Photon-efficient optical tweezers via wavefront shaping

Given the widespread use of optical trapping, it may seem surprising that after more than 30 years, the most commonly employed spatial shape of laser beams used to create an optical tweezer is still the conventional Gaussian beam profile first suggested by Ashkin ashkin1986Observation . Indeed, Gaussian beams do come with many advantages: they are straight-forward to create and highly versatile – operating in a broadly similar manner over a wide range of micro-particle sizes and shapes jones2015Optical ; simpson2014inhomogeneous .

However, this versatility comes at a cost: a Gaussian beam is typically not the optimal shape of light field to tightly trap a given micro-particle. This drawback is exacerbated for larger particles whose size is greater than the trapping wavelength. Such objects, which include many types of biological cells, are typically only weakly optically tweezed, suffering from a low trapping stiffness in a Gaussian beam. A straightforward way to overcome this issue is to crank up the laser power: doubling the intensity of the trapping light in turn doubles the trapping stiffness in all dimensions, thus confining a particle more tightly. Unfortunately, increasing the power focused onto the trapped object can lead to a variety of undesirable effects: excess photons can damage photosensitive biological systems blazquez2019optical , heat the particle and its local environment peterman2003laser , and increase decoherence effects in quantum ground state experiments tebbenjohanns2021quantum .

In this article we explore an alternate paradigm: we demonstrate to what extent it is possible to enhance 3D trap stiffness without increasing laser power, by instead tailoring the spatial profile of the laser beam. Optical fields have been previously shaped in a plethora of different ways to exert targeted optical forces and torques on micro-particles friese1998optical ; arlt2000generation ; friese2001optically ; ladavac2004sorting ; roichman2008optical ; dholakia2011shaping ; padgett2011tweezers ; brzobohaty2013experimental ; woerdemann2013advanced ; taylor2015enhanced ; bezryadina2016optical ; stilgoe2022controlled ; hu2022structured . As far as enhancing trapping stiffness is concerned, various beam shapes have been tried in the past friese1996determination ; simpson1998optical ; bowman2010particle ; phillips2011optimizing ; mcalinden2014accurate ; singh2017particle . An important step forward was taken by Taylor et al. in 2015, who demonstrated that carefully sculpting the incident optical field can deliver impressive one-dimensional lateral stiffness enhancements of trapped micro-spheres – by up to a factor of $\sim$ 30 compared to conventional optical tweezers of the same power taylor2015enhanced . It is possible to identify globally optimum structured fields that accomplish these 1D stiffness enhancements using eigenvalue-based approaches mazilu2011optical ; taylor2017optimizing ; agate2004femtosecond ; ambichl2017focusing ; horodynski2020optimal . However, as we show in Supplementary Information (SI) §1, these 1D trapping enhancements place no constraints on stiffness in other dimensions, and so do not guarantee that a particle is tightly trapped in 3D, or even stably trapped at all volpe2023roadmap .

It is not straightforward to extend such eigenvalue-based strategies to enhance multi-dimensional optical trapping, since the stiffnesses along different directions are not independent. Consequently, the optimum 3D trapping field will not simply be a superposition of optimal trapping fields for the individual axes. Thus, despite multiple decades of research into optical trapping, an understanding of how to calculate the shape of ‘optimal’ three-dimensional optical traps, or the level of multi-dimensional stiffness enhancement they may deliver, has remained out of reach volpe2023roadmap .

Here, we overcome these difficulties by designing bespoke trapping beams using an integrated multi-parameter optimisation strategy, guided efficiently to a solution by the information held within the Generalised Wigner Smith (GWS) operator ambichl2017focusing ; horodynski2020optimal . Our approach allows all three dimensions to be considered simultaneously – in terms of both stiffness enhancement and trap stability. Remarkably, we predict that custom-tailored trap shapes can hold micro-spheres up to a factor of 200 times more tightly than a Gaussian trap of equivalent power. To validate this concept experimentally, we develop a real-time optimisation routine that iteratively adapts the trapping wavefront to the shape of the particle. We experimentally demonstrate order-of-magnitude improvements in how tightly it is possible to hold micro-spheres in three dimensions. Our work establishes that dramatic gains in 3D optical trapping efficiency are possible by judiciously structuring light fields, and presents new theoretical and experimental routes to achieve them.

Thermal motion of the fluid thus drives the particle’s centre-of-mass (CoM) to stochastically explore a small region around its equilibrium position. To quantify the size of this region, we utilise the concept of the confinement volume $V_{\text{c}}$ – see Fig. 1(a), also referred to as the thermal ellipsoid phillips2012force . The shape of the confinement volume is typically a prolate ellipsoid with its long axis parallel to the optical axis of the trapping beam – reflecting the lower axial trapping stiffness arising from weaker intensity gradients in this direction. $V_{\text{c}}$ is given by:

and is such that the probability of finding the CoM inside it is $p\sim 0.99$. Here $k_{\text{B}}$ is Boltzmann’s constant, $T$ is the absolute temperature of the surrounding fluid, and $\kappa_{x,y,z}$ are the eigenvalues of $\bm{\kappa}$, which represent the stiffnesses of the optical trap along the principal axes of the thermal ellipsoid (see SI §2). Our aim in this work is to find the shape of light fields that minimise the particle’s confinement volume by maximising all eigenvalues of the stiffness matrix simultaneously.

We begin by investigating the level of 3D optical trapping enhancement possible from a theoretical perspective. We use the T-matrix formalism waterman1971symmetry ; nieminen2007optical ; barton1989theoretical (see SI §3) to model the interaction of a shaped incident light field $\bm{u}$ with a microscopic particle in an optical tweezers setup. The incident field coefficients held in the column vector $\bm{u}$ are defined at the pupil plane of the objective lens, expressed in the Bessel beam basis (see SI §4), enabling us to explore the consequences of spatially shaping the intensity and wavefront of the trapping beam.

We seek the power-normalised complex values of $\bm{u}$ that correspond to a structured light field that maximises the trapping stiffness in three dimensions, while also stably holding the particle. To this end, we employ a constrained numerical optimisation routine, where we treat the desired relative stiffness in each dimension as a tunable parameter – yielding flexibility in the final aspect ratio of the particle’s confinement volume (see Methods and SI §5). A large number of optimisation variables $N$ (i.e. elements of $\bm{u}$) is required in order to shape the light effectively. This makes the problem computationally demanding, with previous examples of numerical 1-D trapping field optimisation taking on the order of days to converge taylor2015enhanced ; butaite2020enhanced . Therefore we hone the efficiency of our optimisation routine to significantly speed up this process.

The cornerstone of our approach is the GWS operators for optical force ($\bm{Q}$) and trapping stiffness ($\bm{K}$) wigner1955lower ; smith1960lifetime ; ambichl2017focusing ; horodynski2020optimal . These operators reduce the evaluation of force and stiffness along a particular direction to a single matrix equation. For example, along the $x$-dimension we have:

where $\bm{S}$ is the scattering matrix describing how the light interacts with the particle (i.e. which incident light modes are scattered into which outgoing modes), $\partial_{x}$ indicates a partial derivative with respect to the $x$-position of the particle, and ^† indicates a conjugate transpose. Equivalent expressions can be written down for any other degree of freedom, including rotations. Crucially, $\bm{Q}$ and $\bm{K}$ only encapsulate properties of the particle, and are independent of the incident field. As such, they only need to be evaluated once, before the optimisation routine commences, significantly cutting down on computational time. Expressions for $f^{\mathrm{opt}}$ and $\kappa$ are also readily differentiable with respect to $\bm{u}$, providing access to analytical expressions for gradients and Hessians, thus further speeding up the optimisation. See SI §1,6-8 for more detail on $\bm{S}$, $\bm{Q}$, and $\bm{K}$, including our derivations of analytical expressions for calculating the derivatives in Eqs. 2–3. This approach reduces the time-scale for a single trap design from days to minutes (see SI §9), allowing us to explore the enhancements achievable over a wide range of different particles.

Our simulations indicate that it is possible to achieve very substantial 3D trapping enhancements – in some cases of up to a 200-fold reduction in the confinement volume, and with stiffness enhancements exceeding $\sim$20 in all dimensions simultaneously for a broad range of micro-sphere sizes and refractive indices. Equivalently, these reductions in particle confinement volume can be translated into improvements in relative trapping efficiency $\eta^{\text{rel}}$: the factor of reduction in trapping power needed to hold a particle as tightly as a conventional optical trap, where $\eta^{\text{rel}}=\left(V^{\text{rel}}_{\text{c}}\right)^{2/3}$ (see SI §11). In Fig. 2(e) we see that $\eta^{\text{rel}}$ is over a factor of ten for a wide range of particles – predicting that the same trapping performance as a conventional optical tweezer can be achieved using less than one-tenth of the laser power. We find that these enhancements are highly repeatable, irrespective of the choice of initial values used for the complex optimisation variables $\bm{u}$ (see SI §12). This suggests we may be finding solutions that are close to the global optimum, given the set of constraints that we have enforced.

A notable feature of our method is that it predicts an extension of the range of particle parameters over which optical trapping is possible: conventional optical tweezers fail to create stable $z$-equilibria for high refractive index particles, while we see in Fig. 2(f) that stiff and stable optical trapping becomes possible in this high-index regime by shaping the structure of the trapping field. Another dominant trend in these results is that large particles with low refractive indices benefit the most. The enhancements are lower for small dielectric particles ($<$$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ radii) because the diffraction limit restricts the extent to which the trapping light can be shaped within the footprint of the particle.

Driven by typical experimental capabilities, we also investigate enhanced optical trapping using phase-only beam shaping, rather than the intensity and phase control employed so far. We reformulate our numerical optimisation routine to incorporate this constraint – see SI §13. We observe that the thermal volume compression achieved by phase-only optimised traps is typically $\sim$ 60 % of that achieved with full-field light shaping, across the 900 scenarios considered. Nonetheless, our analysis shows substantial trapping enhancements over conventional optical tweezers are still possible with phase-only light modulation.

A number of specific challenges arise when seeking to apply a numerically pre-designed trapping field in a real-world experiment. Commercially available colloids exhibit a statistical variation in size, and our modelling shows (SI §14) that the optimal trap shape is highly dependent on the exact physical properties of the target particle. The volumetric trapping fields must also be generated with extremely high fidelity, since the mechanical response of the particle is highly sensitive to small variations in the applied trapping field. The creation of pre-designed fields is influenced by factors including: the imperfect response of the SLM moser2019model ; kupianskyi2023high , the precise action of the objective lens gu1996effect ; sheppard1988aberrations ; he2021vectorial , and the configuration of the oil-glass-water interfaces that light passes through on its way to the target particle iwaniuk2011correcting – even after in situ aberration correction is performed with the SLM vcivzmar2010situ . The combined effect of the above factors means that employing pre-designed optimised traps is very challenging.

We therefore devise a new experimental strategy compatible with these challenges: we take inspiration from the field of wavefront shaping which has emerged as a powerful way to optimise coherent light transport in unknown complex scattering environments vellekoop2007focusing ; gigan2022roadmap ; cao2022shaping . We implement a live iterative optimisation routine that automatically tailors the trapping field to the (unknown) properties of a target particle. The optimisation is conducted using real-time measurements of the three-dimensional trapping strength, as inferred from the stochastic trajectory of the particle. Crucially, our approach does not require a priori knowledge of the properties of the target particle, or the optical characteristics of optical tweezers platform itself, since the optimisation is conducted based purely on experimentally-measured metrics. See Methods for a detailed description of our algorithm.

Figure 3(b) displays measured point clouds representing the confinement volume explored by the CoM of a $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ radius silica micro-sphere trapped in water using conventional optical tweezers (left) and an optimised trap (right). We see that the thermal ellipsoid has shrunk in all dimensions – in this case reducing in volume by a factor of $V^{\text{rel}}_{\text{c}}\sim 13$. We validate our live optimisation strategy on silica micro-spheres ranging in size from 2.5-$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in radius. Figure 3(c) shows a typical progression of the live optimiser.

Figure 3(d) shows examples of trapping stiffness enhancements for these different micro-sphere sizes. In all cases we observe that stiffness enhancement factors in the $x$ and $y$ dimensions ($\sim 6-8$) exceed those in the $z$ dimension ($\sim 2-3$). Cases where the $z$ stiffness enhancement is larger result in a concomitant reduction in the $x$ and $y$ enhancements, as highlighted for the trap optimisation targeting the $4.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ radius micro-sphere (points enclosed by a dashed ellipse).

Figure 3(e) shows the 3D confinement volume reduction factors ($V^{\text{rel}}_{\text{c}}$) for each micro-sphere size. Our experimentally optimised traps deliver about an order of magnitude reduction in the confinement volume across all tested particles. This corresponds to an improvement in trapping efficiency by a factor of $\sim 5$: i.e. the same 3D trapping stiffness as conventional optical tweezers can be achieved using only $\sim 20\%$ of the laser intensity. Traps optimised for larger micro-spheres tend to yield higher enhancements, consistent with our expectations from theory and simulations. For the smallest and largest micro-spheres, we perform five optimisation runs on different particles within the sample to demonstrate the repeatability of our live-optimisation strategy (see SI §15), despite variation in particle size, and the presence of significant measurement noise.

In order to better understand the limitations of our live optimisation strategy, we simulate its performance when stiffness measurements are subjected to the level of noise found in our experiments (for experimental noise analysis see SI §17.). Figure 3(e) shows the anticipated reduction in confinement volume based on these simulations (grey circles). By comparison, our experiments reach $\sim$ 70-90 % of these values. Residual differences are due to real-world SLM diffraction losses that occur as the complexity of the displayed patterns increases (see SI §20), and the presence of minor system aberrations that break the circular symmetry of the trapping field, neither of which feature in our simulations.

In the future it may be possible to develop faster and more sophisticated live-optimisation algorithms with improved resilience to noise mastiani2021noise ; mididoddi2023threading . Furthermore, we anticipate it will become feasible to directly experimentally deploy pre-optimised traps following high-fidelity system and sample calibration – so that the trap design algorithm accurately captures the capabilities of the experimental platform. SI §16 shows a direct comparison of our experimental results with all simulated optimisation approaches, highlighting the future potential of such customised 3D optical trapping.

We now consider the advantages and trade-offs of enhanced optical trapping more broadly. Our optimised trap design algorithm is efficient and versatile: capable of full-field or phase-only optimisation, compatible with microscopes of any numerical aperture up to a solid angle of $4\pi$ (thus including counter-propagating dual-beam trapping ashkin1970acceleration ; thalhammer2011optical ), and can be readily extended to optimise beams with spatially varying polarisation, and to multi-spectral light control. Our strategy also allows the aspect-ratio of the confinement volume to be freely tuned bowman2010particle (as shown in SI §5.). Additional optimisation criteria, such as prescribed optical forces, or optical torques about any axis, can also be specified using the GWS operators horodynski2020optimal ; stilgoe2022controlled ; hu2022structured .

Here we have focused on the discovery of new traps for homogeneous and isotropic micro-spheres. However, these design techniques can be applied to optimise traps for particles of arbitrary geometry by first pre-calculating the shape’s T-matrix – in these cases, the optimised trapping fields will depend upon particle orientation. We have shown how high refractive index particles, previously considered ‘untrappable’, have the potential to be stably held using appropriately shaped beams. Furthermore, our design algorithm can also generate new forms of optimised ‘bottle-beam’ or ‘dark’ optical tweezers arlt2000generation ; xiao2021efficient , capable of stably holding objects of lower refractive index than the surrounding medium. Current dark trap design methods struggle with extended low-index particles that are significantly larger than the diffraction limit melo2020optical – a challenge our optimiser can overcome, as shown in SI §18.

It is important to note that trapping stiffness enhancements tend to come at the expense of a reduction in the energy barrier preventing a particle from escaping the trap – as also observed previously for 1D optimised traps taylor2015enhanced . This effect is noticeable in Fig. 1(b-c), where the stiffness at the origin is many times higher for the optimised trap (red curve), yet the trapping range, as well as the maximum restoring force, are reduced (see also extended plots in SI §10.) In practice, this means that the beam must carry enough power to counteract the thermal motion of the particle and prevent it from ‘jumping out’ of the trap, and this effect becomes more acute for traps optimised for particles of high refractive index (see SI §19).

Our concept relies on being able to shape the incident field across the footprint of the target object. Therefore, dielectric particles of a diameter close to the diffraction limit and smaller do not benefit from dramatic 3D stiffness enhancements over conventional optical tweezers – although we note that sub-diffraction limited super-oscillating beams may offer new opportunities for tightly trapping smaller particles singh2017particle . In addition, lossy metallic particles cannot presently be treated by our fast optimiser in its current form, due to its reliance on flux-conservation. Recently, efficient single-parameter numerical optimisation of force or torque on metallic nano-particles was demonstrated lee2017computational , pointing to a way forward in this regime.

Finally, we note that holographic beam shaping has previously been used to adapt the 3D intensity of light to match the shape of trapped particles kim2017tomographic – an approach capable of holding irregularly shaped objects at desired positions and orientations, which can be viewed as optimising the trapping stability. Our concept is fundamentally different – we aim to optimise trapping stiffness, which entails shaping light fields to create high intensity gradients on the boundaries of a homogeneous particle, rather than projecting uniform intensity throughout its volume. Recent work has also begun to explore new ways to identify regions of high refractive index gradients within large inhomogeneous particles, in order to exert higher optical forces by focusing light onto these areas landenberger2021nonblind .