Abstract

The general case of monochromatic plane-wave reflection and refraction at oblique incidence from a planar biaxial–biaxial interface is presented for arbitrary principal-axes orientation in each region. A complete, systematic methodology is provided for calculating all properties of the transmitted and reflected light waves. The singularities that arise when one or both regions are taken to be isotropic are addressed, to our knowledge for the first time. Example calculations are presented for all cases. Finally, the methodology yields all wave parameters, including phase-velocity indices of refraction, angles of refraction and reflection, polarization angles, walk-off angles, and Poynting-vector relative magnitudes. Some common applications of this theory include multilayer structure analysis and the determination of internal angles for second-harmonic generation.

© 1995 Optical Society of America

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References

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  1. A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 64, 447–452 (1974).
    [CrossRef]
  2. S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
    [CrossRef]
  3. P. J. Valle, F. Moreno, “Theoretical study of birefringent filters as intercavity wavelength selectors,” Appl. Opt. 31, 528–535 (1992).
    [CrossRef] [PubMed]
  4. D. R. Preuss, J. L. Gole, “Three-stage birefringent filter tuning smoothly over the visible region: theoretical treatment and experimental design,” Appl. Opt. 19, 702–710 (1980).
    [CrossRef] [PubMed]
  5. X. Wang, J. Yao, “Transmitted and tuning characteristics of birefringent filters,” Appl. Opt. 31, 4505–4508 (1992).
    [CrossRef] [PubMed]
  6. J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum Electron. QE-18, 1253–1258 (1982).
    [CrossRef]
  7. F. K. von Wilson, “A tunable birefringent filter,” Appl. Opt. 5, 97–104 (1966).
    [CrossRef]
  8. J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
    [CrossRef]
  9. R. S. Weis, T. K. Gaylord, “Magnetooptic multilayered memory structure with a birefringent superstrate: a rigorous analysis,” Appl. Opt. 28, 1926–1930 (1989).
    [CrossRef] [PubMed]
  10. Z. M. Li, B. T. Sullivan, R. R. Parsons, “Use of the 4 × 4 matrix method in the optics of multilayer magnetooptic recording media,” Appl. Opt. 27, 1334–1338 (1988).
    [CrossRef] [PubMed]
  11. See for example, Organic Thin Films for Photonic Applications, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993).
  12. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  13. P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
    [CrossRef]
  14. R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North–Holland, Amsterdam, 1977).
  15. F. Bréhat, B. Wyncke, “Calculation of the refractive indices and direction of refracted rays as functions of the angle of incidence, in uniaxial and biaxial crystals,” J. Phys. D 26, 293–301 (1993).
    [CrossRef]
  16. M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
    [CrossRef]
  17. M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
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  18. C. E. Curry, Electromagnetic Theory of Light (MacMillan, New York, 1905), Chap. 8, pp. 330–409.
  19. G. Svizzey, “Licht als Wellenbewegung” in Handbuch der Physik, R. von H. Konen, ed. (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 698–717.
  20. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
    [CrossRef]
  21. D. A. Holmes, D. L. Feucht, “Electromagnetic wave propagation in birefringent multilayers,” J. Opt. Soc. Am. 56, 1763–1769 (1966).
    [CrossRef]
  22. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1981).
  23. See, for example, M. Born, E. Wolf, Principals of Optics (Pergamon, New York, 1980), Chap. 14, pp. 684–690.
  24. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate Free Approach (McGraw-Hill, New York, 1983).
  25. F. I. Fedorov, Optics of Anisotropic Media (Izd. AN BSSR, Minsk, Byelorussia, 1958).
  26. T. A. Maldonado, T. K. Gaylord, “Light propagation characteristics for arbitrary wavevector directions in biaxial media by a coordinate-free approach,” Appl. Opt. 30, 2465–2480 (1991).
    [CrossRef] [PubMed]
  27. mathematica®, Version 2.2.1 (Wolfram Research, Inc., Champaign, Ill., 1993).
  28. J. H. Jellet, S. Haughton, eds., The Collected Works of James MacCullagh (Hodges, Figgis, Dublin, 1880).

1993

F. Bréhat, B. Wyncke, “Calculation of the refractive indices and direction of refracted rays as functions of the angle of incidence, in uniaxial and biaxial crystals,” J. Phys. D 26, 293–301 (1993).
[CrossRef]

1992

1991

1989

1988

1987

1986

1985

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
[CrossRef]

1982

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum Electron. QE-18, 1253–1258 (1982).
[CrossRef]

1980

1974

1972

1966

Azzam, R. M. A.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North–Holland, Amsterdam, 1977).

Bashara, N. H.

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North–Holland, Amsterdam, 1977).

Berreman, D. W.

Bloom, A. L.

Born, M.

See, for example, M. Born, E. Wolf, Principals of Optics (Pergamon, New York, 1980), Chap. 14, pp. 684–690.

Bréhat, F.

F. Bréhat, B. Wyncke, “Calculation of the refractive indices and direction of refracted rays as functions of the angle of incidence, in uniaxial and biaxial crystals,” J. Phys. D 26, 293–301 (1993).
[CrossRef]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate Free Approach (McGraw-Hill, New York, 1983).

Curry, C. E.

C. E. Curry, Electromagnetic Theory of Light (MacMillan, New York, 1905), Chap. 8, pp. 330–409.

Echarri, R. M.

Fedorov, F. I.

F. I. Fedorov, Optics of Anisotropic Media (Izd. AN BSSR, Minsk, Byelorussia, 1958).

Feucht, D. L.

Gaylord, T. K.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1981).

Gole, J. L.

Henderson, D. M.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Holmes, D. A.

Killinger, D. K.

S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
[CrossRef]

Knoesen, A.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Li, Z. M.

Lotspeich, J. F.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Løvold, S.

S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
[CrossRef]

Maldonado, T. A.

Menyuk, N.

S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
[CrossRef]

Moharam, M. G.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Moreno, F.

Moulton, P. F.

S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
[CrossRef]

Parsons, R. R.

Preuss, D. R.

Simon, M. C.

Staromlynska, J.

J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
[CrossRef]

Stephens, R. R.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum Electron. QE-18, 1253–1258 (1982).
[CrossRef]

Sullivan, B. T.

Svizzey, G.

G. Svizzey, “Licht als Wellenbewegung” in Handbuch der Physik, R. von H. Konen, ed. (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 698–717.

Valle, P. J.

von Wilson, F. K.

Wang, X.

Weis, R. S.

Wolf, E.

See, for example, M. Born, E. Wolf, Principals of Optics (Pergamon, New York, 1980), Chap. 14, pp. 684–690.

Wyncke, B.

F. Bréhat, B. Wyncke, “Calculation of the refractive indices and direction of refracted rays as functions of the angle of incidence, in uniaxial and biaxial crystals,” J. Phys. D 26, 293–301 (1993).
[CrossRef]

Yao, J.

Yeh, P.

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Appl. Opt.

Appl. Phys. B

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

IEEE J. Quantum Electron.

S. Løvold, P. F. Moulton, D. K. Killinger, N. Menyuk, “Frequency tuning characteristics of a Q-switched Co:MgF2laser,” IEEE J. Quantum Electron. QE-21, 202–208 (1985).
[CrossRef]

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electrooptic tunable filters for infrared wavelengths,” IEEE J. Quantum Electron. QE-18, 1253–1258 (1982).
[CrossRef]

J. Staromlynska, “A double-element broad-band liquid crystal tunable filter—factors affecting contrast ratio,” IEEE J. Quantum Electron. 28, 501–506 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

F. Bréhat, B. Wyncke, “Calculation of the refractive indices and direction of refracted rays as functions of the angle of incidence, in uniaxial and biaxial crystals,” J. Phys. D 26, 293–301 (1993).
[CrossRef]

Surf. Sci.

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Other

R. M. A. Azzam, N. H. Bashara, Ellipsometry and Polarized Light (North–Holland, Amsterdam, 1977).

C. E. Curry, Electromagnetic Theory of Light (MacMillan, New York, 1905), Chap. 8, pp. 330–409.

G. Svizzey, “Licht als Wellenbewegung” in Handbuch der Physik, R. von H. Konen, ed. (Springer-Verlag, Berlin, 1928), Vol. 20, Chap. 11, pp. 698–717.

See for example, Organic Thin Films for Photonic Applications, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1981).

See, for example, M. Born, E. Wolf, Principals of Optics (Pergamon, New York, 1980), Chap. 14, pp. 684–690.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate Free Approach (McGraw-Hill, New York, 1983).

F. I. Fedorov, Optics of Anisotropic Media (Izd. AN BSSR, Minsk, Byelorussia, 1958).

mathematica®, Version 2.2.1 (Wolfram Research, Inc., Champaign, Ill., 1993).

J. H. Jellet, S. Haughton, eds., The Collected Works of James MacCullagh (Hodges, Figgis, Dublin, 1880).

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Figures (13)

Fig. 1
Fig. 1

Definition of the coordinate system for a single planar interface between two general dielectric media. The interface lies in the (x, y) plane, and the plane of incidence is the (x, z) plane. If region 1 or region 2 is anisotropic, two waves will be reflected or transmitted, respectively. The vectors that meet at the origin are the incident, reflected, and transmitted wave vectors. The vectors normal to the wave vectors are the electric displacement polarization vectors D.

Fig. 2
Fig. 2

Relationship between the (x′, y′, z′) principal dielectric axes and the (x, y, z) laboratory coordinate systems. The transformation between the coordinate systems is specified by the Euler angles (x convention): ϕ, θ, and ψ.

Fig. 3
Fig. 3

Wave-vector reflection and refraction at a planar interface for various degrees of anisotropy: (a) isotropic–isotropic, (b) biaxial–biaxial, (c) isotropic–biaxial, (d) biaxial–isotropic. These diagrams also apply to uniaxial media.

Fig. 4
Fig. 4

Graphic-solution method of phase matching for reflected and transmitted wave vectors across two generally oriented biaxial media. The wave-vector surface cross section is shown in each region. Both optic axes for region 1 lie in the plane of incidence. The principal indices of refraction are nx = 1.2, ny = 1.7, and nz = 2.2 in both regions. The Euler angles for regions 1 and 2 are ϕ1 = 90°, θ1 = 70°, and ψ1 = −90° and ϕ2 = θ2 = ψ2 = 30°, respectively.

Fig. 5
Fig. 5

Relationship of k, D, and E for a forward-traveling wave in bulk media. The direction of k is specified by angle υ only, the direction of D by υ and θ, and the direction of E by υ, θ, and η.

Fig. 6
Fig. 6

Relationship of k, D, and E for a reverse-traveling wave in bulk media. As for forward-traveling waves, the direction of k is specified by angle υ only, the direction of D by υ and θ, and the direction of E by υ, θ, and η.

Fig. 7
Fig. 7

Transmission and reflection coefficients at a biaxial–biaxial interface versus incidence angle υi. The dashed (solid) curves specify the incident-wave definition coming from the inner (outer) sheet of the wave-vector surface. Both regions have indices of refraction nx = 1.2, ny = 1.7 and nz = 2.2. The Euler angles for regions 1 and 2 are ϕ1 = 90°, θ1 = 70°, and ψ1 = −90° and ϕ2 = θ2 = ψ2 = 30°, respectively. (a) Transmission coefficient 1, (b) transmission coefficient 2, (c) reflection coefficient 1, (d) reflection coefficient 2. The point near 12° corresponds to the incident-wave definition changing between an o wave and an e wave.

Fig. 8
Fig. 8

Transmission and reflection coefficients at an isotropic (air)–biaxial (KTP with indices of refraction of nx = 1.73863, ny = 1.74580, nz = 1.82986 at Euler angles ϕ = 10°, θ = 20°, and ψ = 30°) interface versus incidence angle υi and polarization angle θi. (a) Transmission coefficient of the first e wave, (b) transmission coefficient of the second e wave, (c) reflection coefficient of the o wave.

Fig. 9
Fig. 9

Uniradial azimuth polarization angle calculations for transmitted wave 1 for the air–biaxial (KTP at ϕ = 10°, θ = 20°, and ψ = 30°) interface depicted in Fig. 8(a): (a) contour plot of the transmission coefficient for wave 1; dark shades have the lowest values, (b) input polarization angle required for extinguishing transmitted wave 1 calculated from Eq. (50).

Fig. 10
Fig. 10

Uniradial azimuth polarization angle calculations for transmitted wave 2 for the air–biaxial (KTP at ϕ = 10°, θ = 20°, and ψ = 30°) interface depicted in Fig. 8(b): (a) contour plot of transmission coefficient for wave 2; dark shades have the lowest values, (b) input polarization angle required for extinguishing transmitted wave 2 calculated from Eq. (50).

Fig. 11
Fig. 11

Transmission and reflection coefficients at a biaxial (nx = 1.2, ny = 1.7, nz = 2.2 at Euler angles ϕ = θ = 75° and ψ = −75°)–isotropic (air) interface. (a)–(c) correspond to an incident wave defined by the inner sheet of the wave-vector surface: (a) transmission coefficient, (b) reflection coefficient 1, (c) reflection coefficient 2; (d)–(f) correspond to an incident wave defined by the outer sheet of the wave-vector surface: (d) transmission coefficient, (e) reflection coefficient 1, (f) reflection coefficient 2.

Fig. 12
Fig. 12

Transmission and reflection coefficients at an isotropic (air)–isotropic (n = 1.7) interface. The independent variables are incidence angle θi and polarization angle υi. (a) Transmission coefficient, (b) reflection coefficient. Unlike in the biaxial cases, the coefficients are symmetric about θi = 0° and υi = 0°. The Brewster angle (rTM = 0) for TM polarization (θi = ±90°) is υi = 59.5345°.

Fig. 13
Fig. 13

Effects of varying birefringence at an isotropic (air)–biaxial [nx = 1.2, ny = (nx + nz)/2, nz = 2.2 → 1.2 at Euler angles ϕ = θ = ψ = 17°] interface with angle of incidence υi = 25° and polarization angle θi = 10°: (a) reflection coefficient, (b) transmission coefficient for each e wave and superposition of the two, (c) reflected polarization angle, (d) transmitted angles of refraction.

Equations (89)

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n i 2 = - B ± B 2 - 4 A C 2 A ,
A = k ^ T · ɛ ¯ · k ^ , B = k ^ T · [ Adj ɛ ¯ - Tr ( Adj ɛ ¯ ) I ¯ ] · k ^ , C = ɛ ¯ ,
Adj ( M ¯ ) = M ¯ · M ¯ - Tr ( M ¯ ) M ¯ + ½ [ Tr ( M ¯ ) 2 - Tr ( M ¯ · M ¯ ) ] I ¯ .
A k z 4 + B k z 3 + C k z 2 + D k z + E = 0 ,
A = ɛ z z , B = 2 k x ɛ x z , C = k x 2 ( ɛ x x + ɛ z z ) + k 0 2 [ ɛ x z 2 + ɛ y z 2 - ɛ z z ( ɛ x x + ɛ y y ) ] , D = 2 k x [ k 0 2 ( ɛ x y ɛ y z - ɛ x z ɛ y y ) + k x 2 ɛ x z ] , E = k x 4 ɛ x x + k x 2 k 0 2 [ ɛ x z 2 + ɛ x y 2 - ɛ x x ( ɛ y y + ɛ z z ) ] + k 0 4 ( ɛ x x ɛ y y ɛ z z + 2 ɛ x y ɛ x z ɛ y z - ɛ x x ɛ y z 2 - ɛ y y ɛ x z 2 - ɛ z z ɛ x y 2 ) , k x = k 0 n i sin ( υ i ) .
k α = k x x ^ + k z 2 : α z ^ , α = t 1 , t 2 , k β = k x x ^ + k z 1 : β z ^ , β = r 1 , r 2.
n α = k x 2 + k z 2 : α 2 k 0 , n β = k x 2 + k z 1 : β 2 k 0 .
υ α = tan - 1 ( k x k z 2 : α ) , υ β = tan - 1 ( k x k z 2 : β ) .
e ^ = [ ɛ ¯ - n 2 I ¯ ] · e ^ ,
J ¯ = [ ɛ ¯ - n 2 I ¯ ] .
J ¯ is nonsingular ( J ¯ 0 ) ,
J ¯ is singular ) ( J ¯ = 0 ) and planar [ Adj ( J ¯ ) 0 ¯ ] ,
J ¯ is singular ( J ¯ = 0 ) and linear [ Adj ( J ¯ ) = 0 ¯ ] .
e ^ = Adj ( J ¯ ) · k ^ ,
e ^ is any nonzero column of Adj ( J ¯ ) ,
e ^ is any nonzero column of J ¯ .
θ = sgn ( k ^ · z ^ ) sgn [ ( d ^ × y ^ ) · k ^ ] cos - 1 ( d ^ · y ^ ) ,
η = - sgn ( k ^ · e ^ ) cos - 1 ( d ^ · e ^ ) .
e ^ f = [ cos ( υ ) 0 sin ( υ ) 0 1 0 - sin ( υ ) 0 cos ( υ ) ] [ cos ( θ ) sin ( θ ) 0 - sin ( θ ) cos ( θ ) 0 0 0 1 ] × [ 1 0 0 0 cos ( η ) sin ( η ) 0 - sin ( η ) cos ( η ) ] ( 0 1 0 ) = ( - sin ( η ) sin ( υ ) + cos ( η ) cos ( υ ) sin ( θ ) cos ( η ) cos ( θ ) - sin ( η ) cos ( υ ) - cos ( η ) sin ( υ ) sin ( θ ) ) .
h ^ f = k ^ f × d ^ f = | x ^ y ^ z ^ sin ( υ ) 0 cos ( υ ) cos ( υ ) sin ( θ ) cos ( θ ) - sin ( υ ) sin ( θ ) | = ( - cos ( υ ) cos ( θ ) sin ( θ ) sin ( υ ) cos ( θ ) ) .
Region 1 E · x ^ = Region 2 E · x ^ ,
Region 1 E · y ^ = Region 2 E · y ^ ,
Region 1 H · x ^ = Region 2 H · x ^ ,
Region 1 H · y ^ = Region 2 H · y ^ .
H = / μ 0 k ^ × E = 0 / μ 0 n k ^ × E ,
H = ( 0 / μ 0 ) n cos ( η ) E ,
[ - sin ( η i ) sin ( υ i ) + cos ( η i ) cos ( υ i ) sin ( θ i ) ] + E r 1 [ - sin ( η r 1 ) sin ( υ r 1 ) + cos ( η r 1 ) cos ( υ r 1 ) sin ( θ r 1 ) ] + E r 2 [ - sin ( η r 2 ) sin ( υ r 2 ) + cos ( η r 2 ) cos ( υ r 2 ) sin ( θ r 2 ) ] = E t 1 [ - sin ( η t 1 ) sin ( υ t 1 ) + cos ( η t 1 ) cos ( υ t 1 ) sin ( θ t 1 ) ] + E t 2 [ - sin ( η t 2 ) sin ( υ t 2 ) + cos ( η t 2 ) cos ( υ t 2 ) sin ( θ t 2 ) ] .
[ cos ( η i ) cos ( θ i ) ] + E r 1 [ cos ( η r 1 ) cos ( θ r 1 ) ] + E r 2 [ cos ( η r 2 ) cos ( θ r 2 ) ] = E t 1 [ cos ( η t 1 ) cos ( θ t 1 ) ] + E t 2 [ cos ( η t 2 ) cos ( θ t 2 ) ] .
n i cos ( η i ) [ - cos ( υ i ) cos ( θ i ) ] + E r 1 n r 1 cos ( η r 1 ) × [ cos ( υ r 1 ) cos ( θ r 1 ) ] + E r 2 n r 2 cos ( η r 2 ) [ cos ( υ r 2 ) cos ( θ r 2 ) ] = E t 1 n t 1 cos ( η t 1 ) [ - cos ( υ t 1 ) cos ( θ t 1 ) ] + E t 2 n t 2 cos ( η t 2 ) [ - cos ( υ t 2 ) cos ( θ t 2 ) ] .
n i cos ( η i ) [ sin ( θ i ) ] + E r 1 n r 1 cos ( η r 1 ) [ - sin ( θ r 1 ) ] + E r 2 n r 2 cos ( η r 2 ) [ - sin ( θ r 2 ) ] = E t 1 n t 1 cos ( η t 1 ) [ sin ( θ t 1 ) ] + E t 2 n t 2 cos ( η t 2 ) [ sin ( θ t 2 ) ] .
AX = B ,
A ¯ = [ A ¯ 11 A ¯ 12 A ¯ 21 A ¯ 22 ] ,
A ¯ 11 = [ - sin ( η r 1 ) sin ( υ r 1 ) + cos ( η r 1 ) cos ( υ r 1 ) sin ( θ r 1 ) - sin ( η r 2 ) sin ( υ r 2 ) + cos ( η r 2 ) cos ( υ r 2 ) sin ( θ r 2 ) cos ( η r 1 ) cos ( θ r 1 ) cos ( η r 2 ) cos ( θ r 2 ) ] , A ¯ 12 = [ sin ( η t 1 ) sin ( υ t 1 ) - cos ( η t 1 ) cos ( υ t 1 ) sin ( θ t 1 ) sin ( η t 2 ) sin ( υ t 2 ) - cos ( η t 2 ) cos ( υ t 2 ) sin ( θ t 2 ) - cos ( η t 1 ) cos ( θ t 1 ) - cos ( η t 2 ) cos ( θ t 2 ) ] , A ¯ 21 = [ n r 1 cos ( η r 1 ) cos ( υ r 1 ) cos ( θ r 1 ) n r 2 cos ( η r 2 ) cos ( υ r 2 ) cos ( θ r 2 ) n r 1 cos ( η r 1 ) sin ( θ r 1 ) n r 2 cos ( η r 2 ) sin ( θ r 2 ) ] , A ¯ 22 = [ n t 1 cos ( η t 1 ) cos ( υ t 1 ) cos ( θ t 1 ) n t 2 cos ( η t 2 ) cos ( υ t 2 ) cos ( θ t 2 ) n t 1 cos ( η t 1 ) sin ( θ t 1 ) n t 2 cos ( η t 2 ) sin ( θ t 2 ) ] ,
X = ( E r 1 E r 2 E t 1 E t 2 ) ,
B = ( sin ( η i ) sin ( υ i ) - cos ( η i ) cos ( υ i ) sin ( θ i ) - cos ( η i ) cos ( θ i ) n i cos ( η i ) cos ( υ i ) cos ( θ i ) n i cos ( η i ) sin ( θ i ) ) .
X = A ¯ - 1 B .
v max = sin - 1 ( n min / n max ) .
s ^ f = e ^ f × h ^ f = ( cos ( η ) sin ( υ ) + sin ( η ) cos ( υ ) sin ( θ ) sin ( η ) cos ( θ ) cos ( η ) cos ( υ ) - sin ( η ) sin ( υ ) sin ( θ ) ) ,
s ^ r = e ^ r × h ^ r = ( cos ( η ) sin ( υ ) + sin ( η ) cos ( υ ) sin ( θ ) sin ( η ) cos ( θ ) - cos ( η ) cos ( υ ) + sin ( η ) sin ( υ ) sin ( θ ) ) .
S = n cos ( η ) E 2 s ^ ,
S t 1 = n t 1 cos ( η t 1 ) n i cos ( η i ) E t 1 2 .
Region 1 S · z ^ = Region 2 S · z ^ ,
n i cos ( η i ) [ cos ( η i ) cos ( υ i ) - sin ( η i ) sin ( υ i ) sin ( θ i ) ] - n r 1 cos ( η r 1 ) [ cos ( η r 1 ) cos ( υ r 1 ) - sin ( η r 1 ) sin ( υ r 1 ) × sin ( θ r 1 ) ] E r 1 2 - n r 2 cos ( η r 2 ) [ cos ( η r 2 ) cos ( υ r 2 ) - sin ( η r 2 ) sin ( υ r 2 ) sin ( θ r 2 ) ] E r 2 2 = n t 1 cos ( η t 1 ) [ cos ( η t 1 ) cos ( υ t 1 ) - sin ( η t 1 ) sin ( υ t 1 ) × sin ( θ t 1 ) ] E t 1 2 + n t 2 cos ( η t 2 ) [ cos ( η t 2 ) cos ( υ t 2 ) - sin ( η t 2 ) sin ( υ t 2 ) sin ( θ t 2 ) ] E t 2 2 .
ɛ 1 = [ 4.44228 0 1.09274 0 2.89 0 1.09274 0 1.83772 ] ,
ɛ 2 = [ 2.59918 - 0.83615 0.22880 - 0.83615 2.30894 - 1.02415 0.22880 - 1.02415 4.26188 ] .
θ OA = tan - 1 [ n z 2 ( n y 2 - n x 2 ) n x 2 ( n z 2 - n y 2 ) ] 1 / 2 = tan - 1 [ 2.2 2 ( 1.7 2 - 1.2 2 ) 1.2 2 ( 2.2 2 - 1.7 2 ) ] 1 / 2 = 57.6848° .
k i = k 0 n i k ^ i = ( 1 ) ( 1.42439 ) ( 0.50000 0 0.86603 ) = ( 0.71219 0 1.23355 ) .
k z 1 = - 2.08052 , - 1.54363 , 1.23355 , 1.54363 , k z 2 = - 1.76224 , - 0.97115 , 1.11170 , 1.54522.
k t 1 = k x x ^ + k z 2 : t 1 z ^ = [ 0.71219 , 0 , 1.54522 ] T , k t 2 = k x x ^ + k z 2 : t 2 z ^ = [ 0.71219 , 0 , 1.11170 ] T , k r 1 = k x x ^ + k z 1 : r 1 z ^ = [ 0.71219 , 0 , - 1.54363 ] T , k r 2 = k x x ^ + k z 1 : r 2 z ^ = [ 0.71219 , 0 , - 2.08052 ] T .
e ^ i = [ - 0.55944 , 0 , 0.82887 ] T , e ^ t 1 = [ 0.80250 , - 0.43916 , - 0.40390 ] T , e ^ t 2 = [ 0.45372 , 0.88335 , 0.11763 ] T , e ^ r 2 = [ - 0.95982 , 0 , - 0.28062 ] T .
e ^ r 1 = 0 , - 1.00000 , 0 ] T .
d ^ i = [ - 0.86603 , 0 , 0.50000 ] T , d ^ t 1 = [ 0.81582 , - 0.43937 , - 0.37601 ] T , d ^ t 2 = [ 0.28568 , 0.94069 , - 0.18302 ] T , d ^ r 1 = [ 0 , - 1.00000 , 0 ] T , d ^ r 2 = [ - 0.94610 , 0 , - 0.32387 ] T .
E i = 1 , E t 1 = - 0.60853 , E t 2 = - 0.35193 , E r 1 = 0.043632 , E r 2 = 0.092288.
[ cos ( υ i ) sin ( θ i ) ] + E r [ cos ( υ r ) sin ( θ r ) ] = E t 1 [ - sin ( η t 1 ) sin ( υ t 1 ) + cos ( η t 1 ) cos ( υ t 1 ) sin ( θ t 1 ) ] + E t 2 [ - sin ( η t 2 ) sin ( υ t 2 ) + cos ( η t 2 ) cos ( υ t 2 ) sin ( θ t 2 ) ] .
[ cos ( θ i ) ] + E r [ cos ( θ r ) ] = E t 1 [ cos ( η t 1 ) cos ( θ t 1 ) ] + E t 2 [ cos ( η t 2 ) cos ( θ t 2 ) ] .
[ - n i cos ( υ i ) cos ( θ i ) ] + E r [ n i cos ( υ r ) cos ( θ r ) ] = E t 1 [ - n t 1 cos ( η t 1 ) cos ( υ t 1 ) cos ( θ t 1 ) ] + E t 2 [ - n t 2 cos ( η t 2 ) cos ( υ t 2 ) cos ( θ t 2 ) ] .
[ n i sin ( θ i ) ] + E r [ - n i sin ( θ r ) ] = E t 1 [ n t 1 cos ( η t 1 ) sin ( θ t 1 ) ] + E t 2 [ n t 2 cos ( η t 2 ) sin ( θ t 2 ) ] ,
X = ( E r TM E r TE E t 1 E t 2 ) ,
B = ( - cos ( υ i ) sin ( θ i ) - cos ( θ i ) n i cos ( υ i ) cos ( θ i ) n i sin ( θ i ) ) .
A ¯ 11 = [ cos ( υ i ) 0 0 1 ] , A ¯ 21 = [ 0 n i cos ( υ i ) n i 0 ] .
E r = [ ( E r TM ) 2 + ( E r TE ) 2 ] 1 / 2 ,
θ r = { sgn ( E r TM ) cos - 1 { E r TE [ ( E r TM ) 2 + ( E r TE ) 2 ] 1 / 2 } E r TM 0 0 E r TM = 0 ,             E r TE > 0 π E r TM = 0 ,             E r TE < 0 .
θ i = tan - 1 { n α cos ( η α ) cos ( υ i ) sin ( θ α ) - n i [ sin ( η α ) sin ( υ α ) - cos ( η α ) cos ( υ α ) sin ( θ α ) ] n α cos ( η σ ) cos ( υ α ) sin ( θ α ) + n i cos ( η α ) cos ( υ i ) cos ( θ α ) } ,
X = ( E r 1 E r 2 E t TM E t TE ) ,
A ¯ 12 = [ - cos ( υ t ) 0 0 - 1 ] , A ¯ 22 = [ 0 n t cos ( υ t ) n t 0 ] .
E t = [ ( E t TM ) 2 + ( E t TE ) 2 ] 1 / 2 ,
θ t = { sgn ( E t TM ) cos - 1 { E t TE [ ( E t TM ) 2 + ( E t TE ) 2 ] 1 / 2 } E t TM 0 0 E t TM = 0 ,             E t TE > 0 π E t TM = 0 ,             E t TE < 0 .
X = ( E r TM E r TE E t TM E t TE ) ,
A ¯ = [ cos ( υ i ) 0 - cos ( υ t ) 0 0 1 0 - 1 0 n i cos ( υ i ) 0 n t cos ( υ t ) n i 0 n t 0 ] .
r TE = E r TE E i TE = n i cos ( υ i ) - n t cos ( υ t ) n i cos ( υ i ) + n t cos ( υ t ) ,
r TM = E r TM E i TM = n i cos ( υ t ) - n t cos ( υ i ) n i cos ( υ t ) + n t cos ( υ i ) ,
t TE = E t TE E i TE = 2 n i cos ( υ i ) n i cos ( υ i ) + n t cos ( υ t ) ,
t TM = E t TM E i TM = 2 n i cos ( υ i ) n i cos ( υ t ) + n t cos ( υ i ) .
E r = { [ r TE cos ( θ i ) ] 2 + [ r TM sin ( θ i ) ] 2 } 1 / 2 ,
E t = { [ t TE cos ( θ i ) ] 2 + [ t TM sin ( θ i ) ] 2 } 1 / 2 ,
θ r = tan - 1 [ r TM cos ( θ i ) r TE sin ( θ i ) ] ,
θ t = tan - 1 [ t TM cos ( θ i ) t TE sin ( θ i ) ] .
v B = tan - 1 ( n t / n i ) = tan - 1 ( 1.7 / 1 ) = 59.5345° .
( x y z ) = M ¯ ( x y z ) .
M ¯ = [ cos ( ψ ) sin ( ψ ) 0 - sin ( ψ ) cos ( ψ ) 0 0 0 1 ] [ 1 0 0 0 cos ( θ ) sin ( θ ) 0 - sin ( θ ) cos ( θ ) ] × [ cos ( ϕ ) sin ( ϕ ) 0 - sin ( ϕ ) cos ( ϕ ) 0 0 0 1 ] .
M ¯ = [ M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33 ] ,
M 11 = cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) , M 12 = cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) , M 13 = sin ( ψ ) sin ( θ ) , M 21 = - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) , M 22 = - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) , M 23 = cos ( ψ ) sin ( θ ) , M 31 = sin ( θ ) sin ( ϕ ) , M 32 = - sin ( θ ) cos ( ϕ ) , M 33 = cos ( θ ) .
ψ = tan - 1 ( M 13 / M 23 ) , θ = sin - 1 [ sin ( ψ ) M 13 + cos ( ψ ) M 23 ] , ϕ = sin - 1 [ cos ( ψ ) M 13 - sin ( ψ ) M 23 ] .
( D x D y D z ) = [ x 0 0 0 y 0 0 0 z ] ( E x E y E z ) ,
x = 0 ɛ x = 0 n x 2 ,
y = 0 ɛ y = 0 n y 2 ,
z = 0 ɛ z = 0 n z 2 .
ɛ ¯ ɛ ¯ laboratory = M ¯ - 1 ɛ ¯ principal M ¯ .
ɛ x x = ɛ x [ cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) ] 2 + ɛ y [ - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) ] 2 + ɛ z [ sin ( θ ) sin ( ϕ ) ] 2 , ɛ x y = ɛ x [ cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) ] × [ cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) ] + ɛ y [ - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) ] × [ - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) ] + ɛ z [ sin ( θ ) sin ( ϕ ) ] [ - sin ( θ ) cos ( ϕ ) ] , ɛ x z = ɛ x [ cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) ] [ sin ( ψ ) sin ( θ ) ] + ɛ y [ - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) ] × [ cos ( ψ ) sin ( θ ) ] + ɛ z [ sin ( θ ) sin ( ϕ ) ] [ cos ( θ ) ] , ɛ y x = ɛ x y , ɛ y y = ɛ x [ cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) ] 2 + ɛ y [ - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) ] 2 + ɛ z [ - sin ( θ ) cos ( ϕ ) ] 2 , ɛ y z = ɛ x [ cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) ] [ sin ( ψ ) sin ( θ ) ] + ɛ y [ - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) ] × [ cos ( ψ ) sin ( θ ) ] + ɛ z [ - sin ( θ ) cos ( ϕ ) ] [ cos ( θ ) ] , ɛ z x = ɛ x z , ɛ z y = ɛ y z , ɛ z z = ɛ x [ sin ( ψ ) sin ( θ ) ] 2 + ɛ y [ cos ( ψ ) sin ( θ ) ] 2 + ɛ z [ cos ( θ ) ] 2 .

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