Least-Squares Analysis of the Film-Substrate Problem in Ellipsometry*†

D. H. Loescher; R. J. Detry; M. J. Clauser

doi:10.1364/JOSA.61.001230

Least-Squares Analysis of the Film-Substrate Problem in Ellipsometry

D. H. Loescher, R. J. Detry, and M. J. Clauser

Journal of the Optical Society of America
Vol. 61,
Issue 9,
pp. 1230-1235
(1971)
•https://doi.org/10.1364/JOSA.61.001230

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D. H. Loescher, R. J. Detry, and M. J. Clauser, "Least-Squares Analysis of the Film-Substrate Problem in Ellipsometry*†," J. Opt. Soc. Am. 61, 1230-1235 (1971)

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History
- Original Manuscript: January 8, 1971
- Published: September 1, 1971

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Abstract

We have found that the thickness and refractive indices for the film–substrate problem in ellipsometry can frequently be calculated with a least-squares technique. It is shown that one algorithm can be used with a variety of data to solve problems that were formerly solved with step-by-step searches through parameter space. Moreover, the least-squares technique leads to a statistically meaningful estimate of the variances of the parameters. It is shown that the variances obtained from a correct error analysis can be at least one hundred times as large as those obtained from a naive analysis. It is also shown how the error analysis can provide information that is helpful in deciding if a given experiment is worthwhile.

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	Parameters

Type	d (nm)	n_f	n_s	k_s	Sum of error
Intercept d	26.007	1.8000	1.8000	1.8000	1.0
Intercept n_f	26.000	1.8002	1.8000	1.8000	0.7
Intercept n_s	26.000	1.8000	1.8005	1.8000	1.2
Intercept k_s	26.000	1.8000	1.8000	1.7995	1.2

Minimum	26.000	1.800	1.800	1.800	0.0

Max. d	26.5	1.783	1.807	1.784	1.9
Test	26.5	1.800	1.800	1.800	5.0 × 10³
Max. n_f	25.6	1.823	1.808	1.823	6.3
Test	26.0	1.823	1.800	1.800	9.2 × 10³
Max. n_s	26.7	1.809	1.822	1.812	1.0
Test	26.0	1.800	1.822	1.800	2.3 × 10³
Max. k_s	26.3	1.777	1.789	1.776	2.4
Test	26.0	1.800	1.800	1.776	2.7 × 10³

Case	Iteration	d(nm)	n_f	n_s	k_s
A-1	Initial	30.5	1.60	0.90	6.2
	50	30.0 ± 1.7	1.65 ± 0.4	1.0 ± 0.7	6.0 ± 2.6
A-2	Initial	29.7	1.70	1.10	5.8
	21	30.0 ± 1.7	1.65 ± 0.4	1.0 ± 0.7	6.0 ± 2.6
B-1	Initial	30.5	1.60	0.90	6.2
	10	30.0 ± 0.9	1.65 ± 0.10	1.00 ± 0.2	6.0 ± 0.7
B-2	Initial	29.7	1.70	1.10	5.8
	11	30.0 ± 0.9	1.65 ± 0.10	1.00 ± 0.2	6.0 ± 0.7
True	Minimum	30.0	1.65	1.00	6.0

	Parameters

Type	d (nm)	n_f	n_s	k_s	Sum of error
Intercept d	26.007	1.8000	1.8000	1.8000	1.0
Intercept n_f	26.000	1.8002	1.8000	1.8000	0.7
Intercept n_s	26.000	1.8000	1.8005	1.8000	1.2
Intercept k_s	26.000	1.8000	1.8000	1.7995	1.2

Minimum	26.000	1.800	1.800	1.800	0.0

Max. d	26.5	1.783	1.807	1.784	1.9
Test	26.5	1.800	1.800	1.800	5.0 × 10³
Max. n_f	25.6	1.823	1.808	1.823	6.3
Test	26.0	1.823	1.800	1.800	9.2 × 10³
Max. n_s	26.7	1.809	1.822	1.812	1.0
Test	26.0	1.800	1.822	1.800	2.3 × 10³
Max. k_s	26.3	1.777	1.789	1.776	2.4
Test	26.0	1.800	1.800	1.776	2.7 × 10³

Case	Iteration	d(nm)	n_f	n_s	k_s
A-1	Initial	30.5	1.60	0.90	6.2
	50	30.0 ± 1.7	1.65 ± 0.4	1.0 ± 0.7	6.0 ± 2.6
A-2	Initial	29.7	1.70	1.10	5.8
	21	30.0 ± 1.7	1.65 ± 0.4	1.0 ± 0.7	6.0 ± 2.6
B-1	Initial	30.5	1.60	0.90	6.2
	10	30.0 ± 0.9	1.65 ± 0.10	1.00 ± 0.2	6.0 ± 0.7
B-2	Initial	29.7	1.70	1.10	5.8
	11	30.0 ± 0.9	1.65 ± 0.10	1.00 ± 0.2	6.0 ± 0.7
True	Minimum	30.0	1.65	1.00	6.0

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Equations (12)

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