Closed equation for the normal incidence reflectance of thin films on absorbing substrates (original) (raw)
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Applied Optics, 2006
A method for determining the optical constants and the thickness of weakly absorbing thin films on substrates is proposed. In this method only the reflectance and transmittance spectra obtained at a single arbitrary angle of incidence are used, provided that the former reveals several interference extrema. The calculation procedure is based on relatively simple relations suitable for the programmed realization and does not call for the prescription of the initial values of the parameters to be determined. The method proposed is fairly accurate and allows one to uniquely solve the inverse problem of spectrophotometry. The optical constants and the thickness of an As x Se y film formed on a glass substrate have been determined by the proposed method in the visible region of the spectrum.
Applied Optics, 1984
We describe a three-stage process for the determination of the optical constants n and k of thin absorbing films on the basis of reflectance, transmittance, and thickness measurements. The first stage uses a bivariate optimization based on the functions (1 i R)/T, R denoting a reflectance and T the transmittance. The second stage uses a phase-variate approach based on an algorithm for locating the complex zeros of analytic functions. The third stage uses an evolution curve, giving the single-wavelength reflectance as a function of film thickness. We illustrate the process using practical examples drawn from recent studies of films of amorphous silicon, hydrogenated carbon, and hydrogenated silicon.
Journal of Applied Physics, 2005
The present work considers the problem of estimating the thickness and the optical constants (extinction coefficient and refractive index) 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 (n,k) for absorbing thin films using reflectance measurements
Applied Optics, 1988
We propose a method for determination of the complex refractive index of absorbing materials either in bulk or film geometry by measuring its reflectivity when coated with a well-characterized transparent dielectric at two specific optical thicknesses: ndj = XO/4 and njdj = X 0 /2. The complex refractive index of the sample n =
Applied Optics, 2001
In this contribution a new efficient modification of a method that enables us to perform the optical characterization of nonabsorbing and weakly absorbing thin films without using the absolute values of the reflectances measured is presented. Namely, this modification is based on determining the values of the wavelengths corresponding to touching the spectral dependences of the reflectances of the studied films measured for several angles of incidence with the envelopes of maxima and minima of these spectral dependences. By means of combining the explicit formulas containing the wavelengths mentioned and the suitable iteration procedure one can evaluate the values of the thicknesses and spectral dependences of the refractive indices of the films analyzed in reliable and precise ways.
Journal of Applied Physics, 2012
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Optical peculiarities of thin absorbing films
OPTIKA '98: 5th Congress on Modern Optics, 1998
Optical parameters of thin semiconductor films in visible region are investigated with two different methods: use the new analogues of Fresnel's formulas for absorbing media for calculation of reflection and transmission [1] and, use the calculation of an index of refraction on a base of a phase change on reflection. Results of these methods agree well. Thickness dependencies of n(d) and k(d) of Ge,Si,Se, Te were obtained for the experimental data [2]. Curves n(d) have resonance maxima at thicknesses, which are less than light wavelength in a medium in it times. Curves
The optical constants of the metal thin films of Rhodium have been determined, the phase angles was determined using the measured spectral reflectance R(λ) by Kramers-Kronig relations. Then, the real part of the refractive index is calculated by the approach of Heavens when the film is highly absorbing, in that range the real refractive was found to be in order of the extinction coefficient k. The interference reflectance spectra at normal incidence for different thicknesses of amorphous metal films deposited by thermal evaporation have been obtained in the spectral range 400-800 nm. We propose a method for determination of the refractive index and extinction coefficients of highly absorbing films. This method is based on measurements of reflectance of the film at normal incidence alone, simulations of the theoretical accuracy and the effect of the error of the spectral reflectance measurements in the determination of the optical constants of the film are analyzed.