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.

💡 Research Summary

This paper presents a comprehensive analytical framework for the free‑space propagation of products of displaced Laguerre‑Gaussian (LG) beams, referred to as p_dLGBs. The authors begin by recalling that standard LG modes, characterized by radial index p and azimuthal index ℓ, form an orthonormal basis of solutions to the paraxial wave equation and carry orbital angular momentum (OAM) ℏℓ per photon. When each LG beam is laterally displaced by a vector (x_j, y_j) in the input plane, the resulting displaced beam can be expressed exactly as a superposition of centered LG modes. This expansion exploits the optical analogue of displaced number states: the displacement is encoded in complex parameters α_j^{±} = (x_j ± i y_j)/(√2 w₀), and the expansion coefficients c_{u_j,v_j} involve exponential factors and confluent hypergeometric functions U(a,b,z). The coefficients satisfy ∑|c|² = 1, guaranteeing normalization, and decay rapidly, allowing practical truncation.

For a product of n displaced beams, the authors substitute each beam’s modal expansion into the product, yielding a multiple sum over all index pairs (u_j, v_j). By collecting terms with identical total radial and azimuthal indices (P, L), the product collapses into a single double series \