AbstractA new method of solving Maxwell's equations is introduced for electromagnetic waves propagating in the steady state along the z axis of a cylindrical coordinate system (r, θ, z) in free space. We solve the wave equation for the electric field E completely for two degrees of freedom and find two methods for obtaining the magnetic field by rotE. Considering completeness of the solution, the electromagnetic fields are classified and summarized in two tables. As an application example, we study complete guided modes of a clad optical fiber. It is found that the electric field preferred boundary conditions of a circular dielectric structure, complete guidance of singular (cutoff) modes with angular index m ≧ 3 and analytical expressions physicially accurate of linearly polarized remote‐cutoff HEIn modes.In one of the two new methods, the wave equation for Ez is solved first. Next, from the Er equation, Eθ is eliminated in divE = 0 and variable coefficient linear partial differential equations for unknown Er with inhomogeneous term Ez are derived. After these equations are solved for Er in both homogeneous and inhomogeneous cases, Eθ derived from divE = 0 is evaluated by using the Eθ equation. In another method, we derive simultaneous angular equations from the Er and Eθ wave equations. After solving these, we evaluate Ez derived from divE = 0.