Abstract

The Yeh’s 4×4 matrix formalism is applied to determine the electromagnetic wave response in multilayers with arbitrary magnetization. With restriction to magneto–optic (MO) effects linear in the off–diagonal permittivity tensor elements, a simplified characteristic matrix for a magnetic layer is obtained. For a magnetic film–substrate system analytical representations of the MO response expressed in terms of the Jones reflection matrix are provided. These are numerically evaluated for cases when the magnetization develops in three mutually perpendicular planes.

© Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. Z. Q. Qiu and S. D. Bader, "Surface magneto-optic Kerr effect (SMOKE)," J. Magn. Magn. Mat. 200, 664-78 (1999).
    [CrossRef]
  2. M. Bauer, R. Lopusnik, J. Fassbender and B. Hillebrands, "Suppression of magnetic field pulse induced magnetization precession by pulse tailoring," Appl. Phys. Lett. 76, 2758-60 (2000).
    [CrossRef]
  3. A. Berger, M. R. Pufall, "Generalized magneto-optical ellipsometry," Appl. Phys. Lett. 71, 965-967 (1997).
    [CrossRef]
  4. A. Berger, M. R. Pufall, "Quantitative vector magnetometry using generalized magneto-optical ellipsometry," J. Appl. Phys. 85, 4583-4585 (1999).
    [CrossRef]
  5. M. Schubert, T. E. Tiwald, J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
    [CrossRef]
  6. P. Yeh, "Optics of anisotropic layered media: a new 4 x 4 matrix algebra," Surface Sci. 96, 41-53 (1980).
    [CrossRef]
  7. S. Visnovsky, "Magneto-optical ellipsometry," Czech. J. Phys. B 36, 625-650 (1986).
    [CrossRef]
  8. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri and K. Driss-Khodja, "Effective medium equivalent to a symmetric multilayer at oblique incidence," Appl. Opt. 29, 2460-2465 (1990).
    [CrossRef] [PubMed]
  9. K. Rokushima and J. Yamakita, "Analysis of anisotropic dielectric gratings," J. Opt. Soc. Am. 73 901-908 (1983).
    [CrossRef]
  10. K. Postava, J. Pistora, D. Ciprian, D. Hrabovsky, M. Lesnak and A.R. Fert, "Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization," in 11th Slovak--Czech--Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE 3820, 412-422 (1999).
  11. I. Kopriva, D. Hrabovsky, K. Postava, D. Ciprian, J. Pistora and A.R.Fert, "Anisotropy of the quadratic magneto-optical effects in a cubic crystal," in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, M. Miler, eds., Proc. SPIE 4016, 54-59 (2000).
  12. J. Zak, E. R. Moog, C. Liu and S. D. Bader, "Magneto-optics of multilayers with arbitrary magnetization directions," Phys. Rev. B 43, 6423-6429 (1991).
    [CrossRef]
  13. H. F. Ding, S. Putter, H. P. Oepen and J. Kirschner, "Experimental method for separating longitudinal and polar Kerr signals," J. Magn. Magn. Mat. 212, L5-L11 (2000).
    [CrossRef]
  14. W. A. McGahan, Liang--Yao Chen, J.A.Woollam, "Variable angle of incidence analysis of magneto-optic multilayers," J. Appl. Phys. 67, 4801-4802 (1990).
    [CrossRef]
  15. S. Visnovsky, M. Nyvlt, V. Prosser, J. Ferre, G. Penissard, D. Renard and G. Sczigel, "Magnetooptical effects in Au/Co/Au ultrathin film sandwiches," J. Magn. Magn. Mater. 128, 179-189 (1993).
    [CrossRef]
  16. S. Visnovsky, M. Nyvlt, V. Prosser, R. Lopusnik, R. Urban, J. Ferre, G. Penissard, D. Renard and R. Krishnan, "Polar magneto-optics in simple ultrathin-magnetic-film structures," Phys. Rev. B 52, 1090-1106 (1995).
    [CrossRef]
  17. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Elsevier, Amsterdam--Lausanne-New York--Oxford--Shannon--Tokyo, 1987).
  18. G. E. Jellison, Jr., "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
    [CrossRef]
  19. F. Abeles, "Recherches sur la propagation des ondes electromagnetiques sinuso�dales dans les milieux stratifies. Application aux couches minces," Ann. Phys., Paris 5, 596-640 (1950). M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), pp.51-70.
  20. W. Wettling, "Magneto-optics of ferrites," J. Magn. Magn. Mat. 3, 147-160 (1976).
    [CrossRef]
  21. S. Visnovsky, "Magneto--optical permittivity tensor in crystals," Czech. J. Phys. B 36, 1424-1433 (1986).
  22. S. Visnovsky, "Magneto-optical longitudinal and transverse Kerr and birefringence effects in orthorhombic crystals," Czech. J. Phys. B 34, 969-980 (1984).
    [CrossRef]
  23. S. Visnovsky, "Magneto-optic effects in ultrathin structures at longitudinal and polar magnetizations," Czech. J. Phys. 48, 1083-1104 (1998).
    [CrossRef]
  24. K.Postava, J. Pistora and S. Visnovsky, "Magneto-optical effects in ultrathin structures at transversal magnetization," Czech. J. Phys. 49 1185-1204 (1999).
    [CrossRef]
  25. V. Doormann, J.-P. Krumme and H. Lenz, "Optical and magneto-optical tensor spectra of bismuth-substituted yttrium-iron-garnet films," J. Appl. Phys. 68, 3544-3553 (1990).
    [CrossRef]
  26. W. Gunsser, U. Wolfmeier and J. Fleischhauer, "Non-iron garnets," in Landolt--Bornstein Numerical Data and Functional Relationship in Science and Technology, vol12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege, A.M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p.307.

Other (26)

Z. Q. Qiu and S. D. Bader, "Surface magneto-optic Kerr effect (SMOKE)," J. Magn. Magn. Mat. 200, 664-78 (1999).
[CrossRef]

M. Bauer, R. Lopusnik, J. Fassbender and B. Hillebrands, "Suppression of magnetic field pulse induced magnetization precession by pulse tailoring," Appl. Phys. Lett. 76, 2758-60 (2000).
[CrossRef]

A. Berger, M. R. Pufall, "Generalized magneto-optical ellipsometry," Appl. Phys. Lett. 71, 965-967 (1997).
[CrossRef]

A. Berger, M. R. Pufall, "Quantitative vector magnetometry using generalized magneto-optical ellipsometry," J. Appl. Phys. 85, 4583-4585 (1999).
[CrossRef]

M. Schubert, T. E. Tiwald, J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
[CrossRef]

P. Yeh, "Optics of anisotropic layered media: a new 4 x 4 matrix algebra," Surface Sci. 96, 41-53 (1980).
[CrossRef]

S. Visnovsky, "Magneto-optical ellipsometry," Czech. J. Phys. B 36, 625-650 (1986).
[CrossRef]

J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri and K. Driss-Khodja, "Effective medium equivalent to a symmetric multilayer at oblique incidence," Appl. Opt. 29, 2460-2465 (1990).
[CrossRef] [PubMed]

K. Rokushima and J. Yamakita, "Analysis of anisotropic dielectric gratings," J. Opt. Soc. Am. 73 901-908 (1983).
[CrossRef]

K. Postava, J. Pistora, D. Ciprian, D. Hrabovsky, M. Lesnak and A.R. Fert, "Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization," in 11th Slovak--Czech--Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE 3820, 412-422 (1999).

I. Kopriva, D. Hrabovsky, K. Postava, D. Ciprian, J. Pistora and A.R.Fert, "Anisotropy of the quadratic magneto-optical effects in a cubic crystal," in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, M. Miler, eds., Proc. SPIE 4016, 54-59 (2000).

J. Zak, E. R. Moog, C. Liu and S. D. Bader, "Magneto-optics of multilayers with arbitrary magnetization directions," Phys. Rev. B 43, 6423-6429 (1991).
[CrossRef]

H. F. Ding, S. Putter, H. P. Oepen and J. Kirschner, "Experimental method for separating longitudinal and polar Kerr signals," J. Magn. Magn. Mat. 212, L5-L11 (2000).
[CrossRef]

W. A. McGahan, Liang--Yao Chen, J.A.Woollam, "Variable angle of incidence analysis of magneto-optic multilayers," J. Appl. Phys. 67, 4801-4802 (1990).
[CrossRef]

S. Visnovsky, M. Nyvlt, V. Prosser, J. Ferre, G. Penissard, D. Renard and G. Sczigel, "Magnetooptical effects in Au/Co/Au ultrathin film sandwiches," J. Magn. Magn. Mater. 128, 179-189 (1993).
[CrossRef]

S. Visnovsky, M. Nyvlt, V. Prosser, R. Lopusnik, R. Urban, J. Ferre, G. Penissard, D. Renard and R. Krishnan, "Polar magneto-optics in simple ultrathin-magnetic-film structures," Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Elsevier, Amsterdam--Lausanne-New York--Oxford--Shannon--Tokyo, 1987).

G. E. Jellison, Jr., "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

F. Abeles, "Recherches sur la propagation des ondes electromagnetiques sinuso�dales dans les milieux stratifies. Application aux couches minces," Ann. Phys., Paris 5, 596-640 (1950). M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), pp.51-70.

W. Wettling, "Magneto-optics of ferrites," J. Magn. Magn. Mat. 3, 147-160 (1976).
[CrossRef]

S. Visnovsky, "Magneto--optical permittivity tensor in crystals," Czech. J. Phys. B 36, 1424-1433 (1986).

S. Visnovsky, "Magneto-optical longitudinal and transverse Kerr and birefringence effects in orthorhombic crystals," Czech. J. Phys. B 34, 969-980 (1984).
[CrossRef]

S. Visnovsky, "Magneto-optic effects in ultrathin structures at longitudinal and polar magnetizations," Czech. J. Phys. 48, 1083-1104 (1998).
[CrossRef]

K.Postava, J. Pistora and S. Visnovsky, "Magneto-optical effects in ultrathin structures at transversal magnetization," Czech. J. Phys. 49 1185-1204 (1999).
[CrossRef]

V. Doormann, J.-P. Krumme and H. Lenz, "Optical and magneto-optical tensor spectra of bismuth-substituted yttrium-iron-garnet films," J. Appl. Phys. 68, 3544-3553 (1990).
[CrossRef]

W. Gunsser, U. Wolfmeier and J. Fleischhauer, "Non-iron garnets," in Landolt--Bornstein Numerical Data and Functional Relationship in Science and Technology, vol12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege, A.M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p.307.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

The magnetization M displayed as a cartesian vector sum of polar, M P , longitudinal, M L , and transverse M T . In the spherical coordinates M is specified by its magnitude | M | and the angles θM and ϕM .

Fig. 2.
Fig. 2.

The geometry used in the modelling. The magnetization vector evolves on cone shaped surfaces about polar (a), longitudinal (b), and transverse (c) axes.

Fig. 3.
Fig. 3.

Magneto–optical Kerr effect for an interface between vacuum and Bi0.96Lu2.04Fe5O12 magnetic garnet expressed in terms of the real part of the ratio r ps/r ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕM (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕy (b), and normal to the plane of incidence specified by an angle ϕx (c). The initial position of M is given by an angle θM between M and the interface normal at a fixed azimuth ϕM =0 deg (a), ϕM =90 deg (b), and ϕM =0 deg (c). The incident radiation is s-polarized.

Fig. 4.
Fig. 4.

Magneto–optical reflection characteristics at a film/substrate system consisting of a Bi0.96Lu2.04Fe5O12 magnetic garnet film 1.5 µm thick deposited on a Gd3Ga5O12 substrate expressed in terms of the real part of the ratio r ps/r ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕM (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕy (b), and normal to the plane of incidence specified by an angle ϕx (c). The initial position of M is given by an angle θM between M and the interface normal at a fixed azimuth ϕM =0 deg (a), ϕM =90 deg (b), and ϕM =0 deg (c). The incident radiation is s-polarized. Note that the MO effect values are of two orders in magnitude higher than in the case of a single vacuum/BiLuIG interface.

Fig. 5.
Fig. 5.

The effect on ℜ(r ps/r ss) of the angle of incidence, θ 0, ranging from -90 deg to +90 deg at an interface between vacuum and Bi0.96Lu2.04Fe5O12 magnetic garnet (a) and in a film/substrate system consisting of a Bi0.96Lu2.04Fe5O12 magnetic garnet film 1.5 µm thick deposited on a Gd3Ga5O12 substrate (b). The magnetization vector M is restricted to the plane of incidence. Its orientation is specified by an angle θM between M and interface normal. The incident radiation is s-polarized.

Tables (1)

Tables Icon

Table 1. The permittivity tensor elements of the materials used in modelling.

Equations (62)

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

ε ( n ) = ( ε 0 ( n ) i ε 1 ( n ) cos θ M ( n ) i ε 1 ( n ) sin θ M ( n ) sin ϕ M ( n ) i ε 1 ( n ) cos θ M ( n ) ε 0 ( n ) i ε 1 ( n ) sin θ M ( n ) cos ϕ M ( n ) i ε 1 ( n ) sin θ M ( n ) sin ϕ M ( n ) i ε 1 ( n ) sin θ M ( n ) cos ϕ M ( n ) ε 0 ( n ) )
γ ( n ) 2 E 0 ( n ) γ ( n ) ( γ ( n ) · E 0 ( n ) ) = ω 2 c 2 ε ( n ) E 0 ( n )
ε 0 ( n ) N z ( n ) 4 ( 2 ε 0 ( n ) N z 0 ( n ) 2 ε 1 ( n ) 2 sin 2 θ M ( n ) ) N z ( n ) 2 2 ε 1 ( n ) 2 sin θ M ( n ) cos θ M ( n ) sin ϕ M ( n ) N y N z ( n )
+ ε 0 ( n ) ( N z 0 ( n ) 4 ε 1 ( n ) 2 ) + N y 2 ε 1 ( n ) 2 ( 1 sin 2 θ M ( n ) sin 2 ϕ M ( n ) ) = 0
N z 1,3 ( n ) = N z 0 ( n ) ( 1 ε 1 ( n ) 2 4 ε 0 ( n ) N z 0 ( n ) 2 ) ± ε 1 ( n ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) ( N z 0 ( n ) cos θ M ( n ) + N y sin θ M ( n ) sin ϕ M ( n ) )
+ ε 1 ( n ) 2 8 ε 0 ( n ) N z 0 ( n ) 3 ( N z 0 ( n ) 2 cos 2 θ M ( n ) N y 2 sin θ M ( n ) 2 sin ϕ M ( n ) 2 )
N z 2,4 ( n ) = N z 0 ( n ) ( 1 ε 1 ( n ) 2 4 ε 0 ( n ) N z 0 ( n ) 2 ) ε 1 ( n ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) ( N z 0 ( n ) cos θ M ( n ) N y sin θ M ( n ) sin ϕ M ( n ) )
ε 1 ( n ) 2 8 ε 0 ( n ) N z 0 ( n ) 3 ( N z 0 ( n ) 2 cos 2 θ M ( n ) N y 2 sin θ M ( n ) 2 sin ϕ M ( n ) 2 )
E 0 ( 0 ) = M E 0 ( 𝓝 + 1 )
M = [ D ( 0 ) ] 1 D ( 1 ) P ( 1 ) [ ( D ) 1 ] 1 D ( 𝓝 ) P ( 𝓝 ) [ D ( 𝓝 ) ] 1 [ D ( 𝓝 + 1 ) ]
D 1 j ( n ) = i ε 1 ( n ) N z 0 ( n ) 2 cos θ M ( n ) i ε 1 ( n ) N y N zj ( n ) sin θ M ( n ) sin ϕ M ( n )
ε 1 ( n ) 2 sin 2 θ M ( n ) cos ϕ M ( n ) sin ϕ M ( n )
D 2 j ( n ) = N zj ( n ) D 1 j ( n )
D 3 j ( n ) = N z 0 ( n ) 2 ( N z 0 ( n ) 2 N zj ( n ) 2 ) ε 1 ( n ) 2 sin 2 θ M ( n ) sin 2 ϕ M ( n )
D 4 j ( n ) = ( ε 0 ( n ) N zj ( n ) i ε 1 ( n ) N y sin θ M ( n ) cos ϕ M ( n ) ) ( N z 0 ( n ) 2 N zj ( n ) 2 )
+ ε 1 ( n ) 2 sin θ M ( n ) sin ϕ M ( n ) ( N zj ( n ) sin θ M ( n ) sin ϕ M ( n ) N y cos θ M ( n ) )
P ( n ) = [ exp ( i ω c N z 1 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 2 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 3 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 4 ( n ) d ( n ) ) ]
D ( n ) = [ 1 1 0 0 N z 0 ( n ) N z 0 ( n ) 0 0 0 0 N z 0 ( n ) ( ε 0 ( n ) ) 1 / 2 N z 0 ( n ) ( ε 0 ( n ) ) 1 / 2 0 0 ( ε 0 ( n ) ) 1 / 2 ( ε 0 ( n ) ) 1 / 2 ]
[ E 0 s ( r ) E 0 p ( r ) ] = [ r ss r sp r ps r pp ] [ E 0 s ( i ) E 0 p ( i ) ]
r ss = [ E 0 s ( r ) E 0 s ( i ) ] E 0 p ( i ) = 0 = M 21 M 33 M 23 M 31 M 11 M 33 M 13 M 31
r ps = [ E 0 p ( r ) E 0 s ( i ) ] E 0 p ( i ) = 0 = M 41 M 33 M 43 M 31 M 11 M 33 M 13 M 31
r sp = [ E 0 s ( r ) E 0 p ( i ) ] E 0 s ( i ) = 0 = M 11 M 23 M 13 M 21 M 11 M 33 M 13 M 31
r pp = [ E 0 p ( r ) E 0 p ( i ) ] E 0 s ( i ) = 0 = M 11 M 43 M 13 M 41 M 11 M 33 M 13 M 31
E 0 ( 0 ) = [ E 01 ( 0 ) E 02 ( 0 ) E 03 ( 0 ) E 04 ( 0 ) ] T = [ E 0 s ( i ) E 0 s ( r ) E 0 p ( i ) E 0 p ( r ) ] T
E 0 ( 𝓝 + 1 ) = [ E 01 ( 𝓝 + 1 ) E 02 ( 𝓝 + 1 ) E 03 ( 𝓝 + 1 ) E 04 ( 𝓝 + 1 ) ] T = [ E 0 s ( t ) 0 E 0 p ( t ) 0 ] T
S ( n ) = D ( n ) P ( n ) ( D ( n ) ) 1
S ( n ) = [ S 11 ( n ) S 12 ( n ) S 13 ( n ) S 14 ( n ) S 21 ( n ) S 11 ( n ) S 23 ( n ) S 24 ( n ) S 24 ( n ) S 14 ( n ) S 33 ( n ) S 34 ( n ) S 23 ( n ) S 13 ( n ) S 43 ( n ) S 44 ( n ) ]
S 11 ( n ) = cos β ( n )
S 12 ( n ) = i N z 0 ( n ) 1 sin β ( n )
S 21 ( n ) = i N z 0 ( n ) sin β ( n )
S 34 ( n ) = i N z 0 ( n ) ε 0 ( n ) 1 sin β ( n )
S 43 ( n ) = i N z 0 ( n ) 1 ε 0 ( n ) sin β ( n )
S 33 ( n ) = cos β ( n ) q ( n ) sin β ( n )
S 44 ( n ) = cos β ( n ) + q ( n ) sin β ( n )
S 13 ( n ) = N z 0 ( n ) 1 ε 0 ( n ) 1 / 2 ( l ( n ) sin β ( n ) + i a n )
S 14 ( n ) = ε 0 ( n ) 1 / 2 ( p ( n ) sin β ( n ) + i b n )
S 23 ( n ) = ε 0 ( n ) 1 / 2 ( p ( n ) sin β ( n ) i b n )
S 24 ( n ) = N z 0 ( n ) ε 0 ( n ) 1 / 2 ( l ( n ) sin β ( n ) i a n )
a n = i 2 ( e i β ( n ) Δ ( n ) + e i β ( n ) Δ ( n ) )
b n = i 2 ( e i β ( n ) Δ ( n ) + + e i β ( n ) Δ ( n ) )
Δ ( n ) ± = ω 2 c d ( n ) ε 1 ( n ) ε 0 ( n ) 1 / 2 N z 0 ( n ) 1 ( N z 0 ( n ) cos θ M ( n ) ± N y sin θ M ( n ) sin ϕ M ( n ) )
β ( n ) = ω c d ( n ) N z 0 ( n )
p ( n ) = ε 1 ( n ) ( N z 0 ( n ) cos θ M ( n ) ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) 2
l ( n ) = ε 1 ( n ) ( N y sin θ M ( n ) sin ϕ M ( n ) ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) 2
q ( n ) = ε 1 ( n ) ( N y sin θ M ( n ) cos ϕ M ( n ) ) ε 0 ( n ) N z 0 ( n )
ω c d ( n ) N z 0 ( n ) 1
M = ( D ( 0 ) ) 1 S ( 1 ) D ( 2 )
r ss = r ss ( 01 ) + r ss ( 12 ) e 2 i β ( 1 ) 1 + r ss ( 01 ) r ss ( 12 ) e 2 i β ( 1 )
r ps , sp = t ss ( 01 ) t pp ( 10 ) { β 1 e 2 i β ( 1 ) [ p ( 1 ) ( r ss ( 12 ) + r pp ( 12 ) ) + l ( 1 ) ( r ss ( 12 ) r pp ( 12 ) ) ]
+ i 2 ( 1 e 2 i β ( 1 ) ) [ ± p ( 1 ) ( 1 + r ss ( 12 ) r pp ( 12 ) e 2 i β ( 1 ) ) l ( 1 ) ( 1 r ss ( 12 ) r pp ( 12 ) e 2 i β ( 1 ) ) ] }
× [ ( 1 + r ss ( 01 ) r ss ( 12 ) e 2 i β ( 1 ) ) ( 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( i ) ) ] 1
r pp = r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) + i 2 q ( 1 ) ( 1 e 2 i β ( 1 ) ) t pp ( 01 ) t pp ( 10 ) 1 r pp ( 12 ) 2 e 2 i β ( 1 ) ( 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) ) 2
r ss ( ij ) = N z 0 ( i ) N z 0 ( j ) N z 0 ( i ) + N z 0 ( j )
r pp ( ij ) = ε 0 ( i ) N z 0 ( j ) ε 0 ( j ) N z 0 ( i ) ε 0 ( i ) N z 0 ( j ) + ε 0 ( j ) N z 0 ( i )
t ss ( ij ) = 1 + r ss ( ij )
t pp ( ij ) = ( ε 0 ( i ) / ε 0 ( j ) ) 1 / 2 ( 1 r pp ( ij ) )
r ps ( 01 , pol ) cos θ M ( 1 ) = i ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) cos θ M ( 1 ) N z 0 ( 1 ) N z 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 1 ) ) ( N ( 0 ) N z 0 ( 1 ) + ε 0 ( 1 ) cos θ ( 0 ) )
= i 2 p ( 1 ) t ss ( 01 ) t pp ( 10 )
r ps ( 01 , lon ) sin θ M ( 1 ) sin ϕ M ( 1 ) = i ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) sin θ M ( 1 ) sin ϕ M ( 1 ) N y N z 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 1 ) ) ( N ( 0 ) N z 0 ( 1 ) + ε 0 ( 1 ) cos θ ( 0 ) )
= i 2 l ( 1 ) t ss ( 01 ) t pp ( 10 )
r ps , sp = 2 ω c d ( 1 ) ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) ( ε 0 ( 1 ) N z 0 ( 2 ) cos θ M ( 1 ) + ε 0 ( 2 ) N y sin θ M ( 1 ) sin ϕ M ( 1 ) ) ε 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 2 ) ) ( N ( 0 ) N z 0 ( 2 ) + ε 0 ( 2 ) cos θ ( 0 ) )
r pp = r pp ( ε 1 ( 1 ) = 0 ) t pp ( 02 ) t pp ( 20 ) ω c d ( 1 ) ε 1 ( 1 ) ε 0 ( 1 ) 1 N y sin θ M ( 1 ) cos ϕ M ( 1 )

Metrics