Abstract

Analytic expressions for the amplitude reflection and transmission coefficients for light incident on a biaxial anisotropic material are presented for the special case where one of the principal axes of the material is parallel to the surface of the material. It is assumed that either the material or the light source can be oriented so that the principal axis that is parallel to the surface is also perpendicular to the plane of incidence of the light.

© 1987 Optical Society of America

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References

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  1. A. B. Winterbottom, “Optical Studies of Metal Surfaces. Especially on the Use of the Reflected-Polarized-Light Method in Investigating Surface Films,” in Det Kongelige Norske Videnskabers Selskabs Skrifter, F. Bruns Bokhandel, Ed. (Trondehim, 1955), pp. 25–29.
  2. R. H. W. Graves, “Determination of the Optical Constants of Anisotropic Crystals,” J. Opt. Soc. Am. 59, 1225 (1969).
    [CrossRef]
  3. D. den Engelsen, “Ellipsometry of Anisotropic Films,” J. Opt. Soc. Am. 61, 1460 (1971).
    [CrossRef]
  4. M. J. Dignam, M. Moskovits, R. W. Stobie, “Specular Reflectance and Ellipsometric Spectroscopy of Oriented Molecular Layers,” Trans. Faraday Soc. 67, 3306 (1971).
    [CrossRef]
  5. T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
    [CrossRef]
  6. D. J. De Smet, “Ellipsometry of a Biaxial Surface,” J. Opt. Soc. Am. 65, 542 (1975).
    [CrossRef]
  7. D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4-Matrix Formalism,” J. Opt. Soc. Am. 62, 502 (1972).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 708.
  9. F. Wooten, Optical Properties of Solids (Academic, New York, 1972), p. 35.
  10. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
    [CrossRef]
  11. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, & Winston, New York, 1968), p. 50.
  12. B. Rossi, Optics (Addison-Wesley, Reading MA, 1957), p. 373.
  13. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1974), p. 74.

1975

1972

D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4-Matrix Formalism,” J. Opt. Soc. Am. 62, 502 (1972).
[CrossRef]

T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
[CrossRef]

1971

M. J. Dignam, M. Moskovits, R. W. Stobie, “Specular Reflectance and Ellipsometric Spectroscopy of Oriented Molecular Layers,” Trans. Faraday Soc. 67, 3306 (1971).
[CrossRef]

D. den Engelsen, “Ellipsometry of Anisotropic Films,” J. Opt. Soc. Am. 61, 1460 (1971).
[CrossRef]

1969

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 708.

De Smet, D. J.

den Engelsen, D.

Dignam, M. J.

M. J. Dignam, M. Moskovits, R. W. Stobie, “Specular Reflectance and Ellipsometric Spectroscopy of Oriented Molecular Layers,” Trans. Faraday Soc. 67, 3306 (1971).
[CrossRef]

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, & Winston, New York, 1968), p. 50.

Graves, R. H. W.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1974), p. 74.

Moskovits, M.

M. J. Dignam, M. Moskovits, R. W. Stobie, “Specular Reflectance and Ellipsometric Spectroscopy of Oriented Molecular Layers,” Trans. Faraday Soc. 67, 3306 (1971).
[CrossRef]

Muller, R. H.

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
[CrossRef]

Rossi, B.

B. Rossi, Optics (Addison-Wesley, Reading MA, 1957), p. 373.

Sosnowski, T. P.

T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
[CrossRef]

Stobie, R. W.

M. J. Dignam, M. Moskovits, R. W. Stobie, “Specular Reflectance and Ellipsometric Spectroscopy of Oriented Molecular Layers,” Trans. Faraday Soc. 67, 3306 (1971).
[CrossRef]

Winterbottom, A. B.

A. B. Winterbottom, “Optical Studies of Metal Surfaces. Especially on the Use of the Reflected-Polarized-Light Method in Investigating Surface Films,” in Det Kongelige Norske Videnskabers Selskabs Skrifter, F. Bruns Bokhandel, Ed. (Trondehim, 1955), pp. 25–29.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 708.

Wooten, F.

F. Wooten, Optical Properties of Solids (Academic, New York, 1972), p. 35.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1974), p. 74.

J. Opt. Soc. Am.

Opt. Commun.

T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
[CrossRef]

Surf. Sci.

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14 (1969).
[CrossRef]

Trans. Faraday Soc.

M. J. Dignam, M. Moskovits, R. W. Stobie, “Specular Reflectance and Ellipsometric Spectroscopy of Oriented Molecular Layers,” Trans. Faraday Soc. 67, 3306 (1971).
[CrossRef]

Other

A. B. Winterbottom, “Optical Studies of Metal Surfaces. Especially on the Use of the Reflected-Polarized-Light Method in Investigating Surface Films,” in Det Kongelige Norske Videnskabers Selskabs Skrifter, F. Bruns Bokhandel, Ed. (Trondehim, 1955), pp. 25–29.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 708.

F. Wooten, Optical Properties of Solids (Academic, New York, 1972), p. 35.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, & Winston, New York, 1968), p. 50.

B. Rossi, Optics (Addison-Wesley, Reading MA, 1957), p. 373.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA, 1974), p. 74.

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Figures (1)

Fig. 1
Fig. 1

Schematic diagram of the electric and magnetic fields for the incident medium and the anisotropic medium for the p polarization.

Equations (49)

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ɛ = ( ɛ x x 0 ɛ x z 0 ɛ y y 0 ɛ z x 0 ɛ z z ) ,
× E = - 1 / c B / t ,
× H = 1 / c D / t ,
B = H and D = ɛ E ,
E y = 0 ,
H x = H z = 0.
E x / z - E z / x = - 1 / c B y / t
H y / x = 1 / c D z / t ,
- H y / z = 1 / c / D x / t .
E x / d z = - i ( ω / c ) ( ɛ E z + H y ) ,
H y / d z = - i ( ω / c ) D x ,
- ξ H y = D z .
D x = ɛ x x E x + ɛ z z E z ,
D z = ɛ z x E x + ɛ z z E z .
H y / d z = - i ω / c [ E x ( ɛ x x - ɛ x z ɛ z x / ɛ z z ) - ɛ x z ξ / ɛ z z H y ] ,
E x / d z = - i ω / c [ H y ( 1 - ξ 2 / ɛ z z ) - ɛ z x ξ / ɛ z z E x ] .
ψ p = ( E x H y ) ,
ψ p / d z = - i ( ω / c ) M p ψ p ,
M p = ( - ɛ z x ξ / ɛ z z 1 - ξ 2 / ɛ z z ɛ x x - ɛ x z ɛ z x / ɛ z z - ɛ x z ξ / ɛ z z ) ,
- i ( ω / c ) ζ p ψ p = - i ( ω / c ) M p ψ p ,
ζ p = - ɛ x z ξ / ɛ z z ± [ ( ɛ x x - ɛ x z 2 / ɛ z z ) ( 1 - ξ 2 / ɛ z z ) ] 1 / 2 .
ψ p + = ( ( 1 - ξ 2 / ɛ z z ) 1 / 2 ( ɛ x x - ɛ x z 2 / ɛ z z ) 1 / 2 ) ,
ψ p - = ( ( 1 - ξ 2 / ɛ z z ) 1 / 2 - ( ɛ x x - ɛ x z 2 / ɛ z z ) 1 / 2 ) .
E x , i = E p , i cos ( θ 1 ) ,
E x , r = - E p , r cos ( θ 1 ) = - r p E p , i cos ( θ 1 ) ,
H y , i = n 1 E p , i ,
H y , r = n 1 E p , r = n 1 r p E p , i ,
ψ p = ( ( 1 - r p ) cos ( θ 1 ) ( 1 + r p ) n 1 ) E p , i .
ξ = n 1 sin ( θ 1 ) .
( ( 1 - r p ) cos ( θ 1 ) ( 1 + r p ) n 1 ) E p , i = ( ( 1 - ξ 2 / ɛ z z ) 1 / 2 ( ɛ x x - ɛ x z 2 / ɛ z z ) 1 / 2 ) K ,
r p = ( ɛ x x - ɛ x z 2 / ɛ z z ) 1 / 2 cos ( θ 1 ) - n 1 ( 1 - ξ 2 / ɛ z z ) 1 / 2 ( ɛ x x - ɛ x z 2 / ɛ z z ) 1 / 2 cos ( θ 1 ) + n 1 ( 1 - ξ 2 / ɛ z z ) 1 / 2 .
E x = t p E p , i / ( 1 + E z 2 / E x 2 ) 1 / 2 .
E x / H y = ± [ ( 1 - ξ 2 / ɛ z z ) / ( ɛ x x - ɛ x z 2 / ɛ z z ) ] 1 / 2 ,
E z / E x = - { ɛ x z ± ξ [ ( ɛ x x - ɛ x z 2 / ɛ z z ) / ( 1 - ξ 2 / ɛ z z ) ] 1 / 2 } / ɛ z z ,
t p = 2 n 1 cos ( θ 1 ) [ ( 1 - ξ 2 / ɛ z z ) ( 1 + E z 2 / E x 2 ) ] 1 / 2 ( ɛ x x - ɛ x z 2 / ɛ z z ) 1 / 2 cos ( θ 1 ) + n 1 ( 1 - ξ 2 / ɛ z z ) 1 / 2 ,
ψ s = ( E y - H x ) ,
ψ s / d z = - i ( ω / c ) M s ψ s ,
M s = ( 0 1 ɛ y y - ξ 2 0 ) .
ζ s = ± ( ɛ y y - ξ 2 ) 1 / 2 .
ψ s + = ( 1 - ( ɛ y y - ξ 2 ) 1 / 2 ) ,
ψ s - = ( 1 + ( ɛ y y - ξ 2 ) 1 / 2 ) .
ψ s = ( ( 1 + r s ) ( 1 - r s ) n 1 cos ( θ 1 ) ) E s , i .
( ( 1 + r s ) ( 1 - r s ) n 1 cos ( θ 1 ) ) = ( t s t s ( ɛ y y - ξ 2 ) 1 / 2 ) ,
r s = n 1 cos ( θ 1 ) - ( ɛ y y - ξ 2 ) 1 / 2 n 1 cos ( θ 1 ) + ( ɛ y y - ξ 2 ) 1 / 2 .
t s = 2 n 1 cos ( θ 1 ) n 1 cos ( θ 1 ) + ( ɛ y y - ξ 2 ) 1 / 2 .
tan ( θ p ) = ξ / ζ p
n 1 sin ( θ 1 ) = n p sin ( θ p ) .
tan ( θ p ) = - D z / D x .
E z / E x = - [ ɛ x x tan ( θ p ) - ɛ x z ] / [ ɛ z z + ɛ x z tan ( θ p ) ] .

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