Vortex structure of elegant Laguerre–Gaussian beams of fractional order

Abstract

The transition of the vortex structure of fractional elegant Laguerre-Gaussian beams is discussed in detail as the angular mode index of the beam is continuously varied between integer values. Under this kind of variation, the vortices can be classified into five groups. Contrary to the behavior of the vortices of the nondiffracting Bessel beams of fractional order, the nodal lines of the vortices in the case of the fractional eLG beams exhibit intricate shapes.

Hybrid topological evolution of multi-singularity vortex beams: generalized nature for helical-Ince–Gaussian and Hermite–Laguerre–Gaussian modes

“…It is noted that HLG is included as a subfamily in HIG family, therefore, HIG description is more felicitous for the actual singularities evolution details in AMC experiment. Further considering the hybrid properties of HIG and HLG modes, a more generalized model is established to elucidate the singularities hybrid evolution in AMC, based on the facts: (a) one usually cannot precisely control an exact phase difference of π/2 in AMC but an arbitrary β ∈ (−π , π]; (b) the ellipticity can be evaluated by a parameter γ ∈ [0, π/2] as = 2/tan 2 γ and the intrinsic elliptic coordinates interpolate between Cartesian coordinates and circular coordinates in a harmonic way: (x, y) = (w cot γ cosh ξ cos η, w cot γ sinh ξ sin η) (14) Thus, a more generalized family of SSGMs called as SHEN family can be established as:…”

“…As distinctive structured light fields with phase singularities, optical vortices carrying orbital angular momentum (OAM) have hatched plenty of modern scientific applications in optical tweezers [1][2][3], optical communications [4,5], quantum entanglement [6][7][8] and nonlinear optics [9][10][11]. Besides the classical Laguerre-Gaussian (LG) beams carrying integer OAM with a single phase singularity, the multi-singularity vortex beams carrying fractional OAM were also reported [12][13][14][15][16][17]. The unique characteristics of multi-singularity beams and fractional OAM can be utilized to significantly increase capacity in optical communication systems [5,18,19], scale multiparticle manipulation technologies in optical tweezers [2,3,20,21], drive advanced micro-opto-mechanics [22], flexibly shape light beam in 3-dimensional [23][24][25], and explore novel optical phenomena such as optical vortex knots [26][27][28] and spin-to-orbital conversion [29].…”

Analytical solution for multi-singular vortex Gaussian beams: the mathematical theory of scattering modes

“…, permit to obtain recursively the generalized F -polynomial of order p and angular momentum |l|, F |l| p (x), from the corresponding fundamental F -polynomial F p (x) previously determined by means of the recurrence relations (18). It is simple to deduce that F |l| 0 (x) = F 0 (x) = 1.…”

Discrete-Gauss states and the generation of focusing dark beams

“…In all cases, both functions tend to a quantity proportional to some f lp function evaluated at w = 0, being its corresponding proportionality constant always different from zero if τ = 0. However, according to the form of f lp functions (19), when τ = 0 f lp (0) is non-vanishing since the polynomial F |l| p (x) always shows non-zero values for its zero order terms.…”

“…(23) and (28) and find the form of F mkN v functions by taking into account that in this regime we can neglect the O(w 2N ) terms and that f lp functions in Eq. (19) can be approximated as f lp ∼ (−iτ ) p F |l| p (0). For k ≥ 1 (l ≥ 0), we find that a for a given DG state characterized by the indices (l, N ) ⇔ (m, k, N ), the F mkN v = 0 condition becomes near the origin:…”