Unified Analysis of Ellipsometry Errors Due to Imperfect Components, Cell-Window Birefringence, and Incorrect Azimuth Angles*

R. M. A. Azzam; N. M. Bashara

doi:10.1364/JOSA.61.000600

Journal of the Optical Society of America
Vol. 61,
Issue 5,
pp. 600-607
(1971)
•https://doi.org/10.1364/JOSA.61.000600

Unified Analysis of Ellipsometry Errors Due to Imperfect Components, Cell-Window Birefringence, and Incorrect Azimuth Angles

R. M. A. Azzam and N. M. Bashara

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R. M. A. Azzam and N. M. Bashara, "Unified Analysis of Ellipsometry Errors Due to Imperfect Components, Cell-Window Birefringence, and Incorrect Azimuth Angles*," J. Opt. Soc. Am. 61, 600-607 (1971)

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Abstract

Coupling constants are introduced that determine the extent to which a component imperfection, window birefringence, or azimuth-angle error couple to cause an error of the specimen reflectance ratio. A unified scheme for treating these sources of error is presented. By expressing the coupling constants as functions of the nulling angles, any of the nulling methods can be considered. Results are presented for the compensator at ±π/4 and nulling by the polarizer and analyzer, which show the contributions of the different sources of error to ψ and Δ in the four ellipsometry zones. The limitations of one- and two-zone measurements are discussed. A procedure is described and demonstrated experimentally for determining the quantities characterizing the different sources of error in the ellipsometer. These results can be used in correcting data from one-, two-, or four-zone measurements.

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	Coupling constants (γ_i, γ_i′)	(γ_i, γ_i′) for ρ_C =−j, C = +π/4	(γ_i, γ_i′) for ρ_C = −j, C = −π/4
Polarizer	$γ_{P} = - \frac{ρ_{C} tan A {sec}^{2} C {sec}^{2} P }{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_P^2,4 = −2 tanAe⁻^j²^P	γ_P^1,3 = −2 tanAe^j²^P
Polarizer	${γ_{P}}^{'} = - \frac{ρ_{C} tan A {sec}^{2} P * {sec}^{2} C}{{(1 - ρ_{C} tan C tan P *)}^{2}}$	γ_P′^2,4 = −2 tanAe⁻^j²^P	γ_P′^1,3 = −2 tanAe^j²^P
Compensator	$γ_{C} = - \frac{tan A {sec}^{2} C tan P }{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_C^2,4 = j tanAcos2Pe⁻^j²^P	γ_C^1,3 = −j tanA cos2Pe^j²^P
Compensator	${γ_{C}}^{'} = - \frac{tan A {sec}^{2} C (1 - {tan}^{2} P * - ρ_{C} {sec}^{2} P )}{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_C′^2,4 = −2tanA(−1+j sin2P)e⁻^j²^P	γ_C′^1,3 = −2 tanA(−1−j sin2P)e^j²^P
Analyzer	γ_A = 0	γ_A^2,4 = 0	γ_A^1,3 = 0
Analyzer	${γ_{A}}^{'} = - {sec}^{2} A \frac{(tan C + ρ_{C} tan P )}{(1 - ρ_{C} tan C tan P )}$	γ_A′^{2 4} = −j sec²Ae⁻^j²^P	γ_A′^1,3 = j sec²Ae^j²^P
1st Window	$γ_{W} = \frac{tan A}{2 {cos}^{2} C {cos}^{2} P } \frac{(sin 2 W - ρ_{C} sin 2 P * cos 2 W )}{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_W^2,4 = tanA(sin2W cos2P−j cos2W)e⁻^j²^P	γ_W^1,3 = tanA(sin2W cos2P+j cos2W)e^j²^P
2nd Window	$\begin{array}{l} {γ_{W}}^{'} & = m \frac{(tan C + ρ_{C} tan P )}{(1 - ρ_{C} tan C tan P )} \\ m & = (sin W^{'} - tan A cos W^{'}) (cos W^{'} + tan A sin W^{'}) \end{array}$	γ_W′^2,4 = jme⁻^j²^P	γ_W′^1,3 = −jme^j²^P

Zone 1	$ψ = A_{1} - t_{2 P} sin 2 A_{1} - \frac{1}{2} t_{1 C} sin 2 A_{1} cos 2 P_{1} + sin 2 A_{1} sin 2 P_{1} δ C + δ A + \frac{1}{2} t_{2 W} sin 2 W cos 2 P_{1} sin 2 A_{1}$
2	$ψ = A_{2} + t_{2 P} sin 2 A_{2} - \frac{1}{2} t_{1 C} sin 2 A_{2} cos 2 P_{2} + sin 2 A_{2} sin 2 P_{2} δ C + δ A - \frac{1}{2} t_{2 W} sin 2 W cos 2 P_{2} sin 2 A_{2}$
3	$ψ = - A_{3} - t_{2 P} sin 2 A_{3} + \frac{1}{2} t_{1 C} sin 2 A_{3} cos 2 P_{3} - sin 2 A_{3} sin 2 P_{3} δ C - δ A - \frac{1}{2} t_{2 W} sin 2 W cos 2 P_{3} sin 2 A_{3}$
4	$ψ = - A_{4} - t_{2 P} sin 2 A_{4} + \frac{1}{2} t_{1 C} sin 2 A_{4} cos 2 P_{4} - sin 2 A_{4} sin 2 P_{4} δ C - δ A + \frac{1}{2} t_{2 W} sin 2 W cos 2 P_{4} sin 2 P_{4}$
1	Δ = 2P₁+π/2+2t₁_P+2δP−t₂_Ccos2P₁−2δC+t₂_Wcos2W+t₂_W′cos2W′−t₂_W′ sin2W′cot2A₁
2	Δ = −2P₂−π/2−2t₁_P−2δP−t₂_Ccos2P₂+2δC+t₂_Wcos2W+t₂_W′cos2W′−t₂_W′ sin2W′cot2A₂
3	Δ = 2P₃−π/2+2t₁_P+2δP−t₂_Ccos2P₃−2δC+t₂_Wcos2W+t₂_W′cos2W′−t₂_W′sin2W′cot2A₃
4	Δ = −2P₄+π/2−2t₁_P−2δP−t₂_Ccos2P₄+2δC+t₂_Wcos2W+t₂_W′cos2W′−t₂_W′sin2W′cot2A₄

Zone 1	$A_{1} = ψ + t_{2 P} sin 2 ψ + \frac{1}{2} t_{1 C} sin 2 ψ sin Δ + sin 2 ψ cos Δ δ C - δ A - \frac{1}{2} t_{2 W} sin 2 W sin Δ sin 2 ψ$
2	$A_{2} = ψ - t_{2 P} sin 2 ψ - \frac{1}{2} t_{1 C} sin 2 ψ sin Δ + sin 2 ψ cos Δ δ C - δ A - \frac{1}{2} t_{2 W} sin 2 W sin Δ sin 2 ψ$
3	$A_{3} = - ψ - t_{2 P} sin 2 ψ + \frac{1}{2} t_{1 C} sin 2 ψ sin Δ + sin 2 ψ cos Δ δ C - δ A - \frac{1}{2} t_{2 W} sin 2 W sin Δ sin 2 ψ$
4	$A_{4} = - ψ + t_{2 P} sin 2 ψ - \frac{1}{2} t_{1 C} sin 2 ψ sin Δ + sin 2 ψ cos Δ δ C - δ A - \frac{1}{2} t_{2 W} sin 2 W sin Δ sin 2 ψ$
1	$P_{1} = Δ / 2 - π / 4 - t_{1 P} - δ P + \frac{1}{2} t_{2 C} sin Δ + δ C - \frac{1}{2} t_{2 W} cos 2 W - \frac{1}{2} t_{2 W^{'}} cos 2 W^{'} + \frac{1}{2} t_{2 W^{'}} sin 2 W^{'} cot 2 ψ$
2	$P_{2} = - Δ / 2 - π / 4 - t_{1 P} - δ P + \frac{1}{2} t_{2 C} sin Δ + δ C + \frac{1}{2} t_{2 W} cos 2 W + \frac{1}{2} t_{2 W^{'}} cos 2 W^{'} - \frac{1}{2} t_{2 W^{'}} sin 2 W^{'} cot 2 ψ$
3	$P_{3} = Δ / 2 + π / 4 - t_{1 P} - δ P - \frac{1}{2} t_{2 C} sin Δ + δ C - \frac{1}{2} t_{2 W} cos 2 W - \frac{1}{2} t_{2 W^{'}} cos 2 W^{'} - \frac{1}{2} t_{2 W^{'}} sin 2 W^{'} cot 2 ψ$
4	$P_{4} = - Δ / 2 + π / 4 - t_{1 P} - δ P - \frac{1}{2} t_{2 C} sin Δ + δ C + \frac{1}{2} t_{2 W} cos 2 W + \frac{1}{2} t_{2 W^{'}} cos 2 W^{'} + \frac{1}{2} t_{2 W^{'}} sin 2 W^{'} cot 2 ψ$

	η_P,C (deg)	t₂_P (deg)	t₁_C* (deg)	t₂_C⁺ (rad)	t₂_C⁻ (rad)
Ellipsometer I	−0.628±0.032	−0.011±0.002	0.307±0.016	0.0019±0.0006	0.0018±0.0005
			(0.175±0.046)^a
Ellipsometer II	−0.352±0.017	0.011±0.004	0.453±0.101	0.0018±0.0005	0.0009±0.0002
			${\begin{array}{l} {t_{1 C}}^{+} & = 2.089 \pm 1.300 \\ {t_{1 C}}^{-} & = 1.180 \pm 1.327 \end{array}}$ ^b

	Coupling constants (γ_i, γ_i′)	(γ_i, γ_i′) for ρ_C =−j, C = +π/4	(γ_i, γ_i′) for ρ_C = −j, C = −π/4
Polarizer	$γ_{P} = - \frac{ρ_{C} tan A {sec}^{2} C {sec}^{2} P }{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_P^2,4 = −2 tanAe⁻^j²^P	γ_P^1,3 = −2 tanAe^j²^P
Polarizer	${γ_{P}}^{'} = - \frac{ρ_{C} tan A {sec}^{2} P * {sec}^{2} C}{{(1 - ρ_{C} tan C tan P *)}^{2}}$	γ_P′^2,4 = −2 tanAe⁻^j²^P	γ_P′^1,3 = −2 tanAe^j²^P
Compensator	$γ_{C} = - \frac{tan A {sec}^{2} C tan P }{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_C^2,4 = j tanAcos2Pe⁻^j²^P	γ_C^1,3 = −j tanA cos2Pe^j²^P
Compensator	${γ_{C}}^{'} = - \frac{tan A {sec}^{2} C (1 - {tan}^{2} P * - ρ_{C} {sec}^{2} P )}{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_C′^2,4 = −2tanA(−1+j sin2P)e⁻^j²^P	γ_C′^1,3 = −2 tanA(−1−j sin2P)e^j²^P
Analyzer	γ_A = 0	γ_A^2,4 = 0	γ_A^1,3 = 0
Analyzer	${γ_{A}}^{'} = - {sec}^{2} A \frac{(tan C + ρ_{C} tan P )}{(1 - ρ_{C} tan C tan P )}$	γ_A′^{2 4} = −j sec²Ae⁻^j²^P	γ_A′^1,3 = j sec²Ae^j²^P
1st Window	$γ_{W} = \frac{tan A}{2 {cos}^{2} C {cos}^{2} P } \frac{(sin 2 W - ρ_{C} sin 2 P * cos 2 W )}{{(1 - ρ_{C} tan C tan P )}^{2}}$	γ_W^2,4 = tanA(sin2W cos2P−j cos2W)e⁻^j²^P	γ_W^1,3 = tanA(sin2W cos2P+j cos2W)e^j²^P
2nd Window	$\begin{array}{l} {γ_{W}}^{'} & = m \frac{(tan C + ρ_{C} tan P )}{(1 - ρ_{C} tan C tan P )} \\ m & = (sin W^{'} - tan A cos W^{'}) (cos W^{'} + tan A sin W^{'}) \end{array}$	γ_W′^2,4 = jme⁻^j²^P	γ_W′^1,3 = −jme^j²^P

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Journal of the Optical Society of America