AM transfer

When a birefringent uniaxial crystal, such as vaterite (see Methods for details), is trapped in a linearly polarized (LP) beam, the optical axis of the crystal will align with the electric field, which is perpendicular to the propagation direction of the beam. In contrast to LP light, circularly polarized (CP) light incident on a birefringent particle will induce a rotation of the particle17,18. This is caused by the transfer of spin AM from the beam to the microparticle. Indeed, the birefringent particle induces a phase retardation between the ordinary and extraordinary components of the beam reversing the photon AM (±ħ per photon) of the incident CP light. If the particle acts as a half-wave (λ/2) plate, the maximum spin AM will be transferred to the particle, with a spin AM change of 2 ħ per photon. Considering the conservation of energy and AM in this process, a frequency shift arises owing to the angular Doppler effect as (ω 1 −ω 2 )/2=Ω rot =2πf rot , where ω 1 and ω 2 are the angular frequencies of the light before and after passing through the particle, repectively, and Ω rot is the angular rotation frequency of the particle19. As a result, we observe optical beating at 2Ω rot when the particle rotates at Ω rot .

Power spectra

Figure 1 shows typical power spectral density (PSD) signals (measured by PD i=1 as defined in the Methods section) of the scattered light from a trapped particle. In Fig. 1a, the particle experiences a rotational trapping torque owing to an LP light field. Figure 1b, c show the cases when the particle is trapped by a CP light field at pressures of 1 kPa and 15.1 Pa, respectively. In the following, we consider the PSD defined using the Fourier transform of the signal measured by PD i and denoted as S(f). The PSD measured at low pressures exhibits resonance peaks at the translational oscillation frequencies of f xy and f z corresponding to the periodic motion of the particle in the lateral x–y and axial z directions. Figure 1c shows the oscillation frequencies of f xy at 660 Hz and f z at 420 Hz for this specific particle. Further, the PSD shows an optical beating at 2f rot when the particle rotates at a rate of f rot (Fig. 1b, c). In practice, f rot is also detected because of the variation in the photodiode signal induced by small optical asymmetry of the particle. We remark that higher harmonics of the fundamental rotation frequency of f rot and translational oscillation frequencies of f xy and f z are also observed in the PSD signal (Fig. 1c).

Figure 1: PSD signals of the scattered light from a trapped particle. (a) Optical trapping by an LP light field (that is, no induced rotation) at 1 kPa. (b) Optical trapping by CP light at 1 kPa showing an optical beating frequency at 2f rot together with f rot . (c) PSD at 15.1 Pa showing translational (f xy , f z ) and rotational (f rot ) frequencies of a trapped particle. Full size image

Rotation rate

A trapped spinning birefringent microparticle will reach a terminal rotation frequency owing to the Stokes drag torque, dependent upon particle size and gas viscosity. The effective gas viscosity, μ e experienced by a spherical microparticle can be empirically estimated as20

where μ 0 is the viscosity coefficient at the reference pressure P 0 , and K n =λ a /d is the Knudsen number. Here is the mean free path (mfp) of air molecules at a pressure P a relative to the mfp of λ 0 at the reference pressure P 0 , and d is the diameter of the particle. As the pressure decreases, the mfp of the surrounding gas molecules becomes comparable to the particle diameter implying K n ≈1. This marks a transition to a regime where radiometric forces are negligible and the viscosity becomes proportional to the pressure. Figure 2 shows the rotation rate f rot of a trapped particle as a function of pressure measured by observation of the PSD (Fig. 2 inset). An initial rotation rate of 110 Hz is recorded at atmospheric pressure, which increases to a stable rotation rate of 5 MHz for a pressure of 0.1 Pa. Decreasing the pressure further can lead to rotation rates of up to 10 MHz, although, at such rates, the particle is lost in a short period of time. We remark that this represents, to date, the largest measured rotation rate for a ‘man-made’ object21. The model in Fig. 2 is calculated using equation (1), implying a pressure-dependent Stokes rotational drag coefficient. The discrepancy between the model and experimental values at pressures below 10 Pa could be attributed to particle instability induced by heat because of light absorption22,23, while the low pressure particle loss might be due to the large inertial forces experienced by the particle at high rotation rates.

Figure 2: Rotation rate of a trapped particle at different gas pressures. The model fits to the experimental data for 0≤K n ≤880. Inset shows the PSD at a rotation rate of 2.45 MHz at a pressure of 1 Pa. Full size image

Parametric coupling

An optically trapped and simultaneously rotated particle in vacuum offers original perspectives on particle dynamics. It is of particular interest to study the dynamics when the rotation frequency f rot coincides with the oscillation frequency f xy of the trapped particle. A series of power spectra tracking the major peaks of the PSD at each pressure from 1 kPa to 100 Pa are shown in Fig. 3a. The rotation frequency signal exhibits kinks at f rot ≈f xy (at 380 Pa) and at f rot ≈2f xy (at 210 Pa). Figure 3b shows the amplitude of the PSD peak at the oscillation frequency f xy as the rotation frequency of the trapped particle changes. We observe an enhanced signal at f rot ≈f xy corresponding to a driven resonance. A second resonance occurs when f rot ≈2f xy , suggesting a parametric resonance24. The coupling between the oscillatory motion and rotational motion of the particle can also be observed in Fig. 3c,d. The photodiode signal (PD 1 ) in time domain measured at a pressure of 13.6 Pa exhibits fine rotational modulation (Fig. 3c inset), which is further modulated by slower frequency components (Fig. 3c). The power spectrum of these modulations reveals the rotation frequency f rot accompanied with sidebands separated by f xy (Fig. 3d).

Figure 3: Coupling of the rotational and translational motion of a trapped particle. (a) Major peaks of the PSD signal around the resonance frequency at f rot ≈f xy . In red are frequency peaks associated with rotation and in blue the ones associated with translational oscillations. (b) Resonances found at f xy and 2f xy when f rot scans across these frequencies. (c) Photodiode signal in time domain showing mixed frequency components at 13.6 Pa. Inset shows the expanded view of the selected region. (d) Fourier transform of the time-domain signal (Fig. 3c) showing the rotation frequency of f rot with sidebands separated by f xy . (e) Simulated PSD signal at a high rotation frequency showing the appearance of sidebands and their harmonics due to the modulation of the trapping frequency. (f) Simulated PSD signal for two different gas viscosities corresponding to the resonant (blue) and non-resonant (red) cases. Full size image

We approximatively model the dynamics of a birefringent particle in an optical potential and subject to position- and orientation-dependent torque. To simplify the system while maintaining its main optical properties, we consider the induced polarization of an anisotropic dipole25 corrected for the anisotropic radiative process26. The optical forces and torques are calculated by generalizing the cycle-averaged Lorentz force27 and torque28 to account for the anisotropy owing to birefringence. Optically, the spherical aberration introduced by the total internal reflection at the glass–vacuum interface is taken into account by using angular spectral decomposition of the incident beam29 (Supplementary Note 1). We remark that at the levitated equilibrium position of the particle, the trapping forces can be approximated with an optical harmonic potential originating from a CP Gaussian beam. In the simulations, we use this potential and adjust the beam parameters such that the transversal trap oscillation is ≈660 Hz for a vaterite microparticle whose diameter is 4.40 μm. The rotation of the microparticle is modelled by the Euler equations for a solid sphere with a slight asymmetry of 0.1% in one of its moments of inertia (Supplementary Note 2). This mechanical asymmetry introduces a principal momentum axis that does not overlap with the optical axis of the particle.

For these mechanically anisotropic particles, the Brownian stochastic torques are introduced in the rotational Langevin equation30, which are generalized to include an external torque31 and contributions from the Euler equations in the body frame of reference (Supplementary Note 3). Finally, the detection is simulated by Fourier transforming the dipole polarization intensity along a fixed direction. This Fourier transform corresponds to the PSD signal observed. Figure 3e shows the simulated PSD of a rotating dipole particle oscillating periodically in the beam. The central resonance corresponds to the polarization change due to rotation, whereas the multiple sidebands correspond to periodic variations of the electric field strength as the particle oscillates in the trap. These simulations indicate that the coupling behaviour observed in Fig. 3b occurs when the rotational frequency is resonant with the fundamental and the second harmonic of the translational frequency corresponding, respectively, to a driven oscillator resonance and a parametric resonance. Indeed, parametric resonance occurs as the trap stiffness varies slightly for different orientations of the particle, whereas this orientation changes as the particle rotates. Figure 3f shows the enhancement of the PSD signal at the transversal oscillation frequency f xy peak as the rotational frequency becomes resonant.

Cooling

Our trapped microparticle has three rotational and three translational degrees of freedom. These are coupled due to the optical anisotropy of the microparticle, which can be seen in Fig. 3b,d–f. We now progress to investigate the impact of the rotation rate on the rotational degrees of freedom. In this case, we observe the effective cooling of microparticles through rotation in the absence of any ‘active’ feedback method. Akin to the motion of a spinning top, a rotating body offers inertial stiffness, which prevents the body from drifting from its desired orientation. Here the high rotation rates achieved using the microgyroscope lead to the intrinsic stabilization of its axis of rotations with respect to perturbations. This effect is similar to stabilizing the rigid-body dynamics of an oval football32 or spin-stabilized satellite33. Our numerical simulations indicate that for increasing rotation rate, the distribution width of the fixed-frame transversal angular velocities ν x and ν y decrease (Fig. 4a,b). This effect can be seen as a cooling effect on the two rotational degrees of freedom of the microgyroscope at the expense of the third rotational degree of freedom around which the rotation occurs, supporting our conclusions from the experimental data. Owing to rotational–translational coupling mentioned above, this rotational stabilization also enables cooling for all the three translational motion degrees of freedom. More specifically, when the laser is switched on, it heats the transversal motion of the particle due to the stochastic fluctuation of the microparticle orientation. As the particle rotates faster, its orientation is stabilized by the gyroscopic effect and as such the particle experiences relative cooling.