Abstract

The light scattering by a spherical particle with radial anisotropic permittivity ε and permeability μ are discussed in detail by expanding Mie theory. With the modified vector potential formulation, the electric anisotropy effects on scattering efficiency are addressed by studying the extinction, scattering, absorption and radar cross sections following the change of the transverse permittivity εt, the longitudinal permittivity εr and the particle size q. The huge scattering cross sections are shown by considering the possible coupling between active medium and plasmon polaritons and this will be possible to result in spaser from the active plasmons of small particle.

© 2010 OSA

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2010 (2)

Y. Fu, J. Zhang, and J.R. Lakowicz, “Plasmon-enhanced fluorescence from single fluorophores end-linked to gold nanorods,” J. Am. Chem. Soc. 132, 5540–5541 (2010).
[CrossRef] [PubMed]

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]

2009 (1)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

2008 (5)

M.I. Stockman, “Spasers explained,” Nat. Photonics 2, 327–329 (2008).
[CrossRef]

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[CrossRef] [PubMed]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008)
[CrossRef]

B. S. Luk’yanchuk and C.-W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A 92,773 (2008)
[CrossRef]

M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express 16, 1385–1392 (2008).
[CrossRef] [PubMed]

2007 (6)

J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007).
[CrossRef] [PubMed]

C.-W. Qiu, L. W. Li, T.-S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[CrossRef]

B. S. Luk’yanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[CrossRef]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

M. T. Hill and et al., “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

2006 (5)

2005 (4)

J. Seidel, S. Grafstroum, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608 (2005).
[CrossRef] [PubMed]

A. J. Haes, L. Chang, W. L. Klein, and R. P. Van Duyne, “Detection of a biomarker for alzheimer’s disease from synthetic and clinical samples using a nanoscale optical biosensor,” J. Am. Chem. Soc. 127, 2264–2271(2005).
[CrossRef] [PubMed]

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sutherland, and M. Kall, “Plasmonic sensing characteristics of single nanometric holes,” Nano Lett. 5, 2335–2339 (2005).
[CrossRef] [PubMed]

2004 (6)

P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science 306, 1002–1005 (2004).
[CrossRef] [PubMed]

E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120, 5444–5454 (2004)
[CrossRef] [PubMed]

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85, 5040–5042 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

R. J. Tarento, K. H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

M. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072–4079 (2004)
[CrossRef] [PubMed]

2003 (2)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302419–422, (2003).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

2001 (1)

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

P. Andrew and W.L. Barnes, “Förster energy transfer in an optical microcavity,” Science 290, 785–788 (2000).
[CrossRef] [PubMed]

1999 (1)

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

1997 (1)

S. Nie and S. R. Emony, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

1996 (1)

C. A. Mirkin, R. L. Letsinger, R. C. Mucic, and J.J. Storhoff, “A DNA-based method for rationally assembling nanoparticles into macroscopic materials,” Nature 382, 607–609 (1996).
[CrossRef] [PubMed]

1993 (1)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664–673 (1993).
[CrossRef]

1991 (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. 26, 13931401(1991).
[CrossRef]

1989 (2)

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800–802 (1989).
[CrossRef]

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

1983 (1)

B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators 4, 299–304 (1983).
[CrossRef]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 12421246 (1951).
[CrossRef]

Adegoke, J.

Adegoke, J. A.

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 12421246 (1951).
[CrossRef]

Alaverdyan, Y.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sutherland, and M. Kall, “Plasmonic sensing characteristics of single nanometric holes,” Nano Lett. 5, 2335–2339 (2005).
[CrossRef] [PubMed]

Andrew, P.

P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science 306, 1002–1005 (2004).
[CrossRef] [PubMed]

P. Andrew and W.L. Barnes, “Förster energy transfer in an optical microcavity,” Science 290, 785–788 (2000).
[CrossRef] [PubMed]

Anger, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

Bahoura, M.

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Barber, P. W.

P. W. Barber and S. C. Hill, Light scattering by particles: computational methods (World Scientific, Singapore, 1990).
[CrossRef]

Barnes, W. L.

P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science 306, 1002–1005 (2004).
[CrossRef] [PubMed]

Barnes, W.L.

P. Andrew and W.L. Barnes, “Förster energy transfer in an optical microcavity,” Science 290, 785–788 (2000).
[CrossRef] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Bennemann, K. H.

R. J. Tarento, K. H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Bharadwaj, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).

Born, M.

M. Born and E. Wolf, Principles of optics, 7th ed. (Cambridge University Press, Cambridge, 1999).

Brongersma, M. L.

M. L. Brongersma and P. G. Kik, Surface plasmon nanophotonics (Springer Series in Optical Sciences, Springer, 2007), Vol. 131.
[CrossRef]

Chang, L.

A. J. Haes, L. Chang, W. L. Klein, and R. P. Van Duyne, “Detection of a biomarker for alzheimer’s disease from synthetic and clinical samples using a nanoscale optical biosensor,” J. Am. Chem. Soc. 127, 2264–2271(2005).
[CrossRef] [PubMed]

Chew, W. C.

W. C. Chew, Waves and fields in inhomogeneous media (Van Nostrand, New York, 1990).

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]

Chong, T. C.

B. S. Luk’yanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[CrossRef]

Cloud, Michael J.

Edward J. Rothwell and Michael J. Cloud, Electromagnetics, 2nd ed. (CRC Press, Taylor & Francis Group, 2009).

Dadap, J. I.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Dahlin, A.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sutherland, and M. Kall, “Plasmonic sensing characteristics of single nanometric holes,” Nano Lett. 5, 2335–2339 (2005).
[CrossRef] [PubMed]

Drachev, V. P.

Ebbesen, T. W.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

Eisenthal, K. B.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Eisler, H.-J.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608 (2005).
[CrossRef] [PubMed]

Emony, S. R.

S. Nie and S. R. Emony, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

Eng, L.

J. Seidel, S. Grafstroum, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Fainman, Y.

Fu, Y.

Y. Fu, J. Zhang, and J.R. Lakowicz, “Plasmon-enhanced fluorescence from single fluorophores end-linked to gold nanorods,” J. Am. Chem. Soc. 132, 5540–5541 (2010).
[CrossRef] [PubMed]

Genet, C.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

Geng, Y. L.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008)
[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]

Gordon, J. A.

Grafstroum, S.

J. Seidel, S. Grafstroum, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Graglia, R. D.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Guan, B. R.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Haes, A. J.

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Appl. Phys. A (2)

B. S. Luk’yanchuk and C.-W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A 92,773 (2008)
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A. J. Haes, L. Chang, W. L. Klein, and R. P. Van Duyne, “Detection of a biomarker for alzheimer’s disease from synthetic and clinical samples using a nanoscale optical biosensor,” J. Am. Chem. Soc. 127, 2264–2271(2005).
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Y. Fu, J. Zhang, and J.R. Lakowicz, “Plasmon-enhanced fluorescence from single fluorophores end-linked to gold nanorods,” J. Am. Chem. Soc. 132, 5540–5541 (2010).
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J. Opt. Soc. Am. A (2)

Nano Lett. (1)

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sutherland, and M. Kall, “Plasmonic sensing characteristics of single nanometric holes,” Nano Lett. 5, 2335–2339 (2005).
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Nanotechnology (1)

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

Nat. Mater. (1)

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).
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Nat. Photonics (2)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008)
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M.I. Stockman, “Spasers explained,” Nat. Photonics 2, 327–329 (2008).
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Nature (4)

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
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Nature Photon. (1)

M. T. Hill and et al., “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
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Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. E (4)

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Figures (8)

Fig. 1
Fig. 1

The maximal value of Qext for each resonance mode as a function of the dissipative damping Im[εD] for the size q = 1 (A) and q = 0.5 (B). The Qext as a function of both frequency ω and Im[εD] for dipole and quadrupole modes of the particle with size q = 1 shown in the inset of (A).

Fig. 2
Fig. 2

The maximal value of Qext as a function of Re[εD] with different Im[εD] (A), the maximal or minimal values of Qsca and Qabs as a function of Im[εD] with Re[εD] = −2.2 (B), maximal value of Qext as a function of Im[εr] with different Re[εr] and the ratio εt/εr (C and D).

Fig. 4
Fig. 4

Log[Qext] as a function of permittivity εD and size q under the nondissipative limit (Im[εD] = 0) (A), the maximal value of Log[Qext] as a function of Re[εD] and Im[εD] (B), the maximal value of Log[Qsca] as a function of Im[εr] and the ratio |εt|/|εr| for Re[εr] = −2.5 and εt = (−2.5 + i Im[εr])|εt|/|εr|(C), Log[Qsca] as a function of the ratio |εt|/|εr| and size q for εr = −2.5 −0.1i and εt = (−2.5 −0.1i)|εt|/|εr|(D).

Fig. 3
Fig. 3

Log[Qmax ext] as a function of both εr and εt/εr (A), Qext as a function of q and εt/εr with fixed longitudinal permittivity εr = −2.5 (B) and Qext as a function of q and εr with fixed ratio εt/εr = 0.75 (C), with the nondissipative limit (Im[εr] = 0 and Im[εt] = 0).

Fig. 5
Fig. 5

the resonance cross section Log[Qsca] as a function of transverse permittivity (Re[εt] and Re[εt]) with the fixed longitudinal permittivity εr = 2.5 – 0.05i (A), the maximal value of scattering amplitude | b 1 e| as a function of Re[εt] and Im[εt] with the fixed εr = 2.5 – 0.05i (B).

Fig. 6
Fig. 6

the maximal value of scattering amplitude | b 1 e| as a function of Re[εr] and Re[εt] with fixed Im[εr](= −0.05) and Im[εt](= 0.008) for εr as active medium (A), with fixed Im[εr](= 0.001) and Im[εt](= −0.02) for εt as active medium (B).

Fig. 7
Fig. 7

the scattering amplitude Log [ | b 1 e | ] as a function of Re[εr] and q with fixed Im[εr](= −0.05) and Im[εt](= 0.008) (Re[εt] is chosen by the formula Re[εt] = −2.62 + 0.608Re[εr] − 0.110Re[εr]2 + 0.008Re[εr]3)(A), the scattering amplitude Log [ | b 1 e | ] as a function of Re[εr] and q with fixed Im[εr](= 0.001) and Im[εt](= −0.02) (Re[εt] is chosen by the formula Re[εt] = 0.389 – 0.154Re[εr])(B), the maximal value of scattering amplitude Log [ | b 1 e | ] as a function of f(Im[εr]) and Im[εt] with εr = 3 – f(Im[εt])Im[εt] i and Re[εt] = −1.5604 (C) and the maximal value of scattering amplitude Log [ | b 1 e | ] as a function of f(Im[εr]) and Im[εr] with Re[εr] = −12 and εt = 2.237 – f(Im[εr])Im[εr]i (D).

Fig. 8
Fig. 8

High scattering efficiencies with huge radar backscattering cross section. Qrbs as the function of size q for εr with the property of energy-gain (A and B) and for εt with the property of energy-gain (C and D).

Equations (17)

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H = i k 0 ɛ E and E = ik 0 μ H ,
ɛ = ( ɛ r 0 0 0 ɛ t 0 0 0 ɛ t ) and μ = ( μ r 0 0 0 μ t 0 0 0 μ t )
ɛ r ɛ t 2 Π TM r 2 + 1 r 2 sin θ θ ( sin θ Π TM θ ) + 1 r 2 sin 2 θ 2 Π TM φ 2 + k 0 2 ɛ r μ t Π TM = 0 ,
μ r μ r 2 Π TE r 2 + 1 r 2 sin θ θ ( sin θ Π TE θ ) + 1 r 2 sin 2 θ 2 Π TE φ 2 + k 0 2 ɛ t μ r Π TE = 0 ,
B l e = i l + 1 2 l + 1 l ( l + 1 ) b l e and B l m = i l + 1 2 l + 1 l ( l + 1 ) b l m ,
b l e = ɛ t Φ l ( k 0 a ) Φ v 1 ( k t a ) μ t Φ l ( k 0 a ) Φ v 1 ( k t a ) ɛ t ξ l ( k 0 a ) Φ v 1 ( k t a ) μ t ξ l ( k 0 a ) Φ v 1 ( k t a ) ,
b l m = ɛ t Φ l ( k 0 a ) Φ v 2 ( k t a ) μ t Φ l ( k 0 a ) Φ v 2 ( k t a ) ɛ t ξ l ( k 0 a ) Φ v 2 ( k t a ) μ t ξ l ( k 0 a ) Φ v 2 ( k t a ) ,
Φ l ( x ) = π x 2 J l + 1 2 ( x ) ,
ξ l ( x ) = π x 2 ( J l + 1 2 ( x ) + i N l + 1 2 ( x ) )
v 1 = [ l ( l + 1 ) ɛ t ɛ r + 1 4 ] 1 / 2 1 2 ,
v 2 = [ l ( l + 1 ) μ t μ r + 1 4 ] 1 / 2 1 2 .
Q ext = 2 k 0 2 a 2 l = 1 ( 2 l + 1 ) Re ( b l e + b l m ) , Q sca = 2 k 0 2 a 2 l = 1 ( 2 l + 1 ) [ | b l e | 2 + | b l m | 2 ] , Q rbs = 1 k 0 a Re | l = 1 ( 1 ) l ( 2 l + 1 ) ( b l e b l m ) | 2 .
b l e = F b e ( l ) F b e ( l ) + i G b e ( l ) and b l m = F b m ( l ) F b m ( l ) + i G b m ( l ) ,
F b e = ɛ t Φ l ( k 0 a ) Φ v 1 ( k t a ) μ t Φ l ( k 0 a ) Φ v 1 ( k t a ) , G b e = ɛ t χ l ( k 0 a ) Φ v 1 ( k t a ) μ t χ l ( k 0 a ) Φ v 1 ( k t a ) , F b m = ɛ t Φ l ( k 0 a ) Φ v 2 ( k t a ) μ t Φ l ( k 0 a ) Φ v 2 ( k t a ) , G b m = ɛ t χ l ( k 0 a ) Φ v 2 ( k t a ) μ t χ l ( k 0 a ) Φ v 2 ( k t a ) ,
ɛ D = 1 ω p 2 ω 2 + i γ ω with ω p = ( ne 2 ɛ 0 m 0 ) 1 / 2 ,
ɛ D = 1 3 ω R 2 + γ R 2 + i γ R ω R 3 ω R 2 + γ R 2 .
γ l = q 2 l + 1 ( l + 1 ) [ l ( 2 l 1 ) ! ! ] 2 ( d ɛ D / d ω ) l

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