Spin Hall Effect of Double-Index Cylindrical Vector Beams in a Tight Focus

Abstract

We investigate the spin angular momentum (SAM) of double-index cylindrical vector beams in tight focus. Such a set of beams is a generalization of the conventional cylindrical vector beams since the polarization order is different for the different transverse field components. Based on the Richards-Wolf theory, we obtain an expression for the SAM distribution and show that if the polarization orders are of different parity, then the spin Hall effect occurs in the tight focus, which is there are alternating areas with positive and negative spin angular momentum, despite linear polarization of the initial field. We also analyze the orbital angular momentum spectrum of all the components of the focused light field and determine the overwhelming angular harmonics. Neglecting the weak harmonics, we predict the SAM distribution and demonstrate the ability to generate the focal distribution where the areas with the positive and negative spin angular momentum reside on a ring and are alternating in pairs, or separated in different semicircles. Application areas of the obtained results are designing micromachines with optically driven elements.

Optical Helicity of Light in the Tight Focus

“…A comparison of Equations ( 22) and (23) indicates that the helicity changes sign in approximately the same areas of the focal spot at which the spin density changes sign, since both quantities depend on sin(mϕ). In any case, with the terms with I 0,0 I 0,m contribute the most [13], and the magnitudes h and S z have the same sign. The difference is, in particular, that the expression for the helicity in (22) includes the longitudinal component of the electric field, whereas the spin density in (23) includes only the transverse components of the electric field.…”

“…It can be shown that the full helicity, averaged over the beam cross-section, equals zero and conserves upon focusing. Indeed, as shown in [13], the following expression holds:…”

“…As seen from Equation (28), the helicity, as the SAM (26), changes sign 4m times along a certain-radius circle around the optical axis. Since the terms with I 2 0,m contribute the most [13], approximate expressions for the SAM (26) and for the helicity (28) are almost identical and equal to S z ≈ h ≈ I 2 0,m sin(2mϕ). From Equation (28), it also follows that the full helicity at the focus is zero, like the full spin in Equation ( 27).…”

“…Due to the circular polarization of the initial field, both the intensity distribution (12) and the spin density distribution (13) have circular symmetry at the focus. The helicity distribution (5) at the focus of field ( 10) is equal to the intensity:…”

“…We consider the following initial fields: a linearly polarized optical vortex, a circularly polarized optical vortex, superposition of a cylindrical vector beam and a linearly polarized beam, and a hybrid light field with circular-azimuthal polarization. Previously, based on the Richards-Wolf theory [11], expressions for all components of the electric and magnetic field vectors at the tight focus of all fields of interest have been derived [12][13][14][15][16]. Using these expressions for the light field components, we can derive analytical expressions for the helicity of these fields at the focus.…”