Abstract
D.Ing. (Electrical and Electronic Engineering)
A theory was developed for the investigation of the optical properties of inhomogeneous
layered media. Reflectivity and transmissivity analysis of multi-layered structures was
realized by utilizing flow graph representations and by employing Mason's rule. This
study served as a base for the development of analytical expressions in integral form for
reflectivity, transmissivity, reflectance, bilinear transformed reflectance and transmittance
of materials possessing inhomogeneous refractive index profiles. These proposed
formulas were derived for both normal and oblique incidence and contemplate
nonabsorbing, as well as, absorbing materials. An ellipsometric expression for
inhomogeneous layers was also derived by employing the developed theory.
Several hypothetical examples that emulate refractive index profiles in ionimplanted
semiconductors were investigated, including a buried layer with a gaussian
refractive index profile, and two homogeneous layers with a half-gaussian transition
region between them. Curves of reflectance versus wave number were simulated using
the derived formulations in two different ways: (i) employing numerical methods (ii)
applying analytical solutions.
The performance of these simulations was compared to standard techniques such
as the matrix method and the Wentzel-Kramers-Brillouin (WKB) approximation. Very
good agreement between the proposed theory and the matrix technique was found. The
developed formulations were appropriate even at wave numbers where the WKB
approximation was not valid. It must be stressed that the analysis of the reflectance at
these wave numbers is important in the study of processed semiconductors. In
comparison to the matrix technique, the integral formulation led to substantial time saving,
which, depending on the particular application, was between one and two orders
of magnitude faster. This fact indicated that the developed expressions for reflectance
and transmittance can be used to great advantage in least-square curve-fit ...