Time-resolved ellipsometry

Gerald E. Jellison; Douglas H. Lowndes

doi:10.1364/AO.24.002948

Applied Optics
Vol. 24,
Issue 18,
pp. 2948-2955
(1985)
•https://doi.org/10.1364/AO.24.002948

Time-resolved ellipsometry

Gerald E. Jellison and Douglas H. Lowndes

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Gerald E. Jellison and Douglas H. Lowndes, "Time-resolved ellipsometry," Appl. Opt. 24, 2948-2955 (1985)

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Abstract

An ellipsometer is described which is capable of measuring one of the related ellipsometric parameters N, S, or C as a function of time. The ellipsometer is of the standard polarizer–compensator–sample–analyzer type, in which the analyzer is a Wollaston prism. An analysis is presented of the errors resulting from misalignment of the azimuths of the various elements from an incorrect phase shift of the compensator and from sample surface effects. The time resolution of the ellipsometer is limited only by the rise time of the photodetector and by the digitization rate of the data acquisition system. Picosecond time resolution is possible, in principle, using a streak camera as both detector and digitizer. Submicrosecond operation of the time-resolved ellipsometer is demonstrated in a study of pulsed excimer laser cleaning of a silicon surface in air.

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θ_a	Case I			+90°	Case II
	Linearly polarized light				Circularly polarized light
	−45°	0°	+45°		θ_b =	−45°	+45°
−45°	1 + C	1 + N	1 − C	1 − N		1 − S	1 + S
0°	1 + N	2(1 + N)	1 + N	0		1 + N	1 + N
+45°	1 − C	1 + N	1 + C	1 − N		1 + S	1 − S
+90°	1 − N	0	1 − N	2(1 − N)		1 − N	1 − N

Symbol	θ_b	θ_p	θ_c	F₀	$\frac{\partial F}{\partial ξ}$	$\frac{\partial F_{0}}{\partial θ_{a}}$	$\frac{{\partial F}_{0}}{\partial θ_{b}}$ ^b	$\frac{\partial F_{0}}{\partial θ_{c}}$	$\frac{\partial F_{0}}{\partial δ}$	K
(a) θ_a = 0°, 90°
$F_{01}^{1}$	—	0°	—	1	0	0	0	—	—	0
$F_{02}^{1}$	—	90°	—	−1	0	0	0	—	—	0
$F_{03}^{1}$	—	−45°	—	N	(1 − N²)	−2C	+2(1 − N²)	—	—	−K_Nl
$F_{04}^{1}$	—	+45°	—	N	(1 − N²)	+2C	−2(1 − N²)	—	—	+K_Nl
$F_{11}^{1}$	0°	0°	0°	1	0	0	0	0	0	0
$F_{12}^{1}$	0°	90°	90°	−1	0	0	0	0	0	0
$F_{13}^{1}$	0°	−45°	−45°	N	(1 − N²)	−2C	0	+2(1 − N²)	0	−K_Nl
$F_{14}^{1}$	0°	+45°	+45°	N	(1 − N²)	+2C	0	−2(1 − N²)	0	+K_Nl
$F_{21}^{1}$	90°	0°	90°	1	0	0	0	0	0	0
$F_{22}^{1}$	90°	90°	0°	−1	0	0	0	0	0	0
$F_{23}^{1}$	90°	−45°	+45°	N	(1 − N²)	−2C	0	+2(1 − N²)	0	−K_Nl
$F_{24}^{1}$	90°	+45°	−45°	N	(1 − N²)	+2C	0	−2(1 − N²)	0	+K_Nl
$F_{31}^{1}$	−45°	0°	+45°	N	(1 − N²)	+2S	0	0	−(1 − N²)	+K_Nc
$F_{32}^{1}$	−45°	90°	−45°	N	(1 − N²)	+2S	0	0	+(1 − N²)	+K_Nc
$F_{33}^{1}$	−45°	−45°	0°	N	(1 − N²)	+2S	+2(1 − N²)	0	0	+K_Nc
$F_{34}^{1}$	−45°	+45°	90°	N	(1 − N²)	+2S	−2(1 − N²)	0	0	+K_Nc
$F_{41}^{1}$	+45°	0°	−45°	N	(1 − N²)	−2S	0	0	−(1 − N²)	−K_Nc
$F_{42}^{1}$	+45°	90°	+45°	N	(1 − N²)	−2S	0	0	+(1 − N²)	−K_Nc
$F_{43}^{1}$	+45°	−45°	90°	N	(1 − N²)	−2S	+2(1 − N²)	0	0	−K_Nc
$F_{44}^{1}$	+45°	+45°	0°	N	(1 − N²)	−2S	−2(1 − N²)	0	0	−K_Nc
(b) θ_a = +45°, −45°
$F_{01}^{4}$	—	0°	—	0	1	−2	+2C/(1 + N)	—	—	K₀₁
$F_{02}^{4}$	—	90°	—	0	1	+2	−2C/(1 − N)	—	—	K₀₂
$F_{03}^{4}$	—	−45°	—	−C	(1 − C²)	−2N	+2NC	—	—	K_C
$F_{04}^{4}$	—	+45°	—	+C	(1 − C²)	−2N	+2NC	—	—	K_C
$F_{11}^{4}$	0°	0°	0°	0	1	−2	−2S/(1 + N)	+2C/(1 + N)	0	K₀₁
$F_{12}^{4}$	0°	90°	90°	0	1	+2	−2S/(1 − N)	+2C/(1 − N)	0	K₀₂
$F_{13}^{4}$	0°	−45°	−45°	−C	(1 − C²)	−2N	−2S	+2NC	0	K_C
$F_{14}^{4}$	0°	+45°	+45°	+C	(1 − C²)	−2N	−2S	+2NC	0	K_C
$F_{21}^{4}$	90°	0°	90°	0	1	−2	+2S/(1 + N)	+2C/(1 + N)	0	K₀₁
$F_{22}^{4}$	90°	90°	0°	0	1	+2	+2S/(1 − N)	−2C/(1 − N)	0	K₀₂
$F_{23}^{4}$	90°	−45°	+45°	−C	(1 − C²)	−2N	+2S	+2NC	0	K_C
$F_{24}^{4}$	90°	+45°	−45°	+C	(1 − C²)	−2N	+2S	+2NC	0	K_C
$F_{31}^{4}$	−45°	0°	+45°	+S	(1 − S²)	−2N	+2C	0	+SN	K_S
$F_{32}^{4}$	−45°	90°	−45°	+S	(1 − S²)	−2N	−2C	0	−SN	K_S
$F_{33}^{4}$	−45°	−45°	0°	+S	(1 − S²)	−2N	−2SN	0	+C	K_S
$F_{34}^{4}$	−45°	+45°	90°	+S	(1 − S²)	−2N	+2SN	0	−C	K_S
$F_{41}^{4}$	+45°	0°	−45°	−S	(1 − S²)	−2N	+2C	0	−SN	K_S
$F_{42}^{4}$	+45°	90°	+45°	−S	(1 − S²)	−2N	−2C	0	+SN	K_S
$F_{43}^{4}$	+45°	−45°	90°	−S	(1 − S²)	−2N	+2SN	0	+C	K_S
$F_{44}^{4}$	+45°	+45°	0°	−S	(1 − S²)	−2N	−2SN	0	−C	K_S

θ_a	θ_b	θ_p	F₀	K
45°	±_b45°	—	∓_bS	K_S = β₂[(1 + N − S²) cosδ₂ + SC sinδ₂]
45°	−,0°,90°	0°	0	K₀₁ = 2β₂ cosδ₂
45°	−,0°,90°	90°	0	K₀₂ = 2β₁(C cosδ₁ + S sinδ₂)/(1 − N)
45°	−,0°,90°	±_p45°	±_pC	K_C = β₂[(1 + N − S²) cosδ₂ − SC sinδ₂] + β₁(S sinδ₁ − CN cosδ₁)
0°	±_b45°	—	N	∓_bK_Nc = (1 + N)[β₁(1 − N) sinδ₁ + β₂(C sinδ₂ − S cosδ₂)]
0°	−,0°,90°	0°	1	0
0°	−,0°,90°	90°	−1	0
0°	−,0°,90°	±_p45°	N	±pK_Nl = (1 + N)[β₁(1 − N) cosδ₁ + β₂(C cosδ₂ + S sinδ₂)]

θ_a	Case I			+90°	Case II
	Linearly polarized light				Circularly polarized light
	−45°	0°	+45°		θ_b =	−45°	+45°
−45°	1 + C	1 + N	1 − C	1 − N		1 − S	1 + S
0°	1 + N	2(1 + N)	1 + N	0		1 + N	1 + N
+45°	1 − C	1 + N	1 + C	1 − N		1 + S	1 − S
+90°	1 − N	0	1 − N	2(1 − N)		1 − N	1 − N

Symbol	θ_b	θ_p	θ_c	F₀	$\frac{\partial F}{\partial ξ}$	$\frac{\partial F_{0}}{\partial θ_{a}}$	$\frac{{\partial F}_{0}}{\partial θ_{b}}$ ^b	$\frac{\partial F_{0}}{\partial θ_{c}}$	$\frac{\partial F_{0}}{\partial δ}$	K
(a) θ_a = 0°, 90°
$F_{01}^{1}$	—	0°	—	1	0	0	0	—	—	0
$F_{02}^{1}$	—	90°	—	−1	0	0	0	—	—	0
$F_{03}^{1}$	—	−45°	—	N	(1 − N²)	−2C	+2(1 − N²)	—	—	−K_Nl
$F_{04}^{1}$	—	+45°	—	N	(1 − N²)	+2C	−2(1 − N²)	—	—	+K_Nl
$F_{11}^{1}$	0°	0°	0°	1	0	0	0	0	0	0
$F_{12}^{1}$	0°	90°	90°	−1	0	0	0	0	0	0
$F_{13}^{1}$	0°	−45°	−45°	N	(1 − N²)	−2C	0	+2(1 − N²)	0	−K_Nl
$F_{14}^{1}$	0°	+45°	+45°	N	(1 − N²)	+2C	0	−2(1 − N²)	0	+K_Nl
$F_{21}^{1}$	90°	0°	90°	1	0	0	0	0	0	0
$F_{22}^{1}$	90°	90°	0°	−1	0	0	0	0	0	0
$F_{23}^{1}$	90°	−45°	+45°	N	(1 − N²)	−2C	0	+2(1 − N²)	0	−K_Nl
$F_{24}^{1}$	90°	+45°	−45°	N	(1 − N²)	+2C	0	−2(1 − N²)	0	+K_Nl
$F_{31}^{1}$	−45°	0°	+45°	N	(1 − N²)	+2S	0	0	−(1 − N²)	+K_Nc
$F_{32}^{1}$	−45°	90°	−45°	N	(1 − N²)	+2S	0	0	+(1 − N²)	+K_Nc
$F_{33}^{1}$	−45°	−45°	0°	N	(1 − N²)	+2S	+2(1 − N²)	0	0	+K_Nc
$F_{34}^{1}$	−45°	+45°	90°	N	(1 − N²)	+2S	−2(1 − N²)	0	0	+K_Nc
$F_{41}^{1}$	+45°	0°	−45°	N	(1 − N²)	−2S	0	0	−(1 − N²)	−K_Nc
$F_{42}^{1}$	+45°	90°	+45°	N	(1 − N²)	−2S	0	0	+(1 − N²)	−K_Nc
$F_{43}^{1}$	+45°	−45°	90°	N	(1 − N²)	−2S	+2(1 − N²)	0	0	−K_Nc
$F_{44}^{1}$	+45°	+45°	0°	N	(1 − N²)	−2S	−2(1 − N²)	0	0	−K_Nc
(b) θ_a = +45°, −45°
$F_{01}^{4}$	—	0°	—	0	1	−2	+2C/(1 + N)	—	—	K₀₁
$F_{02}^{4}$	—	90°	—	0	1	+2	−2C/(1 − N)	—	—	K₀₂
$F_{03}^{4}$	—	−45°	—	−C	(1 − C²)	−2N	+2NC	—	—	K_C
$F_{04}^{4}$	—	+45°	—	+C	(1 − C²)	−2N	+2NC	—	—	K_C
$F_{11}^{4}$	0°	0°	0°	0	1	−2	−2S/(1 + N)	+2C/(1 + N)	0	K₀₁
$F_{12}^{4}$	0°	90°	90°	0	1	+2	−2S/(1 − N)	+2C/(1 − N)	0	K₀₂
$F_{13}^{4}$	0°	−45°	−45°	−C	(1 − C²)	−2N	−2S	+2NC	0	K_C
$F_{14}^{4}$	0°	+45°	+45°	+C	(1 − C²)	−2N	−2S	+2NC	0	K_C
$F_{21}^{4}$	90°	0°	90°	0	1	−2	+2S/(1 + N)	+2C/(1 + N)	0	K₀₁
$F_{22}^{4}$	90°	90°	0°	0	1	+2	+2S/(1 − N)	−2C/(1 − N)	0	K₀₂
$F_{23}^{4}$	90°	−45°	+45°	−C	(1 − C²)	−2N	+2S	+2NC	0	K_C
$F_{24}^{4}$	90°	+45°	−45°	+C	(1 − C²)	−2N	+2S	+2NC	0	K_C
$F_{31}^{4}$	−45°	0°	+45°	+S	(1 − S²)	−2N	+2C	0	+SN	K_S
$F_{32}^{4}$	−45°	90°	−45°	+S	(1 − S²)	−2N	−2C	0	−SN	K_S
$F_{33}^{4}$	−45°	−45°	0°	+S	(1 − S²)	−2N	−2SN	0	+C	K_S
$F_{34}^{4}$	−45°	+45°	90°	+S	(1 − S²)	−2N	+2SN	0	−C	K_S
$F_{41}^{4}$	+45°	0°	−45°	−S	(1 − S²)	−2N	+2C	0	−SN	K_S
$F_{42}^{4}$	+45°	90°	+45°	−S	(1 − S²)	−2N	−2C	0	+SN	K_S
$F_{43}^{4}$	+45°	−45°	90°	−S	(1 − S²)	−2N	+2SN	0	+C	K_S
$F_{44}^{4}$	+45°	+45°	0°	−S	(1 − S²)	−2N	−2SN	0	−C	K_S

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Equations (34)

Applied Optics