Abstract

The types of error produced by beam deviation in the optical elements of an ellipsometer are examined. It is shown that there are two types of error that may be significant—systematic errors due to a variation in the plane of incidence and in the angle of incidence at the specimen and errors due to the combined effects of beam displacement and nonuniformities in either the detector response or the optical properties of the specimen, the compensator, the polarizer, the analyzer, or the specimen cell. Analytic expressions for the variation in the plane of incidence and in the angle of incidence are given in terms of parameters that can be determined experimentally. A method by which these parameters can be measured is described. It is shown that the azimuthal variation in the angle of incidence produces fundamental errors in conventional zone averaging techniques because the values of ψ and Δ are averaged at different angles of incidence in different zones. Methods of experimentally predetermining when such errors are likely to be significant are discussed, and a procedure that cancels most systematic errors due to beam deviation in each zone is described. The combined effects of beam deviation in the polarizer, the compensator, the cell windows, and the analyzer are considered in several commonly used configurations, and the configurations that minimize beam deviation errors are described.

© 1974 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 61, 118, 600, 1236, 1380, (1971); J. Opt. Soc. Am. 62, 700 (1972).
    [CrossRef]
  2. D. E. Aspnes, J. Opt. Soc. Am. 61, 1077 (1971).
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  3. D. E. Aspnes, A. A. Studna, Appl. Opt. 10, 1024 (1971).
    [CrossRef] [PubMed]
  4. F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970).
    [CrossRef]
  5. W. R. Hunter, D. H. Eaton, C. T. Sah, Surf. Sci. 20, 355 (1970).
    [CrossRef]
  6. W. R. Hunter, J. Opt. Soc. Am. 63, 951 (1973).
    [CrossRef]
  7. J. R. Zeidler, R. B. Kohles, N. M. Bashara, Appl. Opt. 13, 1115 (1974).
    [CrossRef] [PubMed]
  8. A. B. Winterbottom, Roy. Norwegian Sci. Soc. Rep. No. 1 (F, Trondheim, 1955), pp. 71–72.
  9. J. M. Bennett, H. E. Bennett, in Handbook of Optics (W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, to be published).
  10. J. R. Zeidler, R. B. Kohles, N. M. Bashara, Appl. Opt. 13, 1591 (1974).
    [CrossRef] [PubMed]
  11. F. Lukeš, Surf. Sci. 16, 74 (1969).
    [CrossRef]
  12. P. H. Smith, Surf. Sci. 16, 34 (1969).
    [CrossRef]
  13. K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).
  14. J. F. Archard, J. Sci. Instrum. 26, 188 (1949).
    [CrossRef]
  15. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964); D. A. Holmes, D. L. Feucht, J. Opt. Soc. Am. 57, 466 (1967).
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  16. W. G. Oldham, J. Opt. Soc. Am. 57, 617 (1967).
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  17. S. S. So, W. H. Knausenberger, K. Vedam, J. Opt. Soc. Am. 61, 124 (1971).
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  18. O. Hunderi, Appl. Opt. 11, 1572 (1972).
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  19. Ref. 1, Paper 3.
  20. J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).
  21. D. E. Aspnes, Opt. Commun. 8, 222 (1973).
    [CrossRef]

1974 (2)

1973 (3)

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).

W. R. Hunter, J. Opt. Soc. Am. 63, 951 (1973).
[CrossRef]

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

1972 (1)

1971 (4)

1970 (2)

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

W. R. Hunter, D. H. Eaton, C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

1969 (2)

F. Lukeš, Surf. Sci. 16, 74 (1969).
[CrossRef]

P. H. Smith, Surf. Sci. 16, 34 (1969).
[CrossRef]

1967 (1)

1964 (1)

1955 (1)

J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).

1949 (1)

J. F. Archard, J. Sci. Instrum. 26, 188 (1949).
[CrossRef]

Archard, J. F.

J. F. Archard, J. Sci. Instrum. 26, 188 (1949).
[CrossRef]

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 61, 118, 600, 1236, 1380, (1971); J. Opt. Soc. Am. 62, 700 (1972).
[CrossRef]

Bashara, N. M.

Bennett, H. E.

J. M. Bennett, H. E. Bennett, in Handbook of Optics (W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, to be published).

Bennett, J. M.

J. M. Bennett, H. E. Bennett, in Handbook of Optics (W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, to be published).

Eaton, D. H.

W. R. Hunter, D. H. Eaton, C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Hodgson, J. N.

J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).

Holmes, D. A.

Hunderi, O.

Hunter, W. R.

W. R. Hunter, J. Opt. Soc. Am. 63, 951 (1973).
[CrossRef]

W. R. Hunter, D. H. Eaton, C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Knausenberger, W. H.

Kohles, R. B.

Lukeš, F.

F. Lukeš, Surf. Sci. 16, 74 (1969).
[CrossRef]

McCrackin, F. L.

Oldham, W. G.

Sah, C. T.

W. R. Hunter, D. H. Eaton, C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Semenenko, A. I.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).

Semenenko, L. V.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).

Smith, P. H.

P. H. Smith, Surf. Sci. 16, 34 (1969).
[CrossRef]

So, S. S.

Sokolov, V. K.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).

Studna, A. A.

Svitashev, K. K.

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).

Vedam, K.

Winterbottom, A. B.

A. B. Winterbottom, Roy. Norwegian Sci. Soc. Rep. No. 1 (F, Trondheim, 1955), pp. 71–72.

Zeidler, J. R.

Appl. Opt. (4)

J. Opt. Soc. Am. (7)

J. Sci. Instrum. (1)

J. F. Archard, J. Sci. Instrum. 26, 188 (1949).
[CrossRef]

Opt. Commun. (1)

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

Opt. Spectrosc. (1)

K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973).

Proc. Phys. Soc. (London) (1)

J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955).

Surf. Sci. (3)

W. R. Hunter, D. H. Eaton, C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

F. Lukeš, Surf. Sci. 16, 74 (1969).
[CrossRef]

P. H. Smith, Surf. Sci. 16, 34 (1969).
[CrossRef]

Other (3)

A. B. Winterbottom, Roy. Norwegian Sci. Soc. Rep. No. 1 (F, Trondheim, 1955), pp. 71–72.

J. M. Bennett, H. E. Bennett, in Handbook of Optics (W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, to be published).

Ref. 1, Paper 3.

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Figures (4)

Fig. 1
Fig. 1

The ellipse of incidence at the specimen formed by rotating the polarizer 360°.

Fig. 2
Fig. 2

Determination of the angle of incidence at the sample for an arbitrary polarized setting, ϕ(P). δV and δH are the projections of the deviation angle of the polarizer δ into the vertical and horizontal planes of the ellipsometer, respectively. ϕH and PH are the projections of ϕ(P) and P into the horizontal plane of the ellipsometer.

Fig. 3
Fig. 3

Determination of the coordinates of the point of incidence, P, at an arbitrary setting of the polarizer. r(θ) is a distance between P and the center of the cone of the deviated beam. θ is the angle between the horizontal plane of the allipsometer and r(θ). x(θ) and y(θ) are components of r(θ) in the horizontal and vertical planes of the ellipsometer, respectively, dO and d1 are the distances between the apex of the cone of the deviated beam and the points O and Px, respectively.

Fig. 4
Fig. 4

Measurement of the radius of the cone of the deviated beam and the reference angle θ. The X’s are the points obtained on an arbitrarily positioned target by varying the angle of incidence at the sample with the polarizer removed. The polarizer is then reinserted. The ●’s are the points of incidence on the target for various azimuthal settings of the polarizer. If the target is normal to the beam, these points should form a circle of radius r. PO is the azimuthal setting of the polarizer at which the cone of the deviated beam intersects the horizontal plane of the ellipsometer at point D.

Equations (15)

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ϕ C = ϕ O + δ ,
ϕ D = ϕ O δ ,
cos ϕ A = cos ϕ B = cos ϕ O cos δ .
ϕ A = ϕ B ϕ O .
cos δ = cos δ V cos δ H ,
cos ϕ ( P ) = cos δ V cos ϕ H ( P ) ,
ϕ ( P ) ϕ H ( P ) .
ϕ ( P ) ϕ O ± δ H .
x 2 cos 2 ϕ O + y 2 = r 2 .
y = x tan θ ,
ϕ ( P ) = ϕ O ± sin 1 [ sin δ cos ϕ O / ( cos 2 ϕ O + tan 2 θ ) 1 / 2 ] ,
θ ( P ) = P P O
δ = tan 1 [ ( r 2 r 1 ) / ( d 2 d 1 ) ] .
( θ / θ ) max = ( 1 ± Δ x / r ) ,
ψ ( ϕ , y ) = ψ O ( ϕ O , 0 ) + θ ( ϕ , y ) .

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