Abstract

A procedure is outlined in which the symmetry of the ellipsometer is used to provide the information needed for its own alignment. Alignment is based upon four null measurements taken on a transparent reflecting surface. These are related to the tilt angles of the polarizer and analyzer telescope arms, and reference angles for which the transmitted polarization vectors of the analyzer and polarizer prisms lie in the plane of incidence. The alignment is not affected by the presence of small parasitic ellipticities induced by defects in either polarizer or analyzer prisms. A step-by-step procedure for ellipsometer alignment, which requires only equipment necessary for normal operation of the instrument, is given.

© 1971 Optical Society of America

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References

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  1. F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Nat. Bur. Stand. 67A, 363 (1963).
    [CrossRef]
  2. M. Ghezzo, Brit. J. Appl. Phys. (J. Phys. D) 2, 1483 (1969).
    [CrossRef]
  3. W. R. Hunter, D. H. Eaton, C. T. Sah, Surface Sci. 20, 355 (1970).
    [CrossRef]
  4. D. E. Aspnes, A. A. Studna, Rev. Sci. Instrum. 41, 966 (1970).
    [CrossRef]
  5. D. E. Aspnes, to be published.
  6. The preferred angles of incidence are θi = ±θB, where θB is the Brewster angle of the transparent reflecting surface. The Brewster angle θB is equal to tann, where n is the index of refraction of the transparent reflecting surface. The surface must be thick enough to prevent multiple reflections from entering the analyzer telescope.
  7. Note that this a purely geometric operation which is in no way influenced by the ellipticity parameters discussed in Sec. III.A.
  8. Strictly speaking, the latter two conditions state that both telescope axes intersect the z axis, which is usually obtained only after lateral adjustment of the telescopes. By extending the following analysis, it may be shown that small deviations of this type introduce a first-order error into θi, but affect Aref, Pref, θA′, and θP′ only in second order. They are of no consequence to the results of this and the next section, and will be discussed later.
  9. The effect of s is contained completely in Δ. Since Δ can be absorbed by defining effective telescope tilt angles θA′, θP′, as in Eqs. (9), this shows that the determination of the plane of incidence is not affected to first order by a small difference between the heights of the two telescope pivots. The effect of s being nonzero is to reduce system transmission by vertically eclipsing the transmitted beam. It can be detected by visual inspection of the transmitted image and corrected as discussed later.
  10. W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge Mass., 1962).

1970 (2)

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

D. E. Aspnes, A. A. Studna, Rev. Sci. Instrum. 41, 966 (1970).
[CrossRef]

1969 (1)

M. Ghezzo, Brit. J. Appl. Phys. (J. Phys. D) 2, 1483 (1969).
[CrossRef]

1963 (1)

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

Aspnes, D. E.

D. E. Aspnes, A. A. Studna, Rev. Sci. Instrum. 41, 966 (1970).
[CrossRef]

D. E. Aspnes, to be published.

Eaton, D. H.

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

Ghezzo, M.

M. Ghezzo, Brit. J. Appl. Phys. (J. Phys. D) 2, 1483 (1969).
[CrossRef]

Hunter, W. R.

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

McCrackin, F. L.

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

Passaglia, E.

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

Sah, C. T.

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

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge Mass., 1962).

Steinberg, H. L.

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

Stromberg, R. R.

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

Studna, A. A.

D. E. Aspnes, A. A. Studna, Rev. Sci. Instrum. 41, 966 (1970).
[CrossRef]

Brit. J. Appl. Phys. (J. Phys. D) (1)

M. Ghezzo, Brit. J. Appl. Phys. (J. Phys. D) 2, 1483 (1969).
[CrossRef]

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

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

Rev. Sci. Instrum. (1)

D. E. Aspnes, A. A. Studna, Rev. Sci. Instrum. 41, 966 (1970).
[CrossRef]

Surface Sci. (1)

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

Other (6)

D. E. Aspnes, to be published.

The preferred angles of incidence are θi = ±θB, where θB is the Brewster angle of the transparent reflecting surface. The Brewster angle θB is equal to tann, where n is the index of refraction of the transparent reflecting surface. The surface must be thick enough to prevent multiple reflections from entering the analyzer telescope.

Note that this a purely geometric operation which is in no way influenced by the ellipticity parameters discussed in Sec. III.A.

Strictly speaking, the latter two conditions state that both telescope axes intersect the z axis, which is usually obtained only after lateral adjustment of the telescopes. By extending the following analysis, it may be shown that small deviations of this type introduce a first-order error into θi, but affect Aref, Pref, θA′, and θP′ only in second order. They are of no consequence to the results of this and the next section, and will be discussed later.

The effect of s is contained completely in Δ. Since Δ can be absorbed by defining effective telescope tilt angles θA′, θP′, as in Eqs. (9), this shows that the determination of the plane of incidence is not affected to first order by a small difference between the heights of the two telescope pivots. The effect of s being nonzero is to reduce system transmission by vertically eclipsing the transmitted beam. It can be detected by visual inspection of the transmitted image and corrected as discussed later.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge Mass., 1962).

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Figures (3)

Fig. 1
Fig. 1

A diagram of the ellipsometer, indicating the conventions and mode of operation assumed in this paper. The light is assumed to pass through the polarizer telescope P, reflect off the surface S, and emerge through the analyzer telescope A. Indexing heads carrying the polarizing prisms are mounted on the interior ends of the telescopes for reading analyzer and polarizer azimuth angles A and P. The primary reference, or z axis, is defined as that axis about which the analyzer telescope arm pivots.

Fig. 2
Fig. 2

A schematic representation of the ellipsometer geometry defining angles and dimensions which locate the apertures at the ends of each telescope. The polarizer and analyzer telescopes taken to lie in the x-z and x′-z planes, respectively. In this figure, the planes x-z and x′-z, in general separated by an angle 2θi, are displayed in the same plane for simplicity (as if θi were equal to 90°). LA and LP represent the distances between the pivot points of the analyzer and polarizer telescopes and the vertical z axis, and s is their relative vertical displacement. The telescope axes pass a length d above these pivot points, and the exterior and interior apertures are located at lengths d1 and d2 from the pivot point projection, as shown. θA and θP are the actual tilt angles of the analyzer and polarizer telescopes, respectively, taken positive when the exterior ends lie higher than the interior ends. The two lines labeled Î represent the central ray of the transmitted beam, passing equidistant from the center of the four apertures, and making angles β and α with the normal to the z axis in the x-z and x′-z planes, respectively. The plane of incidence contains the central ray.

Fig. 3
Fig. 3

The relation between the plane of incidence, represented by the dashed lines, and the angles of the indexing heads. The unit vectors β ^ and α ^ are along the directions of propagation of the transmitted beam in the analyzer and polarizer telescopes, respectively. The unit vector z ^ lies in the direction of the rotation (z) axis of the analyzer arm. The unit vector n ^ is normal to the plane of incidence. The angular reference shown in the inset conforms to the standard convention, where angles are measured from their reference value and the light propagates in the local positive z direction.

Equations (54)

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Δ A = Δ P tan ψ cos Δ - α P tan ψ sin Δ ,             ( 0 , π / 2 , A ) ,
Δ P tan ψ = Δ A cos Δ - β A sin Δ ,             ( 0 , π / 2 , P ) ,
Δ A tan ψ = Δ P cos Δ + α P sin Δ ,             ( π / 2 , 0 , A ) ,
Δ P = Δ A tan ψ cos Δ + β A tan ψ sin Δ ,             ( π / 2 , 0 , P ) .
R = r / r = tan ψ e i Δ .
Δ A = ( A - A ref i ) - A 0 ,
Δ P = ( P - P ref i ) - P 0 ,
Δ A = ( A - A ref ) - A 0 ,
Δ P = ( P - P ref ) - P 0 ,
( A 0 , P 0 ) = ( 0 , π / 2 ) , ( A 0 , P 0 ) = ( π / 2 , 0 ) .
Δ A 0 ,             ( 0 , π / 2 , A ) ,
Δ P 0 ,             ( π / 2 , 0 , P ) ,
Δ A = ± Δ P tan ψ , θ i θ B ,             ( 0 , π / 2 , A ) ,
Δ P = ± Δ A tan ψ , θ i θ B ,             ( π / 2 , 0 , P ) .
s A = 1 2 ( d 1 + d 2 ) ( θ A - α ) ,
s P = 1 2 ( d 1 + d 2 ) ( θ P - β ) .
s A = - s P .
s + d + θ A · 1 2 ( d 1 - d 2 ) - α [ 1 2 ( d 1 - d 2 ) + L A ] = d + θ P · 1 2 ( d 1 - d 2 ) - β [ 1 2 ( d 1 - d 2 ) + L P ] .
γ A = 1 + 2 L A d 1 - d 2 ,
γ P = 1 + 2 L P d 1 - d 2 ,
Δ = 2 s d 1 - d 2 ,
θ A = θ A + Δ 2 = α ( 1 + γ A 2 ) + β ( 1 - γ P 2 ) ,
θ P = θ P - Δ 2 = α ( 1 - γ A 2 ) + β ( 1 + γ P 2 ) .
α ^ = x ^ cos 2 θ i + y ^ sin 2 θ i + z ^ α ,
β ^ = x ^ + β z ^ .
n ^ = ( β ^ × α ^ ) β ^ × α ^ - 1 = - x ^ β + y ^ ( α - β cos 2 θ i ) csc 2 θ i + z ^ .
t ^ A = α ^ × z ^ ,
t ^ P = z ^ × β ^ .
ι ^ A = α ^ × n ^ ,
ι ^ P = n ^ × β ^ .
A ref i - A ref = ι ^ A × t ^ A = ( α cos 2 θ i - β ) csc 2 θ i ,
P ref i - P ref = ι ^ P × t ^ P = ( β cos 2 θ i - α ) csc 2 θ i .
( A ref + - A ref ) sin 2 θ B = α cos 2 θ B - β ,
( A ref - - A ref ) sin 2 θ B = - α cos 2 θ B + β ,
( P ref + - P ref ) sin 2 θ B = - α + β cos 2 θ B ,
( P ref - - P ref ) sin 2 θ B = - α - β cos 2 θ B ,
A ref = 1 2 ( A ref + + A ref - ) ,
P ref = 1 2 ( P ref + + P ref - )
α = - 1 sin 2 θ B [ 1 2 ( A ref + - A ref - ) cos 2 θ B + 1 2 ( P ref + - P ref - ) ] ,
β = - 1 sin 2 θ B [ 1 2 ( A ref + - A ref - ) + 1 2 ( P ref + - P ref - ) cos 2 θ B ] .
( E T x E T y ) A = ( 1 - i β A 0 0 ) ( cos a sin a - sin a cos a ) ( e i Δ tan ψ 0 0 1 ) × ( cos p - sin p sin p cos p ) ( 1 0 i α P 0 ) ( E I x E I y ) P ,
( E T x ) A = [ ( e i Δ tan ψ cos a cos p + sin a sin p ) + i α P ( - e i Δ tan ψ cos a sin p + sin a cos p ) + i β A ( e i Δ tan ψ sin a cos p - cos a sin p ) ] ( E I x ) P ,
( E T y ) A = 0.
a = A - A ref ,
p = P - P ref ,
( a , p ) = ( A 0 , P 0 ) = ( 0 , π / 2 ) ,
( a , p ) = ( A 0 , P 0 ) = ( π / 2 , 0 ) .
1 2 A I = 1 2 A ( E T x * E T x ) = Re [ ( A E T x * ) E T x ] = 0 ( analyzer ) ;
= Re [ ( P E T x * ) E T x ] = 0 ( polarizer ) .
Δ a = Δ A = ( A - A ref ) - A 0 ,
Δ p = Δ P = ( P - P ref ) - P 0 ,
Δ a = Δ A = ( A - A ref ) - A 0 ,
Δ p = Δ P = ( P - P ref ) - P 0 ,
0 = 1 ( - Δ P tan ψ cos Δ + Δ A + α P tan ψ sin Δ ) .

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