Abstract

In this work we analyze the propagation of a plane wave that passes from an isotropic transparent medium to a uniaxial absorbing medium. Detailed expressions that give the real directions of propagation of the wave and the energy of the reflected and refracted ordinary and extraordinary waves are obtained. These expressions are valid for every orientation of the optic axis of the uniaxial medium and for every direction of propagation of the incident wave. Expressions are tested in the case of an interface between a transparent and an absorbing isotropic media and for the air–rutile (TiO2) interface. The effect of absorption has been evaluated by comparing the results obtained in rutile with the results obtained in a transparent uniaxial medium with the same real refractive indices. Results are presented for different values of the angle of incidence and the orientation of the plane of incidence.

© 2008 Optical Society of America

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References

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  1. J. M. Cabrera, F. Agulló, and F. J. López, Optica Electromagnética. Vol. II: Materiales y Aplicaciones (Addison Wesley Iberoamericana Española S.A., 2000).
  2. M. F. Al-kuhalaili, E. E. Khawaja, and S. M. A. Durrani, “A method for the determination of the optical constants (n and k) of thin films with large optical inhomogeneities,” J. Mod. Opt. 54, 1453-1465 (2007).
    [CrossRef]
  3. C. Jenkins, R. Bingham, K. Moore, and G. D. Love, “Ray equation for a spatially variable uniaxial crystal and its use in the optical design of liquid-crystal lenses,” J. Opt. Soc. Am. A 24, 2089-2096 (2007).
    [CrossRef]
  4. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys.: Condens. Matter 3, 6121-6133 (1991).
    [CrossRef]
  5. D. J. De Smet, “Ellipsometry of anistropic surfaces,” J. Opt. Soc. Am. 63, 958-964 (1973).
    [CrossRef]
  6. M. C. Simon, K. V. Gottschlak, and J. M. Simon, “The coincidence of the ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959-974 (2007).
    [CrossRef]
  7. L. I. Perez and C. E. Vanney, “Non-absorbing isotropic-uniaxial interfaces: refraction in ordinary and extraordinary total reflection,” J. Mod. Opt. 52, 1981-2000 (2005).
    [CrossRef]
  8. C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553-1566 (2002).
    [CrossRef]
  9. Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1999), Vol. 39, pp. 1-62.
    [CrossRef]
  10. R. A. Egorchenkov and Y. A. Kravtsov, “Complex ray-tracing algorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650-656 (2001).
    [CrossRef]
  11. J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media,” J. Opt. Soc. Am. A 24, 1767-1775 (2007).
    [CrossRef]
  12. H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109-112 (2003).
    [CrossRef]
  13. R. M. A. Azzam and E. Ericsson “Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence,” J. Opt. Soc. Am. A 19, 112-115 (2002).
    [CrossRef]
  14. L. P. Mosteller, Jr. and F. Wooten, “Optical properties and reflectance of uniaxial absorbing crystals,” J. Opt. Soc. Am. 58, 511-518 (1968).
    [CrossRef]
  15. Halevi and A. Mendoza-Hernandez, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. 71, 1238-1242 (1981).
    [CrossRef]
  16. S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in absorbing isotropic media,” J. Opt. Soc. Am. A 21, 1776-1784 (2004).
    [CrossRef]
  17. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).
  18. G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, “Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2),” Opt. Lett. 22, 1808-1810 (1997).
    [CrossRef]
  19. C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
    [CrossRef]
  20. M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
    [CrossRef]
  21. T. E. Tiwlad and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148to33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19-29 (2000).
    [CrossRef]
  22. G. E. Jellison, Jr. and J. S. Baba, “Pseudodielectric functions of uniaxial materials in certain symmetry directions,” J. Opt. Soc. Am. A 33, 468-475 (2006).
    [CrossRef]

2007 (4)

M. F. Al-kuhalaili, E. E. Khawaja, and S. M. A. Durrani, “A method for the determination of the optical constants (n and k) of thin films with large optical inhomogeneities,” J. Mod. Opt. 54, 1453-1465 (2007).
[CrossRef]

M. C. Simon, K. V. Gottschlak, and J. M. Simon, “The coincidence of the ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959-974 (2007).
[CrossRef]

J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media,” J. Opt. Soc. Am. A 24, 1767-1775 (2007).
[CrossRef]

C. Jenkins, R. Bingham, K. Moore, and G. D. Love, “Ray equation for a spatially variable uniaxial crystal and its use in the optical design of liquid-crystal lenses,” J. Opt. Soc. Am. A 24, 2089-2096 (2007).
[CrossRef]

2006 (1)

G. E. Jellison, Jr. and J. S. Baba, “Pseudodielectric functions of uniaxial materials in certain symmetry directions,” J. Opt. Soc. Am. A 33, 468-475 (2006).
[CrossRef]

2005 (1)

L. I. Perez and C. E. Vanney, “Non-absorbing isotropic-uniaxial interfaces: refraction in ordinary and extraordinary total reflection,” J. Mod. Opt. 52, 1981-2000 (2005).
[CrossRef]

2004 (1)

2003 (1)

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109-112 (2003).
[CrossRef]

2002 (2)

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553-1566 (2002).
[CrossRef]

R. M. A. Azzam and E. Ericsson “Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence,” J. Opt. Soc. Am. A 19, 112-115 (2002).
[CrossRef]

2001 (1)

2000 (2)

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

T. E. Tiwlad and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148to33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19-29 (2000).
[CrossRef]

1997 (1)

1996 (1)

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys.: Condens. Matter 3, 6121-6133 (1991).
[CrossRef]

1981 (1)

1973 (1)

1968 (1)

Agulló, F.

J. M. Cabrera, F. Agulló, and F. J. López, Optica Electromagnética. Vol. II: Materiales y Aplicaciones (Addison Wesley Iberoamericana Española S.A., 2000).

Alberdi, C.

Alfonso, S.

Al-kuhalaili, M. F.

M. F. Al-kuhalaili, E. E. Khawaja, and S. M. A. Durrani, “A method for the determination of the optical constants (n and k) of thin films with large optical inhomogeneities,” J. Mod. Opt. 54, 1453-1465 (2007).
[CrossRef]

Asatryan, A. A.

Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1999), Vol. 39, pp. 1-62.
[CrossRef]

Audier, M.

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

Azzam, R. M. A.

Baba, J. S.

G. E. Jellison, Jr. and J. S. Baba, “Pseudodielectric functions of uniaxial materials in certain symmetry directions,” J. Opt. Soc. Am. A 33, 468-475 (2006).
[CrossRef]

Berrogui, M.

Bingham, R.

Boatner, L. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Cabrera, J. M.

J. M. Cabrera, F. Agulló, and F. J. López, Optica Electromagnética. Vol. II: Materiales y Aplicaciones (Addison Wesley Iberoamericana Española S.A., 2000).

Coutier, C.

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

De Smet, D. J.

Diñeiro, J. M.

Durrani, S. M. A.

M. F. Al-kuhalaili, E. E. Khawaja, and S. M. A. Durrani, “A method for the determination of the optical constants (n and k) of thin films with large optical inhomogeneities,” J. Mod. Opt. 54, 1453-1465 (2007).
[CrossRef]

Egorchenkov, R. A.

Ericsson, E.

Fick, J.

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

Forbes, G. W.

Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1999), Vol. 39, pp. 1-62.
[CrossRef]

Gottschlak, K. V.

M. C. Simon, K. V. Gottschlak, and J. M. Simon, “The coincidence of the ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959-974 (2007).
[CrossRef]

Greiner, H.

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109-112 (2003).
[CrossRef]

Halevi,

Hernandez, B.

Herzinger, C. M.

Jellison, G. E.

G. E. Jellison, Jr. and J. S. Baba, “Pseudodielectric functions of uniaxial materials in certain symmetry directions,” J. Opt. Soc. Am. A 33, 468-475 (2006).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, “Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2),” Opt. Lett. 22, 1808-1810 (1997).
[CrossRef]

Jenkins, C.

Johs, B.

Khawaja, E. E.

M. F. Al-kuhalaili, E. E. Khawaja, and S. M. A. Durrani, “A method for the determination of the optical constants (n and k) of thin films with large optical inhomogeneities,” J. Mod. Opt. 54, 1453-1465 (2007).
[CrossRef]

Kravtsov, Y. A.

R. A. Egorchenkov and Y. A. Kravtsov, “Complex ray-tracing algorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650-656 (2001).
[CrossRef]

Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1999), Vol. 39, pp. 1-62.
[CrossRef]

Langet, M.

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

Lekner, J.

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys.: Condens. Matter 3, 6121-6133 (1991).
[CrossRef]

López, F. J.

J. M. Cabrera, F. Agulló, and F. J. López, Optica Electromagnética. Vol. II: Materiales y Aplicaciones (Addison Wesley Iberoamericana Española S.A., 2000).

Love, G. D.

Mendoza-Hernandez, A.

Modine, F. A.

Moore, K.

Mosteller, L. P.

Perez, L. I.

L. I. Perez and C. E. Vanney, “Non-absorbing isotropic-uniaxial interfaces: refraction in ordinary and extraordinary total reflection,” J. Mod. Opt. 52, 1981-2000 (2005).
[CrossRef]

Rheinlander, B.

Rimet, R.

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

Sáenz, C.

Schubert, M.

Simon, J. M.

M. C. Simon, K. V. Gottschlak, and J. M. Simon, “The coincidence of the ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959-974 (2007).
[CrossRef]

Simon, M. C.

M. C. Simon, K. V. Gottschlak, and J. M. Simon, “The coincidence of the ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959-974 (2007).
[CrossRef]

Tiwlad, T. E.

T. E. Tiwlad and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148to33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19-29 (2000).
[CrossRef]

Vanney, C. E.

L. I. Perez and C. E. Vanney, “Non-absorbing isotropic-uniaxial interfaces: refraction in ordinary and extraordinary total reflection,” J. Mod. Opt. 52, 1981-2000 (2005).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Woollam, J. A.

Wooten, F.

J. Mod. Opt. (4)

M. F. Al-kuhalaili, E. E. Khawaja, and S. M. A. Durrani, “A method for the determination of the optical constants (n and k) of thin films with large optical inhomogeneities,” J. Mod. Opt. 54, 1453-1465 (2007).
[CrossRef]

M. C. Simon, K. V. Gottschlak, and J. M. Simon, “The coincidence of the ordinary and extraordinary rays in a uniaxial birefringent crystal,” J. Mod. Opt. 55, 959-974 (2007).
[CrossRef]

L. I. Perez and C. E. Vanney, “Non-absorbing isotropic-uniaxial interfaces: refraction in ordinary and extraordinary total reflection,” J. Mod. Opt. 52, 1981-2000 (2005).
[CrossRef]

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553-1566 (2002).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (7)

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in absorbing isotropic media,” J. Opt. Soc. Am. A 21, 1776-1784 (2004).
[CrossRef]

R. A. Egorchenkov and Y. A. Kravtsov, “Complex ray-tracing algorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650-656 (2001).
[CrossRef]

J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media,” J. Opt. Soc. Am. A 24, 1767-1775 (2007).
[CrossRef]

C. Jenkins, R. Bingham, K. Moore, and G. D. Love, “Ray equation for a spatially variable uniaxial crystal and its use in the optical design of liquid-crystal lenses,” J. Opt. Soc. Am. A 24, 2089-2096 (2007).
[CrossRef]

M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

R. M. A. Azzam and E. Ericsson “Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence,” J. Opt. Soc. Am. A 19, 112-115 (2002).
[CrossRef]

G. E. Jellison, Jr. and J. S. Baba, “Pseudodielectric functions of uniaxial materials in certain symmetry directions,” J. Opt. Soc. Am. A 33, 468-475 (2006).
[CrossRef]

J. Phys.: Condens. Matter (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys.: Condens. Matter 3, 6121-6133 (1991).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109-112 (2003).
[CrossRef]

Proc. SPIE (1)

T. E. Tiwlad and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148to33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19-29 (2000).
[CrossRef]

Thin Solid Films (1)

C. Coutier, M. Audier, J. Fick, R. Rimet, and M. Langet, “Aerosol-gel preparation of optically active layers in the system Er/SiO2-TiO2,” Thin Solid Films 372, 177-189 (2000).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

J. M. Cabrera, F. Agulló, and F. J. López, Optica Electromagnética. Vol. II: Materiales y Aplicaciones (Addison Wesley Iberoamericana Española S.A., 2000).

Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1999), Vol. 39, pp. 1-62.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometrical elements used to describe the interface between the transparent isotropic medium with refractive index n 0 and the absorbing uniaxial medium with principal complex refractive indices n ̃ ω and n ̃ ϵ ; X Y Z are the principal axes, and X m Y m Z m is the coordinate system in the interface. The plane of incidence π that contains the direction of propagation of the incident wave s 0 and the Z m axis is also shown.

Fig. 2
Fig. 2

Representation of the unitary vector u ̃ x y , the projection of s ̃ in the X Y plane.

Fig. 3
Fig. 3

Air–rutile interface with the optic axis perpendicular to the surface. The dependence on the angle of incidence α 0 of (a) refractive angle α 2 of the real direction of propagation of the ordinary wave, (b) refractive angle α 1 of the real direction of propagation of the extraordinary wave, (c) refractive index n 1 of the extraordinary wave, (d) absorption index χ 1 of the extraordinary wave, (e) angle between the direction of propagation of the energy and Z m for the ordinary wave, (f) angle between the direction of propagation of the energy and Z m for the extraordinary wave.

Fig. 4
Fig. 4

Air–rutile interface with the optic axis contained in the surface. The dependence on the angle of incidence α 0 and the orientation δ of the plane of incidence of (a) refractive angle α 2 of the real direction of propagation of the ordinary wave, (b) refractive angle α 1 of the real direction of propagation of the extraordinary wave, (c) refractive index n 1 of the extraordinary wave, (d) absorption index χ 1 of the extraordinary wave, (e) angle between the direction of propagation of the energy and Z m for the ordinary wave, (f) angle between the direction of propagation of the energy and Z m for the extraordinary wave.

Fig. 5
Fig. 5

Comparison between the transparent and the absorbing cases when γ = 0 . The following quantities have been plotted as a function of the α 0 : (a) the angle of refraction α 2 of the direction of propagation of the ordinary wave; (b) the angle of refraction α 1 of the direction of propagation of the extraordinary wave; (c) the refractive index n 1 ; (d) the angle between the direction of propagation of the energy and Z m for the ordinary wave; (e) the angle between the direction of propagation of the energy and Z m for the extraordinary wave.

Fig. 6
Fig. 6

Comparison between the transparent and the absorbing cases when γ = π 2 . The following quantities have been plotted as a function of the α 0 and the orientation δ of the plane of incidence. (a) Angle of refraction α 2 of the direction of propagation of the ordinary wave. (b) Angle between the direction of propagation of the energy and Z m for the ordinary wave. (c) Angle of refraction α 1 of the direction of propagation of the extraordinary wave: absorbing (c1) and transparent (c2). (d) Refractive index n 1 : absorbing (d1) and transparent (d2). (e) Angle between Z m and the direction of propagation of the energy of extraordinary wave: absorbing (e1) and transparent (e2).

Equations (74)

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D ̃ = ϵ ̃ E ̃ ,
n ̃ x , y , z = ( ϵ ̃ x , y , z ϵ 0 ) 1 2 ,
n ̃ x , y = n ̃ ω = n ω i k ω = n ω ( 1 i χ ω ) = n ̃ ω exp ( i β ω ) ,
n ̃ z = n ̃ ϵ = n ϵ i k ϵ = n ϵ ( 1 i χ ϵ ) = n ̃ ϵ exp ( i β ϵ ) .
tan β ω , ϵ = χ ω , ϵ , n ̃ ω , ϵ = n ω , ϵ ( 1 + χ ω , ϵ 2 ) 1 2 ,
F ̃ = F u ̃ F exp [ i ω ( t n ̃ c r s ̃ ) ] exp ( i θ F ) = F u ̃ F exp ω c n r ( χ s R s I ) exp i ω ( t n c r ( s R + χ s I ) ) exp ( i θ F ) = F u ̃ F exp ( i θ F ) A t P r ,
( v x v y v z ) = ( cos γ 0 sin γ 0 1 0 sin γ 0 cos γ ) ( v x m v y m v z m ) .
Incident : E ̃ 0 = E 0 u ̃ 0 exp [ i ω 0 ( t n 0 c r s 0 ) ] exp ( i θ n ) ,
Reflected : E ̃ r = E r u ̃ r exp [ i ω r ( t n 0 c r s r ) ] exp ( i θ r ) ,
Refracted 1 : E ̃ 1 = E 1 u ̃ 1 exp [ i ω 1 ( t n ̃ 1 c r s ̃ 1 ) ] exp ( i θ 1 ) ,
Refracted 2 : E ̃ 2 = E 2 u ̃ 2 exp [ i ω 2 ( t n ̃ 2 c r s ̃ 2 ) ] exp ( i θ 2 ) ,
( E ̃ 0 ) x m , y m + ( E ̃ r ) x m , y m = ( E ̃ 1 ) x m , y m + ( E ̃ 2 ) x m , y m .
ω 0 ( t n 0 c r s 0 ) = ω r ( t n 0 c r s r ) = ω 1 ( t n ̃ 1 c r s ̃ 1 ) = ω 2 ( t n ̃ 2 c r s ̃ 2 ) .
ω 0 = ω r = ω 1 = ω 2 ,
n 0 r s 0 = n 0 r s r = n ̃ 1 r s ̃ 1 = n ̃ 2 r s ̃ 2 .
( n 0 s 0 n 0 s r ) r = 0 ,
( n 0 s 0 n ̃ 1 , 2 s ̃ 1 , 2 ) r = 0 .
( n 0 s 0 n 0 s r ) × u z m = 0 ,
( n 0 s 0 n ̃ 1 , 2 s ̃ 1 , 2 ) × u z m = 0 .
n 0 s 0 × u z m = n 0 s r × u z m = n ̂ 1 , 2 s ̃ 1 , 2 × u z m .
n 0 s 0 × u z m = n 1 , 2 ( s R + χ s I ) 1 , 2 × u z m ,
( χ s R s I ) 1 , 2 × u z m = 0 .
n 0 sin α 0 = n 1 , 2 s R + χ s I 1 , 2 sin α 1 , 2 = ( s I 1 , 2 2 n ̃ 1 , 2 2 + n 1 , 2 2 ) 1 2 sin α 1 , 2 = n 0 sin α r ,
s R + χ s I 1 , 2 = { [ s R 2 ( 1 + χ 2 ) χ 2 ] 1 2 } 1 , 2 = { [ 1 + s I 2 ( 1 + χ 2 ) ] 1 2 } 1 , 2 .
n 0 s 0 y m = n 0 s r y m = n ̃ 1 , 2 s ̃ 1 , 2 y m = n 1 , 2 ( s R y m + χ s I y m ) 1 , 2 ,
n 0 s 0 x m = n 0 s r x m = n ̃ 1 , 2 s ̃ 1 , 2 x m = n 1 , 2 ( s R x m + χ s I x m ) 1 , 2 ,
( χ s R x m s I x m ) 1 , 2 = 0 ,
( χ s R y m s I y m ) 1 , 2 = 0 .
s 0 = ( sin α 0 sin δ , sin α 0 cos δ , cos α 0 ) .
s r = ( sin α 0 sin δ , sin α 0 cos δ , cos α 0 ) ,
s ̃ 1 , 2 = [ n 0 n ̃ 1 , 2 sin α 0 sin δ , n 0 n ̃ 1 , 2 sin α 0 cos δ , ( 1 n 0 2 n ̃ 1 , 2 2 sin 2 α 0 ) 1 2 ] .
1 n ̃ 1 2 = 1 s ̃ 1 z 2 n ̃ ϵ 2 + s ̃ 1 z 2 n ̃ ω 2 ,
1 n ̃ 2 2 = 1 n ̃ ω 2 ,
s ̃ 2 = [ n 0 n ̃ ω sin α 0 sin δ , n 0 n ̃ ω sin α 0 cos δ , ( 1 n 0 2 n ̃ ω 2 sin 2 α 0 ) 1 2 ] .
s 2 R = 1 ( 1 + χ ω 2 ) ( n 0 n ω sin α 0 sin δ , n 0 n ω sin α 0 cos δ , m 1 4 cos φ 2 ) ,
s 2 I = 1 ( 1 + χ ω 2 ) ( n 0 χ ω n ω sin α 0 sin δ , n 0 χ ω n ω sin α 0 cos δ , m 1 4 sin φ 2 ) ,
tg φ = 2 χ ω n 0 2 n ω 2 sin 2 α 0 ( 1 + χ ω 2 ) 2 ( 1 χ ω 2 ) n 0 2 n ω 2 sin 2 α 0 ,
m = ( 1 + χ ω 2 ) 2 [ ( 1 + χ ω 2 ) 2 + n 0 4 n ω 4 sin 4 α 0 2 ( 1 χ ω 2 ) n 0 2 n ω 2 sin 2 α 0 ] .
s 2 R + χ ω s 2 1 = [ n 0 n ω sin α 0 sin δ , n 0 n ω sin α 0 cos δ , m 1 4 cos β cos ( β + φ 2 ) ] ,
χ ω s 2 R s 2 I = [ 0 , 0 , m 1 4 cos β sin ( β + φ 2 ) ] ,
s ̃ 1 = n 0 n ̃ 1 sin α 0 sin δ , n 0 n ̃ 1 sin α 0 cos δ , ( 1 n 0 2 n ̃ 1 2 sin 2 α 0 ) 1 2 .
s ̃ 1 x = cos γ s ̃ 1 x m sin γ s ̃ 1 z m ,
s ̃ 1 y = s ̃ 1 y m .
s ̃ 1 z 2 = 1 ( cos γ s ̃ 1 x m sin γ s ̃ 1 z m ) 2 s ̃ 1 y m 2 .
s ̃ 1 z 2 = 1 ( cos γ s ̃ 1 x m sin γ s ̃ 1 z m ) 2 s ̃ 1 y m 2 = cos 2 γ n 0 2 n ̃ 1 2 ( s 0 m x 2 cos 2 γ + s 0 m y 2 cos 2 γ ) + sin 2 γ n 0 n ̃ 1 s 0 m x [ 1 n 0 2 n ̃ 1 2 ( s 0 m x 2 + s 0 m y 2 ) ] 1 2 .
M ̃ 1 = [ 1 + ( 1 n ̃ ω 2 1 n ̃ ϵ 2 ) n 0 2 ( s 0 m x 2 cos 2 γ + s 0 m y 2 cos 2 γ ) ] 2 + n 0 4 sin 2 2 γ s 0 m x 2 ( 1 n ̃ ω 2 1 n ̃ ϵ 2 ) 2 ( s 0 m x 2 + s 0 m y 2 ) ,
M ̃ 2 = 2 [ 1 + ( 1 n ̃ ω 2 1 n ̃ ϵ 2 ) n 0 2 ( s 0 m x 2 cos 2 γ + s 0 m y 2 cos 2 γ ) ] [ cos 2 γ n 0 2 n ̃ ϵ 2 ( s 0 m x 2 cos 2 γ + s 0 m y 2 cos 2 γ ) ] n 0 2 s 0 m x 2 sin 2 2 γ ( 1 n ̃ ω 2 1 n ̃ ϵ 2 ) 1 2 n 0 2 ( s 0 m x 2 + s 0 m y 2 ) n ̃ ϵ 2 ,
M ̃ 3 = [ cos 2 γ n 0 2 n ̃ ϵ 2 ( s 0 m x 2 cos 2 γ + s 0 m y 2 cos 2 γ ) ] 2 n 0 2 s 0 m x 2 sin 2 2 γ n ̃ ϵ 2 [ 1 n 0 2 ( s 0 m x 2 + s 0 m y 2 ) n ̃ ϵ 2 ] ,
M ̃ 1 s ̃ 1 z 4 + M ̃ 2 s ̃ 1 z 2 + M ̃ 3 = 0 .
s ̃ 1 z 2 = M ̃ 2 ± ( M ̃ z 2 4 M ̃ 1 M ̃ 3 ) 1 2 2 M ̃ 1 ,
s ̃ 1 z = s ̃ 1 x m sin γ + s ̃ 1 z m cos γ .
I 2 = Re { exp ( i β ω ) [ s ̃ x y ( u R x y u x i u I x y u y ) + s x ( u 2 R x y + u 2 I x y ) u x ] } ,
I 1 = Re { 1 n ̃ 1 [ ( s ̃ z n ̃ ω 2 ) * ( u 2 R x y + u 2 I x y ) u z + u ̃ x y ( s ̃ x y n ̃ ϵ 2 ) * ] } ,
s ̃ = s ̃ x y u ̃ x y + s ̃ z u z ,
u ̃ x y = u R x y u x + i u I x y u y .
s ̃ x y 1 , 2 = [ n 0 n ̃ 1 , 2 sin α 0 sin δ cos γ + sin γ ( 1 n 0 2 n ̃ 1 , 2 2 sin 2 α 0 ) 1 2 ] u x + n 0 n ̃ 1 , 2 sin α 0 cos δ u y .
s 0 = ( 0 , sin α 0 , cos α 0 ) ,
s r = ( 0 , sin α 0 , cos α 0 ) ,
s ̃ z = 0 , n 0 n ̃ ω sin α 0 , ( 1 n ω 2 n ̃ ω 2 sin 2 α 0 ) 1 2 .
s ̃ 1 z 4 2 ( 1 n 0 2 n ̃ ω 2 sin 2 α 0 ) s ̃ 1 z 2 + ( 1 n 0 2 n ̃ ω 2 sin 2 α 0 ) 2 = 0 ,
s ̃ 1 z = ( 1 n 0 2 n ̃ ω 2 sin 2 α 0 ) 1 2 .
s ̃ 1 = 0 , n 0 n ̃ 1 sin α 0 , ( 1 n 0 2 n ̃ 0 2 sin 2 α 0 ) 1 2 .
u ̃ x y = u R x y + i u I x y = ( 0 , 1 , 0 ) .
I = Re [ exp ( i β ω ) ( s ̃ x y u x + s z u z ) ] = Re [ exp ( i β ω ) s ̃ ] = 1 ( 1 + χ 2 ) 1 2 ( s R + χ s I ) .
s ̃ 2 = s 2 R + i s 2 I = [ n 0 n ̃ ω sin α 0 sin δ , n 0 n ̃ ω sin α 0 cos δ , ( 1 n 0 2 n ̃ ω 2 sin 2 α 0 ) 1 2 ] ,
s ̃ 1 = s 1 R + i s 1 I = [ n 0 n ̃ 1 sin α 0 sin δ , n 0 n ̃ 1 sin α 0 cos δ , ( 1 n 0 2 n ̃ ϵ 2 sin 2 α 0 1 + ( 1 n ̃ ω 2 1 n ̃ ϵ 2 ) n 0 2 sin 2 α 0 ) 1 2 ] ,
1 n ̃ 1 2 = 1 n ̃ ϵ 2 n ̃ ω 2 n ̃ ϵ 2 + ( 1 n ̃ ω 2 n ̃ ϵ 2 ) n 0 2 sin 2 α 0 n ̃ ω 2 n ̃ ϵ 2 + ( n ̃ ϵ 2 n ̃ ω 2 ) n 0 2 sin 2 α 0 .
s ̃ x y 1 , 2 = n 0 n ̃ 1 , 2 sin α 0 ( sin δ u x + cos δ u y ) = s ̃ x y 1 , 2 u ̃ x y ,
I 2 = cos β ω ( s R + χ ω s 1 ) ,
I 1 = Re ( 1 n ̃ 1 { [ ( 1 n 0 2 n ̃ ϵ 2 sin 2 α 0 n ̃ ω 4 + n ̃ ω 2 ( n ̃ ϵ 2 n ̃ ω 2 n ̃ ϵ 2 ) n 0 2 sin 2 α 0 ) 1 2 ] u z + ( sin δ u x + cos δ u y ) n 0 sin α 0 ( n ̃ ϵ 2 n ̃ 1 ) } ) .
s ̃ 2 = s 2 R + i s 2 I = [ n 0 n ̃ ω sin α 0 sin δ , n 0 n ̃ ω sin α 0 cos δ , ( 1 n 0 2 n ̃ ω 2 sin 2 α 0 ) 1 2 ] ,
s ̃ 1 = s 1 R + i s 1 I = [ n 0 n ̃ 1 sin α 0 sin δ , n 0 n ̃ 1 sin α 0 cos δ , ( 1 n 0 2 n ̃ 1 2 sin 2 α 0 ) 1 2 ] ,
1 n ̃ 1 2 = 1 n ̃ ϵ 2 1 n 0 2 sin 2 α 0 sin 2 δ ( 1 n ̃ ω 2 1 n ̃ ϵ 2 ) ,
s ̃ x y 1 , 2 = ( 1 n 0 2 n ̃ 1 , 2 2 sin 2 α 0 ) 1 2 u x + n 0 n ̃ 1 , 2 sin α 0 cos δ u y = s ̃ x y 1 , 2 u x y 1 , 2 .

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