Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data

Abstract

The present work considers the problem of estimating the thickness and the optical constants ͑extinction coefficient and refractive index͒ of thin films from the spectrum of normal reflectance R. This is an ill-conditioned highly underdetermined inverse problem. The estimation is done in the spectral range where the film is not opaque. The idea behind the choice of this particular spectral range is to compare the film characteristics retrieved from transmittance T and from reflectance data. In the first part of the paper a compact formula for R is deduced. The approach to deconvolute the R data is to use well-known information on the dependence of the optical constants on photon energy of semiconductors and dielectrics and to formulate the estimation as a nonlinear optimization problem. Previous publications of the group on the subject provide the guidelines for designing the new procedures. The consistency of the approach is tested with computer-generated thin films and also with measured R and T spectral data of an a-Si: H film deposited onto glass. The algorithms can handle satisfactorily the problem of a poor photometric accuracy in reflectance data, as well as a partial linearity of the detector response. The results on gedanken films and on a-Si: H indicate a very good agreement between expected and retrieved values.

Determination of Optical Properties of Undoped Amorphous Selenium (a-Se) Films by Dielectric Modeling of Their Normal-Incidence Transmittance Spectra

“…Therefore, in principle, two independent optical measurements are necessary at each spectral wavelength to solve for the unknowns and but, in practice, the true geometric thickness of the film is not known in prior and is thus to be determined at the same time-that is, three unknowns to be retrieved from two independent optical measurements at the same spectral wavelength, which demands adequate model suppositions or an otherwise optical analytical approach (Tan, 2006;Tan et al, 2007;Tan et al, 2006;Vautier et al, 1988;Bettstelleret al, 1993;Mulama et al, 2014;Benkhedir, 2006;Solieman et al, 2014;Adachi & Kao1980;Kotkata & Abdel-Wahab, 1990;Kotkata et al, 2009;Navarrete et al, 1990;Solieman & Abu-Sehly, 2010;Nagels et al, 1995;Tichy et al, 1996;Nagels et al, 1997;Tauc, 1972;Tauc, 1979;Heavens, 1991;Azzam & Bashara, 1987;Ward, 1994;Dragoman & Dragoman, 2002;Stenzel, 2005;Joo et al, 1999;Dobrowolski et al, 1983;Klein et al, 1990;Kukinyi et al, 1996;Chiao et al, 1995;Case, 1983;Peng & Desu, 1994;Birgin et al, 1999;Mulato et al, 2000;Erarslan & Gungor, 2010;Ventura et al, 2005;Manifacier et al, 1976;Swanepoel, 1983;Minkov, 1990;El-Naggar et al, 2009;…”

“…Only when a global minimum solution for the curve-fitting problem is achieved, one obtains reliable and informative dielectric and optical parameters for the studied film. If the substrate in the {air/thin uniform film/thick substrate/air}-stacking is transparent ( ≅ 0) in the spectral region above the absorption edge of the film, the -formula given in Equation (4) reduces to a wieldy expression that forms the basis of the analytical envelope and PUMA methods (Birgin et al, 1999;Mulato et al, 2000;Erarslan & Gungor, 2010;Ventura et al, 2005;Manifacier et al, 1976;Swanepoel, 1983;Minkov, 1990;El-Naggar et al, 2009;Epstein et al, 1987;González-Leal et al, 1998;Tigau, 2006;Poelman & Smet, 2003;Swanepoel, 1984;Chambouleyron & Martínez, 2001;Truong & Tanemura, 2006;Kasap & Capper, 2006), the application of which do not necessitate in prior dispersion functions for the film's optical constants.…”

“…When a global optimum solution of the fitting problem is realized, the output parameters will be evocative and physically meaningful; otherwise, local multi solutions are obtained, and hence the output parameters must be identified with utmost care to reject vulnerable ones and select those yield reliable solution, a job that is virtually challenging unless it is assisted by related independent findings. The Pointwise Unconstrained Minimization Approach (PUMA) numerical method (Birgin et al, 1999;Mulato et al, 2000;Erarslan & Gungor, 2010;Ventura et al, 2005) and algebraic methods (Peng & Desu, 1994;Birgin et al, 1999), based on approximate equations derived from reasonable assumptions, were widely used to extract spectral dispersion of optical constants of a film and its geometric thickness from normal-incidence -and/or -spectra of its air-supported {thin film/thick substrate}-stack, both of which do not require in prior a dispersion functional model. An algebraic method in this category is the envelope method, developed and refined by a number of authors ( Manifacier et al, 1976;Swanepoel, 1983;Minkov, 1990;El-Naggar et al, 2009;Epstein et al, 1987;González-Leal et al, 1998;Tigau, 2006;Poelman & Smet, 2003), whose viability is based on the presence of many interference-fringe maxima and minima on transmittance (reflectance) spectra in transparent and weak absorption regions of the film, and cannot be applied to spectra displaying few interference fringes.…”

Determination of thin film refractive index and thickness by means of film phase thickness

“…Another disadvantage of the envelope methods is the usage of interpolated points (not really measured) between two successive measured maxima or minima [1,2]. Curve fitting methods apply numerical optimisation to fit a theoretical model to measured data in order to find dispersion (λ) and thickness [9][10][11][12][13][14][15]. These methods need specific and flexible software environment for the calculations because of the variety of dispersion equations in literature [9].…”

“…Due to interference of light, an oscillating structure with maxima and minima is typical for thin films optical transmittance/reflectance spectra. Optical parameters of thin films can be extracted from optical spectra using several conventional methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Most precise information for a thin film may be obtained from transmittance spectra measurements at normal incidence of light [9], because these measurements do not depend on the properties of etalon mirrors and angle of incidence of light (as in case of reflectance measurements).…”