Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures

We present an analytical method for solution of one-dimensional optical systems, based on the differential transfer matrices. This approach can be used for exact calculation of various functions including reflection and transmission coefficients, band structures, and bound states. We show the consistency of the WKB method with our approach and discuss improvements for even symmetry and infinite periodic structures. Moreover, a general variational representation of bound states is introduced. As application examples, we consider the reflection from a sinusoidal grating and the band structure of an infinite exponential grating. An excellent agreement between the results from our differential transfer-matrix method with other methods is observed. The method can be equally applied to one-dimensional time-harmonic quantum-mechanical systems.