Abstract

The propagation of inhomogeneous and elliptically polarized plane waves in absorbing uniaxial anisotropic media is described using complex unitary vectors to represent the direction of propagation and the direction of polarization. Detailed expressions for electric displacement, electric field, and magnetic field vectors are obtained for the ordinary and extraordinary waves, and their geometry is discussed. According to the complex direction of propagation, three particular cases are studied: the real case (homogeneous wave), the case perpendicular to the optical axis, and the case coplanar with the optic axis. The case of isotropic media is also analyzed.

© 2007 Optical Society of America

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  1. M. Berek, "Optische Messmethoden in polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer Theorie der Optick absorbierender Kristalle," Fortschr. Mineral. 22, 1-104 (1937).
  2. B. D. Cervelle, R. Caye, and J. Billard, "Determination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale," Bull. Soc. Fr. Mineral. Cristallogr. 93, 72-82 (1970).
  3. G. N. Borzdov, "Waves with quadratic amplitude dependence on coordinates in uniaxial crystals," J. Mod. Opt. 37, 281-284 (1990).
    [CrossRef]
  4. C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, "Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media," J. Mod. Opt. 49, 1553-1566 (2002).
    [CrossRef]
  5. J. M. Cabrera, F. Agulló, and F. J. López, Optica Electromagnética. Vol. II: Materiales y Aplicaciones (Addison Wesley, 2000).
  6. 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 indicence," J. Opt. Soc. Am. A 19, 112-115 (2002).
    [CrossRef]
  7. R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
    [CrossRef]
  8. M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
    [CrossRef]
  9. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  10. M. C. Simon and L. I. Perez, "Goos-Hanchen effect of an ordinary refracted beam," J. Mod. Opt. 52, 515-528 (2005).
    [CrossRef]
  11. L. I. Perez, "Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic-uniaxial crystals interfaces," J. Opt. A, Pure Appl. Opt. 4, 640-649 (2002).
    [CrossRef]
  12. 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]
  13. T. Saastamoinen and J. Tervo, "Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields," presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30-July 3, 2003.
  14. H. Greiner, "Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption," Optik (Stuttgart) 114, 109-112 (2003).
    [CrossRef]
  15. Y. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, "Theory and applications of complex rays," in Progress in Optics, Vol. 39, E.Wolf, ed. (Elsevier Science, 1999), pp. 1-62.
    [CrossRef]
  16. 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]
  17. M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.
  18. S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

2005 (2)

M. C. Simon and L. I. Perez, "Goos-Hanchen effect of an ordinary refracted beam," J. Mod. Opt. 52, 515-528 (2005).
[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]

2004 (1)

2003 (2)

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

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

2002 (4)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
[CrossRef]

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

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, "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 indicence," J. Opt. Soc. Am. A 19, 112-115 (2002).
[CrossRef]

2001 (1)

1994 (1)

R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
[CrossRef]

1990 (1)

G. N. Borzdov, "Waves with quadratic amplitude dependence on coordinates in uniaxial crystals," J. Mod. Opt. 37, 281-284 (1990).
[CrossRef]

1970 (1)

B. D. Cervelle, R. Caye, and J. Billard, "Determination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale," Bull. Soc. Fr. Mineral. Cristallogr. 93, 72-82 (1970).

1937 (1)

M. Berek, "Optische Messmethoden in polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer Theorie der Optick absorbierender Kristalle," Fortschr. Mineral. 22, 1-104 (1937).

Agulló, F.

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

Alberdi, C.

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

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

Alfonso, S.

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

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

Asatryan, A. A.

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

Azzam, R. M. A.

Berek, M.

M. Berek, "Optische Messmethoden in polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer Theorie der Optick absorbierender Kristalle," Fortschr. Mineral. 22, 1-104 (1937).

Berrogui, M.

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

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

Berry, M. V.

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Billard, J.

B. D. Cervelle, R. Caye, and J. Billard, "Determination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale," Bull. Soc. Fr. Mineral. Cristallogr. 93, 72-82 (1970).

Bornatici, M.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.

Borzdov, G. N.

G. N. Borzdov, "Waves with quadratic amplitude dependence on coordinates in uniaxial crystals," J. Mod. Opt. 37, 281-284 (1990).
[CrossRef]

Cabrera, J. M.

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

Caye, R.

B. D. Cervelle, R. Caye, and J. Billard, "Determination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale," Bull. Soc. Fr. Mineral. Cristallogr. 93, 72-82 (1970).

Cervelle, B. D.

B. D. Cervelle, R. Caye, and J. Billard, "Determination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale," Bull. Soc. Fr. Mineral. Cristallogr. 93, 72-82 (1970).

Dennis, M. R.

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Diñeiro, J. M.

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

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

Echarri, R.

R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
[CrossRef]

Egorchenkov, R. 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]

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.

Ericsson, E.

Forbes, G. W.

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

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Garea, M. T.

R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
[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]

Hernández, B.

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

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

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
[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]

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.

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

López, F. J.

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

Maj, O.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.

Perez, L. I.

M. C. Simon and L. I. Perez, "Goos-Hanchen effect of an ordinary refracted beam," J. Mod. Opt. 52, 515-528 (2005).
[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]

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

Poli, E.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.

Saastamoinen, T.

T. Saastamoinen and J. Tervo, "Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields," presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30-July 3, 2003.

Sáenz, C.

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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]

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

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Simon, M. C.

M. C. Simon and L. I. Perez, "Goos-Hanchen effect of an ordinary refracted beam," J. Mod. Opt. 52, 515-528 (2005).
[CrossRef]

Tervo, J.

T. Saastamoinen and J. Tervo, "Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields," presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30-July 3, 2003.

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]

Bull. Soc. Fr. Mineral. Cristallogr. (1)

B. D. Cervelle, R. Caye, and J. Billard, "Determination de l'ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale," Bull. Soc. Fr. Mineral. Cristallogr. 93, 72-82 (1970).

Fortschr. Mineral. (1)

M. Berek, "Optische Messmethoden in polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer Theorie der Optick absorbierender Kristalle," Fortschr. Mineral. 22, 1-104 (1937).

J. Mod. Opt. (4)

G. N. Borzdov, "Waves with quadratic amplitude dependence on coordinates in uniaxial crystals," J. Mod. Opt. 37, 281-284 (1990).
[CrossRef]

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

M. C. Simon and L. I. Perez, "Goos-Hanchen effect of an ordinary refracted beam," J. Mod. Opt. 52, 515-528 (2005).
[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]

J. Opt. A, Pure Appl. Opt. (1)

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

J. Opt. Soc. Am. A (3)

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]

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Pure Appl. Opt. (1)

R. Echarri and M. T. Garea, "Behaviour of the Poynting vector in uniaxial absorbing media," Pure Appl. Opt. 3, 931-941 (1994).
[CrossRef]

Other (4)

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

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

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, and E. Poli, "Exact beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams," presented at the 28th European Physical Society Conference on Controlled Fusion and Plasma Physics, Funchal, Madeira, 18-22 June,2001.

T. Saastamoinen and J. Tervo, "Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields," presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30-July 3, 2003.

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Figures (7)

Fig. 1
Fig. 1

The complex direction of propagation s ̃ and the definition of the coordinate system XYZ.

Fig. 2
Fig. 2

Complex unitary vectors defining the polarization state (and polarization ellipses) of the displacement vector and electric field u ̃ D o , and of the magnetic field u ̃ H o , for the ordinary wave. The elliptical cylinder in which u ̃ H o is inscribed (parallel to the optic axis) and the complex direction of propagation s ̃ with its real ( s R ) and imaginary ( s I ) parts are also shown.

Fig. 3
Fig. 3

Geometrical relation between the complex unitary vectors defining the polarization state (and related ellipses) of the displacement vectors of the ordinary and extraordinary waves, u ̃ D o and u ̃ D e , and the planes that contain them (XY and Λ, respectively). The projection of u ̃ D e in the XY plane ( u ̃ D e ) x y , and its unitary vector u ̃ D e x y are also shown.

Fig. 4
Fig. 4

Complex unitary vectors defining the polarization state (and polarization ellipses) of the displacement vector u ̃ D e , electric field u ̃ E e , and magnetic field u ̃ H e , of the extraordinary wave and the planes defined by them (Λ, Φ, and XY, respectively). The complex direction of propagation s ̃ , its real and imaginary components, and the plane Γ are also shown.

Fig. 5
Fig. 5

Field geometry in the case of a wave propagating in the real direction of propagation.

Fig. 6
Fig. 6

Field geometry in the case of a wave propagating in the complex direction of propagation perpendicular to the optic axis.

Fig. 7
Fig. 7

Field geometry in the case of a wave propagating in the complex direction of propagation with its real and imaginary parts contained in the same plane as the optic axis.

Equations (69)

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

D ̃ = ε ̃ E ̃ ,
n ̃ x , y , z = ( ε ̃ x , y , z ε 0 ) 1 2 ,
n ̃ x , y = n ̃ o = n o i k 0 = n o ( 1 i χ o ) = n ̃ o exp ( i β o ) ,
n ̃ z = n ̃ e = n e i k e = n e ( 1 i χ e ) = n ̃ e exp ( i β e ) ,
t g β o , e = χ o , e , n ̃ o , e , = n o , e ( 1 + χ o , e 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 ,
s ̃ = s ̃ x u x + s ̃ y u y + s ̃ z u z = s ̃ x y u ̃ x y + s ̃ z u z = s x y exp ( i φ x y ) u ̃ x y + s z exp ( i φ z ) u z .
H ̃ × s ̃ = c n ̃ D ̃ ,
s ̃ × E ̃ = c μ o n ̃ H ̃ ,
D ̃ s ̃ = 0 ,
H ̃ s ̃ = 0 ,
D ̃ = ε 0 n ̃ 2 ( E ̃ s ̃ ( E ̃ s ̃ ) ) .
D ̃ x , y , z = ε 0 ( E ̃ s ̃ ) s ̃ x , y , z 1 n ̃ x , y , z 2 1 n ̃ 2 .
D ̃ x D ̃ y = s ̃ x s ̃ y .
s ̃ x 2 1 n ̃ o 2 1 n ̃ 2 + s ̃ y 2 1 n ̃ o 2 1 n ̃ 2 + s ̃ z 2 1 n ̃ e 2 1 n ̃ 2 = 0 .
1 n ̃ 1 2 = 1 s ̃ z 2 n ̃ e 2 + s ̃ z 2 n ̃ o 2 ,
1 n ̃ 2 2 = 1 n ̃ o 2 .
D ̃ o = D o u ̃ D o exp [ i ω ( t n ̃ o c r s ̃ ) ] exp ( i θ D o ) = D o u ̃ D o A t P r .
D ̃ o x , y , z [ 1 n ̃ x , y , z 2 1 n ̃ o 2 ] = ε 0 ( E ¯ o s ̃ ) s ̃ x , y , z .
u ̃ D o = ( i u I x y , u R x y , 0 ) = i u I x y u x + u R x y u y = u z × u ̃ x y .
E ̃ o = D ̃ o ε 0 n ̃ o 2 = E o u ̃ E o exp ( i θ E o ) A t P r ,
E o = D o ε 0 n o 2 ( 1 + χ o 2 ) ,
θ E o = 2 β o ,
u ̃ E o = u R E o + i u I E o = u ̃ D o = ( i u I x y , u R x y , 0 ) = u z × u ̃ x y .
H ̃ o = c ε 0 n ̃ o ( s ̃ × E ̃ o ) = H o exp ( i θ H o ) u ̃ H o A t P r ,
H o = c D o n o ( 1 + χ o 2 ) ,
θ H o = β o ,
u ̃ H o = s ̃ × u ̃ D o = s ̃ x y u z s ̃ z u ̃ x y .
I o = 1 2 E o H o A t 2 Re [ exp ( i θ E o ) exp ( i θ H o ) ( u ̃ E o * × u ̃ H o ) ] = D o 2 c A t 2 2 ε 0 n o 3 ( 1 + χ 0 2 ) 3 2 Re [ exp ( i β o ) ( u ̃ E o * × ( s ̃ × u ̃ E o ) ) ] = D o 2 c A t 2 2 ε 0 n o 3 ( 1 + χ 0 2 ) 3 2 Re [ exp ( i β o ) ( s ̃ x y [ u R x y u x i u l x y u y ] + s z ( u R x y 2 + u l x y 2 ) u z ) ] .
D ̃ e x , y , z = ε 0 ( E ̃ e s ̃ ) s ̃ x , y , z 1 n ̃ x , y , z 2 1 n ̃ 1 2 .
D ̃ e x y = ε 0 ( E ̃ e s ̃ ) s ̃ x y ( 1 s z 2 ̃ ) ( 1 n ̃ o 2 1 n ̃ e 2 ) ,
D ̃ e z = ε 0 ( E ̃ e s ̃ ) s ̃ z s ̃ z 2 ( 1 n ̃ o 2 1 n ̃ e 2 ) .
D ̃ e = ε 0 ( E ̃ e s ̃ ) ( 1 s ̃ z 2 ) ( 1 n ̃ o 2 1 n ̃ e 2 ) ( s ̃ x y u ̃ x y 1 s ̃ z 2 s ̃ z 2 s ̃ z u z ) = ε 0 ( E ̃ e s ̃ ) ( 1 s ̃ z 2 ) ( 1 n ̃ o 2 1 n ̃ e 2 ) s ̃ x y s ̃ z ( s ̃ z u ̃ x y s ̃ x y u z ) .
D e A t P r = ε 0 ( E ̃ e s ̃ ) ( 1 s ̃ z 2 ) ( 1 n ̃ o 2 1 n ̃ e 2 ) s ̃ x y s ̃ z ,
u ̃ D e = s ̃ z u ̃ x y s ̃ x y u ̃ z = u ̃ H o .
cos δ
= s z u R x y u l x y s x y 2 [ u I x y 2 cos 2 ( φ x y φ z ) + u R x y 2 sin 2 ( φ x y φ z ) ] + s z 2 u R x y 2 u I x y 2 .
E ̃ e x y = D ̃ e x y ε 0 n ̃ o 2 ,
E ̃ e z = D ̃ e z ε 0 n ̃ e 2 .
E ̃ e = D e ε 0 ( s ̃ z n ̃ o 2 u ̃ x y s ̃ x y n ̃ e 2 u z ) A t P r = E e exp ( i δ E ) u ̃ E e A t P r ,
E e exp ( i δ E ) = D e ε 0 ( s ̃ z 2 n ̃ o 4 + s ̃ x y 2 n ̃ e 4 ) 1 2 ,
u ̃ E e = ( s ̃ z 2 n ̃ o 4 + s ̃ x y 2 n ̃ e 4 ) 1 2 ( s ̃ z n ̃ o 2 u ̃ x y s ̃ x y n ̃ e 2 u z ) .
cos γ = s z u R x y u I x y n ̃ o 2 { s x y 2 n ̃ e 4 [ u I x y 2 cos 2 ( γ 1 ) + u R x y 2 sin 2 ( γ 1 ) ] + s z 2 n ̃ o 4 u R x y 2 u I x y 2 } 1 2 ,
γ 1 = 2 ( β o β e ) ( φ x y φ z ) .
H ̃ e = c ε 0 n ̃ 1 ( s ̃ × E ̃ e ) = c ε 0 n ̃ 1 E e exp ( i δ E ) ( s ̃ × u ̃ E e ) A t P r
= c ε 0 n ̃ 1 E e exp ( i δ E ) ( u ̃ x y × u z ) 1 n ̃ 1 2 ( s ̃ z 2 n ̃ o 4 + s ̃ x y 2 n ̃ e 4 ) 1 2 A t P r .
H ̃ e = H e exp ( i θ e H ) u ̃ H e A t P r ,
H e exp ( i θ e H ) = c ε 0 E e exp ( i δ E ) 1 n ̃ 1 ( s ̃ z 2 n ̃ o 4 + s ̃ x y 2 n ̃ e 4 ) 1 2 = c 1 n ̃ 1 D e ,
u ̃ H e = u z × u ̃ x y .
I e = D e 2 c A t 2 2 ε 0 Re [ 1 n ̃ 1 ( s ̃ z n ̃ o 2 u ̃ x y s ̃ x y n ̃ e 2 u ̃ z ) * × ( u z × u ̃ x y ) ] = D e 2 c A t 2 2 ε 0 Re [ 1 n ̃ 1 ( { s ̃ z n ̃ o 2 } * { u R x y 2 + u I x y 2 } u z + u ̃ x y { s ̃ x y n ̃ e 2 } * ) ] .
u D o = u E o = u y ,
u H o = s x u z s z u x ,
I O = D o 2 c A t 2 2 ε 0 n o 3 ( 1 + χ o 2 ) 3 2 Re [ exp ( i β o ) ( s x u x + s z u z ) ] = D o 2 c A t 2 2 ε 0 n ̃ o 4 n o s .
u D e = s x u z + s z u x ,
u ̃ E e = ( s z 2 n ̃ o 4 + s x 2 n ̃ e 4 ) 1 2 ( s z n ̃ o 2 u x s x n ̃ e 2 u z ) ,
u H e = u y ,
I e = D e 2 c A t 2 2 ε 0 Re [ 1 n ̃ 1 ( s x n ̃ o 2 u x + s z n ̃ e 2 u z ) ]
= D e 2 c A t 2 2 ε 0 1 n ̃ 1 ( s x cos ( 2 β e β 1 ) n ̃ e 2 u x + s z cos ( 2 β o β 1 ) n ̃ o 2 u z ) .
u ̃ E o = u ̃ D o = ( i u l x y , u R x y , 0 ) = u z × s ̃ ,
u ̃ H o = s ̃ × u ̃ D o = u z ,
I 0 = D o 2 c A t 2 x ε 0 n o 3 ( 1 χ o 2 ) 3 2 Re [ exp ( i β o ) ( u R x y u x i u I x y u y ) ] = D o 2 c A t 2 n o 2 ε 0 n o 4 ( s R + χ o s I ) .
u ̃ D e = u ̃ E e = u z ,
u ̃ H e = ( i u I x y , u R x y , 0 ) = u z × s ̃ ,
I e = D e 2 c A t 2 2 ε 0 Re 1 n ̃ e u ̃ x y ( s ̃ x y n ̃ e 2 ) * = D e 2 c A t 2 2 ε 0 Re exp ( i β e ) n ̃ e 3 u ̃ x y = D e 2 c A t 2 2 ε 0 n ̃ e 4 n e ( s R + χ e s I ) .
u ̃ D o = u ̃ E o = u H e = u y ,
u ̃ H o = u ̃ D e = s ̃ x y u z s ̃ z u x ,
u ̃ E e = ( s ̃ z 2 n ̃ o 4 + s ̃ x y 2 n ̃ e 4 ) 1 2 ( s ̃ z n ̃ o 2 u x s ̃ x y n ̃ e 2 u z ) ,
I O = D o 2 c A t 2 2 ε 0 n o 3 ( 1 + χ o 2 ) 3 2 Re [ exp ( i β o ) ( s ̃ x y u x + s ̃ z u z ) ] = D o 2 c A t 2 2 ε 0 n ̃ o 4 n o ( s R + χ o s I ) ,
I e = D e 2 c A t 2 2 ε 0 Re [ 1 n ̃ 1 ( { s ̃ z n ̃ o 2 } u z + u x { s ̃ x y n ̃ e 2 } * ) ] .

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