Application of quaternions to the analysis of electromagnetic propagation through curved waveguide

Summary form only given. The notion of quaternions was conceived in the 19th century by the Irish theoretical Physicist G. Hamilton. The initial motivation was a mathematically compact expression of vector rotations in a four-dimensional space. Quaternions became a subject of theoretical algebra, but they were also used in problems of theoretical physics. In electromagnetics, quaternions have been used very rarely, and always in the form of bicomplex numbers. Apparently they are very useful in the analysis of curved waveguides, since they facilitate the decoupling of the associated vector Helmholtz equations. The standard decoupling technique invokes the definition of two new vector fields employing the second imaginary unit j, apart from the standard imaginary unit i. In more detail, if the components of the electric and magnetic field are respectively E₁, E₂, E₃ and H ₁, H₂, H₃, where index 3 corresponds to the longitudinal component, then the following transverse bicomplex fields can be defined: E_t≡E₁+jE₂, H_t≡H₁+jH₂ along with the differential operator D≡1/h₁ ∂/∂u₁+j1/h₂ ∂/∂u₂ where u₁ and u₂ are the transverse coordinates in the geometry and h₁, h₂ are the corresponding metric coefficients. A modification of this method invokes the bicomplex field F, which includes components of both the electric and magnetic fields, defined by F≡E+jZH where Z is the medium intrinsic impedance. The F field technique has been successfully utilized in the analysis of toroidal dielectric waveguides with elliptical cross-section. In this paper, both versions of the quaternion methodology are extended to metallic or dielectric curved waveguides with various cross-sections and bending schemes