Calculation of thin-film optical constants by transmittance-spectra fitting

Milen Nenkov; Tamara Pencheva

doi:10.1364/JOSAA.15.001852

Journal of the Optical Society of America A
Vol. 15,
Issue 7,
pp. 1852-1857
(1998)
•https://doi.org/10.1364/JOSAA.15.001852

Calculation of thin-film optical constants by transmittance-spectra fitting

Milen Nenkov and Tamara Pencheva

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Milen Nenkov and Tamara Pencheva, "Calculation of thin-film optical constants by transmittance-spectra fitting," J. Opt. Soc. Am. A 15, 1852-1857 (1998)

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Abstract

Our investigation uses the film-interference transmittance spectrum in the computer calculation of thin-film thickness $t$ and complex refractive index $n^{*} = n - ik .$ Titanium oxide films of different thicknesses are studied, and a new approach to determine the film thickness and optical constants is proposed. This new approach is based on the use of numerical optimization methods in transmittance-spectra fitting. Various dispersion equations of the finite-power-series type are used to obtain the best fit between transmittance measurements and calculations. The best-fit results for $n (λ)$ and $k (λ)$ are found to agree with dispersion relations that follow from a quantum theory of light absorption.

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$λ_{extr}$ (µm)	Initial Data			Final Results
$λ_{extr}$ (µm)	$T_{S}$	$T_{max}$	$T_{min}$	$m$	$n$	$k$
0.415	0.920	0.650	0.491	4.5	2.430	0.0251
0.465	0.923	0.816	0.570	4.0	2.375	0.0099
0.507	0.925	0.870	0.655	3.5	2.340	0.0054
0.585	0.925	0.921	0.670	3.0	2.295	0.0005
0.700	0.909	0.906	0.685	2.5	2.254	0.0004

$λ_{extr}$ (µm)	Initial Data			Final Results
$λ_{extr}$ (µm)	$T_{S}$	$T_{max}$	$T_{min}$	$m$	$n$	$k$
0.490	0.924	0.910	0.610	2.0	2.393	0.0027
0.640	0.919	0.900	0.670	1.5	2.277	0.0026
0.795	0.908	0.900	0.720	1.0	2.220	0.0023

Constants in Eqs. (4)	Step Number $j$
Constants in Eqs. (4)	3	4	5	6
$a_{nj}$	2.19876	2.20606	2.20652	2.20653
$b_{nj} (μ m^{2})$	$- 0.02626$	$- 0.00584$	$- 0.00579$	$- 0.00579$
$c_{nj} (μ m^{4})$	0.01159	0.00259	0.00303	0.00303
$d_{nj} (μ m^{6})$	—	0.00094	0.00068	0.00068
$e_{nj} (μ m^{8})$	—	—	0.00003	0.00003
$f_{nj} (μ m^{10})$	—	—	—	$- {3.010}^{- 11}$
$a_{kj}$	0.03906	0.00270	0.00229	0.00229
$b_{kj} (μ m^{2})$	$- 0.02619$	0.00395	0.00333	0.00333
$c_{kj} (μ m^{4})$	0.00432	$- 0.00343$	$- 0.00275$	$- 0.00275$
$d_{kj} (μ m^{6})$	—	0.00062	0.00044	0.00044
$e_{kj} (μ m^{8})$	—	—	0.00001	0.00001
$f_{kj} (μ m^{10})$	—	—	—	${2.810}^{- 8}$
$t_{j}$ a (µm)	0.3926	0.3939	0.3924	0.3924
$δ T$ b (%)	1.503	0.998	0.984	0.984

Result	Parameter
Result	$A$	$B$ (eV)	$C$ $({eV}^{2})$	$E_{g}$ (eV)	$n (\infty)$	$t$ (µm)	$δ T$ (%)
Initial	0.060	7.69	15.6	2.10	2.24	0.3926	5.9
Final	0.0467	8.191	16.93	2.047	0.020	0.3877	1.24

$λ_{extr}$ (µm)	Initial Data			Final Results
$λ_{extr}$ (µm)	$T_{S}$	$T_{max}$	$T_{min}$	$m$	$n$	$k$
0.415	0.920	0.650	0.491	4.5	2.430	0.0251
0.465	0.923	0.816	0.570	4.0	2.375	0.0099
0.507	0.925	0.870	0.655	3.5	2.340	0.0054
0.585	0.925	0.921	0.670	3.0	2.295	0.0005
0.700	0.909	0.906	0.685	2.5	2.254	0.0004

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Journal of the Optical Society of America A