Self-similar rigid rotation in anisotropic squeezed Laguerre-Gaussian beams

We introduce anisotropic squeezed Laguerre--Gaussian beams as an analytically complete family of structured paraxial scalar beams with Gouy-phase-controlled rigid rotation. Our construction organizes the Laguerre--Gaussian transverse modal Hilbert space into circular modal subspaces carrying parity-resolved single-mode $\mathrm{su}(1,1)$ representations. The resulting optical Bargmann bases support squeezed states whose lowest-weight forms factor into Gaussian beam phases, parity-dependent polynomials, and elliptically anisotropic Gaussian envelopes. Excited seeds are polynomial deformations of the same squeezed envelope, extending the analytic construction to the full family. The Gouy-dressed squeezing parameter sets the semi-axis widths, aspect ratio, handedness, and orientation, while propagation rotates the profile by the accumulated Gouy phase without changing the scaled transverse shape. We generate these fields with synthetic phase holograms and experimentally confirm their self-similar rigid rotation. Our scalar-beam family provides a tunable anisotropic structured-light platform whose transverse orientation encodes propagation distance and may support axial localization, range finding, optical manipulation, and modal diagnostics.