Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry

Abstract

Taking into account the characteristics of a free scalar field in elliptic coordinates, a new dynamical variable is found for the free electromagnetic field. The conservation law associated to this variable cannot be obtained by direct application of standard Noether theorem since the symmetry generator is of second order. Consequences on the expected mechanical behavior of an atomic system interacting with electromagnetic waves exhibiting such a symmetry are also discussed.

Control of atomic transition rates via laser-light shaping

“…Propagation invariant beams are nonlocalized waves (the electric fields do not become zero fast enough as |x| → ∞) so that the orthonormalization requirement (A1) gives rise to delta functions on the modulus of k ⊥ . Explicit results for TE and TM modes have been reported for Bessel [9], Mathieu [12], and Weber [15] modes.…”

“…(15) Notice that the superpositions depend on k. Quantization of structured modes in terms of vectorial beams [9] is usually more direct than with scalar modes due to the evident orthogonality between TE and TM modes, in contrast to the overlap between the scalar modes defined by Eqs. (11) and (12).…”

“…(b) In elliptic coordinates with an ellipticity parameter q and order m, they are known as Mathieu beams [11,12]: (20) where ce m (ϕ; q) and se m (ϕ; q) are Mathieu functions of order m in standard notation [13].…”

Dynamical constants of structured photons with parabolic-cylindrical symmetry

“…Another example corresponds to Mathieu modes which exhibit elliptical-cylindrical symmetry. Mathieu photons carry constant values for the balanced composition of the orbital angular momentum with respect to the foci of the elliptical coordinate system [6].…”

Single structured light beam as an atomic cloud splitter

“…Light beams with special intensity or phase structure yielding peculiar dynamical properties have been extensively used for the manipulation of microparticles [8] and cold atoms [9]. Most theoretical and experimental studies for cold atoms focus on using such light beams as wave guides for atom transport with the possibility of exchanging orbital angular momentum [9,10,11,12,13,14,15,16] or other dynamical properties [17] between the light and the atoms. We shall show that experimentally accessible structured light beams can also be used efficiently as atomic thermal cloud splitters.…”