Estimation for a complex index of refraction based on the Stokes vector of the reflected light under active polarized detection

Zhiyong Yang; Zhiwei Zhang; Zhili Zhang; Shun Li

doi:10.1364/AO.464539

Applied Optics
Vol. 61,
Issue 27,
pp. 7830-7837
(2022)
•https://doi.org/10.1364/AO.464539

Estimation for a complex index of refraction based on the Stokes vector of the reflected light under active polarized detection

Zhiyong Yang, Zhiwei Zhang, Zhili Zhang, and Shun Li

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Zhiyong Yang, Zhiwei Zhang, Zhili Zhang, and Shun Li, "Estimation for a complex index of refraction based on the Stokes vector of the reflected light under active polarized detection," Appl. Opt. 61, 7830-7837 (2022)

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Abstract

An estimation method for a complex index of refraction based on the Stokes vector of the reflected light under active polarized detection is presented. In this method, the difference of the Stokes vector of the reflected light is used to introduce the three new estimation indices for the L–M nonlinear least squares algorithm to estimate the complex index of refraction $n$ and $k$ of the target, which does not require either interpretation of the diffuse reflection part or calculation of every polarized bidirectional reflection distribution function (pBRDF) matrix element. Aluminum is chosen to be the sample to prove the principle of this method, and the result shows that this method can estimate the complex index of refraction more precisely compared with the Hyde method. The result further shows that the estimation index containing elements related to circular polarization has better performance than the index only containing elements related to linear polarization.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Polarization State of Incident Light	Stokes Vector of Incident Light	Stokes Vector of Reflected Light
Horizontal polarized light	$S_{0^{\circ}}^{i n} = I_{0^{\circ}}^{i n} (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix})^{T}$	$S_{0^{\circ}}^{o u t} = I_{0^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} + f_{01}^{s} & f_{01}^{s} + f_{00}^{s} & 0 & 0 \end{matrix})_{0^{\circ}}^{T}$
Vertical polarized light	$S_{90^{\circ}}^{i n} = I_{90^{\circ}}^{i n} (\begin{matrix} 1 & - 1 & 0 & 0 \end{matrix})^{T}$	$S_{90^{\circ}}^{o u t} = I_{90^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} - f_{01}^{s} & f_{01}^{s} - f_{00}^{s} & 0 & 0 \end{matrix})_{90^{\circ}}^{T}$
45° linear polarized light	$S_{45^{\circ}}^{i n} = I_{45^{\circ}}^{i n} (\begin{matrix} 1 & 0 & 1 & 0 \end{matrix})^{T}$	$S_{45^{\circ}}^{o u t} = I_{45^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & f_{22}^{s} & - f_{23}^{s} \end{matrix})_{45^{\circ}}^{T}$
$- 45^{\circ}$ linear polarized light	$S_{- 45^{\circ}}^{i n} = I_{- 45^{\circ}}^{i n} (\begin{matrix} 1 & 0 & - 1 & 0 \end{matrix})^{T}$	$S_{- 45^{\circ}}^{o u t} = I_{- 45^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & - f_{22}^{s} & f_{23}^{s} \end{matrix})_{- 45^{\circ}}^{T}$
Right-rotated circularly polarized light	$S_{R}^{i n} = I_{R}^{i n} (\begin{matrix} 1 & 0 & 0 & 1 \end{matrix})^{T}$	$S_{R}^{o u t} = I_{R}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & f_{23}^{s} & f_{22}^{s} \end{matrix})_{R}^{T}$
Left-rotated circularly polarized light	$S_{L}^{i n} = I_{L}^{i n} (\begin{matrix} 1 & 0 & 0 & - 1 \end{matrix})^{T}$	$S_{L}^{o u t} = I_{L}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & - f_{23}^{s} & - f_{22}^{s} \end{matrix})_{L}^{T}$

	Hyde Method		Method Given in This Paper
Incident Angle	$n$	$k$	$n$	$k$
$θ_{i} = 10^{\circ}$	1.9969	10.3016	2.3415	9.7664
$θ_{i} = 20^{\circ}$	5.0401	13.8891	1.6160	7.8903
$θ_{i} = 30^{\circ}$	0.4668	4.5402	0.8894	6.0672
$θ_{i} = 40^{\circ}$	0.2875	3.3982	1.2469	7.1743
$θ_{i} = 50^{\circ}$	0.2919	3.3414	1.5799	8.0742
$θ_{i} = 60^{\circ}$	0.5474	4.6578	1.5004	7.8614

	Hyde Method		Method Given in This Paper
Incident Angle	$n$	$k$	$n$	$k$
$θ_{i} = 10^{\circ}$	1.0551	7.7260	1.2500	7.5960
$θ_{i} = 20^{\circ}$	0.4836	4.4520	1.1137	7.7009
$θ_{i} = 30^{\circ}$	0.8914	6.1427	1.9613	7.3469
$θ_{i} = 40^{\circ}$	0.4184	4.2215	0.9532	7.7505
$θ_{i} = 50^{\circ}$	0.3057	3.5238	1.5580	7.5414
$θ_{i} = 60^{\circ}$	3.5698	12.1681	1.4043	7.5969

	Hyde Method		Method Given in This Paper
Incident Angle	$n$	$k$	$n$	$k$
$θ_{i} = 10^{\circ}$	0.8316	7.7895	1.1152	7.6179
$θ_{i} = 20^{\circ}$	0.1503	2.34806	1.9641	7.32157
$θ_{i} = 30^{\circ}$	0.8566	6.0206	1.6467	7.4925
$θ_{i} = 40^{\circ}$	0.43065	4.2862	1.6893	7.4731
$θ_{i} = 50^{\circ}$	0.4808	4.5077	0.9036	7.7413
$θ_{i} = 60^{\circ}$	5.0560	14.23172	1.39492	7.6008

Polarization State of Incident Light	Stokes Vector of Incident Light	Stokes Vector of Reflected Light
Horizontal polarized light	$S_{0^{\circ}}^{i n} = I_{0^{\circ}}^{i n} (\begin{matrix} 1 & 1 & 0 & 0 \end{matrix})^{T}$	$S_{0^{\circ}}^{o u t} = I_{0^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} + f_{01}^{s} & f_{01}^{s} + f_{00}^{s} & 0 & 0 \end{matrix})_{0^{\circ}}^{T}$
Vertical polarized light	$S_{90^{\circ}}^{i n} = I_{90^{\circ}}^{i n} (\begin{matrix} 1 & - 1 & 0 & 0 \end{matrix})^{T}$	$S_{90^{\circ}}^{o u t} = I_{90^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} - f_{01}^{s} & f_{01}^{s} - f_{00}^{s} & 0 & 0 \end{matrix})_{90^{\circ}}^{T}$
45° linear polarized light	$S_{45^{\circ}}^{i n} = I_{45^{\circ}}^{i n} (\begin{matrix} 1 & 0 & 1 & 0 \end{matrix})^{T}$	$S_{45^{\circ}}^{o u t} = I_{45^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & f_{22}^{s} & - f_{23}^{s} \end{matrix})_{45^{\circ}}^{T}$
$- 45^{\circ}$ linear polarized light	$S_{- 45^{\circ}}^{i n} = I_{- 45^{\circ}}^{i n} (\begin{matrix} 1 & 0 & - 1 & 0 \end{matrix})^{T}$	$S_{- 45^{\circ}}^{o u t} = I_{- 45^{\circ}}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & - f_{22}^{s} & f_{23}^{s} \end{matrix})_{- 45^{\circ}}^{T}$
Right-rotated circularly polarized light	$S_{R}^{i n} = I_{R}^{i n} (\begin{matrix} 1 & 0 & 0 & 1 \end{matrix})^{T}$	$S_{R}^{o u t} = I_{R}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & f_{23}^{s} & f_{22}^{s} \end{matrix})_{R}^{T}$
Left-rotated circularly polarized light	$S_{L}^{i n} = I_{L}^{i n} (\begin{matrix} 1 & 0 & 0 & - 1 \end{matrix})^{T}$	$S_{L}^{o u t} = I_{L}^{i n} (\begin{matrix} f_{00}^{s} + f^{d} & f_{01}^{s} & - f_{23}^{s} & - f_{22}^{s} \end{matrix})_{L}^{T}$

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Applied Optics