Asymmetric Cauchy-Riemann beams

We investigate, theoretically and experimentally, the evolution of a paraxial beam propagating in free space when its initial transverse structure is characterized by an asymmetric Gaussian modulation combined with an entire function. Utilizing a quantum optics operator approach, our study specifically examines the effects of parameter variations within the Gaussian modulation on two entire functions: the complex-valued Bessel function and the Airy function. Through this investigation, we aim to elucidate how these parameter variations influence the beam's propagation dynamics and the role played by the asymmetry of the Gaussian modulation in the propagation of such paraxial beams. Additionally, we derive an integral representation of the propagated field that enables efficient numerical computation of these optical fields. As a representative example, we numerically propagate the complex-valued Hermite polynomial. The method requires only that the input field modulation be an entire function and provides exact solutions to the paraxial wave equation with a reduced computational cost compared to standard approaches.