Abstract

The general case of obliquely incident plane-wave propagation in periodic anisotropic layered media is presented. Arbitrary permittivity tensors of the two alternating anisotropic layers are considered. An immersion model is used with the assumption that each layer is embedded between two isotropic regions that have the same index of refraction as the isotropic medium of incidence and a thickness that is set equal to zero. Then explicit relations are presented for normally incident plane waves in periodic structures that consist of alternating biaxial layers of arbitrary principal-axis orientation. Specific cases of alternating isotropic and biaxial layers are also considered. Unit cell translation matrices are presented for both traveling directions, from the left to the right and vice versa. Dispersion relations that contain information regarding the propagation bands and the forbidden gaps in periodic anisotropic structures are presented.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
    [CrossRef]
  4. K.-K. Law, D. I. Babic, “Effect of layer thickness variations on propagation delay and penetration depth of a quarter-wave distributed Bragg reflector,” IEEE Photon. Technol. Lett. 5, 1294–1300 (1993).
    [CrossRef]
  5. J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
    [CrossRef]
  6. M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
    [CrossRef]
  7. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
    [CrossRef]
  8. F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
    [CrossRef] [PubMed]
  9. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
    [CrossRef]
  10. I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
    [CrossRef]
  11. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
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  12. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
    [CrossRef]
  13. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
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  14. G. D. Landry, T. A. Maldonado, “Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial media,” J. Opt. Soc. Am. A 12, 2048–2063 (1995).
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  16. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
    [CrossRef]
  17. I. Abdulhalim, “Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media,” Opt. Commun. 157, 265–272 (1998).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.
  19. D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
    [CrossRef]
  20. E. Cojocaru, “Simple expressions for transmission and reflection matrix elements of a biaxial thin layer at normal incidence,” Appl. Opt. 38, 2053–2058 (1999).
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2000 (1)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

1999 (1)

1998 (1)

I. Abdulhalim, “Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media,” Opt. Commun. 157, 265–272 (1998).
[CrossRef]

1997 (2)

F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

1996 (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

1995 (2)

1994 (2)

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

1993 (3)

1992 (1)

D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

1990 (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

1988 (1)

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1979 (1)

1977 (1)

Abdulhalim, I.

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

I. Abdulhalim, “Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media,” Opt. Commun. 157, 265–272 (1998).
[CrossRef]

Babic, D. I.

K.-K. Law, D. I. Babic, “Effect of layer thickness variations on propagation delay and penetration depth of a quarter-wave distributed Bragg reflector,” IEEE Photon. Technol. Lett. 5, 1294–1300 (1993).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Cojocaru, E.

Dalichaouch, R.

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Gu, C.

Haus, H. A.

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Heine, C.

Hong, C. S.

Kärtner, F. X.

Keller, U.

Kroll, N.

Landry, G. D.

Law, K.-K.

K.-K. Law, D. I. Babic, “Effect of layer thickness variations on propagation delay and penetration depth of a quarter-wave distributed Bragg reflector,” IEEE Photon. Technol. Lett. 5, 1294–1300 (1993).
[CrossRef]

Maldonado, T. A.

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Mansuripur, M.

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

Matuschek, N.

McCall, S. L.

Morf, R.

Platzman, P. M.

Roberts, D. A.

D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Scheuer, V.

Schibli, T.

Schubert, M.

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Schultz, S.

Smith, D. R.

Spink, D. M.

Thomas, C. B.

Tilsch, M.

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Tschudi, T.

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

Yeh, P.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. A. Roberts, “Simplified characteristics of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K.-K. Law, D. I. Babic, “Effect of layer thickness variations on propagation delay and penetration depth of a quarter-wave distributed Bragg reflector,” IEEE Photon. Technol. Lett. 5, 1294–1300 (1993).
[CrossRef]

J. Appl. Phys. (3)

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

I. Abdulhalim, “Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media,” Opt. Commun. 157, 265–272 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Phys. Rev. B (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

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Figures (7)

Fig. 1
Fig. 1

Reflection and refraction of an obliquelly incident plane wave upon the ith anisotropic layer, which is surrounded by the 0ith and the 0i + 1th isotropic (open dots) regions of index of refraction n 0. Layer interfaces are parallel to the xy plane. The propagation directions of the incident, reflected, forward-, and backward-propagating α and β waves are defined by wave vectors k 0 +, k 0 -, k αi +, k βi +, k αi -, and k βi -. All the wave vectors lie in the xz plane.

Fig. 2
Fig. 2

Relationship between the orthogonal triplet (E vi σ, H vi σ, S vi σ) and the (x, y, z) laboratory coordinate systems. The orientation of the (E vi σ, H vi σ, S vi σ) triplet is specified by three angles: φ vi σ, η vi σ, and χ vi σ.

Fig. 3
Fig. 3

(a) Periodic anisotropic layered medium that consists of alternating anisotropic layers 1 and 2. (b) Illustration of the immersion model. Each layer in the stack is imaginatively embedded between two isotropic (open dots) regions of refractive index n 0 and thickness d 0, where d 0 is set equal to zero.

Fig. 4
Fig. 4

Normally incident plane waves from the left side upon periodic layered media that consist of alternating biaxial and isotropic layers. The anisotropic biaxial regions of phase velocity indices n α and n β and thickness d a are hatched by thick lines, whereas the isotropic regions of refractive index n d and thickness d are hatched by thin lines. Two cases are considered: (a) the first layer encountered by the wave is biaxial and the second is isotropic and (b) the first layer is isotropic and the second is biaxial.

Fig. 5
Fig. 5

Variation of R ps = R sp against d/λ (a) in case I and (b) in case II. (c) Variation of T ps = T sp against d/λ in both cases I and II.

Fig. 6
Fig. 6

Variation (a) of R ts and (b) of R tp against d/λ in both cases I and II.

Fig. 7
Fig. 7

Variation of cos(KΛ) against d/λ in both cases I and II. Curves that correspond to waves α and β are marked with crosses and pluses, respectively. The horizontal line represents cos(KΛ) = -1.

Equations (98)

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k0σ=ω/cξxˆ+σζ0zˆ, σ=±,
kviσ=ω/cξxˆ+ζviσzˆ,  v=α, β, σ=±,
E0iσ=E0siσsˆ+E0piσpˆσexpjk0σr-ωt,
eˆviσevxiσ, evyiσ, evziσT=cos χviσ cos ηviσ cos φviσ-sin ηviσ sin φviσ, sin χviσ cos ηviσ, -cos χviσ cos ηviσ sin φviσ-sin ηviσcos φviσT,
hˆviσhvxiσ, hvyiσ, hvziσT=-sin χviσ cos φviσ, cos χviσ, sin χviσ sin φviσT.
Eiσ=v=α,β Eviσeˆviσ expjkviσrexp-jωt,
Hiσ=v=α,β nviσ/cEviσhˆviσ cos ηviσ expjkviσr×exp-jωt.
0si+0si-0pi+0pi-=bαi-+bαi--bβi-+bβi--bαi++bαi+-bβi++bβi+--aαi++-aαi+--aβi++-aβi+-aαi-+aαi--aβi-+aβi--αi+αi-βi+βi-.
0qiσ=expjk0zσziE0qiσ,  q=p, s, σ=±,
viσ=expjkvziσziEviσ,  v=α, β, σ=±;
avi±σ=n0evxiσ±nviσhvyiσ cos φ0 cos ηviσ/2ζ0,
bvi±σ=ζ0evyiσ±nviσhvxiσ cos ηviσ/2ζ0.
¯0i=C˜i¯i,
¯0i+1=C˜iX˜i¯i,
Xi11=expjkαzi+di,  Xi22=expjkαzi-di,
Xi33=expjkβzi+di,  Xi44=expjkβzi-di.
¯0i+1=C˜iX˜iC˜i-1¯0i.
M˜i=C˜iX˜iC˜i-1.
Ci11=Ci22=1+ζi/ζ0/2,  Ci12=Ci21=1-ζi/ζ0/2,
Ci33=Ci44=ni/n0+n0ζi/niζ0/2,  Ci34=Ci43=ni/n0-n0ζi/niζ0/2,
¯02N+1=M˜2M˜1N¯01.
¯1N=˜¯1N-1.
˜=C˜1-1C˜2X˜2C˜2-1C˜1X˜1.
˜=C˜2-1C˜1X˜1C˜1-1C˜2X˜2.
EKr, t=EKzexpjKzexpjω/cξ-ωt.
EKz+Λ=EKz.
¯1N=expjKΛ¯1N-1.
˜¯1N=expjKΛ¯1N.
det˜-γĨ4=0,
ω=ωK, φ0, Λ, .
xpi, ypi, zpiT=P˘ix, y, zT,
ε˘i=P˘i-1ε˘piP˘i,  i=1, 2.
M˜i=Mi11Mi12Mi13Mi14Mi12*Mi11*-Mi14*-Mi13*Mi13-Mi14Mi33Mi34Mi14*-Mi13*Mi34*Mi33*,
ρi=tan χαi,  τvi=ω/cnvidi, v=α, β, i=1, 2
Aiv0=cos τvi+j/2nvi/n0+n0/nvisin τvi,
Biv0=nvi/n0-n0/nvisin τvi,  v=α, β, i=1, 2.
Mi11=ρi2Aiα0+Aiβ0/1+ρi2,
Mi12=j/2ρi2Biα0+Biβ0/1+ρi2,
Mi13=-ρiAiα0-Aiβ0/1+ρi2,
Mi14=j/2ρiBiα0-Biβ0/1+ρi2,
Mi33=Aiα0+ρi2Aiβ0/1+ρi2,
Mi34=-j/2Biα0+ρi2Biβ0/1+ρi2.
δ=tanχα1-χα2, g=cos χβ1/sin χα1cos ηβ1/cos ηα1;
Avα=cos τv2+j/2nα1/nv2+nv2/nα1sin τv2,
Avβ=cos τv2+j/2nβ1/nv2+nv2/nβ1sin τv2,
Bvα=nv2/nα1-nα1/nv2sin τv2,
Bvβ=nv2/nβ1-nβ1/nv2sin τv2,  v=α, β,
F±=nα1±nβ1cos τα2-cos τβ2,
G±=nα2±nα1nβ1/nα2sin τα2-nβ2±nα1nβ1/nβ2sin τβ2.
˜=1112131412*11*14*13*3132333432*31*34*33*,
11=expjτα1Aαα+δ2Aβα/1+δ2,
12=-j/2 exp-jτα1Bαα+δ2Bβα/1+δ2,
13=gδ/2nα1expjτβ1F++jG+/1+δ2,
14=-gδ/2nα1exp-jτβ1F-+jG-/1+δ2,
31=δ/2gnβ1expjτα1F++jG+/1+δ2,
32=δ/2gnβ1exp-jτα1F--jG-/1+δ2,
33=expjτβ1δ2Aαβ+Aββ/1+δ2,
34=-j/2 exp-jτβ1δ2Bαβ+Bββ/1+δ2.
γ4-c1γ3+c2γ2-c1γ+1=0,
c1=2fα+fβ+δ2f12+f21/1+δ2,
c2=2+4fαfβ+δ2f12f21-δ21-fαβ11-fαβ2/1+δ2/1+δ2;
fv=cos τv1 cos τv2-1/2nv1/nv2+nv2/nv1sin τv1 sin τv2,  v=α, β,
fαβi=cos ταi cos τβi+1/2nαi/nβi+nβi/nαisin ταi sin τβi,  i=1, 2,
f12=cos τα1 cos τβ2-1/2nα1/nβ2+nβ2/nα1sin τα1 sin τβ2,
f21=cos τα2 cos τβ1-1/2nα2/nβ1+nβ1/nα2sin τα2 sin τβ1.
γ2-c+γ+1γ2-c-γ+1=0,
c±=c1±c12-4c2+81/2/2.
a2invi4+a1invi2+a0i=0,
a2i=εxpiPi132+εypiPi232+εzpiPi332,
a1i=-εxpiεypi1-Pi332-εxpiεzpi1-Pi232-εypiεzpi1-Pi132,
a0i=εxpiεypiεzpi,
tan 2χvi=2Pi11Pi12-Pi31Pi32 cot2 Ωi/Pi112-Pi122+cot2 ΩiPi322-Pi312,
zˆpi=sin θi sin ϕi, -sin θi cos ϕi, cos θiT,
M˜d=Md11Md1200Md12*Md11*0000Md11Md12*00Md12Md11*,
Md11=nd+n02 expjτd-nd-n02×exp-jτd/4n0nd,
Md12=nd2-n02expjτd-exp-jτd/4n0nd.
˜=11120012*11*000033340034*33*,
11=expjταAdα,  12=-j/2 exp-jταBdα,
33=expjτβAdβ,  34=-j/2 exp-jτβBdβ,
Adv=cos τd+j/2nd/nv+nv/ndsin τd,
Bdv=nd/nv-nv/ndsin τd,  v=α, β.
˜=1112131412*11*-14*-13*13-14333414*-13*34*33*,
11=expjτdρ2Aαd+Aβd/1+ρ2,
12=j/2 exp-jτdρ2Bαd+Bβd/1+ρ2,
13=-ρ/2ndexpjτdF++jG+/1+ρ2,
14=jρ/2ndexp-jτdG-/1+ρ2,
33=expjτdAαd+ρ2Aβd/1+ρ2,
34=-j/2 exp-jτdBαd+ρ2Bβd/1+ρ2,
Avd=cos τv+j/2nv/nd+nd/nvsin τv,
Bvd=nv/nd-nd/nvsin τv,  v=α, β,
F+=2ndcos τα-cos τβ,
G±=nα±nd2/nαsin τα-nβ±nd2/nβsin τβ.
cosKΛ=cos τd cos τα-1/2nα/nd+nd/nαsin τd sin τα,
cosKΛ=cos τd cos τβ-1/2nβ/nd+nd/nβsin τd sin τβ.
ηv=-signzˆ·eˆvarcoseˆv·dˆv,
χv=signzˆ·yˆ×hˆvarcosyˆ·hˆv,
rrssrpsrsprpp=-M22M24M42M44-1M21M23M41M43.
tsstpstsptpp=M11M13M31M33+M12M14M32M34rssrpsrsprpp.

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