Improved determination of the optical constants of anisotropic thin films by ellipsometry using ant colony fitting algorithms

Jean-François Bisson; Gabriel Cormier; Ange Busungu; Koffi N. Amouzou

doi:10.1364/JOSAB.34.001957

Journal of the Optical Society of America B
Vol. 34,
Issue 9,
pp. 1957-1964
(2017)
•https://doi.org/10.1364/JOSAB.34.001957

Improved determination of the optical constants of anisotropic thin films by ellipsometry using ant colony fitting algorithms

Jean-François Bisson, Gabriel Cormier, Ange Busungu, and Koffi N. Amouzou

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Jean-François Bisson, Gabriel Cormier, Ange Busungu, and Koffi N. Amouzou, "Improved determination of the optical constants of anisotropic thin films by ellipsometry using ant colony fitting algorithms," J. Opt. Soc. Am. B 34, 1957-1964 (2017)

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Abstract

Birefringent ${TiO}_{2}$ thin films were characterized using spectroscopic ellipsometry. Ellipsometric data were fitted to a model, taking into account the anisotropic nature of the film and the coherence length of the source being shorter than the thickness of the substrate. Using a conventional gradient-based algorithm often produced a wrong set of optical constants corresponding to a local minimum of the objective function, unless the initial guess was carefully chosen. Using an optimization algorithm inspired by the foraging behavior of ants makes it possible to systematically find that small region in the parameter space where the best solution lies, without a priori knowledge of the optical properties of the sample. The proposed method is especially suitable for samples with unknown optical properties or complex materials, such as anisotropic thin films, for which a large number of parameters are needed to describe them.

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Equations (29)

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Parameter	Symbol	Min. Value	Max. Value	Typical Value
Parameters of Cauchy’s model	$A_{x}$ , $A_{y}$ , $A_{z}$	1	4	Not applicable
	$B_{x}$ , $B_{y}$ , $B_{z}$	−0.05	0.05	Not applicable
	$C_{x}$ , $C_{y}$ , $C_{z}$	−0.05	0.05	Not applicable
Film thickness (nm)	$d_{1}$	15	500	Not applicable
Archive size	$k$	10	40	20
Speed of convergence	$ξ$	0.1	2	0.75
Search locality	$q$	0.01	10	0.1
Number of ants per iteration	$N_{ants}$	1	5	3

Starting Thickness (nm)	Initial $χ^{2}$ Value	Final $χ^{2}$ Value	Estimated Thickness (nm)	Estimated $n_{1 x}; n_{1 y}$ $λ = 450 nm$	Estimated $n_{1 x}; n_{1 y}$ $λ = 650 nm$	Estimated $n_{1 x}; n_{1 y}$ $λ = 850 nm$
25	538	36	55.5	1.71; 1.72	1.86; 1.79	2.32; 2.19
50	949	36	55.5	1.71; 1.72	1.86; 1.79	2.32; 2.19
75	1172	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
100	531	111	123.3	1.69; 1.68	2.19; 2.16	2.06; 2.05
125	65	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
150	219	36	55.5	1.71; 1.72	1.86; 1.79	2.32; 2.19
175	872	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
200	1392	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
225	1210	44	303.6	1.83; 1.79	1.87; 1.82	2.09; 2.04
250	593	51	295.1	1.83; 1.79	1.97; 1.90	2.11; 2.06
300	461	45	295.7	1.83; 1.79	1.96; 1.90	2.11; 2.06

Parameter	Symbol	Min. Value	Max. Value	Typical Value
Parameters of Cauchy’s model	$A_{x}$ , $A_{y}$ , $A_{z}$	1	4	Not applicable
	$B_{x}$ , $B_{y}$ , $B_{z}$	−0.05	0.05	Not applicable
	$C_{x}$ , $C_{y}$ , $C_{z}$	−0.05	0.05	Not applicable
Film thickness (nm)	$d_{1}$	15	500	Not applicable
Archive size	$k$	10	40	20
Speed of convergence	$ξ$	0.1	2	0.75
Search locality	$q$	0.01	10	0.1
Number of ants per iteration	$N_{ants}$	1	5	3

Starting Thickness (nm)	Initial $χ^{2}$ Value	Final $χ^{2}$ Value	Estimated Thickness (nm)	Estimated $n_{1 x}; n_{1 y}$ $λ = 450 nm$	Estimated $n_{1 x}; n_{1 y}$ $λ = 650 nm$	Estimated $n_{1 x}; n_{1 y}$ $λ = 850 nm$
25	538	36	55.5	1.71; 1.72	1.86; 1.79	2.32; 2.19
50	949	36	55.5	1.71; 1.72	1.86; 1.79	2.32; 2.19
75	1172	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
100	531	111	123.3	1.69; 1.68	2.19; 2.16	2.06; 2.05
125	65	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
150	219	36	55.5	1.71; 1.72	1.86; 1.79	2.32; 2.19
175	872	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
200	1392	1.66	125.2	2.19; 2.13	2.08; 2.03	2.06; 1.99
225	1210	44	303.6	1.83; 1.79	1.87; 1.82	2.09; 2.04
250	593	51	295.1	1.83; 1.79	1.97; 1.90	2.11; 2.06
300	461	45	295.7	1.83; 1.79	1.96; 1.90	2.11; 2.06

Abstract

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Journal of the Optical Society of America B