Transverse Angular Momentum and Geometric Spin Hall Effect of Light

Abstract

We present a novel fundamental phenomenon occurring when a polarized beam of light is observed from a reference frame tilted with respect to the direction of propagation of the beam. This effect has a purely geometric nature and amounts to a polarization-dependent shift or split of the beam intensity distribution evaluated as the time-averaged flux of the Poynting vector across the plane of observation. We demonstrate that such a shift is unavoidable whenever the beam possesses a nonzero transverse angular momentum. This latter result has general validity and applies to arbitrary systems such as, e.g., electronic and atomic beams.

Transverse angular momentum of photons

“…Then, we have calculated the commutation relations between the Cartesian components ofĴ ,L,Ŝ and we have found that they differ from the standard one. In particular, whileL z is still a bona fide generator of rotations around the propagation axis z, the transverse componentsL x andL y commute and, as it was already found at a classical level [12], they are strictly connected with the transverse coordinates y and −x of center of the beam, respectively. Finally, as a realistic example illustrating the above mentioned connection, we calculated the expectation value ofĴ between multi-mode coherent states of the electromagnetic field.…”

“…In physical terms, the essence is that any well collimated beam is basically an eigenstate of the zcomponent of the linear momentum operator which, for such beams, practically reduces to a c-number. Thus, in order to recover canonical commutation relations, it is necessary to deal with beams with either high angular aperture θ 0 for whose our first-order approximation breaks down, or with a direction of propagation that deviates from the reference axis z by an angle grater than θ 0 [12]. As a final remark, it should be noticed that violations of canonical commutation relations for the angular momentum of an electromagnetic field of arbitrary shape, were already reported by van Enk and Nienhuis [10].…”

“…However, it is well known that a superposition with real coefficients of the fundamental mode ψ 00 with either ψ 10 or ψ 01 describes approximatively a displaced Gaussian beam along the x-or the y-axis, respectively [24]. In other words, a lateral displacement of a Gaussian beam changes its transverse angular momentum or, vice versa, the occurrence of non zero transverse components of the angular momentum cause a transverse displacement of the beam [12]. On the other hand, it is also known that a transverse displacement cannot affect the longitudinal angular momentum J z [25,26], as it is confirmed by Eq.…”

“…This was probably due to the fact that the spin angular momentum of photons can only be defined along the direction of propagation. However, it was very recently noticed that transverse, as opposed * Electronic address: andrea.aiello@mpl.mpg.de to longitudinal, components of angular momentum may be responsible for interesting phenomena as, e.g., the socalled geometric spin Hall effect of light [12,13]. A classical theory of transverse optical angular momentum was developed in [12].…”

Tighter spots of light with superposed orbital-angular-momentum beams

“…In the limit of tight focusing of light beams, the electromagnetic field distribution can become highly complex in the focal plane [5][6][7][8]29]. In this context, the occurrence of longitudinally oscillating field components or, more generally, of three-dimensional field distributions gives rise to a variety of interesting effects and phenomena, including spin-to-orbit coupling [30], transverse angular momentum [31][32][33][34], the creation of complex polarization topologies [35] at the nanoscale, and also the possibility of focusing light more tightly, as observed for instance for radially polarized light [4,7].…”