Products of displaced Laguerre-Gaussian beams

We present a framework for propagating products of displaced Laguerre-Gaussian beams under the free-space paraxial wave equation. Using an optical analogy to quantum coherent states, we express the product as a linear superposition of Laguerre-Gaussian beams, with complex coefficients that encode displacements and mode overlaps. Our formalism enables direct calculation of the net orbital angular momentum and reduces propagation to tracking the evolution of independent weighted modes. These components share a common spherical phase and radial coordinate scaling, while acquiring distinct Gouy phases set by their total modal numbers. By absorbing the Gouy phase into the displacement coefficients, we make the transverse rotational dynamics governed by orbital angular momentum explicit. Our framework supports the engineering of structured light beams from the product of multiple optical vortices for applications that require controlled three-dimensional positioning.