Abstract

The equations of ellipsometry are derived relating the specimen properties, ρ=tanψeiΔ, to the instrument settings P, A, and Q. The treatment differs from previous works in that both of the important parameters of the retardation plate are considered, namely, the retardation δ and the transmission ratio T, of the slow to the fast axis. Equations are developed which determine δ and T from the instrument readings. Conventional averaging procedures to obtain ψ from readings of A are shown to be valid; however, it is demonstrated that the usual procedures for obtaining Δ from readings of P are invalid and a correction is derived. Furthermore, an unsuspected source of error, namely, orientation of the retardation plate oblique to the light beam, is shown to produce large errors in some measurements.

© 1967 Optical Society of America

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References

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  1. A collection of typical recent papers is available in Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Std. (U. S.) Misc. Publ. 256 (1964); a recent review paper is K. H. Zaininger and A. G. Revesz, RCA Review 25, 85 (1964). See also Refs. 2 and 4.
  2. F. L. McCrackin, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
    [CrossRef]
  3. Reference 2, Table 2.
  4. A. B. Winterbottom, Optical Studies of Metal Surfaces, Royal Norwegian Sci. Soc. Rept. 1 (F. Bruns, Trondheim, Norway1955), p. 68.
  5. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964).
    [CrossRef]
  6. H. H. Landolt and R. Boernstein, Phys. Chem. Tabellen, II Band, 8 Teil, 2-123 (Julius Springer-Verlag, Berlin, 1962).
  7. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
    [CrossRef]
  8. We note from Fig. 1 that for a given plate of arbitrary thickness at an arbitrary angle, there is a 16% chance that |T−1|<0.025.

1964 (2)

A collection of typical recent papers is available in Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Std. (U. S.) Misc. Publ. 256 (1964); a recent review paper is K. H. Zaininger and A. G. Revesz, RCA Review 25, 85 (1964). See also Refs. 2 and 4.

D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964).
[CrossRef]

1963 (1)

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

1962 (1)

Archer, R. J.

Boernstein, R.

H. H. Landolt and R. Boernstein, Phys. Chem. Tabellen, II Band, 8 Teil, 2-123 (Julius Springer-Verlag, Berlin, 1962).

Holmes, D. A.

Landolt, H. H.

H. H. Landolt and R. Boernstein, Phys. Chem. Tabellen, II Band, 8 Teil, 2-123 (Julius Springer-Verlag, Berlin, 1962).

McCrackin, F. L.

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

Steinberg, H. L.

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

Stromberg, R. R.

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

Winterbottom, A. B.

A. B. Winterbottom, Optical Studies of Metal Surfaces, Royal Norwegian Sci. Soc. Rept. 1 (F. Bruns, Trondheim, Norway1955), p. 68.

Ellipsometry in the Measurement of Surfaces and Thin Films (1)

A collection of typical recent papers is available in Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Std. (U. S.) Misc. Publ. 256 (1964); a recent review paper is K. H. Zaininger and A. G. Revesz, RCA Review 25, 85 (1964). See also Refs. 2 and 4.

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Std. (U. S.) (1)

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

Other (4)

Reference 2, Table 2.

A. B. Winterbottom, Optical Studies of Metal Surfaces, Royal Norwegian Sci. Soc. Rept. 1 (F. Bruns, Trondheim, Norway1955), p. 68.

We note from Fig. 1 that for a given plate of arbitrary thickness at an arbitrary angle, there is a 16% chance that |T−1|<0.025.

H. H. Landolt and R. Boernstein, Phys. Chem. Tabellen, II Band, 8 Teil, 2-123 (Julius Springer-Verlag, Berlin, 1962).

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Figures (7)

Fig. 1
Fig. 1

The retardation δ and transmittance ratio T vs thickness of a mica retardation plate for normal incidence at 6328 Å. The assumed optical const. are nF=1.592, nS=1.596, nZ=1.556. The dashed lines correspond to a tilt of the plate of 5° with respect to the propagation direction.

Fig. 2
Fig. 2

The physical arrangement of the ellipsometer.

Fig. 3
Fig. 3

The orientation of the retardation plate with respect to the light beam.

Fig. 4
Fig. 4

The orientation of the retardation plate mounting and the retardation plate with respect to the light beam.

Fig. 5
Fig. 5

Resolution of light into components. F refers to the fast axis, S the slow axis, x the plane of incidence, and y the plane of the surface.

Fig. 6
Fig. 6

Construction in Zone 1 of elliptically polarized light with an ideal quarter-wave plate.

Fig. 7
Fig. 7

Construction of elliptically polarized light of the same ellipticity as in Fig. 6 with a quarter-wave plate for which T>1.

Tables (1)

Tables Icon

Table I Relationships between Q, P, and A for T=1, δ=90°.

Equations (58)

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δ = tan - 1 ( K S tan β S d - K F tan β F d 1 + K S K F tan β S d tan β F d ) ,
K = ( n 2 + 1 ) / 2 n ,
β = 2 π n / λ .
T = ( cos 2 β F d + K F 2 sin 2 β F d cos 2 β S d + K S 2 sin 2 β S d ) 1 2 .
tan Δ = T sin δ sin ( 90 ° + 2 P ) cos 2 ( 45 ° + P ) - T 2 sin 2 ( 45 ° + P ) ;
tan Δ = T sin δ sin ( 90 ° - 2 P ) cos 2 ( 45 ° - P ) - T 2 sin 2 ( 45 ° - P ) .
P 1 = - P 4             P 2 = - P 3
Δ = tan - 1 [ sin δ tan ( 2 P 1 + 90 ° ) ] = tan - 1 [ sin δ tan ( P 1 + P 3 ) ] .
Δ a ( P 1 + P 3 )
Δ - Δ a             for             δ = 90 ° .
Δ tan - 1 [ sin δ tan ( Δ a ) ] + .
[ ( T - 1 ) / 2 ] 2 sin 2 Δ
tan ψ = tan A [ T sin δ cot ( 45 ° + P ) sin Δ + T sin ( Δ - δ ) ] .
tan ψ = ± tan A [ cos 2 ( 45 ° + P ) + T 2 sin 2 ( 45 ° + P ) - T sin ( 90 ° + 2 P ) cos δ cos 2 ( 45 ° + P ) + T 2 sin 2 ( 45 ° + P ) + T sin ( 90 ° + 2 P ) cos δ ] 1 2 ,
tan ψ = ± tan A [ cos 2 ( 45 ° - P ) + T 2 sin 2 ( 45 ° - P ) - T sin ( 90 ° - 2 P ) cos δ cos 2 ( 45 ° - P ) + T 2 sin 2 ( 45 ° - P ) + T sin ( 90 ° - 2 P ) cos δ ] 1 2 ,
A 1 = - A 4 ,             A 2 = - A 3 .
ψ = tan - 1 [ tan A 1 tan ( - A 3 ) ] 1 2 .
ψ ( A 1 - A 3 ) / 2
T 2 = sin 2 U 3 cos 2 U 1 - sin 2 U 1 cos 2 U 3 sin 2 U 3 sin 2 U 1 - sin 2 U 1 sin 2 U 3 ,
U 1 = P 1 + 45 ° , U 3 = P 3 + 45 ° .
E P 1 - P 3 + 90 °
E ( 1 - T ) sin Δ .
T 1 + ( P 3 - P 1 + 90 ° ) / sin Δ .
tan A = tan ψ T sin δ [ cos U sin Δ + T sin U sin ( Δ - δ ) sin U ]
sin ( Δ - δ ) sin Δ = tan A 1 sin U 1 cos U 3 - tan A 3 sin U 3 cos U 1 T sin U 3 sin U 1 ( tan A 3 - tan A 1 ) .
sin ( Δ - δ ) / sin Δ = cos δ - sin δ / tan Δ .
cos δ = tan A 1 sin U 1 cos U 3 - tan A 3 sin U 3 cos U 1 T sin U 3 sin U 1 ( tan A 3 - tan A 1 ) + cos 2 U - T 2 sin 2 U T sin 2 U .
K S = 1 2 { [ cos i / ( n S 2 - sin 2 i ) 1 2 ] + [ ( n S 2 - sin 2 i ) 1 2 / cos i ] } , K F = 1 2 { [ n F n Z cos i / ( n Z 2 - sin 2 i ) 1 2 ] + [ ( n Z 2 - sin 2 i ) 1 2 / n F n Z cos i ] } , β S = ( 2 π / λ ) ( n S 2 - sin 2 i ) 1 2 , β F = ( 2 π / λ ) ( n F / n Z ) ( n Z 2 - sin 2 i ) 1 2 ,
E F = cos ( Q - P )             E S = sin ( Q - P ) .
E F = E F ,             E S = T E S e - j δ .
E x = E F cos Q + E S sin Q E y = E F sin Q - E S cos Q .
E x = E x tan ψ e j Δ             E y = E y .
A ± π / 2 = tan - 1 ( E y / E x ) .
phase ( E x ) = phase ( E y ) - Δ .
E x = K 1 + K 2 e - j δ , E y = K 3 + K 4 e - j δ ,
K 1 = cos ( Q - P ) cos Q , K 2 = T sin ( Q - P ) sin Q , K 3 = cos ( Q - P ) sin Q , K 4 = - T sin ( Q - P ) cos Q .
tan Δ = sin δ ( K 2 K 3 - K 1 K 4 ) K 1 K 3 + K 2 K 4 + cos δ ( K 2 K 3 + K 1 K 4 ) .
tan Δ = T sin δ sin 2 ( Q - P ) sin 2 Q [ cos 2 ( Q - P ) - T 2 sin 2 ( Q - P ) ] - T cos 2 Q cos δ sin 2 ( Q - P ) .
tan Δ = T sin δ sin ( 90 - 2 P ) cos 2 ( 45 ° - P ) - T 2 sin 2 ( 45 ° - P ) ;
tan A = - tan ψ [ ( K 1 + K 2 e - j δ ) e j Δ / ( K 3 + K 4 e - j δ ) ] .
tan ψ = tan A { K 4 sin δ / [ K 1 sin Δ + K 2 sin ( Δ - δ ) ] } .
tan ψ = ± tan A [ ( K 3 + K 4 cos δ ) 2 + ( K 4 sin δ ) 2 ( K 1 + K 2 cos δ ) 2 + ( K 2 sin δ ) 2 ] 1 2 .
tan ψ = ± tan A [ [ tan Q cos ( Q - P ) - T sin ( Q - P ) cos δ ] 2 + T 2 sin 2 ( Q - P ) sin 2 δ [ cos ( Q - P ) + T tan Q sin ( Q - P ) cos δ ] 2 + T 2 tan 2 Q sin 2 ( Q - P ) sin 2 δ ] 1 2 ,
tan ψ = ± tan A [ cos 2 ( 45 ° - P ) + T 2 sin 2 ( 45 ° - P ) - 2 T cos ( 45 ° - P ) sin ( 45 ° - P ) cos δ cos 2 ( 45 ° - P ) + T 2 sin 2 ( 45 ° - P ) + 2 T cos ( 45 ° - P ) sin ( 45 ° - P ) cos δ ] 1 2 ,
E ( 1 - T ) sin Δ .
χ = 45 + ( 90 ° - Δ ) / 2 = 90 ° - Δ / 2.
tan χ = ( 1 / T ) tan ( 45 ° - P 1 ) .
( tan χ ) / T = tan ( P 2 - 45 ° ) .
E 90 ° + P 1 - P 3 = 90 ° + P 1 + P 2 ,
tan E = tan [ tan - 1 ( tan δ / T ) - tan - 1 ( T tan χ ) ] .
E = tan - 1 [ - sin Δ · ( T 2 - 1 ) / 2 T ] ,
E ( 1 - T ) sin Δ .
= Δ - Δ a = Δ - P 1 - P 3 = Δ - P 1 + P 2 .
= 180 ° + tan - 1 ( T tan χ ) + tan - 1 ( tan χ / T ) - 2 χ ,
tan ( + 2 χ ) = [ T tan χ + ( 1 / T ) tan χ ] / ( 1 - tan 2 χ ) .
tan = - [ ( T 2 + 1 ) / 2 T - 1 ] tan Δ 1 + [ ( T 2 + 1 ) / 2 T ] tan 2 Δ .
- ( sin 2 Δ / 2 ) [ ( T 2 + 1 ) / ( 2 T ) - 1 ]
- [ ( T - 1 ) / 2 ] 2 sin 2 Δ .