Solutions to the wave equation and Maxwell’s equations in cylindrical coordinates were found that satisfy the boundary conditions at the interface between an infinitely long absorbing or active cylinder and an arbitrary outside medium. These solutions are exponentially decreasing or increasing TE, TM, or hybrid waves traveling along the direction of the axis. The components of the complex Poynting vector make it possible to determine the time average of the excess of the inflow or outflow of radiant flux along each of the coordinate directions and over the surfaces of an arbitrary cylindrical volume element. These equations show for the first time, in the simpler case of TE and TM waves, that cylinders of large radii possess unique modes, which simultaneously exhibit stimulated emission and absorption along different coordinate directions. In each case, the pumped-in radiant flux either reappears as stimulated emission along other coordinate directions or disappears by absorption introduced by the conductivity and complex dielectric constant. The theory is of a linear, resonant type, and is therefore limited to weak pumping fields for which fluorescence is absent.
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