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This study establishes a mathematical equivalence between two distinct classes of laser beams: twisted Gaussian Schell-model (TGSM) beams, which are partially coherent, and coherent orbital angular momentum (OAM) eigenmodes, which carry quantized rotational phase structure. The authors demonstrate that both beam types share an identical covariance matrix structure at the second-order moment level, meaning they are statistically indistinguishable in terms of beam-width evolution, far-field divergence, and beam quality factor (M²) under any optical transformation described by ABCD matrices. This equivalence is worked out explicitly for three important beam families — Laguerre-Gaussian, Bessel-Gaussian, and perfect vortex beams — and includes a proof that perfect vortex beam modes form a complete orthonormal basis in a specific limiting regime.
Why it matters
This result allows the well-established mathematical toolkit developed for TGSM beam propagation to be directly applied to coherent OAM beams, potentially simplifying analysis and design in optical communications, quantum information systems, and free-space beam shaping applications that exploit orbital angular momentum.
arXiv:2605.15408v1 Announce Type: new
Abstract: We show that the covariance matrix of any cylindrically symmetric coherent orbital angular momentum (OAM) eigenmode with quantum number $ell$ takes a universal form depending only on $langle r^2rangle$, $langle k_r^2rangle$, and $ell$, independently of the radial profile, and that this form is identical to the covariance matrix of a twisted Gaussian Schell-model (TGSM) beam.} More specifically, both matrices share the same pattern of zero and nonzero entries, with the off-diagonal blocks proportional to $ell$ and the TGSM twist parameter $u$, respectively. This result holds for an arbitrary radial profile and provides direct term-by-term identification of parameters between the two sets of beams. We work out the correspondence in detail for three important families: Laguerre–Gaussian (LG), Bessel–Gaussian, and perfect vortex beams (PVBs), and derive the conditions under which each coherent OAM mode maps onto a physically realizable TGSM beam. {Because the covariance matrix governs second-moment evolution under arbitrary ABCD (symplectic) transformations, any two beams sharing the same covariance matrix are second-order indistinguishable at every propagation plane. In particular, the matched TGSM and coherent OAM beams share identical beam-width evolution, far-field divergence, and $M^2$ beam-quality factor.} In particular, the well-developed TGSM propagation toolbox applies directly to the second-order moment evolution of the three coherent families. We further show that within each beam family the covariance matrix uniquely determines the beam parameters, with exact uniqueness established for LG modes. Additional results include cross-family second-moment equivalence conditions and a proof that PVB modes form a complete orthonormal basis in the limit $wto 0$.