Asymptotic Solutions Of The Leaky Modes And The Berenger Modes In Several Typical Waveguides Terminated By Perfectly Matched Layers

This thesis focuses on the solutions of the leaky modes and the Berenger modes in several typical waveguides terminated by perfectly matched layers. S-ince familiar numerical methods, such as the finite difference method and the finite element, method, will face a severe challenge in computing the propagation constants of which the absolute values are large, a method based on asymptotic expansions is employed. Once the absolute values of the propagation constants are large enough, these obtained asymptotic solutions are highly accurate, and get more accurate when the absolute values become larger.In Chapter 2 , a systematic asymptotic approach is developed to reobtain asymptotic solutions of the leaky modes and the Berenger modes in an isotropic circular waveguide terminated by a perfectly matched layer, and the asymptotic accuracies are improved from the leading order to the first and the second orders.In the derivations, the dispersion relation is firstly treated asymptotically using the asymptotic expansions of the Bessel functions for large arguments and the asymptotic properties of the phase angles of the propagation constants corre-sponding to the leaky modes and the Berenger modes. After that, the results for the HE leaky modes with and without magnetic contrast axe derived by the inverse power series, respectively. Then similar derivations for the HE Berenger modes and the EH Berenger modes are given and the main differences between the derivations for the Berenger modes and the leaky modes are also pointed out.At the end of this chapter, two examples are considered to compare the leading,the first and the second order asymptotic solutions with the exact propagation constants.In Chapter 3 , the method developed in Chapter 2 is extended to deriving the asymptotic solutions of the leaky modes and the PML modes in a three-layer anisotropic planar waveguide. Since the considered waveguide only supports the TE modes and the TM modes and the derivations for the TE modes are the same as in an isotropic waveguide, only the TM modes are derived asymptoti-cally. Because of the anisotropy, three cases should be considered to obtain the asymptotic solutions of the TM leaky modes. The derivations in the first two case are similar to the ones for the TE modes and the TM modes in an isotropic waveguide, and the methods both used there are employed together in the third case. Again, due to the anisotropy, two cases should be considered in the deriva-tions for both two sequences of the Berenger modes. In this chapter, the derived asymptotic solutions for the TM leaky modes and the TM Berenger modes are up to the fourth order or the sixth order. Three numerical examples are given to verify the effectiveness and the high accuracies of our results.In Chapter 4, the asymptotic approach given in Chapter 2 is also extended to an anisotropic circular waveguide terminated by a perfectly matched layer.According to the conditions of a perfect electric conductor resulted from the perfectly matched layer and the continuity conditions in the interface between the core and the cladding, the dispersion relation is obtained. Then the leading and the first order asymptotic approximations for the leaky modes and the Berenger modes are derived by using the asymptotic dispersion relations. In the derivations of the EH modes, two cases should be considered due to the anisotropy, one of which is similar and the other is totally different to the one in a isotropic waveguide. At the end of this chapter two numerical examples are considered.