# Magnetooptics in Cylindrical Structures
## 1 Introduction  
Electromagnetic waves in magnetized media depend on magnetization and often show nonreciprocal propagation. The phenomenon is exploited in sensors and devices such as waveguide isolators, phase shifters, circulators, and modulators. Their operation can be explained by considering circularly polarized (CP) transverse electromagnetic waves of opposite handedness (\(\pm\)) propagating in infinite lossless uniformly magnetized media. In such media, the propagation vectors, \(\mathbf{k}_{\pm}\), slightly differ in magnitude and are both oriented parallel to the medium magnetization \(\mathbf{M}\), i.e., \(\mathbf{k}\|\mathbf{M}\). Then the CP waves represent eigenmodes, one pair in forward direction and another one in reversed direction [1]. At oblique wave incidence in perpendicularly magnetized multilayers or in planar, cylindrical or channel waveguides with axial \(\mathbf{M}\), the eigenmodes are no more CP waves [2; 3; 4; 5; 6; 7]. In circular cylindrical waveguides, the eigenmodes can only be approximated by decoupled CP waves, particularly in the weak guidance limit [8; 9; 10; 11; 12; 13]. The analysis of magnetic waveguide structures is often complex and, in most cases, requires numerical methods [14; 15; 16]. In the microwave region, the effect of \(\mathbf{M}\) is deduced from the tensor nature of magnetic permeability with the scalar electric permittivity [17] while in the near infrared and visible regions, the analysis assumes the tensor nature of electric permittivity with magnetic permeability reduced to its vacuum value [18; 19]. In both cases, the diagonal and off-diagonal tensor elements are even and odd functions of \(\mathbf{M}\), respectively [18; 19; 20; 21; 22; 23]. In the infrared and visible spectral regions, a more rigorous treatment would account for the tensorial nature of both material parameters [24; 25; 26; 27; 28; 29]. The present work provides an analytical approach to the problems of circular cylindrical structures displaying magnetooptic (MO) activity induced by axial magnetization. The approach covers the frequency range from microwave to optical frequencies. The \(i\)-th medium in a cylindrically layered structure is characterized by the magnetic permeability, \(\bar{\mu}^{(i)}\), and electric permittivity, \(\bar{\varepsilon}^{(i)}\), tensorsfor axial \(\mathbf{M}^{(i)}\). The work represents an extension of a previous study on the Helmholtz equation in isotropic cylindrical structures [30][PERSON] The final expressions provide insight into the trends and the symmetry which are difficult to appreciate in the use of numerical methods. To the best of the author's knowledge, the problem was not treated before with the exception of a previous study on the Faraday effect in optical fibers characterized by the gyroelectric dielectric permittivity tensors published by [PERSON] [12]. The study assumed CP eigenmodes, an approximation justified to some extent only in the weak guiding limit. The presentation is organized into five sections. The Helmholtz vector wave equations deduced from the Maxwell equations in transverse circularly polarized (TCP) representation provide electric, \(\mathbf{E}^{(i)}\), and magnetic, \(\mathbf{H}^{(i)}\), fields (Sections 2 and 3). With the restriction to terms linear in the off-diagonal elements of \(\tilde{\mathbf{z}}^{(i)}\) and \(\tilde{\mathbf{\mu}}^{(i)}\), these can be expressed analytically as functions of circular cylindrical coordinates \(\varrho\), \(\varphi\), and \(z\). The analysis results are applied to the simplest cylindrically layered structure, an optical fiber with step refractive index profile, formed by a uniform core and a uniform cladding. The conditions for waveguiding in the structure follow from the boundary conditions for \(\mathbf{E}^{(i)}\) and \(\mathbf{H}^{(i)}\) at the core--cladding interface. These provide eigenvalue equations for pairs of longitudinal propagation constants, \(\beta_{\pm}\). The difference \(\beta_{+}-\beta_{-}\) characterizes the nonreciprocity.