Abstract

The active layer materials used in organic photovoltaic (OPV) cells often self-assemble into highly ordered morphologies, resulting in significant optical anisotropies. However, the impact of these anisotropies on light trapping in nanophotonic OPV architectures has not been considered. In this paper, we show that optical anisotropies in a canonical OPV material, P3HT, strongly affect absorption enhancements in ultra-thin textured OPV cells. In particular we show that plasmonic and gap-mode solar cell architectures redistribute electromagnetic energy into the out-of-plane field component, independent of the active layer orientation. Using analytical and numerical calculations, we demonstrate how the absorption in these solar cell designs can be significantly increased by reorienting polymer domains such that strongly absorbing axes align with the direction of maximum field enhancement.

© 2013 Optical Society of America

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2013 (1)

J. A. Schuller, S. Karaveli, T. Schiros, K. He, S. Yang, I. Kymissis, J. Shan, and R. Zia, “Orientation of luminescent excitons in layered nanomaterials,” Nature Nanotech. 8, 271–276 (2013).
[Crossref]

2012 (9)

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

D. E. Johnston, K. G. Yager, C.-Y. Nam, B. M. Ocko, and C. T. Black, “One-volt operation of high-current vertical channel polymer semiconductor field-effect transistors,” Nano Lett. 12, 4181–4186 (2012).
[Crossref] [PubMed]

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

D. M. O’Carroll, J. S. Fakonas, D. M. Callahan, M. Schierhorn, and H. A. Atwater, “Metal-polymer-metal split-dipole nanoantennas,” Adv. Mater. 24, OP136–OP142 (2012).
[Crossref]

T. Z. Oo, N. Mathews, G. Xing, B. Wu, B. Xing, L. H. Wong, T. C. Sum, and S. G. Mhaisalkar, “Ultrafine gold nanowire networks as plasmonic antennae in organic photovoltaics,” J. Phys. Chem. C 116, 6453–6458 (2012).
[Crossref]

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

J. N. Munday, D. M. Callahan, and H. A. Atwater, “Light trapping beyond the 4n2 limit in thin waveguides,” Appl. Phys. Lett. 100, 121121 (2012).
[Crossref]

T. U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100, 161103 (2012).
[Crossref]

X. Sheng, J. Hu, J. Michel, and L. C. Kimerling, “Light trapping limits in plasmonic solar cells: an analytical investigation,” Opt. Express 20, A496–A501 (2012).
[Crossref] [PubMed]

2011 (8)

S. A. Mann, R. R. Grote, R. M. Osgood, and J. A. Schuller, “Dielectric particle and void resonators for thin film solar cell textures,” Opt. Express 19, 25729–25740 (2011).
[Crossref]

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

S. B. Mallick, N. P. Sergeant, M. Agrawal, J.-Y. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bulletin 36, 453–460 (2011).
[Crossref]

E. A. Schiff, “Thermodynamic limit to photonic-plasmonic light-trapping in thin films on metals,” J. Appl. Phys. 110, 104501 (2011).
[Crossref]

D.-H. Ko, J. R. Tumbleston, A. Gadisa, M. Aryal, Y. Liu, R. Lopez, and E. T. Samulski, “Light-trapping nano-structures in organic photovoltaic cells,” J. Mater. Chem. 21, 16293–16303 (2011).
[Crossref]

K. S. Nalwa, J.-M. Park, K.-M. Ho, and S. Chaudhary, “On realizing higher efficiency polymer solar cells using a textured substrate platform,” Adv. Mater. 23, 112–116 (2011).
[Crossref]

M. R. Hammond, R. J. Kline, A. A. Herzing, L. J. Richter, D. S. Germack, H.-W. Ro, C. L. Soles, D. A. Fischer, T. Xu, L. Yu, M. F. Toney, and D. M. DeLongchamp, “Molecular order in high-efficiency polymer/fullerene bulk heterojunction solar cells,” ACS Nano 5, 8248–8257 (2011).
[Crossref] [PubMed]

J. Liu, Y. Sun, X. Gao, R. Xing, L. Zheng, S. Wu, Y. Geng, and Y. Han, “Oriented poly(3-hexylthiophene) nanofibril with the π- π stacking growth direction by solvent directional evaporation,” Langmuir 27, 4212–4219 (2011).
[Crossref] [PubMed]

2010 (5)

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. De-Longchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43, 3828–3836 (2010).
[Crossref]

M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010).
[Crossref]

D. M. O’Carroll, C. E. Hofmann, and H. A. Atwater, “Conjugated polymer/metal nanowire heterostructure plasmonic antennas,” Adv. Mater. 22, 1223–1227 (2010).
[Crossref]

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

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

2009 (4)

F.-C. Chen, J.-L. Wu, C.-L. Lee, Y. Hong, C.-H. Kuo, and M. H. Huang, “Plasmonic-enhanced polymer photovoltaic devices incorporating solution-processable metal nanoparticles,” Appl. Phys. Lett. 95, 013305 (2009).
[Crossref]

J. Jo, S.-I. Na, S.-S. Kim, T.-W. Lee, Y. Chung, S.-J. Kang, D. Vak, and D.-Y. Kim, “Three-dimensional bulk heterojunction morphology for achieving high internal quantum efficiency in polymer solar cells,” Adv. Funct. Mater. 19, 2398–2406 (2009).
[Crossref]

G. F. Burkhard, E. T. Hoke, S. R. Scully, and M. D. McGehee, “Incomplete exciton harvesting from fullerenes in bulk heterojunction solar cells,” Nano Lett. 9, 4037–4041 (2009).
[Crossref] [PubMed]

S.-Y. Chuang, H.-L. Chen, W.-H. Lee, Y.-C. Huang, W.-F. Su, W.-M. Jen, and C.-W. Chen, “Regioregularity effects in the chain orientation and optical anisotropy of composite polymer/fullerene films for high-efficiency, large-area organic solar cells,” J. Mater. Chem. 19, 5554–5560 (2009).
[Crossref]

2008 (2)

2007 (4)

N. C. Panoiu and R. M. Osgood, “Enhanced optical absorption for photovoltaics via excitation of waveguide and plasmon-polariton modes,” Opt. Lett. 32, 2825–2827 (2007).
[Crossref] [PubMed]

D. M. DeLongchamp, R. J. Kline, E. K. Lin, D. A. Fischer, L. J. Richter, L. A. Lucas, M. Heeney, I. McCulloch, and J. E. Northrup, “High carrier mobility polythiophene thin films: structure determination by experiment and theory,” Adv. Mater. 19, 833–837 (2007).
[Crossref]

M. C. Gurau, D. M. Delongchamp, B. M. Vogel, E. K. Lin, D. A. Fischer, S. Sambasivan, and L. J. Richter, “Measuring molecular order in poly(3-alkylthiophene) thin films with polarizing spectroscopies,” Langmuir 23, 834–842 (2007).
[Crossref] [PubMed]

A. C. Mayer, S. R. Scully, B. E. Hardin, M. W. Rowell, and M. D. McGehee, “Polymer-based solar cells,” Mater. Today 10, 28–33 (2007).
[Crossref]

2006 (2)

B. O’Connor, K. H. An, K. P. Pipe, Y. Zhao, and M. Shtein, “Enhanced optical field intensity distribution in organic photovoltaic devices using external coatings,” Appl. Phys. Lett. 89, 233502 (2006).
[Crossref]

U. Zhokhavets, T. Erb, H. Hoppe, G. Gobsch, and N. S. Sariciftci, “Effect of annealing of poly(3-hexylthiophene)/fullerene bulk heterojunction composites on structural and optical properties,” Thin Solid Films 496, 679–682 (2006).
[Crossref]

2005 (1)

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[Crossref]

2004 (1)

O. D. Gordan, M. Friedrich, and D. R. T. Zahn, “The anisotropic dielectric function for copper phthalocyanine thin films,” Org. Electron. 5, 291–297 (2004).
[Crossref]

2003 (1)

P. J. Brown, D. S. Thomas, A. Köhler, J. S. Wilson, J.-S. Kim, C. M. Ramsdale, H. Sirringhaus, and R. H. Friend, “Effect of interchain interactions on the absorption and emission of poly(3-hexylthiophene),” Phys. Rev. B 67, 064203 (2003).
[Crossref]

2000 (2)

C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000).
[Crossref] [PubMed]

P. Peumans, V. Bulović, and S. R. Forrest, “Efficient photon harvesting at high optical intensities in ultrathin organic double-heterostructure photovoltaic diodes,” Appl. Phys. Lett. 76, 2650–2652 (2000).
[Crossref]

1999 (1)

H. Sirringhaus, P. J. Brown, R. H. Friend, M. M. Nielsen, K. Bechgaard, B. M. W. Langeveld-Voss, A. J. H. Spiering, R. A. J. Janssen, E. W. Meijer, P. Herwig, and D. M. de Leeuw, “Two-dimensional charge transport in self-organized, high-mobility conjugated polymers,” Nature 401, 685–688 (1999).
[Crossref]

1997 (1)

1992 (1)

M. K. Debe, “Variable angle spectroscopic ellipsometry studies of oriented phthalocyanine films. II. copper phthalocyanine,” J. Vac. Sci. Technol., A 10, 2816–2821 (1992).
[Crossref]

1982 (1)

Agrawal, M.

S. B. Mallick, N. P. Sergeant, M. Agrawal, J.-Y. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bulletin 36, 453–460 (2011).
[Crossref]

An, K. H.

B. O’Connor, K. H. An, K. P. Pipe, Y. Zhao, and M. Shtein, “Enhanced optical field intensity distribution in organic photovoltaic devices using external coatings,” Appl. Phys. Lett. 89, 233502 (2006).
[Crossref]

Aryal, M.

D.-H. Ko, J. R. Tumbleston, A. Gadisa, M. Aryal, Y. Liu, R. Lopez, and E. T. Samulski, “Light-trapping nano-structures in organic photovoltaic cells,” J. Mater. Chem. 21, 16293–16303 (2011).
[Crossref]

Atwater, H. A.

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

D. M. O’Carroll, J. S. Fakonas, D. M. Callahan, M. Schierhorn, and H. A. Atwater, “Metal-polymer-metal split-dipole nanoantennas,” Adv. Mater. 24, OP136–OP142 (2012).
[Crossref]

J. N. Munday, D. M. Callahan, and H. A. Atwater, “Light trapping beyond the 4n2 limit in thin waveguides,” Appl. Phys. Lett. 100, 121121 (2012).
[Crossref]

D. M. O’Carroll, C. E. Hofmann, and H. A. Atwater, “Conjugated polymer/metal nanowire heterostructure plasmonic antennas,” Adv. Mater. 22, 1223–1227 (2010).
[Crossref]

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

Bechgaard, K.

H. Sirringhaus, P. J. Brown, R. H. Friend, M. M. Nielsen, K. Bechgaard, B. M. W. Langeveld-Voss, A. J. H. Spiering, R. A. J. Janssen, E. W. Meijer, P. Herwig, and D. M. de Leeuw, “Two-dimensional charge transport in self-organized, high-mobility conjugated polymers,” Nature 401, 685–688 (1999).
[Crossref]

Berini, P.

Black, C. T.

D. E. Johnston, K. G. Yager, C.-Y. Nam, B. M. Ocko, and C. T. Black, “One-volt operation of high-current vertical channel polymer semiconductor field-effect transistors,” Nano Lett. 12, 4181–4186 (2012).
[Crossref] [PubMed]

Bradley, D. D. C.

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[Crossref]

Brown, P. J.

P. J. Brown, D. S. Thomas, A. Köhler, J. S. Wilson, J.-S. Kim, C. M. Ramsdale, H. Sirringhaus, and R. H. Friend, “Effect of interchain interactions on the absorption and emission of poly(3-hexylthiophene),” Phys. Rev. B 67, 064203 (2003).
[Crossref]

H. Sirringhaus, P. J. Brown, R. H. Friend, M. M. Nielsen, K. Bechgaard, B. M. W. Langeveld-Voss, A. J. H. Spiering, R. A. J. Janssen, E. W. Meijer, P. Herwig, and D. M. de Leeuw, “Two-dimensional charge transport in self-organized, high-mobility conjugated polymers,” Nature 401, 685–688 (1999).
[Crossref]

Bulovic, V.

P. Peumans, V. Bulović, and S. R. Forrest, “Efficient photon harvesting at high optical intensities in ultrathin organic double-heterostructure photovoltaic diodes,” Appl. Phys. Lett. 76, 2650–2652 (2000).
[Crossref]

Burkhard, G. F.

G. F. Burkhard, E. T. Hoke, S. R. Scully, and M. D. McGehee, “Incomplete exciton harvesting from fullerenes in bulk heterojunction solar cells,” Nano Lett. 9, 4037–4041 (2009).
[Crossref] [PubMed]

Callahan, D. M.

J. N. Munday, D. M. Callahan, and H. A. Atwater, “Light trapping beyond the 4n2 limit in thin waveguides,” Appl. Phys. Lett. 100, 121121 (2012).
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ACS Nano (1)

M. R. Hammond, R. J. Kline, A. A. Herzing, L. J. Richter, D. S. Germack, H.-W. Ro, C. L. Soles, D. A. Fischer, T. Xu, L. Yu, M. F. Toney, and D. M. DeLongchamp, “Molecular order in high-efficiency polymer/fullerene bulk heterojunction solar cells,” ACS Nano 5, 8248–8257 (2011).
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Adv. Funct. Mater. (1)

J. Jo, S.-I. Na, S.-S. Kim, T.-W. Lee, Y. Chung, S.-J. Kang, D. Vak, and D.-Y. Kim, “Three-dimensional bulk heterojunction morphology for achieving high internal quantum efficiency in polymer solar cells,” Adv. Funct. Mater. 19, 2398–2406 (2009).
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Figures (7)

Fig. 1
Fig. 1 (a) Surface plasmon (SP) and gap mode (GM) OPV architectures and |Ez|2 mode profiles, (b) molecular orientation of P3HT and a model material, zP3HT, which consists of P3HT molecules oriented such that the polymer chains are aligned out-of-plane, (c) anisotropic real and imaginary parts of the P3HT and zP3HT dielectric functions versus wavelength. The P3HT dielectric function is measured by spectroscopic ellipsometry, while zP3HT is modeled by switching the in-plane (||) and out-of-plane (⊥) optical coefficients of P3HT.

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Fig. 2
Fig. 2 (a) Real part of the SP effective index for P3HT (black), zP3HT (red), and air (blue, dashed), and the real part of the out-of-plane top dielectric material index for P3HT (black) and zP3HT (red). (b) Imaginary part of the SP effective index and out-of-plane dielectric index for the same materials, showing that out-of-plane absorption is the dominant contributor to SP modal loss. The dashed line indicates λp, the wavelength at which the plasma resonance ε ′ d , | | ε ′ d , ⊥ = ε ′ m 2 occurs.

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Fig. 3
Fig. 3 (a) Fraction of power absorbed in the polymer for an SP mode. The dashed line indicates λp. (b) Integrated absorption weighted by the AM1.5 solar spectrum [41] for increasing thickness, h, of the polymer thin film atop a Ag substrate (shown in inset). Single pass absorption through each material is plotted with dashed lines for comparison.

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Fig. 4
Fig. 4 (a) Effective indices of a gap mode with P3HT and zP3HT in the gap as a function of gap height and wavelength, (b) integrated absorbance of the gap-mode structure for both P3HT (black) and zP3HT (red) as a function of gap height using the following methods: transfer matrix method (TMM, left panel, dashed-dotted lines), electrostatic limit (ES, left panel, solid lines), rigorous coupled wave analysis (RCWA, right panel, solid lines), and single pass absorption (right panel, dashed lines). (inset) Out-of-plane field profile at 550 nm in the active layer beneath the numerically optimized coupling grating from [7]. (c) Side view of the grating structure used for RCWA calculations and indices used for TMM calculations where the scattering layer has been replaced with an average index of ns = 1.87.

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Fig. 5
Fig. 5 (a) Real part of the SP effective index for P3HT, zP3HT, P3HT(rotated, y-direction), and zP3HT(max). (b) Imaginary part of the SP effective index for the same set of materials showing the theoretical bounds of losses with reorientation.

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Fig. 6
Fig. 6 Geometry and propagation vector definitions for anisotropic SP dispersion derivation.

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Fig. 7
Fig. 7 (a),(c) RCWA calculated absorption versus wavelength and active layer thickness, (b),(d) field intensities and absorption density inside the active area at λ0 = 550 nm and h = 5 nm for P3HT and zP3HT, indicated by white circles in (a), (c).

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Equations (42)

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D → ( ω ) = ε 0 ε ↔ ( ω ) E → ( ω ) = ε 0 [ ε ˜ | | ( ω ) 0 0 0 ε ˜ | | ( ω ) 0 0 0 ε ˜ ⊥ ( ω ) ] E → ( ω )
A ( ω ) = ω ε 0 2 ∫ V E → ⋅ Im { ε ↔ } E → * d V = ω ε 0 2 ∫ V ε ″ | | | E x | 2 + ε ″ | | | E y | 2 + ε ″ ⊥ | E z | 2 d V
β ˜ SP = k 0 [ ε ˜ d , ⊥ ε ˜ m ( ε ˜ d , | | − ε ˜ m ) ε ˜ d , | | ε ˜ d , ⊥ − ε ˜ m 2 ] 1 2
A SP , total = 1 1 + [ 4 π n SP κ SP ] − 1 .
A SP , polymer = ξ a A SP , total .
F ES = 4 n H n L ( 1 3 + 1 3 + n H 4 3 n L 4 ) .
A ES = α L , | | h α L , | | h + [ 4 n H n L , | | ( 2 3 + 1 3 ε ″ L , ⊥ ε ″ L , | | n H 4 | n ˜ L , ⊥ | 4 ) ] − 1 .
ε ↔ P 3 HT = [ ε ˜ | | 0 0 0 ε ˜ | | 0 0 0 ε ˜ ⊥ ] = [ ε ˜ SA 0 0 0 ε ˜ SA 0 0 0 ε ˜ WA ]
ε ↔ P 3 HT ( rotated ) = [ ε ˜ WA 0 0 0 ε ˜ SA 0 0 0 ε ˜ SA ] .
ε ↔ z P 3 HT = [ ε ˜ WA 0 0 0 ε ˜ WA 0 0 0 ε ˜ SA ] .
χ eff = χ ∫ 0 π cos ( θ ) ⋅ cos ( θ ) d θ ∫ 0 π d θ = χ 2 .
ε ˜ SA = 1 + χ eff = 1 + χ 2 .
ε ˜ z P 3 HT ( max ) , ⊥ = 1 + χ = 2 ε ˜ S A − 1 .
ε ↔ z P 3 HT ( max ) = [ 1 0 0 0 1 0 0 0 2 ε ˜ SA − 1 ] .
H y = H 0 { exp   ( − j β ˜ x + j k ˜ z I z ) Region   I exp   ( − j β ˜ x − j k ˜ z I z ) Region   II .
ε ↔ d = ( ε ˜ d , | | 0 0 0 ε ˜ d , | | 0 0 0 ε ˜ d , ⊥ ) ,
E x = { k ˜ z I ω ε 0 ε ˜ d , | | H y Region   I − k ˜ z II ω ε 0 ε ˜ m H y Region   II
E z = { β ˜ ω ε 0 ε ˜ d , ⊥ H y Region   I β ˜ ω ε 0 ε ˜ m H y Region   II .
∇ ⋅ ε ↔ d ( ω ) E → ( ω ) = ∂ ∂ x ε ˜ d , | | E x + ∂ ∂ z ε ˜ d , ⊥ E z I = 0 ,
E x = k ˜ z I ε ˜ d , ⊥ β ˜ ε ˜ d , | | E z I = − k ˜ z II β ˜ E z II
k ˜ z II = − ε ˜ m ε ˜ d , | | k ˜ z I ,
k ˜ z I 2 ε ˜ d , | | + β ˜ 2 ε ˜ d , ⊥ = k 0 2
ε m k ˜ z I 2 ε ˜ d , | | + β ˜ 2 ε ˜ m = k 0 2 .
A w g = α eff h α eff h + [ 4 ( ρ t o t ν g ρ 0 c ) ] − 1 .
A w g = α eff h α eff h + [ n eff λ 0 h ] − 1 = 1 1 + [ 4 π n eff κ eff ] − 1 ,
F w g = 2 π γ eff α 0 h d ω M w g N ,
N = 2 π k 0 2 ( L 2 π ) 2 ,
m w g = π β ′ w g 2 ( L 2 π ) 2 .
M w g = d m w g = 2 π β ′ w g 2 ν g ( L 2 π ) 2 d ω
F w g = α eff n eff λ 0 α 0 h ,
A w g = α 0 h α 0 h + F w g − 1 = α eff h α eff h + [ n eff λ 0 h ] − 1 = 1 1 + [ 4 π n eff κ eff ] − 1 .
A = ω ε 0 2 [ ∫ 0 ∞ ε ″ d , | | | E x | 2 + ε ″ d , ⊥ | E z | 2 d z + ε ″ m ∫ − ∞ 0 | E x | 2 + | E z | 2 d z ] .
A I = 1 4 ω ε 0 [ ε ″ d , | | | k ˜ z I | 2 | ε ″ d , | | | 2 k ˜ z I + ε ″ d , ⊥ | β ˜ | 2 | ε ″ d , ⊥ | 2 k ˜ z I ] H 0 2 exp   ( − 2 β ″ x ) ,
A II = 1 4 ω ε 0 [ ε ″ m | k ˜ z II | 2 | ε ″ m | 2 k ˜ z II + ε ″ m | β ˜ | 2 | ε ″ m | 2 k ˜ z II ] H 0 2 exp   ( − 2 β ″ x ) .
ξ a = A I A I = A II = ε ″ d , | | | k ″ z I | 2 | ε ˜ d , | | | 2 k ″ z I + ε ″ d , ⊥ | β ˜ | 2 | ε ˜ d , ⊥ | 2 k ″ z I ε ″ d , | | | k ″ z I | 2 | ε ˜ d , | | | 2 k ″ z I + ε ″ d , ⊥ | β ˜ | 2 | ε ˜ d , ⊥ | 2 k ″ z I + ε ″ m | k ″ z II | 2 | ε ˜ m | 2 k ″ z II + ε ″ m | β ˜ | 2 | ε ˜ m | 2 k ″ z II ,
γ m = A m W m = ω ∫ gap ε ″ L , | | | E x , m | 2 + ε ″ L , | | | E y , m | 2 + ε ″ L , ⊥ | E z , m | 2 d V ∫ n 2 ( r → ) | E → | 2 d V .
∫ n 2 ( r → ) | E → | 2 d V ≈ n H 2 | E 0 | 2 L 2 D .
γ m = ω ( ε ″ L , | | | E x , m 0 | 2 + ε ″ L , | | | E y , m 0 | 2 + ε ″ L , ⊥ n H 4 | n ˜ L , ⊥ | 4 | E z , m 0 | 2 ) L 2 h n H 2 | E 0 | 2 L 2 D
∑ m | E x , m 0 | 2 = ∑ m | E y , m 0 | 2 = ∑ m | E z , m 0 | 2 = 1 3 M | E 0 | 2 ,
a ES = ∑ m 2 π γ m N d ω = 4 n H n L , | | ( 2 3 + 1 3 ε ″ L , ⊥ ε ″ L , | | n H 4 | n ˜ L , ⊥ | 4 ) α L , | | h .
F ES ≡ a ES α L , | | h = 4 n H n L , | | ( 2 3 + 1 3 ε ″ L , ⊥ ε ″ L , | | n H 4 | n ˜ L , ⊥ | 4 )
A E S = α L , | | h α L , | | h + F ES − 1 = α L , | | h α L , | | h + [ 4 n H n L , | | ( 2 3 + 1 3 ε ″ L , ⊥ ε ″ L , | | n H 4 | n ˜ L , ⊥ | 4 ) ] − 1 .

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