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Abstract
Orbital dynamics of a dielectric microparticle in air using a lensed counter-propagating dual-beam trap was studied experimentally and by numerical simulations. Relationships between the dynamic parameters, trap geometry, and optical power were examined both experimentally and computationally. We found that this scheme can provide narrow bandwidth ($\delta \nu /\nu \approx {10^{- 3}}$) detection that is at least two orders of magnitude below typical values attainable with previously studied geometries. We predict that this characteristic makes the approach suitable for ultrasensitive in-situ detection of particle mass changes. In our experimental conditions, silica microspheres orbited on trajectories spanning tens of µm, at frequencies of up to ${\sim}2\,\,\rm kHz$, at atmospheric pressure. With the help of simulations, we briefly discuss how the dual-beam lensed orbital trap approach can be further enhanced to gain unmatched capabilities to measure changes in the physical parameters associated with a particle interacting with its surrounding medium.
© 2022 Optica Publishing Group
1. INTRODUCTION
Since its 1970 demonstration [1], optical trapping has been widely used as a noncontact, noninvasive method to separate and characterize individual microscopic particles for a broad range of applications [2–4]. Optical traps, when coupled with other techniques or applied independently, have proven to be powerful tools to identify and quantify the changes in the physical parameters associated with a particle’s interaction with its surrounding medium for a broad range of phenomena ranging from biomolecular affinity and quantum friction to ultraprecise torque sensors [5–9]. Quantitative measurements of torque and other dynamic parameters can be performed with exquisite accuracy if trapping is carried out in a vacuum [9,10]. However, because of its relevance for atmospheric processes, there is also great interest in techniques that would allow resolving the state of aerosol particles in an interaction with a vapor phase [11].
In this work, we isolate and manipulate single particles at atmospheric pressure by employing an orbital optical trap, where the particle moves on a closed trajectory in a field of two counter-propagating, focused laser beams. While dual beam traps with optical fibers have been recently examined [12,13], it is not clear what conditions are needed to realize the balancing of optical forces and frictional drag required for a stable trajectory that would allow precise measurement of the orbital period, much like a suspension spring in a pendulum clock allows for the accurate measurement of the pendulum period.
Through numerical and experimental analyses of particle dynamics in a lensed dual-beam trapping scheme, we lay down a formalism to disentangle the key parameters leading to a particle’s dynamics. The emerging theoretical framework suggests that such orbital optical traps carry significant promise for real-time, accurate measurements of, for instance, in-situ kinetics of aerosol phenomena such as accretion, desorption, and aging, phenomena for which discrepancies between various models point to the need for additional experimental control [14]. In this context, it is worth mentioning that control of the rotational motion of a particle have led to a series of interesting microrheological findings [15–18].
A much less explored optomechanical effect is the orbital momentum that can be imparted to a particle by a pair of counter-propagating laser beams, where neither beam carries orbital angular momentum. The first experimental demonstration of a dielectric particle moving around a periodic orbit inside a dual-beam trap was acquired with cleaved optical fibers aligned to have an angular offset between them, with water as the medium surrounding the particle [19]. Later, this phenomenon was also observed in fiber-optic traps with a transverse offset between parallel beams [20–22]. However, in these experiments, the period was only a few Hz and the quality factor associated with the periodic motion was very low because of the high viscosity of an aqueous medium.
Subsequently, orbital motion of dielectric microparticles was achieved in both air [12] and low vacuum [13], with counter-propagating, dual-beam offset fiber optic traps. These studies used strongly divergent beams for particle trapping. Even though optical trapping has been demonstrated with lensed optical fiber [22,23], its applicability is limited by relatively short working distances even when combined with diffractive optical elements [24]. Hence, while convenient, fiber-optic-based traps have physical limitations related to geometric constraints, which limit spatial control. In addition, the fiber shaft precludes the realization of extended closed orbits, and may induce spurious aerodynamic effects. To address these challenges and more widely explore the possibilities of a free space experiment, we have studied trapping in air with a lensed, dual-beam assembly employing a pair of focused laser beams for particle control. This new trapping scheme adds another degree of freedom to the toolbox of optical manipulation. Using lenses instead of fibers allows the particle to move through and around either focus. Thus, our modified approach to dual-beam orbital trapping provides an additional mechanism to selectively control the distribution of optical forces in the trap. Importantly, as it will be shown later, a practical advantage of the lensed dual-beam scheme is access to orbits that are largely insensitive to the initial conditions (insertion location and initial velocity).
Fig. 1. Experimental setup for counter-propagating dual-beam trapping in air. Two counter-propagating laser beams of equal power and orthogonal polarization states are focused inside a sample chamber where trapping occurs. CW, continuous wave; BS, beam splitter; M, mirror; L, lens; NA, numerical aperture; $\lambda /2$, half-wave plate; QPD, quadrant photodiode; EMCCD, electron-multiplying CCD.
We analyze the dynamics of orbiting particles by measuring the period, local velocities around the trajectory, and the shape of the orbit. Under our experimental conditions, the power spectral distribution of particle’s motion exhibits sharp peaks with narrow widths, at least two orders of magnitude narrower than those associated with the Brownian oscillator motion of a trapped particle in air. The apparent quality factors are also considerably higher than what was reported for trapping with fiber optic traps in a low vacuum [13]. Since high mechanical quality factors are desirable for accurate determination of change in the orbital period upon particle size and/or mass change [25,26], our results have promising implications for the use of orbital optical trapping to probe the kinetics of surface reactions on levitated particles in gaseous environments at a broad range of pressures.
In addition, with the help of numerical simulations, we study the trajectory stability and shape as a function of the particle insertion location in the optical field, and examine why having control over the point of entry might be beneficial for the implementation of this system as a precision sensor.
2. EXPERIMENTAL DETAILS
A schematic of the experimental apparatus used for dual-beam orbital trapping of particles in air is shown in Fig. 1. Optical trapping of silica microspheres was performed with a 1064 nm, linearly polarized cw laser. The nearly Gaussian trapping beam was split into two halves of equal power using a nonpolarizing beamsplitter cube and focused by two identical microscope objectives (${40\times}$ S Plan Fluor ELWD, Nikon) of working distance 3.5 mm, inside the sample chamber made of glass with a wall thickness ${\sim}150\,\,\unicode{x00B5}{\rm m}$. The Gaussian beam waist radius in the focal plane ${{w}_0}$ measured by the knife edge method was 1.5 µm for both beams. The counter-propagating laser beams were parallel, and their linear polarization states were set to be orthogonal to each other with the help of a half-wave plate to avoid interference. The microscope objectives were placed on $xyz$ manual translation mounts with differential micrometers to aid the displacement of beams along the direction of laser propagation ($z$) and in one of the perpendicular directions ($x$). Gravity acts along the $-y$ direction. A low power, visible (632.8 nm) laser for imaging and detection purposes was aligned along the optical axis ($z$). Silica spheres of diameter $4.82\pm 0.38\,\,\unicode{x00B5}\rm m$ were initially dispersed on a silica glass coverslip. To overcome the attractive van der Waals and capillary interactions between particles and the glass substrate, the coverslip was oscillated with a piezoelectric transducer (70-2221, APC International, Ltd.). To achieve the high accelerations necessary to detach the particles from the glass surface, the piezoelectric transducer was driven with a high-power (${\sim}70\,\,\rm W$) square wave at the resonance frequency (${\sim}340\,\,\rm kHz$) of the piezoelectric ring for 1 millisecond. The ultrasonic launcher assembly was placed above the optical trap, as shown in Fig. 1. The coverslip holding the particles and the glass chamber formed a sealed enclosure to ensure protection from the air currents. The terminal speed of a descending particle was calculated to be ${\sim}2\,\,\rm mm/s$.
A long working distance objective (Mitutouyo Plan Apo 20X) was aligned perpendicular to the optical trap axis (along $y$ axis) to collect scattered light from the trapped particle. Half of the scattered light was directed to a quadrant photodiode to measure the displacement of particle along the axis of propagation of the laser beams ($z$), and along one of the transverse axes ($x$) as a function of time. The other half of scattered light was projected onto an EMCCD camera (iXon $+897$, Andor Technology) to visualize particle motion and, using short exposure times, to determine the local speed of the particle along its trajectory.
3. SIMULATION OF PARTICLE DYNAMICS
The optical force acting on a particle in a dual-beam optical trap was calculated using a formalism based on the ray (geometric) optics approximation, which is valid when the size of the particle is much larger than the wavelength of illumination. For our case of sphere diameters ${\sim}5\,\,\unicode{x00B5}{\rm m}$, this constraint is marginally satisfied. Under the ray optics approximation, a focused laser beam is considered to consist of a collection of rays and the forces acting on the particle due to individual rays reflecting and/or refracting at surfaces are calculated [27,28]. The optical force from a single ray was divided into two orthogonal components calculated using [27]
In Eq. (1), $dP$ denotes the differential power in an incident ray hitting a small patch of the surface area ($dS$) on the sphere and is given by
where $I$ is the intensity of the beam, which we describe with a Gaussian profile.Although most of the equations we use and present here are from [27], their version of our Eq. (3) is different. Instead of $\cos({\theta _i})$, they have the cosine of the sphere’s polar angle (measured from the $z$ axis), where it is struck by the incident ray. Since this angle can exceed ${90^ \circ}$, its cosine may be negative, which has the unphysical implication $dP \lt 0$. Another difference is that the expression for the parameter ${R_z}$ is attributed to ${R_c}$.
Optical forces from individual rays from both beams were integrated over the whole surface area of the particle illuminated by the beam and their projections along the $z$ and transverse $x$ directions were calculated. The total forces when the sphere’s center is at $\vec r$ are called axial $\hat z{F_z}(\vec r)$ and transverse $\hat x{F_x}(\vec r)$.
A spherical particle moving in an optical trap at the location $\vec r$ is also influenced by a viscous drag force from the surrounding medium:
Where $\eta = 1.81 \times {10^{- 5}}\,\,{\rm Pa}\cdot \rm s$ is the dynamic viscosity of air [30], ${r_0}$ is the radius of the silica sphere, and $\vec v(r,t)$ is its velocity. The trajectory of a trapped particle was determined by solving Newton’s equation of motion in two dimensions: where $m$ is the mass of the particle. Others have gone further than our simple treatment. For instance, one can allow for random Langevin forces in the medium in addition to viscous drag [12,13]. A full 3D calculation of forces can also be done, including forces and motion in the $y$ direction [12,13]. We have done a few 3D calculations but find that the motion in the $xz$ plane is scarcely changed, with the excursions along $y$ being a small fraction of a micron. The results presented here were all done in 2D within the $xz$ plane.The particle positions at different times separated by small increments ($\delta t$) were computed by numerically solving the finite-difference form of Eq. (5) using a Runge–Kutta fourth-order approach. Typically, $\delta t$ was taken to be 10 µs or less. The initial velocity of the particle was kept below 1 µm/ms in the $xz$ plane to roughly match its magnitude under our experimental conditions. When the beam powers and focusing lenses are identical, the optical forces vanish half way between both the two focal planes and the two beam axes. This point is chosen as the coordinate system origin. We call $d$ the separation between the axes and $s$ the separation between the focal planes. For $d \gt 0$, the axis of the beam coming from the right (left) crosses the $x$ axis at $x = d/2(- d/2)$. We assume the beam on the right (left) has its focal plane at $z = s/2(- s/2)$, where $s \gt 0$. These choices suggest that the motion away from the origin should be counterclockwise. This qualitative behavior occurs in all our calculations, after the decay of initial transients. A further consequence of our choices of common beam powers and identical lenses is that the optical forces are antisymmetric under inversion through the origin. This means that
The reason for this antisymmetry is that inverting the 2D coordinates of the sphere switches which focal plane and beam axis will be the closest. In all our calculations, $d$ will be positive and, except where explicitly noted, $s$ will be positive too, and usually 35 µm. The two beam powers are identical, ${P_1} = P = {P_2}$ with $P$ usually 0.25 W. In most of our calculations, the silica sphere’s radius (${r_0}$) is 2.5 µm.
Fig. 2. Calculated values of optical forces acting on a silica sphere of radius 2.5 µm in the (a) transverse and (b) axial directions in the vicinity of the trapping geometry center [$(x,z)=(0,0)$], at different laser powers ${P}$. The dashed lines indicate the position of the focal planes.
4. RESULTS AND DISCUSSION
The optical forces acting on a silica sphere in a trapping configuration with two collinear ($d = 0$, $s = 35\,\,\unicode{x00B5}{\rm m}$) and counter-propagating beams are presented in Fig. 2. As expected, raising the common laser power $P$ smoothly increases these forces. When the sphere is on the $x$ or $z$ axis, it feels a restoring force only in the transverse or axial direction in Figs. 2(a) or 2(b). Close to the origin, these forces are linear in the displacement from the origin. Their limiting slopes give an estimate of the trap stiffness, which here is ${\approx} {10^{- 5}}\,\,\rm N/m$ for either direction. If a sphere is released from rest on an axis, it moves along the axis like a damped oscillator, coming to rest at the origin. If the sphere starts instead at an off-axis point, it follows a decaying Lissajous path to the origin. Figures S1(a) and S1(b) in Supplement 1 illustrate these behaviors. Further away from the origin, the magnitude of the transverse force reaches a maximum more quickly than the axial force, which does not have a maximum until $z$ has moved past a focal plane. As the sphere, moving along the $z$ axis and away from the origin, goes through a focal plane, the rays striking it switch from diverging to converging, resulting in additional structure in the axial force profile. Finally, we note that the maximum restoring force along the axial direction is several times higher than the restoring force along the transverse direction, in contrast to single beam optical traps [31].
Figure 3 compares the distribution of optical forces over two dimensions for a zero or nonzero beam transverse offset. In both cases, the antisymmetry under inversion of the optical force field is obvious. When the beams are collinear in Fig. 3(a), the optical force is directed mostly toward the center, which leads to the motions discussed in the previous paragraph. However, when a transverse offset is present as in Fig. 3(b), a “swirling” of the force field becomes evident, suggesting that rotational orbits will be possible. In this case, the curl of the total force is nonzero around the trap center. Thus, the force cannot be generated by a Hamiltonian of the familiar isotropic kinetic energy ${+}$ scalar potential type [32]. A line integral of the optical force on a closed path that surrounds the origin will be nonzero, which means that the forces are nonconservative and that the viscous drag is necessary to stabilize an orbit. Another subtle feature of Fig. 3(b) is that, contrary to the impression gained by looking at the red arrows, the beam axes are not at $x=\pm 3\,\,\unicode{x00B5}\rm m$ but, in fact, are at $x=\pm\frac{d}{2}=\pm 1.5\,\,\unicode{x00B5}{\rm m}$. Only by going to larger values of $d$ do the separate effects of the two beams become clear. See Fig. S2(a) in Supplement 1 for the case with $d=6\,\,\unicode{x00B5}\rm m$ along with the force field profile of a single beam in Fig. S2(b) in Supplement 1.
Fig. 3. 2D distribution of the optical force field in a dual-beam optical trap with (a) zero transverse offset and (b) for a transverse offset of 3 µm. Axial foci separation, $s=+35\,\,\unicode{x00B5}\rm m$, radius of particle = 2.5 µm. Arrows indicate the direction of force vectors. The optical power from each laser was 0.25 W.
Examples illustrating different stable closed trajectories, after transients (typically less than 5 ms), associated with different transverse offsets are shown in Fig. 4. The common beam power is again 0.25 W. The calculations were all started with the sphere stationary at $(x,z)=(0,1\,\,\unicode{x00B5}\rm m)$. We have plotted the results in different ways to emphasize different features. For Figs. 4(b) and 4(c), we used identical ranges for the $z$ and $x$ axes, adapted to the size of the orbit. This prevents the force vectors from appearing to be rotated due to a stretching of one axis. The force vectors then better appear to be “guiding” the sphere along its trajectory. We have used quotes here because forces only directly produce accelerations, not velocities or locations. In addition, the plotted force field omits the drag force. The orbit in Fig. 4(b) is confined well inside the focal planes and its frequency is 1587 Hz. The projections of the sphere’s location along the $x$ and $z$ axes were analyzed as a function of time, after it had settled into the stable orbit (Fig. S3b). The trajectory in Fig. 4(c) is not a stable orbit for our viscosity. However, if the viscosity is reduced by a factor of 2, a stable, nearly elliptical orbit is obtained.
Fig. 4. Simulated trajectories of a silica sphere of radius 2.5 µm along with the optical force distribution in a counter-propagating dual-beam trap with an axial offset of ${+}{35}\;{\unicode{x00B5}{\rm m}}$, and a transverse offset of (a) 6 µm, (b) 4 µm, and (c) 2.3 µm. The trajectory in (c) corresponds to the dynamics of the particle for a duration of 10 ms. The direction of the particle’s motion is counterclockwise in all three cases.
For the case of Fig. 4(a), we have stretched the axes to include the whole orbit. The ratio of the orbit’s length/width is nearly 10 and the orbit extends past each focal plane. Such an orbit is physically impossible in a dual fiber trap. There are noticeable dips in the trajectory when the sphere is close to the focal points of the beams. These can be understood by zooming in on sections of the orbit, as shown $i$ Fig. S4 in Supplement 1. The orbit’s frequency is 474 Hz. The time-domain signal for the sphere’s location around this orbit is shown in Fig. S3(a) in Supplement 1. The $x$ component shows a fine structure when the sphere is in the region of the dips. The electric field intensity distribution inside the trap corresponding to the trapping schemes shown in Figs. 4(a) and 4(b) can be found in Fig. S5 in Supplement 1.
Fig. 5. (a) EMCCD-acquired experimental trajectory [illustrative of case (a) in Fig. 4]. (b) Distribution of orbital velocity estimated from short-exposure snapshots. (c) Power spectrum derived from particle’s motion along the axial direction ($z$) by the position-sensitive detector for the same trajectory.
There are some common features of the stable, single-loop orbits in Figs. 4(a) and 4(b). The sphere moves counterclockwise around its orbit and each orbit is centered about the origin. Indeed, the orbits have an inversion symmetry about the origin in that, if a point $(x,z)$ is on the orbit then so too is the point $(- x, - z)$, and its velocity there is the negative of that at $(x,z)$. These symmetries arise from the antisymmetry of the optical forces in Eq. (6), but do not always occur. Orbits that lack such inversion symmetry will be shown later.
Experimental results for the particle dynamics in a dual-beam trap with a transverse offset of 6 µm and an axial offset of ${+}35\,\,\unicode{x00B5}\rm m$ are presented in Fig. 5. The electron-multiplying CCD (EMCCD) image of particle’s orbit acquired at an exposure time of 3 ms/frame is shown in Fig. 5(a). The maximum displacements of the particle along the axial and transverse axes agree well with the simulation results for the same experimental parameters [Fig. 4(a)]. The speed of the particle at different locations on its trajectory is mapped along the orbit in Fig. 5(b). According to our calculations, the particle attains maximum speed in the vicinity of the laser foci, while moving along the dips in the trajectory shown in Fig. 4(a). Even though these fine features of the orbit are not well resolved in the EMCCD image, it is evident from the speed distribution that the particle moves the fastest on the quasi-linear segments of the trajectory where the dips are expected. We note that the maximum measured velocity (10 s of cm/s), and the particle diameter are similar to values encountered in the updraft of cloud aerosol particles [33], which makes lensed orbital trapping interesting for controlled laboratory experiments of atmospheric relevance.
The scattered light power spectrum result of the microsphere motion along $z$ axis is shown in Fig. 5(c). The power spectral distribution was calculated using
To further test the computational approach against the experiment, we have examined how the orbital frequency and spatial extent of the trajectories change separately with transverse offset, axial offset, and laser power, as shown in Fig. 6. Figures 6(a) and 6(b) show the changes in orbital frequency and maximum axial displacement of the particle (${z_{{\max}}} = {z_{{\rm largest}}} \gt 0 - {z_{{\rm smallest}}} \lt 0$), respectively, upon varying $d$ with $s$ and $P$ held constant.
Fig. 6. (a) Orbital frequency and (b) maximum axial displacement as a function of the transverse offset between the beams ($s=35\,\,\unicode{x00B5}\rm m$ and ${P=0.25\,\,\rm W}$). (c) Change in orbital frequency and (d) maximum axial displacement upon varying the axial offset between the laser foci ($P=0.25\,\,\rm W$ and $d=3.5\,\,\unicode{x00B5}\rm m$). Experimental and numerical analyses of change in (e) orbital frequency and (f) maximum displacement upon changing the total laser power ($d=4\,\,\unicode{x00B5}\rm m$ and $s=35\,\,\unicode{x00B5}\rm m$).
On increasing $d$, the orbital frequency decreases with a more rapid drop around $d=5\,\,\unicode{x00B5}\rm m$. In contrast, the orbit size increases with $d$ with a more rapid rise near $d=5\,\,\unicode{x00B5}\rm m$. The typical speed of the sphere slightly increases over the range of $d$, so the key to the frequency decrease is the large increase in the orbit size, as shown in Figs. 4(a) and 4(b) and below.
Next, we explored how particle dynamics are changed by variations in $s$ at fixed $d$ and $P$, as shown in Figs. 6(c) and 6(d). The changes are not monotonic in $s$: The orbital frequency first increases and then decreases after $s=30$, while ${z_{{\max}}}$ has a shallow minimum near $s=30$. Again, the rationale is based on considering the orbit size. When $s$ is small, ${z_{{\max}}}$ is considerably larger than $s$ and a large orbit implies (with only a modest increase in speed) that the orbital frequency will be smaller. In the limit of large $s$, the optical forces near the origin become very weak and the viscous drag leads to lower speeds in orbits of nearly constant size, which implies a decrease in orbital frequency. In a lensed, dual-beam trap, both negative and positive axial offsets can be realized. Crossing the two foci for negative axial offsets destabilizes the optical trap leading to particle escape. This result was also predicted by the theoretical model (Fig. S6).
Finally, we studied the change in particle response when the common laser power of the beams $P$ is varied while keeping $d$ and $s$ fixed, as shown in Figures 6(e) and 6(f). When increasing the optical power, the particle experiences an increased magnitude of optical force resulting in an increase in the orbital frequency. Interestingly, the orbit size quickly reaches a saturation value while the frequency continues to grow. To maintain a balance between the optical and drag forces, the speeds in an orbit must increase when $P$ increases, leading to greater frequencies since ${z_{{\max}}}$ is saturating. With reasonable laser powers, kHz frequencies can be attained inside a dual-beam lensed orbital trap.
To summarize our results so far, there is good agreement between experiments and simulations for the dependencies on $d$, $P$, and $s$, which suggests that the several approximations made in constructing our theoretical model are reasonable.
Fig. 7. (a) Scattered light power spectral distribution (${S_z}(\omega)$) for the particle’s oscillation inside the trap when the laser beams are perfectly collinear. (b) Power spectrum showing the orbital frequency ($\nu$) of trapped particle when the beams have a transverse offset of 3.5 µm. Peaks corresponding to higher harmonics ($2\nu$, $3\nu$) of the orbital frequency also are present in the spectrum. For both the cases, $s=35\,\,\unicode{x00B5}\rm m$ and $P=0.25\,\,\rm W$.
Since, for a given pressure, the dynamics of the particle in the trap depends on its size, mass, and composition, changes in the dynamics, measurable as frequency shifts and/or orbit shape can be indicative of particular changes in the particle properties. Stimulated by this idea, we now turn our attention to the question of whether orbital trapping could qualify as an analytical technique for the in-situ characterization of the chemical and physical changes occurring on single particles in interaction with a gaseous medium. Another interesting prospect would be to apply this technique to perform quantitative measurements on levitated micro-droplets. In this regard, good control over the rotational frequencies is necessary since a rotating droplet might stay intact, deform, or even break-up due to competition between surface tension and centrifugal forces [34–36]. So far, we have seen that one can obtain sharp frequency peaks under experimental conditions that are relevant to atmospheric ones, which have not been explored much yet (for instance, accretion at drift velocities relevant to updrafts in clouds).
A comparison of the power spectra of scattered light collected from a particle under two different trapping scenarios is given in Fig. 7. The power spectrum for periodic oscillation of particle in a dual-beam trap with coaxial beams, as shown in Fig. 7(a) was fit with [37,38]
Compared to the oscillatory motion in a traditional trap, the sustained orbital rotation of a particle results in scattered light power spectra with much sharper peaks. Comparable spectral linewidths for conventionally trapped microspheres have only been attained in vacuum conditions [38].
From the point of view of atmospheric aerosol experimental studies, observation of reaction kinetics on levitated particles often require broad time scales, from ms to hundreds of seconds. At the maximum bandwidth of the acquisition system, the observation time could be extended to 8 s. At these extended data acquisition times, the Fourier spectrum of a particle in an orbital trap with a transverse offset of 5.5 µm showed an intriguing form of broadening. The band corresponding to the fundamental frequency contained a collection of distinct peaks spanning a frequency range of less than 8 Hz, as shown in Fig. 8. The linewidth associated with the individual peaks from this distribution was about 0.2 Hz. The existence of multiple, well-separated peaks in the power spectrum suggests that there are different metastable dynamic states, and the actual trajectory might not be described by a simple single orbit as the calculations suggest. Later on, we will discuss a possible explanation for this result, and a pathway to an even higher quality factor (and thus, sensitivity) for the orbital optical trap. For now, it is worth noting that, if a trapping scheme could be engineered to stabilize the particle in one of the states corresponding to a single narrow peak in Fig. 8, it should be possible to obtain very narrow linewidths, thus enhancing sensitivity to changes in mass or size of the particle. In this regard, the orbit size saturation with power, as shown in Fig. 6(f), suggests that it might be possible to use a phase-lock loop detection with feedback control on the trapping power to stabilize the fundamental frequency and select a single peak (0.2 Hz wide) in the Fourier spectrum.
Fig. 8. (a) Scattered light power spectrum (${S_z}(\omega)$) for a particle in a dual beam trap with transverse offset ($d=5.5\,\,\unicode{x00B5}\rm m$, $s=35\,\,\unicode{x00B5}\rm m$, and ${P=0.25\,\,\rm W}$), for the maximum number of data points (total time: 8 s) at a maximum sampling rate. (b) Power spectrum corresponding to the dynamics of a particle for the next 8 s. A set of narrow peaks can be seen within a frequency range of 10 Hz in both the spectra.
The scattered light power spectrum under the same experimental parameters, recorded for a wider frequency range can be found in Fig. S7 in Supplement 1.
Calculations based on the ray optics approximation, as shown in Fig. S8 in Supplement 1, indicate that the orbital frequency varies nearly linearly with respect to the particle radius. For the aforementioned experimental parameters, a change in frequency of 0.2 Hz would correspond to a change in the particle radius of 2 nm. This looks extremely promising from the point of view of growth/accretion kinetics studies.
However, when the transverse offset between the beams was decreased to $3\,\,\unicode{x00B5}\rm m$, the linewidth associated with the sharpest peak in the distribution (power spectrum given in Fig. S9 in Supplement 1) was adequate to resolve a change in radius of 0.5 nm (Fig. S8b in Supplement 1) or about a 0.06% mass change of particle under consideration. Thus, by tuning the laser parameters, orbital trapping can be made extremely sensitive toward deposition or loss of materials on the surface of a trapped particle.
To discern the appearance of multiple peaks in the power spectrum, we investigated whether it is feasible for the particle to progress along more than one trajectory under the same experimental conditions. To this end, we simulated the dynamics of a particle by systematically varying its point of entry in the optical field in a dual-beam trap with a transverse offset of 5 µm, as shown in Fig. 9. A similar analysis for a trapping scheme with $d=4\,\,\unicode{x00B5}\rm m$ can be found in Fig. S10 in Supplement 1. Our survey results indicate that, depending on where it starts, the particle could be advancing along one of the three distinct periodic orbits, as shown in Fig. 9, color-coded in red, yellow, and pale blue. The regions of instability in the trap are highlighted in indigo. The red orbit has an inversion symmetry and a frequency of 621 Hz. The pale blue and yellow trajectories are degenerate orbits. They both have an asymmetric structure, but are related to each other by inversion through the origin. This means that if a point ($x,z$) is on a yellow orbit, the point (${-}x, - z$) is on a blue orbit. They also differ from the rest of the orbits we have seen so far in that they are double-loop orbits. The particle spends the same amount of time in the individual loops of the orbit. The orbital frequency (associated with a single loop) in this case is 1266 Hz. The $z$ projection of these orbits as a function of time are presented in Fig. S11 in Supplement 1. If the particle is initially launched in the area ranging from ${-}{30}\;{\unicode{x00B5}{\rm m}}$ to ${+}{30}\;{\unicode{x00B5}{\rm m}}$ in the axial ($z$) direction and ${-}{10}\;{\unicode{x00B5}{\rm m}}$ to ${+}10\,\,\unicode{x00B5}{\rm m}$ in the transverse ($x$) direction, it will be difficult to predict the exact orbit along which the particle moves during the course of an experiment since even a mere 2 µm difference in the point of entry along either axis can entirely change the course of the particle. Unfortunately, all three orbits lie in this area and, as evident from Fig. 9, there are multiple locations at which the orbits crossover. Hence, even a slight external perturbation, such as thermal fluctuations, could result in the particle migrating randomly from one orbit to another. Thus, we propose that the presence of multiple peaks in the power spectrum can be attributed to orbit cross-over. The real trajectory of the particle may not be a simple, closed orbit as predicted by our theory, which at this time does not include any random, Langevin forces.
Fig. 9. Map of the dependence of the adopted trajectory on the starting point of a microsphere in the optical field. Bottom graph: Typical orbits along which the particle would advance, if it were to start in the yellow, pale blue, or red patches in the block diagram. There are multiple crossovers between the orbits. The regions of instability, where the optical forces are not strong enough to trap the particle, are color-coded in indigo. The yellow and pale blue orbits are asymmetric, and correlated by inversion through the origin ($d=5\,\,\unicode{x00B5}\rm m$, $s=35\,\,\unicode{x00B5}\rm m$, and $P=0.25\,\,\rm W$).
Our analysis suggests that, to obtain superior resolving power, particle motion must be stabilized along a single orbit. Although there are various sites in the trap where the orbits overlap, the orbital frequency associated with the red trajectory in Fig. 9 is nearly half of that of the other orbits. Therefore, we hypothesize that once the particle’s dynamics are initiated along the red orbit, shifting to another path might be unfavorable. To attain selectivity toward the red orbit, the particle must be loaded at least 30 µm away from the origin in the axial direction. With our current experimental setup, preferential launching of particles into specific regions in the trap cannot be achieved. Future experiments will address this challenge.
The possibility of more than one type of particle trajectory for the same laser parameters complicates the trends shown in Fig. 6. However, the orbits we observe in our experiments always closely match with one of the orbits predicted by the theory in its shape and frequency. Whether other choices of $s\!$, $d$, $P$, and ${r_0}$ could lead to different types of orbits is yet to be fully explored. A complete optimization of orbital trapping is worth pursuing, in view of the potential of this technology to study systems mimicking atmospheric aerosol particles in reactive environments. Previously, quantitative measurements of atmospherically relevant reactions on levitated, solid particles were achieved by monitoring the time evolution of Raman signatures of the trapped particles [39–41]. Kinetics of evaporation, hydration, and coagulation of levitated micro-droplets were studied in general by combining optical trapping with other optical techniques [42–45]. The approach presented here is amenable to in-situ simultaneous spectroscopic and precise physical properties measurements.
5. CONCLUSION
Orbital optical trapping of a microparticle in air was demonstrated with a lensed, dual-beam trap with a transverse offset. The characteristics of particle dynamics were analyzed as a function of the optical power, the transverse offset between the beams, and the axial offsets between the laser foci, both experimentally and computationally. It was shown that the insertion point of the particle in the optical field determines the nature of the trajectory in general, but there are areas that are insensitive, in terms of the trajectory, to where the particle enters. Under our experimental conditions, spectral features associated with orbital trapping exhibit narrower orbital frequencies than the resonant frequency characteristic of conventional optical trapping schemes. Considering that the changes in orbital frequencies can be correlated to changes in the physicochemical properties of the particle in real time, this approach could be useful in the future as an analytical tool to detect chemical transformations in single particles and levitated microdroplets held in a controlled environment.
Funding
Army Research Office (W911NF1310490).
Acknowledgment
The authors thank Dr. Irina Tsvetkova for helpful discussions and reviewing the manuscript in its early form.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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