Simple derivative-free method of zero extraction by phase-based enclosure for determination of complex propagation constants in planar multilayer waveguides.

A rigorous and simple way to implement a numerical method for extracting the propagation constants and modal characteristics of lossless, lossy, and active planar multilayer waveguides is presented. The method can accurately determine the complex propagation constants of both guided and leaky modes. The method does not utilize the derivative of the dispersion function nor any integrals of it. It is based on finding the estimates of the zeros of the dispersion function by a systematic successively tighter enclosure of the zeros with exclusive use of only the phase of the dispersion function along varying rectangular contours in the complex plane. The zero estimates are then refined by Müller's iterative scheme employing deflation. The method was named derivative-free zero extraction by phase-based enclosure (DFZEPE) for brevity. The DFZEPE method has been applied to lossless, lossy, active, and antiresonant reflecting optical waveguides published in the literature and analyzed with similar methods, and the results are excellent. The proposed DFZEPE method has global convergence and can be applied to a plethora of electromagnetic (and other) problems where the zeros of an analytic function are sought.