Abstract

The rigorous magneto-optical response coefficients of the oblique-incidence Faraday and Kerr effects in anisotropic monolayer graphene interfaced with bulk media in general are presented for the first time, to the author’s knowledge. The dependencies of magneto-optical rotation angle and ellipticity on the incident angle, incident polarization, and refractive indices of interfaced media are numerically analyzed and characteristic features are discussed in detail. It is shown that the Kerr effect has the substantially same off-diagonal response coefficient as the Faraday effect, in contrast with the case of conventional media. It is found that supergiant Kerr rotation angles of ±90° are produced in the graphene with a transparent substrate when illuminated with p-polarized light in the vicinity of the substrate’s Brewster angle.

© 2013 Optical Society of America

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  1. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
    [CrossRef]
  2. A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
    [CrossRef]
  3. I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
    [CrossRef]
  4. E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, “Magneto-optical and optical probes of gapped ground states of bilayer graphene,” arXiv:1205.2361 [cond-mat.str-el] (May2012).
  5. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
    [CrossRef]
  6. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
    [CrossRef]
  7. T. Yoshino and S. Tanaka, “Theory of polar magneto-optic effect in thin films,” Oyo Buturi 34, 572–581 (1965).
  8. T. Yoshino and S. Tanaka, “Longitudinal magneto-optic effect in ferromagnetic thin films. I,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
    [CrossRef]
  9. T. Yoshino and S. Tanaka, “Polar and longitudinal magneto-optical Kerr and Faraday coefficients of bi-gyrotropic thin films,” Jpn. J. Appl. Phys. 9, 1282–1283 (1970).
    [CrossRef]
  10. C. C. Robinson, “Polar and longitudinal magneto-optic effects in a planar geometry,” J. Opt. Soc. Am. 58, 1342–1347 (1968).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964), pp. 6–40.
  12. S. Tanaka, T. Yoshino, and T. Takahashi, “Longitudinal magneto-optic effect in ferromagnetic thin films. II,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
    [CrossRef]
  13. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
    [CrossRef]
  14. Y. Tomita and T. Yoshino, “Optimum design of multilayer-medium structures in a magneto-optical readout system,” J. Opt. Soc. Am. A 1, 809–817 (1984).
    [CrossRef]
  15. T. Yoshino and S. Tanaka, “Longitudinal magneto-optical effect in Ni and nickel-rich Ni-Fe films in visible and near infrared regions,” Opt. Commun. 1, 149–152 (1969).
    [CrossRef]

2011

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

2010

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[CrossRef]

2007

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

1984

1970

T. Yoshino and S. Tanaka, “Polar and longitudinal magneto-optical Kerr and Faraday coefficients of bi-gyrotropic thin films,” Jpn. J. Appl. Phys. 9, 1282–1283 (1970).
[CrossRef]

1969

T. Yoshino and S. Tanaka, “Longitudinal magneto-optical effect in Ni and nickel-rich Ni-Fe films in visible and near infrared regions,” Opt. Commun. 1, 149–152 (1969).
[CrossRef]

1968

1966

T. Yoshino and S. Tanaka, “Longitudinal magneto-optic effect in ferromagnetic thin films. I,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

S. Tanaka, T. Yoshino, and T. Takahashi, “Longitudinal magneto-optic effect in ferromagnetic thin films. II,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

1965

T. Yoshino and S. Tanaka, “Theory of polar magneto-optic effect in thin films,” Oyo Buturi 34, 572–581 (1965).

Bludov, Yu. V.

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964), pp. 6–40.

Bostwick, A.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

Crassee, I.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[CrossRef]

Ferreira, A.

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

Gorbar, E. V.

E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, “Magneto-optical and optical probes of gapped ground states of bilayer graphene,” arXiv:1205.2361 [cond-mat.str-el] (May2012).

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, “Magneto-optical and optical probes of gapped ground states of bilayer graphene,” arXiv:1205.2361 [cond-mat.str-el] (May2012).

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[CrossRef]

Kuzmenko, A. B.

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, “Magneto-optical and optical probes of gapped ground states of bilayer graphene,” arXiv:1205.2361 [cond-mat.str-el] (May2012).

Levallois, J.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

Neto, A. H. C.

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

Novoselov, K. S.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

Ostler, M.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

Pereira, V. M.

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

Peres, N. M. R.

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

Robinson, C. C.

Rotenberg, E.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

Seyller, Th.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, “Magneto-optical and optical probes of gapped ground states of bilayer graphene,” arXiv:1205.2361 [cond-mat.str-el] (May2012).

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[CrossRef]

Takahashi, T.

S. Tanaka, T. Yoshino, and T. Takahashi, “Longitudinal magneto-optic effect in ferromagnetic thin films. II,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

Tanaka, S.

T. Yoshino and S. Tanaka, “Polar and longitudinal magneto-optical Kerr and Faraday coefficients of bi-gyrotropic thin films,” Jpn. J. Appl. Phys. 9, 1282–1283 (1970).
[CrossRef]

T. Yoshino and S. Tanaka, “Longitudinal magneto-optical effect in Ni and nickel-rich Ni-Fe films in visible and near infrared regions,” Opt. Commun. 1, 149–152 (1969).
[CrossRef]

T. Yoshino and S. Tanaka, “Longitudinal magneto-optic effect in ferromagnetic thin films. I,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

S. Tanaka, T. Yoshino, and T. Takahashi, “Longitudinal magneto-optic effect in ferromagnetic thin films. II,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

T. Yoshino and S. Tanaka, “Theory of polar magneto-optic effect in thin films,” Oyo Buturi 34, 572–581 (1965).

Tomita, Y.

van der Marel, D.

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

Viana-Gomes, J.

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

Walter, A. L.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964), pp. 6–40.

Yoshino, T.

Y. Tomita and T. Yoshino, “Optimum design of multilayer-medium structures in a magneto-optical readout system,” J. Opt. Soc. Am. A 1, 809–817 (1984).
[CrossRef]

T. Yoshino and S. Tanaka, “Polar and longitudinal magneto-optical Kerr and Faraday coefficients of bi-gyrotropic thin films,” Jpn. J. Appl. Phys. 9, 1282–1283 (1970).
[CrossRef]

T. Yoshino and S. Tanaka, “Longitudinal magneto-optical effect in Ni and nickel-rich Ni-Fe films in visible and near infrared regions,” Opt. Commun. 1, 149–152 (1969).
[CrossRef]

T. Yoshino and S. Tanaka, “Longitudinal magneto-optic effect in ferromagnetic thin films. I,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

S. Tanaka, T. Yoshino, and T. Takahashi, “Longitudinal magneto-optic effect in ferromagnetic thin films. II,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

T. Yoshino and S. Tanaka, “Theory of polar magneto-optic effect in thin films,” Oyo Buturi 34, 572–581 (1965).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. Condens. Matter

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

Jpn. J. Appl. Phys.

T. Yoshino and S. Tanaka, “Longitudinal magneto-optic effect in ferromagnetic thin films. I,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

T. Yoshino and S. Tanaka, “Polar and longitudinal magneto-optical Kerr and Faraday coefficients of bi-gyrotropic thin films,” Jpn. J. Appl. Phys. 9, 1282–1283 (1970).
[CrossRef]

S. Tanaka, T. Yoshino, and T. Takahashi, “Longitudinal magneto-optic effect in ferromagnetic thin films. II,” Jpn. J. Appl. Phys. 5, 994–1000 (1966).
[CrossRef]

Nat. Mater.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

Nat. Photonics

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[CrossRef]

Nat. Phys.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, Th. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multi-layer graphene,” Nat. Phys. 7, 48–51 (2011).
[CrossRef]

Opt. Commun.

T. Yoshino and S. Tanaka, “Longitudinal magneto-optical effect in Ni and nickel-rich Ni-Fe films in visible and near infrared regions,” Opt. Commun. 1, 149–152 (1969).
[CrossRef]

Oyo Buturi

T. Yoshino and S. Tanaka, “Theory of polar magneto-optic effect in thin films,” Oyo Buturi 34, 572–581 (1965).

Phys. Rev. B

A. Ferreira, J. Viana-Gomes, Yu. V. Bludov, V. M. Pereira, N. M. R. Peres, and A. H. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[CrossRef]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, Th. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductance of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[CrossRef]

Other

E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, “Magneto-optical and optical probes of gapped ground states of bilayer graphene,” arXiv:1205.2361 [cond-mat.str-el] (May2012).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964), pp. 6–40.

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Figures (6)

Fig. 1.
Fig. 1.

Optical configuration of polar magneto-optic effect in monolayer graphene subject to perpendicular ( z axis) magnetic field H .

Fig. 2.
Fig. 2.

Magneto-optical off-diagonal responsivity | ORC | calculated as a function of incident angle θ u in SG of different refractive indices N l . Solid lines, | r s p | = | r p s | = | t s p | ; dashed lines, | t p s | .

Fig. 3.
Fig. 3.

Magneto-optical diagonal responsivity | DRC | (solid lines) calculated as a function of incident angle θ u in (a) FG and (b) SG of refractive index of 3 together with Fresnel transmissivity and reflectivity (dashed lines).

Fig. 4.
Fig. 4.

Faraday rotation angle φ F , [rad] (heavy solid line), phase-compensated Faraday rotation angle γ F , [rad] (thin solid line), and ellipticity ε F , (dashed line) calculated as a function of incident angle θ u for s - and p -polarization incidence in (a) FG and (b) SG of refractive index of 3.

Fig. 5.
Fig. 5.

Kerr rotation angle φ K , [rad] (heavy solid line), phase-compensated rotation angle γ K , [rad] (thin solid line), and ellipticity ε K , (dashed line) in FG, calculated as a function of incident angle θ u for (a)  s -polarization incidence and (b)  p -polarization incidence, where γ K , p is very close to φ K , p .

Fig. 6.
Fig. 6.

Kerr rotation angle φ K , [rad] (heavy solid line), phase-compensated rotation angle γ K , [rad] (thin solid line) and ellipticity ε K , (dashed line) in SG of refractive index of 3, calculated as a function of incident angle θ u for (a)  s -polarization incidence and (b)  p -polarization incidence.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

E⃗ i = ( 0 , E s , 0 ) ,
E⃗ r = ( R p cos θ u , R s , R p sin θ u ) ,
E⃗ t = ( T P cos θ , T s , T P sin θ ) ,
N u sin θ u = N sin θ .
H⃗ i = Z 0 1 ( N u E s cos θ u , 0 , N u E s sin θ u ) ,
H⃗ r = Z 0 1 ( N u R s cos θ u , N u R p , N u R s sin θ u ) ,
H⃗ t = Z 0 1 ( N T s cos θ , N T p , N T s sin θ ) ,
E i , x + E r , x = E t , x ,
E i , y + E r , y = E t , y .
J x = σ x x E t , x + σ x y E t , y ,
J y = σ y x E t , x + σ y y E t , y .
J x = H⃗ i , y + H⃗ r , y H⃗ t , y ,
J y = H⃗ i , x H⃗ r , x + H⃗ t , x .
R p cos θ u = T p cos θ ,
E s + R s = T s .
N u R p N T p = Z 0 σ x x T p cos θ + Z 0 σ x y T s ,
N u ( E s R s ) cos θ u N T s cos θ = Z 0 σ y y T s + Z 0 σ y x T p cos θ .
E⃗ i = ( E p cos θ u , 0 , E p sin θ u ) ,
H⃗ i = Z 0 1 ( 0 , N u E p , 0 ) ,
( E p R p ) cos θ u = T p cos θ ,
R s = T s ,
N u ( E p + R p ) N T p = Z 0 σ x x T p cos θ + Z 0 σ x y T s ,
N u R s cos θ u N T s cos θ = Z 0 σ y y T s + Z 0 σ y x T p cos θ .
[ T s T p ] = [ t s s t s p t p s t p p ] E s E p ,
[ R s R p ] = [ r s s r s p r p s r p p ] E s E p ,
r p s = t p s cos θ cos θ u = 2 N u Z 0 σ x y cos θ u cos θ Δ 1 ,
t s p = r s p = 2 N u Z 0 σ y x cos θ u cos θ Δ 1 ,
Δ = ( N cos θ u + N u cos θ + Z 0 σ x x cos θ u cos θ ) ( N u cos θ u + N cos θ + Z 0 σ y y ) Z 0 2 σ x y σ y x cos θ u cos θ .
t s s = 2 N u cos θ u ( N cos θ u + N u cos θ + Z 0 σ x x cos θ u cos θ ) Δ 1 ,
t p p = 2 N u cos θ u ( N u cos θ u + N cos θ + Z 0 σ y y ) Δ 1 ,
r s s = [ ( N cos θ u + N u cos θ + Z 0 σ x x cos θ u cos θ ) ( N u cos θ u N cos θ Z 0 σ y y ) + Z 0 2 σ x y σ x y cos θ u cos θ ] Δ 1 ,
r p p = [ ( N cos θ u N u cos θ + Z 0 σ x x cos θ u cos θ ) ( N u cos θ u + N cos θ + Z 0 σ y y ) Z 0 2 σ x y σ y x cos θ u cos θ ] Δ 1 ,
t s s r s s = 1 ,
t p p cos cos θ u + r p p = 1 .
Θ F , s = t p s t s s = Z 0 σ x y cos θ u N cos θ u + N u cos θ + Z 0 σ x x cos θ u cos θ ,
Θ F , p = t s p t p p = Z 0 σ y x cos θ N u cos θ u + N cos θ + Z 0 σ y y ,
Θ K , s = r p s r s s = 2 N u Z 0 σ x y cos θ u cos θ ( N cos θ u + N u cos θ + Z 0 σ x x cos θ u cos θ ) ( N u cos θ u N cos θ Z 0 σ y y ) + Z 0 2 σ x y σ x y cos θ u cos θ ,
Θ K , p = r s p r p p = 2 N u Z 0 σ y x cos θ u cos θ ( N cos θ u N u cos θ + Z 0 σ x x cos θ u cos θ ) ( N u cos θ u + N cos θ + Z 0 σ y y ) Z 0 2 σ x y σ x y cos θ u cos θ .
sin ( 2 φ M , ) = 2 [ Θ M , ] r { ( 1 | Θ M , | 2 ) 2 + 4 [ Θ M , ] r 2 } 1 2 ,
cos ( 2 φ M , ) = 1 | Θ M , | 2 [ ( 1 | Θ M , | 2 ) 2 + 4 [ Θ M , ] r 2 ] 1 2 ,
ε M , = tan ( 1 2 sin 1 2 [ Θ M , ] i 1 + | Θ M , | 2 ) ,
γ M , = ± tan 1 | Θ M , | ( | γ M , | | φ M , | ) ,
r p s = r s p = t s p = cos θ cos θ u t p s g ,
g = 2 N u Z 0 σ 1 cos θ u cos θ ( N cos θ u + N u cos θ + Z 0 σ 0 cos θ u cos θ ) ( N u cos θ u + N cos θ + Z 0 σ 0 ) + Z 0 2 σ 1 2 cos θ u cos θ .
g 0 = 2 N u Z 0 σ 1 ( N u + N + Z 0 σ 0 ) 2 + Z 0 2 σ 1 2 ,
t 0 = 2 N u ( N u + N + Z 0 σ 0 ) ( N u + N + Z 0 σ 0 ) 2 + Z 0 2 σ 1 2 ,
r 0 = ( N 2 N u 2 ) + 2 N Z 0 σ 0 + ( Z 0 2 σ 0 2 + Z 0 2 σ 1 2 ) ( N u + N + Z 0 σ 0 ) 2 + Z 0 2 σ 1 2 ,
Θ F , 0 = Z 0 σ 1 Z 0 σ 0 + N u + N ,
Θ K , 0 = 2 N u Z 0 σ 1 ( N 2 N u 2 ) + 2 N Z 0 σ 0 + ( Z 0 2 σ 0 2 + Z 0 2 σ 1 2 ) ,

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