AbstractsPhysics

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 ...