When introducing a nanoparticle into an optical trap, its mass and shape are not immediately apparent. We combine a charge-based mass measurement with a shape determination method based on light scattering and an analysis of the damping rate anisotropy, all on the same set of silica nanoparticles, trapped using optical tweezers in vacuum. These methods have previously only been used separately, and the mass determination method has not been applied to asymmetric particles before. We demonstrate that the combination of these classification techniques is required to distinguish particles with similar mass but different shape, and vice versa. The ability to identify these parameters is a key step for a range of experiments on precision measurements and sensing using optically levitated nanoparticles.

With a rapidly increasing number of developments over the recent years, levitated nanospheres have evolved into an exciting platform for innovative measurement opportunities and applications. Demonstrated applications span from the manipulation of microscopic biological systems1–4 to ultra-sensitive accelerometers and force-sensors,5–9 torque detectors,10–13 hyper-fast mechanical rotors,14–16 and measurements on thermal diffusion.17–21 Numerous proposals in recent years have explored the potential of using isolated nanometer-sized particles for probing gravitational waves,22,23 to observe quantum gravity,24–26 to employ in the dark matter scattering experiments,27,28 or to trial quantum collapse models.29 For numerous of these8–13,27,28 and other30–34 applications, knowing the precise mass and morphology of the levitated particles is essential. When introducing a particle into an optical tweezer, however, its shape and mass are not immediately apparent. Particles from a monodisperse solution of spheres have been observed to regularly carry some non-negligible ellipticity35 or they may aggregate to form composite structures.10 

While mass determination of charged particles and shape determination through the anisotropy of light scattering and the damping rate have been demonstrated on an individual basis, a comprehensive correlative study of these techniques applied to a set of differently shaped and sized particles remains unexplored. In this work, we present the shape and mass determination for optically levitated silica nanoparticles of various sizes and shapes in vacuum. We demonstrate that the combination of these classification techniques is required to discriminate between different particles. We manage to capture both single particles as well as aggregated structures in an optical tweezer trap by adjusting the concentration of a monodisperse solution of nanospheres.10 As illustration of some of the typical shapes that we encounter, Fig. 1 shows a scanning electron microscope picture of our solution. To unambiguously determine the mass and shape of the optically levitated nanoparticles, we combine the above-mentioned in situ classification methods.

FIG. 1.

Two pictures revealing possible shapes of the (composite) nanoparticles used for this paper, taken using a scanning electron microscope. The black patches represent holes in the holey carbon substrate. The left picture shows some of the nanoparticles used for trapping, where each sphere has a diameter of 142 ± 4 nm, according to the manufacturer. The panels on the right show a close-up of several composite structures.

FIG. 1.

Two pictures revealing possible shapes of the (composite) nanoparticles used for this paper, taken using a scanning electron microscope. The black patches represent holes in the holey carbon substrate. The left picture shows some of the nanoparticles used for trapping, where each sphere has a diameter of 142 ± 4 nm, according to the manufacturer. The panels on the right show a close-up of several composite structures.

Close modal

Specifically, we extend the mass determination that was previously demonstrated for nanospheres36,37 to asymmetric compositions of nanospheres, using similar and smaller sizes. We combine this method with the shape determination from a non-isotropic damping rate due to residual background gas.10,38 In addition, by employing a secondary probe laser, we find the particle's morphology through its angle-resolved Rayleigh scattering profile.35 We discuss the reliability of each method individually as well as their combined results. Our approach does not rely on precise calculations of the moment of inertia or the polarizability of the particle.

The structure of this paper is as follows. Following an introduction of the experimental setup, we present the methods and results from each individual classification technique to determine the properties of a fixed set of particles. After that we correlate and discuss the combined results.

We combine a number of standard techniques39 to optically trap nanoparticles in a controlled pressure environment. A schematic of our experimental setup is shown in Fig. 2. We use the light of a Coherent Nd:YAG Mephisto MOPA trapping laser ( λ trap = 1064 nm) and a spatial filter (SF) to obtain a Gaussian trapping beam. Using a half-waveplate (HWP), we control its polarization. The light then enters the vacuum chamber through a window and is focused using a microscope objective (MO; NA = 0.8), establishing the dipole trap at its focus point. The strong and inhomogeneous electric field at the focus creates a gradient force on the dielectric nanoparticles, allowing them to be trapped at the focus. We evaporate silica ( SiO 2) nanospheres (diameters 103 ± 6 , 142 ± 4 nm; Microparticles GmbH) from an ethanol solution using a medical nebuliser40 at ambient pressure near the trapping region. When a droplet from the cloud of ethanol, carrying a nanoparticle, passes through the optical trap, the particle can be caught. Once the particle is trapped, we reduce the pressure toward the operational domain between 20 and 0.1 mbar.

FIG. 2.

A schematic representation of the experimental setup. Details are presented in the main text.

FIG. 2.

A schematic representation of the experimental setup. Details are presented in the main text.

Close modal

Surrounding the trapping region, we have placed two copper electrodes (CE), with which we create a controlled electric field at the location of the trapped particle. We placed a discharge electrode (QE) to be able to charge/discharge the trapped nanoparticle.36,37 The electrode setup is used in the mass measurement of the nanoparticle. To visualize the trapped nanoparticle and to measure its angular Rayleigh scattering profile, we use the light of a probe diode laser ( λ probe = 660 nm) and overlap this beam with the trapping beam using a dichroic mirror (DM). A fraction of the scattered light at λ probe is collected using a CMOS camera. To track the dynamics of the particle inside the trap, we collect the transmitted trapping light using a collection lens (CL) and guide it toward a series of beamsplitters. We first use 10% of the remaining light for the angular detection, by employing a polarizing beam splitter (PBS) and a differential photodiode (PD θ), which is balanced by another HWP. The final 90% is used for the X and Y detection, which include a D-shaped mirror and a differential photodiode (PDX and PDY), to measure the spatial intensity differences within the beam profile.

For each particle whose data are represented in this paper, the same experimental procedure was conducted. A detailed description on this procedure is given in Appendix A of the supplementary material. The particle is essentially cleaned from possible residual water in the porous internal silica structure37,41,42 and charged to about 6–10 charges in preparation of the mass measurement. Once ready, we perform the mass measurements and record the angular Rayleigh scattering profile in the harmonic regime at around 20 mbar. We then periodically record the particle's signal during a slow pump-down, to determine its damping rate as a function of pressure.

When the translational damping of the particle is sufficiently high ( 3 mbar), its motion is primarily described by the (single-sided) linear power spectral density (PSD)43,44
(1)
which can be directly obtained from the signals recorded by PDX and PDY. Here, Ω 0 represents the natural oscillation frequency and Γ 0 is the damping rate of the nanoparticle due to the background gas, both in rad / s. The mass of the particle is denoted with m, and kB and T are the Boltzmann constant and the heatbath temperature, respectively. The subscript F is used to denote that Eq. (1) is driven by the thermal fluctuation force. Figure 3(a) shows a typical translational PSD from a spherical nanoparticle. The cyclic frequencies of the X and Y channels are not degenerate due to some ellipticity in the optical trap.45 We obtain the natural frequency Ω 0 and the damping rates Γ 0 by fitting Eq. (1) to Fig. 3(a). To obtain an impression of the ellipticity of the particle's morphology, we also use the torsional PSD, recorded by PD θ. Typical torsional PSDs for three different particles are shown in Fig. 3(b). The three particles were classified as a nanosphere, a dumbbell, and a triangle trimer, using the techniques described in this paper. We observe similar trap frequencies Ω 0 (and, therefore, similar trap stiffness) for the translational motion of these three particles.
FIG. 3.

Typical power spectral densities (PSD) of the trapped nanoparticles. (a) The transverse (X,Y) motion of a spherical nanoparticle (particle 15 from Fig. 7). The dashed/dotted lines indicate the best-fit of Eq. (1); (b) PSDs of the signal recorded by PD θ for three particles (15, 18, and 6 from Fig. 7); The additional peaks in both insets correspond to the Z, X, and Y motions of the particle. (See Appendix C of the supplementary material for further details.) The highlighted parts in both insets indicate the regions where the signal of interest dominates, which are used for the analysis.

FIG. 3.

Typical power spectral densities (PSD) of the trapped nanoparticles. (a) The transverse (X,Y) motion of a spherical nanoparticle (particle 15 from Fig. 7). The dashed/dotted lines indicate the best-fit of Eq. (1); (b) PSDs of the signal recorded by PD θ for three particles (15, 18, and 6 from Fig. 7); The additional peaks in both insets correspond to the Z, X, and Y motions of the particle. (See Appendix C of the supplementary material for further details.) The highlighted parts in both insets indicate the regions where the signal of interest dominates, which are used for the analysis.

Close modal

In linearly polarized light, the asymmetric susceptibility tensor of a non-spherical particle will introduce a torque to the system that causes the particle to align its major axis of polarizability with the polarization of the trapping light. One of the consequences of this laser-induced alignment is that the measured damping rates in the X and Y channels become unequal. This anisotropy in the damping rates can be used as a measure of the asymmetry of the particle's shape.10,38 Figure 4(a) shows the damping rates of a nanoparticle, which was classified as a dumbbell, as a function of the pressure. Figure 4(b) shows the corresponding ratio Γ 0 ( y ) / Γ 0 ( x ), as well as that of a sphere and a triangle trimer. The nanodumbbell shows a high degree of asymmetry, with a ratio Γ 0 ( y ) / Γ 0 ( x ) 1.27, which is in agreement with that of a dumbbell with a length-to-diameter ratio close to 1.7 (Ref. 10). Meanwhile, the sphere ( Γ 0 ( y ) / Γ 0 ( x ) 1.03) and trimer ( Γ 0 ( y ) / Γ 0 ( x ) 1.11) appear to be much more symmetric in the XY plane of the setup, as is in agreement with simulated results,38 and they require additional classification techniques to be unambiguously distinguished from one another.

FIG. 4.

(a) The damping rate of a nanoparticle as a function of pressure, corresponding to particle 3 from Fig. 7; (b) the ratio Γ 0 ( y ) / Γ 0 ( x ) for a set of different particles, which were classified as a sphere, a dumbbell, and a triangle trimer. The particles correspond to 15, 3, and 6 from Fig. 7, respectively.

FIG. 4.

(a) The damping rate of a nanoparticle as a function of pressure, corresponding to particle 3 from Fig. 7; (b) the ratio Γ 0 ( y ) / Γ 0 ( x ) for a set of different particles, which were classified as a sphere, a dumbbell, and a triangle trimer. The particles correspond to 15, 3, and 6 from Fig. 7, respectively.

Close modal

In addition to the asymmetric damping rates, we also record the angle-resolved Rayleigh scattering profile of the particle, which was recently demonstrated to allow the detection of asymmetries in the particle's morphology down to a few nanometers35 in the XY-plane of our setup. To realize this procedure, we use the probe laser (co-propagating with the trapping laser along the Z axis) to illuminate the nanoparticle inside the optical trap at a different wavelength ( λ probe = 660 nm) and lower power ( 2 mW) from the trapping laser ( λ trap = 1064 nm; 330 mW). A HWP is used to rotate the linear polarization of the trapping laser, while keeping that of the probe laser fixed. The laser induced optical alignment of an asymmetric particle then allows us to effectively rotate it about the longitudinal Z axis of our setup. The scattered light is then detected at a right angle using a 12-bit CMOS camera, from which we track the averaged intensity over a closed pixel region around the particle. To improve the signal-to-noise ratio, we image the nanoparticle slightly out of focus, such that we could adjust the gain of the camera without causing the pixels to immediately saturate. A dichroic mirror is used to prevent the scattered light at the trapping wavelength from reaching the camera.

Theoretically, the expected scattering intensity can be obtained from the particle's susceptibility tensor as follows:35 
(2)
where χ 0 represents the susceptibility tensor in the particle's eigenframe, and R is the rotation matrix to map to the laboratory frame. For the case of a dumbbell specifically, values for χ 0 ( y y ) / χ 0 ( x x ) have been computed10 and suggest I min / I max 0.75 when the length to diameter ratio reaches close to 1.7. Figure 5 shows the resulting data for three particles, which were classified as a nanosphere, a dumbbell, and a triangle trimer. The susceptibility tensor of the particle in Fig. 5(a) appears to be almost fully symmetrical in the XY-plane, however, the data reveal some small ellipticity of the particle at I min / I max 0.99. In contrast, the particle in Fig. 5(b) shows a high degree of asymmetry between χ 0 ( y y ) and χ 0 ( x x ), with I min / I max 0.72, and most likely resulted from a dumbbell. The particle in Fig. 5(c), however, shows far less contrast and could not be distinguished from a single elliptical nanosphere based on the scattering data alone. By including its mass and damping rate measurements, this particle was classified as a triangle trimer.
FIG. 5.

Measured Rayleigh scattering profile for several different particles. Each particle shown in this figure consisted of a number of nanospheres, with an individual size of d = 103 ± 6 nm. The data were taken from particles 1, 3, and 6 from Fig. 7, respectively.

FIG. 5.

Measured Rayleigh scattering profile for several different particles. Each particle shown in this figure consisted of a number of nanospheres, with an individual size of d = 103 ± 6 nm. The data were taken from particles 1, 3, and 6 from Fig. 7, respectively.

Close modal
The instantaneous inertial response of the nanoparticle to a well-defined externally applied force may be used to directly determine the mass of the particle. A protocol for the active control over the nanoparticle's electric charge has already been developed37 and is also elaborated on in Appendix A of the supplementary material. We use the CE to harmonically drive the charged particle using a quasi-static electric field near its natural oscillation frequency. Due to the zero correlation between the electrical driving force and the random Brownian fluctuation force, which is responsible for Eq. (1), the PSD of this driven system can be written by simply appending the additional term36 
(3)
to Eq. (1). In this new term, F0 represents the amplitude of the applied sinusoidal force, ΩM is the frequency of this force, and τ is the duration of the finitely recorded signal used to compute the Fourier transform. The subscript M is used to denote that Eq. (3) results from the modulation force that is applied to the particle. While Eq. (1) scales with the inverse of the particle's mass m, Eq. (3) does with its square. Therefore, the ratio,
(4)
may be used to extract the particle's mass.

Figures 6(a) and 6(b) show two examples of the PSD following two harmonically driven nanoparticles, which were classified as a sphere and a dumbbell. The inset in both figures shows a close-up of the PSD around the driving frequency ΩM, revealing the sinc-shape as described by Eq. (3). The dashed lines on both show the corresponding fits to Eq. (1) for the large-scale figure and Eq. (3) for the inset. Both particles are driven using the exact same force, and the presented data were recorded at the same pressure ( 15 mbar). Nonetheless, it can be seen that Fig. 6(b) has a lower peak and a smaller overall width than Fig. 6(a), the latter of which relates to a lower damping rate Γ 0. Both properties indicate that Fig. 6(b) was taken from a heavier particle than Fig. 6(a). Below, in Fig. 6(c), the resulting mass is shown for both particles in femtograms. To improve our statistics, we measure each particle for a series of driving frequencies ΩM around the particle's natural frequency Ω 0. From the results, it can be seen that the mass of the dumbbell is about twice as much as that of a single sphere.

FIG. 6.

Mass data for two particles. (a) and (b) show the PSDs following two harmonically driven particles, alongside an enlarged version of the driving peak in the inset, following Eq. (3). The dashed lines in both figures represent the best-fit to Eqs. (1) and (3). In (b), one may observe that part of the torsional signal is leaked into the data starting around 180 kHz. (a) and (b) correspond to particles 15 and 17 in Fig. 7. (c) shows the resulting mass measurements for both particles, measured for a series of driving frequencies ΩM around the natural frequency Ω 0.

FIG. 6.

Mass data for two particles. (a) and (b) show the PSDs following two harmonically driven particles, alongside an enlarged version of the driving peak in the inset, following Eq. (3). The dashed lines in both figures represent the best-fit to Eqs. (1) and (3). In (b), one may observe that part of the torsional signal is leaked into the data starting around 180 kHz. (a) and (b) correspond to particles 15 and 17 in Fig. 7. (c) shows the resulting mass measurements for both particles, measured for a series of driving frequencies ΩM around the natural frequency Ω 0.

Close modal

We will now consider the combined results of the damping rate, scattering, and mass measurements. Figure 7 shows an overview of a set of 18 different nanoparticles. For each particles, we have performed the three classification methods, which leads us to categorize as indicated at the bottom of the figure. The horizontal dashed lines represent the average results and the standard deviation of the spread, for each classification category.

FIG. 7.

The combined results from the mass, damping rate, and scattering measurements, sorted by their classified categories. The data on each column correspond to the same particles. The horizontal dashed lines represent the average result per category (excluding other, which represents compositions of four or more spheres), and the colored band shows the standard deviation of the data points.

FIG. 7.

The combined results from the mass, damping rate, and scattering measurements, sorted by their classified categories. The data on each column correspond to the same particles. The horizontal dashed lines represent the average result per category (excluding other, which represents compositions of four or more spheres), and the colored band shows the standard deviation of the data points.

Close modal

As a single classification technique, the mass measurements appear to show the best resolution. We classify particles whose masses are approximately twice as much as that of the corresponding spheres as dumbbells and those which are three times as much as trimers. We find that the mass of a nanoparticle is typically a bit high compared to the specifications of the manufacturer. One likely reason for this is the presence of residual water in the silica structure.37,41,42 However, also uncertainties in the mass density of the nanoparticles play a role.37,42 (See also Appendix A in the supplementary material for more details.) Our results on the mass determination are in line with those obtained in other publications.36,37

The measurements on the scattering profile and the damping rates do not show a significant difference between the nanoparticles composed of 103 or 142 nm spheres, as these methods focus on the morphology of the particle. Neither does either of the methods single-handedly give a satisfactory resolution that allows for an unambiguous distinction between single spheres, triangle trimers, and particles composed of four or more nanospheres (denoted as “other” in Fig. 7). In this case, the combination of multiple classification techniques is required to classify the particle.

On a single particle level, the measurements on the mass and shape using the three methods (PSD analysis, scattering anisotropy, and mass determination) are reproducible and show consistent results. When comparing multiple particles of the same category with each other, however, we see significant scattering of the data beyond the statistical error bars, which lead us to perform an analysis of the main systematic effects that influence these measurements. Here, we present the main conclusions, while the full details are given in the supplementary material.

Concerning the mass measurement, the accuracy with which we can determine the distance between the copper electrodes limits the accuracy with which we can quantify the mass of the particle. In our results, this is given as a symmetric uncertainty of 5.6% in the value of the electric field. The fitting of the PSD, which is used to assess the morphology of the particle, is influenced by the crosstalk between the different motional modes of the particle. At pressure below a few mbar, where damping by background gas is reduced, the anharmonic part of the potential is explored, which leads to a reduced quality of the fits. We find an overestimation of the damping rate of a given motional degree of freedom in the pressure range we use by at most 5%. Regarding the scattering anisotropy, we have evaluated detector linearity and alignment as potential systematic effects, which play a minor role. In this paper, we have primarily demonstrated the classification of spheres, dumbbells, and triangle trimers, as these appear to be some of the simplest configurations that we most often trap from our solutions. However, nanospheres can also aggregate into different shapes, such as chain trimers, or obtuse configurations. These shapes were not observed in our measurements, which we suspect is due to the stability of our trap and the rate at which these configurations form in our solution.

We have combined a charge-based mass measurement with a shape determination method based on light-scattering and an analysis of the damping rate anisotropy, all on the same set of trapped nanoparticles. We observe a large variation in the shapes and sizes within a set of nanoparticles. This is not only caused by the variation in the shape and mass of the individual spheres but also by the way in which they combine to form composite particles. We have demonstrated that the combination of these classification techniques enables us to obtain an unambiguous conclusion on the particle's shape, size, and mass.

See the supplementary material for a detailed description of the measurement protocol, the data analysis (specifically for obtaining the damping rate Γ 0), and the consideration of systematic effects in all three classification techniques.

We acknowledge the support from Gert ten Brink, Leo Huisman, and George Palasantzas. This project has received funding from NWO through NWA Startimpuls (No. 400.17.608/4303).

The authors have no conflicts to disclose.

Bart Schellenberg: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mina Morshed Behbahani: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nithesh Balasubramanian: Conceptualization (equal); Formal analysis (supporting); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (supporting); Writing – review & editing (equal). Ties Hendrik Fikkers: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (supporting); Writing – review & editing (equal). Steven Hoekstra: Conceptualization (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material