Products of displaced Laguerre-Gaussian beams

We study the free-space propagation of products of displaced Laguerre--Gaussian beams. Each displaced factor admits an exact representation as a superposition of standard Laguerre--Gaussian beam modes through the optical analog of displaced number states. We reduce the resulting product to a single modal expansion with closed-form weight coefficients and explicit azimuthal selection rules. Working in the reference frame defined by the centroid of the transverse displacements, we evaluate the net orbital angular momentum directly from the modal weights, which provides a criterion to predict transverse rotation. We identify three propagation regimes: no transverse rotation for zero net orbital angular momentum, rigid rotation for products of identical factors with zero radial index, and nonrigid rotation with intensity redistribution otherwise. Our framework enables engineering structured light beams whose transverse rotation encodes propagation information, relevant to depth-sensitive optical microscopy, imaging, and tracking.