Exponential beams of electromagnetic radiation**We dedicate this paper to the memory of Edwin Power who recognized in his book [1] the power of Riemann–Silberstein vector although he has not been aware of its early history.

Abstract

We show that in addition to well-known Bessel, Hermitte–Gauss and Laguerre–Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of ρ. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann–Silberstein vector and the Whittaker representation of the solutions of the Maxwell equations in terms of just one complex function. The connection between the Bessel beams and the exponential beams is made explicit by constructing the exponential beams as wave packets of Bessel beams.