Abstract

An exact expression for calculating the complex reflectance ratio of a surface, from data obtained with a rotating-analyzer ellipsometer system using optically active quartz Rochon prisms, shows that optical activity affects relative values of measured quantities by an amount of the order of 1%. For component settings near the normal modes of the system, these effects can be much greater. By contrast to null ellipsometry, there is no surface for which these effects vanish in calibration. Therefore, corrections of the order of 1% (0.5°) are necessary in the calibration of the azimuth scales of these ellipsometer systems, even in the most-favorable cases.

© 1974 Optical Society of America

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References

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  1. E. O. Ammann and G. A. Massey, J. Opt. Soc. Am. 58, 1427 (1968).
    [Crossref]
  2. B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
    [Crossref]
  3. R. Greef, Rev. Sci. Instr. 41, 532 (1970).
    [Crossref]
  4. D. E. Aspnes, Optics Commun. 8, 222 (1973).
    [Crossref]
  5. P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
    [Crossref]
  6. R. C. O’Handley, J. Opt. Soc. Am. 63, 523 (1973).
    [Crossref]
  7. J. F. Nye, Physical Properties of Crystals (Oxford U. P., London, 1957), pp. 260ff.
  8. D. E. Aspnes, J. Opt. Soc. Am. 61, 1077 (1971).
    [Crossref]
  9. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 1236 (1971); J. Opt. Soc. Am. 62, 700 (1972).
    [Crossref]
  10. D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974).
    [Crossref]
  11. S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instr. 40, 761 (1969); S. N. Jasperson, D. K. Burge, and R. C. O’Handley, Surf. Sci. 37, 548 (1973).
    [Crossref]
  12. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 1118 (1971).
    [Crossref]
  13. M. Ghezzo, Brit. J. Appl. Phys. 2, 1483 (1969).
  14. R. M. A. Azzam and N. H. Bashara, J. Opt. Soc. Am. 61, 600 (1971); J. Opt. Soc. Am. 61, 773 (1971); J. Opt. Soc. Am. 61, 1380 (1971).
    [Crossref]
  15. G. Szivessy and C. Munster, Ann. Phys. 20, 703 (1934); G. Bruhat and P. Grivet, J. Phys. Radium 6, 12 (1935); C. Munster and G. Szivessy, Phys. Z. 36, 101 (1935).
    [Crossref]
  16. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice–Hall, Englewood Cliffs, N. J., 1972), pp. 288ff.
  17. C. V. Kent and J. Lawson, J. Opt. Soc. Am. 27, 117 (1937).
    [Crossref]
  18. D. E. Aspnes, Appl. Opt. 10, 2545 (1971).
    [Crossref] [PubMed]
  19. M. J. Dignam and M. Moskovits, Appl. Opt. 9, 1868 (1970).
    [PubMed]
  20. N. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966), p. 129.

1974 (1)

1973 (3)

D. E. Aspnes, Optics Commun. 8, 222 (1973).
[Crossref]

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[Crossref]

R. C. O’Handley, J. Opt. Soc. Am. 63, 523 (1973).
[Crossref]

1971 (5)

1970 (2)

1969 (3)

M. Ghezzo, Brit. J. Appl. Phys. 2, 1483 (1969).

S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instr. 40, 761 (1969); S. N. Jasperson, D. K. Burge, and R. C. O’Handley, Surf. Sci. 37, 548 (1973).
[Crossref]

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

1968 (1)

1937 (1)

1934 (1)

G. Szivessy and C. Munster, Ann. Phys. 20, 703 (1934); G. Bruhat and P. Grivet, J. Phys. Radium 6, 12 (1935); C. Munster and G. Szivessy, Phys. Z. 36, 101 (1935).
[Crossref]

Ammann, E. O.

Aspnes, D. E.

Azzam, R. M. A.

Bashara, N. H.

Bashara, N. M.

Cahan, B. D.

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

Dignam, M. J.

Dill, F. H.

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[Crossref]

Draper, N.

N. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966), p. 129.

Ghezzo, M.

M. Ghezzo, Brit. J. Appl. Phys. 2, 1483 (1969).

Greef, R.

R. Greef, Rev. Sci. Instr. 41, 532 (1970).
[Crossref]

Hauge, P. S.

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[Crossref]

Jasperson, S. N.

S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instr. 40, 761 (1969); S. N. Jasperson, D. K. Burge, and R. C. O’Handley, Surf. Sci. 37, 548 (1973).
[Crossref]

Kent, C. V.

Lawson, J.

Massey, G. A.

Meyer-Arendt, J. R.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice–Hall, Englewood Cliffs, N. J., 1972), pp. 288ff.

Moskovits, M.

Munster, C.

G. Szivessy and C. Munster, Ann. Phys. 20, 703 (1934); G. Bruhat and P. Grivet, J. Phys. Radium 6, 12 (1935); C. Munster and G. Szivessy, Phys. Z. 36, 101 (1935).
[Crossref]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U. P., London, 1957), pp. 260ff.

O’Handley, R. C.

Schnatterly, S. E.

S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instr. 40, 761 (1969); S. N. Jasperson, D. K. Burge, and R. C. O’Handley, Surf. Sci. 37, 548 (1973).
[Crossref]

Smith, H.

N. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966), p. 129.

Spanier, R. F.

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

Szivessy, G.

G. Szivessy and C. Munster, Ann. Phys. 20, 703 (1934); G. Bruhat and P. Grivet, J. Phys. Radium 6, 12 (1935); C. Munster and G. Szivessy, Phys. Z. 36, 101 (1935).
[Crossref]

Ann. Phys. (1)

G. Szivessy and C. Munster, Ann. Phys. 20, 703 (1934); G. Bruhat and P. Grivet, J. Phys. Radium 6, 12 (1935); C. Munster and G. Szivessy, Phys. Z. 36, 101 (1935).
[Crossref]

Appl. Opt. (2)

Brit. J. Appl. Phys. (1)

M. Ghezzo, Brit. J. Appl. Phys. 2, 1483 (1969).

IBM J. Res. Dev. (1)

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[Crossref]

J. Opt. Soc. Am. (8)

Optics Commun. (1)

D. E. Aspnes, Optics Commun. 8, 222 (1973).
[Crossref]

Rev. Sci. Instr. (2)

R. Greef, Rev. Sci. Instr. 41, 532 (1970).
[Crossref]

S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instr. 40, 761 (1969); S. N. Jasperson, D. K. Burge, and R. C. O’Handley, Surf. Sci. 37, 548 (1973).
[Crossref]

Surf. Sci. (1)

B. D. Cahan and R. F. Spanier, Surf. Sci. 16, 166 (1969).
[Crossref]

Other (3)

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice–Hall, Englewood Cliffs, N. J., 1972), pp. 288ff.

J. F. Nye, Physical Properties of Crystals (Oxford U. P., London, 1957), pp. 260ff.

N. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966), p. 129.

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Figures (1)

Fig. 1
Fig. 1

Residuals calculated from data, and best-fit quadratic function for Ni film at λ = 300 nm. Each point represents an average over 20 cycles (10 analyzer rotations) at the given polarizer azimuth. The minimum azimuth P1 and the calculated azimuth PS of the plane of incidence are indicated explicitly.

Tables (1)

Tables Icon

Table I Evaluation of PS and AS, the azimuths of plane of incidence in polarizer and analyzer frames of reference, respectively. The quantities P1 and A1 are data, and ΔP1 and ΔA1 represent first-order correction terms calculated from Eqs. (16a) and (16c), respectively. The columns (raw) and (corr) refer to values of the dielectric function calculated without and with optical-activity corrections, respectively.

Equations (63)

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Ɛ u = u ( x ˆ A + i γ A ŷ A ) ,
Ɛ v = v ( ŷ A + i γ A x ˆ A ) ,
γ A = 0.0010 ( ω / eV ) ,
I Ɛ 2 ,
Ɛ = ( u v ) = ( 1 - i γ A 0 0 ) × ( cos ( A - Q ) sin ( A - Q ) - sin ( A - Q ) cos ( A - Q ) ) ( 1 i a ) ,
I 1 + ( 1 - a 2 ) ( 1 - γ A 2 ) ( 1 + a γ A ) 2 + ( a + γ A ) 2 × [ cos 2 Q cos 2 A + sin 2 Q sin 2 A ]
= 1 + α cos 2 A + β sin 2 A ,
Q = 1 2 tan - 1 ( β / α ) + π 2 u ( - α ) sgn ( β ) ,
a = - 2 γ A ζ ± ( 1 - γ A 2 ) ( 1 - ζ 2 ) 1 2 ( 1 + ζ ) - γ A 2 ( 1 - ζ ) ,
ζ = [ α 2 + β 2 ] 1 2 0.
( x y ) = ( r 0 0 r ) × ( cos ( P - P S ) - sin ( P - P S ) sin ( P - P S ) cos ( P - P S ) ) ( 1 i γ P ) ,
x = [ cos ( Q - A S ) - i a sin ( Q - A S ) ] ,
y = [ sin ( Q - A S ) + i a cos ( Q - A S ) ] ,
ρ = r / r = ( tan ψ ) e i Δ
= [ cot ( Q - A S ) - i a ] [ tan ( P - P S ) + i γ P ] [ 1 + i a cot ( Q - A S ) ] [ 1 - i γ P tan ( P - P S ) ] .
( x y ) = ( r 0 0 r ) ( cos ( P C - P S ) - sin ( P C - P S ) sin ( P C - P S ) cos ( P C - P S ) ) × ( 1 + γ C 2 e i δ - i γ C ( 1 - e i δ ) i γ C ( 1 - e i δ ) e i δ + γ C 2 ) × ( cos ( P - P C ) - sin ( P - P C ) sin ( P - P C ) cos ( P - P C ) ) ( 1 i γ P ) ,
ρ = [ cot ( Q - A S ) - i a ] [ tan ( P C - P S ) + z ] [ 1 + i a cot ( Q - A S ) ] [ 1 - z tan ( P C - P S ) ] ;
z = { [ 1 + γ C 2 e i δ ] [ cos ( P - P C ) - i γ P sin ( P - P C ) ] - i γ C ( 1 - e i δ ) [ sin ( P - P C ) + i γ P cos ( P - P C ) ] } × { i γ C ( 1 - e i δ ) [ cos ( P - P C ) - i γ P sin ( P - P C ) ] + [ e i δ + γ C 2 ] [ sin ( P - P C ) + i γ P cos ( P - P C ) ] } - 1 .
R ( P ) = 1 - α 2 - β 2 .
( u v ) = ( 1 - i γ A 0 0 ) × ( cos ( A - A S ) sin ( A - A S ) - sin ( A - A S ) cos ( A - A S ) ) ( r 0 0 r ) × ( cos ( P - P S ) - sin ( P - P S ) sin ( P - P S ) cos ( P - P S ) ) ( 1 i γ P ) .
I 1 + α cos 2 ( A - A S ) + β sin 2 ( A - A S ) ,
α = [ tan 2 ψ cos 2 ( P - P S ) - sin 2 ( P - P S ) ] / D ,
β = tan ψ [ cos Δ sin 2 ( P - P S ) + 2 γ P sin Δ ] / D ,
D = tan 2 ψ cos 2 ( P - P S ) + sin 2 ( P - P S ) - 2 γ A tan ψ sin Δ sin 2 ( P - P S ) .
R ( P ) = 1 - α 2 - β 2 .
R ( P ) P P S 4 ( P - P S ) cot 2 ψ sin Δ × [ ( P - P S ) sin Δ - 2 ( γ A tan ψ + γ P cos Δ ) ] ,
P S = P 1 - ( γ A tan ψ + γ P cos Δ ) / sin Δ P P S ,
A 1 = 1 2 tan - 1 ( β / α ) P = P 1 .
A S = A 1 - ( γ P cot ψ + γ A cos Δ ) / sin Δ .
R ( P ) 4 ( P - P S ) tan 2 ψ sin Δ [ ( P - P S ) sin Δ + 2 ( γ A cot ψ + γ P cos Δ ) ] ,
P S P 2 + ( γ A cot ψ + γ P cos Δ ) / sin Δ P P S + π / 2 ,
A 2 = 1 2 tan - 1 ( β / α ) P = P 2 + π / 2 ,
A S = A 2 + ( γ P tan ψ + γ A cos Δ ) / sin Δ .
R ( P ) = c 0 + c 1 P + c 2 P 2 ,
c 0 = [ r 0 ( p 2 p 4 - p 3 2 ) + r 1 ( p 2 p 3 - p 1 p 4 ) + r 2 ( p 1 p 3 - p 2 2 ) ] / d ,
c 1 = [ r 0 ( p 2 p 3 - p 1 p 4 ) + r 1 ( p 0 p 4 - p 2 2 ) + r 2 ( p 1 p 2 - p 0 p 3 ) ] / d ,
c 2 = [ r 0 ( p 1 p 2 - p 0 p 3 ) + r 1 ( p 1 p 2 - p 0 p 3 ) + r 2 ( p 0 p 2 - p 1 2 ) ] / d ,
d = p 0 p 2 p 4 + 2 p 1 p 2 p 3 - p 2 3 - p 0 p 3 2 - p 1 2 p 4 ,
p k = N - 1 j = 1 N P j k ,
r k = N - 1 j = 1 N R j P j k .
p 0 = N - 1 j = 1 N 1 = 1.
P 1 = - c 1 / ( 2 c 2 ) ,             { P j }             near P S ;
P 2 = - c 1 / ( 2 c 2 ) - π 2 ,             { P j }             near P S + π 2 .
A S = ( A 1 + A 2 ) / 2 - γ P cot 2 ψ / sin Δ ,
P S = ( P 1 + P 2 ) / 2 + γ A cot 2 ψ / sin Δ .
γ A = [ ( A 1 - A 2 ) cos Δ - ( P 1 - P 2 ) cos 2 ψ ] / D 1 ,
γ P = [ ( P 1 - P 2 ) cos Δ - ( A 1 - A 2 ) csc 2 ψ ] / D 1 ,
D 1 = 2 ( cos 2 Δ - csc 2 2 ψ ) .
I 1 + cos 2 [ P - ( A - A P ) ] ,
A P = 1 2 tan - 1 ( β / α ) P + π 2 u ( - α ) sgn ( β ) P - P .
( u v ) = ( 1 - i γ A 0 0 ) ( cos ( A - A C ) sin ( A - A C ) - sin ( A - A C ) cos ( A - A C ) ) × ( 1 - i γ C ( 1 - e - i δ ) i γ C ( 1 - e - i δ ) e i δ ) × ( cos ( P - P C ) - sin ( P - P C ) sin ( P - P C ) cos ( P - P C ) ) ( 1 i γ P ) ,
A 1 = 1 2 tan - 1 ( β / α ) P = P 1 - A P 1
A 2 = 1 2 tan - 1 ( β / α ) P = P 2 + π / 2 - A P 2
P C = ( A 1 + P 1 + A 2 + P 2 ) / 4 ,
γ C = ( cot δ 2 ) [ A 1 - P 1 + A 2 - P 2 ] / 4 ,
γ A = [ ( A 1 - A 2 ) cot δ - ( P 1 - P 2 ) csc δ ] / 2 ,
γ P = [ ( P 1 - P 2 ) cot δ - ( A 1 - A 2 ) csc δ ] / 2.
δ = 2 π ( n s - n f ) d / λ
π λ Q / ( 2 λ ) ,
λ Q = 2 λ 1 λ 2 / ( λ 1 + λ 2 ) ,
P C = ( P 1 + P 2 ) / 2 ,
γ C = π / 4 - ( P 2 - P 1 ) / 2 ,
γ P = π λ Q ( λ 1 - λ 2 ) / ( 8 λ 1 λ 2 ) ,