Optomechanical Kerker Effect

Tunable directional scattering is of paramount importance for operation of antennas, routing of light, and design of topologically protected optical states. For visible light scattered on a nanoparticle, the directionality could be provided by the Kerker effect, exploiting the interference of electric and magnetic dipole emission patterns. However, magnetic optical resonances in small sub-100-nm particles are relativistically weak. Here, we predict inelastic scattering with the unexpectedly strong tunable directivity up to 5.25 driven by a trembling of a small particle without any magnetic resonance. The proposed optomechanical Kerker effect originates from the vibration-induced multipole conversion. We also put forward an optomechanical spin-Hall effect, the inelastic polarization-dependent directional scattering. Our results uncover an intrinsically multipolar nature of the interaction between light and mechanical motion and apply to a variety of systems from cold atoms to two-dimensional materials to superconducting qubits. An application for engineering of chiral optomechanical coupling and nonreciprocal transmission at nanoscale is proposed.

Popular Summary

The ability to route light in any direction is essential for optical devices. A standard approach to achieve directional emission in nano-optics is based on the Kerker effect: Light scatters off a particle whose electric and magnetic dipole radiation patterns interfere. However, this does not work when the particles are smaller than the wavelength of light in the medium because of the absence of magnetic optical resonances. Here, we propose a new paradigm of tunable directional light scattering that exploits mechanical trembling. This approach does not require a magnetic response and works for subwavelength objects.

Our proposed optomechanical Kerker effect originates from the vibration-induced multipole conversion. It allows a small particle, whose electric dipole can be polarized when at rest, to yield both electric and magnetic dipole radiation patterns when it trembles. For a particle with resonant permittivity, this enables control of the scattering direction by detuning the light frequency from resonance. Namely, when excited at resonance, interference of electric and magnetic dipole emission yields forward scattering, while for off-resonant excitation backscattering is enhanced. We also put forward an optomechanical spin Hall effect—directional inelastic scattering of light depending on its circular polarization.

Our results apply to a variety of optomechanical systems based on objects with resonant response such as quantum dots, 2D semiconductors, cold atoms, and superconducting qubits. The directional scattering can be used to engineer chiral optomechanical coupling, nonreciprocal transmission, and topologically protected optical states at nanoscale.

Figure 1

(a) A sketch of the motion-induced multipole conversion. Trembling of an oscillating dipole in the space $\mathit{u}$ leads to appearance of an electric quadrupole and a current curl. The latter yields a magnetic dipole $\mathit{m}$. (b) A sketch directional inelastic light scattering on a trembling particle. (c) A sketch of the trembling resonant layer, an optomechanical analogue of Huygens surface.

Figure 2

(a) A sketch of light scattering on a trembling particle. The incident and elastically scattered light are shown by yellow color, inelastically scattered light is shown by blue color. Inelastic scattering is caused by the temporal modulation of the optical path (cyan and magenta arrows) due to particle displacement. (b),(c) Diagrammatic representation for the inelastic light scattering on a trembling particle. Wavy lines denote photons, bubbles correspond to the dressed polarization operator of the particle at rest, dashed lines represent mechanical displacement, solid dot stands for the optomechanical interaction given by Eq. (a5).

Figure 3

Radiation pattern for (a) elastic light scattering and (b)–(e) inelastic scattering by a trembling particle for different ratios of the polarizabilities at initial and scattered frequencies $\alpha (\omega )$ and $\alpha ({\omega }^{\prime })$. The interference of the electric dipole (c) and magnetic dipole and electric quadrupole (d) patterns results in directional inelastic forward (e) and backward (b) scattering. The light is incident from the bottom (yellow arrow) and is linearly polarized (green arrow), the particle trembles along the light propagation direction (black arrow). Red and blue colors indicate the sign of electric field.

Figure 4

(a) Backward and (b) forward directivity of nonpolarized light inelastically scattered by a trembling resonant particle depending on the incident light frequency $\omega$ and trembling frequency $\mathrm{\Omega }$. Solid line shows the directivity equal to 3 that limits usual Kerker effect. (c) Degree of directivity $[D({\mathit{n}}_0)-D(-{\mathit{n}}_0)]/[D({\mathit{n}}_0)+D(-{\mathit{n}}_0)]$.

Figure 5

The anti-Stokes light scattering pattern with the degree of circular polarization marked by red and blue colors. Light is incident from the bottom and linearly polarized, see yellow and green arrows on the left, respectively. Rows correspond to different trembling directions indicated by black arrows on the left. Columns correspond to different relations between particle polarizabilities at the frequencies of incident and scattered light, $\alpha (\omega )$ and $\alpha ({\omega }^{\prime })$, indicated in the top. The colored areas represent different origins of the optomechanical spin-Hall effect: phase difference between the polarizabilities $\alpha (\omega )$ and $\alpha ({\omega }^{\prime })$ (blue color), the phase difference between components of the displacement vector $\mathit{u}$ (yellow color), or their combined action (purple color).

Figure 6

(a) A sketch of light scattering on a resonant layer with flexural vibrations. Color maps of (b) backward- and (c) forward-scattered power for the case of normal incidence as a function of incident light frequency and scattering angle ${\theta }^{\prime }$. Dashed lines indicate the resonances for incident and scattered light. The reflection coefficient of resonant layer was taken in the form $r_{s(p)}(\omega ,\theta )=-i{\mathrm{\Gamma }}_{s(p)}/(\omega -{\omega }_x+i{\mathrm{\Gamma }}_{s(p)})$, with ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }/\cos \theta$ and ${\mathrm{\Gamma }}_p=\mathrm{\Gamma }\cos \theta$ [31]. The linear dispersion of phonons was assumed, $\mathrm{\Omega }=s|\mathit{q}|$, with $(s/c)({\omega }_x/\mathrm{\Gamma })=5$.

Figure 7

(a) A sketch of an optomechanical system consisting of a waveguide and a small particle that can oscillate in space along its axis. Color plots of the optomechanically induced probe light (b) nonreciprocal transmission and (c) asymmetrical reflection as a function of pump and probe light frequencies. Calculation is made for a resonant particle with the reflection coefficient $r(\omega )=-i\mathrm{\Gamma }/(\omega -{\omega }_x+i\mathrm{\Gamma })$, ${\mathrm{\Omega }}_r=\mathrm{\Gamma }$, and $\gamma =0.2\mathrm{\Gamma }$.

Figure 8

Tunability of various resonant optomechanical systems operating in different spectral ranges. When $\mathrm{\Omega }\ll \mathrm{\Gamma }$ (blue area), the directional scattering can occur only in the backward direction. When $\mathrm{\Omega }\gg \mathrm{\Gamma }$ (red area), one can realize both directional forward and backward scattering by a proper tuning of the incident light frequency and trembling frequency. Numerical parameters and references are given in Table 1.

Figure 9

(a),(b) Diagrammatic representation for the optomechanical interaction of light (wavy line), medium polarization (bubble), and mechanical vibration (dashed line). (c),(d) The contributions to the amplitude of light scattering by a trembling medium that together with contributions shown in Figs. 2 and 2 give the full amplitude of inelastic light scattering. Wavy line denotes photon and corresponds to the Green’s function for vector potential $\mathit{D}(\mathit{k},\omega )=-4\pi (1-\mathit{k}\otimes \mathit{k}/{\omega }^2)/({\omega }^2-{\mathit{k}}^2)$, empty and filled bubbles correspond to the bare and dressed polarization operators of the medium at rest, $\mathbf{\Pi }$ and $\mathcal{P}=\mathbf{\Pi }(1-\mathit{D}\mathbf{\Pi }{)}^{-1}$, respectively, dashed line represents mechanical displacement ${\mathit{u}}_{\mathit{q}}$, solid dot is the optomechanical interaction $\mathbf{\Lambda }$ given by Eq. (a5).

Figure 10

Diagrammatic representation for the optomechanical correction to the mechanical motion self-energy. Star stands for the optical pump. In the four upper diagrams, the solid dot denotes the first-order optomechanical interaction, Eq. (b3). In the four lower diagrams, the solid dot represents the second-order optomechanical interaction described by ${\mathrm{\Lambda }}^{(2)}=-|u{|}^2{k}^2$, where $k$ is the wave vector of the involved photon, and $u$ is the displacement.