1. INTRODUCTION

Optical manipulation, whereby light is used to exert a mechanical force on microscopic particles, is a powerful quantitative tool in fields ranging from molecular biology [1] to fundamental physics [2, 3]. The key component of the optical manipulation toolbox is optical tweezers [4], which is a highly flexible device easily integrated into conventional microscope systems. Typically optical tweezers experiments are carried out with samples dispersed in water or some other liquid. In this case the particles within the trap act as classical overdamped oscillators. Access to underdamped dynamics [5] is restricted to airborne samples or samples in vacuum. This is an emerging area of study [6] and is opening up new avenues of research in applied fields such as the basic physical and chemical properties of aerosols [7] as well as in more fundamental areas [8].

In order to make quantitative measurements using these tools it is important to have an understanding of the processes that underpin them. In optical tweezers the key processes are the optically generated forces and the Brownian motion of the trapped particle. In the context of optically trapped airborne particles, we have previously looked in detail at the Brownian motion [5], and in this paper we consider the corresponding optical forces.

Airborne particles, or aerosols, are of interest as a significant constituent of the atmosphere and a major factor in determining its chemical balance, for example, the ozone hole and acid rain [9], impairing visibility and in their contribution to radiative balance [10]. Furthermore, how they enter and interact with the body is relevant for both the effects of pollution on the human population and the efficacy of medicinal drugs [11]. Also, aerosols offer new possibilities for looking at fundamental physics, such as cooling processes [12].

In terms of surface-area-to-volume ratio accumulation mode aerosol, with radii between 0.05 and $1\mathrm{\mu m}$ [10], constitutes the largest proportion of atmospheric aerosol. However, most aerosol optical manipulation to date has concentrated on coarse mode liquid phase aerosols, with radii greater than $1\mathrm{\mu m}$ [10], thus limiting the study of the important accumulation mode regime. By understanding the optical processes, this paper also aims to investigate these limits.

Modeling of optical forces is used extensively in the field of optical manipulation to understand, for example, force mapping [13] and optical binding [14] or to extract physical parameters not otherwise obtainable from measurements [15]. The standard technique is to computationally model momentum transfer from focused beam to particle. For a sphere in an axially symmetric beam, a large body of work is already available, but for nonspherical objects the computation becomes more complex with the T-matrix approach the favored method [16].

A key problem with many of the approaches available is their inappropriateness for the size scale we are studying (around $1\mathrm{\mu m}$) and the lack of a description of the true trapping beam. A lucid introduction to the inadequacies is given by Viana et al. [17]. We will explore, to a first approximation, a geometrical optics (GO) model [18] before moving on to a more accurate, integral representation of focused light that incorporates refractive index interfaces and plane-wave scattering from spheres.

Our previous work has shown [5] that the motion of optically trapped droplets can be described using a simple harmonic oscillator model, in which inertia cannot be ignored. This inclusion of inertia makes the behavior of a trapped aerosol quantitatively different from a classically overdamped particle trapped in a liquid. When a comparison between trapping in air and liquid (by ourselves and others) is made, there are several conspicuous phenomena that are easily seen through video microscopy observation alone and could easily be overlooked in colloidal cases. These include the following. (1) As the trapping laser power increases so does the height above the water layer or coverslip at which the droplet is trapped [19]. (2) With further increases of power the droplet is lost from the trap. This does not always occur and is more pronounced for smaller droplets [5]. (3) After first capture, the droplet can undergo significant growth or evaporation coupled with large axial oscillations. These oscillations can occur significantly after capture but are far slower [20, 21]. (4) There is a linear dependence of “captured” droplet radius with trapping power. Small droplets cannot be trapped with high power [22, 23].

These effects are not due to the existence of the aerosol within an underdamped regime, so the hypothesis is that they are due to the optical forces on the particles. Therefore, together with understanding the optical force processes and the limits of the techniques, we aim to investigate if the isolated physics of optical forces lead to the phenomena observed.

2. THEORY

The forces exerted on spheres are decomposed into lateral and axial directions. The simulations, programmed in MATLAB, calculate, for a given point, the efficiency with which momentum is transferred to the object, Q. The optical force, F, can be determined through $F=Q{n}_{m}P/c$, where P is the trapping power, $n_m$ is the refractive index of the medium, and c is the speed of light. From plots of Q as a function of displacement, parameters can be taken or calculated that describe the system under study [24].

A single beam gradient force trap is achieved when an object is placed at a point where it experiences zero total force and is surrounded (within a certain proximity) by a negative gradient. The maximum negative magnitude, $Q_{z,\max }^{-}$, is a good measure of the axial optical trap strength [24], with its sign indicating whether a stable equilibrium position exists.

2A. System Geometry

The geometry of our system is different to those normally studied [25, 26], as an additional discrete step in refractive index is located between microscope objective and trapped particle due to the way in which liquid aerosols are loaded into our traps [7, 23]. The detailed system is illustrated in Fig. 1 and makes use of a laser beam of wavelength λ, waist w, incident upon the back aperture of an objective lens of focal length f ($\gamma =f/\omega$), aperture radius ρ, and focused to the diffraction limit at a converging angle ${\theta }_0$, which is quantified in terms of numerical aperture, $\mathrm{NA}=n_m\sin {\theta }_0$. Our system assumes matching of objective lens, oil, and glass coverslip refractive indices resulting in three consecutive layers of media and two refractive index mismatched interfaces.

We initially utilize results from GO, using the approach of Mazolli et al. [27], to see how it performs in our environment. We find that GO is not applicable for our system, so we rely heavily on the Mie–Debye model given by Mazolli et al. [27] and its extension to Mie–Debye spherical aberration (MDSA) theory described by Viana et al. [17]. We present an extension of this theory to include two interfaces of mismatched refractive index, accurately describing the geometry of our experiment.

2B. Focused Beam Description

When focused into a medium of refractive index $n_g$, in our case glass, the beam occupies a conical region in space with an electric field given by an integral representation of electromagnetic diffraction [28]. Our beam representation must include the effects of propagation through media of stratified refractive index [29]. Spherical aberration that is induced due to the additional interfaces is quantified in terms of an aberration function as will be given later in Subsection 2C.

Having focused the beam through two mismatched refractive index interfaces, the height of the paraxial focal plane above the water layer is found from the objective displacement, X, through

Using Török and Varga [29], we calculate the profiles of beams focused in our system and compare them to the ideal beam assumed in most cases, giving some insight into the physics.

Figure 2 displays the $yz$-plane beam profiles for focusing in water through a glass–water interface and through glass– aqueous–air interfaces. The axial displacement zero point is the paraxial focus, ${\mathit{r}}_f$, position when no refractive index interfaces are present.

The beam focused in water with no preceding interfaces varies smoothly at the focus compared to those focused in water and air having first travelled through glass coverslips. The beam focused to a point in air has a large number of oscillations in intensity along the beam axis. Previous work has shown such landscapes can interact with particles in nontrivial manners [30]. Finally, the maximum intensity is less in an airborne trap than for other traps given the same input power.

2C. Force Calculation

To calculate the force, F, we take the approach described by Mazolli et al. [27] where the vector electric and magnetic fields are given in terms of scalar Debye potentials, also known as Hertz vectors [31], and the optical forces are calculated by following Farsund and Felderhof [32] and then converted to trapping efficiency.

The general form of the theory remains the same as described in Viana et al. [17] and Török et al. [33] except we extend it using Török and Varga [29]. Because of the differences in our system, the relative locations of the planes in Fig. 1 are $z_f=1/N_1(\mathrm{\Delta }h+L_2/N_2)-L_1-z$, giving the aberration function [17, 29] to be

The efficiencies are given for the lateral and axial components each with two separate contributions, one for the rate of removal of momentum from the incident beam, $Q_e$, and the other for minus the rate of momentum transfer to the scattered field, $Q_s$, so the total efficiency $Q_{\text{tot}}^{\rho ,z}=Q_s^{\rho ,z}+Q_e^{\rho ,z}$. The forces are calculated for circularly polarized light but can equally be done for linear polarizations [34]. The axial component of the trapping efficiencies are given in [17, 27, 34, 35], except for the new aberration function.

In the limiting case where $\mathrm{\Delta }h=0$ and $n_w=n_g$, the results return to those of Viana et al. [17] for a glass-to-water interface without an aqueous layer. For $\mathrm{\Delta }h=0$, $n_g=n_a=n_w$, and $X=0$, the results of Mazolli et al. [27] and Maia Neto and Nussenzveig [35] are matched.

3. RESULTS AND DISCUSSION

It is the axial efficiency and force curves that govern whether a particle is trapped or not; thus, it will be the axial direction we consider as it determines the unusual phenomena observed. The results we present should be considered a set of typical examples that can be produced using the theory and is by no means exhaustive. It is noted that the data presented assume a wavelength of $532\mathrm{nm}$.

3A. Comparison of GO and Mie Scattering

Figure 3 plots the axial efficiency for a $250\mathrm{nm}$, $1\mathrm{\mu m}$, and $5\mathrm{\mu m}$ water droplet ($n_p=1.342$) as a function of $z/R$ (R is the particle radius) trapped with $532\mathrm{nm}$ light in air ($n_m=1.00$) calculated through GO and Mie–Debye theory with a reduced NA (${\theta }_0={\theta }_c\simeq 41.2°\Rightarrow \mathrm{NA}\simeq 0.66$).

GO predicts the droplet will “just” not obtain an axial equilibrium position, allowing it a brief reprieve in matching the more rigorous Mie theory. However, this disappears quite quickly when noting the drastic curve change as the efficiencies on three sizes of droplet are computed using Mie scattering. The largest sphere, $5\mathrm{\mu m}$, enters the beginning of the GO regime, yet the theory completely fails to indicate the occurrence of a second minimum, predicted by Mie theory.

The discrepancies in the curves predicted by the GO and Mie models show that GO is not an appropriate description of airborne trapping with the wild variation as a function of radius and the inability to predict important features. Next we use Mie theory to accommodate variations in the wavefront phase profile.

Spherical aberration must be considered in our system where there are two interfaces with mismatched refractive indices. In Fig. 4 we plot for 1 and $5\mathrm{\mu m}$ spheres the axial force curves with and without aberration included.

The inclusion of spherical aberration in the description greatly affects the efficiency. There is a reduction in $Q_{z,\max }^{-}$, and for the larger droplet a general “smoothing” of the curve occurs, with smaller local minima created. The efficiency curves in Fig. 5, for the same system and objective displacement but with and without the aqueous layer, shows a change in the axial curves indicating its significance and necessary inclusion in the theory.

The results in this section establish that a Mie-based model must be used, unlike in some optical tweezers models where a GO model is broadly applicable. We also see the necessary physics to include, so we can now attempt to explain the observed phenomena described in the introduction.

3B. Predicting Experimental Observations

For colloidal systems, neglecting particle weight is a reasonable approximation since the density of trapped objects is similar to that of the medium. Thus, there is only a multiplicative factor between efficiency and force graphs, allowing the axial Q curves to be treated as scaled force curves. However, this is a very poor assumption when considering water droplets suspended in air with the large density contrast: ${\rho }_{\text{water}}\simeq 1000{\rho }_{\text{air}}$. In order to fully appreciate what the theory predicts, we must calculate the force experienced by the microsphere and subtract its weight. In the previous section neglecting particle weight did not hinder our understanding, but for the next section weight will be included.

We now examine if the theory predicts the behavior observed during aerosol trapping experiments, starting with points 1 and 2 in Section 1. In Fig. 6a we plot for a $4\mathrm{\mu m}$ radius water droplet, trapped in the experimental system depicted in Fig. 1, the predicted axial force curves for increasing trapping powers. Repeating for several droplet radii, the height above the water layer that a droplet is trapped can be plotted as a function of power as shown in Fig. 6b.

Figure 6a successfully predicts two physical observations from experiments. As the trapping power increases, $z_{\text{eq}}$ and hence height above the underlying water layer increases, and with enough power the droplet eventually falls from the trap [5]. The curves of 2, 3, and $4\mathrm{\mu m}$ droplets in Fig. 6b stop at certain power values, as above these the droplet no longer has an equilibrium position. However, the $5.5\mathrm{\mu m}$ droplet curve would continue indefinitely, although not plotted here, as the respective efficiency curve crosses the zero efficiency axis with negative gradient. Clearly, if an equilibrium position exists in the efficiency curves alone, then the droplet will always remain trapped. If no such position exists, then the force curve may eventually lose the equilibrium position with increasing power. This qualitatively explains our own experimentally observed results and the power gradients of Knox et al. [19], indicating the authors measured a segment of the extended curves in Fig. 6b.

Figure 6b may also explain point 4 from Section 1. The power gradients show that, above certain powers, depending on droplet radius, it is possible for no equilibrium position to exist. Therefore, although a “large” droplet may be trapped at relatively large powers, smaller droplets cannot be trapped at the same power.

An important parameter that governs the magnitude of the spherical aberration induced by the interfaces is the depth into the sample which the beam is focused. For example, a lower focus has less aberration. In Fig. 7 the beam is simulated to focus at several depths into the sample chamber and the force curve calculated again for a $4\mathrm{\mu m}$ water droplet. The decrease in aberration not only shifts $z_{\text{eq}}$ closer to the paraxial focus but also increases $F_{z,\max }^{-}$ and deepens the potential well.

Now consider point 3 from the introduction. Liquid aerosols will establish a stable size once in equilibrium with their surrounding environment, namely the relative humidity [7]. Although the process of growth and evaporation is relatively fast, it is at times clear that one of these occurs just after the droplet becomes trapped. Investigating how the height at which the droplet is trapped varies with droplet radius we plot Fig. 8.

There is a clear, near-sinusoidal oscillation in droplet height as a function of radius. A change in trap height of $\sim 2\mathrm{\mu m}$ occurs due to only a $50\mathrm{nm}$ change in droplet radius. When observing a droplet just after capture, the change in size is clear, far above the limit of resolution, so must be greater than $50\mathrm{nm}$. Knowing that the oscillations are most frequent just after capture, we conclude the multiple oscillations that occur in experiments are due to changing particle radius and hence $z_{\text{eq}}$.

Measuring this oscillation experimentally would be a challenge. The droplet height would need to be obtained precisely and also coupled with a high precision sizing technique such as cavity enhanced Raman spectroscopy [36]. Finally, the droplet radius would have to be varied by altering ambient relative humidity or varying droplet temperature.

3C. Limits of Techniques

To explore the limits of these airborne trapping techniques, $Q_{z,\max }^{-}$ is calculated and plotted in Fig. 10 as a function of both particle radius and relative refractive index $n_{\text{rel}}$ [24], first for particles suspended in water, $n_{\text{rel}}=n_p/n_w$, Fig. 9, as means of comparison, and then for airborne particles, $n_{\text{rel}}=n_p/n_a$, where $n_p$ is the particle refractive index.

The white areas on the plots represent parameter space where a negative $Q_{\text{axial}}$ value does not exist and hence no stable trap position is possible [24, 37, 38]. Of immediate note are the “spikes” in the contour plots indicating resonances in the optical efficiency experienced by the particles. The effect is more pronounced as a function of radius although at the high refractive index end of the spikes there are rapid resonances in force as a function of refractive index also.

These resonances can be explained by interference effects due to the sphere’s increased reflectivity at high relative refractive index and its associated variation with radius [24]. The decreased frequency of the resonances in air is due to the lower medium refractive index.

The plots are functions of relative refractive index, so for a given particle refractive index the horizontal line of interest is higher up the refractive index axis in air than in water. For example, a silica sphere in water exists along the line defined by $n_{\text{rel}}\simeq 1.09$ in Fig. 9, and for a water droplet in air the line is at $n_{\text{rel}}\simeq 1.34$ in Fig. 10.

From Figs. 9, 10 the continuous region of stability for optical tweezers in air is over a smaller range of relative refractive indices ($\mathrm{\Delta }{n}_{\text{rel}}\simeq 1–1.25$) than when trapping in water ($\mathrm{\Delta }{n}_{\text{rel}}\simeq 1.33–1.65$). Also, $Q_{z,\max }^{-}$ is smaller overall for trapping in air than in water. This is understandable because the larger relative refractive index between particle and medium for aerosols increases the Fresnel reflection coefficients, hence increases scattering forces. The minimum radius possible to trap is smaller in water than air probably due to the increased spherical aberration in the focused beam, induced by the coverslip interface, which has a larger refractive index contrast in airborne traps.

Next we include the weight of the particle to calculate $F_{z,\max }^{-}$, the truly relevant quantity that will allow the determination of whether the spheres are isolated in three dimensions or not. Unfortunately, this poses a problem, as F is dependent on laser power, and with this additional variable not all param eter space can be easily displayed. Instead $F_{z,\max }^{-}$ is plotted for a single power, $P=10\mathrm{mW}$, in Fig. 11.

Comparing Figs. 10, 11 we come to an interesting conclusion. For droplets with certain particle parameters, indicated in Fig. 10, traps are created through the transfer of optical momentum alone: a single beam gradient force trap or optical tweezers. However, Fig. 11 indicates that, with the assistance of gravity, a larger range of droplets can be “trapped,” although not with momentum transfer alone. Consider a droplet that evolves in size; as the radius varies, the “path” of the particle in the parameter space of Fig. 10 may cross through a nontweezing region but due to its weight remains trapped (Fig. 11). This difficulty in deciding whether a droplet is tweezed or levitated leads to the conclusion that, for airborne particles in a particular size regime, what we are seeing is really a quasi optical tweezers.

In Fig. 12 we superimpose the tweezing and trapping areas of Figs. 10, 11. Areas of parameter space truly optically tweezed are colored gray, areas that are only trapped with the assistance of gravity are colored red, and the area that would be truly tweezed if the droplets had neutral buoyancy is colored blue. White areas retain the same meaning of regions of no equilibrium position.

Although confined laterally in both cases, it is clear then that the choice of inverted or noninverted tweezers is critical to the success of optically trapping a given aerosol and the range over which this can be accomplished. Having established the true nature of the technique, the next section discusses increasing the area of parameter space in the tweezing regime, trapping through momentum transfer alone.

3D. Optimization and Extension of Limits

We have demonstrated several points that stop airborne traps from reaching their optimum performance. These include spherical aberration, a high refractive index contrast between particle and medium causing large scattering forces, and reduced NA due to total internal reflection at the coverslip interface.

Total internal reflection is not easily circumvented, but it could be possible to correct for spherical aberration or possibly remove the large scattering forces, which we shall now explore.

3D1. Spherical Aberration Correction

It is feasible to correct for the spherical aberration using wavefront modifying elements [39]. Any correction would clearly be advantageous, creating better localization of aerosols and hopefully moving into the important accumulation mode size regime. In Fig. 13 we plot the same parameters as Fig. 10 but an additional spherical aberration of magnitude $0.08\lambda$ is placed on the beam input to the objective.

Figure 13 shows an improvement in the axial strength of the optical tweezers, an increase in the overall range of parameter space that can be tweezed but also an increase in the minimum particle radius that can be tweezed.

3D2. Removal of Beam Center Intensity

Increasing $Q_{z,\max }^{-}$ is by far the most difficult problem in airborne tweezers, as shown and discussed, so next we predict the effects of removing the central portion of a Gaussian beam as it enters the objective [40]. In Fig. 14 we plot $Q_{z,\max }^{-}$ against both radius and relative refractive index for a Gaussian beam where $\sim 57\%$ of the beam area is removed, leaving an annulus, yet keeping the total power the same.

Figure 14 shows that the parameter space over which a true optical tweezers can be created is greatly increased by removing the central core of the beam, although the minimum sphere radius tweezable has increased.

The predicted increase in parameter space over which aerosols can be tweezed is of great promise to the field of aerosol optical manipulation in which it is difficult to trap high refractive index aerosols, specifically solid microspheres. It is hoped that a definitive experiment can be performed in the future to verify this increase in optical tweezers parameter space.

3E. Capture Volume

In order for droplets to become trapped, they must enter a capture volume, so it would be pertinent to calculate how this volume varies with trapping power and droplet radius. This volume extends between the maximum and minimum force points in the axial and lateral directions simultaneously. At these locations there is a complex interplay between axial and lateral efficiencies [17, 27]. This will require more study to ascertain a suitable description and answer to the question of whether the capture volume plays a significant role in the linear dependence of captured droplet size on trapping power.

4. CONCLUSION

We have outlined a modified MDSA model of optical traps to describe our experimental system and have shown that it must be used over a simpler GO approximation in order to fully describe the beam profile and scattering. Investigation of the MDSA parameters indicate spherical aberration must be included, and we have discussed that gravity must not be naïvely ignored. However, within the Mie scattering theory outlined above, we make the assumption that the Fresnel transmission coefficients, $t_s$ and $t_p$, for TE and TM modes of polarization, respectively, are equal and take $t_s=t_p$ [13]. We propose that, in future work, it should be attempted to place both TE and TM modes into the theory.

Once correctly described we use our extension to MDSA theory to qualitatively predict and explain four unusual phenomena observed only in airborne optical traps. It has also allowed us to explore limitations of current experiments and approaches before investigating what can be improved.

Obviously, it is not easy to prove the nonresult of being unable to trap certain objects, but the results here give some indication as to why it is hard to trap small spheres, with relatively high refractive index, that are easily trapped in water.

This work has lead to many new insights into how aerosols are trapped in single beam gradient force traps, namely that an optical trap in the inverted geometry behaves as a quasi optical tweezers, at times tweezing droplets and at others only levitating them. We have given qualitative predictions that explain physical phenomena observed experimentally and explored the theoretical limits of trapping aerosols, both of which help to define the parameters of the current tools at our disposal. The challenge for the future is to produce quantitative agreement between experiment and theory.

ACKNOWLEDGMENTS

D. R. B. thanks the Lindemann Trust and Engineering and Physical Sciences Research Council UK (EPSRC) for funding. D. M. thanks the Royal Society for support.

 

Fig. 1 (Inset) Beam of waist w enters objective lens of focal length f with back aperture radius ρ. It is focused to a point $f_p$ having propagated through two mismatched refractive index interfaces. The first interface is glass to water and the second is water to air. $n_g$, $n_w$, and $n_a$ are the refractive indices of glass, water, and air, respectively. Without interfaces the light would be focused to point $r_f$. (Left) Expanded view of focal region. Light is incident on the first interface, $z_1$, at ${\theta }_g$ and refracted to ${\theta }_w$, and is incident on the second interface $z_2$, a distance $\mathrm{\Delta }h$ away where it is refracted to ${\theta }_a$ and focused to its paraxial focus point $f_p$. The height of the paraxial focus above the second and first interfaces is $L_2$ and $L_1$, respectively. The droplet is trapped a distance h above the first interface, z above the paraxial focus, and $z_f$ below the point $r_f$.

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Fig. 2 Intensity profile of a focused $532\mathrm{nm}$ Gaussian beam taken from a $y\text{-}z$ slice through the beam axis. (a) The beam is focused into water ($n_w=1.33$). (b) The beam is focused into water having crossed a glass ($n_g=1.517$)-to-water ($n_w=1.33$) interface after the objective lens. (c) The beam is focused into air ($n_a=1.00$) across glass-to-water ($n_w=1.342$) and water-to-air interfaces. The objective displacement $X=40\mathrm{\mu m}$, the water layer is $10\mathrm{\mu m}$ thick, $\gamma =1$ and ${\theta }_0=41.23°$. Zero on the axial axis is the position of the paraxial focus had there been no interfaces.

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Fig. 3 (a) Axial trapping efficiencies calculated through Mie scattering for a $5\mathrm{\mu m}$ (purple dashed) water droplet ($n_p=1.342$) and through GO (black solid). (b) Axial trapping efficiencies for Mie calculation of $250\mathrm{nm}$ (blue dashed) and $1\mathrm{\mu m}$ (red solid) water droplets. All are trapped with $532\mathrm{nm}$ light in air ($n_m=1.000$) with $\gamma =1$ and ${\theta }_0=41.23°$ in a system like Fig. 1 with no refractive index interfaces. The four curves are plotted on two separate graphs for clarity.

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Fig. 4 Axial efficiency curves for 1 and $5\mathrm{\mu m}$ water droplets ($n_p=1.342$) trapped in air ($n_a=1.000$) above a glass coverslip ($n_g=1.517$) and thin water layer ($n_w=1.342$) as depicted in Fig. 1. $X=40\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$, $\gamma =1$ and ${\theta }_0=41.23°$. (a) For a $1\mathrm{\mu m}$ sphere the blue solid line is without aberration and the purple dashed line with aberration. (b) For a $5\mathrm{\mu m}$ sphere the red solid line is without aberration and the black dashed line with aberration.

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Fig. 5 Axial efficiency curves as a function of $z/R$ for 1 and $5\mathrm{\mu m}$ water droplets ($n_p=1.342$) trapped in air ($n_a=1.000$) above a glass coverslip ($n_g=1.517$) with and without a thin water layer ($n_w=1.342$). $X=40\mathrm{\mu m}$, $\gamma =1$, ${\theta }_0=41.23°$, and if present when the thin water layer is $10\mathrm{\mu m}$ thick. In (a) the blue solid and purple dashed curves are calculated without and with the thin water layer, respectively. In (b) the red solid and black dashed curves are calculated without and with the thin water layer, respectively.

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Fig. 6 (a) Variation of axial force for a $4\mathrm{\mu m}$ water droplet ($n_p=1.342$) trapped in air ($n_a=1.000$) at trapping powers of $10\mathrm{mW}$ (blue dotted–dashed), $20\mathrm{mW}$ (green dotted), $50\mathrm{mW}$ (purple dashed), and $200\mathrm{mW}$ (red solid). The force has been normalized to unity for clarity. (b) Height variation above the water layer droplets of radius $2\mathrm{\mu m}$ (blue dotted), $3\mathrm{\mu m}$ (red dotted–dashed), $4\mathrm{\mu m}$ (purple dashed), and $5.5\mathrm{\mu m}$ (black solid) are trapped as a function of power. All but the $5.5\mathrm{\mu m}$ droplet curve stop due to the loss of axial equilibrium position at high powers as in (a). $X=40\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$ ($n_w=1.342$), $\gamma =1$, ${\theta }_0=41.23°$, and the coverslip refractive index $n_g=1.517$ for both (a) and (b).

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Fig. 7 Variation of axial force for a $4\mathrm{\mu m}$ water droplet ($n_p=1.342$) trapped in air ($n_a=1.000$) with $8\mathrm{mW}$ of power for microscope objective displacements of $25\mathrm{\mu m}$ (red dotted–dashed), $30\mathrm{\mu m}$ (solid black), and $35\mathrm{\mu m}$ (dashed blue). The water layer ($n_w=1.342$) is $10\mathrm{\mu m}$ thick, $\gamma =1$, ${\theta }_0=41.23°$, and the coverslip refractive index $n_g=1.517$.

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Fig. 8 Plot of the height a water droplet ($n_p=1.342$) in air ($n_a=1.000$) that is trapped above the underlying water layer ($n_w=1.342$) as a function of radius. $X=25\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$, $\gamma =1$, ${\theta }_0=41.23°$, the coverslip refractive index $n_g=1.517$, and the trap power is $10\mathrm{mW}$.

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Fig. 9 $Q_{z,\max }^{-}$ as a function of relative refractive index and radius for spheres trapped in a water medium ($n_w=1.33$). $X=40\mathrm{\mu m}$, $\gamma =1$, ${\theta }_0=61.25°$, and the coverslip refractive index $n_g=1.517$.

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Fig. 10 $Q_{z,\max }^{-}$ as a function of relative refractive index and radius for spheres trapped in an air medium ($n_a=1.000$). $X=40\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$ ($n_w=1.342$), $\gamma =1$, ${\theta }_0=41.23°$, and the coverslip refractive index $n_g=1.517$.

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Fig. 11 $F_{z,\max }^{-}$ as a function of relative refractive index and radius for spheres trapped in an air medium ($n_a=1.000$). $X=40\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$ ($n_w=1.342$), $\gamma =1$, ${\theta }_0=41.23°$, and the coverslip refractive index $n_g=1.517$.

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Fig. 12 Superposition of Figs. 10, 11 highlighting the areas of parameter space as a function of relative refractive index and radius, where water droplets are truly optically tweezed (high R, low $n_{\text{rel}}$, gray), only trapped with the assistance of gravity (high R, high $n_{\text{rel}}$, red), and optically tweezed if the droplet had neutral buoyancy (low R, low $n_{\text{rel}}$, blue). The white area represents areas where neither optical tweezing nor levitation occurs. The parameters for these plots are the same as the respective figures.

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Fig. 13 $Q_{z,\max }^{-}$ as a function of relative refractive index and radius for spheres trapped in an air medium ($n_a=1.00$) with an additional spherical aberration placed on the Gaussian beam entering the objective back aperture of magnitude $0.08\lambda$. $X=40\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$ ($n_w=1.342$), $\gamma =1$, ${\theta }_0=41.23°$, and the coverslip refractive index $n_g=1.517$.

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Fig. 14 $Q_{z,\max }^{-}$ as a function of relative refractive index and radius for spheres trapped in an air medium ($n_a=1.000$) with a Gaussian beam entering the back aperture of the microscope objective with 57% of its central area removed. The objective axial displacement, $X=40\mathrm{\mu m}$, $\mathrm{\Delta }h=10\mathrm{\mu m}$ ($n_w=1.342$), $\gamma =1$, ${\theta }_0=41.23°$, and the coverslip refractive index $n_g=1.517$.

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