Abstract

Eigenmodes of a chiral sphere placed in a dielectric medium were investigated in details. Excitation of these eigenmodes by a plane wave and a chiral molecule radiation was studied both analytically and numerically. It was found that decay rates of “right” and “left” enantiomers are different in the presence of the chiral sphere. Strong dependence of radiation pattern of the chiral molecule placed in the vicinity of the chiral sphere on chirality strength was also demonstrated. An interesting correlation between chirality of sphere and spatial spirality (helicity, vorticity ...) of the electromagnetic fields in the presence of chiral sphere was observed for the first time.

© 2014 Optical Society of America

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2013 (1)

D. V. Guzatov, V. V. Klimov, and N. S. Poprukailo, “Spontaneous radiation of a chiral molecule located near a half-space of a bi-isotropic material,” J. Exp. Theor. Phys. 116(4), 531–540 (2013).
[CrossRef]

2012 (2)

V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97(4), 47004 (2012).
[CrossRef]

D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14(12), 123009 (2012).
[CrossRef]

2011 (1)

S. A. R. Horsley, “Consistency of certain constitutive relations with quantum electromagnetism,” Phys. Rev. A 84(6), 063822 (2011).
[CrossRef]

2010 (1)

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

2009 (4)

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009).
[CrossRef]

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009).
[CrossRef]

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009).
[CrossRef]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

2005 (1)

V. Klimov and M. Ducloy, “Quadrupole transitions near an interface: general theory and application to an atom inside a planar cavity,” Phys. Rev. A 72(4), 043809 (2005).
[CrossRef]

2004 (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

L. Li, D. You, M. Leong, and T. Kong, “Electromagnetic scattering by multilayered chiral-media structures: A scattering-to-radiation transform,” Prog. Electromagnetics Res. 26, 249–291 (2000).
[CrossRef]

1999 (1)

L. Li, Y. Dan, M. S. Leong, and J. A. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” J. Electromagn. Waves Appl. 13(9), 1203–1205 (1999).
[CrossRef]

1998 (1)

R. Sharma and N. Balakrishnan, “Scattering of electromagnetic waves from chirally coated cylinders,” Smart Mater. Struct. 7(4), 512–521 (1998).
[CrossRef]

1994 (1)

A. Lakhtakia and W. S. Weiglhofer, “Constraint on linear, homogeneous, constitutive relations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(6), 5017–5019 (1994).
[CrossRef] [PubMed]

1990 (4)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation by a point electric dipole embedded in a chiral sphere,” J. Phys. D Appl. Phys. 23(5), 481–485 (1990).
[CrossRef]

N. Engheta and M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 67(2), 639 (1990).
[CrossRef]

M. W. Kowarz and N. Engheta, “Spherical chirolenses,” Opt. Lett. 15(6), 299–301 (1990).
[CrossRef] [PubMed]

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10(1-2), 201–211 (1990).
[CrossRef]

1989 (1)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Eigenmodes of a chiral sphere with a perfectly conducting coating,” J. Phys. D Appl. Phys. 22(6), 825–828 (1989).
[CrossRef]

1985 (1)

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface. II,” Phys. Rev. A 32(4), 2030–2043 (1985).
[CrossRef] [PubMed]

1984 (1)

J. Wylie and J. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30(3), 1185–1193 (1984).
[CrossRef] [PubMed]

1978 (2)

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

C. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66(1), 105–109 (1978).
[CrossRef]

1975 (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62(4), 1566 (1975).
[CrossRef]

1974 (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29(3), 458–462 (1974).
[CrossRef]

1971 (1)

B. V. Bokut’, A. N. Serdyukov, and F. I. Fedorov, “Phenomenological theory of optically active crystals,” Sov. Phys. Crystallogr. 15, 871–874 (1971).

1929 (1)

L. Rosenfeld, “Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen,” Z. Phys. 52(3-4), 161–174 (1929).
[CrossRef]

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Balakrishnan, N.

R. Sharma and N. Balakrishnan, “Scattering of electromagnetic waves from chirally coated cylinders,” Smart Mater. Struct. 7(4), 512–521 (1998).
[CrossRef]

Bohren, C.

C. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66(1), 105–109 (1978).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62(4), 1566 (1975).
[CrossRef]

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29(3), 458–462 (1974).
[CrossRef]

Bokut’, B. V.

B. V. Bokut’, A. N. Serdyukov, and F. I. Fedorov, “Phenomenological theory of optically active crystals,” Sov. Phys. Crystallogr. 15, 871–874 (1971).

Chance, R.

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Dan, Y.

L. Li, Y. Dan, M. S. Leong, and J. A. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” J. Electromagn. Waves Appl. 13(9), 1203–1205 (1999).
[CrossRef]

Decker, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Dong, J.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009).
[CrossRef]

Ducloy, M.

V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97(4), 47004 (2012).
[CrossRef]

V. Klimov and M. Ducloy, “Quadrupole transitions near an interface: general theory and application to an atom inside a planar cavity,” Phys. Rev. A 72(4), 043809 (2005).
[CrossRef]

Engheta, N.

M. W. Kowarz and N. Engheta, “Spherical chirolenses,” Opt. Lett. 15(6), 299–301 (1990).
[CrossRef] [PubMed]

N. Engheta and M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 67(2), 639 (1990).
[CrossRef]

Fan, Z.

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

Fedorov, F. I.

B. V. Bokut’, A. N. Serdyukov, and F. I. Fedorov, “Phenomenological theory of optically active crystals,” Sov. Phys. Crystallogr. 15, 871–874 (1971).

Fedotov, V.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009).
[CrossRef]

Gansel, J. K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Govorov, A. O.

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10(7), 2580–2587 (2010).
[CrossRef] [PubMed]

Guzatov, D. V.

D. V. Guzatov, V. V. Klimov, and N. S. Poprukailo, “Spontaneous radiation of a chiral molecule located near a half-space of a bi-isotropic material,” J. Exp. Theor. Phys. 116(4), 531–540 (2013).
[CrossRef]

D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14(12), 123009 (2012).
[CrossRef]

V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97(4), 47004 (2012).
[CrossRef]

V. V. Klimov and D. V. Guzatov, “Focussing dipole radiation by chiral layer with negative refraction: case of thin slab,” Quantum Electron.submitted.

V. V. Klimov and D. V. Guzatov, “Focussing dipole radiation by chiral layer with negative refraction: case of thick slab,” Quantum Electron.44 (2014) (in print).

He, S.

Horsley, S. A. R.

S. A. R. Horsley, “Consistency of certain constitutive relations with quantum electromagnetism,” Phys. Rev. A 84(6), 063822 (2011).
[CrossRef]

Klimov, V.

V. Klimov and M. Ducloy, “Quadrupole transitions near an interface: general theory and application to an atom inside a planar cavity,” Phys. Rev. A 72(4), 043809 (2005).
[CrossRef]

Klimov, V. V.

D. V. Guzatov, V. V. Klimov, and N. S. Poprukailo, “Spontaneous radiation of a chiral molecule located near a half-space of a bi-isotropic material,” J. Exp. Theor. Phys. 116(4), 531–540 (2013).
[CrossRef]

V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97(4), 47004 (2012).
[CrossRef]

D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14(12), 123009 (2012).
[CrossRef]

V. V. Klimov and D. V. Guzatov, “Focussing dipole radiation by chiral layer with negative refraction: case of thick slab,” Quantum Electron.44 (2014) (in print).

V. V. Klimov and D. V. Guzatov, “Focussing dipole radiation by chiral layer with negative refraction: case of thin slab,” Quantum Electron.submitted.

Kong, J. A.

L. Li, Y. Dan, M. S. Leong, and J. A. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” J. Electromagn. Waves Appl. 13(9), 1203–1205 (1999).
[CrossRef]

Kong, T.

L. Li, D. You, M. Leong, and T. Kong, “Electromagnetic scattering by multilayered chiral-media structures: A scattering-to-radiation transform,” Prog. Electromagnetics Res. 26, 249–291 (2000).
[CrossRef]

Koschny, T.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009).
[CrossRef]

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009).
[CrossRef]

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009).
[CrossRef]

Kowarz, M. W.

N. Engheta and M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 67(2), 639 (1990).
[CrossRef]

M. W. Kowarz and N. Engheta, “Spherical chirolenses,” Opt. Lett. 15(6), 299–301 (1990).
[CrossRef] [PubMed]

Lakhtakia, A.

A. Lakhtakia and W. S. Weiglhofer, “Constraint on linear, homogeneous, constitutive relations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(6), 5017–5019 (1994).
[CrossRef] [PubMed]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation by a point electric dipole embedded in a chiral sphere,” J. Phys. D Appl. Phys. 23(5), 481–485 (1990).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Eigenmodes of a chiral sphere with a perfectly conducting coating,” J. Phys. D Appl. Phys. 22(6), 825–828 (1989).
[CrossRef]

Leong, M.

L. Li, D. You, M. Leong, and T. Kong, “Electromagnetic scattering by multilayered chiral-media structures: A scattering-to-radiation transform,” Prog. Electromagnetics Res. 26, 249–291 (2000).
[CrossRef]

Leong, M. S.

L. Li, Y. Dan, M. S. Leong, and J. A. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” J. Electromagn. Waves Appl. 13(9), 1203–1205 (1999).
[CrossRef]

Li, L.

L. Li, D. You, M. Leong, and T. Kong, “Electromagnetic scattering by multilayered chiral-media structures: A scattering-to-radiation transform,” Prog. Electromagnetics Res. 26, 249–291 (2000).
[CrossRef]

L. Li, Y. Dan, M. S. Leong, and J. A. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” J. Electromagn. Waves Appl. 13(9), 1203–1205 (1999).
[CrossRef]

Linden, S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004).
[CrossRef] [PubMed]

Plum, E.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009).
[CrossRef]

Poprukailo, N. S.

D. V. Guzatov, V. V. Klimov, and N. S. Poprukailo, “Spontaneous radiation of a chiral molecule located near a half-space of a bi-isotropic material,” J. Exp. Theor. Phys. 116(4), 531–540 (2013).
[CrossRef]

Prock, A.

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Rosenfeld, L.

L. Rosenfeld, “Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen,” Z. Phys. 52(3-4), 161–174 (1929).
[CrossRef]

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Serdyukov, A. N.

B. V. Bokut’, A. N. Serdyukov, and F. I. Fedorov, “Phenomenological theory of optically active crystals,” Sov. Phys. Crystallogr. 15, 871–874 (1971).

Sharma, R.

R. Sharma and N. Balakrishnan, “Scattering of electromagnetic waves from chirally coated cylinders,” Smart Mater. Struct. 7(4), 512–521 (1998).
[CrossRef]

Silbey, R.

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Sipe, J.

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[CrossRef]

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[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Possible realization of a chiral sphere: (a) a gold core covered by a sugar shell; (b) radial helices forming the sphere; (c) effective description of a nanoparticle.

Fig. 2
Fig. 2

Dependence of the inverse determinant 1 / | Δ 1 | (see Eq. (15)) for a chiral sphere with the radius a = 70 nm and ε = 2 + 0.04 j on the dimensionless chirality parameter χ and the wavenumber normalized imaginary part in free space Im k m / Re k m . The real part of the wavenumber is fixed and defined by the wavelength λ = 570 nm , Re k m = 2 π / λ .

Fig. 3
Fig. 3

Projection of eigenmodes curves of a chiral sphere with ε = 2 + 0.04 j in the coordinates χ , Re k m , Im k m onto the Re ( k m a ) and Im ( k m a ) plane for the orbital quantum number n = 1 . Different colors of the curves correspond to different radial quantum numbers ν = 1...5 . Colored marks near points indicate the value of the dimensionless parameter of chirality χ in them (the 3rd coordinate).

Fig. 4
Fig. 4

Distribution of the Re E φ of the eigenmode on the surface of a chiral sphere corresponding to the mode with n = 1 4 , m = 1 , ν = 1 . Parameters of the sphere are a = 70 nm , ε = 2 + 0.04 j . Values of the dimensionless parameter of chirality χ are 0.4914, 0.5630, 0.5923, 0.6105 correspondingly. Red and blue colors show extreme (positive and negative) values of the field. Near-zero values were removed for a better perception of the field structure. One can clearly see a spiral spatial structure, where number of turns increases with the orbital quantum number n .

Fig. 5
Fig. 5

Extinction efficiency of a chiral sphere with ε = 2 + 0.04 j and a = 70 nm irradiated by a plane linearly polarized (black), right-handed polarized (red), and left-handed polarized waves with the wavelength λ = 570 nm depending on the dimensionless parameter of chirality. Solid lines stand for analytical solution [38], while circles show results of numerical simulation. The geometry of the problem is shown in the inset. Red numbers near peaks indicate radial numbers ν = 1 , 2 , 3 , 4 .

Fig. 6
Fig. 6

Geometry of the problem and distribution of the z component of the electric field E z in the plane z = 0 (a shadowed plane in the geometry inset) for different values of the dimensionless parameter of chirality corresponding to eigenmodes with n = 1 , m = 1 and ν = 1 , 2 , 3 , 4 . The case χ = 0 is also presented for comparison.

Fig. 7
Fig. 7

Distribution of the z-component of the electric field in different planes z = 0 , 150 , 300 n m for a chiral dielectric sphere with χ = 0.363 , ε = 3 + 0.1 j , a = 70 nm exited by a linearly polarized plane wave.

Fig. 8
Fig. 8

Total decay rate of a chiral molecule situated in the vicinity of a chiral dielectric sphere ε = 2 + 0.04 j with the radius a = 70 nm versus the dimensionless parameter of chirality χ . Solid lines correspond to analytical results [25], circles denote numerical calculation according to the Eq. (18). From the top down, lines correspond to: black – radially oriented (along the sphere radius) right molecule, blue – radially oriented left molecule, red – tangentially oriented right molecule, and green – tangentially oriented left molecule.

Fig. 9
Fig. 9

Far-field pattern Eq. (20) of a radially oriented molecule (along OZ – see inset Fig. 8) (which excite modes with m = 0 ) situated in the vicinity of a chiral dielectric sphere ε = 2 + 0.04 j in the y = 0 plane for three different values of the dimensionless parameter of chirality: χ = 0 corresponds to the absence of chirality and χ = 0.563 , 0.5925 corresponds to the second and third resonances in Fig. 8 (black and blue lines). Since the field in this case is axisymmetric, the distribution in the x = 0 plane will be the same.

Fig. 10
Fig. 10

Far-field pattern Eq. (20) of a tangentially oriented molecule (which excite modes with m = 1 ) situated in the vicinity of a chiral dielectric sphere with ε = 2 + 0.04 j in two different projections for four values of the dimensionless parameter of chirality χ : 0, 0.495, 0.562, 0.5925. For better understanding of the projections, colored planes are shown: z = 0– red, y = 0 – blue, x = 0 – green. The bottom set of pictures corresponds to the z = 0 view from negative z.

Equations (20)

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D = ε ( E + η rot E ) , B = μ ( H + η rot H ) ,
D = ε E + j κ c B , H = 1 μ B + j κ c E ,
rot { rot ( E ( 1 k 0 2 η 2 ε μ ) ) / μ } k 0 2 rot ( ε η E ) k 0 2 ε η rot E k 0 2 ε E = 0 ,
j = k 0 2 ( ε 2 η 2 ε 1 η 1 ) [ n 1 2 × E ] ,
H = { rot ( E ( 1 k 0 2 η 2 ε μ ) ) / μ k 0 2 η ε E } / j k 0
E in ( r ) = n = 1 m = n n ( A m n ( L ) Q m n ( L ) + A m n ( R ) Q m n ( R ) ) , H in ( r ) = j ( ε / μ ) 1 / 2 n = 1 m = n n ( A m n ( L ) Q m n ( L ) A m n ( R ) Q m n ( R ) ) ,
Q m n ( L ) = N ψ m n ( L ) + M ψ m n ( L ) , Q m n ( R ) = N ψ m n ( R ) M ψ m n ( R ) ,
M ψ m n ( J ) = rot ( r Ψ n m ( J ) ) , N ψ m n ( J ) = rot ( M ψ m n ( J ) ) / k J .
Ψ n m ( J ) = exp ( j m φ ) P n m ( cos θ ) j n ( k J r ) ,
k L = k 0 ( ε μ ) 1 / 2 ( 1 χ ( ε μ ) 1 / 2 ) 1 , k R = k 0 ( ε μ ) 1 / 2 ( 1 + χ ( ε μ ) 1 / 2 ) 1 ,
E out = n = 1 m = n n ( C m n N ζ m n + D m n M ζ m n ) , H out = j n = 1 m = n n ( D m n N ζ m n + C m n M ζ m n ) ,
M ζ m n = rot ( r Ζ m n ) , N ζ m n = rot ( M ζ m n ) / k J .
Ζ m n = exp ( j m φ ) P n m ( cos θ ) h n ( 1 ) ( k 0 r ) ,
[ E out × u r ] | r = a = [ E in × u r ] | r = a ; [ H out × u r ] | r = a = [ H in × u r ] | r = a ,
Δ n = W n ( L ) V n ( R ) + W n ( R ) V n ( L ) , W n ( J ) = ( ε / μ ) 1 / 2 ψ n ( k J a ) ζ n ( 1 ) ( k 0 a ) ψ n ( k J a ) ζ n ( 1 ) ( k 0 a ) , V n ( J ) = ψ n ( k J a ) ζ n ( 1 ) ( k 0 a ) ( ε / μ ) 1 / 2 ψ n ( k J a ) ζ n ( 1 ) ( k 0 a ) ,
A m n ( R ) = k R / k L α n , C m n = k R / k L [ A n ( L ) + α n A n ( R ) ] / G n , D m n = k R / k L [ B n ( L ) α n B n ( R ) ] / G n ,
α n = V n ( L ) / V n ( R ) = W n ( L ) / W n ( R ) , A n ( J ) = ( ε / μ ) 1 / 2 ψ n ( k J a ) ψ n ( k 0 a ) ψ n ( k J a ) ψ n ( k 0 a ) , B n ( J ) = ψ n ( k J a ) ψ n ( k 0 a ) ( ε / μ ) 1 / 2 ψ n ( k J a ) ψ n ( k 0 a ) , G n = ψ n ( k 0 a ) ζ n ( 1 ) ( k 0 a ) ψ n ( k 0 a ) ζ n ( 1 ) ( k 0 a ) .
γ tot γ 0 = 1 + 3 2 Im [ d 0 * E sc ( r 0 ) + j m 0 * H sc ( r 0 ) k 0 3 ( | d 0 | 2 + | m 0 | 2 ) ] ,
γ 0 = 4 / 3 k 0 3 ( | d 0 | 2 + | m 0 | 2 ) / .
D ( θ , φ ) = 4 π P r a d ( θ , φ ) / P r a d ,

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