicon solar cell design based on photonic crystal and diffractive grating structures,” Opt. Express 16, 15238–15248 (2008).
[Crossref]

C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

M. A. Green, “Self-consistent optical parameters of intrinsic silicon at 300 k including temperature coefficients,” Solar Energy Mater. Sol. Cells 92, 1305–1310 (2008).
[Crossref]

2007 (1)

2004 (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

1997 (1)

1986 (1)

P. Campbell and M. A. Green, “The limiting efficiency of silicon solar cells under concentrated sunlight,” IEEE Trans. Electron. Devices 33, 234 (1986).

1983 (1)

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[Crossref]

1982 (2)

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

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Dev. 29, 300–350 (1982).
[Crossref]

Abdel-Latif, G. Y.

G. Y. Abdel-Latif, M. F. O. Hameed, M. Hussein, M. A. Razzak, and S. S. Obayya, “Electrical characteristics of funnel-shaped silicon nanowire solar cells,” J. Photonics Energy 7, 047501 (2017).
[Crossref]

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, 2790–2797 (2012).
[Crossref]

Andreani, L. C.

Askarov, D.

R. A. Pala, J. S. Q. Liu, E. S. Barnard, D. Askarov, E. C. Garnett, S. Fan, and M. L. Brongersma, “Optimization of non-periodic plasmonic light-trapping layers for thin-film solar cells,” Nat. Commun. 4, 2095 (2013).
[Crossref]

Atwater, H. A.

E. D. Kosten, B. K. Newman, J. V. Lloyd, A. Polman, and H. A. Atwater, “Limiting light escape angle in silicon photovoltaics: ideal and realistic cells,” IEEE J. Photovoltaics 5, 61–69 (2015).

E. D. Kosten, J. H. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

D. M. Callahan, J. N. Munday, and H. A. Atwater, “Solar cell light trapping beyond the ray optic limit,” Nano Lett. 12, 214–218 (2012).
[Crossref]

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11, 174–177 (2012).
[Crossref]

Atwater, J. H.

E. D. Kosten, J. H. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

Ballif, C.

A. Naqavi, F.-J. Haug, C. Battaglia, H. P. Herzig, and C. Ballif, “Light trapping in solar cells at the extreme coupling limit,” J. Opt. Soc. Am. B 30, 13–20 (2013).
[Crossref]

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, 2790–2797 (2012).
[Crossref]

Bao, X.

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

Barnard, E. S.

R. A. Pala, J. S. Q. Liu, E. S. Barnard, D. Askarov, E. C. Garnett, S. Fan, and M. L. Brongersma, “Optimization of non-periodic plasmonic light-trapping layers for thin-film solar cells,” Nat. Commun. 4, 2095 (2013).
[Crossref]

Barnett, A.

Barybin, A. A.

A. A. Barybin and V. A. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory: Application to Guided-Wave Optics (Rinton Press, 2002).

Battaglia, C.

A. Naqavi, F.-J. Haug, C. Battaglia, H. P. Herzig, and C. Ballif, “Light trapping in solar cells at the extreme coupling limit,” J. Opt. Soc. Am. B 30, 13–20 (2013).
[Crossref]

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, 2790–2797 (2012).
[Crossref]

Bermel, P.

Bloch, A. N.

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
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[Crossref]

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C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

Polman, A.

E. D. Kosten, B. K. Newman, J. V. Lloyd, A. Polman, and H. A. Atwater, “Limiting light escape angle in silicon photovoltaics: ideal and realistic cells,” IEEE J. Photovoltaics 5, 61–69 (2015).

E. D. Kosten, J. H. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11, 174–177 (2012).
[Crossref]

Prather, D. W.

Priolo, F.

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
[Crossref]

Raman, A.

K. X. Wang, Z. Yu, V. Liu, A. Raman, Y. Cui, and S. Fan, “Light trapping in photonic crystals,” Energy Environ. Sci. 7, 2725–2738 (2014).
[Crossref]

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

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

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

Rau, U.

C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

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G. Y. Abdel-Latif, M. F. O. Hameed, M. Hussein, M. A. Razzak, and S. S. Obayya, “Electrical characteristics of funnel-shaped silicon nanowire solar cells,” J. Photonics Energy 7, 047501 (2017).
[Crossref]

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K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. 11, 1017–1022 (2012).
[Crossref]

Rockstuhl, C.

C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

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Z. Yu, S. Sandhu, and S. Fan, “Efficiency above the Shockley-Queisser limit by using nanophotonic effects to create multiple effective bandgaps with a single semiconductor,” Nano Lett. 14, 66–70 (2014).
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Santbergen, R.

H. Tan, R. Santbergen, A. H. Smets, and M. Zeman, “Plasmonic light trapping in thin-film silicon solar cells with improved self-assembled silver nanoparticles,” Nano Lett. 12, 4070–4076 (2012).
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G. Gomard, R. Peretti, E. Drouard, X. Meng, and C. Seassal, “Photonic crystals and optical mode engineering for thin film photovoltaics,” Opt. Express 21, A515–A527 (2013).
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R. Peretti, G. Gomard, C. Seassal, X. Letartre, and E. Drouard, “Modal approach for tailoring the absorption in a photonic crystal membrane,” J. Appl. Phys. 111, 123114 (2012).
[Crossref]

Sheng, P.

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[Crossref]

Sheng, X.

Shi, S.

Smets, A. H.

H. Tan, R. Santbergen, A. H. Smets, and M. Zeman, “Plasmonic light trapping in thin-film silicon solar cells with improved self-assembled silver nanoparticles,” Nano Lett. 12, 4070–4076 (2012).
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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, 2790–2797 (2012).
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M. Zeman, O. Isabella, S. Solntsev, and K. Jäger, “Modelling of thin-film silicon solar cells,” Sol. Energ. Mat. Sol. Cells 119, 94–111 (2013).
[Crossref]

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P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[Crossref]

Stuart, H. R.

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

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C. Wang, S. Yu, W. Chen, and C. Sun, “Highly efficient light-trapping structure design inspired by natural evolution,” Sci. Rep. 3, 1025 (2013).
[Crossref]

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H. Tan, R. Santbergen, A. H. Smets, and M. Zeman, “Plasmonic light trapping in thin-film silicon solar cells with improved self-assembled silver nanoparticles,” Nano Lett. 12, 4070–4076 (2012).
[Crossref]

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D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

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C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

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C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
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S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Vynck, K.

K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. 11, 1017–1022 (2012).
[Crossref]

Wang, C.

C. Wang, S. Yu, W. Chen, and C. Sun, “Highly efficient light-trapping structure design inspired by natural evolution,” Sci. Rep. 3, 1025 (2013).
[Crossref]

Wang, K. X.

K. X. Wang, Z. Yu, V. Liu, M. L. Brongersma, T. F. Jaramillo, and S. Fan, “Nearly total solar absorption in ultrathin nanostructured iron oxide for efficient photoelectrochemical water splitting,” ACS Photon. 1, 235–240 (2014).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, A. Raman, Y. Cui, and S. Fan, “Light trapping in photonic crystals,” Energy Environ. Sci. 7, 2725–2738 (2014).
[Crossref]

S. Wang, B. D. Weil, Y. Li, K. X. Wang, E. Garnett, S. Fan, and Y. Cui, “Large-area free-standing ultrathin single-crystal silicon as processable materials,” Nano Lett. 13, 4393–4398 (2013).
[Crossref]

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, Y. Cui, and S. Fan, “Absorption enhancement in ultrathin crystalline silicon solar cells with antireflection and light-trapping nanocone gratings,” Nano Lett. 12, 1616–1619 (2012).
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K. X. Wang, Y. Guo, and Z. Yu, Photonic Crystal Metasurface Optoelectronics (Elsevier, 2019), chap. Light trapping in photonic structures.

Wang, S.

S. Wang, B. D. Weil, Y. Li, K. X. Wang, E. Garnett, S. Fan, and Y. Cui, “Large-area free-standing ultrathin single-crystal silicon as processable materials,” Nano Lett. 13, 4393–4398 (2013).
[Crossref]

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
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Z. Wang, Z. Wang, and Z. Yu, “Photon management with index-near-zero materials,” Appl. Phys. Lett. 109, 051101 (2016).
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Z. Wang, Z. Wang, and Z. Yu, “Photon management with index-near-zero materials,” Appl. Phys. Lett. 109, 051101 (2016).
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W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

Wang, Z. M.

P. Yu, J. Wu, S. Liu, J. Xiong, C. Jagadish, and Z. M. Wang, “Design and fabrication of silicon nanowires towards efficient solar cells,” Nano Today 11, 704–737 (2016).
[Crossref]

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C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

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S. Wang, B. D. Weil, Y. Li, K. X. Wang, E. Garnett, S. Fan, and Y. Cui, “Large-area free-standing ultrathin single-crystal silicon as processable materials,” Nano Lett. 13, 4393–4398 (2013).
[Crossref]

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K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. 11, 1017–1022 (2012).
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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

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D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

Wu, J.

P. Yu, J. Wu, S. Liu, J. Xiong, C. Jagadish, and Z. M. Wang, “Design and fabrication of silicon nanowires towards efficient solar cells,” Nano Today 11, 704–737 (2016).
[Crossref]

Xiong, J.

P. Yu, J. Wu, S. Liu, J. Xiong, C. Jagadish, and Z. M. Wang, “Design and fabrication of silicon nanowires towards efficient solar cells,” Nano Today 11, 704–737 (2016).
[Crossref]

Yablonovitch, E.

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4, 175 (2014).
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E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907 (1982).
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E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Dev. 29, 300–350 (1982).
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Yao, Y.

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

Ye, Y.

D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd ed. (Springer, 2008).

Yu, P.

P. Yu, J. Wu, S. Liu, J. Xiong, C. Jagadish, and Z. M. Wang, “Design and fabrication of silicon nanowires towards efficient solar cells,” Nano Today 11, 704–737 (2016).
[Crossref]

Yu, S.

C. Wang, S. Yu, W. Chen, and C. Sun, “Highly efficient light-trapping structure design inspired by natural evolution,” Sci. Rep. 3, 1025 (2013).
[Crossref]

Yu, Z.

Z. Wang, Z. Wang, and Z. Yu, “Photon management with index-near-zero materials,” Appl. Phys. Lett. 109, 051101 (2016).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, A. Raman, Y. Cui, and S. Fan, “Light trapping in photonic crystals,” Energy Environ. Sci. 7, 2725–2738 (2014).
[Crossref]

Z. Yu, S. Sandhu, and S. Fan, “Efficiency above the Shockley-Queisser limit by using nanophotonic effects to create multiple effective bandgaps with a single semiconductor,” Nano Lett. 14, 66–70 (2014).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, M. L. Brongersma, T. F. Jaramillo, and S. Fan, “Nearly total solar absorption in ultrathin nanostructured iron oxide for efficient photoelectrochemical water splitting,” ACS Photon. 1, 235–240 (2014).
[Crossref]

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

K. X. Wang, Z. Yu, V. Liu, Y. Cui, and S. Fan, “Absorption enhancement in ultrathin crystalline silicon solar cells with antireflection and light-trapping nanocone gratings,” Nano Lett. 12, 1616–1619 (2012).
[Crossref]

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

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

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

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

K. X. Wang, Y. Guo, and Z. Yu, Photonic Crystal Metasurface Optoelectronics (Elsevier, 2019), chap. Light trapping in photonic structures.

Zeman, M.

M. Zeman, O. Isabella, S. Solntsev, and K. Jäger, “Modelling of thin-film silicon solar cells,” Sol. Energ. Mat. Sol. Cells 119, 94–111 (2013).
[Crossref]

H. Tan, R. Santbergen, A. H. Smets, and M. Zeman, “Plasmonic light trapping in thin-film silicon solar cells with improved self-assembled silver nanoparticles,” Nano Lett. 12, 4070–4076 (2012).
[Crossref]

Zeng, L.

Zhang, X.

A. Niv, M. Gharghi, C. Gladden, O. D. Miller, and X. Zhang, “Near-field electromagnetic theory for thin solar cells,” Phys. Rev. Lett. 109, 138701 (2012).
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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, 2790–2797 (2012).
[Crossref]

ACS Photon. (1)

K. X. Wang, Z. Yu, V. Liu, M. L. Brongersma, T. F. Jaramillo, and S. Fan, “Nearly total solar absorption in ultrathin nanostructured iron oxide for efficient photoelectrochemical water splitting,” ACS Photon. 1, 235–240 (2014).
[Crossref]

Adv. Energy Mater. (1)

D. Liang, Y. Huo, Y. Kang, K. X. Wang, A. Gu, M. Tan, Z. Yu, S. Li, J. Jia, X. Bao, S. Wang, Y. Yao, H.-S. P. Wong, S. Fan, Y. Cui, and J. S. Harris, “Optical absorption enhancement in freestanding GaAs thin film nanopyramid arrays,” Adv. Energy Mater. 2, 1254–1260 (2012).
[Crossref]

Appl. Phys. Lett. (3)

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

Z. Wang, Z. Wang, and Z. Yu, “Photon management with index-near-zero materials,” Appl. Phys. Lett. 109, 051101 (2016).
[Crossref]

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[Crossref]

Energy Environ. Sci. (1)

K. X. Wang, Z. Yu, V. Liu, A. Raman, Y. Cui, and S. Fan, “Light trapping in photonic crystals,” Energy Environ. Sci. 7, 2725–2738 (2014).
[Crossref]

IEEE J. Photovolt. (1)

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4, 175 (2014).
[Crossref]

IEEE J. Photovoltaics (1)

E. D. Kosten, B. K. Newman, J. V. Lloyd, A. Polman, and H. A. Atwater, “Limiting light escape angle in silicon photovoltaics: ideal and realistic cells,” IEEE J. Photovoltaics 5, 61–69 (2015).

IEEE J. Quantum Electron. (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

IEEE Trans. Electron Dev. (1)

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron Dev. 29, 300–350 (1982).
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IEEE Trans. Electron. Devices (1)

P. Campbell and M. A. Green, “The limiting efficiency of silicon solar cells under concentrated sunlight,” IEEE Trans. Electron. Devices 33, 234 (1986).

J. Appl. Phys. (2)

S. Mokkapati and K. R. Catchpole, “Nanophotonic light trapping in solar cells,” J. Appl. Phys. 112, 101101 (2012).
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R. Peretti, G. Gomard, C. Seassal, X. Letartre, and E. Drouard, “Modal approach for tailoring the absorption in a photonic crystal membrane,” J. Appl. Phys. 111, 123114 (2012).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Photonics Energy (1)

G. Y. Abdel-Latif, M. F. O. Hameed, M. Hussein, M. A. Razzak, and S. S. Obayya, “Electrical characteristics of funnel-shaped silicon nanowire solar cells,” J. Photonics Energy 7, 047501 (2017).
[Crossref]

Light Sci. Appl. (1)

E. D. Kosten, J. H. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
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Nano Lett. (7)

M. A. Green, “Time-asymmetric photovoltaics,” Nano Lett. 12, 5985–5988 (2012).
[Crossref]

Z. Yu, S. Sandhu, and S. Fan, “Efficiency above the Shockley-Queisser limit by using nanophotonic effects to create multiple effective bandgaps with a single semiconductor,” Nano Lett. 14, 66–70 (2014).
[Crossref]

K. X. Wang, Z. Yu, V. Liu, Y. Cui, and S. Fan, “Absorption enhancement in ultrathin crystalline silicon solar cells with antireflection and light-trapping nanocone gratings,” Nano Lett. 12, 1616–1619 (2012).
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D. M. Callahan, J. N. Munday, and H. A. Atwater, “Solar cell light trapping beyond the ray optic limit,” Nano Lett. 12, 214–218 (2012).
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S. Wang, B. D. Weil, Y. Li, K. X. Wang, E. Garnett, S. Fan, and Y. Cui, “Large-area free-standing ultrathin single-crystal silicon as processable materials,” Nano Lett. 13, 4393–4398 (2013).
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S. A. Mann and E. C. Garnett, “Extreme light absorption in thin semiconductor films wrapped around metal nanowires,” Nano Lett. 13, 3173–3178 (2013).
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H. Tan, R. Santbergen, A. H. Smets, and M. Zeman, “Plasmonic light trapping in thin-film silicon solar cells with improved self-assembled silver nanoparticles,” Nano Lett. 12, 4070–4076 (2012).
[Crossref]

Nano Today (1)

P. Yu, J. Wu, S. Liu, J. Xiong, C. Jagadish, and Z. M. Wang, “Design and fabrication of silicon nanowires towards efficient solar cells,” Nano Today 11, 704–737 (2016).
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Nanophotonics (1)

V. K. Narasimhan and Y. Cui, “Nanostructures for photon management in solar cells,” Nanophotonics 2, 187–210 (2013).
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Nat. Commun. (1)

R. A. Pala, J. S. Q. Liu, E. S. Barnard, D. Askarov, E. C. Garnett, S. Fan, and M. L. Brongersma, “Optimization of non-periodic plasmonic light-trapping layers for thin-film solar cells,” Nat. Commun. 4, 2095 (2013).
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Nat. Mater. (3)

K. Vynck, M. Burresi, F. Riboli, and D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nat. Mater. 11, 1017–1022 (2012).
[Crossref]

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11, 174–177 (2012).
[Crossref]

M. L. Brongersma, Y. Cui, and S. Fan, “Light management for photovoltaics using high-index nanostructures,” Nat. Mater. 13, 451–460 (2014).
[Crossref]

Nat. Nanotechnol. (1)

F. Priolo, T. Gregorkiewicz, M. Galli, and T. F. Krauss, “Silicon nanostructures for photonics and photovoltaics,” Nat. Nanotechnol. 9, 19–32 (2014).
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Opt. Express (6)

Phys. Rev. B (1)

S. Buddhiraju and S. Fan, “Theory of solar cell light trapping through a nonequilibrium Green’s function formulation of Maxwell’s equations,” Phys. Rev. B 96, 035304 (2017).
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Phys. Rev. Lett. (2)

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

A. Niv, M. Gharghi, C. Gladden, O. D. Miller, and X. Zhang, “Near-field electromagnetic theory for thin solar cells,” Phys. Rev. Lett. 109, 138701 (2012).
[Crossref]

Phys. Status Solidi A (1)

C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz, C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer, and U. Rau, “Directional selectivity and ultra-light-trapping in solar cells,” Phys. Status Solidi A 205, 2831–2843 (2008).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
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Prog. Photovolt. (1)

M. A. Green, “Enhanced evanescent mode light trapping in organic solar cells and other low index optoelectronic devices,” Prog. Photovolt. 19, 473–477 (2011).
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Sci. Rep. (1)

C. Wang, S. Yu, W. Chen, and C. Sun, “Highly efficient light-trapping structure design inspired by natural evolution,” Sci. Rep. 3, 1025 (2013).
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Sol. Energ. Mat. Sol. Cells (1)

M. Zeman, O. Isabella, S. Solntsev, and K. Jäger, “Modelling of thin-film silicon solar cells,” Sol. Energ. Mat. Sol. Cells 119, 94–111 (2013).
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M. A. Green, “Self-consistent optical parameters of intrinsic silicon at 300 k including temperature coefficients,” Solar Energy Mater. Sol. Cells 92, 1305–1310 (2008).
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S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd ed. (Springer, 2008).

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K. X. Wang, Y. Guo, and Z. Yu, Photonic Crystal Metasurface Optoelectronics (Elsevier, 2019), chap. Light trapping in photonic structures.

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Figures (6)

Fig. 1.
Fig. 1. Ultimate light-trapping limit F in c-Si versus wavelength for d = 5 μm and L = 10 μm .
Fig. 2.
Fig. 2. Ultimate light-trapping limit F in c-Si versus d for a wavelength of 1100 nm and L = 10 μm .
Fig. 3.
Fig. 3. Ultimate light-trapping limit F in c-Si versus L for a wavelength of 1100 nm and d = 5 μm .
Fig. 4.
Fig. 4. Ultimate light-trapping limit F in c-Si versus L and d for a wavelength of 1100 nm.
Fig. 5.
Fig. 5. Photocurrent limits in c-Si versus d for L = 10 μm . From low to high, the four curves correspond to the single-pass absorption, Yablonovitch 4 n 2 limit, the ultimate light-trapping limit F , and full absorption.
Fig. 6.
Fig. 6. Photocurrent limits in c-Si versus L for d = 5 μm . From low to high, the four curves correspond to the single-pass absorption, Yablonovitch 4 n 2 limit, the ultimate light-trapping limit F , and full absorption. Only F depends on L .

Equations (61)

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A m , n ( ω ) = γ ˜ m γ m , n ( ω ω m ) 2 + ( γ ˜ m + γ m ) 2 / 4 ,
N = 2 π ω 2 c 2 ( L 2 π ) 2
A n ( ω ) = 1 Δ ω ω ω + Δ ω m = 1 M A m , n ( ω ) d ω = 1 Δ ω m = 1 M ω ω + Δ ω A m , n ( ω ) d ω = 1 Δ ω m = 1 M A m , n ( ω ) d ω ,
M = 2 4 π n c 3 Δ ( ω 3 ) 3 c 3 ( L 2 π ) 2 ( d 2 π )
m = 1 M γ m L 2 ω 2 4 π 2 c 2 Δ ω .
M N .
A n ( ω ) = 2 π Δ ω m = 1 M γ ˜ m γ m , i γ ˜ m + γ m .
A ( ω ) = 2 π Δ ω max γ m , n m = 1 M γ ˜ m γ m , i γ ˜ m + n = 1 N γ m , n
m = 1 M γ m , n Δ ω 2 π , n ,
γ m , n 0 , m , n ,
γ ˜ m > 0 , m ,
γ ˜ m = α c n c ,
L = m = 1 M γ ˜ m γ m , i γ ˜ m + n = 1 N γ m , n n = 1 N λ n ( m = 1 M γ m , n Δ ω 2 π ) + m = 1 M n = 1 N μ m , n γ m , n ,
λ n 0 , n ,
μ m , n 0 , m , n ,
L γ m , n = γ ˜ m γ m , i ( γ ˜ m + n = 1 N γ m , n ) 2 λ n + μ m , n = 0 , m , n i ,
L γ m , i = γ ˜ m ( γ ˜ m + n = 1 N γ m , n γ m , i ) ( γ ˜ m + n = 1 N γ m , n ) 2 λ i + μ m , i = 0 , m ,
λ n ( m = 1 M γ m , n Δ ω 2 π ) = 0 , n ,
μ m , n γ m , n = 0 , m , n .
m = 1 M γ m , i = Δ ω 2 π .
A ( ω ) = 2 π Δ ω m M γ ˜ m γ m , i γ ˜ m + γ m , i .
A ( ω ) < 2 π Δ ω m M γ m , i = 2 π Δ ω m = 1 M γ m , i = 1 .
A ( ω ) = 2 π Δ ω max γ m , i m M γ ˜ m γ m , i γ ˜ m + γ m , i
m M γ m , i = Δ ω 2 π
L = m M γ ˜ m γ m , i γ ˜ m + γ m , i λ ( m M γ m , i Δ ω 2 π ) + m M μ m γ m , i ,
μ m 0 , m M .
L γ m , i = ( γ ˜ m γ ˜ m + γ m , i ) 2 λ + μ m = 0 , m M ,
μ m γ m , i = 0 , m M .
γ m , i = γ ˜ m ( 1 λ 1 ) .
λ = 1 1 + Δ ω 2 π 1 m M γ ˜ m .
A ( ω ) = max γ m , i 1 1 + Δ ω 2 π 1 m M γ ˜ m .
A ( ω ) = 1 1 + Δ ω 2 π 1 m = 1 M γ ˜ m
A ( ω ) = 2 π Δ ω max γ m , n n = 1 N ξ n m = 1 M γ ˜ m γ m , n γ ˜ m + n = 1 N γ m , n
L = n = 1 N ξ n m = 1 M γ ˜ m γ m , n γ ˜ m + n = 1 N γ m , n n = 1 N λ n ( m = 1 M γ m , n Δ ω 2 π ) + m = 1 M n = 1 N μ m , n γ m , n ,
L γ m , n = γ ˜ m ξ n ( γ ˜ m + n = 1 N γ m , n ) n = 1 N ξ n γ m , n ( γ ˜ m + n = 1 N γ m , n ) 2 λ n + μ m , n = 0 , m , n ,
λ n ( m = 1 M γ m , n Δ ω 2 π ) = 0 , n ,
μ m , n γ m , n = 0 , m , n .
ξ n ( γ ˜ m + n = 1 N γ m , n ) > n = 1 N ξ n γ m , n
ξ n ( γ ˜ m + n = 1 N γ m , n ) < n = 1 N ξ n γ m , n
A ( ω ) = 2 π Δ ω 1 N max γ m , n m = 1 M γ ˜ m n = 1 N γ m , n γ ˜ m + n = 1 N γ m , n
m = 1 M γ m , n = Δ ω 2 π , n ,
L = m = 1 M γ ˜ m n = 1 N γ m , n γ ˜ m + n = 1 N γ m , n n = 1 N λ n ( m = 1 M γ m , n Δ ω 2 π ) + m = 1 M n = 1 N μ m , n γ m , n ,
L γ m , n = ( γ ˜ m γ ˜ m + n = 1 N γ m , n ) 2 λ n + μ m , n = 0 , m , n ,
μ m , n γ m , n = 0 , m , n .
λ n = ( γ ˜ m γ ˜ m + n = 1 N γ m , n ) 2 , m M n .
A ( ω ) = 2 π Δ ω 1 N λ m = 1 M n = 1 N γ m , n ,
A ( ω ) = λ = 1 1 + Δ ω 2 π N m M γ ˜ m ,
A ( ω ) = 1 1 + Δ ω 2 π N m = 1 M γ ˜ m .
A ( ω ) = 1 1 + 1 4 n c 2 α d ,
F ( ω ) = 1 α d + π 2 ( c n c ω L ) 2 .
d a d t = ( j Ω Γ Γ ˜ ) a + K s + ,
s = C s + + D a .
D D = 2 Γ ,
C K * = D ,
K = D .
a = G ( ω ) K s + ,
G ( ω ) = ( j ω I j Ω + Γ + Γ ˜ ) 1
p = 2 a Γ ˜ a .
A ( ω ) = p | s + | 2 .
A ( ω ) = s + K * G ( ω ) ( 2 Γ ˜ ) G ( ω ) K s + | s + | 2 .
2 L γ m , n γ m , n = { 0 , m m ; 2 γ ˜ m γ m , i ( γ ˜ m + n = 1 N γ m , n ) 3 , m = m , n i , n i ; γ ˜ m ( γ ˜ m + n = 1 N γ m , n 2 γ m , i ) ( γ ˜ m + n = 1 N γ m , n ) 3 , m = m , n i , n = i ; γ ˜ m ( γ ˜ m + n = 1 N γ m , n 2 γ m , i ) ( γ ˜ m + n = 1 N γ m , n ) 3 , m = m , n = i , n i ; 2 γ ˜ m ( γ ˜ m + n = 1 N γ m , n γ m , i ) ( γ ˜ m + n = 1 N γ m , n ) 3 , m = m , n = i , n = i .

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