Abstract

We analyze the first-order nonspecular effects of three-dimensional beams in interfaces formed by two lineal dielectric media. We also analyze the dependence of the transverse effects on polarization and compare the results with those obtained for isotropic interfaces. In particular, we determine analytically the complex transverse lateral displacements of a beam with Gaussian distribution of intensity when it is reflected on an interface formed by an isotropic medium and a uniaxial anisotropic one and the mean direction of propagation of the beam is contained in each of the characteristic planes of the crystal.

© 2003 Optical Society of America

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References

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  1. F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).
  2. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [CrossRef]
  3. W. Nasalski, “Longitudinal and transverse effects of nonspecular reflection,” J. Opt. Soc. Am. A 13, 172–181 (1996).
    [CrossRef]
  4. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  5. F. I. Baida, D. Van Labeke, J.-M. Vigoureux, “Numerical study of the displacement of a three-dimensional Gaussian beam transmitted at total internal reflection. Near-field application,” J. Opt. Soc. Am. A 17, 858–866 (2000).
  6. J. Alda, “Transverse angular shift in the reflection of light beams,” Opt. Commun. 182, 1–10 (2000).
    [CrossRef]
  7. M. A. Porras, “Nonspecular reflection of general light beams at a dielectric interface,” Opt. Commun. 135, 369–377 (1997).
    [CrossRef]
  8. W. Nasalski, “Three-dimensional beam reflection at dielectric interfaces,” Opt. Commun. 197, 217–233 (2001).
    [CrossRef]
  9. O. N. Stavroudis, “Ray tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962).
    [CrossRef]
  10. W. Swindell, “Extraordinary ray and wave-tracing in uniaxial crystals,” Appl. Opt. 14, 2298–2301 (1975).
    [CrossRef] [PubMed]
  11. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef] [PubMed]
  12. M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [CrossRef] [PubMed]
  13. J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at a planar interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
    [CrossRef]
  14. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  15. M. Barabas, G. Szarvas, “Fourier description of the propagation and focusing of an extraordinary beam in a planar uniaxial medium,” Appl. Opt. 34, 11–21 (1995).
    [CrossRef] [PubMed]
  16. L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).
  17. A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  18. M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals II: coefficients of transmission and reflection for an ordinary incident wave,” J. Mod. Opt. 37, 131–144 (1990).
    [CrossRef]
  19. M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals III: coefficients of transmission and reflection for an extraordinary incident wave,” J. Mod. Opt. 37, 1139–1148 (1990).
    [CrossRef]
  20. L. I. Perez, “Reflexión y refracción en cristales birrefringentes monoaxiales” Chap. II, doctoral thesis (University of Buenos Aires, Buenos Aires, Argentina, 1990).
  21. M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
    [CrossRef]
  22. M. C. Simon, D. Farı́as, “Reflection and refraction in uniaxial crystals with dielectric and magnetic anisotropy,” J. Mod. Opt. 41, 413–429 (1994).
    [CrossRef]
  23. J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
    [CrossRef]
  24. J. Lekner, “Normal incidence transmission ellipsometry of anisotropic layers,” J. Opt. A Pure Appl. Opt. 3, 307–321 (1994).
    [CrossRef]
  25. J. Lekner, “Reflection ellipsometry of uniaxial crystals,” J. Opt. Soc. Am. A 14, 1359–1362 (1997).
    [CrossRef]
  26. G. D. Landry, T. A. Maldonado, “Gaussian beam transmission and reflection from a general anisotropic multiplayer structure,” Appl. Opt. 35, 5870–5879 (1996).
    [CrossRef] [PubMed]
  27. D. Dhayalan, J. J. Stamnes, “Focusing of electromag-netic waves into a dielectric slab: I. Exact and asymptotic results,” J. Opt. A Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  28. L. I. Perez, “Reflection and nonspecular effects of 2D Gaussian beams in interfaces between isotropic and uniaxial anisotropic media,” J. Mod. Opt. 47, 1645–1658 (2000).
    [CrossRef]
  29. J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
    [CrossRef]
  30. L. I. Perez, “Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial interfaces,” J. Opt. A Pure Appl. Opt. 4, 640–649 (2002).
    [CrossRef]
  31. A. Ciattoni, G. Cincotti, C. Palma, “Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1431 (2002).
    [CrossRef]
  32. C. E. Vanney, “Displacement of the refracted wave in presence of inhibited reflection,” J. Mod. Opt. 47, 685–700 (2000).
    [CrossRef]
  33. L. I. Perez, “Nonabsorbent uniaxial crystals: Poynting vectors for the evanescent waves,” Optik 97, 142–148 (1994).
  34. L. I. Perez, “Transverse angular shift of linearly polarised beams reflected on isotropic interfaces,” in Optics for the Quality of Life, Proceedings of the 19th Congress of the International Commission for Optics (International Commission for Optics, 2002 www.ico-optics.org ), pp. 781–782.
  35. L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” in Proc. Asoc. Fı́s. Argentina, Anales AFA5, 224–228 (1993).
  36. J. M. Simon, L. I. Perez, V. A. Presa, “Surface electromagnetic waves in the interface of an absorbing medium with an uniaxial crystal: comparison between closed solutions and attenuated total reflection,” J. Opt. Soc. Am. A 13, 1249–1257 (1996).
    [CrossRef]
  37. M. C. Simon, L. I. Perez, “Total reflection in uniaxial crystals,” Optik 82, 37–42 (1989).

2002

L. I. Perez, “Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial interfaces,” J. Opt. A Pure Appl. Opt. 4, 640–649 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1431 (2002).
[CrossRef]

2001

2000

J. Alda, “Transverse angular shift in the reflection of light beams,” Opt. Commun. 182, 1–10 (2000).
[CrossRef]

C. E. Vanney, “Displacement of the refracted wave in presence of inhibited reflection,” J. Mod. Opt. 47, 685–700 (2000).
[CrossRef]

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

L. I. Perez, “Reflection and nonspecular effects of 2D Gaussian beams in interfaces between isotropic and uniaxial anisotropic media,” J. Mod. Opt. 47, 1645–1658 (2000).
[CrossRef]

F. I. Baida, D. Van Labeke, J.-M. Vigoureux, “Numerical study of the displacement of a three-dimensional Gaussian beam transmitted at total internal reflection. Near-field application,” J. Opt. Soc. Am. A 17, 858–866 (2000).

1998

D. Dhayalan, J. J. Stamnes, “Focusing of electromag-netic waves into a dielectric slab: I. Exact and asymptotic results,” J. Opt. A Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

1997

M. A. Porras, “Nonspecular reflection of general light beams at a dielectric interface,” Opt. Commun. 135, 369–377 (1997).
[CrossRef]

J. Lekner, “Reflection ellipsometry of uniaxial crystals,” J. Opt. Soc. Am. A 14, 1359–1362 (1997).
[CrossRef]

1996

1995

1994

M. C. Simon, D. Farı́as, “Reflection and refraction in uniaxial crystals with dielectric and magnetic anisotropy,” J. Mod. Opt. 41, 413–429 (1994).
[CrossRef]

J. Lekner, “Normal incidence transmission ellipsometry of anisotropic layers,” J. Opt. A Pure Appl. Opt. 3, 307–321 (1994).
[CrossRef]

L. I. Perez, “Nonabsorbent uniaxial crystals: Poynting vectors for the evanescent waves,” Optik 97, 142–148 (1994).

1993

1991

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

1990

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals II: coefficients of transmission and reflection for an ordinary incident wave,” J. Mod. Opt. 37, 131–144 (1990).
[CrossRef]

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals III: coefficients of transmission and reflection for an extraordinary incident wave,” J. Mod. Opt. 37, 1139–1148 (1990).
[CrossRef]

1989

M. C. Simon, L. I. Perez, “Total reflection in uniaxial crystals,” Optik 82, 37–42 (1989).

1986

1983

1977

1975

1972

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

1962

1955

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).

Alda, J.

J. Alda, “Transverse angular shift in the reflection of light beams,” Opt. Commun. 182, 1–10 (2000).
[CrossRef]

Baida, F. I.

Barabas, M.

Ciattoni, A.

Cincotti, G.

Crosignani, B.

Dhayalan, D.

D. Dhayalan, J. J. Stamnes, “Focusing of electromag-netic waves into a dielectric slab: I. Exact and asymptotic results,” J. Opt. A Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Dhayalan, V.

Di Porto, P.

Echarri, R. M.

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals II: coefficients of transmission and reflection for an ordinary incident wave,” J. Mod. Opt. 37, 131–144 (1990).
[CrossRef]

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals III: coefficients of transmission and reflection for an extraordinary incident wave,” J. Mod. Opt. 37, 1139–1148 (1990).
[CrossRef]

M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
[CrossRef] [PubMed]

Fari´as, D.

M. C. Simon, D. Farı́as, “Reflection and refraction in uniaxial crystals with dielectric and magnetic anisotropy,” J. Mod. Opt. 41, 413–429 (1994).
[CrossRef]

Fedorov, F. I.

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).

Feit, M. D.

Fleck, J. A.

Garea, M. T.

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

Imbert, C.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Landry, G. D.

Lekner, J.

Maldonado, T. A.

Nasalski, W.

W. Nasalski, “Three-dimensional beam reflection at dielectric interfaces,” Opt. Commun. 197, 217–233 (2001).
[CrossRef]

W. Nasalski, “Longitudinal and transverse effects of nonspecular reflection,” J. Opt. Soc. Am. A 13, 172–181 (1996).
[CrossRef]

Palma, C.

Perez, L. I.

L. I. Perez, “Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial interfaces,” J. Opt. A Pure Appl. Opt. 4, 640–649 (2002).
[CrossRef]

L. I. Perez, “Reflection and nonspecular effects of 2D Gaussian beams in interfaces between isotropic and uniaxial anisotropic media,” J. Mod. Opt. 47, 1645–1658 (2000).
[CrossRef]

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

J. M. Simon, L. I. Perez, V. A. Presa, “Surface electromagnetic waves in the interface of an absorbing medium with an uniaxial crystal: comparison between closed solutions and attenuated total reflection,” J. Opt. Soc. Am. A 13, 1249–1257 (1996).
[CrossRef]

L. I. Perez, “Nonabsorbent uniaxial crystals: Poynting vectors for the evanescent waves,” Optik 97, 142–148 (1994).

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

M. C. Simon, L. I. Perez, “Total reflection in uniaxial crystals,” Optik 82, 37–42 (1989).

L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” in Proc. Asoc. Fı́s. Argentina, Anales AFA5, 224–228 (1993).

L. I. Perez, “Reflexión y refracción en cristales birrefringentes monoaxiales” Chap. II, doctoral thesis (University of Buenos Aires, Buenos Aires, Argentina, 1990).

Porras, M. A.

M. A. Porras, “Nonspecular reflection of general light beams at a dielectric interface,” Opt. Commun. 135, 369–377 (1997).
[CrossRef]

Presa, V. A.

Sherman, G. C.

Simon, J. M.

Simon, M. C.

M. C. Simon, D. Farı́as, “Reflection and refraction in uniaxial crystals with dielectric and magnetic anisotropy,” J. Mod. Opt. 41, 413–429 (1994).
[CrossRef]

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals II: coefficients of transmission and reflection for an ordinary incident wave,” J. Mod. Opt. 37, 131–144 (1990).
[CrossRef]

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals III: coefficients of transmission and reflection for an extraordinary incident wave,” J. Mod. Opt. 37, 1139–1148 (1990).
[CrossRef]

M. C. Simon, L. I. Perez, “Total reflection in uniaxial crystals,” Optik 82, 37–42 (1989).

M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
[CrossRef] [PubMed]

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
[CrossRef] [PubMed]

Stamnes, J. J.

Stavroudis, O. N.

Swindell, W.

Szarvas, G.

Tamir, T.

Van Labeke, D.

Vanney, C. E.

C. E. Vanney, “Displacement of the refracted wave in presence of inhibited reflection,” J. Mod. Opt. 47, 685–700 (2000).
[CrossRef]

Vigoureux, J.-M.

Appl. Opt.

Dokl. Akad. Nauk SSSR

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).

J. Mod. Opt.

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals II: coefficients of transmission and reflection for an ordinary incident wave,” J. Mod. Opt. 37, 131–144 (1990).
[CrossRef]

M. C. Simon, R. M. Echarri, “Internal reflection in uniaxial crystals III: coefficients of transmission and reflection for an extraordinary incident wave,” J. Mod. Opt. 37, 1139–1148 (1990).
[CrossRef]

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

M. C. Simon, D. Farı́as, “Reflection and refraction in uniaxial crystals with dielectric and magnetic anisotropy,” J. Mod. Opt. 41, 413–429 (1994).
[CrossRef]

L. I. Perez, “Reflection and nonspecular effects of 2D Gaussian beams in interfaces between isotropic and uniaxial anisotropic media,” J. Mod. Opt. 47, 1645–1658 (2000).
[CrossRef]

C. E. Vanney, “Displacement of the refracted wave in presence of inhibited reflection,” J. Mod. Opt. 47, 685–700 (2000).
[CrossRef]

J. Opt. A Pure Appl. Opt.

L. I. Perez, “Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial interfaces,” J. Opt. A Pure Appl. Opt. 4, 640–649 (2002).
[CrossRef]

J. Lekner, “Normal incidence transmission ellipsometry of anisotropic layers,” J. Opt. A Pure Appl. Opt. 3, 307–321 (1994).
[CrossRef]

D. Dhayalan, J. J. Stamnes, “Focusing of electromag-netic waves into a dielectric slab: I. Exact and asymptotic results,” J. Opt. A Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

F. I. Baida, D. Van Labeke, J.-M. Vigoureux, “Numerical study of the displacement of a three-dimensional Gaussian beam transmitted at total internal reflection. Near-field application,” J. Opt. Soc. Am. A 17, 858–866 (2000).

J. Lekner, “Reflection ellipsometry of uniaxial crystals,” J. Opt. Soc. Am. A 14, 1359–1362 (1997).
[CrossRef]

T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
[CrossRef]

J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
[CrossRef]

W. Nasalski, “Longitudinal and transverse effects of nonspecular reflection,” J. Opt. Soc. Am. A 13, 172–181 (1996).
[CrossRef]

J. M. Simon, L. I. Perez, V. A. Presa, “Surface electromagnetic waves in the interface of an absorbing medium with an uniaxial crystal: comparison between closed solutions and attenuated total reflection,” J. Opt. Soc. Am. A 13, 1249–1257 (1996).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[CrossRef]

J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1431 (2002).
[CrossRef]

Opt. Commun.

J. Alda, “Transverse angular shift in the reflection of light beams,” Opt. Commun. 182, 1–10 (2000).
[CrossRef]

M. A. Porras, “Nonspecular reflection of general light beams at a dielectric interface,” Opt. Commun. 135, 369–377 (1997).
[CrossRef]

W. Nasalski, “Three-dimensional beam reflection at dielectric interfaces,” Opt. Commun. 197, 217–233 (2001).
[CrossRef]

Optik

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

L. I. Perez, “Nonabsorbent uniaxial crystals: Poynting vectors for the evanescent waves,” Optik 97, 142–148 (1994).

M. C. Simon, L. I. Perez, “Total reflection in uniaxial crystals,” Optik 82, 37–42 (1989).

Phys. Rev. D

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Other

L. I. Perez, “Reflexión y refracción en cristales birrefringentes monoaxiales” Chap. II, doctoral thesis (University of Buenos Aires, Buenos Aires, Argentina, 1990).

L. I. Perez, “Transverse angular shift of linearly polarised beams reflected on isotropic interfaces,” in Optics for the Quality of Life, Proceedings of the 19th Congress of the International Commission for Optics (International Commission for Optics, 2002 www.ico-optics.org ), pp. 781–782.

L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” in Proc. Asoc. Fı́s. Argentina, Anales AFA5, 224–228 (1993).

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Figures (8)

Fig. 1
Fig. 1

Representation of two of the rays of an incident beam that are symmetric with respect to the plane of incidence xz. R is the mean direction of propagation of the beam, and the isotropic interface is in the yz plane.

Fig. 2
Fig. 2

Coordinate systems. The angle ϑ (positive in the figure) determines the direction of the optical axis z 3 of the crystal; ε is the dielectric constant of the isotropic medium, ε o the ordinary principal dielectric constant, and ε e the extraordinary one. The coordinates x i and x r correspond to the mean direction of the incident and the specularly reflected beams, respectively (determined by the angle α); z i and z r are in the incidence plane, and the coordinates y i and y r are perpendicular to them.

Fig. 3
Fig. 3

Incident and reflected tridimensional Gaussian beams considering first-order effects. P and P are polaroids perpendicular to the mean direction of incidence x i and of specular reflection x r , respectively. The optical axis is contained in the plane of incidence.

Fig. 4
Fig. 4

  δ = 0 ° . (a) Transverse lateral displacement (in vacuum wavelengths units) and (b) transverse angular shift (in minutes) of the reflected beam with p polarization for two interfaces. Solid curves: = 1.7550 , o = 1.6584 , e = 1.4865 , and ϑ = 30 ° . ( α T = 70.9 ° , α T = 67.1 ° , α B = 49.6 ° ). Dotted curves: = 1.7550 , o = 1.6584 , e = 1.4865 , and ϑ = 60 ° . ( α T = 70.9 ° , α T = 60.8 ° , α B = 35.4 ° ). The incident beam is linearly polarized with γ = 45 ° .

Fig. 5
Fig. 5

The same as Fig. 4 but for the s-reflected beam.

Fig. 6
Fig. 6

  δ = 90 ° . Transverse lateral displacement (in vacuum wavelengths units) and transverse angular shift (in minutes) of the reflected beam with (a) p and (b) s polarization for two interfaces. Dotted curves: = 1.7550 , o = 1.6584 , e = 1.4865 , and ϑ = 30 ° . Solid curves, = 1.7550 , o = 1.6584 , e = 1.4865 , and ϑ = 60 ° . The incident beam is linearly polarized with γ = 45 ° , α T = 70.9 ° and α T = 57.9 ° .

Fig. 7
Fig. 7

  δ = 90 ° . Transverse lateral displacement (in vacuum wavelength units) and transverse angular shift (in minutes) of the reflected beam with (a) p and (b) s polarization for a glass–calcite interface with ϑ = 40 ° for two angles of polarization of the incident beam. Solid curves, γ = 30 ° ; dotted curves, γ = - 30 ° .

Fig. 8
Fig. 8

  δ = 90 ° . Transverse lateral displacement (in vacuum wavelength units) and transverse angular shift (in minutes) of the reflected beams for a glass–calcite interface with ϑ = 40 ° for an unpolarized incident beam.

Equations (60)

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

E * ( x ,   y ,   z ) = 1 2 π - d k y r * - E * ( k x ,   k y ,   k z ) × exp w 2 ( k y i 2 + k z i 2 ) 4 × exp [ i ( k x * x + k y * y + k z * z ) ] d k z r * ,
E y r * ( x r ,   y r ,   z r ) = 1 2 π - d k y r * - E y r * ( k x r * ,   k y r * ,   k z r * ) × exp - w 2 ( k y r * 2 + k z r * 2 ) 4 × exp [ i ( k x r * x r + k y r *   y r + k z r * z r ) ] d k z r * .
E y r * = M yy E y i + M yz E z i , E z r * = M zy E y i + M zz E z i
 
M yy = cos   δ ( R yy   cos   δ - R yz   sin   δ ) - k y r * k x r *   sin   α ( R yy   sin   δ + R yz   cos   δ ) + sin   δ ( R zz   sin   δ - R zy   cos   δ ) + k y r * k x r *   sin   α ( R zz   cos   δ + R zy   sin   δ ) ,
M yz = cos   δ ( R yy   sin   δ + R yz   cos   δ ) cos   α + sin   α   k z r * k x r * - sin   δ ( R zz   cos   δ + R zy   sin   δ ) cos   α + sin   α   k z r * k x r * ,
M zy = - sin   δ k x r * + sin   α   cos   δ k y r * cos   α k x r * + sin   α k z r * ( R yy   cos   δ - R yz   sin   δ ) - k y r * k x r *   sin   α ( R yy   sin   δ + R yz   cos   δ ) + cos   δ k x r * - sin   α   sin   δ k y r * cos   α k x r * + sin   α k z r * ( R zz   sin   δ - R zy   cos   δ ) + k y r * k x r *   sin   α ( R zz   cos   δ + R zy   sin   δ ) ,
M zz = - sin   δ k x r * + sin   α   cos   δ k y r * cos   α k x r * + sin   α k z r * × ( R yy sin   δ + R yz cos   δ ) cos   α + sin   α   k z r * k x r * - cos   δ k x r * - sin   α   sin   δ k y r * cos   α k x r * + sin   α k z r * × ( R zz cos   δ + R zy   sin   δ ) cos   α + sin   α   k z r * k x r * ,
R yy = 2   T 1 + k y 2 T 2 G 1 + k y 2 G 2 - 1 , R yz = - 2 k y   S 1 + k y 2 S 2 G 1 + k y 2 G 2 ,
R zz = 2   P 1 + k y 2 P 2 G 1 + k y 2 G 2 - 1 , R zy = - 2 k y   Q 1 + k y 2 Q 2 G 1 + k y 2 G 2 ,
T 1 = k x k z 2 ( ε o k x + ε k x ) ( x     z 3 ) 2 + k x k x ( ε k x 2 + ε o k x k x ) ( z     z 3 ) 2 - k x k z ( k x + k x ) ( ε o k x + ε k x ) ( x     z 3 ) ( z     z 3 ) ,
T 2 = ε k x ( k x + k x ) ( x     z 3 ) 2 + [ k x ( ε k x + ε o k x ) + ( ε - ε o ) k z 2 ] ( z     z 3 ) 2 - k z [ 2 ε o k x - ε ( k x + k x ) ] ( x     z 3 ) ( z     z 3 ) ,
S 1 = k z ( ε o k x 2 - ε k x k x ) ( x     z 3 ) 2 + k z k x ( ε k x - ε o k x ) ( z     z 3 ) 2 + [ k z 2 ( ε o k x - ε k x ) + k x ( ε k x k x - ε o k x 2 ) ] ( x     z 3 ) ( z     z 3 ) ,
S 2 = k z ( ε - ε o ) ( z     z 3 ) 2 + k x ( ε - ε o ) ( x     z 3 ) ( z z 3 ) ,
P 1 = ( k x + k x ) [ ε k x k z 2 ( x     z 3 ) 2 + ε k x 3 ( z     z 3 ) 2 - ε k x k z ( k x + k x ) ( x     z 3 ) ( z     z 3 ) ] ,
P 2 = k x ( ε k x + ε o k x ) ( x     z 3 ) 2 + [ ( ε - ε o ) ( k x 2 + k y 2 ) + ε k x ( k x + k x ) ] ( z     z 3 ) 2 + k z [ ( ε o - ε ) k x + ( ε o k x - ε k x ) ] ( x     z 3 ) ( z     z 3 ) ,
Q 1 = k z ( ε o k x 2 - ε k x k x ) ( x     z 3 ) 2 + k z k x 2 ( ε - ε o ) × ( z     z 3 ) 2 + [ ε k x 2 ( k x + k x ) - 2 ε o k x 2 k x ] × ( x     z 3 ) ( z     z 3 ) ,
Q 2 = k z ( ε - ε o ) ( z     z 3 ) 2 + [ ε ( k x + k x ) - 2 ε o k x ] ( x     z 3 ) ( z     z 3 ) ,
G 1 = ( k x + k x ) [ k z 2 ( ε o k x + ε k x ) ( x     z 3 ) 2 + k x ( ε k x 2 + ε o k x k x ) ( z     z 3 ) 2 - k z ( k x + k x ) ( ε o k x + ε k x ) ( x     z 3 ) ( z     z 3 ) ] ,
G 2 = ( k x + k x ) [ ( ε o k x + ε k x ) ( x     z 3 ) 2 + [ ( ε o k x + ε k x ) + ε o ( k x - k x ) ] ( z     z 3 ) 2 ] ,
k x = [ μ ω 2 ε - k y 2 - k z 2 ] 1 / 2
k x = [ μ ω 2 ε o - k y 2 - k z 2 ] 1 / 2
k x 2 [ ε o ( z z 3 ) 2 + ε e ( x     z 3 ) 2 ] + 2 k x k z ( ε e - ε o ) × ( z     z 3 ) ( x     z 3 ) + ε o k y 2 + k z 2 [ ε o ( x     z 3 ) 2 + ε e ( z     z 3 ) 2 ] = μ ω 2 ε o ε e .
E y r * ( x r ,   y r ,   z r )
= 1 π   E y r * | 0 w 2 + i   2 x r k   × exp - ( z r - L s lo ) 2 w 2 + i   2 x r k   exp - ( y r - L s tr ) 2 w 2 + i   2 x r k   ,
E z r * ( x r ,   y r ,   z r )
= 1 π   E z r * | 0 w 2 + i   2 x r k   × exp - ( z r - L p lo ) 2 w 2 + i   2 x r k   exp - ( y r - L p tr ) 2 w 2 + i   2 x r k   ,
L s lo = i     ln   E y r * k z r * 0 , L p lo = i     ln   E z r * k z r * 0 ,
L s tr = i     ln   E y r * k y r * 0 , L p tr = i     ln   E z r * k y r * 0 .
E y i / E z i = R   exp ( i χ ) ,
( Re   E y r * ) 2 = E z i 2 ( R 2 M yy 2 sin 2   ω t + M yz 2 cos 2   ω t + 2 R   M yy M yz cos   χ   cos 2   ω t ) ,
( Re   E z r * ) 2 = E z i 2 ( R 2 M zy 2 sin 2   ω t + M zz 2 cos 2   ω t + 2 R   M zz M zy cos   χ   cos 2   ω t ) .
E y r * ( k x ,   k y ,   k z ) = R yy - k y   sin   α k x i   R yz × E y i + k x k x i   R yz E z i ,
E z r * ( k x ,   k y ,   k z ) = k y   sin   α k x   ( R zz - R yy ) + k y 2   sin 2   α k x k x i   R yz - k x i k x   R zy E y i - R zz + k y   sin   α k x i   R yz E z i ,
L z lo = - i   2   sin   α ( μ ω 2 ) 1 / 2 ( ε o - ε   sin 2   α ) 1 / 2 ,
L p lo = - i   2   sin   α ( μ ω 2 ) 1 / 2
× ε o ε e ( ε * - ε ) ( ε * - ε   sin 2   α ) 1 / 2 [ ε ( ε * - ε   sin 2   α ) - ε e ε o   cos 2   α ] ,
L s tr = - 2 i   cos   α ( k x - k x )     ( ε o k x 2 - ε k x k x ) ( x     z 3 ) + ( ε o k x - ε k x ) k z ( z     z 3 ) ( ε k x + ε o k x ) k z ( x     z 3 ) - ( ε k x 2 + ε o k x k x ) ( z     z 3 )     1 R   exp ( i χ ) ,
L p tr = - 2 i   cos   α ( k x + k x )     ( ε o k x 2 - ε k x k x ) ( x     z 3 ) + ( ε o k x - ε k x ) k z ( z     z 3 ) ( ε k x - ε o k x ) k z ( x     z 3 ) - ( ε k x 2 - ε o k x k x ) ( z     z 3 ) + 2 μ ω 2 ε ε o ( k x - k x ) ( x     z 3 ) ( z     z 3 ) [ k z ( x     z 3 ) - k x ( z     z 3 ) ] [ ( ε k x + ε o k x ) k z ( x     z 3 ) - ( ε k x 2 + ε o k x k x ) ( z     z 3 ) ] R     exp ( i χ ) ,
k x = μ ω 2 ε cos   α ,
k x = μ ω 2 ε o - ε   sin 2   α
k x = μ ω 2     sin   α ε ( ε o - ε e ) ( x     z 3 ) ( z     z 3 ) + ε o ε e ε * - ε   sin 2   α ε * .
E y r * ( k x ,   k y ,   k z ) = R zz - R zy   sin   α   k y i k x i E y i + k z i k x i   sin   α - cos   α R yz E z i ,
E z r * ( k x ,   k y ,   k z )
= sin   α   k y k x + cos   α   sin   α   k y i k x i   R yy + R yz + sin   α   k z k x   R zz + R zy   k y i k x i   sin   α E y i + sin   α   k z i k x i - cos   α   sin   α   k y k x i + cos   α R yy + sin   α   k z k x   R zy E z i ,
k x i = k x   cos   α + k y   sin   α ,
k y i = - k z ,
k z i = - k x   sin   α + k y   cos   α ,
  L s tr = - 2 i μ ω 2 ε   sin   α   cos   α [ ( ε o k x 2 - ε k x k x ) ( x z 3 ) 2 + μ ω 2 ε o ( ε - ε o ) ( z     z 3 ) 2 ] [ k y 2 ( ε o k x + ε k x ) ( k x - k x ) ( x z 3 ) 2 + μ ω 2 ε o ( k x - k x ) ( ε k x + ε o k x ) ( z z 3 ) 2 ] R   exp ( i χ ) - 2 ε o μ ω 2 ε k x k y ( k x - k x ) ( x z 3 ) ( z z 3 ) ,
L p tr = - 2 i μ ω 2 ε   sin   α   cos   α [ ( ε o k x 2 - ε k x k x ) ( x     z 3 ) 2 + μ ω 2 ε o ( ε - ε o ) ( z z 3 ) 2 ] R   exp ( i χ ) [ k y 2 ( ε k x - ε o k x ) ( k x + k x ) ( x z 3 ) 2 + μ ω 2 ε o ( k x + k x ) ( ε k x - ε o k x ) ( z z 3 ) 2 ] + 2 ε o μ ω 2 ε k x k y ( k x - k x ) ( x z 3 ) ( z z 3 ) R   exp ( i χ ) ,
k x = μ ω 2 ε cos   α ,
k x = μ ω 2 ε o - ε   sin 2   α ,
k x = μ ω 2 ε o / ε * ε e - ε   sin 2   α .
    L s tr = - 2 i μ ω 2 ε   sin   α   cos   α [ ( ε o k x 2 - ε k x k x ) ( x ˘ z ˘ 3 ) 2 + μ ω 2 ε o ( ε - ε o ) ( z ˘ z ˘ 3 ) 2 ] [ k y 2 ( ε o k x + ε k x ) ( k x - k x ) ( x ˘ z ˘ 3 ) 2 + μ ω 2 ε o ( k x - k x ) ( ε k x + ε o k x ) ( z ˘ z ˘ 3 ) 2 ] tan   γ - 2 ε o μ ω 2 ε k x k y ( k x - k x ) ( x ˘ z ˘ 3 ) ( z ˘ z ˘ 3 ) ,
    L p tr = - 2 i μ ω 2 ε   sin   α   cos   α [ ( ε o k x 2 - ε k x k x ) ( x ˘ z ˘ 3 ) 2 + μ ω 2 ε o ( ε - ε o ) ( z ˘ z ˘ 3 ) 2 ] tan   γ [ k y 2 ( ε k x - ε o k x ) ( k x + k x ) ( x ˘ z ˘ 3 ) 2 + μ ω 2 ε o ( k x + k x ) ( ε k x - ε o k x ) ( z ˘ z ˘ 3 ) 2 ] + 2 ε o μ ω 2 ε k x k y ( k x - k x ) ( x ˘ z ˘ 3 ) ( z ˘ z ˘ 3 ) tan   γ .
ln   E y r * ln ( M yz E z i ) | 0 , 0 + ( ln   M yz E z i ) k y r * 0 , 0 k y r * + ( ln   M yz E z i ) k z r * 0 , 0 k z r * ,
ln   E z r * ln ( M yz E z i ) | 0 , 0 + ( ln   M zz E z i ) k z r * 0 , 0 k z r * .
L s tr = - L p tr = i   [ ( ε o k x 2 - ε k x k x ) ( x z 3 ) 2 + μ ω 2 ε o ( ε - ε o ) ( z z 3 ) 2 ] μ ω 2 ε ε o ( k x - k x ) ( x z 3 ) ( z z 3 ) ,
E y * ( x r ,   y ,   z r ) ( E y * ) 0 , 0 2 π   -   exp - w 2 k y 2 4 exp ( ik y y ) d k y × -   exp - w 2 k z r * 2 4 exp ( ik z r * z r ) × exp ( ln   R yy ) k z r * 0 d k z r * ,
E z r * ( x r ,   y ,   z r ) = E y 2 π   - d k y - M zy × exp - w 2 ( k y 2 + k z i * 2 ) 4 exp [ i ( k x r * x r + k y y + k z r * z r ) ] d k z r * ,

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