Waveguides with continuously varying cross-sections are often used in modern microwave, millimeter and terahertz wave technologies as phase changers, tapers, transformers, couplers, transitions and so forth. Theoretical foundations of such waveguides have been developed a long time ago [1]–[3]. They exploit vector expansions (two dimensional [1], [2] or three-dimensional [3]) in which fields of a regular waveguide (reference waveguide) serve as basis functions. This reduces the problem to the infinite set of one-dimensional differential equations. Such a methodology has a very evident physical interpretation and is very convenient for analysis both propagation and excitation problems in irregular waveguides. Therefore it is still widely used in both numerical codes [4] and separate specific calculations [5] along with purely numerical approaches developed latter [6]. However, significant theoretical and practical difficulties appear for irregular waveguides with lossy walls. In this case basis functions used in [1]–[3] (which are fields of regular lossless waveguides) do not satisfy boundary conditions resulting in non-uniform convergence and Gibbs instability of numerical calculations. Due to recent progress in generation of terahertz radiation [7,8] analysis of lossy waveguide structures becomes particularly relevant. For instance, Ohmic quality factors of terahertz gyrotron cavities are comparable with diffraction ones and the level of energy dissipation inside the cavities can achieve 65–85 %. Therefore, to get an adequate theoretical analysis one should take wall and radiation losses in to account. From the other hand, in the terahertz range δk ≪ 1 still holds (δ is the skin depth of the wall material and k is the free space wavenumber) and one should expect that use of fields of regular lossless waveguides as basis functions can be still efficient. The attempt to extend the abovementioned methodology [1]–- [3] to lossy waveguide structures was made in [4]. If was found that the direct application of vector expansions similar [1]–[3] is associated with significant mathematical difficulties. Practical results were obtained only in the simplest case of a circular waveguide provided for losses are small or the waveguide of constant radius. The alternative approach to the analysis of irregular lossy waveguides is developed in [9]. It based on using the curvilinear coordinate system (which transforms the irregular waveguide to the regular one but with a non-uniform anisotropic filling). It is rather cumbersome both from analytical and numerical points of view. Besides, it is applicable only in cases of simplest cross-sections (circular, coaxial and some others).
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