In this paper, we present the formalism of the transverse operator method (TOM). A rigorous study of propagation in anisotropic and multi-layer medium using the tensor character of the permeability and permittivity is presented. With the application of the Galerkin method, the propagating modes in metallic rectangular waveguides filled with anisotropic metamaterial are exploited. The results are compared with those previously published and show a good agreement. The complex modes have been obtained.The advantages of the techniques used in this paper lie in the proper analytical formulation of the problem studied, on the one hand, and the speed of convergence, on the other hand. TOM offers a fast convergence of the propagation constant. This shows the effectiveness of our numerical model. As such, the formulation of the transverse operator could be a useful tool for microwave engineers. This type of materials known as metamaterial is widely used and needed by industries and information technology, especially in microwave and radiofrequency devices such as patch antennas, antennas waveguides, resonators, circulators, isolators, phase shifters, and filters.

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This paper presents an extension of the formulation of wave propagation in transverse electric (TE) and transverse magnetic (TM) modes in the case of metallic circular waveguides filled with anisotropic metamaterials. The determined higher-order modes have been analyzed and exploited to the design of filters. Among the particularities of anisotropic material, the backward waves can propagate below the cut-off frequency. The numerical results for TE and TM modes have been compared with theoretical predictions. Good agreements were obtained. We analyzed a periodic structure containing waveguides filled with anisotropic metamaterial using the mode-matching technique. By using modal analysis, our approach reduced considerably the computation time compared to HFSS.