Abstract

The reflection properties of a uniaxial thin film on an isotropic substrate have been calculated. The optic axis of the film is assumed to be parallel to the film surface. The reflection properties are summarized in terms of a reflection matrix suitable for use in the Jones calculus. A method is given for relating the reflection properties of this film-covered surface to the null settings of the polarizer and analyzer of a conventional ellipsometer. Results of calculations for a few specific cases are discussed.

© 1974 Optical Society of America

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References

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  1. Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. Stromberg, and J. Kruger, Natl. Bur. Stand. (U.S.) Misc. Publ. No. 256 (U. S. Government Printing Office, Washington, D. C., 1961).
  2. Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall, Surf. Sci.16,(1969).
  3. A. B. Winterbottom, Optical Studies of Metal Surfaces (Bruns, Trondheim, 1955), Vol. 1, pp. 37–38.
  4. J. V. Cathcart and G. F. Petersen, in Ref. 1, p. 201.
  5. D. den Engelsen, J. Opt. Soc. Am. 61, 1460 (1971).
    [CrossRef]
  6. D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
    [CrossRef]
  7. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
    [CrossRef]
  8. R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
    [CrossRef]
  9. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [CrossRef]
  10. R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
    [CrossRef]
  11. D. J. De Smet, J. Opt. Soc. Am. 63, 958 (1973).
    [CrossRef]
  12. See, for example, M. V. Klein, Optics (Wiley, New York, 1970), pp. 596–605.
  13. F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw–Hill, New York, 1957), pp. 513 and 514.
  14. F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970).
    [CrossRef]
  15. Equations (23) and (24) are correctly stated here. They were stated incorrectly in Ref. 11; however, all numerical calculations in Ref. 11 were done with the correct equations. Also, note that tan−1(G/F) must be taken in the correct quadrant.
  16. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964).
    [CrossRef]
  17. W. G. Oldham, J. Opt. Soc. Am. 57, 617 (1967).
    [CrossRef]
  18. H. T. Yolken, R. M. Waxler, and J. Kruger, J. Opt. Soc. Am. 57, 283 (1967).
    [CrossRef]
  19. F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
    [CrossRef]
  20. This is necessarily a vague term, because the quality of the optical components of the ellipsometer, and in particular the compensator, also affect the zone differences.

1973 (2)

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[CrossRef]

D. J. De Smet, J. Opt. Soc. Am. 63, 958 (1973).
[CrossRef]

1972 (4)

1971 (1)

1970 (1)

1967 (2)

1964 (1)

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
[CrossRef]

Bundy, T. L.

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[CrossRef]

Cathcart, J. V.

J. V. Cathcart and G. F. Petersen, in Ref. 1, p. 201.

De Smet, D. J.

den Engelsen, D.

D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[CrossRef]

D. den Engelsen, J. Opt. Soc. Am. 61, 1460 (1971).
[CrossRef]

Holmes, D. A.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw–Hill, New York, 1957), pp. 513 and 514.

Klein, M. V.

See, for example, M. V. Klein, Optics (Wiley, New York, 1970), pp. 596–605.

Kruger, J.

McCrackin, F. L.

F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970).
[CrossRef]

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
[CrossRef]

Oldham, W. G.

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
[CrossRef]

Petersen, G. F.

J. V. Cathcart and G. F. Petersen, in Ref. 1, p. 201.

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
[CrossRef]

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
[CrossRef]

Waxler, R. M.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw–Hill, New York, 1957), pp. 513 and 514.

Winterbottom, A. B.

A. B. Winterbottom, Optical Studies of Metal Surfaces (Bruns, Trondheim, 1955), Vol. 1, pp. 37–38.

Yolken, H. T.

J. Opt. Soc. Am. (8)

J. Phys. Chem. (1)

D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[CrossRef]

J. Res. Natl. Bur. Stand. (U.S.) A (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stand. (U.S.) A 67, 363 (1963).
[CrossRef]

Opt. Commun. (2)

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[CrossRef]

Other (8)

Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. Stromberg, and J. Kruger, Natl. Bur. Stand. (U.S.) Misc. Publ. No. 256 (U. S. Government Printing Office, Washington, D. C., 1961).

Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall, Surf. Sci.16,(1969).

A. B. Winterbottom, Optical Studies of Metal Surfaces (Bruns, Trondheim, 1955), Vol. 1, pp. 37–38.

J. V. Cathcart and G. F. Petersen, in Ref. 1, p. 201.

See, for example, M. V. Klein, Optics (Wiley, New York, 1970), pp. 596–605.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw–Hill, New York, 1957), pp. 513 and 514.

This is necessarily a vague term, because the quality of the optical components of the ellipsometer, and in particular the compensator, also affect the zone differences.

Equations (23) and (24) are correctly stated here. They were stated incorrectly in Ref. 11; however, all numerical calculations in Ref. 11 were done with the correct equations. Also, note that tan−1(G/F) must be taken in the correct quadrant.

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Figures (5)

Fig. 1
Fig. 1

Light with wave normal s propagating through a uniaxial medium. The wave normal is in the x,z plane and the optic axis is in the x,y plane.

Fig. 2
Fig. 2

Schematic of the reflection and transmission of polarized light incident from an isotropic medium of index nm on an anisotropic thin film of thickness d and refractive indices no and ne. The film is on an isotropic substrate of index n3. All wave normals are in the x,z plane.

Fig. 3
Fig. 3

Points represent the calculated variation of ellipsometer null settings as a function of angle of rotation about the surface normal for an anisotropic film 250 Å thick with refractive indices no = 1.30 and ne = 1.60. The film is on a substrate with a refractive index 3.5 − 3.66i. The lines represent calculated values for a film-free anisotropic surface with refractive indices n0 = 1.7867 − 3.1258i and ne = 1.8682 − 2.19376i.

Fig. 4
Fig. 4

Calculated null settings of P and A for the growth of an anisotropic film with refractive indices no = 1.30 and ne = 1.60 on a substrate of index 3.5 − 3.66i. The points arc spaced at 50-Å increments of thickness. Open squares: ζ = 0°; filled circles: ζ = 45°, P1, A1; filled squares: ζ = 45°, P2′, A2′; open circles: ζ = 90° (P2′ = P2 − 90° and A2′ = 180° − A2). The lines are calculated values of P and A for isotropic films with indicated refractive indices with 50-Å increments of thickness marked.

Fig. 5
Fig. 5

Calculated null settings of P and A at ζ = 45° for the growth of an anisotropic film with refractive indices no = 1.5 − 0.03i, and ne = 1.6, on a substrate with index 3.5 − 3.66i. The points are spaced at 200-Å increments of thickness. Circles: P1, A1; squares: P2′, A2′, where P2′ = P2 − 90° and A2° = 180° − A2′; triangles: zone averages. The solid line represents the null settings for the growth of an isotropic film with refractive index 1.52 − 0.03i and 200-Å increments marked. P = 32.29, A = 34.13 for the bare substrate.

Equations (47)

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n = n o
1 n 2 = 1 - cos 2 ζ sin 2 θ e n e 2 + cos 2 ζ sin 2 θ e n o 2 ,
1 / n 2 = sin 2 ξ / n e 2 + cos 2 ξ / n o 2 ,
α o = sin ζ cos θ o ,             β o = - cos ζ cos θ o , and γ o = - sin ζ sin θ o .
α e = cos ζ ( N sin 2 ξ + cos 2 ξ - sin 2 θ e ) , β e = sin ζ ( N sin 2 ξ + cos 2 ξ ) , and γ e = - sin θ e cos θ e cos ζ ,
n m sin ϕ = n o sin θ o = n sin θ e ,
n 3 sin ϕ 3 = n o sin θ o
n = n e [ 1 + n m 2 cos 2 ζ sin 2 ϕ ( 1 - / n e 2 - 1 / n o 2 ) ] 1 2 .
cos ϕ 3 = [ 1 - ( n m sin ϕ / n 3 ) 2 ] 1 2 .
α o E o , u ,             β o E o , u ,             γ o E o , u ,
α e E e , u ,             β e E e , u ,             γ e E e , u ,
- α o E o , u ,             - β o E o , v ,             γ o E o , v ,
α e E e , v ,             β e E e , v ,             - γ e E e , v ,
E x , i = + E p , i cos ϕ ,             E y , i = E s , i , E z , i = - E p , i sin ϕ ,
E x , r = - E p , r cos ϕ ,             E y , r = E s , r , E z , r = - E p , r sin ϕ ,
E x , t = + E p , t cos ϕ 3 ,             E y , t = E s , t , E z , t = - E p , t sin ϕ 3 .
+ E p , i cos ϕ - E p , r cos ϕ - α o E o , u - α e E e , u + α o E o , v - α e E e , v = 0.
E s , i + E s , r - β o E o , u - β e E e , u + β o E o , v - β e E e , v = 0.
- n m cos ϕ E s , i + n m cos ϕ E s , r + n o β o cos θ o E o , u + n β e cos θ e E e , u + n o β o cos θ o E o , v - n β e cos θ e E e , v = 0.
+ n m E p , i + n m E p , r - n o ( α o cos θ 0 - γ o sin θ o ) E o , u - n ( α e cos θ e - γ e sin θ e ) E e , u - n 0 ( α o cos θ o - γ o sin θ o ) E o , v + n ( α e cos θ e - γ e sin θ e ) E e , v = 0.
δ o = 2 π d n o cos θ o / λ
δ e = 2 π d n cos θ e / λ .
α o e - i δ o E o , u + α e e - i δ e E e , u - α o e i δ o E o , v + α e e i δ e E e , v - E p , t cos ϕ 3 = 0.
β o e - i δ o E o , u + β e e - i δ e E e , u - β o e i δ o E o , v + β e e i δ e E e , v - E s , t = 0.
- n o β o cos θ o e - i δ o E o , u - n β e cos θ e e - i δ e E e , u - n o β o cos θ o e i δ o E o , v + n β e cos θ e e i δ e E e , v + n 3 cos ϕ 3 E s , t = 0.
n o ( α o cos θ o - γ o sin θ o ) e - i δ o E o , u + n ( a e cos θ e - γ e sin θ e ) e - i δ e E e , u + n o ( α o cos θ o - γ o sin θ o ) e i δ o E o , v - n ( α e cos θ e - γ e sin θ e ) e i δ e E e , v - n 3 E p , t = 0.
AX = 0 ,
X = ( Z Y 1 Y 2 ) ,
Y 1 = ( E p , r E s , r ) ,             Y 2 = ( E p , i E s , i ) ,
A = ( A 1 ( 6 × 6 ) A 2 ( 6 × 2 ) A 3 ( 6 × 2 ) A 4 ( 2 × 6 ) A 5 ( 2 × 2 ) A 6 ( 2 × 2 ) ) ,
Y 1 = RY 2 ,
( E p , r E s , r ) = ( R 11 R 12 R 21 R 22 ) ( E p , i E s , i ) ,
R = ( A 5 - A 4 A 1 - 1 A 2 ) - 1 ( A 4 A 1 - 1 A 3 - A 6 ) .
( E p , i E s , i ) = ( cos Q cos ( P - Q ) + i sin Q sin ( P - Q ) sin Q sin ( P - Q ) - i cos Q sin ( P - Q ) ) .
F sin ( 2 P - 2 Q ) + G cos ( 2 P - 2 Q ) + H = 0 ,
P 1 = 1 2 { sin - 1 [ H / ( F 2 + G 2 ) 1 2 ] - tan - 1 ( G / F ) } + Q
P 2 = 1 2 { π - sin - 1 [ H / ( F 2 + G 2 ) 1 2 ] - tan - 1 ( G / F ) } + Q
F = Re ( R 11 R 22 * - R 12 R 21 * ) ,
G = [ Im ( R 11 R 22 * + R 12 R 21 * ) ] sin 2 Q + [ Im ( R 11 R 21 * - R 12 R 22 * ) ] cos 2 Q ,
H = - Im ( R 11 R 21 * + R 12 R 22 * ) .
A = tan - 1 ( - R 11 E p , i + R 12 E s , i R 21 E p , i + R 22 E s , i ) ,
X = ( Z Y 1 Y 2 ) ,
Y 1 = ( E p , t E s , t )
Y 1 = T Y 2 ,
T = ( B 5 - B 4 B 1 - 1 B 2 ) - 1 ( B 4 B 1 - 1 B 3 - B 6 ) .
tan ψ e i Δ = E p , r / E p , i E s , r / E s , i ,
tan ψ e i Δ = R 11 + R 12 ( E s , i / E p , i ) R 21 ( E p , i / E s , i ) + R 22 .