Second-order systematic errors in Mueller matrix dual rotating compensator ellipsometry

Laurent Broch; Aotmane En Naciri; Luc Johann

doi:10.1364/AO.49.003250

Applied Optics
Vol. 49,
Issue 17,
pp. 3250-3258
(2010)
•https://doi.org/10.1364/AO.49.003250

Second-order systematic errors in Mueller matrix dual rotating compensator ellipsometry

Laurent Broch, Aotmane En Naciri, and Luc Johann

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Laurent Broch, Aotmane En Naciri, and Luc Johann, "Second-order systematic errors in Mueller matrix dual rotating compensator ellipsometry," Appl. Opt. 49, 3250-3258 (2010)

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Abstract

We investigate the systematic errors at the second order for a Mueller matrix ellipsometer in the dual rotating compensator configuration. Starting from a general formalism, we derive explicit second- order errors in the Mueller matrix coefficients of a given sample. We present the errors caused by the azimuthal inaccuracy of the optical components and their influences on the measurements. We demonstrate that the methods based on four-zone or two-zone averaging measurement are effective to vanish the errors due to the compensators. For the other elements, it is shown that the systematic errors at the second order can be canceled only for some coefficients of the Mueller matrix. The calibration step for the analyzer and the polarizer is developed. This important step is necessary to avoid the azimuthal inaccuracy in such elements. Numerical simulations and experimental measurements are presented and discussed.

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Error	Special Configuration	$\frac{1}{2} \frac{\partial^{2} M}{\partial x^{2}}$
$δ A^{2}$	Without	$- 2 (\begin{matrix} 0 & 0 & 0 & 0 \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{matrix})$
$δ P^{2}$	Without	$- 2 (\begin{matrix} 0 & M_{12} & M_{13} & M_{14} \\ 0 & M_{22} & M_{23} & M_{24} \\ 0 & M_{32} & M_{33} & M_{34} \\ 0 & M_{42} & M_{43} & M_{44} \end{matrix})$
$δ C_{S 1}^{2}$	Four-zone averaging: $\begin{matrix} C_{S 1} = C_{S 2} = 0 \end{matrix}$	$- 2 (\begin{matrix} 0 & 4 M_{12} & 4 M_{13} & M_{14} \\ 0 & 4 M_{22} & 4 M_{23} & M_{24} \\ 0 & 4 M_{32} & 4 M_{33} & M_{34} \\ 0 & 4 M_{42} & 4 M_{43} & M_{44} \end{matrix})$
$δ C_{S 2}^{2}$	Four-zone averaging: $\begin{matrix} C_{S 1} = C_{S 2} = 0 \end{matrix}$	$- 2 (\begin{matrix} 0 & 0 & 0 & 0 \\ 4 M_{21} & 4 M_{22} & 4 M_{23} & 4 M_{24} \\ 4 M_{31} & 4 M_{32} & 4 M_{33} & 4 M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{matrix})$

Coefficient	Value
$δ^{2} M_{12}$	$f (- M_{13} + M_{23} B_{1} + M_{33} B_{2}, - M_{12} + M_{22} B_{1} + M_{32} B_{2}, 2 C_{S 1})$
$δ^{2} M_{13}$	$f (M_{12} - M_{22} B_{1} - M_{32} B_{2}, - M_{13} + M_{23} B_{1} + M_{33} B_{2}, 2 C_{S 1})$
$δ^{2} M_{14}$	$- \frac{M_{14}}{4} \cos 2 C_{S 1} - \frac{1}{4} c_{2} f (M_{34}, M_{24}, A) \cos 2 C_{S 1} + \frac{1}{4} c_{2} f (M_{34}, M_{24}, A - C_{S 1} + 2 C_{S 2})$
$δ^{2} M_{21}$	$c_{1} f (f (M_{33}, M_{32}, P + 2 C_{S 1}), f (M_{23}, M_{22}, P + 2 C_{S 1}), 2 C_{S 2})$
$δ^{2} M_{22}$	$- f (M_{32}, M_{22}, 2 C_{S 2}) \cos 4 C_{S 1} - f (M_{33}, M_{23}, 2 C_{S 2}) \sin 4 C_{S 1}$
$δ^{2} M_{23}$	$f (f (M_{32}, M_{22}, 2 C_{S 2}), - f (M_{33}, M_{23}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{24}$	$\frac{1}{4} f (M_{34}, - M_{24}, C_{S 1} - 2 C_{S 2})$
$δ^{2} M_{31}$	$c_{1} f (- f (M_{23}, M_{21}, P + 2 C_{S 1}), f (M_{33}, M_{32}, P + 2 C_{S 1}), 2 C_{S 2})$
$δ^{2} M_{32}$	$f (f (M_{23}, - M_{33}, 2 C_{S 2}), f (M_{22}, - M_{32}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{33}$	$f (f (- M_{22}, M_{32}, 2 C_{S 2}), f (M_{23}, - M_{33}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{34}$	$- \frac{1}{4} f (M_{24}, M_{34}, C_{S 1} - 2 C_{S 2})$
$δ^{2} M_{41}$	$c_{1} f (M_{43}, M_{42}, P + 2 C_{S 1} - C_{S 2})$
$δ^{2} M_{42}$	$- f (M_{43}, M_{42}, 2 C_{S 1} - C_{S 2})$
$δ^{2} M_{43}$	$f (M_{42}, - M_{43}, 2 C_{S 1} - C_{S 2})$
$δ^{2} M_{44}$	$- \frac{1}{4} M_{44} \cos (2 C_{S 1} - 2 C_{S 2})$

Coefficient	Value
$δ^{2} M_{12}$	$c_{2} f (f (M_{33}, M_{23}, A + 2 C_{S 2}), f (M_{32}, M_{22}, A + 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{13}$	$c_{2} f (- f (M_{32}, M_{22}, A + 2 C_{S 2}), f (M_{33}, M_{23}, A + 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{14}$	$c_{2} f (M_{34}, M_{24}, A - C_{S 1} + 2 C_{S 2})$
$δ^{2} M_{21}$	$f (- M_{31} + B_{3} M_{32} + B_{4} M_{33}, - M_{21} + B_{3} M_{22} + B_{4} M_{23}, 2 C_{S 2})$
$δ^{2} M_{22}$	$- f (f (M_{33}, M_{23}, 2 C_{S 2}), f (M_{32}, M_{22}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{23}$	$f (f (M_{32}, M_{22}, 2 C_{S 2}), - f (M_{33}, M_{23}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{24}$	$f (M_{34}, - M_{24}, C_{S 1} - 2 C_{S 2})$
$δ^{2} M_{31}$	$f (M_{21} - B_{3} M_{22} - B_{4} M_{23}, - M_{31} + B_{3} M_{32} + B_{4} M_{33}, 2 C_{S 2})$
$δ^{2} M_{32}$	$f (f (M_{23}, - M_{33}, 2 C_{S 2}), f (M_{22}, - M_{32}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{33}$	$f (f (- M_{22}, M_{32}, 2 C_{S 2}), f (M_{23}, - M_{33}, 2 C_{S 2}), 2 C_{S 1})$
$δ^{2} M_{34}$	$- f (M_{24}, M_{34}, C_{S 1} - 2 C_{S 2})$
$δ^{2} M_{41}$	$\frac{1}{4} [- (M_{41} + M_{42} B_{3} - M_{43} B_{4}) \cos 2 C_{S 2} + c_{1} M_{42} \sin (2 P + 4 C_{S 1}) \sin 2 C_{S 2} + c_{1} M_{43} \sin (2 P + 4 C_{S 1}) \cos 2 C_{S 2}]$
$δ^{2} M_{42}$	$- \frac{1}{4} f (M_{43}, M_{42}, 2 C_{S 1} - C_{S 2})$
$δ^{2} M_{43}$	$\frac{1}{4} f (M_{42}, - M_{43}, 2 C_{S 1} - C_{S 2})$
$δ^{2} M_{44}$	$- \frac{1}{4} M_{44} \cos (2 C_{S 1} - 2 C_{S 2})$

Origin	$δ M$
$δ A$	$2 (\begin{matrix} 0 & 0 & 0 & 0 \\ N (δ A)^{2} & - (δ A)^{2} & - C δ A & - S δ A \\ - N δ A & δ A & - C (δ A)^{2} & - S (δ A)^{2} \\ 0 & - S δ A & S (δ A)^{2} & - C (δ A)^{2} \end{matrix})$
$δ P$	$2 (\begin{matrix} 0 & N (δ P)^{2} & - N δ P & 0 \\ 0 & - (δ P)^{2} & δ P & S δ P \\ 0 & - C δ P & - C (δ P)^{2} & - S (δ P)^{2} \\ 0 & S δ P & S (δ P)^{2} & - C (δ P)^{2} \end{matrix})$
$δ C_{S 1}$	$2 (\begin{matrix} 0 & 4 N (δ C_{S 1})^{2} & 2 N δ C_{S 1} & 0 \\ 0 & - 4 (δ C_{S 1})^{2} & - 2 δ C_{S 1} & - S δ C_{S 1} \\ 0 & 2 C δ C_{S 1} & 4 C (δ C_{S 1})^{2} & - S (δ C_{S 1})^{2} \\ 0 & 2 S δ C_{S 1} & 4 S (δ C_{S 1})^{2} & C (δ C_{S 1})^{2} \end{matrix})$
$δ C_{S 2}$	$2 (\begin{matrix} 0 & 0 & 0 & 0 \\ 4 N (δ C_{S 2})^{2} & - 4 (δ C_{S 2})^{2} & 2 C δ C_{S 2} & 2 S δ C_{S 2} \\ 2 N δ C_{S 2} & - 2 δ C_{S 2} & 4 C (δ C_{S 2})^{2} & - 4 S (δ C_{S 2})^{2} \\ 0 & - S δ C_{S 2} & S (δ C_{S 2})^{2} & C (δ C_{S 2})^{2} \end{matrix})$
$δ δ_{1}$	$(\begin{matrix} 0 & - N δ δ_{1} & 0 & \approx 0 \\ 0 & δ δ_{1} & 0 & \approx 0 \\ 0 & 0 & C δ δ_{1} & - S (δ δ_{1})^{2} / 2 \\ 0 & 0 & S δ δ_{1} & - C (δ δ_{1})^{2} / 2 \end{matrix})$
$δ δ_{2}$	$(\begin{matrix} 0 & 0 & 0 & 0 \\ - N δ δ_{2} & δ δ_{2} & 0 & 0 \\ 0 & 0 & C δ δ_{2} & S δ δ_{2} \\ \approx 0 & \approx 0 & S (δ δ_{2})^{2} / 2 & - C (δ δ_{2})^{2} / 2 \end{matrix})$

Sample	Systematic Errors
Vacuum $\begin{matrix} (A = P = 0 °) \end{matrix}$	$(\begin{matrix} 0 & \begin{matrix} \approx 0 \pm 0.011 \end{matrix} & \begin{matrix} 0 \pm 0.030 \end{matrix} & 0 \\ \begin{matrix} \approx 0 \pm 0.011 \end{matrix} & \begin{matrix} - 0.002 \pm 0.015 \end{matrix} & \begin{matrix} 0 \pm 0.067 \end{matrix} & 0 \\ \begin{matrix} 0 \pm 0.021 \end{matrix} & \begin{matrix} 0 \pm 0.067 \end{matrix} & \begin{matrix} - 0.002 \pm 0.015 \end{matrix} & 0 \\ 0 & 0 & 0 & \begin{matrix} - 0.001 \pm 0.0004 \end{matrix} \end{matrix})$
Vacuum (four-zone averaging: $\begin{matrix} A, P = \pm 45 °) \end{matrix}$	$(\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & \begin{matrix} - 0.002 \pm 0.015 \end{matrix} & \begin{matrix} 0 \pm 0.067 \end{matrix} & 0 \\ 0 & \begin{matrix} 0 \pm 0.067 \end{matrix} & \begin{matrix} 0.001 \pm 0.015 \end{matrix} & 0 \\ 0 & 0 & 0 & \begin{matrix} \approx 0 \pm 0.0004 \end{matrix} \end{matrix})$
Isotropic $106 nm - Si O_{2} / Si (A = P = 0 °)$	$(\begin{matrix} 0 & \begin{matrix} \approx 0 \pm 0.011 \end{matrix} & \begin{matrix} 0 \pm 0.007 \end{matrix} & \begin{matrix} 0 \pm 0.037 \end{matrix} \\ \begin{matrix} \approx 0 \pm 0.011 \end{matrix} & \begin{matrix} - 0.001 \pm 0.015 \end{matrix} & \begin{matrix} 0 \pm 0.048 \end{matrix} & \begin{matrix} 0 \pm 0.055 \end{matrix} \\ \begin{matrix} \approx 0 \pm 0.007 \end{matrix} & \begin{matrix} 0 \pm 0.048 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.003 \end{matrix} & \begin{matrix} - 0.001 \pm 0.010 \end{matrix} \\ \begin{matrix} 0 \pm 0.033 \end{matrix} & \begin{matrix} 0 \pm 0.055 \end{matrix} & \begin{matrix} 0.001 \pm 0.010 \end{matrix} & \begin{matrix} \approx - 0 \pm 0 \end{matrix} \end{matrix})$
Isotropic $106 nm - Si O_{2} / Si (A, P = \pm 45 °)$ (four-zone averaging: $\begin{matrix} A, P = \pm 45 °) \end{matrix}$	$(\begin{matrix} 0 & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} & \begin{matrix} 0 \pm 0.003 \end{matrix} & 0 \\ \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} & \begin{matrix} - 0.002 \pm 0.015 \end{matrix} & \begin{matrix} 0 \pm 0.048 \end{matrix} & \begin{matrix} 0 \pm 0.055 \end{matrix} \\ \begin{matrix} 0 \pm 0.003 \end{matrix} & \begin{matrix} 0 \pm 0.048 \end{matrix} & \begin{matrix} \approx 0 \pm 0 \end{matrix} & \begin{matrix} - 0.002 \pm 0.010 \end{matrix} \\ 0 & \begin{matrix} 0 \pm 0.055 \end{matrix} & \begin{matrix} 0.002 \pm 0.010 \end{matrix} & \begin{matrix} \approx 0 \pm 0 \end{matrix} \end{matrix})$
Anisotropic dielectric S $(A = P = 0 °)$	$(\begin{matrix} 0 & \begin{matrix} \approx 0 \pm 0.011 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.030 \end{matrix} & \begin{matrix} \approx 0 \pm 0.0006 \end{matrix} \\ \begin{matrix} \approx 0 \pm 0.011 \end{matrix} & \begin{matrix} - 0.002 \pm 0.015 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.067 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} \\ \begin{matrix} \approx 0 \pm 0.030 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.067 \end{matrix} & \begin{matrix} 0.002 \pm 0.015 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} \\ \begin{matrix} \approx - 0 \pm 0.0005 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} & \begin{matrix} \approx 0 \pm 0.001 \end{matrix} & \begin{matrix} 0.001 \pm 0.004 \end{matrix} \end{matrix})$
Anisotropic dielectric S (four-zone averaging: $\begin{matrix} A, P = \pm 45 °) \end{matrix}$	$(\begin{matrix} 0 & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.0004 \end{matrix} & \begin{matrix} \approx - 0 \pm 0 \end{matrix} \\ \begin{matrix} 0 \pm 0.001 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.015 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.067 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} \\ \begin{matrix} \approx 0 \pm 0 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.067 \end{matrix} & \begin{matrix} 0.003 \pm 0.015 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.0011 \end{matrix} \\ \begin{matrix} \approx 0 \pm 0 \end{matrix} & \begin{matrix} \approx - 0 \pm 0.001 \end{matrix} & \begin{matrix} \approx 0 \pm 0.001 \end{matrix} & \begin{matrix} 0.0015 \pm 0.0006 \end{matrix} \end{matrix})$

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Applied Optics