Spin Hall Effect before and after the Focus of a High-Order Cylindrical Vector Beam

Abstract

It is known that in the cross-section of a high-order cylindrical vector beam (CVB), polarization is locally linear. The higher the beam order, the higher the number of full rotations of the vector of local linear polarization when passing along a contour around the optical axis. It is also known that both in the input and in the focal planes, the CVB has neither the spin angular momentum (SAM), nor the orbital angular momentum (OAM). We demonstrate here that near the focal plane of the CVB (before and after the focus), an even number of local subwavelength areas is generated, where the polarization vector in each point is rotating. In addition, in the neighboring areas, polarization vectors are rotating in different directions, so that the longitudinal component of SAM vectors in these neighboring areas is of the opposite sign. In addition, after the beam passes the focus, the rotation direction of the polarization vector in each point of the beam cross-section is changed to the opposite one. Such spatial separation of the left and right rotation of the polarization vectors manifests so that the optical spin Hall effect takes place.

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

“…As seen in Figure 4 , indeed, the SAM distribution consists of alternating pairs of spots with positive or negative SAM. This is different from the patterns we obtained earlier near the focus when the spots with positive and negative SAM were alternating in singles rather than in pairs [ 11 ].…”

“…Since the areas with negative and positive SAM are spatially separated, this phenomenon is a manifestation of the optical spin Hall effect. Later, the same effect was discovered for tightly focused high-order cylindrical vector beams [ 11 ]. In [ 11 ], the SAM is distributed mostly on a ring and consists of alternating areas with positive and negative values.…”

“…Later, the same effect was discovered for tightly focused high-order cylindrical vector beams [ 11 ]. In [ 11 ], the SAM is distributed mostly on a ring and consists of alternating areas with positive and negative values. The cylindrical vector beams have the Jones vector J = [cos m φ, sin m φ] with φ being the angular polar coordinate and m being the polarization order (for m = 1, radial polarization).…”

Focusing a vortex beam with circular polarization: angular momentum

“…Отметим, что первая работа по угловому моменту света [24] содержит выражение (13) без вывода из выражения (11).…”

“…В этом параграфе мы рассмотрим некоторое противоречие, которое нам не удалось полностью разрешить и которому мы не нашли объяснения в литературе [6 -10]. Дело в том, что уравнение (13) не может быть получено из уравнения (11). Для начального светового поля (1) формула (16) указывает на то, что УМ в фокусе равен нулю при n = -1.…”