Unified fractional Laguerre-Gaussian beams

We introduce a Fourier-Laplace fractional framework that unifies the standard and elegant Laguerre-Gaussian families through exact paraxial solutions. By imposing a Gaussian-type envelope constraint, we recover the integer-indexed standard modes and simultaneously generate a bounded, elegant branch with fractional-radial order. The azimuthal dependence is explored with Chebyshev structures and realized physically through a Shannon-sinc superposition of integer orbital-angular-momentum (OAM) modes, which preserves single-valuedness yet enables continuously tunable mean OAM. We derive closed-form normalized fields, numerically validate their free-space propagation, and uncover an OAM degeneracy between the standard and elegant branches, each exhibiting distinct transverse robustness.