Imaging of localized whispering-gallery-modes in a cylindrical fiber

Modes of a dielectric cylindrical waveguide are characterized by three numbers: azimuthal number m, radial number, s, and a continuous propagation constant β.¹ The azimuthal number determines the field dependence of the azimuthal angle φ of the cylindrical coordinate system with the polar axis along the axis of the cylinder in the form of exp (imφ). The radial number characterizes the behavior of the field in the radial direction and in an ideal cylinder is given by a cylindrical Bessel function J_m (κ_m,sr). Discrete values of the parameter κ_s characterizing the radial dependence of the field is determined by Maxwell boundary conditions. The propagation constant β describes the propagation of the mode along the axis of the fiber (Z-axis of the cylindrical coordinate system). The frequency of the modes ω is related to the parameters β and κ_m,s via a standard relation.