Abstract

A self-consistent analytical approach was developed to describe the optical properties of an assembly of magneto-plasmonic composite nanoparticles distributed on a dielectric substrate. The proposed theory is based on the effective susceptibility concept in the frame of Green’s function method and accounts for the local-field effects in the system. The analytical expressions for effective susceptibility of a single particle, as well as for assemblies of particles, on the substrate were derived to treat the scattered electromagnetic field by an assembly of composite nanoparticles. The developed formalism was applied to describe the polar Kerr effect in the Au/Co-based magneto-plasmonic system. The results of the simulation suggest that the magneto-optical properties of the considered magneto-plasmonic composite are determined by the interplay between “particle–particle” and “particle–surface” interactions, which are mainly given by the geometry of the particles, their concentration, and the optical properties of the substrate.

© 2010 Optical Society of America

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2010 (1)

V. Lozovski, “The effective susceptibility concept in the electrodynamics of nanosystems,” J. Comput. Theor. Nanosci. 7, 2077–2093 (2010).
[CrossRef]

2009 (2)

N. A. Mirin and N. J. Halas, “Light-bending nanoparticles,” Nano Lett. 9, 1255–1259 (2009).
[CrossRef] [PubMed]

G. Armelles, A. Cebollada, A. García-Martín, J. M. García-Martín, M. U. González, J. B. González-Díaz, E. Ferreiro-Vila, and J. F. Torrado, “Magnetoplasmonic nanostructures: systems supporting both plasmonic and magnetic properties,” J. Opt. A, Pure Appl. Opt. 11, 114023 (2009).
[CrossRef]

2008 (6)

R. Ruiz, H. Kang, F. A. Detcheverry, E. Dobisz, D. S. Kercher, T. R. Albrecht, J. J. de Pablo, and P. F. Nealey, “Density multiplication and improved lithography by directed block copolymer assembly,” Science 321, 936–939 (2008).
[CrossRef] [PubMed]

I. Bita, J. K. W. Yang, J. S. Jung, C. A. Ross, E. L. Thomas, and K. K. Berggren, “Graphoepitaxy of self-assembled block copolymers on two-dimensional periodic patterned templates,” Science 321, 939–943 (2008).
[CrossRef] [PubMed]

M. De, P. S. Ghosh, and V. M. Rotello, “Applications of nanoparticles in biology,” Adv. Mater. 20, 4225–4241 (2008).
[CrossRef]

P. Weinberger, “John Kerr and his effects found in 1877 and 1878,” Philos. Mag. Lett. 88, 897–907 (2008).
[CrossRef]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au nanosandwiches with enhanced magneto-optical activity,” Small 4, 202–205 (2008).
[CrossRef] [PubMed]

G. Armelles, J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, M. U. González, S. Acimovic, J. Cesario, R. Quidant, and G. Badenes, “Localized surface plasmon resonance effects on the magneto-optical activity of continuous Au/Co/Au trilayers,” Opt. Express 16, 16104–16112 (2008).
[CrossRef] [PubMed]

2007 (2)

C. Clavero, G. Armelles, J. Margueritat, J. Gonzalo, M. G. del Muro, A. Labarta, and X. Batlle, “Interface effects in the magneto-optical properties of Co nanoparticles in dielectric matrix,” Appl. Phys. Lett. 90, 182506 (2007).
[CrossRef]

J. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. Wehrspohn, and U. Gösele, “Enhanced magneto-optics and size effects in ferromagnetic nanowire arrays,” Adv. Mater. 19, 2643–2647 (2007).
[CrossRef]

2005 (2)

A. García-Martín, G. Armelles, and S. Pereira, “Light transport in photonic crystals composed of magneto-optically active materials,” Phys. Rev. B 71, 205116 (2005).
[CrossRef]

M. Albrecht, G. Hu, I. L. Guhr, T. C. Ulbrich, J. Boneberg, P. Leiderer, and G. Schatz, “Magnetic multilayers on nanospheres,” Nature Mater. 4, 203–206 (2005).
[CrossRef]

2004 (4)

P. H. M. Hoet, I. Brüske-Hohlfeld, and O. V. Salata, “Nanoparticles—known and unknown health risks,” J. Nanobiotechnology 2, 12 (2004).
[CrossRef] [PubMed]

M. Diwekar, V. Kamaev, J. Shi, and Z. V. Vardeny, “Optical and magneto-optical studies of two-dimensional metallodielectric photonic crystals on cobalt films,” Appl. Phys. Lett. 84, 3112–3114 (2004).
[CrossRef]

M. Abe and T. Suwa, “Surface plasma resonance and magneto-optical enhancement in composites containing multicore-shell structured nanoparticles,” Phys. Rev. B 70, 235103 (2004).
[CrossRef]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: the concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
[CrossRef]

2003 (2)

S. Melle, J. L. Menéndez, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Magneto-optical properties of nickel nanowire arrays,” Appl. Phys. Lett. 83, 4547–4549 (2003).
[CrossRef]

J. Williams, R. Lansdown, R. Sweitzer, M. Romanowski, R. LaBell, R. Ramaswami, and E. Unger, “Nanoparticle drug delivery system for intravenous delivery of topoisomerase inhibitors,” J. Controlled Release 91, 167–172 (2003).
[CrossRef]

2001 (2)

V. Lozovski, “Susceptibilities of nano-particles at the surface of a solid,” Physica E (Amsterdam) 9, 642–651 (2001).
[CrossRef]

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC, 2001).
[CrossRef]

2000 (3)

Ch. Girard, C. Joachim, and S. Gauthier, “The physics of the near-field,” Rep. Prog. Phys. 63, 893–938 (2000).
[CrossRef]

G. D. Mahan, Many-Particle Physics (Kluwer Academic/Plenum, 2000).

S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, “Monodisperse FePt nanoparticles and ferromagnetic FePt nanocrystal superlattices,” Science 287, 1989–1992 (2000).
[CrossRef] [PubMed]

1996 (2)

C. Y. You and S. C. Shin, “Derivation of simplified analytic formulae for magneto-optical Kerr effects,” Appl. Phys. Lett. 69, 1315–1317 (1996).
[CrossRef]

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

1993 (1)

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[CrossRef]

1992 (1)

M. L. Bah, A. Akjouj, and L. Dobrzhynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 97–131 (1992).
[CrossRef]

1988 (1)

T. Katayama, Y. Suzuki, H. Awano, Y. Nishihara, and N. Koshizuka, “Enhancement of the magneto-optical Kerr rotation in Fe/Cu bilayered films,” Phys. Rev. Lett. 60, 1426–1429 (1988).
[CrossRef] [PubMed]

1985 (1)

1983 (1)

1980 (2)

E. M. Lifshitz and L. P. Pitaevskii, in Statistical Physics (Part 2), Course of Theoretical Physics, E.M.Lifshitz and L.P.Pitaevskii, eds. (Pergamon, 1980), Vol. 9.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1979 (1)

V. M. Kolomietz and P. J. Siemens, “Self-consistent field approximation to the linear response function for nuclear dissipation,” Nucl. Phys. A 314, 141–160 (1979).
[CrossRef]

1975 (1)

A. A. Maradudin and D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

1974 (1)

R. Carey and B. W. J. Thomas, “The theory of the Voigt effect in ferromagnetic materials,” J. Phys. D: Appl. Phys. 7, 2362–2368 (1974).
[CrossRef]

1965 (1)

A. A. Abrikosov, L. P. Gor’kov, and I. Ye. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics (Pergamon, 1965).

1964 (1)

G. S. Krinchik, “Ferromagnetic Hall effect at optical frequencies and inner effective magnetic field of ferromagnetic metals,” J. Appl. Phys. 35, 1089–1092 (1964).
[CrossRef]

Abe, M.

M. Abe and T. Suwa, “Surface plasma resonance and magneto-optical enhancement in composites containing multicore-shell structured nanoparticles,” Phys. Rev. B 70, 235103 (2004).
[CrossRef]

Abrikosov, A. A.

A. A. Abrikosov, L. P. Gor’kov, and I. Ye. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics (Pergamon, 1965).

Acimovic, S.

Akjouj, A.

M. L. Bah, A. Akjouj, and L. Dobrzhynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 97–131 (1992).
[CrossRef]

Alaverdyan, Y.

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au nanosandwiches with enhanced magneto-optical activity,” Small 4, 202–205 (2008).
[CrossRef] [PubMed]

Albrecht, M.

M. Albrecht, G. Hu, I. L. Guhr, T. C. Ulbrich, J. Boneberg, P. Leiderer, and G. Schatz, “Magnetic multilayers on nanospheres,” Nature Mater. 4, 203–206 (2005).
[CrossRef]

Albrecht, T. R.

R. Ruiz, H. Kang, F. A. Detcheverry, E. Dobisz, D. S. Kercher, T. R. Albrecht, J. J. de Pablo, and P. F. Nealey, “Density multiplication and improved lithography by directed block copolymer assembly,” Science 321, 936–939 (2008).
[CrossRef] [PubMed]

Alexander, R. W.

Armelles, G.

G. Armelles, A. Cebollada, A. García-Martín, J. M. García-Martín, M. U. González, J. B. González-Díaz, E. Ferreiro-Vila, and J. F. Torrado, “Magnetoplasmonic nanostructures: systems supporting both plasmonic and magnetic properties,” J. Opt. A, Pure Appl. Opt. 11, 114023 (2009).
[CrossRef]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au nanosandwiches with enhanced magneto-optical activity,” Small 4, 202–205 (2008).
[CrossRef] [PubMed]

G. Armelles, J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, M. U. González, S. Acimovic, J. Cesario, R. Quidant, and G. Badenes, “Localized surface plasmon resonance effects on the magneto-optical activity of continuous Au/Co/Au trilayers,” Opt. Express 16, 16104–16112 (2008).
[CrossRef] [PubMed]

C. Clavero, G. Armelles, J. Margueritat, J. Gonzalo, M. G. del Muro, A. Labarta, and X. Batlle, “Interface effects in the magneto-optical properties of Co nanoparticles in dielectric matrix,” Appl. Phys. Lett. 90, 182506 (2007).
[CrossRef]

J. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. Wehrspohn, and U. Gösele, “Enhanced magneto-optics and size effects in ferromagnetic nanowire arrays,” Adv. Mater. 19, 2643–2647 (2007).
[CrossRef]

A. García-Martín, G. Armelles, and S. Pereira, “Light transport in photonic crystals composed of magneto-optically active materials,” Phys. Rev. B 71, 205116 (2005).
[CrossRef]

S. Melle, J. L. Menéndez, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Magneto-optical properties of nickel nanowire arrays,” Appl. Phys. Lett. 83, 4547–4549 (2003).
[CrossRef]

Awano, H.

T. Katayama, Y. Suzuki, H. Awano, Y. Nishihara, and N. Koshizuka, “Enhancement of the magneto-optical Kerr rotation in Fe/Cu bilayered films,” Phys. Rev. Lett. 60, 1426–1429 (1988).
[CrossRef] [PubMed]

Badenes, G.

Bah, M. L.

M. L. Bah, A. Akjouj, and L. Dobrzhynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 97–131 (1992).
[CrossRef]

Batlle, X.

C. Clavero, G. Armelles, J. Margueritat, J. Gonzalo, M. G. del Muro, A. Labarta, and X. Batlle, “Interface effects in the magneto-optical properties of Co nanoparticles in dielectric matrix,” Appl. Phys. Lett. 90, 182506 (2007).
[CrossRef]

Bell, R. J.

Bell, R. R.

Berggren, K. K.

I. Bita, J. K. W. Yang, J. S. Jung, C. A. Ross, E. L. Thomas, and K. K. Berggren, “Graphoepitaxy of self-assembled block copolymers on two-dimensional periodic patterned templates,” Science 321, 939–943 (2008).
[CrossRef] [PubMed]

Bita, I.

I. Bita, J. K. W. Yang, J. S. Jung, C. A. Ross, E. L. Thomas, and K. K. Berggren, “Graphoepitaxy of self-assembled block copolymers on two-dimensional periodic patterned templates,” Science 321, 939–943 (2008).
[CrossRef] [PubMed]

Boneberg, J.

M. Albrecht, G. Hu, I. L. Guhr, T. C. Ulbrich, J. Boneberg, P. Leiderer, and G. Schatz, “Magnetic multilayers on nanospheres,” Nature Mater. 4, 203–206 (2005).
[CrossRef]

Bonn, M.

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: the concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004).
[CrossRef]

Bozhevolnyi, S.

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[CrossRef]

Brüske-Hohlfeld, I.

P. H. M. Hoet, I. Brüske-Hohlfeld, and O. V. Salata, “Nanoparticles—known and unknown health risks,” J. Nanobiotechnology 2, 12 (2004).
[CrossRef] [PubMed]

Carey, R.

R. Carey and B. W. J. Thomas, “The theory of the Voigt effect in ferromagnetic materials,” J. Phys. D: Appl. Phys. 7, 2362–2368 (1974).
[CrossRef]

Cebollada, A.

G. Armelles, A. Cebollada, A. García-Martín, J. M. García-Martín, M. U. González, J. B. González-Díaz, E. Ferreiro-Vila, and J. F. Torrado, “Magnetoplasmonic nanostructures: systems supporting both plasmonic and magnetic properties,” J. Opt. A, Pure Appl. Opt. 11, 114023 (2009).
[CrossRef]

G. Armelles, J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, M. U. González, S. Acimovic, J. Cesario, R. Quidant, and G. Badenes, “Localized surface plasmon resonance effects on the magneto-optical activity of continuous Au/Co/Au trilayers,” Opt. Express 16, 16104–16112 (2008).
[CrossRef] [PubMed]

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au nanosandwiches with enhanced magneto-optical activity,” Small 4, 202–205 (2008).
[CrossRef] [PubMed]

Cesario, J.

Clavero, C.

C. Clavero, G. Armelles, J. Margueritat, J. Gonzalo, M. G. del Muro, A. Labarta, and X. Batlle, “Interface effects in the magneto-optical properties of Co nanoparticles in dielectric matrix,” Appl. Phys. Lett. 90, 182506 (2007).
[CrossRef]

Cloud, M. J.

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC, 2001).
[CrossRef]

De, M.

M. De, P. S. Ghosh, and V. M. Rotello, “Applications of nanoparticles in biology,” Adv. Mater. 20, 4225–4241 (2008).
[CrossRef]

de Pablo, J. J.

R. Ruiz, H. Kang, F. A. Detcheverry, E. Dobisz, D. S. Kercher, T. R. Albrecht, J. J. de Pablo, and P. F. Nealey, “Density multiplication and improved lithography by directed block copolymer assembly,” Science 321, 936–939 (2008).
[CrossRef] [PubMed]

del Muro, M. G.

C. Clavero, G. Armelles, J. Margueritat, J. Gonzalo, M. G. del Muro, A. Labarta, and X. Batlle, “Interface effects in the magneto-optical properties of Co nanoparticles in dielectric matrix,” Appl. Phys. Lett. 90, 182506 (2007).
[CrossRef]

Detcheverry, F. A.

R. Ruiz, H. Kang, F. A. Detcheverry, E. Dobisz, D. S. Kercher, T. R. Albrecht, J. J. de Pablo, and P. F. Nealey, “Density multiplication and improved lithography by directed block copolymer assembly,” Science 321, 936–939 (2008).
[CrossRef] [PubMed]

Diwekar, M.

M. Diwekar, V. Kamaev, J. Shi, and Z. V. Vardeny, “Optical and magneto-optical studies of two-dimensional metallodielectric photonic crystals on cobalt films,” Appl. Phys. Lett. 84, 3112–3114 (2004).
[CrossRef]

Dobisz, E.

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Figures (5)

Fig. 1
Fig. 1

Schematics of the reflection of light by an assembly of composite nanoparticles each consisting of magnetic (thickness h b ) and nonmagnetic (thickness h a ) layers. The total thickness of a composite particle is h = h a + h b .

Fig. 2
Fig. 2

(a) Kerr rotation as a function of the incident angle θ for p-polarized light. The dielectric function of the substrate, ε 1 , is a parameter. (b) Change in the self-energy term A with the dielectric function of the substrate, ε 1

Fig. 3
Fig. 3

Kerr rotation as a function of the incident angle θ: (a) s-polarized light; (b) p-polarized light. The geometric properties of the nanoparticles are varied. Geometric properties of the models are given in Table 1. The strength of the external magnetic field is 0.2 T. Concentration of the particles is n = 2.5 × 10 13 m 2 . Please note the different scales in (a) and (b).

Fig. 4
Fig. 4

Kerr rotation as a function of the incident angle θ: (a) s-polarized light; (b) p-polarized light. Concentration of nanoparticles is a parameter. The concentration of 2.50 × 10 13 m 2 corresponds to the average interparticle distance of 200 nm, the concentration of 2.78 × 10 13 m 2 corresponds to the average interparticle distance of 190 nm, and the concentration of 6.95 × 10 13 m 2 corresponds to the average interparticle distance of 120 nm. The strength of the external magnetic field is 0.2 T. The linear dimensions of the nanoparticles were chosen according to model 1 (Table 1). Note that legend in (a) corresponds to (b) as well.

Fig. 5
Fig. 5

Kerr rotation as a function of the frequency of the p-polarized incident light. (a) Concentration of the particles is a parameter. Note that the concentration of 2.50 × 10 13 m 2 corresponds to the average interparticle distance of 200 nm, while the concentration of 2.78 × 10 13 m 2 corresponds to the average interparticle distance of 190 nm. (b) The geometric properties of the nanoparticles are varied. Geometric properties of the models are given in Table 1. The strength of the external magnetic field is 0.2 T.

Tables (1)

Tables Icon

Table 1 Geometric Parameters of the Composite Cylindrical Nanoparticles (Radius r 0 ) Consisting of Nonmagnetic (Total Thickness h a ) and Magnetic Components (Total Thickness h b )

Equations (24)

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χ i j eff ( ω , B ) = χ 1 ( ω ) ( χ 2 ( ω ) i Q ( ω , B ) 0 i Q ( ω , B ) χ 2 ( ω ) 0 0 0 χ 2 ( ω ) ) ,
χ 1 ( ω ) = ω p b 2 ω 2 + i ω γ b
χ 2 ( ω ) = ρ b + ρ a ω p a 2 ω p b 2 ω 2 + i ω γ b ω 2 + i ω γ a
Q ( ω , B ) = ω c / ω ,
X i j S ( R , ω , B ) = χ i l eff ( ω , B ) [ δ j l + k 0 2 d R G j k ( R , R , ω ) χ k l eff ( ω , B ) ] 1 ,
A j k ( ω , ξ ) = k 0 2 V d R G j k NF ( R c , R , ω ) = ( A ( ω , ξ ) 0 0 0 A ( ω , ξ ) 0 0 0 A ( ω , ξ ) ) ,
A ( ω , ξ ) = 1 2 ( 1 f 1 ( ξ ) + 1 2 ε 1 ( ω ) 1 ε 1 ( ω ) + 1 [ f 2 ( ξ ) f 1 ( ξ ) ] ) ,
A ( ω , ξ ) = 1 f 1 ( ξ ) + ε 1 ( ω ) 1 ε 1 ( ω ) + 1 ( 1 2 ξ 2 [ f 2 ( ξ ) f 1 ( ξ ) ] 5 2 ( 1 f 1 ( ξ ) ) 7 18 ( 1 f 2 ( ξ ) ) ) ,
f 1 ( ξ ) = ( 1 + ( 2 ξ ) 2 ) 1 / 2 ,     f 2 ( ξ ) = ( 1 + ( 2 ξ / 3 ) 2 ) 1 / 2 ,     ξ = r 0 / h .
X i j S ( R c , ω , B ) = χ i k eff ( ω , B ) [ δ j k + A j l ( ω , ξ ) χ l k eff ( ω , B ) ] 1 = χ i k eff ( ω , B ) Φ k j ( ω , B ) / Ξ ( ω , B ) ,
Ξ ( ω , B ) = det [ δ j k + A j l ( ω ) χ l k eff ( ω , B ) ]
G y y ( θ ) = i ( 1 R S ( θ ) ) / b 2 ( θ ) ,     G x x ( θ ) = i b 2 ( θ ) ( 1 R P ( θ ) ) / b 2 ( θ ) ,     G x z ( θ ) = i   sin   θ ( 1 + R P ( θ ) ) ,
G z x ( θ ) = i   sin   θ ( 1 R P ( θ ) ) ,     G z z ( θ ) = i   sin 2 θ ( 1 + R P ( θ ) ) / b 2 ( θ ) ,
X i j L ( k , l , ω , B ) = X i l S ( l , ω , B ) [ δ j l + n k 0 2 π r 0 2 h G j k ( k , l , l , ω ) X k l S ( l , ω , B ) ] 1 ,
X i j L ( θ , l , ω , B ) = χ i k eff ( ω , B ) Φ k l ( ω , B ) Δ ( θ , l , ω , B ) Ξ ( ω , B ) Ψ l j ( θ , l , ω , B ) ,
Δ ( θ , l , ω , B ) = det [ δ j l + n k 0 2 π r 0 2 h G j k ( θ , l , l , ω ) X k l S ( l , ω , B ) ]
E i ( k , z , ω , B ) π n r 0 2 G i j ( k , z , ω ) 0 h d z X j l L ( k , z , ω , B ) E l ( 0 ) ( k , ω ) ,
Θ K S = r p s / r s s ,     Θ K P = r s p / r p p
η v u = E v / E u ( 0 )     ( v , u = x , y , z ) .
r s s = | η y y | 2 ,     r p s = | η x y + η z y | 2 .
r s p = | η y x + η y z | 2 ,     r p p = | η x x + η z z | 2 .
Θ K S = | ( G x x + G z x ) Π x y + ( G x z + G z y ) Π z y | 2 | G y y Π y y | 2 ,
Θ K P = | G y y ( Π y x   cos   θ + Π y z   sin   θ ) | 2 | ( G x x + G z x ) ( Π x x   cos   θ + Π x z   sin   θ ) + ( G x z + G z z ) ( Π z x   cos   θ + Π z z   sin   θ ) | 2 ,
Π i j = χ i k eff ( ω , B ) Φ k l ( ω , B ) Ψ l j ( θ , l , ω , B ) .

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