Abstract

We propose a method for calculating the optical response to partially-coherent light based on the coherence length. Using a Fourier transform of a randomly-generated partially-coherent wave, we demonstrate that the reflectance, transmittance, and absorption with the incidence of partially-coherent light can be calculated from the Poynting vector of the incident coherent light. We also demonstrate that the statistical field distribution of partially-coherent light can be obtained from the proposed method using a rigorous coupled wave analysis. The optical characteristics of grating structures in photovoltaic devices are analyzed as a function of coherence length. The method is capable of providing a general procedure for analyzing the incoherent optical characteristics of thick layers or nano particles in photovoltaic devices with the incidence of partially-coherent light.

© 2012 OSA

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    [CrossRef]
  4. I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
    [CrossRef]
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    [CrossRef]
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2012

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012).
[CrossRef]

M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett.12(6), 2894–2900 (2012).
[CrossRef] [PubMed]

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

2011

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

2010

S. C. Kim and I. Sohn, “Simulation of energy conversion efficiency of a solar cell with gratings,” J. Opt. Soc. Kor.14(2), 142–145 (2010).
[CrossRef]

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

M. C. Troparevsky, A. S. Sabau, A. R. Lupini, and Z. Zhang, “Transfer-matrix formalism for the calculation of optical response in multilayer systems: from coherent to incoherent interference,” Opt. Express18(24), 24715–24721 (2010).
[CrossRef] [PubMed]

2009

T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci.2(4), 347–363 (2009).
[CrossRef]

2008

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

2007

2006

G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006).
[CrossRef]

2005

M. A. Green, “Silicon photovoltaic modules: a brief history of the first 50 years,” Prog. Photovolt. Res. Appl.13(5), 447–455 (2005).
[CrossRef]

E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt.44(35), 7532–7539 (2005).
[CrossRef] [PubMed]

2002

2000

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys.33(24), 3139–3145 (2000).
[CrossRef]

1999

J. S. C. Prentice, “Optical generation rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D Appl. Phys.32(17), 2146–2150 (1999).
[CrossRef]

1995

Alivisatos, A. P.

M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett.12(6), 2894–2900 (2012).
[CrossRef] [PubMed]

Ameri, T.

T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci.2(4), 347–363 (2009).
[CrossRef]

Argitis, P.

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

Atwater, H. A.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett.12(6), 2894–2900 (2012).
[CrossRef] [PubMed]

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

Brabec, C. J.

T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci.2(4), 347–363 (2009).
[CrossRef]

Centurioni, E.

Contreras, M. A.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Deceglie, M. G.

M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett.12(6), 2894–2900 (2012).
[CrossRef] [PubMed]

DeHart, C.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Dennler, G.

T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci.2(4), 347–363 (2009).
[CrossRef]

Dunlop, E. D.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012).
[CrossRef]

Egaas, B.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Emery, K.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012).
[CrossRef]

Ferry, V. E.

M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett.12(6), 2894–2900 (2012).
[CrossRef] [PubMed]

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Green, M. A.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012).
[CrossRef]

M. A. Green, “Silicon photovoltaic modules: a brief history of the first 50 years,” Prog. Photovolt. Res. Appl.13(5), 447–455 (2005).
[CrossRef]

Grimes, C. A.

G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006).
[CrossRef]

Günes, S.

S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev.107(4), 1324–1338 (2007).
[CrossRef] [PubMed]

Hishikawa, Y.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012).
[CrossRef]

Jang, J.

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

Jung, S.

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

Katsidis, C. C.

Kim, H.

Kim, J.

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

Kim, K.-Y.

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

Kim, S. C.

S. C. Kim and I. Sohn, “Simulation of energy conversion efficiency of a solar cell with gratings,” J. Opt. Soc. Kor.14(2), 142–145 (2010).
[CrossRef]

Lee, B.

Lee, I.-M.

Lee, Y.-I.

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

Lungenschmied, C.

T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci.2(4), 347–363 (2009).
[CrossRef]

Lupini, A. R.

Moharam, M. G.

Moon, H.-T.

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011).
[CrossRef]

Mor, G. K.

G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006).
[CrossRef]

Neugebauer, H.

S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev.107(4), 1324–1338 (2007).
[CrossRef] [PubMed]

Noufi, R.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Palilis, L. C.

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

Paulose, M.

G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006).
[CrossRef]

Perkins, C. L.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Polman, A.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

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

Pommet, D. A.

Prentice, J. S. C.

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys.33(24), 3139–3145 (2000).
[CrossRef]

J. S. C. Prentice, “Optical generation rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D Appl. Phys.32(17), 2146–2150 (1999).
[CrossRef]

Repins, I.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Sabau, A. S.

Sariciftci, N. S.

S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev.107(4), 1324–1338 (2007).
[CrossRef] [PubMed]

Savaidis, S. P.

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

Scharf, J.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Schropp, R. E. I.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

Shankar, K.

G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006).
[CrossRef]

Siapkas, D. I.

Sohn, I.

S. C. Kim and I. Sohn, “Simulation of energy conversion efficiency of a solar cell with gratings,” J. Opt. Soc. Kor.14(2), 142–145 (2010).
[CrossRef]

Spinelli, P.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

Stathopoulos, N. A.

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

To, B.

I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008).
[CrossRef]

Troparevsky, M. C.

van de Groep, J.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

van Lare, M.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

Varghese, O. K.

G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006).
[CrossRef]

Vasilopoulou, M.

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

Verschuuren, M. A.

P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012).
[CrossRef]

Warta, W.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012).
[CrossRef]

Yesayan, S. R.

N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011).
[CrossRef]

Youn, J.-H.

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Figures (8)

Fig. 1
Fig. 1

(a) Temporal behavior of a pseudo source in the time domain, (b) discrete Fourier transform of the pseudo source in Fourier domain, and (c) the modified partially-coherent source that originates from the data within the effective spectral width in time domain. The red diamonds represent data within the effective spectral width and the black diamonds represent the data outside of it.

Fig. 2
Fig. 2

(a) Schematic diagram of Si thin film. Calculated (b) reflectance and (c) absorption spectra with coherent and partially-coherent lights.

Fig. 3
Fig. 3

Calculated absorption spectra with oblique incidence (a) 30°, (b) 60°, and (c) 70°.

Fig. 4
Fig. 4

(a) Schematic diagram of Si thin film with grating structures. Calculated (b) reflectance and (c) absorption spectra with coherent and partially-coherent lights.

Fig. 5
Fig. 5

The amplitude distributions of the electric fields (a) with 525 nm (coherent), (b) with 525 nm (partially-coherent), (c) with 560 nm (coherent), and (d) with 560 nm (partially-coherent). The coherence time of partially-coherent wave is 5 fs.

Fig. 6
Fig. 6

(a) Device structure of the CIGS solar cell. (b) Calculated absorption spectra with coherent and partially-coherent lights.

Fig. 7
Fig. 7

(a) Device structure of the CIGS solar cell with grating structure. (b) Calculated absorption spectra with the coherent and partially-coherent lights.

Fig. 8
Fig. 8

(a) Spectral current density of CIGS solar cell. (b) Spectral current density of CIGS solar cell with grating structure.

Tables (1)

Tables Icon

Table 1 Total Current Density (mA/cm2) in the Absorption Layer of CIGS Solar Cells

Equations (13)

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L c = c W s ,
x re [n]=sin( 2πf n N T+ ϕ i ),
ϕ i = ϕ i1 + ϕ rand (fori=1,2,3,),
X k = 1 N n= N1 2 N1 2 x re [n] e jk 2π N n .
x re [n]= X 0 +2 k=1 N1 2 Re( X k e jk 2π N n ) ,
x im [n]=2 k=1 N1 2 Im( X k e jk 2π N n ) .
x[n]= x re [n]+j x im [n]= X 0 +2 k=1 N1 2 X k e jk 2π N n .
E= m=1 M E m = m=1 M ( E m,x , E m,y , E m,z ) e j ω m t ,
H= m=1 M H m = m=1 M ( H m,x , H m,y , H m,z ) e j ω m t ,
S z = E x H y * E y H x * =( m=1 M E m,x e j ω m t )( m=1 M H m,y * e j ω m t )( m=1 M E m,y e j ω m t )( m=1 M H m,x * e j ω m t ).
P z = 1 T 0 T S z dt = 1 T 0 T [ m=1 M ( E m,x H m,y * E m,y H m,x * ) ] dt= m=1 M P z,m ,
P z,m = 1 T 0 T S z,m = 1 T 0 T ( E m,x H m,y * E m,y H m,x * ) dt ,
T c = L c c .

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