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In quantum mechanics, the space-fractional Schr\"{o}dinger equation providesa natural extension of the standard Schr\"{o}dinger equation when the Browniantrajectories in Feynman path integrals are replaced by Levy flights. Here anoptical realization of the fractional Schr\"{o}dinger equation, based ontransverse light dynamics in aspherical optical cavities, is proposed. As anexample a laser implementation of the fractional quantum harmonic oscillator ispresented, in which dual Airy beams can be selectively generated under off-axislongitudinal pumping.

Fractional Schr\"{o}dinger equation in optics