Rigorous higher-order Poincaré optical vortex modes

TY - JOUR

T1 - Rigorous higher-order Poincaré optical vortex modes

AU - Babiker, M.

AU - Koksal, K.

AU - Lembessis, Vassilis

N1 - Funding Information: The authors would like to acknowledge helpful discussions with Professor J. Yuan and are grateful to Professors S. Franke-Arnold, E.J. Galvez, Q. Zhan, G. Milione, and P. Banzer for useful correspondence. Publisher Copyright: © 2023 Optica Publishing Group. This is an author-produced version of the published paper. Uploaded in accordance with the University’s Research Publications and Open Access policy.

PY - 2023/12/11

Y1 - 2023/12/11

N2 - The state of polarization of a general form of an optical vortex mode is represented by the vector ε̂m, which is associated with a vector light mode of order m > 0. It is formed as a linear combination of two product terms involving the phase functions e±imφ times the optical spin unit vectors σ∓. Any such state of polarization corresponds to a unique point (ΘP, ΦP) on the surface of the order m unit Poincaré sphere. However, albeit a key property, the general form of the vector potential in the Lorenz gauge A = ε̂mψm, from which the fields are derived, including the longitudinal fields, has neither been considered nor has had its consequences been explored. Here, we show that the spatial dependence of ψm can be found by rigorously demanding that the product ε̂mψm satisfies the vector paraxial equation. For a given order m this leads to a unique ψm, which has no azimuthal phase of the kind eiℓφ, and it is a solution of a scalar partial differential equation with ρ and zas the only variables. The theory is employed to evaluate the angular momentum for a general Poincaré mode of order m yielding the angular momentum for right- and left- circularly polarized, elliptically polarized, linearly polarized and radially and azimuthally polarized higher-order modes. We find that in applications involving Laguerre–Gaussian modes, only the modes of order m ≥ 2 have non-zero angular momentum. All modes have zero angular momentum for points on the equatorial circle for which cos ΘP = 0.

AB - The state of polarization of a general form of an optical vortex mode is represented by the vector ε̂m, which is associated with a vector light mode of order m > 0. It is formed as a linear combination of two product terms involving the phase functions e±imφ times the optical spin unit vectors σ∓. Any such state of polarization corresponds to a unique point (ΘP, ΦP) on the surface of the order m unit Poincaré sphere. However, albeit a key property, the general form of the vector potential in the Lorenz gauge A = ε̂mψm, from which the fields are derived, including the longitudinal fields, has neither been considered nor has had its consequences been explored. Here, we show that the spatial dependence of ψm can be found by rigorously demanding that the product ε̂mψm satisfies the vector paraxial equation. For a given order m this leads to a unique ψm, which has no azimuthal phase of the kind eiℓφ, and it is a solution of a scalar partial differential equation with ρ and zas the only variables. The theory is employed to evaluate the angular momentum for a general Poincaré mode of order m yielding the angular momentum for right- and left- circularly polarized, elliptically polarized, linearly polarized and radially and azimuthally polarized higher-order modes. We find that in applications involving Laguerre–Gaussian modes, only the modes of order m ≥ 2 have non-zero angular momentum. All modes have zero angular momentum for points on the equatorial circle for which cos ΘP = 0.

UR - http://www.scopus.com/inward/record.url?scp=85182396386&partnerID=8YFLogxK

U2 - 10.1364/JOSAB.500511

DO - 10.1364/JOSAB.500511

M3 - Article

AN - SCOPUS:85182396386

SN - 0740-3224

VL - 41

SP - 191

EP - 196

JO - Journal of the Optical Society of America B: Optical Physics

JF - Journal of the Optical Society of America B: Optical Physics

IS - 1

ER -