Abstract

Surface plasmon enhancement has been proposed as a way to achieve higher absorption for thin-film photovoltaics, where surface plasmon polariton(SPP) and localized surface plasmon (LSP) are shown to provide dense near field and far field light scattering. Here it is shown that controlled far-field light scattering can be achieved using successive coupling between surface plasmonic (SP) nano-particles. Through genetic algorithm (GA) optimization, energy transfer between discrete nano-particles (ETDNP) is identified, which enhances solar cell efficiency. The optimized energy transfer structure acts like lumped-element transmission line and can properly alter the direction of photon flow. Increased in-plane component of wavevector is thus achieved and photon path length is extended. In addition, Wood-Rayleigh anomaly, at which transmission minimum occurs, is avoided through GA optimization. Optimized energy transfer structure provides 46.95% improvement over baseline planar cell. It achieves larger angular scattering capability compared to conventional surface plasmon polariton back reflector structure and index-guided structure due to SP energy transfer through mode coupling. Via SP mediated energy transfer, an alternative way to control the light flow inside thin-film is proposed, which can be more efficient than conventional index-guided mode using total internal reflection (TIR).

©2012 Optical Society of America

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2012 (4)

2011 (7)

W. E. I. Sha, W. C. H. Choy, and W. C. Chew, “Angular response of thin-film organic solar cells with periodic metal back nanostrips,” Opt. Lett. 36(4), 478–480 (2011).
[Crossref] [PubMed]

U. W. Paetzold, E. Moulin, B. E. Pieters, R. Carius, and U. Rau, “Design of nanostructured plasmonic back contacts for thin-film silicon solar cells,” Opt. Express 19(S6Suppl 6), A1219–A1230 (2011).
[Crossref] [PubMed]

F. J. Beck, S. Mokkapati, and K. R. Catchpole, “Light trapping with plasmonic particles: beyond the dipole model,” Opt. Express 19(25), 25230–25241 (2011).
[Crossref] [PubMed]

J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett. 11(6), 2195–2201 (2011).
[Crossref] [PubMed]

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

N. Lagos, M. M. Sigalas, and E. Lidorikis, “Theory of plasmonic near-field enhanced absorption in solar cells,” Appl. Phys. Lett. 99(6), 063304 (2011).
[Crossref]

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

2010 (4)

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

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

F. J. Beck, S. Mokkapati, A. Polman, and K. R. Catchpole, “Asymmetry in photocurrent enhancement by plasmonic nanoparticle arrays located on the front or on the rear of solar cells,” Appl. Phys. Lett. 96(3), 033113 (2010).
[Crossref]

V. E. Ferry, M. A. Verschuuren, H. B. T. Li, E. Verhagen, R. J. Walters, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18(S2Suppl 2), A237–A245 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (3)

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

A. Lin and J. D. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[Crossref]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008).
[Crossref] [PubMed]

2007 (2)

J. M. Khoshman and M. E. Kordesch, “Optical constants and band edge of amorphous zinc oxide thin films,” Thin Solid Films 515(18), 7393–7399 (2007).
[Crossref]

S. J. Kang and Y. H. Joung, “Influence of substrate temperature on the optical and piezoelectric properties of ZnO thin films deposited by rf magnetron sputtering,” Appl. Surf. Sci. 253(17), 7330–7335 (2007).
[Crossref]

2004 (4)

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

B. P. Rand, P. Peumans, and S. R. Forrest, “Long-range absorption enhancement in organic tandem thin-film solar cells containing silver nanoclusters,” J. Appl. Phys. 96(12), 7519–7526 (2004).
[Crossref]

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

2003 (3)

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

J. R. Krenn, “Nanoparticle waveguides: Watching energy transfer,” Nat. Mater. 2(4), 210–211 (2003).
[Crossref] [PubMed]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).

2000 (1)

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Abass, A.

Ade, H.

A. Meyer and H. Ade, “The effect of angle of incidence on the optical field distribution within thin film organic solar cells,” J. Appl. Phys. 106(11), 113101 (2009).
[Crossref]

Alù, A.

Atwater, H. A.

J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett. 11(6), 2195–2201 (2011).
[Crossref] [PubMed]

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

V. E. Ferry, M. A. Verschuuren, H. B. T. Li, E. Verhagen, R. J. Walters, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18(S2Suppl 2), A237–A245 (2010).
[Crossref] [PubMed]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).

Bai, W.

Bailat, J.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Bartoli, F.

Beck, F. J.

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

F. J. Beck, S. Mokkapati, and K. R. Catchpole, “Light trapping with plasmonic particles: beyond the dipole model,” Opt. Express 19(25), 25230–25241 (2011).
[Crossref] [PubMed]

F. J. Beck, S. Mokkapati, A. Polman, and K. R. Catchpole, “Asymmetry in photocurrent enhancement by plasmonic nanoparticle arrays located on the front or on the rear of solar cells,” Appl. Phys. Lett. 96(3), 033113 (2010).
[Crossref]

Bienstman, P.

Boltasseva, A.

Bottler, W.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

Cai, L.

Carius, R.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

U. W. Paetzold, E. Moulin, B. E. Pieters, R. Carius, and U. Rau, “Design of nanostructured plasmonic back contacts for thin-film silicon solar cells,” Opt. Express 19(S6Suppl 6), A1219–A1230 (2011).
[Crossref] [PubMed]

Catchpole, K. R.

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

F. J. Beck, S. Mokkapati, and K. R. Catchpole, “Light trapping with plasmonic particles: beyond the dipole model,” Opt. Express 19(25), 25230–25241 (2011).
[Crossref] [PubMed]

F. J. Beck, S. Mokkapati, A. Polman, and K. R. Catchpole, “Asymmetry in photocurrent enhancement by plasmonic nanoparticle arrays located on the front or on the rear of solar cells,” Appl. Phys. Lett. 96(3), 033113 (2010).
[Crossref]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008).
[Crossref] [PubMed]

Chen, C.-Y.

Chew, W. C.

Cheyns, D.

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

Chiu, T. T.

Choy, W. C. H.

Chrisey, D. B.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

Droz, C.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Fan, S.

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

Ferry, V. E.

Forrest, S. R.

B. P. Rand, P. Peumans, and S. R. Forrest, “Long-range absorption enhancement in organic tandem thin-film solar cells containing silver nanoclusters,” J. Appl. Phys. 96(12), 7519–7526 (2004).
[Crossref]

Gan, Q.

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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

Genoe, J.

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

Gilmore, C. M.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Green, M. A.

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

Hagemann, V.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

Heremans, P.

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

Horwitz, J. S.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Hsing, J. Y.

Huang, Y.

Jiang, Y.-W.

Joung, Y. H.

S. J. Kang and Y. H. Joung, “Influence of substrate temperature on the optical and piezoelectric properties of ZnO thin films deposited by rf magnetron sputtering,” Appl. Surf. Sci. 253(17), 7330–7335 (2007).
[Crossref]

Kafafi, Z. H.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Kang, S. J.

S. J. Kang and Y. H. Joung, “Influence of substrate temperature on the optical and piezoelectric properties of ZnO thin films deposited by rf magnetron sputtering,” Appl. Surf. Sci. 253(17), 7330–7335 (2007).
[Crossref]

Khoshman, J. M.

J. M. Khoshman and M. E. Kordesch, “Optical constants and band edge of amorphous zinc oxide thin films,” Thin Solid Films 515(18), 7393–7399 (2007).
[Crossref]

Kiang, Y.-W.

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).

Kildishev, A. V.

Kim, H.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

Kordesch, M. E.

J. M. Khoshman and M. E. Kordesch, “Optical constants and band edge of amorphous zinc oxide thin films,” Thin Solid Films 515(18), 7393–7399 (2007).
[Crossref]

Krenn, J. R.

J. R. Krenn, “Nanoparticle waveguides: Watching energy transfer,” Nat. Mater. 2(4), 210–211 (2003).
[Crossref] [PubMed]

Kroll, U.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Kuo, M. Y.

Kuo, W. T.

Kuo, Y.

Lagos, N.

N. Lagos, M. M. Sigalas, and E. Lidorikis, “Theory of plasmonic near-field enhanced absorption in solar cells,” Appl. Phys. Lett. 99(6), 063304 (2011).
[Crossref]

Lay, T. S.

Le, K. Q.

Lee, J.-Y.

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

Lee, S.-C.

Li, C. N.

Li, H. B. T.

Li, J.

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

Liao, C.-Y.

Lidorikis, E.

N. Lagos, M. M. Sigalas, and E. Lidorikis, “Theory of plasmonic near-field enhanced absorption in solar cells,” Appl. Phys. Lett. 99(6), 063304 (2011).
[Crossref]

Lin, A.

A. Lin and J. D. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[Crossref]

Lin, H.-Y.

Maes, B.

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).

Meier, J.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Meier, M.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

Meyer, A.

A. Meyer and H. Ade, “The effect of angle of incidence on the optical field distribution within thin film organic solar cells,” J. Appl. Phys. 106(11), 113101 (2009).
[Crossref]

Michaelis, D.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

Min, C.

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

Mokkapati, S.

F. J. Beck, S. Mokkapati, and K. R. Catchpole, “Light trapping with plasmonic particles: beyond the dipole model,” Opt. Express 19(25), 25230–25241 (2011).
[Crossref] [PubMed]

F. J. Beck, S. Mokkapati, A. Polman, and K. R. Catchpole, “Asymmetry in photocurrent enhancement by plasmonic nanoparticle arrays located on the front or on the rear of solar cells,” Appl. Phys. Lett. 96(3), 033113 (2010).
[Crossref]

Moulin, E.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

U. W. Paetzold, E. Moulin, B. E. Pieters, R. Carius, and U. Rau, “Design of nanostructured plasmonic back contacts for thin-film silicon solar cells,” Opt. Express 19(S6Suppl 6), A1219–A1230 (2011).
[Crossref] [PubMed]

Munday, J. N.

J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett. 11(6), 2195–2201 (2011).
[Crossref] [PubMed]

Munoz-Sanjose, V.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Munuera, C.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Murata, H.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Naik, G. V.

Ni, X.

Ocal, C.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Ouyang, Z.

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

Paetzold, U. W.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

U. W. Paetzold, E. Moulin, B. E. Pieters, R. Carius, and U. Rau, “Design of nanostructured plasmonic back contacts for thin-film silicon solar cells,” Opt. Express 19(S6Suppl 6), A1219–A1230 (2011).
[Crossref] [PubMed]

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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

Peumans, P.

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

B. P. Rand, P. Peumans, and S. R. Forrest, “Long-range absorption enhancement in organic tandem thin-film solar cells containing silver nanoclusters,” J. Appl. Phys. 96(12), 7519–7526 (2004).
[Crossref]

Phillips, J. D.

A. Lin and J. D. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[Crossref]

Pieters, B. E.

Pillai, S.

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

Pique, A.

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

Polman, A.

V. E. Ferry, M. A. Verschuuren, H. B. T. Li, E. Verhagen, R. J. Walters, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18(S2Suppl 2), A237–A245 (2010).
[Crossref] [PubMed]

F. J. Beck, S. Mokkapati, A. Polman, and K. R. Catchpole, “Asymmetry in photocurrent enhancement by plasmonic nanoparticle arrays located on the front or on the rear of solar cells,” Appl. Phys. Lett. 96(3), 033113 (2010).
[Crossref]

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

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008).
[Crossref] [PubMed]

Poortmans, J.

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

Rand, B. P.

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

B. P. Rand, P. Peumans, and S. R. Forrest, “Long-range absorption enhancement in organic tandem thin-film solar cells containing silver nanoclusters,” J. Appl. Phys. 96(12), 7519–7526 (2004).
[Crossref]

Rau, U.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

U. W. Paetzold, E. Moulin, B. E. Pieters, R. Carius, and U. Rau, “Design of nanostructured plasmonic back contacts for thin-film silicon solar cells,” Opt. Express 19(S6Suppl 6), A1219–A1230 (2011).
[Crossref] [PubMed]

Requicha, A. A. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

Rommeluere, J. F.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Sallet, V.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Sands, T. D.

Schade, H.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Schroeder, J. L.

Schropp, R. E. I.

Sha, W. E. I.

Shah, V.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Shih, M. H.

Sigalas, M. M.

N. Lagos, M. M. Sigalas, and E. Lidorikis, “Theory of plasmonic near-field enhanced absorption in solar cells,” Appl. Phys. Lett. 99(6), 063304 (2011).
[Crossref]

Song, G.

Soria, F.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Triboulet, R.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Tsai, M.-W.

Tzuang, L. D.-C.

Vallat-Sauvain, E.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Vanecek, M.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Verhagen, E.

Veronis, G.

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

Verreet, B.

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

Verschuuren, M. A.

Wächter, C.

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

Walters, R. J.

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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

Wu, Y.-T.

Wyrsch, N.

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Yang, C. C.

Ye, Y.-H.

Zhang, J.

Zuniga-Perez, J.

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Appl. Phys. Lett. (5)

F. J. Beck, S. Mokkapati, A. Polman, and K. R. Catchpole, “Asymmetry in photocurrent enhancement by plasmonic nanoparticle arrays located on the front or on the rear of solar cells,” Appl. Phys. Lett. 96(3), 033113 (2010).
[Crossref]

D. Cheyns, B. P. Rand, B. Verreet, J. Genoe, J. Poortmans, and P. Heremans, “The angular response of ultrathin film organic solar cells,” Appl. Phys. Lett. 92(24), 243310 (2008).
[Crossref]

U. W. Paetzold, E. Moulin, D. Michaelis, W. Bottler, C. Wächter, V. Hagemann, M. Meier, R. Carius, and U. Rau, “Plasmonic reflection grating back contacts for microcrystalline silicon solar cells,” Appl. Phys. Lett. 99(18), 181105 (2011).
[Crossref]

N. Lagos, M. M. Sigalas, and E. Lidorikis, “Theory of plasmonic near-field enhanced absorption in solar cells,” Appl. Phys. Lett. 99(6), 063304 (2011).
[Crossref]

C. Min, J. Li, G. Veronis, J.-Y. Lee, S. Fan, and P. Peumans, “Enhancement of optical absorption in thin-film organic solar cells through the excitation of plasmonic modes in metallic gratings,” Appl. Phys. Lett. 96(13), 133302 (2010).
[Crossref]

Appl. Surf. Sci. (1)

S. J. Kang and Y. H. Joung, “Influence of substrate temperature on the optical and piezoelectric properties of ZnO thin films deposited by rf magnetron sputtering,” Appl. Surf. Sci. 253(17), 7330–7335 (2007).
[Crossref]

J. Appl. Phys. (3)

A. Meyer and H. Ade, “The effect of angle of incidence on the optical field distribution within thin film organic solar cells,” J. Appl. Phys. 106(11), 113101 (2009).
[Crossref]

S. Pillai, F. J. Beck, K. R. Catchpole, Z. Ouyang, and M. A. Green, “The effect of dielectric spacer thickness on surface plasmon enhanced solar cells for front and rear side depositions,” J. Appl. Phys. 109(7), 073105 (2011).
[Crossref]

B. P. Rand, P. Peumans, and S. R. Forrest, “Long-range absorption enhancement in organic tandem thin-film solar cells containing silver nanoclusters,” J. Appl. Phys. 96(12), 7519–7526 (2004).
[Crossref]

J. Cryst. Growth (1)

C. Munuera, J. Zuniga-Perez, J. F. Rommeluere, V. Sallet, R. Triboulet, F. Soria, V. Munoz-Sanjose, and C. Ocal, “Morphology of ZnO grown by MOCVD on sapphire substrates,” J. Cryst. Growth 264(1-3), 70–78 (2004).
[Crossref]

Nano Lett. (1)

J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett. 11(6), 2195–2201 (2011).
[Crossref] [PubMed]

Nat. Mater. (3)

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

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

J. R. Krenn, “Nanoparticle waveguides: Watching energy transfer,” Nat. Mater. 2(4), 210–211 (2003).
[Crossref] [PubMed]

Opt. Express (8)

M. Y. Kuo, J. Y. Hsing, T. T. Chiu, C. N. Li, W. T. Kuo, T. S. Lay, and M. H. Shih, “Quantum efficiency enhancement in selectively transparent silicon thin film solar cells by distributed Bragg reflectors,” Opt. Express 20(S6Suppl 6), A828–A835 (2012).
[Crossref] [PubMed]

U. W. Paetzold, E. Moulin, B. E. Pieters, R. Carius, and U. Rau, “Design of nanostructured plasmonic back contacts for thin-film silicon solar cells,” Opt. Express 19(S6Suppl 6), A1219–A1230 (2011).
[Crossref] [PubMed]

F. J. Beck, S. Mokkapati, and K. R. Catchpole, “Light trapping with plasmonic particles: beyond the dipole model,” Opt. Express 19(25), 25230–25241 (2011).
[Crossref] [PubMed]

K. Q. Le, A. Abass, B. Maes, P. Bienstman, and A. Alù, “Comparing plasmonic and dielectric gratings for absorption enhancement in thin-film organic solar cells,” Opt. Express 20(S1), A39–A50 (2012).
[Crossref] [PubMed]

H.-Y. Lin, Y. Kuo, C.-Y. Liao, C. C. Yang, and Y.-W. Kiang, “Surface plasmon effects in the absorption enhancements of amorphous silicon solar cells with periodical metal nanowall and nanopillar structures,” Opt. Express 20(S1), A104–A118 (2012).
[Crossref] [PubMed]

V. E. Ferry, M. A. Verschuuren, H. B. T. Li, E. Verhagen, R. J. Walters, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18(S2Suppl 2), A237–A245 (2010).
[Crossref] [PubMed]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008).
[Crossref] [PubMed]

Y.-W. Jiang, L. D.-C. Tzuang, Y.-H. Ye, Y.-T. Wu, M.-W. Tsai, C.-Y. Chen, and S.-C. Lee, “Effect of Wood’s anomalies on the profile of extraordinary transmission spectra through metal periodic arrays of rectangular subwavelength holes with different aspect ratio,” Opt. Express 17(4), 2631–2637 (2009).
[Crossref] [PubMed]

Opt. Lett. (2)

Opt. Mater. Express (1)

Phys. Rev. B (1)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).

Prog. Photovolt. Res. Appl. (1)

V. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Sol. Energy Mater. Sol. Cells (1)

A. Lin and J. D. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92(12), 1689–1696 (2008).
[Crossref]

Thin Solid Films (3)

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 and its applications for thin film solar cells,” Thin Solid Films 455–456, 388–392 (2004).

H. Kim, A. Pique, J. S. Horwitz, H. Murata, Z. H. Kafafi, C. M. Gilmore, and D. B. Chrisey, “Effect of aluminum doping on zinc oxide thin films grown by pulsed laser deposition for organic light-emitting devices,” Thin Solid Films 377–378, 798–802 (2000).
[Crossref]

J. M. Khoshman and M. E. Kordesch, “Optical constants and band edge of amorphous zinc oxide thin films,” Thin Solid Films 515(18), 7393–7399 (2007).
[Crossref]

Other (4)

E. D. Palik, Handbook of optical constants of solids (Academic Press, 1985).

P. Bhattacharya, Semiconductor optoelectronic devices, 2nd ed. (Prentice-Hall, 2006).

C. AB, Comsol multiphysics RF module user guide V 3.3 (2006).

Synopsys, “Sentaurus device EMW user manual V. X-2005.10,” (2005), pp. 78–79.

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Figures (11)

Fig. 1
Fig. 1 Illustration of SP mode coupling between discrete metallic nano-particles.

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Fig. 2
Fig. 2 The first attempt of SP energy transfer structure and its parameters to be optimized.

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Fig. 3
Fig. 3 Spectral response for initial attempt of optimized SP energy transfer structure.

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Fig. 4
Fig. 4 The second SP energy transfer structure and its parameters to be optimized.

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Fig. 5
Fig. 5 Spectral response for the Optimized second evolutionary structure with more closely spaced bottom Ag grating

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Fig. 6
Fig. 6 Genetic algorithm statistics for the second evolutionary structure.

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Fig. 7
Fig. 7 Electric field profile of optimized energy transfer structure at λ = 642.4nm (a) for scattering problem (b) for corresponding eigen mode (c) successive mode coupling between adjacent nano-particles.

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Fig. 8
Fig. 8 (a) Vector field plot of Poynting vector for optimized energy transfer structure at λ = 642.4nm. (b) lateral component of Poynting vector at λ = 642.4nm.

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Fig. 9
Fig. 9 (a) Spectral response, (b) angular distribution of Poynting vector, and (c) electric field profile for energy transfer between discrete nano-particles (ETDNP) at λ = 642nm.

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Fig. 10
Fig. 10 (a) Spectral response, (b) angular distribution of Poynting vector, and (c) electric field profile for surface plasmon polariton (SPP) enhancement at λ = 812nm.

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Fig. 11
Fig. 11 (a)Spectral response, (b)angular distribution of Poynting vector, and (c) electric field profile for index-guided (IG) enhancement at λ = 606nm.

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Tables (1)

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Table 1 Comparison of Silicon Absorbance and Metallic Loss for Various Schemes

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Equations (11)

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Ρ ⇀ poynintg = E ⇀ ( r ⇀ )× H ⇀ ( r ⇀ ) Ρ absorption = E ⇀ ( r ⇀ )⋅ J ⇀  ( r ⇀ )= E ⇀ ( r ⇀ )⋅σ(λ) E ⇀  ( r ⇀ )
P ⇀ poynting, avg = P poynting, avg, x a ^ x + P poynting, avg, y a ^ y                                        = 1 2 Re{ E y ( r ⇀ )∗ H z * ( r ⇀ )} a ^ x − 1 2 Re{ E x ( r ⇀ )∗ H z * ( r ⇀ )} a ^ y
∂ Ε EMW,avg ∂t =0
∫ V P Loss, avg dV + ∮ S P ⇀ poynting, avg ⋅d S ⇀ =0
1 2 ∫ V Re{ E ⇀ ( r ⇀ )⋅ J ⇀   * ( r ⇀ ) }dv + 1 2 ∮ S Re{ E ⇀ ( r ⇀ )× H ⇀ * ( r ⇀ ) }⋅d S ⇀ = 1 2 ∫ V E ⇀ ( r ⇀ )⋅σ(λ) E ⇀ * ( r ⇀ )dv + 1 2 ∮ S Re{ E ⇀ ( r ⇀ )× H ⇀ * ( r ⇀ ) }⋅d S ⇀ =0
θ= tan −1 P poynting,  avg,  x P poynting,  avg,  y = tan −1 Re{ E y ( r ⇀ )∗ H z * ( r ⇀ )} −Re{ E x ( r ⇀ )∗ H z * ( r ⇀ )}
θ avg = ∑ θ=0 2π | modulus(θ,π)− π 2 |×‖ P ⇀ poynting,  avg (θ) ‖ ∑ θ=0 2π ‖ P ⇀ poynting,  avg (θ) ‖                  = ∑ θ=0 π | θ− π 2 |×‖ P ⇀ poynting,  avg (θ) ‖+ ∑ θ=π 2π | θ− 3π 2 |×‖ P ⇀ poynting,  avg (θ) ‖ ∑ θ=0 2π ‖ P ⇀ poynting,  avg (θ) ‖
E ⇀ ( r ⇀ )= u ⇀ ( r ⇀ )exp(i k ⇀ InPlane ⋅ r ⇀ )
k x =k ' x +ik " x = ω c ( ε m ε d ε m + ε d ) 1/2
k z,m =k ' z,m +ik " z,m = ω c ( ε m 2 ε m + ε d ) 1/2
∇ 2 E(x,z)+ ω 2 μεE(x,z) = ∇ 2 [ ∑ i b i (z)exp(j k x,i x) ]                            + ω 2 με ∑ i b i (z)exp(j k x,i x)=0

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