Vector Modes of a Ring-Core Optical Fiber: A Maxwell-Based Tutorial Derivation with Physical Interpretation

Ring-core optical fibers support guided electromagnetic fields whose energy can be localized in an annular region rather than near the fiber axis. Such structures are important for higher-order vector modes, annular guided fields, and vortex-compatible optical states. This tutorial presents a step-by-step Maxwell-based derivation of vector modes in a three-layer ring-core optical fiber consisting of an inner cladding, an annular core, and an outer cladding. The purpose is pedagogical rather than to introduce a new eigenmode theory. Starting from Maxwell’s equations in a source-free, lossless, nonmagnetic dielectric medium, we introduce time-harmonic fields, the longitudinal propagation factor exp(iβz), and the azimuthal dependence exp(ilφ). The curl equations are then written explicitly in cylindrical coordinates, yielding six coupled first-order equations for the electromagnetic field components. The system is reduced to the two longitudinal components E_z and H_z, which satisfy Bessel-type radial equations in each homogeneous layer. Regularity at the fiber axis, decay at radial infinity, and continuity of the tangential field components at the two interfaces lead to a homogeneous matrix equation M_l(β)x= 0. The characteristic condition det M_l(β)= 0 determines the allowed propagation constants and effective refractive indices of the guided vector modes. The physical meaning of β, n_eff, l, radial localization, and annular power confinement is discussed. Particular attention is given to the distinction between annular intensity and vortex phase: an l = 0 mode may be ring-shaped but is not a vortex mode, whereas l ≠ 0 modes possess azimuthal phase winding and are vortex-compatible. Representative numerical illustrations and supplementary code are provided to connect the analytical derivation with modal profiles, intensity maps, phase maps, and power localization in the annular core.