Abstract

We derive an exact expression for the time-averaged electromagnetic (EM) energy inside a chiral dispersive sphere irradiated by a plane wave. The dispersion relations correspond to a chiral metamaterial consisting of uncoupled single-resonance helical resonators. Using a field decomposition scheme and a general expression for the EM energy density in bianisotropic media, we calculate the Lorenz–Mie solution for the internal fields in a medium that is simultaneously magnetic and chiral. We also obtain an explicit analytical relation between the internal EM energy and the absorption cross section. This result is applied to demonstrate that strong chirality leads to an off-resonance field enhancement within weakly absorbing spheres.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013

T. J. Arruda, A. S. Martinez, and F. A. Pinheiro, “Unconventional Fano effect and off-resonance field enhancement in plasmonic coated spheres,” Phys. Rev. A 87, 043841 (2013).
[CrossRef]

2012

T. J. Arruda, F. A. Pinheiro, and A. S. Martinez, “Electromagnetic energy within coated spheres containing dispersive metamaterials,” J. Opt. 14, 065101 (2012).
[CrossRef]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nano photonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

2011

2010

2009

P. G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009).
[CrossRef]

N. Zheludev and N. Papasimakis, “Metamaterial-induced transparency: sharp Fano resonances and slow light,” Opt. Photon. News 20(10), 22–27 (2009).
[CrossRef]

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

2008

F. A. Pinheiro, “Statistics of quality factors in three-dimensional disordered magneto-optical systems and its applications to random lasers,” Phys. Rev. A 78, 023812 (2008).
[CrossRef]

2007

2006

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

2005

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef]

S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

2004

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004).
[CrossRef]

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

2002

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

2000

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435–1438 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport velocity and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563–5566 (2000).
[CrossRef]

1996

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996).
[CrossRef]

B. A. van Tiggelen and A. Lagendijk, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

1988

1987

1986

1974

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

1970

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

1937

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Arruda, T. J.

T. J. Arruda, A. S. Martinez, and F. A. Pinheiro, “Unconventional Fano effect and off-resonance field enhancement in plasmonic coated spheres,” Phys. Rev. A 87, 043841 (2013).
[CrossRef]

T. J. Arruda, F. A. Pinheiro, and A. S. Martinez, “Electromagnetic energy within coated spheres containing dispersive metamaterials,” J. Opt. 14, 065101 (2012).
[CrossRef]

T. J. Arruda and A. S. Martinez, “Electromagnetic energy within a magnetic sphere,” J. Opt. Soc. Am. A 27, 992–1001 (2010).
[CrossRef]

T. J. Arruda and A. S. Martinez, “Electromagnetic energy within a magnetic infinite cylinder and scattering properties for oblique incidence,” J. Opt. Soc. Am. A 27, 1679–1687 (2010).
[CrossRef]

Boardman, A. D.

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bott, A.

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Cai, W.

Chettiar, U. K.

Chong, C. T.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nano photonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

de Silva, V. C.

Drachev, V. P.

Economou, E. N.

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef]

Enkrich, C.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Giessen, H.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Halas, N. J.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jacob, Z.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nano photonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Kafesaki, M.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef]

Kildishev, A. V.

Kolinko, P.

Koschny, T.

R. Zhao, T. Koschny, and C. M. Soukoulis, “Chiral metamaterials: retrieval of the effective parameters with and without substrate,” Opt. Express 18, 14553–14567 (2010).
[CrossRef]

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Lagendijk, A.

B. A. van Tiggelen and A. Lagendijk, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Lakhtakia, A.

Lekner, J.

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

Linden, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Luan, P. G.

P. G. Luan, Y. T. Wang, S. Zhang, and X. Zhang, “Electromagnetic energy density in a single-resonance chiral metamaterial,” Opt. Lett. 36, 675–677 (2011).
[CrossRef]

P. G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009).
[CrossRef]

Luk’yanchuk, B.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Maier, S. A.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Marinov, K.

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

Martinez, A. S.

T. J. Arruda, A. S. Martinez, and F. A. Pinheiro, “Unconventional Fano effect and off-resonance field enhancement in plasmonic coated spheres,” Phys. Rev. A 87, 043841 (2013).
[CrossRef]

T. J. Arruda, F. A. Pinheiro, and A. S. Martinez, “Electromagnetic energy within coated spheres containing dispersive metamaterials,” J. Opt. 14, 065101 (2012).
[CrossRef]

T. J. Arruda and A. S. Martinez, “Electromagnetic energy within a magnetic infinite cylinder and scattering properties for oblique incidence,” J. Opt. Soc. Am. A 27, 1679–1687 (2010).
[CrossRef]

T. J. Arruda and A. S. Martinez, “Electromagnetic energy within a magnetic sphere,” J. Opt. Soc. Am. A 27, 992–1001 (2010).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport velocity and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563–5566 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435–1438 (2000).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko, “Off-resonance field enhancement by spherical nanoshells,” Phys. Rev. A 81, 053818 (2010).
[CrossRef]

Molesky, S.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nano photonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Newman, W.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nano photonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Nordlander, P.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Papasimakis, N.

N. Zheludev and N. Papasimakis, “Metamaterial-induced transparency: sharp Fano resonances and slow light,” Opt. Photon. News 20(10), 22–27 (2009).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef]

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Pinheiro, F. A.

T. J. Arruda, A. S. Martinez, and F. A. Pinheiro, “Unconventional Fano effect and off-resonance field enhancement in plasmonic coated spheres,” Phys. Rev. A 87, 043841 (2013).
[CrossRef]

T. J. Arruda, F. A. Pinheiro, and A. S. Martinez, “Electromagnetic energy within coated spheres containing dispersive metamaterials,” J. Opt. 14, 065101 (2012).
[CrossRef]

F. A. Pinheiro, “Statistics of quality factors in three-dimensional disordered magneto-optical systems and its applications to random lasers,” Phys. Rev. A 78, 023812 (2008).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435–1438 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport velocity and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563–5566 (2000).
[CrossRef]

Podolskiy, V. A.

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

Ruppin, R.

R. Ruppin, “Electric and magnetic energies within dispersive metamaterial spheres,” J. Opt. 13, 095101 (2011).
[CrossRef]

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Sampaio, L. C.

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435–1438 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport velocity and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563–5566 (2000).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Schurig, D.

Semchenko, I.

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications (Gordon and Breach Science, 2001).

Serdyukov, A.

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Figures (4)

Fig. 1.
Fig. 1.

EM scattering by a CMM sphere with radius a=5×105m, ω0=ωp=2THz, and Γ=0.05ω0 as a function of the frequency ω. (a) The real part of the LCP refractive index mL for three chirality parameters κ(ω) with filling factors F=0.1, 0.5, and 0.9, and (b) the corresponding LCP extinction efficiency Qext,L. There are no negative refractive indices for these parameters.

Fig. 2.
Fig. 2.

EM scattering by a CMM sphere with radius a=5×105m, ω0=ωp=2THz, and Γ=0.05ω0 as a function of the frequency ω. (a) The real part of the RCP refractive index mR for three chirality parameters κ(ω) with filling factors F=0.1, 0.5, and 0.9, and (b) the corresponding RCP extinction efficiency Qext,R. The negative refractive indices for RCP waves occur for ω>ω0.

Fig. 3.
Fig. 3.

EM scattering by a CMM sphere with radius a=5×105m, ω0=ωp=2THz, and Γ=0.05ω0 as a function of the frequency. (a) Average energy inside the CMM sphere for filling parameters F=0.1, 0.5, 0.9, (b) the corresponding mean extinction efficiency Qext. The decrease in Wt for ω>ω0 coincides with the increase of Qext.

Fig. 4.
Fig. 4.

CMM sphere with radius a=5×105m, filling factor F=0.9, and ω0=ωp=2THz as a function of the frequency and the absorption parameter G=0.01, 0.05, 0.09, where Γ=Gω0. (a) The mean absorption efficiency Qabs, (b) the mean extinction efficiency Qext. The inset shows the mean scattering efficiency Qsca for G=0.01. (c) The internal energy Wt. For ωω0, a decrease in Qabs and Qext does not necessarily coincide with a decrease in Wt.

Equations (54)

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D1=ϵ1(ω)E1+iκ(ω)cH1,
B1=μ1(ω)H1iκ(ω)cE1,
ϵ1(ω)=ϵ0[1ωp2ω2ω02+iΓω],
μ1(ω)=μ0[1Fω2ω2ω02+iΓω],
κ(ω)=Aωω2ω02+iΓω,
ut=14[ϵeff|E1|2+μeff|H1|2+2cRe(κeffE1*·H1)],
ϵeff(ω)=ϵ0[1+ωp2(ω02+ω2)(ω02ω2)2+Γ2ω2],
μeff(ω)=μ0[1+Fω2(3ω02ω2)(ω02ω2)2+Γ2ω2],
κeff(ω)=Aω(Γω+2iω02)(ω02ω2)2+Γ2ω2.
Wt=02πdϕ11d(cosθ)0adrr2ut,
D1=ϵ1(DBF)(E1+β×E1),
B1=μ1(DBF)(H1+β×H1),
ϵ1(DBF)=ϵ1γ2μ1,
μ1(DBF)=μ1γ2ϵ1,
α=γϵ1μ1γ2,
β=12(1kR1kL),
ω[ϵ1(DBF)μ1(DBF)]1/2=[12(1kR+1kL)]1,
K=iωζ(iαϵ1(DBF)μ1(DBF)μ1(DBF)ϵ1(DBF)iαϵ1(DBF)μ1(DBF)).
E1r=n=1Ensinθn(n+1)πn{ψn(ρL)ρL2[fonsinϕ+fencosϕ]αRψn(ρR)ρR2[gonsinϕ+gencosϕ]},
E1θ=n=1En{cosϕρL[fonπnψn(ρL)+fenτnψn(ρL)]+sinϕρL[fonτnψn(ρL)fenπnψn(ρL)]+αRcosϕρR[gonπnψn(ρR)genτnψn(ρR)]αRsinϕρR[gonτnψn(ρR)+genπnψn(ρR)]},
E1ϕ=n=1En{cosϕρL[fonπnψn(ρL)fenτnψn(ρL)]sinϕρL[fonτnψn(ρL)+fenπnψn(ρL)]αRcosϕρR[gonπnψn(ρR)+genτnψn(ρR)]αRsinϕρR[gonτnψn(ρR)genπnψn(ρR)]},
fon=imLWn(R)Wn(L)Vn(R)+Wn(R)Vn(L),
fen=mLVn(R)Wn(L)Vn(R)+Wn(R)Vn(L),
gon=imRWn(L)αLWn(L)Vn(R)+Wn(R)Vn(L),
gen=mRVn(L)αLWn(L)Vn(R)+Wn(R)Vn(L),
Vn(q)=ψn(mqx)ξn(x)m˜ξn(x)ψn(mqx),
Wn(q)=m˜ψn(mqx)ξn(x)ξn(x)ψn(mqx),
m˜=[μ0ϵ1(DBF)ϵ0μ1(DBF)]1/2=μ0μ1(DBF)[12(1mR+1mL)]1,
Wt=WEt+WHt+WEHt,
WErt=ϵeff402πdϕ11d(cosθ)0adrr2|E1r|2=34a3W0tϵeffϵ0n=1n(n+1)(2n+1)0adr{(|fon|2+|fen|2)|jn(ρL)|2|kL|2+|αR|2(|gon|2+|gen|2)|jn(ρR)|2|kR|22Re[αR*(fongon*+fengen*)jn(ρL)jn(ρR*)kLkR*]},
WEθ+WEϕt=34a3W0tϵeffϵ0n=1(2n+1)0adr{(|fon|2+|fen|2)|ψn(ρL)|2+|ψn(ρL)|2|kL|2+|αR|2(|gon|2+|gen|2)|ψn(ρR)|2+|ψn(ρR)|2|kR|22Re[αR*(fongon*+fengen*)ψn(ρL)ψn(ρR*)ψn(ρL)ψn(ρR*)kLkR*]},
(2n+1)ρAρB*[n(n+1)jn(ρA)jn(ρB*)+ψn(ρA)ψn(ρB*)]=njn+1(ρA)jn+1(ρB*)+(n+1)jn1(ρA)jn1(ρB*),
0adrr2jn(ρA)jn(ρB*)=a3[yB*jn(yA)jn(yB*)yAjn(yB*)jn(yA)]yA2yB*2,
limmA±mB*0adrr2jn(mAkr)jn(mB*kr)=a3i(1±1)n2[jn2(yA)jn1(yA)jn+1(yA)].
In(AB)=1a30adrr2jn(mAkr)jn(mB*kr),
Fn,±(AB)=nIn+1(AB)+(n+1)In1(AB)±(2n+1)In(AB),
S(±)=n=1{(|fon|2+|fen|2)Fn,+(LL)+|αR|2(|gon|2+|gen|2)Fn,+(RR)±2Re[αR*(fongon*+fengen*)Fn,(LR)]},
WEtW0t=34ϵeffϵ0SE,
WHtW0t=34|m˜|2μeffμ0SH,
WEHtW0t=32Im(m˜κeffSEH),
SEH=n=1{(|fon|2+|fon|2)Fn,+(LL)|αR|2(|gon|2+|gen|2)Fn,+(RR)2iIm[αR*(fongon*+fengen*)Fn,(LR)]}
ϵeff2ωϵ1(ω)=ϵ0[Γω2ωp2(ω02ω2)2+Γ2ω2],
μeff2ωμ1(ω)=μ0[ΓFω4(ω02ω2)2+Γ2ω2],
κeff2iωκ(ω)=iΓAω3(ω02ω2)2+Γ2ω2,
Qsca,±=2x2n=1(2n+1){|an|2+|bn|2+2|cn|2±2Im[(an+bn)cn*]},
Qext,±=2x2n=1(2n+1)Re(an+bn±2icn),
Qabs,±=Qext,±Qsca,±,
an=Vn(R)An(L)+Vn(L)An(R)Wn(L)Vn(R)+Wn(R)Vn(L),
bn=Wn(R)Bn(L)+Wn(L)Bn(R)Wn(L)Vn(R)+Wn(R)Vn(L),
cn=iWn(R)An(L)Wn(L)An(R)Wn(L)Vn(R)+Wn(R)Vn(L),
dn=cn,
An(q)=m˜ψn(mqx)ψn(x)ψn(x)ψn(mqx),
Bn(q)=ψn(mqx)ψn(x)m˜ψn(x)ψn(mqx).
Qabs=8x3[ϵ1ϵeffWEtW0t+μ1μeffWHtW0t+32Re(m˜κSEH)]=2x[ϵ1ϵ0SE+|m˜|2μ1μ0SH+2Re(m˜κSEH)],

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