Abstract

The rigorous coupled-wave approach was used to compute the plane-wave absorptance of a thin-film tandem solar cell with a metallic surface-relief grating as its back reflector. The absorptance is a function of the angle of incidence and the polarization state of incident light; the free-space wavelength; and the period, duty cycle, the corrugation height, and the shape of the unit cell of the surface-relief grating. The solar cell was assumed to be made of hydrogenated amorphous-silicon alloys and the back reflector of bulk aluminum. The incidence and the grating planes were taken to be identical. The AM1.5 solar irradiance spectrum was used for computations in the 400–1100 nm wavelength range. Inspection of parametric plots of the solar-spectrum-integrated (SSI) absorption efficiency and numerical optimization using the differential evolution algorithm were employed to determine the optimal surface-relief grating. For direct insolation, the SSI absorption efficiency is maximizable by appropriate choices of the period, the duty cycle, and the corrugation height, regardless of the shape of the corrugation in each unit cell of the grating. A similar conclusion also holds for diffuse insolation, but the maximum efficiency for diffuse insolation is about 20% smaller than for direct insolation. Although a tin-doped indium-oxide layer at the front and an aluminum-doped zinc-oxide layer between the semiconductor material and the backing metallic layer change the optimal depth of the periodic corrugations, the optimal period of the corrugations does not significantly change.

© 2013 Optical Society of America

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2012

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
[CrossRef]

S. Xiao, E. Stassen, and N. A. Mortensen, “Ultrathin silicon solar cells with enhanced photocurrents assisted by plasmonic nanostructures,” J. Nanophoton. 6, 061503 (2012).
[CrossRef]

M. Agrawal and M. Frei, “Rigorous optical modeling and optimization of thin-film photovoltaic cells with textured transparent conductive oxides,” Prog. Photovolt. 20, 442–451 (2012).
[CrossRef]

M. Faryad, A. S. Hall, G. D. Barber, T. E. Mallouk, and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves guided by the periodically corrugated interface of a metal and a periodic multilayered isotropic dielectric material,” J. Opt. Soc. Am. B 29, 704–713 (2012).
[CrossRef]

M. Faryad and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves using a compound surface-relief grating,” J. Nanophoton. 6, 061701 (2012).
[CrossRef]

2011

M. Faryad and A. Lakhtakia, “Enhanced absorption of light due to multiple surface-plasmon-polariton waves,” Proc. SPIE 8110, 81100F (2011).
[CrossRef]

I. Dolev, M. Volodarsky, G. Porat, and A. Arie, “Multiple coupling of surface plasmons in quasiperiodic gratings,” Opt. Lett. 36, 1584–1586 (2011).
[CrossRef]

M. Faryad and A. Lakhtakia, “Grating-coupled excitation of multiple surface plasmon-polariton waves,” Phys. Rev. A 84, 033852 (2011).
[CrossRef]

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

2010

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun. 283, 5236–5244(2010).
[CrossRef]

X.-Y. Gao, Y. Liang, and Q.-G. Lin, “Analysis of the optical constants of aluminum-doped zinc oxide films by using the single-oscillator model,” J. Korean Phys. Soc. 57, 710–714 (2010).
[CrossRef]

2009

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009).
[CrossRef]

R. Singh, “Why silicon is and will remain the dominant photovoltaic material,” J. Nanophoton. 3, 032503 (2009).
[CrossRef]

2008

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

A. Gombert and A. Luque, “Photonics in photovoltaic systems,” Phys. Status Solidi A 205, 2757–2765 (2008).
[CrossRef]

2007

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

2005

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

2002

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

2001

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

1998

R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films 313, 394–397 (1998).
[CrossRef]

1997

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

1995

1994

1990

D. L. Flamm, “Mechanisms of silicon etching in fluorine- and chlorine-containing plasmas,” Pure Appl. Chem. 62, 1709–1720 (1990).
[CrossRef]

1983

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

Agrawal, M.

M. Agrawal and M. Frei, “Rigorous optical modeling and optimization of thin-film photovoltaic cells with textured transparent conductive oxides,” Prog. Photovolt. 20, 442–451 (2012).
[CrossRef]

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

Arie, A.

Atwater, H. A.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Ballif, C.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

Barber, G. D.

Bhatnagar, Y.

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

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).
[CrossRef]

Boer, M.

H. Jansen, M. Boer, R. Legtenberg, and M. J. Elwenspoek, “The black silicon method: a universal method for determining the parameter setting of a fluorine-based reactive ion etcher in deep silicon trench etching with profile control,” J. Micromech. Microeng. 5, 115–120 (1995).
[CrossRef]

Borg, H.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Campa, A.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Chateau, N.

Chen, J.

J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun. 283, 5236–5244(2010).
[CrossRef]

Collins, R. W.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Cubero, O.

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

Deng, X.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Devilee, C.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

Diviliansky, I.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Dolev, I.

Dörenkämper, M.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

Eberspacher, C.

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

Egan, G. L.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Elwenspoek, M. J.

H. Jansen, M. Boer, R. Legtenberg, and M. J. Elwenspoek, “The black silicon method: a universal method for determining the parameter setting of a fluorine-based reactive ion etcher in deep silicon trench etching with profile control,” J. Micromech. Microeng. 5, 115–120 (1995).
[CrossRef]

Faryad, M.

M. Faryad and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves using a compound surface-relief grating,” J. Nanophoton. 6, 061701 (2012).
[CrossRef]

M. Faryad, A. S. Hall, G. D. Barber, T. E. Mallouk, and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves guided by the periodically corrugated interface of a metal and a periodic multilayered isotropic dielectric material,” J. Opt. Soc. Am. B 29, 704–713 (2012).
[CrossRef]

M. Faryad and A. Lakhtakia, “Grating-coupled excitation of multiple surface plasmon-polariton waves,” Phys. Rev. A 84, 033852 (2011).
[CrossRef]

M. Faryad and A. Lakhtakia, “Enhanced absorption of light due to multiple surface-plasmon-polariton waves,” Proc. SPIE 8110, 81100F (2011).
[CrossRef]

Feng, B.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Ferlauto, A. S.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Ferreira, G. M.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Ferry, V. E.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Filipic, B.

T. Tušar and B. Filipič, “Differential evolution versus genetic algorithms in multiobjective optimization,” in Evolutionary Multi-Criterion Optimization, S. Obayashi, K. Deb, C. Poloni, T. Hiroyasu, and T. Murata, eds. (Springer, 2007), pp. 257–271.

Flamm, D. L.

D. L. Flamm, “Mechanisms of silicon etching in fluorine- and chlorine-containing plasmas,” Pure Appl. Chem. 62, 1709–1720 (1990).
[CrossRef]

Fonash, S. J.

S. J. Fonash, Solar Cell Device Physics, 2nd ed. (Academic, 2010), pp. 68–102.

Frei, M.

M. Agrawal and M. Frei, “Rigorous optical modeling and optimization of thin-film photovoltaic cells with textured transparent conductive oxides,” Prog. Photovolt. 20, 442–451 (2012).
[CrossRef]

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

Ganguly, G.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Gao, X.-Y.

X.-Y. Gao, Y. Liang, and Q.-G. Lin, “Analysis of the optical constants of aluminum-doped zinc oxide films by using the single-oscillator model,” J. Korean Phys. Soc. 57, 710–714 (2010).
[CrossRef]

Gaylord, T. K.

Geusebroek, M.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

Gombert, A.

A. Gombert and A. Luque, “Photonics in photovoltaic systems,” Phys. Status Solidi A 205, 2757–2765 (2008).
[CrossRef]

Grann, E. B.

Green, M. A.

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
[CrossRef]

Hall, A. S.

Haug, F.-J.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

Heijna, M.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

Heine, C.

Herzig, H. P.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

Hugonin, J.-P.

Isabella, O.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Jansen, H.

H. Jansen, M. Boer, R. Legtenberg, and M. J. Elwenspoek, “The black silicon method: a universal method for determining the parameter setting of a fluorine-based reactive ion etcher in deep silicon trench etching with profile control,” J. Micromech. Microeng. 5, 115–120 (1995).
[CrossRef]

Khoo, I. C.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Krc, J.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Lakhtakia, A.

M. Faryad, A. S. Hall, G. D. Barber, T. E. Mallouk, and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves guided by the periodically corrugated interface of a metal and a periodic multilayered isotropic dielectric material,” J. Opt. Soc. Am. B 29, 704–713 (2012).
[CrossRef]

M. Faryad and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves using a compound surface-relief grating,” J. Nanophoton. 6, 061701 (2012).
[CrossRef]

M. Faryad and A. Lakhtakia, “Enhanced absorption of light due to multiple surface-plasmon-polariton waves,” Proc. SPIE 8110, 81100F (2011).
[CrossRef]

M. Faryad and A. Lakhtakia, “Grating-coupled excitation of multiple surface plasmon-polariton waves,” Phys. Rev. A 84, 033852 (2011).
[CrossRef]

Lampinen, J.

K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization (Springer, 2005).

Legtenberg, R.

H. Jansen, M. Boer, R. Legtenberg, and M. J. Elwenspoek, “The black silicon method: a universal method for determining the parameter setting of a fluorine-based reactive ion etcher in deep silicon trench etching with profile control,” J. Micromech. Microeng. 5, 115–120 (1995).
[CrossRef]

Li, H.

J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun. 283, 5236–5244(2010).
[CrossRef]

Liang, Y.

X.-Y. Gao, Y. Liang, and Q.-G. Lin, “Analysis of the optical constants of aluminum-doped zinc oxide films by using the single-oscillator model,” J. Korean Phys. Soc. 57, 710–714 (2010).
[CrossRef]

Lin, Q.-G.

X.-Y. Gao, Y. Liang, and Q.-G. Lin, “Analysis of the optical constants of aluminum-doped zinc oxide films by using the single-oscillator model,” J. Korean Phys. Soc. 57, 710–714 (2010).
[CrossRef]

Lindquist, N. C.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009).
[CrossRef]

Löffler, J.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

Luque, A.

A. Gombert and A. Luque, “Photonics in photovoltaic systems,” Phys. Status Solidi A 205, 2757–2765 (2008).
[CrossRef]

Mallouk, T. E.

M. Faryad, A. S. Hall, G. D. Barber, T. E. Mallouk, and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves guided by the periodically corrugated interface of a metal and a periodic multilayered isotropic dielectric material,” J. Opt. Soc. Am. B 29, 704–713 (2012).
[CrossRef]

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Mayer, T. S.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Moharam, M. G.

Morf, R. H.

Mortensen, N. A.

S. Xiao, E. Stassen, and N. A. Mortensen, “Ultrathin silicon solar cells with enhanced photocurrents assisted by plasmonic nanostructures,” J. Nanophoton. 6, 061503 (2012).
[CrossRef]

Nagpal, P.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009).
[CrossRef]

Naqavi, A.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

Nishimura, S.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Norris, D. J.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009).
[CrossRef]

Oh, S.-H.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009).
[CrossRef]

Pacifici, D.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Paeder, V.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

Pearce, J. M.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Peeters, P.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Pillai, S.

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
[CrossRef]

Pommet, D. A.

Porat, G.

Price, K.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization (Springer, 2005).

Rakic, A. D.

Repmann, T.

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

Schaadt, D. M.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Scharf, T.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

Schroeder, J.

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

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]

Shishido, A.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Singh, R.

R. Singh, “Why silicon is and will remain the dominant photovoltaic material,” J. Nanophoton. 3, 032503 (2009).
[CrossRef]

Söderström, K.

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

Söderström, T.

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

Soppe, W. J.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

Stassen, E.

S. Xiao, E. Stassen, and N. A. Mortensen, “Ultrathin silicon solar cells with enhanced photocurrents assisted by plasmonic nanostructures,” J. Nanophoton. 6, 061503 (2012).
[CrossRef]

Stepleman, R. S.

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]

Storn, R.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization (Springer, 2005).

Sweatlock, L. A.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Synowicki, R. A.

R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films 313, 394–397 (1998).
[CrossRef]

Terrazzoni-Daudrix, V.

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

Topic, M.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Tušar, T.

T. Tušar and B. Filipič, “Differential evolution versus genetic algorithms in multiobjective optimization,” in Evolutionary Multi-Criterion Optimization, S. Obayashi, K. Deb, C. Poloni, T. Hiroyasu, and T. Murata, eds. (Springer, 2007), pp. 257–271.

Van Aken, B. B.

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

van Erven, R.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Volodarsky, M.

Wang, Q.

J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun. 283, 5236–5244(2010).
[CrossRef]

Witting, K.

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

Wronski, C. R.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

Xiao, S.

S. Xiao, E. Stassen, and N. A. Mortensen, “Ultrathin silicon solar cells with enhanced photocurrents assisted by plasmonic nanostructures,” J. Nanophoton. 6, 061503 (2012).
[CrossRef]

Yu, E. T.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Zeman, M.

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. Shishido, I. Diviliansky, G. L. Egan, I. C. Khoo, T. S. Mayer, S. Nishimura, G. L. Egan, and T. E. Mallouk, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[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]

J. Appl. Phys.

F.-J. Haug, T. Söderström, O. Cubero, V. Terrazzoni-Daudrix, and C. Ballif, “Plasmonic absorption in textured silver back reflectors of thin film solar cells,” J. Appl. Phys. 104, 064509 (2008).
[CrossRef]

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436(2002).
[CrossRef]

J. Global Optim.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

J. Korean Phys. Soc.

X.-Y. Gao, Y. Liang, and Q.-G. Lin, “Analysis of the optical constants of aluminum-doped zinc oxide films by using the single-oscillator model,” J. Korean Phys. Soc. 57, 710–714 (2010).
[CrossRef]

J. Micromech. Microeng.

H. Jansen, M. Boer, R. Legtenberg, and M. J. Elwenspoek, “The black silicon method: a universal method for determining the parameter setting of a fluorine-based reactive ion etcher in deep silicon trench etching with profile control,” J. Micromech. Microeng. 5, 115–120 (1995).
[CrossRef]

J. Nanophoton.

M. Faryad and A. Lakhtakia, “Excitation of multiple surface-plasmon-polariton waves using a compound surface-relief grating,” J. Nanophoton. 6, 061701 (2012).
[CrossRef]

S. Xiao, E. Stassen, and N. A. Mortensen, “Ultrathin silicon solar cells with enhanced photocurrents assisted by plasmonic nanostructures,” J. Nanophoton. 6, 061503 (2012).
[CrossRef]

R. Singh, “Why silicon is and will remain the dominant photovoltaic material,” J. Nanophoton. 3, 032503 (2009).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nano Lett.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Nat. Photonics

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6, 130–132 (2012).
[CrossRef]

Opt. Commun.

J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun. 283, 5236–5244(2010).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. Faryad and A. Lakhtakia, “Grating-coupled excitation of multiple surface plasmon-polariton waves,” Phys. Rev. A 84, 033852 (2011).
[CrossRef]

Phys. Status Solidi A

A. Gombert and A. Luque, “Photonics in photovoltaic systems,” Phys. Status Solidi A 205, 2757–2765 (2008).
[CrossRef]

Proc. SPIE

A. Naqavi, K. Söderström, F.-J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Enhanced light trapping in realistic thin film solar cells using one-dimensional gratings,” Proc. SPIE 8065, 80650A (2011).
[CrossRef]

B. B. Van Aken, C. Devilee, M. Dörenkämper, M. Geusebroek, M. Heijna, J. Löffler, and W. J. Soppe, “PECVD deposition of a-Si:H and μc-Si:H using a linear RF source,” Proc. SPIE 6651, 66510C (2007).
[CrossRef]

M. Faryad and A. Lakhtakia, “Enhanced absorption of light due to multiple surface-plasmon-polariton waves,” Proc. SPIE 8110, 81100F (2011).
[CrossRef]

Prog. Photovolt.

M. Agrawal and M. Frei, “Rigorous optical modeling and optimization of thin-film photovoltaic cells with textured transparent conductive oxides,” Prog. Photovolt. 20, 442–451 (2012).
[CrossRef]

Prog. Photovoltaics

A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovoltaics 18, 160–167 (2010).
[CrossRef]

Pure Appl. Chem.

D. L. Flamm, “Mechanisms of silicon etching in fluorine- and chlorine-containing plasmas,” Pure Appl. Chem. 62, 1709–1720 (1990).
[CrossRef]

Science

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325, 594–597 (2009).
[CrossRef]

Thin Solid Films

R. A. Synowicki, “Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants,” Thin Solid Films 313, 394–397 (1998).
[CrossRef]

Other

http://pvcdrom.pveducation.org/APPEND/Am1_5.htm (accessed 21 July 2012).

We used the DEA code available at: http://www1.icsi.berkeley.edu/storn/code.html .

K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization (Springer, 2005).

T. Tušar and B. Filipič, “Differential evolution versus genetic algorithms in multiobjective optimization,” in Evolutionary Multi-Criterion Optimization, S. Obayashi, K. Deb, C. Poloni, T. Hiroyasu, and T. Murata, eds. (Springer, 2007), pp. 257–271.

M. Agrawal, M. Frei, Y. Bhatnagar, T. Repmann, K. Witting, J. Schroeder, and C. Eberspacher, “Comprehensive experimental and numerical optimization of surface morphology of transparent conductive oxide films for tandem thin film photovoltaic cells,” in Proc. 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010).

S. J. Fonash, Solar Cell Device Physics, 2nd ed. (Academic, 2010), pp. 68–102.

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Figures (15)

Fig. 1.
Fig. 1.

Schematic of the tandem solar cell with a metallic surface-relief grating as the back reflector. The ITO layer is dark blue, the intrinsic semiconductor layers are yellow, the p-type semiconductor layers are blue, the n-type semiconductor layers are red, and the layer in contact with the metal is gray. Specular components of the reflected and transmitted light are identified as of order 0, whereas nonspecular components are identified as of nonzero orders. The function g(x) was chosen so that the fraction ζ=L1/L of each unit cell is a corrugation and sunken while the fraction 1ζ is flat.

Fig. 2.
Fig. 2.

(Left) Real and (right) imaginary parts of the relative permittivities of all semiconductor layers as functions of the free-space wavelength. All p-type layers are made of a-SiC:H with bandgap Eg=1.94eV. All n-type layers are of a-Si:H with bandgap Eg=1.8eV. The top (1i) intrinsic layer is of a-Si1uGeu:H with bandgap Eg=1.4eV, the middle (2i) intrinsic layer is of a-Si1uGeu:H with bandgap Eg=1.6eV, and the bottom (3i) intrinsic layer is of a-Si:H with bandgap Eg=1.69eV.

Fig. 3.
Fig. 3.

Real and imaginary parts of the relative permittivity of bulk aluminum as functions of the free-space wavelength.

Fig. 4.
Fig. 4.

Unit cell of a surface-relief grating with a (top) rectangular corrugation, (bottom left) sinusoidal corrugation, or (bottom right) trapezoidal corrugation, used for optimization.

Fig. 5.
Fig. 5.

ηp versus Lg for L=400nm and θ{0°,15°,30°,45°} for rectangular (blue curves), sinusoidal (red curves), and trapezoidal (black curves) corrugations. The duty cycle ζ=0.3 (dashed curves), 0.5 (curves with crosses), and 0.7 (curves with asterisks). The computations were made for a solar cell without ITO and AZO layers: dT=0, da=Lg, and ϵa=ϵ1n.

Fig. 6.
Fig. 6.

Same as Fig. 5 except that ηs is plotted instead of ηp.

Fig. 7.
Fig. 7.

Same as Fig. 5, except that η is plotted instead of ηp.

Fig. 8.
Fig. 8.

Variations of η with L and θ, the incident light being unpolarized. Top: ζ=0.37 and Lg=66.88nm for the rectangular corrugation. Center: ζ=0.5 and Lg=80nm for the sinusoidal corrugation. Bottom: ζ=0.37 and Lg=70.3nm for the trapezoidal corrugation. The computations were made for a solar cell without ITO and AZO layers: dT=0, da=Lg, and ϵa=ϵ1n.

Fig. 9.
Fig. 9.

Same as Fig. 8 except that (top) Lg=0 and Lm=30nm, and (bottom) Lg=Lm=0.

Fig. 10.
Fig. 10.

η¯ versus Lg for L=400nm for rectangular (blue curves), sinusoidal (red curves), and trapezoidal (black curves) corrugations. The duty cycle ζ=0.3 (dashed curves), 0.5 (curves with crosses), and 0.7 (curves with asterisks). The grating with sinusoidal corrugations with ζ=0.5 and Lg=80nm maximizes η¯ (at 0.5005). The computations were made for a solar cell without ITO and AZO layers: dT=0, da=Lg, and ϵa=ϵ1n.

Fig. 11.
Fig. 11.

η¯ versus L for a surface-relief grating with rectangular (blue dashed curve), sinusoidal (red crossed curve), or trapezoidal (black asterisk curve) corrugations, when ζ=0.5 and Lg=80nm. The computations were made for a solar cell without ITO and AZO layers: dT=0, da=Lg, and ϵa=ϵ1n.

Fig. 12.
Fig. 12.

Angularly averaged local absorptance density α¯ versus λ0 and z for unpolarized incident light in the region 0<z<Ld for a solar cell without an ITO or an AZO layer when the optimum surface-relief grating was used as the backing metallic layer: ζ=0.34, Lg=65nm, L=385nm, dT=0, da=Lg, ϵa=ϵ1n, and the corrugations are rectangular.

Fig. 13.
Fig. 13.

Same as Fig. 10 except that the ITO and AZO layers are also present: dT=200nm and da=Lg+10nm.

Fig. 14.
Fig. 14.

Same as Fig. 11 except that the ITO and AZO layers are also present: dT=200nm, ζ=0.3, Lg=150nm, and da=Lg+10nm.

Fig. 15.
Fig. 15.

Same as Fig. 12 except that L=433nm, dT=200nm, ζ=0.3, Lg=150nm, and da=Lg+10nm.

Tables (6)

Tables Icon

Table 1. Corrugation Shape and Values of the Duty Cycle ζ and the Corrugation Height Lg that Maximize ηp for Four Values of θ, When L=400nma

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Table 2. Same as Table 1 Except That ηs Is Maximized and the Results Were Obtained by Inspection of the Parametric Plots in Fig. 6

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Table 3. Same as Table 1 Except That η Is Maximized and the Results Were Obtained by Inspection of the Parametric Plots in Fig. 7

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Table 4. Optimal Parameters {θopt,ζopt,Lgopt} to Maximize the SSI Absorption Efficiency η Delivered by the DEA for a Solar Cell Without ITO and AZO Layers: dT=0, da=Lg, and ϵa=ϵ1na

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Table 5. Maximum Values of the SSI absorption Efficiency η and Optimal Parameters for Incident Unpolarized Light Obtained by Inspection of the Parametric Plots in Fig. 8a

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Table 6. Optimal Values {ζopt,Lgopt,Lopt} of the Different Corrugation Shapes that Maximize η¯ for Unpolarized Incident Lighta

Equations (26)

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ϵg(x,z)=ϵg(x±L,z)={ϵd(z),z<g(x),ϵm,z>g(x),
Einc(r)=nZ(u^yas(n)+pn+ap(n))exp[i(kx(n)x+kz(n)z)],z0,
Eref(r)=nZ(u^yrs(n)+pnrp(n))exp[i(kx(n)xkz(n)z)],z0,
Etr(r)=nZ(u^yts(n)+pn+tp(n))exp{i[kx(n)x+kz(n)(zLt)]},zLt,
kz(n)={+k02(kx(n))2,k02>(kx(n))2+i(kx(n))2k02,k02<(kx(n))2
pn±=kz(n)k0u^x+kx(n)k0u^z.
ϵ(x,z)=nZϵ(n)(z)exp(inκxx),z[0,Lt],
E(r)=nZE(n)(z)exp(ikx(n)x),z[0,Lt]
H(r)=nZH(n)(z)exp(ikx(n)x),z[0,Lt].
Rp(n)=|rp(n)/ap(0)|2Re[kz(n)]/k0cosθ,
Tp(n)=|tp(n)/ap(0)|2Re[kz(n)]/k0cosθ.
Rs(n)=|rs(n)/as(0)|2Re[kz(n)]/k0cosθ,
Ts(n)=|ts(n)/as(0)|2Re[kz(n)]/k0cosθ.
Ap=1n=NtNt[Rp(n)+Tp(n)],
As=1n=NtNt[Rs(n)+Ts(n)].
g(x)={Ld,x(x1,x2)Ld+Lg,x(0,x1)(x2,L)
g(x)={(Ld+Lg)Lgsin[π(xx1)/L1],x(x1,x2)Ld+Lg,x(0,x1)(x2,L)
g(x)={(Ld+Lg)Lg(xx1)/(x3x1),x(x1,x3)Ld,x(x3,x4)(Ld+Lg)+Lg(xx2)/(x2x4),x(x4,x2)Ld+Lg,x(0,x1)(x2,L)
ηp(L,ζ,Lg,θ)λminλmaxAp(λ0,L,ζ,Lg,θ)λ0S(λ0)dλ0λminλmaxλ0S(λ0)dλ0
ηs(L,ζ,Lg,θ)λminλmaxAs(λ0,L,ζ,Lg,θ)λ0S(λ0)dλ0λminλmaxλ0S(λ0)dλ0
η(ηp+ηs)/2
η¯(L,ζ,Lg)=3π0π/3η(L,ζ,Lg,θ)cosθdθ
dQ(z)dz=ωϵ02Im{ϵd(z)}|E(z)|2,
Pzinc=12ϵ0μ0[|ap(0)|2+|as(0)|2]cosθ
α(z)=dQ(z)/dzPzinc=k0Im{ϵd(z)}|E(z)|2[|ap(0)|2+|as(0)|2]cosθ.
α¯(L,ζ,Lg,λ0,z)=3π0π/3[α(L,ζ,Lg,θ,λ0,z)|ap(0)=as(0)]cosθdθ.

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