Abstract

In this paper, we perform a coated coherent potential approximation method to investigate the transport properties of disordered media consisting of two-layered dielectric spheres whose constituent layer is dispersive. The admixture of quantum dots to polymers to a certain concentration is used as dispersive medium. We find that the dispersive inclusion of the two-layered spheres influences the transport velocities greatly and a resonant scattering taking place in a dilute disordered medium is smeared out in the corresponding densely disordered medium where the correlation effects of multiple scattering are taken into account.

© 2011 Optical Society of America

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2008

D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B 77, 035112 (2008).
[CrossRef]

2007

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

2006

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 065602 (2006).
[CrossRef]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

2005

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

2004

2003

1996

K. Busch, and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

1995

K. Busch, and C. M. Soukoulis, “Transport properties of random media: A new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

1994

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[CrossRef]

1993

1991

Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. 26(6), 1393 (1991).
[CrossRef]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

1989

J. M. Drake, and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. 63, 259–262 (1989).
[CrossRef] [PubMed]

1987

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1986

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

1985

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

M. P. V. Albada, and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P.-E. Wolf, and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

1984

1969

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop. 13, 302–313 (1969).
[CrossRef]

1960

A. Ioffe, and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

1958

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

J. K. Percus, and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

1955

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Aegerter, C. M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 065602 (2006).
[CrossRef]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Albada, M. P. V.

M. P. V. Albada, and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Anderson, P. W.

P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52, 505–509 (1985).
[CrossRef]

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Barber, P. W.

Bertolotti, J.

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Blanco, A.

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Busch, K.

D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B 77, 035112 (2008).
[CrossRef]

K. Busch, and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

K. Busch, and C. M. Soukoulis, “Transport properties of random media: A new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

Conwell, P. R.

Datta, S.

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[CrossRef]

Dave, J. V.

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop. 13, 302–313 (1969).
[CrossRef]

Diem, M.

D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B 77, 035112 (2008).
[CrossRef]

Drake, J. M.

J. M. Drake, and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. 63, 259–262 (1989).
[CrossRef] [PubMed]

Du, H.

Economou, E. N.

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[CrossRef]

Eychmller, A.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Gaponik, N.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Garc’ia, P. D.

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

García-Martín, A.

D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B 77, 035112 (2008).
[CrossRef]

Genack, A. Z.

J. M. Drake, and A. Z. Genack, “Observation of nonclassical optical diffusion,” Phys. Rev. Lett. 63, 259–262 (1989).
[CrossRef] [PubMed]

Gross, P.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Gsele, U.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Hermann, D.

D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B 77, 035112 (2008).
[CrossRef]

Hofmeister, H.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Ioffe, A.

A. Ioffe, and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Lagendijk, A.

B. A. van Tiggelen, and A. Lagendijk, “Rigorous Treatment of the Speed of Diffusing Classical Waves,” Europhys. Lett. 23, 311–316 (1993).
[CrossRef]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

M. P. V. Albada, and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

López, C.

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Maret, G.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 065602 (2006).
[CrossRef]

P.-E. Wolf, and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Martín, M. D.

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef] [PubMed]

Mingaleev, S. F.

D. Hermann, M. Diem, S. F. Mingaleev, A. García-Martín, P. Wölfle, and K. Busch, “Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap,” Phys. Rev. B 77, 035112 (2008).
[CrossRef]

Pecharroman, C.

Percus, J. K.

J. K. Percus, and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Regel, A.

A. Ioffe, and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semiconduct. 4, 237 (1960).

Richter, S.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Rogach, A. L.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Rushforth, C. K.

Sapienza, R.

R. Sapienza, P. D. Garc’ıa, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Schweizer, S. L.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Soukoulis, C. M.

K. Busch, and C. M. Soukoulis, “Transport properties of random media: An energy-density cpa approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

K. Busch, and C. M. Soukoulis, “Transport properties of random media: A new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810 (1994).
[CrossRef]

Steinhart, M.

S. Richter, M. Steinhart, H. Hofmeister, M. Zacharias, U. Gsele, N. Gaponik, A. Eychmller, A. L. Rogach, J. H. Wendorff, S. L. Schweizer, A. von Rhein, and R. B. Wehrspohn, “Quantum dot emitters in two-dimensional photonic crystals of macroporous silicon,” Appl. Phys. Lett. 87, 142107 (2005).
[CrossRef]

Störzer, M.

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 065602 (2006).
[CrossRef]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Tip, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

van Albada, M. P.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

van Tiggelen, B. A.

B. A. van Tiggelen, and A. Lagendijk, “Rigorous Treatment of the Speed of Diffusing Classical Waves,” Europhys. Lett. 23, 311–316 (1993).
[CrossRef]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Viña, L.

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Figures (6)

Fig. 1
Fig. 1

Effective dielectric constants for the dispersive material constructed by polymer doped with quantum dots for three different concentrations η = 0.001, 0.005, 0.01, respectively. r is the radius of the mantel layer.

Fig. 2
Fig. 2

Scheme of the coated CPA as an effective medium theory. Different colorful spheres in (a) indicate different materials with different refractive indices. The effective dielectric constant ɛ̄ is calculated by the effective medium theory, Ψl represents the vectorial field in lth layer, and Ψi represents the vectorial incident field. The dashed volume in (b) indicates the averaged counterpart of the coated spheres.

Fig. 3
Fig. 3

Calculated scattering efficiency factors for two-layered spheres whose core layers are dispersive. The dielectric constant of the core layer, i.e ɛcore is determined by incident wavelengths and the concentration of QDs (a) η = 0, (b) η = 0.001, (c) η = 0.005 and (d) η = 0.01 respectively.

Fig. 4
Fig. 4

Numerical comparisons between the Qscas calculated by the coated CPA method and those calculated by the Maxwell-Garnett theory.

Fig. 5
Fig. 5

Calculated scattering efficiency factors Qsca for two-layered spheres under considerations and transport velocities of electromagnetic energy for optical disordered medium composed of the two-layered spheres with f = 60%. The radia of the core layer are r1 = 0.5r2 in (a) and (c), r1 = 0.9r2 in (b) and (d), respectively. Numerical results for systems with different values of the concentration η, i.e η = 0, η = 0.001, η = 0.005 and η = 0.01 are indicated by the blue, red, black and green lines respectively.

Fig. 6
Fig. 6

(a) Calculated scattering efficiency factor for a single scattering of a two-layered sphere with r1 = 0.5r2 and η = 0.01. Figs. (b–d) show the transport velocities of electromagnetic energy for disordered media composed of two-layered spheres with r1 = 0.5r2, η = 0.01 and volume fractions f = 1%, 30%, 50%, respectively.

Equations (28)

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ɛ c o r e = ɛ m ( 1 + 3 η β ( ω ) 1 η β ( ω ) )
ɛ Q D = 1 + ω p 2 ω 0 2 ω 2 i γ ω
0 R c d 3 r ρ E ( 1 ) ( r ) = 0 R c d 3 r ρ E ( 2 ) ( r )
ρ E ( r ) = 1 2 [ ɛ ( r ) | E ( r ) | 2 + μ | H ( r ) | 2 ]
ɛ ¯ = ( 1 f ) ɛ 3 + f S 1 1 f + f S 2
S 1 = 3 ɛ 3 ɛ 2 ( ɛ 1 ( 1 + 2 ξ ) + 2 ɛ 2 ( 1 ξ ) ) / Θ
S 2 = 3 ɛ 3 ( ɛ 1 ( 1 ξ ) + ɛ 2 ( 2 + ξ ) ) / Θ
Θ = ( ɛ 1 + 2 ɛ 2 ) ( ɛ 2 + 2 ɛ 3 ) + 2 ξ ( ɛ 2 ɛ 3 ) ( ɛ 1 ɛ 2 )
E 1 = n = 1 E n ( c n M o 1 n ( 1 ) i d n N e 1 n ( 1 ) )
H 1 = k 1 ω μ 1 n = 1 E n ( d n M e 1 n ( 1 ) + i c n N o 1 n ( 1 ) )
E n = E 0 i n 2 n + 1 n ( n + 1 )
E l = n = 1 E n ( f n ( l ) M o 1 n ( 1 ) i g n ( l ) N e 1 n ( 1 ) + v n ( l ) M o 1 n ( 2 ) i w n ( l ) N e 1 n ( 2 ) )
H l = k l ω μ l n = 1 E n ( g n ( l ) M e 1 n ( 1 ) + i f n ( l ) N o 1 n ( 1 ) + w n ( l ) M e 1 n ( 2 ) + i v n ( l ) N o 1 n ( 2 ) )
E ( 1 ) = E 1 ( 1 ) + E 2 ( 1 ) + E 3 ( 1 )
E 1 ( 1 ) = 1 2 ω 2 c 2 ɛ 1 n = 1 ( | c n | 2 + | d n | 2 ) 1 k 1 3 0 k 1 r 1 ρ 2 d ρ W n ( j n , j n ; ρ )
W n ( z n , z n ¯ ; r ) = ( 2 n + 1 ) z n ( r ) z n ¯ ( r ) + ( n + 1 ) z n 1 ( r ) z n 1 ¯ ( r ) + n z n + 1 ( r ) z n + 1 ¯ ( r )
E 2 ( 1 ) = 1 2 ω 2 c 2 ɛ 2 n = 1 1 k 2 3 k 2 r 1 k 2 r 2 ρ 2 d ρ [ ( | f n ( 2 ) | 2 + | g n ( 2 ) | 2 ) W n ( j n , j n ; ρ ) + ( | v n ( 2 ) | 2 + | w n ( 2 ) | 2 ) W n ( n n , n n ; ρ ) + 2 ( | f n ( 2 ) v n ( 2 ) | + | g n ( 2 ) w n ( 2 ) | ) W n ( j n , n n ; ρ ) ]
f n ( 2 ) = ψ n ( m 1 x ) χ n ( m 2 x ) [ G 1 n ( m 1 x ) ( m 2 / m 1 ) G 2 n ( m 2 x ) ] c n ϕ n ( m 1 , m 2 ; x ) c n
v n ( 2 ) = ψ n ( m 1 x ) ψ n ( m 2 x ) [ G 1 n ( m 1 x ) ( m 2 / m 1 ) G 1 n ( m 2 x ) ] c n ζ n ( m 1 , m 2 ; x ) c n
g n ( 2 ) = ψ n ( m 1 x ) χ n ( m 2 x ) [ ( m 2 / m 1 ) G 1 n ( m 1 x ) G 2 n ( m 2 x ) ] d n γ n ( m 1 , m 2 ; x ) d n
w n ( 2 ) = ψ n ( m 1 x ) ψ n ( m 2 x ) [ ( m 2 / m 1 ) G 1 n ( m 1 x ) G 1 n ( m 2 x ) ] d n η n ( m 1 , m 2 ; x ) d n
G 1 n ( m 1 x ) = ψ n ( m 1 x ) / ψ n ( m 1 x ) , G 2 n ( m 1 x ) = χ n ( m 1 x ) / χ n ( m 1 x )
E 3 ( 1 ) = 1 2 ω 2 c 3 ɛ 3 n = 1 1 k 3 3 k 3 r 2 k 3 R c ρ 2 d ρ [ ( | f n ( 3 ) | 2 + | g n ( 3 ) | 2 ) W n ( j n , j n ; ρ ) + ( | v n ( 3 ) | 2 + | w n ( 3 ) | 2 ) W n ( n n , n n ; ρ ) + 2 ( | f n ( 3 ) v n ( 3 ) | + | g n ( 3 ) w n ( 3 ) | ) W n ( j n , n n ; ρ ) ]
( f n ( 3 ) v n ( 3 ) g n ( 3 ) w n ( 3 ) ) = ( ϕ n ( m 2 , m 3 ; y ) ς n ( m 2 , m 3 ; y ) 0 0 ζ n ( m 2 , m 3 ; y ) ϑ n ( m 2 , m 3 ; y ) 0 0 0 0 γ n ( m 2 , m 3 ; y ) ϖ n ( m 2 , m 3 ; y ) 0 0 η n ( m 2 , m 3 ; y ) ρ n ( m 2 , m 3 ; y ) ) ( f n ( 2 ) v n ( 2 ) g n ( 2 ) w n ( 2 ) )
ς n ( m 2 , m 3 ; y ) χ n ( m 2 y ) χ n ( m 3 y ) [ ( m 3 / m 2 ) G 2 n ( m 3 y ) G 2 n ( m 2 y ) ]
ϑ n ( m 2 , m 3 ; y ) χ n ( m 2 y ) ψ n ( m 3 y ) [ ( m 3 / m 2 ) G 1 n ( m 3 y ) G 2 n ( m 2 y ) ]
ϖ n ( m 2 , m 3 ; y ) χ n ( m 2 y ) χ n ( m 3 y ) [ G 2 n ( m 3 y ) ( m 3 / m 2 ) G 2 n ( m 2 y ) ]
ρ n ( m 2 , m 3 ; y ) χ n ( m 2 y ) ψ n ( m 3 y ) [ G 1 n ( m 3 y ) ( m 3 / m 2 ) G 2 n ( m 2 y ) ] )

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