abstract
- We establish new expansion formulas connecting parabolic and Bessel waves in both directions, covering both ideal nondiffracting beams and their Gaussian-apodized (Helmholtz¿Gauss) counterparts. Beyond deriving new closed-form expressions for the expansion coefficients in terms of hypergeometric functions, we introduce a simple unified recursive scheme that generates all coefficients, bypassing the numerical evaluation of hypergeometric functions and significantly reducing computational complexity. The recurrence relations not only improve numerical evaluation of the coefficients but also reveal an underlying algebraic structure common to all coefficient families, enabling more transparent derivations and straightforward inversion of the expansion formulas. Additionally, we show that the expansion coefficients admit an equivalent formulation in terms of orthonormal functions whose polynomial component is proportional to continuous Hahn polynomials. For parabolic¿Gauss beams, we further derive the normalization constant in terms of a compact and rapidly convergent series built from the same orthonormal functions. Numerical validations show that the recursive approach enables high-fidelity reconstruction of parabolic beams from the more accessible Bessel basis.