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We describe a new class of propagation-invariant light beams with Fouriertransform given by an eigenfunction of the quantum mechanical pendulum. Thesebeams, whose spectra (restricted to a circle) are doubly-periodic Mathieufunctions in azimuth, depend on a field strength parameter. When the parameteris zero, pendulum beams are Bessel beams, and as the parameter approachesinfinity, they resemble transversely propagating one-dimensional Gaussianwavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of anoperator which interpolates between the squared angular momentum operator andthe linear momentum operator. The analysis reveals connections with Mathieubeams, and insight into the paraxial approximation.

Propagation-invariant beams with quantum pendulum spectra: from Bessel beams to Gaussian beam-beams