Study and synthesis of inhomogeneous optical modes by solving the wave and telegraph equations

We describe the synthesis and physical properties of inhomogeneous optical modes propagating in a dielectric medium. These modes have electric field components transverse and parallel to the propagation coordinate, respectively. The first satisfies the wave equation, and the second satisfies the telegraph operator's equation. The modal structure arises when both solutions couple with the same phase term along the propagation coordinate. An important result of the analysis is that both solutions share the same frequency representation, remaining in a circle, which presents a strong analogy with non-diffracting homogeneous beams. This frequency representation allows their experimental synthesis, which is obtained by illuminating, with interfered plane waves, a transmittance containing a circular slit. The plane waves have s-polarization; this type of illumination has an electric field component that runs parallel to the propagation coordinate, which is necessary to meet the electromagnetic boundary conditions. Experimental results and computational simulations are shown.