Sharp focusing of a light field with polarization and phase singularities of an arbitrary order

Abstract

Using the Richards-Wolf formalism, we obtain general expressions for all components of the electric and magnetic strength vectors near the sharp focus of an optical vortex with the topological charge m and nth-order azimuthal polarization. From these equations, simple consequences are derived for different values of m and n. If m=n>1, there is a non-zero intensity on the optical axis, like the one observed when focusing a vortex-free circularly polarized light field. If n=m+2, there is a reverse flux of light energy near the optical axis in the focal plane. The derived expressions can be used both for simulating the sharp focusing of optical fields with the double singularity (phase and polarization) and for a theoretical analysis of focal distributions of the intensity and the Poynting vector, allowing one to reveal the presence of rotational symmetry or the on-axis reverse energy flux, as well as the focal spot shape (a circle or a doughnut).

Minimal Focal Spot Size Measured Based on Intensity and Power Flow

“…In [ 29 , 30 ], expressions are obtained for the projections of the electric field strength vector at the focus of the aplanatic system. The Jones vector for an initial field with linear polarization directed along the x-axis has the form: and the components of the electric field strength vector near the focus for the initial field in Equation (1) have the form: where where λ is a wavelength, f is a focal length, x = kr sinθ, J μ ( x ) is a Bessel function of the first kind, and NA = sinθ 0 is a numerical aperture.…”

“…The Poynting vector was calculated by the formula in [ 29 ], P = [c/(8π)] Re[ E × H *], where c is the speed of light in a vacuum, Re is a real part of a number, × is the cross product, and * is a complex conjugation (we omit the constant c/(8π)). In [ 30 ], an expression was obtained for the axial projection of the energy flux vector at the focus when focusing light with linear polarization: …”

“…The components of the electric field strength vector at the focus for the initial radial polarization in Equation (13) can be expressed as [ 30 , 32 ]: …”

“…For the initial field in Equation (28), the projections of the electric vector at the focus are given in [ 30 ] as: …”

“…In [29,30], expressions are obtained for the projections of the electric field strength vector at the focus of the aplanatic system. The Jones vector for an initial field with linear polarization directed along the x-axis has the form:…”