Abstract

In employing ellipsometry to determine the complex refractive index of a reflecting surface, a small systematic error in the zero-angle positions of the polarizers is shown under certain circumstances to have a large effect on the values of the optical constants calculated. The effect of superficial optical anisotropy on ellipsometry in general, and on the zero position determination in particular, is analyzed and a test for the presence of such anisotropy presented.

© 1970 Optical Society of America

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References

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  1. A. B. Winterbottom, Kgl. Norske Vidensk. Selsk. Skr. 1, 1 (1955).
  2. A. Vasicek, Optics of Thin Films (North-Holland Publ. Co., Amsterdam, 1960).
  3. J. R. Beattie, C. K. T. Conn, Phil. Mag. 46, 222 (1955).
  4. S. Roberts, Phys. Rev. 114, 104 (1959).
    [CrossRef]
  5. S. Roberts, Phys. Rev., 118, 1509 (1960).
    [CrossRef]
  6. R. C. Plumb, J. Opt. Soc. Amer. 50, 892 (1960).
    [CrossRef]
  7. H. G. Jerrard, J. Opt. Soc. Amer. 44, 289 (1954).
    [CrossRef]
  8. S. J. Williamson, J. M. Weingart, R. D. Andrews, J. Opt. Soc. Amer. 54, 337 (1964).
    [CrossRef]
  9. R. M. Emberson, J. Opt. Soc. Amer. 26, 443 (1936).
    [CrossRef]
  10. A. C. Hall, J. Opt. Soc. Amer. 55, 911 (1965).
  11. E. P. Mertens, R. C. Plumb, J. Opt. Soc. Amer. 54, 1063 (1964).
    [CrossRef]
  12. L. Tronstad, C. G. P. Feachem, Proc. Roy. Soc. (London), A145, 115 (1934).
  13. A. Vasicek, Czech. J. Phys. 1, 190 (1952).
    [CrossRef]
  14. F. P. Mertens, P. Theroux, R. C. Plumb, J. Opt. Soc. Amer. 53, 788 (1963).
    [CrossRef]
  15. L. P. Mosteller, F. Wooten, J. Opt. Soc. Amer. 58, 511 (1968).
    [CrossRef]
  16. J. G. Collins, Appl. Sci. Res., Sec. B 7, 1 (1958).
    [CrossRef]
  17. K. L. Kliewer, R. Fuchs, Phys. Rev. 172, 607 (1968).
    [CrossRef]
  18. A. Rothen, M. Hansen, New Sci. Instrum. 19, 839 (1948).
    [CrossRef]
  19. M. Ghezzo, Brit. J. Appl. Phys., Sec. 2, 2, 1483 (1960).
  20. F. L. McCrackin et al., J. Res. Nat. Bur. Stand. 67A, 363 (1963).
    [CrossRef]
  21. R. C. Jones, J. Opt. Soc. Amer. 37, 110 (1947).
    [CrossRef]

1968 (2)

L. P. Mosteller, F. Wooten, J. Opt. Soc. Amer. 58, 511 (1968).
[CrossRef]

K. L. Kliewer, R. Fuchs, Phys. Rev. 172, 607 (1968).
[CrossRef]

1965 (1)

A. C. Hall, J. Opt. Soc. Amer. 55, 911 (1965).

1964 (2)

E. P. Mertens, R. C. Plumb, J. Opt. Soc. Amer. 54, 1063 (1964).
[CrossRef]

S. J. Williamson, J. M. Weingart, R. D. Andrews, J. Opt. Soc. Amer. 54, 337 (1964).
[CrossRef]

1963 (2)

F. P. Mertens, P. Theroux, R. C. Plumb, J. Opt. Soc. Amer. 53, 788 (1963).
[CrossRef]

F. L. McCrackin et al., J. Res. Nat. Bur. Stand. 67A, 363 (1963).
[CrossRef]

1960 (3)

M. Ghezzo, Brit. J. Appl. Phys., Sec. 2, 2, 1483 (1960).

S. Roberts, Phys. Rev., 118, 1509 (1960).
[CrossRef]

R. C. Plumb, J. Opt. Soc. Amer. 50, 892 (1960).
[CrossRef]

1959 (1)

S. Roberts, Phys. Rev. 114, 104 (1959).
[CrossRef]

1958 (1)

J. G. Collins, Appl. Sci. Res., Sec. B 7, 1 (1958).
[CrossRef]

1955 (2)

A. B. Winterbottom, Kgl. Norske Vidensk. Selsk. Skr. 1, 1 (1955).

J. R. Beattie, C. K. T. Conn, Phil. Mag. 46, 222 (1955).

1954 (1)

H. G. Jerrard, J. Opt. Soc. Amer. 44, 289 (1954).
[CrossRef]

1952 (1)

A. Vasicek, Czech. J. Phys. 1, 190 (1952).
[CrossRef]

1948 (1)

A. Rothen, M. Hansen, New Sci. Instrum. 19, 839 (1948).
[CrossRef]

1947 (1)

R. C. Jones, J. Opt. Soc. Amer. 37, 110 (1947).
[CrossRef]

1936 (1)

R. M. Emberson, J. Opt. Soc. Amer. 26, 443 (1936).
[CrossRef]

1934 (1)

L. Tronstad, C. G. P. Feachem, Proc. Roy. Soc. (London), A145, 115 (1934).

Andrews, R. D.

S. J. Williamson, J. M. Weingart, R. D. Andrews, J. Opt. Soc. Amer. 54, 337 (1964).
[CrossRef]

Beattie, J. R.

J. R. Beattie, C. K. T. Conn, Phil. Mag. 46, 222 (1955).

Collins, J. G.

J. G. Collins, Appl. Sci. Res., Sec. B 7, 1 (1958).
[CrossRef]

Conn, C. K. T.

J. R. Beattie, C. K. T. Conn, Phil. Mag. 46, 222 (1955).

Emberson, R. M.

R. M. Emberson, J. Opt. Soc. Amer. 26, 443 (1936).
[CrossRef]

Feachem, C. G. P.

L. Tronstad, C. G. P. Feachem, Proc. Roy. Soc. (London), A145, 115 (1934).

Fuchs, R.

K. L. Kliewer, R. Fuchs, Phys. Rev. 172, 607 (1968).
[CrossRef]

Ghezzo, M.

M. Ghezzo, Brit. J. Appl. Phys., Sec. 2, 2, 1483 (1960).

Hall, A. C.

A. C. Hall, J. Opt. Soc. Amer. 55, 911 (1965).

Hansen, M.

A. Rothen, M. Hansen, New Sci. Instrum. 19, 839 (1948).
[CrossRef]

Jerrard, H. G.

H. G. Jerrard, J. Opt. Soc. Amer. 44, 289 (1954).
[CrossRef]

Jones, R. C.

R. C. Jones, J. Opt. Soc. Amer. 37, 110 (1947).
[CrossRef]

Kliewer, K. L.

K. L. Kliewer, R. Fuchs, Phys. Rev. 172, 607 (1968).
[CrossRef]

McCrackin, F. L.

F. L. McCrackin et al., J. Res. Nat. Bur. Stand. 67A, 363 (1963).
[CrossRef]

Mertens, E. P.

E. P. Mertens, R. C. Plumb, J. Opt. Soc. Amer. 54, 1063 (1964).
[CrossRef]

Mertens, F. P.

F. P. Mertens, P. Theroux, R. C. Plumb, J. Opt. Soc. Amer. 53, 788 (1963).
[CrossRef]

Mosteller, L. P.

L. P. Mosteller, F. Wooten, J. Opt. Soc. Amer. 58, 511 (1968).
[CrossRef]

Plumb, R. C.

E. P. Mertens, R. C. Plumb, J. Opt. Soc. Amer. 54, 1063 (1964).
[CrossRef]

F. P. Mertens, P. Theroux, R. C. Plumb, J. Opt. Soc. Amer. 53, 788 (1963).
[CrossRef]

R. C. Plumb, J. Opt. Soc. Amer. 50, 892 (1960).
[CrossRef]

Roberts, S.

S. Roberts, Phys. Rev., 118, 1509 (1960).
[CrossRef]

S. Roberts, Phys. Rev. 114, 104 (1959).
[CrossRef]

Rothen, A.

A. Rothen, M. Hansen, New Sci. Instrum. 19, 839 (1948).
[CrossRef]

Theroux, P.

F. P. Mertens, P. Theroux, R. C. Plumb, J. Opt. Soc. Amer. 53, 788 (1963).
[CrossRef]

Tronstad, L.

L. Tronstad, C. G. P. Feachem, Proc. Roy. Soc. (London), A145, 115 (1934).

Vasicek, A.

A. Vasicek, Czech. J. Phys. 1, 190 (1952).
[CrossRef]

A. Vasicek, Optics of Thin Films (North-Holland Publ. Co., Amsterdam, 1960).

Weingart, J. M.

S. J. Williamson, J. M. Weingart, R. D. Andrews, J. Opt. Soc. Amer. 54, 337 (1964).
[CrossRef]

Williamson, S. J.

S. J. Williamson, J. M. Weingart, R. D. Andrews, J. Opt. Soc. Amer. 54, 337 (1964).
[CrossRef]

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Vidensk. Selsk. Skr. 1, 1 (1955).

Wooten, F.

L. P. Mosteller, F. Wooten, J. Opt. Soc. Amer. 58, 511 (1968).
[CrossRef]

Appl. Sci. Res., Sec. B (1)

J. G. Collins, Appl. Sci. Res., Sec. B 7, 1 (1958).
[CrossRef]

Brit. J. Appl. Phys. (1)

M. Ghezzo, Brit. J. Appl. Phys., Sec. 2, 2, 1483 (1960).

Czech. J. Phys. (1)

A. Vasicek, Czech. J. Phys. 1, 190 (1952).
[CrossRef]

J. Opt. Soc. Amer. (9)

F. P. Mertens, P. Theroux, R. C. Plumb, J. Opt. Soc. Amer. 53, 788 (1963).
[CrossRef]

L. P. Mosteller, F. Wooten, J. Opt. Soc. Amer. 58, 511 (1968).
[CrossRef]

R. C. Jones, J. Opt. Soc. Amer. 37, 110 (1947).
[CrossRef]

R. C. Plumb, J. Opt. Soc. Amer. 50, 892 (1960).
[CrossRef]

H. G. Jerrard, J. Opt. Soc. Amer. 44, 289 (1954).
[CrossRef]

S. J. Williamson, J. M. Weingart, R. D. Andrews, J. Opt. Soc. Amer. 54, 337 (1964).
[CrossRef]

R. M. Emberson, J. Opt. Soc. Amer. 26, 443 (1936).
[CrossRef]

A. C. Hall, J. Opt. Soc. Amer. 55, 911 (1965).

E. P. Mertens, R. C. Plumb, J. Opt. Soc. Amer. 54, 1063 (1964).
[CrossRef]

J. Res. Nat. Bur. Stand. (1)

F. L. McCrackin et al., J. Res. Nat. Bur. Stand. 67A, 363 (1963).
[CrossRef]

Kgl. Norske Vidensk. Selsk. Skr. (1)

A. B. Winterbottom, Kgl. Norske Vidensk. Selsk. Skr. 1, 1 (1955).

New Sci. Instrum. (1)

A. Rothen, M. Hansen, New Sci. Instrum. 19, 839 (1948).
[CrossRef]

Phil. Mag. (1)

J. R. Beattie, C. K. T. Conn, Phil. Mag. 46, 222 (1955).

Phys. Rev. (3)

S. Roberts, Phys. Rev. 114, 104 (1959).
[CrossRef]

S. Roberts, Phys. Rev., 118, 1509 (1960).
[CrossRef]

K. L. Kliewer, R. Fuchs, Phys. Rev. 172, 607 (1968).
[CrossRef]

Proc. Roy. Soc. (London) (1)

L. Tronstad, C. G. P. Feachem, Proc. Roy. Soc. (London), A145, 115 (1934).

Other (1)

A. Vasicek, Optics of Thin Films (North-Holland Publ. Co., Amsterdam, 1960).

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Figures (4)

Fig. 1
Fig. 1

Error in the optical constants calculated for a systematic azimuthal error of 1° at the principal angle of incidence. (Broken line for the intensity ratio method, solid line for the retarder method.)

Fig. 2
Fig. 2

Apparent augle-of-iucidence dependence of the optical constants due to a systematic azimuthal error of 1°. [(1) and (2), fractional deviation of n and k, respectively, for a substance with complex refractive index 0.388–2.27i; (3) and (4), fractional deviation of n and k, respectively, for a substance with complex refractive index 4.05i–0.028i.]

Fig. 3
Fig. 3

Intensity vs analyzer setting for a first surface mirror. (Solid line θp = 0°, broken line θp = 0.52°, dotted line θp = −0.48°. Vertical arrows indicate crossed polarizer positions.)

Fig. 4
Fig. 4

Intensity vs θA for an aluminum plate polished unidirectionally. (Group A, one optic axis approximately in the plane of incidence; group B, optic axis rotated 45° from position A. Solid line, θp = 0°; broken line, θp = 0.52°; dotted line, θp = −0.48°. Vertical arrows indicate crossed polarizer positions.)

Equations (30)

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n 2 k 2 = sin 2 θ [ 1 + tan 2 θ ( cos 2 2 ψ sin 2 2 ψ sin 2 Δ ) / ( 1 + sin 2 ψ cos Δ ) 2 ] ,
2 n k = sin 2 θ tan 2 θ sin 4 ψ sin Δ / ( 1 + sin 2 ψ cos Δ ) 2 .
n = sin Ө tan Ө [ ( 1 + cos 4 Ψ ) / 2 ] 1 2 ,
k = sin Ө tan Ө [ ( 1 cos 4 Ψ ) / 2 ] 1 2 ,
2 n d n 2 k d k = sin 2 Ө tan 2 Ө ( 4 sin 4 Ψ d ψ + 2 sin 2 Ψ cos 4 Ψ d Δ ) ,
2 n d k + 2 k d n = sin 2 Ө tan 2 Ө ( 4 cos 4 Ψ d ψ + 2 sin 4 Ψ sin 2 Ψ δ Δ ) .
δ n / n [ 2 cos 4 Ψ ( 1 cos 4 Ψ 1 + cos 4 Ψ ) 1 2 2 sin 4 Ψ ] δ ψ + [ sin 4 Ψ sin 2 Ψ ( 1 cos 4 Ψ 1 + cos 4 Ψ ) 1 2 + sin 2 Ψ cos 4 Ψ ] δ Δ ,
δ k / k [ 2 ( 1 + cos 4 Ψ 1 cos 4 Ψ ) 1 2 cos 4 Ψ + 2 sin 4 Ψ ] δ ψ + [ sin 4 Ψ sin 2 Ψ ( 1 + cos 4 Ψ 1 cos 4 Ψ ) 1 2 sin 2 Ψ cos 4 Ψ ] δ Δ ,
cos 4 Ψ = ( n 2 k 2 ) / ( n 2 + k 2 ) ,
for n k , 1 Ψ > 0 ; for k n , 1 [ ( π / 4 ) Ψ ] > 0.
δ n A Ψ ( δ Δ 2 δ ψ ) δ k A δ ψ } for 1 Ψ > 0 , δ n A δ ψ δ k ( A / 2 ) { δ Δ 4 [ ( π / 4 ) Ψ ] δ Ψ } } for 1 1 4 π ψ > 0 ,
tan 2 2 θ = ( I 1 + I 4 ) / ( I 2 + I 3 ) .
I = ε R * t ε R ,
ε R = V A M R ε i .
ε i = E o ( sin θ p cos θ p ) .
V A = ( cos 2 θ A , sin θ A · cos θ A sin θ A · cos θ A , sin 2 θ A ) .
M R = ( ρ p exp i δ p 0 0 ρ s exp i δ s )
tan ψ = ρ p / ρ s ,
Δ = δ p δ s .
I = I o ( tan 2 ψ sin 2 θ p cos 2 θ A + cos 2 θ p sin 2 θ A 2 tan ψ cos Δ sin θ p cos θ p sin θ A cos θ A ) ,
[ cos 2 η exp ( i δ f ) + sin 2 η exp ( i δ f ) , sin η cos η [ exp ( i δ f ) exp ( i δ f ) ] sin η cos η [ exp ( i δ f ) exp ( i δ f ) ] , sin 2 ρ exp ( i δ f ) + cos 2 η exp ( i δ f ) ] ,
I = I 0 { tan 2 ψ [ cos 2 η sin 2 ( θ p η ) + sin 2 η cos 2 ( θ p η ) ] cos 2 θ A + 2 sin θ A cos θ A tan ψ [ sin 2 ( θ p η ) sin Δ sin 2 η cos Δ cos 2 ( θ p η ) ] + sin 2 θ A [ sin 2 η sin 2 ( θ p η ) cos 2 η cos 2 ( θ p η ) ] } .
tan 2 ψ = I 2 / I 4 ,
2 tan ψ cos Δ / ( 1 + tan 2 ψ ) = ( I 1 I 3 ) / ( I 1 + I 3 ) .
δ ψ = 2 γ tan Ψ / ( 1 + tan 2 Ψ ) ,
δ Δ = ( 1 / tan Ψ tan Ψ ) .
M R = [ ρ p e i δ p cos 2 θ i sin 2 χ ρ e i δ sin χ cos χ cos θ i ρ e i δ sin χ cos χ cos θ i ρ e i δ ρ s e i δ s cos 2 χ ρ e i δ ] .
I = ρ p 2 sin 2 θ p cos 2 θ A 2 ρ p ρ s sin θ A cos θ A sin θ p cos θ p cos Δ + ρ p 2 cos 2 θ p sin 2 θ A + ρ 2 Q 2 R 2 2 Q R ρ ( ρ p cos θ A sin θ p cos 1 ρ s sin θ A cos θ p cos 2 ) ,
I = ρ 2 cos 2 θ i sin 2 χ cos 2 χ .
θ = ρ sin χ cos χ cos θ i ρ p cos 1 ρ s cos 2 .

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