Abstract
Optically levitated micro and nanoparticles offer an ideal playground for investigating photon–phonon interactions over macroscopic distances. Here we report the observation of longrange optical binding of multiple levitated microparticles, mediated by intermodal scattering and interference inside the evacuated core of a hollowcore photonic crystal fibre (HCPCF). Three polystyrene particles with a diameter of 1 µm are stably bound together with an interparticle distance of ~40 μm, or 50 times longer than the wavelength of the trapping laser. The levitated boundparticle array can be translated toandfro over centimetre distances along the fibre. When evacuated to a gas pressure of 6 mbar, the collective mechanical modes of the boundparticle array are able to be observed. The measured interparticle distance at equilibrium and mechanical eigenfrequencies are supported by a novel analytical formalism modelling the dynamics of the binding process. The HCPCF system offers a unique platform for investigating the rich optomechanical dynamics of arrays of levitated particles in a wellisolated and protected environment.
Introduction
Since Ashkin’s first report of the acceleration and trapping of microparticles by optical forces^{1}, the use of optical tweezers has developed into a standard technique for biological manipulation and picoNewton force sensing^{2,3}, to mention just two examples from a wide range of applications. In recent years, the emerging field of “levitated optomechanics” has attracted increasing interest. An optically tweezered particle, especially at low gas pressure, is isolated from the external environment, resulting in very low mechanical damping. This leads to veryhigh mechanical Qfactors and permits particle rotational speeds in the MHz range^{4,5}. Recent advances include the use of feedback or cavity cooling of the centreofmass temperature towards the mechanical ground state^{6,7,8,9,10,11}, permitting experimental tests of fundamental physics^{12,13,14}, and the use of individual levitated particles as point sensors^{15,16}. Optical binding between arrays of trapped particles adds an additional dimension, and will result in rich dynamics, enabling access to collective coupling between highQ mechanical oscillators and, potentially, simultaneous cooling of the mechanical motion of multiple particles.
Multiple trapping sites have previously been created using interference^{17} and holographic tweezers^{18}, allowing the formation of a lattice of trapped microparticles. In these experiments, there is typically verylittle multiple scattering between particles—a necessary prerequisite for optical binding^{19,20,21,22}, which can only occur if the scattered field from one particle strongly interacts with the other particles in the array, and viceversa. In a 1D particle array, such as the one studied here, binding is possible because the optical fields propagate bidirectionally along the array.
The majority of 1D optical binding experiments to date have been performed in free space over distances that are limited by the Rayleigh range of the focusing optics. However, to control and measure the collective binding dynamics of such arrays, it is necessary to manipulate and monitor individual particles within the array without perturbing the other degrees of freedom, which requires extended interparticle distances and, therefore, longrange interactions between trapped particles. In recent years, the optical binding range has been extended using nondiffracting Bessel beams^{21} or by trapping particles in the evanescent field at the surface of a multimode glass microfibre^{23}. These experiments were however conducted in a liquid environment, resulting in strongly damped collective dynamics. In evanescent fieldbased traps, particles are in physical contact with the microfibre (not levitating), giving rise to additional damping terms^{23}. In addition, external spatial light modulators were used in those experiments to create binding sites, limiting the stability and power handling of the system.
Here we report longrange optical binding of a chain of levitated particles inside the core of an evacuated hollowcore photonic crystal fibre (HCPCF). When the fundamental mode is launched into the HCPCF, particleinduced scattering to higherorder guided modes results in an intermodal interference pattern, and a landscape of periodically distributed trapping potentials, within which subsequent particles can be trapped. The binding forces vary as the particles move relative to each other, and stable trapping and binding of a chain of particles results for configurations that locally minimise the free energy of the system. By offering modal fields that are tightly confined over the entire fibre length, i.e., an infinite Rayleigh length, HCPCF permits trapped particles to interact over distances orders of magnitude longer than the trapping wavelength, limited only by fibre and scattering loss. The absence of viscous damping at low gas pressure makes it possible to observe, for the first time, the collective dynamics of a boundparticle chain—an important first step towards “levitated collective optomechanics”.
Materials and methods
The experimental setup is sketched in Fig. 1. Light from a pulsed Ti:sapphire laser (800 nm wavelength, 80 MHz repetition rate, 60 fs 1/e halfwidth pulse duration) was delivered to the setup through a 15 m long single mode fibre (SMF, with group velocity dispersion β_{2} = 40 fs/(m.THz)). This caused the pulses to broaden to 1/e halfwidth durations of ~14.4 ps at the output of the SMF, with a frequency chirp of ~0.5 THz/ps. The pulses were split at a polarising beam splitter (PBS) and coupled into the opposite ends of an 8 cm length of HCPCF with a core diameter of 7.8 µm, which was mounted inside a vacuum chamber. The counter propagating trapping beams were orthogonally polarised to prevent the formation of intensity interference fringes. A 96:4 beam splitter (BS1) and photodiode (PD) were used to monitor the transmitted power at one end of the HCPCF. A medical nebuliser was used to inject polystyrene particles (typical diameters of 1 µm) into a dualbeam trap placed close to the fibre input face^{24} (see Supplementary Information S1). Once trapped, a particle was propelled into the hollow core by adjusting the beam splitter to increase the forward power. Subsequent particles were launched into the fibre using the same technique (see Supplementary Media S1). Once the desired number of particles had been launched, the chamber was sealed and pumped out to a pressure of a few mbar.
Results and discussion
When light is launched into the HCPCF, it is difficult to avoid weak excitation of higherorder modes (HOMs). Even a few percent of HOM power will cause strong intermodal interference, which, however, fades away with distance due to a combination of the group velocity walkoff and pulse chirp. Thus, when a single particle is launched into the core, the transmitted power will fluctuate as the particle passes through the interference pattern, eventually stabilising once the particle reaches a position where the stationary interference between the HOM pulses has faded away. The fadeaway length can be written in the form (see Supplementary Information S2):
where L_{D} = τ_{0}^{2}/β_{2} is the dispersion length in the SMF, τ_{0} is the 1/e halfwidth pulse duration at the laser, L is the SMF length, Δv_{G} is the group velocity difference between the HCPCF modes and \(\bar v_{\mathrm{G}}\) is the mean group velocity (spectral broadening due to selfphase modulation in the SMF has been neglected, but is expected to be small). For the experimental parameters, z_{F} = 1.5 mm.
A very similar effect is seen when two or more particles are close enough to sit in their respective intermodal interference patterns. Under these circumstances longrange optical binding can be observed. To explore these effects, we launched a 1μmdiameter polystyrene particle into the fibre, followed by a second similar particle 0.5 s later (Fig. 2a, b). The power ratio was adjusted so that the second particle moved towards the first one. The transmitted power at the photodiode is plotted against time over a 5 s interval in Fig. 2a. The effects of intermodal interference are seen at 0.5 s and 4.4 s, with magnifications of the responses in each case shown in Fig. 2b, c. As expected, intermodal interference causes the transmitted power to oscillate as the second particle moves away from the input face (Fig. 2b). After 0.55 s, the transmitted power becomes constant. Then, at 4.36 s, the first particle begins to be disturbed by the intermodal interference created by the second particle (Fig. 2c). The oscillations increase in amplitude until the two particles become stably bound, when the modulation in the transmitted signal once again becomes constant. An exponential fit (dashedred curve) to the envelope of the oscillations yields a decay time of ~0.9 s, which, given that the average speed of the particles is ~1.5 cm/s, corresponds to a fadeout distance of ~1.35 mm, in good agreement with the above analysis.
Three particles were then launched into the fibre in sequence, and after several attempts and adjustments, it was found that these particles were optically bound. Figure 3a shows optical images of the boundparticle array at equilibrium, captured through the side of the HCPCF by using a CCD camera. The particles are spaced by 40 ± 3 μm, and the uncertainty is caused by the finite width of the intensity peaks at the CCD. Assuming that there are several elementary charges on each particle^{11}, we estimate the interparticle Coulomb forces to be several attoNewton, which will displace the particles from their equilibrium positions by only a few tens of femtometres, which is negligible compared to the interparticle distance.
Thermally driven vibrations for the boundparticle array (driven by Brownian motion) were resolved at a pressure of 6 mbar by using a highspeed video camera (20,000 frames per sec). Figure 3a shows 9 consecutive frames at time intervals of 50 µs (see Supplementary Media S2 for the video). During the measurement, electronic feedback was used to stabilise the particle array within the field of view of the imaging system using a positionsensing diode (PSD) to generate a signal proportional to the centreofmass motion of the particles. This was then fed back via a PID controller to the EOM.
The “breathing” mode of the bound particles, in which the two outer particles move outofphase while the central particle is stationary, can be directly observed in the video frames (Fig. 3a). The peaktopeak amplitude for the vibrations is ~5 µm according to the scale bar in the video. The spectra for the thermally driven centreofmass motion of particles 1, 2 and 3 are plotted in Fig. 3bd using data extracted from the highspeed video. The redsolid curves are Lorentzian fits to the individual spectral peaks. For particles 1 and 3, a strong peak occurs at 2.3 kHz (the frequency of the breathing mode), while for particle 2, a much weaker peak appears at 3.9 kHz, related to the case in which the central particle moves outofphase with the two outer particles (see the analysis below). Note that these distinct resonances can only be resolved at a low gas pressure, when gasrelated viscous damping is suppressed.
Experimentally up to five particles have been successfully bound to form an array moving toandfro along the fibre (see Supplementary Media S3 for the video). Here, using three bound particles as an example, we develop scattering matrix analysis to calculate the forces acting on each particle, and, thus, identify configurations that result in stable binding. We represent the complex amplitudes of individual modes in the system by a column vector v, which is normalised so that v·v* = 1, meaning that the power in the ith mode is _{ Pi } = v_{ i }^{2}P_{0}, where P_{0} is the total power. The modes are assumed to form a complete orthogonal set. Particleinduced scattering between incident and transmitted modes can then be described by v_{out} = [S]·v_{in}, where [S] is the scattering matrix and v_{in} and v_{out} are column vectors, the elements of which are the complex amplitudes of the incident and scattered modes. Orthogonality allows us to write the scattering coefficient from incident mode i to forwardscattered mode j in the form:
where integration is over the transverse plane, m_{ i }(x, y) is the transverse field distribution of the ith mode and s_{ i }(x, y) is the scattered field distribution immediately after the particle, which we calculate by 3D finite element modelling.
We now assume that backscattering (which is very weak for the experimental parameters) and material absorption are negligible. We also assume that the particle is trapped at the centre of the core (this seems to be a good approximation since the launching beam is Gaussian) so that scattering occurs only to radially symmetric modes and that the Coulomb forces between the particles and core wall are negligible. Figure 4a shows the geometry of the threeparticle system. The phase index, group index and loss of the modes were calculated by finite element modelling of the actual fibre microstructure based on a highresolution scanning electron micrograph. Since linearly polarised LP_{0i} modes of the order i > 3 were found to have a veryhigh propagation loss, contributing negligibly to optical binding, we included only the first three lowestorder modes (i = 1, 2, 3), resulting in a 3 × 3 scattering matrix.
Since binding is observed at ~4 cm from the fibre input and the fadeout distance of intermodal beating is ~1.35 mm, the three incident modes will each independently transfer momentum to the particles because their instantaneous frequencies at each position along the fibre are different and no stationary intermodal beatpattern can form. Additionally, since the pulse durations (ps) are orders of magnitude shorter than the mechanical response time of the particle (hundreds of μs), the particles will respond to the vector sum of the resulting optical momenta. Thus, each incident mode can be treated independently. For the pth incident mode, the modal amplitudes on opposite sides of the first particle are (using bracket notation to indicate left and right, respectively):
for the second particle:
and for the third particle:
The propagation matrix [P_{ kl }] between particle l and particle k is diagonal, with elements (1, exp[iβ_{12}(z_{ k }−z_{ l })], exp[iβ_{12}(z_{ k }−z_{ l })]), where β_{ pq } = β_{ p }–β_{ q } is the propagation constant difference between the LP_{0p} and LP_{0q} modes (p < q). The same analysis applies for the backwardpropagating modes. The launched amplitudes of the three modes at the fibre input were estimated by calculating their overlap with a focused Gaussian beam (beam waist ~2.5 μm), yielding \({\mathbf{v}}_{{\mathrm{in}}}^{(1)} = (0.982,0,0)\), \({\mathbf{v}}_{{\mathrm{in}}}^{(2)} = (0,0.143,0)\) and \({\mathbf{v}}_{{\mathrm{in}}}^{(3)} = (0,0,0.081)\).
To calculate the axial optical force \(\vec F_k^p\) on the kth particle, for the incidence of the pth mode on the first particle, we first need to find the local intensity on each side of the particle. This involves taking the modulus squared of the sum of the local field amplitudes. The total optical force on particle k exerted by forwardtravelling light is then given by:
where P_{0}/2 is the total power incident in the forward direction (an equal amount is incident in the backward direction) and c is the speed of light in vacuo. The threeelement rowmatrix N has elements that correct for the overlap between the particle and individual mode shapes:
where a is the core radius, a_{p} is the particle radius and u_{0i} is the ith zero of the Bessel function J_{0}. For our experimental parameters, N_{1} = 0.250, N_{2} = 0.362 and N_{3} = 0.412. The total optical force on particle k is found by adding the complementary force \(\overleftarrow F_k\) exerted by the backwardpropagating light (see Supplementary Information S3).
A necessary (but insufficient) condition for binding occurs when the optical forces acting on all three particles are simultaneously zero. Figure 4b plots the lines of zero optical force on each particle as a function of the interparticle distances d_{12} and d_{23}. The diagonal line represents the symmetric configuration when d_{12} = d_{23} = d, and Fig. 4c plots the three forces vs. d in this case. All three forces are simultaneously zero at two positions, one of which is unstable (red dot) and the other is stable (green dot). This predicts an interparticle binding distance of d = 44.6 μm, in good agreement with the measurements. These binding positions are found to be relatively insensitive to the launched beam waist at each end of the fibre (in agreement with experiment), i.e., a change in beam waist from 2.5 to 2 μm results in the predicted interparticle distance varying from 44.6 to 48.1 μm.
The stiffness of the optical springs in each particle trap can be straightforwardly calculated by partial differentiation of the forces with respect to the particle displacement, resulting in a stiffness tensor [K]:
The components of the tensor are numerically calculated using the above model for d_{12} = d_{23} = d and P_{0} = 50 mW. The asymmetry of [K], e.g., \(k_{12} \ne k_{21}\), is caused by the fact that particles 1 and 2 scatter differently to particle 3, so that for the same values of displacement (\(\Delta z_1 = \Delta z_2\)), \(\Delta F_2 \ne \Delta F_1\). From Eq. (8) we can obtain the equation of motion for the collective oscillations \(\ddot A(t) + [{\mathbf{M}}]^{  1}[{\mathbf{K}}] \cdot A(t) = {\mathbf{0}}\), where [M] is a diagonal matrix with elements given by the mass of each particle (5.86 × 10^{–16} kg). The mechanical frequencies and shapes of the eigenmodes can then be calculated (Table 1), where A_{k} represents the oscillation amplitude of the kth particle of the corresponding eigenmode. The calculated eigenfrequencies agree with the observations with no free parameters. This theoretical model may be extended to analyse the case with more than three bound particles and to study the collective dynamics with external driving or dissipative terms.
Conclusions
In summary, multiple polystyrene microparticles can be optically bound by intermodal interference within the evacuated core of a HCPCF, where protected from environmental disturbance, their collective vibrational modes can be resolved. By adjusting the trapping pulse chirp and duration as well as the fibre dispersion, the binding length (40 µm in the experiments reported here) can be increased, potentially allowing particle binding over centimetre distances. By further increasing the average power of the pulsed trapping laser, it will be possible to achieve stable trapping and binding of particles tens of nanometre in diameter and to increase the number of bound particles within the array. Currently the lowest gas pressure is set by the increasing instability of optically trapped particles in vacuum^{6,12,13,14}. By using electronic feedback to stabilise the mechanical motion at ultralow pressure, it should be possible to reach higher mechanical Qfactors and to explore cooling to the groundstate, nonlinear coupling and synchronisation of the motion of multiple particles. Finally, the whole assembly of particles can be moved toandfro along the HCPCF, which may enable potential applications in remote sensing, with the boundparticle array being propelled into a region where external perturbations are present. A boundparticle array offers more mechanical degrees of freedom than a single trapped particle in a HCPCF, suggesting more complex forms of flying particle sensors^{15}. In addition, versatile multisensors can be constructed by binding particles with different physical properties.
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Acknowledgements
This work was supported by Max Planck Society. R.Z. acknowledges funding from the Cluster of Excellence “Engineering of Advanced Materials” at the FriedrichAlexander University in Erlangen, Germany.
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Bykov, D.S., Xie, S., Zeltner, R. et al. Longrange optical trapping and binding of microparticles in hollowcore photonic crystal fibre. Light Sci Appl 7, 22 (2018). https://doi.org/10.1038/s413770180015z
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DOI: https://doi.org/10.1038/s413770180015z
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