Numerical Method for Electromagnetic Wave Propagation Problem in a Cylindrical Anisotropic Waveguide with Longitudinal Magnetization

The propagation of monochromatic electromagnetic waves in metal circular cylindrical dielectric waveguide with longitudinal magnetization filled with anisotropic inhomogeneous waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. Numerical results are obtained using a modification of the projecting methods. The comparison with known exact solutions (for particular values of parameters) are made.


Introduction
A large class of vector electromagnetic problem concerns electromagnetic wave propagation. The constitutive parameters ε and µ of standard dielectric and magnetic media are determined by their physical structure. However, the media with unusual properties are often required which can be obtained using either dielectrics that are uniform or partially filled. The parameters of such media depend on the mutual position of the particles and may be anisotropic [1]. It is known also that the permittivity of a dielectric (or the permeability of a magnetic) may depend on the radial coordinate [2]. The primary goal here is to construct a numerical method to determine the spectrum of normal electromagnetic waves that propagate in such structures.
Numerical methods for calculating the parameters of various types of waveguide structures are described in the monographs and review papers [3,4,5,6]. However, it should be said that most of the methods applied to homogeneous waveguides, are not common and are difficult to implement and apply for specific inhomogeneous and/or anisotropic structures.
In this work the wave propagation in inhomogeneous metal-dielectric anisotropic cylindrical waveguides is studied numerically using the modification of the projection methods [7].

Statement of the problem
Consider three-dimensional space R 3 with a cylindrical coordinate system Oρφz filled with isotropic medium having constant permittivity ε = ε 0 (ε 0 > 0 is the permittivity of free space), and constant permeability µ = µ 0 ( where µ 0 > 0 is the permeability of free space).
A metal dielectric circular cylindrical waveguide Σ filled with anisotropic inhomogeneous medium is placed parallel to the axis Oz. The waveguide Σ has a cross section and its generating line (the waveguide axis) is parallel to the axis Oz (see. Fig. 1). We will consider monochromatic waves where ( · ) T denotes the transpose operation. Each component of the field E, H is a function of three spatial variables. Complex amplitudes of the electromagnetic field E, H satisfy the Maxwell equations subject to the following boundary conditions. The tangential components of the electric field vanish on the metal surface ρ = r 0 ; tangential field components are continuous on the media interface ρ = r; the complex amplitudes obey the radiation condition at infinity: the electromagnetic field decays as O(|ρ| −1 ) when ρ → ∞. The permittivity ε inside the waveguide is constant; the permeabilityμ is specified by the expressionμ where µ ρ (ρ), µ φ (ρ) and µ z (ρ) are sufficiently smooth functions which depend on the radial coordinate ρ. The surface waves propagating along the axis Oz of the waveguide Σ have the form [8] where γ is the real propagation constant (spectral parameter of the problem). In what follows we often omit the arguments of functions when it does not lead to misunderstanding.

Differential equations
Inside the waveguide µ =μ and ε = ε. Substituting E and H with components (3) into equations (1), we obtain where the prime denotes differentiation w.r.a ρ. Expressing the functions E ρ , E z , H ρ and H z through E φ and H φ from the 1st, 3rd, 4th and 6th equation of system (4), we find Substituting the expressions for E ρ , E z , H ρ and H z into the 2nd and 5th equations of system (5) and introducing the notation Outside the waveguide, where µ = µ 0 = 1 and ε = ε 0 = 1, system (1) takes the form of Bessel's equations with general solutions where k 2 1 = γ 2 − ω 2 , I 1 is the modified Bessel function and K 1 is the Macdonald function [9] and C 1 , ..., C 4 denote arbitrary constants.
The solutions (7) takes a form where the radiation condition at infinity is taken into account.

Transmission conditions and transmission problem
Tangential components of the electromagnetic field are known to be continuous at the interface. In this case the tangential components are E φ , E z , H φ and H z . Thus we obtain the following transmission conditions for u e and u m The main problem considered in this study is formulated as follows: Problem P : to findγ such that there exist non-trivial functions u e (ρ;γ) and u m (ρ;γ) satisfying system (6), transmission conditions (9), and having the form (8) outside the waveguide.

Variation formulation
Let us give the variational formulation of the problem P . Using the first Green's formula, we obtain where From (11) taking into account (12) and (13), we obtain the variational equation which hold for any test functions v e and v m . The solution of (15) is equivalent to the original problem P .

Projection method
Using the projection method [10] let us reduce the variational equation (15)  These subintervals we call base finite elements.
In accordance with the scheme of the projection method, it is necessary to introduce basis functions ϕ i and ψ j in order to approximate the solution. The basis functions are defined on each subinterval Φ i and Ψ j (ϕ i and ψ j vanishes outside the intervals Φ i and Ψ j , respectively).

The basis functions
The basis functions ψ i defined on Φ i are Such defined basis functions takes into account the physical nature of the problem under consideration.
We assume an approximate solution with real coefficients α i and β j such that Substituting functions u e and u m with representations (16) into the variational equation (15), we obtain a system of linear equations with respect to α i and β j (for fixed value of γ) where matrices A(γ) and x have the form Note 2. If there exists γ = γ such that ∆( γ) = 0, then γ is an approximate spectral parameter of Problem P . In other words, if an interval [γ, γ] is such that ∆(γ) × ∆(γ) < 0, then this means that there exists γ = γ ∈ [γ, γ] which is an spectral parameter of Problem P . This value can be calculated with any prescribed accuracy.

Numerical results
The results of the numerical solution of the problem of propagating electromagnetic waves of an anisotropic magnetic waveguide structure are presented. Numerical results are obtained with the help of the shooting method. Radii of the waveguide (internal and external) r 0 = 2 cm, r = 4 cm, permittivity ε = 4. The values of the tensor componentsμ, are shown in the figure captions.
Numerical analysis of the behavior of dispersion curves (graphs of the dependence of the propagation constant γ on the circular frequency ω) is performed for the different components of tensorμ. In the case of µ φ → 0, the number of hybrid modes coincides with the sum of the "polarized" modes (TE and TM), the dispersion curves for µ φ = 0 (Fig.  2) coincide with the known dispersion curves for problems on propagating TE-and TM-polarized waves of an metaldielectric waveguide [11]. Figure 3 and Figure 4 shows the dispersion curves for the case when components of tensorμ are functions.

Conclusion
This work continues the investigation of the spectrum of metal dielectric waveguides with inhomogeneous filling. The paper [7] presents a numerical method for solving the problem of propagating waves of a dielectric waveguide. This method was used to numerically study the spectrum of a waveguide filled with an inhomogeneous anisotropic magnetic medium (ferrite). The method allows us to determine approximate eigenvalues with any prescribed accu-racy. The approach described in this paper can be applied to other problems, e.g., to multilayered opened waveguides.