Abstract

We find remarkably strong absorption due to magnetic polarization in common colloidal and lithographic metallic nanoparticles. Our analysis is based upon a thorough examination of the dipolar electric and magnetic polarizabilities for representative combinations of nanoparticle composition, size, and morphology. We illustrate this concept by first discussing absorption in metallic spheres and then exploring ellipsoids, disks, and rings. Magnetic polarization reaches ∼ 90% of the total absorption in 100 nm disks and rings for wavelengths above 1 μm under co-linear electric and magnetic irradiation. Our results demonstrate that the magnetic contribution to absorption cannot be naively overlooked, as it can largely exceed the contribution of electric polarization.

© 2012 OSA

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    [CrossRef] [PubMed]

2012 (2)

2011 (5)

S. Karaveli and R. Zia, “Spectral tuning by selective enhancement of electric and magnetic dipole emission,” Phys. Rev. Lett.106, 193004 (2011).
[CrossRef] [PubMed]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, E. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express19, 4815–4826 (2011).
[CrossRef] [PubMed]

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun.2, 451 (2011).
[CrossRef] [PubMed]

J. H. Lee, J. Jang, J. Choi, S. H. Moon, S. Noh, J. Kim, J. Kim, I. S. Kim, K. I. Park, and J. Cheon, “Exchange-coupled magnetic nanoparticles for efficient heat induction,” Nat. Nanotech.6, 418–422 (2011).
[CrossRef]

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

2010 (8)

H. Huang, S. Delikanli, H. Zeng, D. M. Ferkey, and A. Pralle, “Remote control of ion channels and neurons through magnetic-field heating of nanoparticles,” Nat. Nanotech.5, 602–606 (2010).
[CrossRef]

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

A. Manjavacas and F. J. García de Abajo, “Thermal and vacuum friction acting on rotating particles,” Phys. Rev. A82, 063827 (2010).
[CrossRef]

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105, 123902 (2010).
[CrossRef] [PubMed]

M. Burresi, T. Kampfrath, D. van Osten, J. C. Prangsma, B. S. Song, S. Noda, and L. Kuipers, “Magnetic light-matter interactions in photonics crystal nanocavity,” Phys. Rev. Lett.105, 123901 (2010).
[CrossRef] [PubMed]

G. Baffou, R. Quidant, and F. J. García de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano4, 709–716 (2010).
[CrossRef] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

S. Karaveli and R. Zia, “Strong enhancement of magnetic dipole emission in a multilevel electronic system,” Opt. Lett.35, 3318–3320 (2010).
[CrossRef] [PubMed]

2009 (2)

N. I. Grigorchuk and P. M. Tomchuk, “Theory for absorption of ultrashort laser pulses by spheroidal metallic nanoparticles,” Phys. Rev. B80, 155456 (2009).
[CrossRef]

M. S. Yavuz, Y. Cheng, J. Chen, C. M. Cobley, Q. Zhang, M. Rycenga, J. Xie, C. Kim, K. H. Song, A. G. Schwartz, L. V. Wang, and Y. Xia, “Gold nanocages covered by smart polymers for controlled release with near-infrared light,” Nat. Mater.8, 935–939 (2009).
[CrossRef] [PubMed]

2008 (3)

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev.37, 1792–1805 (2008).
[CrossRef] [PubMed]

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Near-field induction heating of metallic nanoparticles due to infrared magnetic dipole contribution,” Phys. Rev. B77, 125402 (2008).
[CrossRef]

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett.93, 201906 (2008).
[CrossRef]

2006 (1)

P. M. Tomchuk and N. I. Grigorchuk, “Shape and size effects on the energy absorption by small metallic particles,” Phys. Rev. B73, 155423 (2006).
[CrossRef]

2005 (1)

C. Loo, A. Lowery, N. J. Halas, J. L. West, and R. Drezek, “Inmunotargeted nanoshells for integrated cancer imaging and therapy,” Nano Lett.5, 709–711 (2005).
[CrossRef] [PubMed]

2004 (1)

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett.209, 171–176 (2004).
[CrossRef]

1998 (2)

M. Wilkinson, B. Mehlig, and P. N. Walker, “Magnetic dipole absorption of radiation in small conducting particles,” J. Phys. Condens. Matter10, 2739–2758 (1998).
[CrossRef]

F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett.80, 5180–5183 (1998).
[CrossRef]

1987 (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, gallium, zinc, and cadmium,” J. Phys. Chem.91, 634–643 (1987).
[CrossRef]

1980 (1)

L. Genzel and U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Physik B37, 93–101 (1980).
[CrossRef]

1973 (1)

T. A. Evans and J. K. Furdyna, “Microwave magnetic dipole interaction in small InSb spheres: induced cyclotron-resonance-like absorption in the Rayleigh limit,” Phys. Rev. B8, 1461–1476 (1973).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig)25, 377–445 (1908).

Aizpurua, J.

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt College Publishers, New York, 1976).

Atwater, H. A.

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

Baffou, G.

G. Baffou, R. Quidant, and F. J. García de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano4, 709–716 (2010).
[CrossRef] [PubMed]

Bak, W. S.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun.2, 451 (2011).
[CrossRef] [PubMed]

Balet, L.

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105, 123902 (2010).
[CrossRef] [PubMed]

Bao, K.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun.2, 451 (2011).
[CrossRef] [PubMed]

Burresi, M.

M. Burresi, T. Kampfrath, D. van Osten, J. C. Prangsma, B. S. Song, S. Noda, and L. Kuipers, “Magnetic light-matter interactions in photonics crystal nanocavity,” Phys. Rev. Lett.105, 123901 (2010).
[CrossRef] [PubMed]

Chantada, L.

Chapuis, P. O.

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Near-field induction heating of metallic nanoparticles due to infrared magnetic dipole contribution,” Phys. Rev. B77, 125402 (2008).
[CrossRef]

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett.93, 201906 (2008).
[CrossRef]

Chen, J.

M. S. Yavuz, Y. Cheng, J. Chen, C. M. Cobley, Q. Zhang, M. Rycenga, J. Xie, C. Kim, K. H. Song, A. G. Schwartz, L. V. Wang, and Y. Xia, “Gold nanocages covered by smart polymers for controlled release with near-infrared light,” Nat. Mater.8, 935–939 (2009).
[CrossRef] [PubMed]

Cheng, Y.

M. S. Yavuz, Y. Cheng, J. Chen, C. M. Cobley, Q. Zhang, M. Rycenga, J. Xie, C. Kim, K. H. Song, A. G. Schwartz, L. V. Wang, and Y. Xia, “Gold nanocages covered by smart polymers for controlled release with near-infrared light,” Nat. Mater.8, 935–939 (2009).
[CrossRef] [PubMed]

Cheon, J.

J. H. Lee, J. Jang, J. Choi, S. H. Moon, S. Noh, J. Kim, J. Kim, I. S. Kim, K. I. Park, and J. Cheon, “Exchange-coupled magnetic nanoparticles for efficient heat induction,” Nat. Nanotech.6, 418–422 (2011).
[CrossRef]

Choi, J.

J. H. Lee, J. Jang, J. Choi, S. H. Moon, S. Noh, J. Kim, J. Kim, I. S. Kim, K. I. Park, and J. Cheon, “Exchange-coupled magnetic nanoparticles for efficient heat induction,” Nat. Nanotech.6, 418–422 (2011).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Cobley, C. M.

M. S. Yavuz, Y. Cheng, J. Chen, C. M. Cobley, Q. Zhang, M. Rycenga, J. Xie, C. Kim, K. H. Song, A. G. Schwartz, L. V. Wang, and Y. Xia, “Gold nanocages covered by smart polymers for controlled release with near-infrared light,” Nat. Mater.8, 935–939 (2009).
[CrossRef] [PubMed]

Curto, A. G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Delikanli, S.

H. Huang, S. Delikanli, H. Zeng, D. M. Ferkey, and A. Pralle, “Remote control of ion channels and neurons through magnetic-field heating of nanoparticles,” Nat. Nanotech.5, 602–606 (2010).
[CrossRef]

Drezek, R.

C. Loo, A. Lowery, N. J. Halas, J. L. West, and R. Drezek, “Inmunotargeted nanoshells for integrated cancer imaging and therapy,” Nano Lett.5, 709–711 (2005).
[CrossRef] [PubMed]

Eah, S. H.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun.2, 451 (2011).
[CrossRef] [PubMed]

Evans, T. A.

T. A. Evans and J. K. Furdyna, “Microwave magnetic dipole interaction in small InSb spheres: induced cyclotron-resonance-like absorption in the Rayleigh limit,” Phys. Rev. B8, 1461–1476 (1973).
[CrossRef]

Ferkey, D. M.

H. Huang, S. Delikanli, H. Zeng, D. M. Ferkey, and A. Pralle, “Remote control of ion channels and neurons through magnetic-field heating of nanoparticles,” Nat. Nanotech.5, 602–606 (2010).
[CrossRef]

Fiore, A.

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105, 123902 (2010).
[CrossRef] [PubMed]

Francardi, M.

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105, 123902 (2010).
[CrossRef] [PubMed]

Froufe-Pérez, L. S.

Funston, A. M.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev.37, 1792–1805 (2008).
[CrossRef] [PubMed]

Furdyna, J. K.

T. A. Evans and J. K. Furdyna, “Microwave magnetic dipole interaction in small InSb spheres: induced cyclotron-resonance-like absorption in the Rayleigh limit,” Phys. Rev. B8, 1461–1476 (1973).
[CrossRef]

García de Abajo, F. J.

L. Shi, E. Xifré-Pérez, F. J. García de Abajo, and F. Meseguer, “Looking through the mirror: Optical microcavity-mirror image photonic interaction,” Opt. Express20, 11247–11255 (2012).
[CrossRef] [PubMed]

A. Manjavacas and F. J. García de Abajo, “Radiative heat transfer between neighboring particles,” Phys. Rev. B86, 075466 (2012).
[CrossRef]

G. Baffou, R. Quidant, and F. J. García de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano4, 709–716 (2010).
[CrossRef] [PubMed]

A. Manjavacas and F. J. García de Abajo, “Thermal and vacuum friction acting on rotating particles,” Phys. Rev. A82, 063827 (2010).
[CrossRef]

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev.37, 1792–1805 (2008).
[CrossRef] [PubMed]

F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett.80, 5180–5183 (1998).
[CrossRef]

García-Etxarri, A.

Genzel, L.

L. Genzel and U. Kreibig, “Dielectric function and infrared absorption of small metal particles,” Z. Physik B37, 93–101 (1980).
[CrossRef]

Gerardino, A.

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105, 123902 (2010).
[CrossRef] [PubMed]

Gómez-Medina, R.

Greffet, J. J.

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Near-field induction heating of metallic nanoparticles due to infrared magnetic dipole contribution,” Phys. Rev. B77, 125402 (2008).
[CrossRef]

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett.93, 201906 (2008).
[CrossRef]

Grigorchuk, N. I.

N. I. Grigorchuk and P. M. Tomchuk, “Theory for absorption of ultrashort laser pulses by spheroidal metallic nanoparticles,” Phys. Rev. B80, 155456 (2009).
[CrossRef]

P. M. Tomchuk and N. I. Grigorchuk, “Shape and size effects on the energy absorption by small metallic particles,” Phys. Rev. B73, 155423 (2006).
[CrossRef]

Gurioli, M.

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105, 123902 (2010).
[CrossRef] [PubMed]

Halas, N. J.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332, 702–704 (2011).
[CrossRef] [PubMed]

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun.2, 451 (2011).
[CrossRef] [PubMed]

C. Loo, A. Lowery, N. J. Halas, J. L. West, and R. Drezek, “Inmunotargeted nanoshells for integrated cancer imaging and therapy,” Nano Lett.5, 709–711 (2005).
[CrossRef] [PubMed]

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett.209, 171–176 (2004).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).
[CrossRef]

Hirsch, L. R.

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ACS Nano (1)

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Ann. Phys. (Leipzig) (1)

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Appl. Phys. Lett. (1)

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[CrossRef]

Cancer Lett. (1)

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett.209, 171–176 (2004).
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Figures (8)

Fig. 1
Fig. 1

Electric vs magnetic absorption in metallic nanospheres. (a) Schematic representation of induced currents leading to electric and magnetic absorption, respectively. (b) Ratio between the imaginary parts of the magnetic and the electric polarizabilities for a 50 nm silver sphere. Exact Mie theory (black solid curve) is compared to various approximations, as explained in the text. Solid curves correspond to a Drude-like dielectric function with ωp = 9.04 eV and γ = 0.021 eV [26], while dotted curves are obtained from the tabulated dielectric function of silver [27]. (c) Conductivity dependence of the magnetic-to-electric absorption ratio for 50 nm spheres described by a Drude dielectric function with parameters appropriate to gold (ωp = 8.89 eV and γ = 0.071 eV [26]) and silver (ωp = 9.04 eV and γ = 0.021 eV [26]), compared to more lossy particles (broken curves, obtained with increased damping γ and similar plasma frequency). (d) Size dependence for silver spheres described by the Drude dielectric function.

Fig. 2
Fig. 2

Spectral dependence of the magnetic contribution to absorption for gold particles of different shape and aspect ratio: (a,d,g) ellipsoids, (b,e,h) rounded disks, and (c,f,i) rings. (a)–(c) Absorption cross-section for two different orientations of the incident electric and magnetic fields relative to the particle symmetry axis (see insets in (a)) and different particle sizes (see upper insets). (d)–(i) Fraction of magnetic losses. Solid curves in (d)–(i) are obtained using a Drude permittivity for gold, while the rest of calculations use tabulated optical data [27]. Parallel and perpendicular orientations in the polarization components are referred to the direction of k.

Fig. 3
Fig. 3

Magnetic absorption under co-linear illumination conditions. We consider two counter-propagating beams, as shown in the left inset, giving rise to aligned external electric and magnetic fields [22]. The spectral dependence of the magnetic contribution to absorption is shown in (a)–(c) for gold ellipsoids, disks, and rings, with the external fields along the axial direction of symmetry. Solid curves and symbols are obtained using the Drude model for gold and a measured dielectric function [27], respectively.

Fig. 4
Fig. 4

Electric and magnetic extinction by a gold nanodisk. We consider two counter-propagating beams, as shown in the inset, leading to only electric or magnetic non-vanishing external field components at the center of the particle. Specifically, we take (φ1, φ2) = (0, 0) for axial E; (π/2, π/2) for radial E; (−π/2, π/2) for axial H; and (0, π) for radial H (see insets). Semi-analytical (broken curves) and fully numerical (solid curves) calculations are compared (see text).

Fig. 5
Fig. 5

Same as Fig. 2 of the main paper for silver particles. The figure shows the spectral dependence of the magnetic contribution to absorption for silver particles of different shape and aspect ratio: (a) ellipsoids, (b) rounded disks, and (c) rings. Two different orientations of the incident electric and magnetic fields are considered (see left insets). Solid curves are obtained using a Drude permittivity for silver, while the rest of calculations use tabulated optical data [27].

Fig. 6
Fig. 6

Same as Fig. 3 of the main paper for silver particles. The figure illustrates the magnetic absorption under co-linear illumination conditions (see left inset). The spectral dependence of the magnetic contribution to absorption is shown in (a), (b) for silver disks and rings, with the external fields along the axial direction of symmetry. Solid curves and symbols are obtained using the Drude model for silver and a measured dielectric function [27], respectively.

Fig. 7
Fig. 7

Relative contribution of magnetic absorption for gold ellipsoids and light incidence along the axis of symmetry. (a) Sketch of the geometry. (b) Fraction of magnetic absorption. (c) Absorption cross-section. A Drude permittivity for gold is used in the solid curves of (b) and a tabulated dielectric function [27] for the rest of the calculations.

Fig. 8
Fig. 8

Relative contribution of magnetic absorption for silver ellipsoids and light incidence along the axis of symmetry. (a) Sketch of the geometry. (b) Fraction of magnetic absorption. A Drude permittivity for silver is used in the solid curves and a tabulated dielectric function [27] for the dots.

Equations (8)

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α E = 3 2 1 k 3 t 1 E ,
α M = 3 2 1 k 3 t 1 M ,
t 1 E = j 1 ( ρ ) [ ρ m j 1 ( ρ m ) ] + ε m j 1 ( ρ m ) [ ρ j 1 ( ρ ) ] h 1 ( + ) ( ρ ) [ ρ m j 1 ( ρ m ) ] ε m j 1 ( ρ m ) [ ρ h 1 ( + ) ( ρ ) ] ,
t 1 M = ρ m j 1 ( ρ ) [ j 1 ( ρ m ) ] + ρ j 1 ( ρ m ) [ j 1 ( ρ ) ] ρ m h 1 ( + ) ( ρ ) [ j 1 ( ρ m ) ] ρ j 1 ( ρ m ) [ h 1 ( + ) ( ρ ) ] .
α E = R 3 ε m 1 ε m + 2 ,
α M = R 3 ( R λ ) 2 2 π 2 15 [ ε m 1 ] ,
ε m = 1 ω p 2 ω ( ω + i γ ) ,
Im { α M } Im { α E } 1 90 ( R c ) 2 ( ω p 2 γ ) 2 = 8 π 2 45 ( R σ 0 c ) 2

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