Abstract

Binary chriped gratings have been empirically found to demonstrate the focusing properties of a lens by many researchers but without rigorous theoretical analysis. This paper presents a theoretical analysis of plane wave diffractions by an arbitrarily chirped binary grating. The solution techniques include successive conformal mappings (SCM) and the vector coupled wave theory. The grating, being perfectly conducting, is uniform in one direction but the grating period varies slowly in the other (the grating vector) direction. Under these assumptions, the orthogonal decomposition of any electromagnetic fields in terms of the fast and the slow polarization is valid. Furthermore, the nonperiodic electromagnetic boundary conditions which have been introduced by the chirped grating profile are mapped into periodic ones with three conformal mappings: (1) the Schwarz-Christoffel transformation, (2) the linear fractional transformation, and (3) the natural logarithmic transformation. This newly mapped domain can be modeled as a smooth, conducting surface loaded by dielectrics with a periodic varying index. The Helmholtz wave equation is treated in this transformed domain where the conventional coupled wave analysis is exploited. As the SCM techniques were previously used for solving periodic grating diffraction problems,¹ this work extends its validity to nonperiodic grating structures.

© 1989 Optical Society of America