First-Order Grating Coupling Coefficients in Asymmetric Three-Layer Waveguides for Transverse Electric Modes

An equivalent surface current method is used to derive an analytic expression that approximates the coupling coefficient, reflectivity bandwidth, and group velocity for coupling between identical and nonidentical TE modes in an asymmetric three-layer waveguide with a sinusoidal grating at one waveguide interface. The analytic expression for the coupling coefficient agrees with expressions derived by two other methods for contradirectional coupling between identical modes. The results from the analytical expressions are compared to results from a numerically accurate Floquet-Bloch solution. The analytical expressions, which do not depend on which interface contains the grating, provide almost identical results obtained by the accurate solution for shallow grating depths and small index changes at the grating interface, a case typical of single-mode distributed feedback lasers. However, the accurate numerical solution, unlike the analytic solution, shows that in waveguides with only a large index step at the grating interface, increasing the grating depth can result in decreasing the coupling coefficient, a case typical of some distributed Bragg reflector lasers. In highly confined silicon photonic waveguides (large index steps at both interfaces), the analytic expression gives accurate results even for deep gratings. The derivation of the analytical expression for the coupling coefficient in this paper using the equivalent surface current method extends the application of the previous analytic formulas to nonidentical mode coupling where the forward and backward modes are not identical, which has application to gratings in multimode broadened waveguide lasers and amplifiers.