APPROXIMATION OF REFLECTION COEFFICIENTS FOR RAPID REAL-TIME CALCULATION OF INHOMOGENEOUS FILMS

Fast computer algorithms for calculating the reflection coefficients o f inhomogeneous films are developed. The applications of such algorith ms are aimed at real-time monitoring and control of the growth of grad ient index optical coatings. Inhomogeneous films with a refractive ind ex varying along the direction normal to the surface are generally tre ated by dividing the film into a large number of homogeneous layers an d using an iterative or transfer-matrix method to calculate reflectivi ties. We develop alternative methods. We show that the elements of the transfer matrix and the reflection coefficients can be written in ter ms of single, double, and higher-dimensional integrals over the films. In most cases it is sufficient to include only the first term, i.e., the single integrals, and the calculation of reflection coefficients i s then between 1 and 2 orders of magnitude faster than that for the tr ansfer-matrix and iteration methods. This gives the possibility of rea l-time determination of optical parameters of inhomogeneous layers for growth monitoring and opens up feedback control. (C) 1997 Optical Soc iety of America.