abstract
- We present a framework to construct generalized fractional Laguerre¿Gaussian (LG) beams as solutions of the paraxial wave equation in cylindrical coordinates. Using a Fourier-based fractionalization with constrained scalings, we unify (i) the standard LG family (exact paraxial solutions for integer indices) and (ii) an elegant LG family that admits fractional radial indices while preserving boundedness. The azimuthal index remains physically quantized; nevertheless, beams with an effective fractional topological charge can be synthesized by sinc-weighted superpositions of integer-OAM modes (a dual Shannon interpolation). We derive closed-form expressions via Fourier/Laplace routes, give recurrence relations that expose the internal mode structure, and validate propagation analytically and numerically. The formalism enables tunable beam shaping and precise control of effective fractional orbital angular momentum.