Cauchy-Riemann beams

Leveraging operator techniques, we address the paraxial wave equation governing a field formed by the multiplication of a Gaussian function and an entire function; notably, the latter adheres to the Laplace equation, 𝛻2⊥𝑓⁡(𝑥+𝑖⁢𝑦)=0, a direct consequence of satisfying the Cauchy-Riemann equations. Our theoretical and experimental exploration brings to light the intrinsic rotation of this field during propagation, elucidated by the incorporation of the quantum (Bohm) potential. This straightforward result holds promise, enabling the analytical deduction of the Fraunhofer or Fresnel diffraction pattern. Essentially, it simplifies the extraction of the Fresnel or Fourier transform from a function satisfying the Cauchy-Riemann equations.