Enhancing gradient force over scattering force for nano-trapping through compensating for aberration

One challenge of optical trapping of nanoparticles is the weak trapping force compared to the destabilizing pushing force. Here we enhance the optical gradient force (GF), which is responsible for trapping, to achieve stable nanoparticle trapping through aberration compensation. The optical forces are calculated using multipole expansion theory and the focused fields are determined using Debye focusing theory accounting for interface aberrations between oil, glass, and water. With typical oil immersion objectives, the glass-water interface aberration reduces the GF relative to the scattering force (SF), leading to unstable trapping. By optimizing the refractive index of the immersion oil, the interface aberrations can be compensated. This significantly enhances the GF while moderately improves the SF, enabling stable nanoparticle trapping. The enhancements are particularly notable for large probe depths. Further improvement can be achieved with a thicker oil layer. With optimized conditions, the GF exceeds the SF by over two-fold. And the minimum axial force and axial stiffness increased approximately three-fold. Our study provides theoretical guidance to improve nanoparticle trapping efficiency through aberration compensation and force optimization.

A prerequisite for successful optical trapping is that the trapping potential should be larger than the thermal energy (∼k B T) in order to overcome Brownian fluctuation [23,24].Typically, optical trapping employs an oil-immersion objective with a high numerical aperture (NA) to tightly focus the light beam.Additionally, metalenses are effective in tightly focusing radially and azimuthally polarized beams into a sub-diffraction spot [25][26][27].The light beams pass through the immersion oil, a cover glass and enter the water solution, creating a stable trapping potential that can hold a microparticle in place.However, for a nanoparticle, the trapping potential is often weak due to its small volume.Moreover, another obstacle in nano-trapping is the common aberration caused by the refractive index mismatch between the cover glass and the water solution in which nanoparticles are suspended.This aberration degrades the focusing [28,29], subsequently reducing the trapping potential and resulting in unstable optical nano-trapping.Without the aberration, the trapping potential of a nanoparticle can be enhanced by simply increasing the input laser power.But with the aberration, the trapping potential cannot be sufficiently enhanced within a reasonable, safe range of input power.Therefore, compensating for the interface aberration is key to achieving stable nano-trapping.
The impact of refractive index mismatch aberration, also known as interface aberration, on optical trapping has been studied [28,[30][31][32][33]. Various methods, such as altering immersion medium [34], changing tube length [30,[35][36][37] and placing a deformable mirror [38], have been utilized to make up for aberration and improve optical trapping performance.However, these studies primarily focused on the total optical force and the corresponding restoring force constant, neglecting the effect of interface aberration on the conservative gradient force (GF) and the non-conservative scattering force (SF) [29,[39][40][41][42][43][44].The concepts of GF and SF are of paramount importance, and have been widely employed in the literature to interpret experimental and theoretical results qualitatively.This force decomposition is the basis to fully understand the overall optical performance with the aberration.
According to Helmholtz theorem, the optical force field acting on a single particle can be decomposed into two parts, i.e.F = F g + F k = − ∇U + ∇ × g.Specifically, F g is the conservative, curl-less GF, while F k is the non-conservative, divergence-less SF [1,4,[45][46][47][48][49]. Roughly speaking, GF is proportional to light intensity gradient, and SF is proportional to intensity and along the direction of light flow [42].Physically, GF is responsible for trapping a particle in the focus region, while SF facilitates transportation and pushes the particle away from the focus.If GF exerted on a nanoparticle is comparable to or even smaller than SF, the trapping potential is so weak that it is easily disrupted by Brownian motion.Therefore, ensuring GF dominates over SF is crucial for achieving a stable optical trapping.
In this study, we present a theoretical investigation aimed at compensating for aberration, enhancing GF, and conducting stable nano-trapping.To compute the focused field, we utilize the Debye-Wolf integral that take aberrations into account [50][51][52].Then, we calculate optical force components using a multipole expansion formula [42,[53][54][55].We simulate the compensation by replacing the conventional immersion oil of the objective with alternative immersion oil and provide a comprehensive analysis of the changes in GF, SF, total force, and the corresponding force constants.Besides, the effect of variation in oil layer thickness is also studied.

Optical gradient, SF formula and the aberrated focused field formula
GF and SF can be calculated by the multipolar expansion theory [42], 9 , where the polarizabilities α ′ , β ′ , γ ′ , γ ′m , Ω ′ and α ′ ′ are calculated from Mie theory [56], k is the wavenumber, and a is the particle radius, E in and B in are the incident focused fields.Equation (1) contains the first few leading multipole terms and is sufficient to accurately calculate the optical forces for a 150 nm-radius nanoparticle concerned in this paper.It is also convenient to use equation ( 1) to calculate GF and SF of a particle with a size less than 40% of wavelength in an arbitrary electromagnetic field.An x-polarized and z-propagating Gaussian beam with a wavelength of 1064 nm is focused by a NA = 1.3 oil-immersion objective (n oil = 1.515) into water (n w = 1.33).To determine the focused field, one must consider the aberrations introduced by refractive index mismatches.In optical tweezer, there may be multiple layers along the light path, as shown in figure 1, and therefore introducing several mismatch aberrations.Then the focused field in the Nth medium can be expressed as [51,52], where f is the focal length of the lens, α is the angular aperture of the objective that is related to NA by NA = n 1 sin α, the angle ϕ N can be determined via Snell law n 1 sin r p is the position vector for the point where the light field is evaluated, κ = k 1 sin ϕ 1 sin ϕ p cos(θ − θ p ), and is the probe depth, representing the distance between the nominal focus and the nearest boundary.T p (N−1) and T s (N−1) are effective transmission coefficients and contain additional phase factors exp[iβ j +1 ] of the aberration, where β j = k j (h j −1 − h j ) cos ϕ j and t s and t p are the Fresnel coefficients.From the initial aberration Ψ i and additional terms exp[iβ j +1 ], the total aberration function is: Along the path of light, there are a layer of immersion oil and a cover glass before the water solution, and it is assumed that the cover glass is homogenous with a refractive index n g and a thickness of t g = 150 µm.In most investigations, n oil and n g were assumed to be the same, leaving no impedance mismatch between the oil and glass.Consequently, there was only one aberration along the optical path, i.e., the glass-water interface aberration (N = 2 in equation ( 2)).If n oil is different from n g , there are four medium layers (N = 4 in equation ( 2)) and thus three aberrations: lens-oil interface aberration, oil-glass interface aberration, and glass-water interface aberration.It is worth noting that the multiple scattering between neighboring interfaces is ignored in equation ( 2), due to the low contrast of the refractive indices between the neighboring media.
By combining equations ( 1) and (2), we can now simultaneously investigate the aberrations effect on GF, SF of a sub-wavelength but non-dipolar particle.

Result
When light passes through the front lens (n lens = 1.515), immersion oil (n oil ̸ = 1.515, with an assumed thickness t o = 100 µm [34]), cover glass (n g = 1.515), and water (n w = 1.33), interface aberrations occur at three impedance-mismatch boundaries, deteriorating the focused light field.For N = 4 and n lens = n g , equation (4) will be Given that n w < n g , the second term on the second line of equation ( 5) is negative.In order to reduce this aberration, the first term should be positive therefore necessitating n oil to be higher than n g .Through this qualitative analysis, one can select an appropriate immersion oil of the objective to significantly reduce the interface aberration and, consequently, improve the optical trapping efficiency.
We calculated the optimal n oil at different probe depths and confirmed the improvement of the optical trapping performance.Figure 2(a) depicts the axial total optical force at a 10 µm probe depth versus the axial displacement for both the widely used immersion oil with n oil = 1.515 (taken as the default case) and the optimum immersion oil with n oil = 1.60.The axial trapping for the optimized case is noticeably better than the default case.The maximum and minimum force (marked by green dots) for the optimized case (solid curve) are significantly larger than the default case (dashed line).This is also true for the trapping depth, which is defined by the area under the force-distance curve between the equilibrium with zero force and minimum axial force [31].For the default case, the force equilibrium is about 2 µms before the nominal focus at z = 0, whereas for the optimized case, it is about 4 µms behind the nominal focus.The mechanisms that determine the equilibrium positions in both cases are tied to the aberration.When the total aberration (equation ( 4)) is positive (negative) throughout the entire focused light cone, the actual focus is behind (before) the nominal focus.The equilibrium associated with the optimized force alters with the probe depth, indicating the trapping position of a nanoparticle varies with the probe depth.For the 10 µm probe depth, the maximum and minimum forces versus the refractive index of the immersion oil n oil are plotted in figure 2(b).As the refractive index n oil increases from 1.515 to 1.60, the optical force is optimized with the compensation between those interface aberrations.However, as the n oil is getting higher and the positive interface aberration surpasses the negative one, the optical force is diminished again.Therefore, it is important to choose an immersion oil with an appropriate refractive index n oil to optimize the optical force.We note that immersion oils with a range of refractive indices covering at least 1.4-1.8 are commercially available.
Figure 2(c) illustrates the maximum and minimum total forces at different probe depths.It is observed that the negative part of the force associated with the 150 nm-radius polystyrene nanoparticle is quite small at the default index n oil = 1.515 (represented by the blue dashed line), especially at large probe depths; thus, the trapping depths are also extremely small.Such shallow trapping depths clearly cannot withstand the Brownian fluctuation, and unfortunately the situation cannot be resolved by simply increasing the laser power, as the induced heating effect can burn the particle and cause irreversible damage.The situation is dramatically changed by optimizing the refractive index of the immersion oil.As presented by the blue solid curve in figure 2(c), the minimum axial forces are substantially enhanced by a factor of 4-10 times across different probe depths.Therefore, trapping a nanoparticle at low input laser power becomes feasible via optimizing the n oil .And for different probe depths, the optimal refractive indices are different.
Figure 2 demonstrates that high-index immersion oil can compensate for the interface aberration.And the resulting enhanced optical force and trapping depth enable the trapping of previously untrappable nanoparticles at a lower input power.Although this phenomenon was also illustrated by some experimental research, there still lacks a detailed explanation for the instability of optical trapping at default setting, as well as an elaborate description of the physical origins leading to the enhancement of overall trapping performance.Our force calculations, which account for aberrations, are enabled by our new theory combining Debye formulation with multipole theory.Proceeding with our investigation, we address these issues by decomposing the optical force into GF and SF, and show quantitatively how GF and SF are affected by different amounts of aberration.
Figure 3(a) shows the maximum GF (red line) and maximum SF (blue line) as a function of n oil for a 10 µm probe depth.For the default case (n oil = 1.515),GF is slightly larger than SF so that the trapping potential created by GF is strongly weakened by SF, which always pushes a particle along the direction of light propagation.In contrast, for the optimal index (n oil = 1.60), maximum GF is more than twice as large as maximum SF.It can also be observed from figure 3(b) that the backward GF, i.e. the force that pulls a nanoparticle back to the trapping position, exceeds the forward SF, thereby yielding a larger trapping depth.Physically, GF is proportional to both the intensity and the corresponding gradient, while SF is proportional to the field intensity.Therefore, the GF is favored by tightly focused light, but also strongly suppressed by the interface aberration that degrades light focusing.This explains the difficulty of nano-trapping with aberration and underscores the importance of compensation for the aberrations from the aspect of force components.
Figure 3(c) shows the maximum and minimum GF and SF versus the probe depth under default n oil and optimal n oil condition.Probe depth h 3 is a critical parameter of interface aberrations.As h 3 increases, the glass-water aberration intensifies, and the optical field becomes less focused along the axial direction [57,58].For the default case, maximum GF is approximately equal to SF, and it gets even smaller than SF at larger probe depths, indicating the difficulty of trapping a nanoparticle across various probe depths.This situation changes when optimal immersion oils are used.GFs become larger than SFs (solid lines in figure 3(c)) and as a result, the backward and forward axial total forces are larger than those in the default condition.The changing behaviors of GF and SF in figure 3(c) are the origin of the change in behavior of axial total force in figure 2(c).We emphasize that the enhancement is more pronounced for GF, especially at a deeper probe depth.As shown in figure 3(d), for probe depths of 10 and 50 µm, GF is enhanced by factors of 2.1 and 3.5, respectively, while the enhancement factors of SF are 1.1 and 1.6.The focus quality becomes worse at the default condition as h 3 increases, and thus there is more room for compensation, yielding a larger enhancement factor.
The trap stiffness is another important parameter that characterizes the of optical trapping.It measures the linear rate of change of the optical force as a nanoparticle deviates from the trapping equilibrium where the optical force is 0. It is demonstrated that the aberration compensation via changing refractive index of the immersion oil leads to improvement of the trap stiffnesses.The stiffnesses in x and z directions, evaluated at the trapping equilibrium (between the positive and negative peaks), are plotted in figure 4. For the default index, the stiffnesses dramatically decline as the probe depth increases from 10 to 50 µm, during which the light field becomes less focused.After using an immersion oil with an optimal refractive index, both k x and k z increase, despite that they still decrease as the probe depth increases.Here, the transverse trap stiffness (k x ) is less affected and k x is larger than the axial one (k z ), which indicates the relatively weak trapping potential along the axial direction.Under similar parameters, such as probe depth, particle size, refractive index, our results agree with those of reported experimental studies [34].
After examining the effects of immersion oil refractive index, we also investigate the influence of oil layer thickness on optical trapping.We found that a thicker immersion oil layer can lead to further compensation for the interface aberration.As shown in figure 5 (probe depth h 3 = 50 µm), a 200 µm-thick oil layer with the optimized refractive index substantially enhances GF, SF and trap stiffnesses.The enhancement factor at a specific thickness is defined as the ratio of quantities with optimal n oil compared to default n oil .And it is shown that GF is improved more than SF with increasing thickness.Consequently, employing a thicker oil layer with an optimized refractive index can further enhance nano-trapping stability.
To expand the scope of this study, we revisit the effect of altering immersion medium, but here it is assumed that the refractive index of front lens is always equal to that of the immersion oil (N = 3 in equation ( 2)).We find that an immersion oil with an index lower than 1.515, rather than higher, compensates for aberration and as shown in figure 6, GF exerted on a 150 nm dielectric nanosphere is improved more than SF.When h 3 = 50 µm, we obtain the optimal n oil = 1.43.This result is consistent with the experimental fact that using silicone immersion objective (n Si = 1.41) to observe thick tissue samples.In other words, it is also suitable to use silicone immersion objective to trap a nanoparticle at a large probe depth.The enhancement factors for kx, kz, and the maximum axial GF and SF, as a function of oil layer thickness.For one thickness, the factor is defined as the quantity at optimal n oil divided by the quantity at default n oil .The probe depth is 50 µm.Figure 6.The enhancement factors as a function of probe depth, with the assumption that n lens is always equal to n oil .Defined as the quantity at optimal n oil divided by the quantity at default n oil (can be referred in dash lines in figures 3(c) and 4).

Conclusion
Our study focused on improving the efficiency of optical trapping of nanoparticles by compensating for glass-water interface aberration.By decomposing the optical force acting on a single nanoparticle into GF and SF, we explained why it is difficult to stably trap a nanoparticle using a common oil-immersion objective.The glass-water interface aberration significantly reduces GF, rendering it roughly equal to SF.Hence, GF and SF cancel each other out and the motion of the nanoparticle is mainly contributed to the thermal motion of medium particles, making optical trapping unattainable.We utilized immersion oil with optimal refractive indices higher than the commonly used 1.515 to reduce the aberration and observed that the maximum GF can be enhanced to twice the maximum SF.This results in significant enhancement of the minimum total force and axial trap stiffness, leading to a stable trapping of nanoparticles.Meanwhile, lateral stiffness and maximum total force are also improved by the aberration compensation.And the improvements in the GF and the optical trapping efficiency are particularly pronounced when trapping is performed at larger probe depths.Further, we also found that a thick layer of immersion oil is benefit for enhancing GF.As a supplement, we re-examined the changing refractive index method, yet assuming that there was no index difference between front lens and immersion oil.An immersion oil with a lower refractive index is favored to enhance GF and a commercial silicone immersion objective can be used to stably trap a nanoparticle at large probe depth.The methods proposed in this study are valuable for achieving a stable and reliable optical nano-trapping.

Figure 1 .
Figure 1.A schematic illustration for a beam focused with an oil-immersion objective through stratified media.The dashed lines represent focusing without any aberration and the solid lines represent the actual focusing with aberrations.

Figure 2 .
Figure 2. (a) The total axial optical forces at a probe depth of 10 µm.Maximum and minimum values are marked by green dots.(b) Maximum and minimum total axial force versus refractive index of the immersion oil at a probe depth of 10 µm.(c) Maximum and minimum total axial force as a function of probe depth under optimal n oil and default n oil conditions.

Figure 3 .
Figure 3. (a) Maximum GF and SF versus refractive index n oil of immersion oil.(b) The axial gradient and SF with n oil = 1.60 (the two green dots corresponding to the arrows on (a)).The probe depth is 10 µm.(c) Maximum axial GF and SF versus probe depth h3 under optimum n oil and default n oil conditions.(d)The enhancement factor defined as the ratio of the maximum force at the optimal index to that at the default index.

Figure 4 .
Figure 4. Trap stiffness in x and z directions versus probe depth, for the default and optimized cases.

Figure 5 .
Figure5.The enhancement factors for kx, kz, and the maximum axial GF and SF, as a function of oil layer thickness.For one thickness, the factor is defined as the quantity at optimal n oil divided by the quantity at default n oil .The probe depth is 50 µm.