Classical emergence of intrinsic spin-orbit interaction of light at the nanoscale

J. Enrique Vázquez-Lozano and Alejandro Martínez

Phys. Rev. A 97, 033804 – Published 7 March 2018

Abstract

Traditionally, in macroscopic geometrical optics intrinsic polarization and spatial degrees of freedom of light can be treated independently. However, at the subwavelength scale these properties appear to be coupled together, giving rise to the spin-orbit interaction (SOI) of light. In this work we address theoretically the classical emergence of the optical SOI at the nanoscale. By means of a full-vector analysis involving spherical vector waves we show that the spin-orbit factorizability condition, accounting for the mutual influence between the amplitude (spin) and phase (orbit), is fulfilled only in the far-field limit. On the other side, in the near-field region, an additional relative phase introduces an extra term that hinders the factorization and reveals an intricate dynamical behavior according to the SOI regime. As a result, we find a suitable theoretical framework able to capture analytically the main features of intrinsic SOI of light. Besides allowing for a better understanding into the mechanism leading to its classical emergence at the nanoscale, our approach may be useful to design experimental setups that enhance the response of SOI-based effects.

Received 25 May 2017
Revised 16 January 2018

DOI:https://doi.org/10.1103/PhysRevA.97.033804

Physics Subject Headings (PhySH)

Angular momentum of light Light-matter interaction Photonics Spin-orbit coupling

Atomic, Molecular & Optical

Authors & Affiliations

J. Enrique Vázquez-Lozano^* and Alejandro Martínez

Nanophotonics Technology Center, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain

^*juavazlo@ntc.upv.es

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Issue

Vol. 97, Iss. 3 — March 2018

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Images

Figure 1
Amplitude ratio (upper panel) and phase delay (lower panel) between the longitudinal and transverse components for different propagating TM multipole fields of $(l, l)$ order with $l \in [1, 9]$ , over the $x y$ plane. The inset shows the locally normalized instantaneous intensity distribution of the $E_{4, 4}^{TM}$ mode, whose spatially varying polarization ellipse along the $x$ axis is schematically depicted at the bottom. For comparison, the evolution of the SoP for the $E_{6, 6}^{TM}$ mode is also plotted. The color coding used in the evolution of the ellipses illustrates the transition from the near- to the far-field zone according to the scale represented in Fig. 3, with the corresponding values for the azimuthal mode order $l$ .
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Figure 2
Main features of the SOI term $Δ_{l}^{(\pm)}$ for different values of $l$ . Upper panel displays the amplitude ratio between the SOI term and the whole $E_{l}^{(Θ)}$ function, i.e., $| Δ_{l}^{} | / | E_{l}^{(Θ)} |$ . Panel at the bottom shows the phase distribution of the SOI term. Dashed red (dark gray) and green (light gray) lines indicate the asymptotic behavior in the near- and far-field regions, respectively. The inset shows the relative phase $ϕ_{l} (k r) - δ_{l} (k r)$ .
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Figure 3
Schematic representation of the near-field features for the $E_{4, 4}^{TM}$ mode. From the SOI term, the near- and the far-field region can be characterized in terms of the relative phase as $d [ϕ_{l} - δ_{l}] / d k r$ . In the near-field region the spatial distribution of the multipole field exhibits a complex shape. This manifests itself by means of a rich and strikingly interesting structure of the local dynamical properties such as the complex Poynting vector, the orbital and the spin momentum given in Eqs. (12, 13, 14) (cf. Ref. [15] for further details on these properties). On the other side, in the far-field limit the field distribution tends to be orthogonal to the direction of propagation.
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Figure 4
Densities of the local dynamical properties for different multipole fields of $(l, m)$ order over the $x y$ plane. Columns 1 to 4 show the real part of the Poynting vector [Eq. (12)], the orbital and spin momentum [Eqs. (13) and (14)], and the locally normalized electric field distribution, respectively. The color coding used indicates the transition from the near- to the far-field zone, according to the scale represented in Fig. 3, for the corresponding value of $l$ . Owing to the complex-shaped spatial field structure of the modes in-plane polarized, the local dynamical properties present streamlines that are sharply twisted in the near-field region. In fact, despite their seemingly planar character, there arises a transverse SAM, which is, in turn, responsible for the occurrence of an abrupt switching on the handedness of the spin-momentum near the source. Insets show a zoom-in view of this feature, underpinned by the nonseparability of the spin-orbit degrees of freedom.
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Figure 5
Separate contributions to the local orbital- and spin-momentum densities for the $E_{1, 1}^{TM}$ mode. Solid and dashed curves in the graphs show, respectively, the evolution of the radial and the azimuthal components normalized with respect to the corresponding local momentum density. Lower panels below the plots provides a better visual representation displaying the trajectories associated to the corresponding curves. In panel (a) the orbital and spin momentum are calculated by considering the whole VSWF, i.e., including both the SOI term and the azimuthal angular dependence given by the phase term $e^{i l φ}$ . In panel (b) the underlying influence of the SOI term is revealed by removing the azimuthal dependence. In this latter case, the plane-like contribution to the orbital-momentum density shows a purely radial behavior. The deviation from the radial direction is due to the nonseparability of the spin-orbit degrees of freedom.
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