## Abstract

The development of optical manipulation techniques focused on the confinement and transport of micro/nano-particles has attracted increased interest in the last decades. In particular the combination of all-optical confinement and propelling forces, respectively arising from high intensity and phase gradients of a strongly focused laser beam, is promising for optical transport. The recently developed freestyle laser trap exploits this manipulation mechanism to achieve optical transport along arbitrary 3D curves. In practice, reconfigurable 3D optical transport of numerous particles is a challenging problem because it requires the ability to easily adapt the trajectory in real time. In this work, we introduce and experimentally demonstrate a strategy for on-task adaptive design of freestyle laser traps based on a dynamic morphing technique. This provides programmable smooth transformation of the 3D shape of the curved laser trap with independent control of the propelling forces along it, that can be configured according to the considered application. Dynamic morphing, proven here on the example of colloidal dielectric micro-particles, significantly simplifies the important problem of real-time reconfigurable 3D optical transport and opens up routes for other sophisticated optical manipulation tasks.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical manipulation tools play a relevant role in micro/nano-science and technology. A well-known example is the optical (laser) tweezers that confine a particle by exploiting high-intensity gradient trapping forces of a strongly focused laser beam [1]. They has been exploited in a large variety of research fields, in particular in biophysics due to their capability of exerting forces on biological and macromolecular systems and measuring their responses [2,3]. The laser tweezers can move the trapped particle by shifting the focal spot. This translation mechanism makes challenging simultaneous manipulation of numerous particles and in particular their optical transport. In the last decade alternative tools suited for all-optical optical transport of numerous particles along tailored trajectories have been investigated [4–10]. For example, it has been proven that numerous particles can be confined by a single-beam laser trap in form of arbitrary 3D curve and simultaneously be optically transported along the curve by exploiting phase gradients of the beam [8]. This new kind of optical manipulation tool has been referred to as freestyle laser trap because the shape of the curve as well as the phase gradient independently prescribed along it can be arbitrary. Note that the particles are continuously propelled along the curve due to optical forces arising from the phase gradient [4], which can be varied to change the particles motion without altering the shape and size of the trajectory [8,10]. These capabilities result particularly useful for simultaneous optical manipulation of multiple particles. The freestyle laser trap has been exploited for optical transport of numerous particles with control of both the trajectory and particles motion, in the case of dielectric micro-particles (silica spheres of 1 *μ*m) [8,10] as well as plasmonic nano-particles (gold and silver, size of 100 nm) [9] and even for transport of biological cells (rod-like bacteria 1 *μ*m wide and 4 *μ*m long) [11]. Other proposals are based on Airy beams [12] that are able to transport multiple particles along a fixed parabolic curve extended in the propagation direction. However, there is a significant lack of control of both the curve shape and the propelling phase gradient force of the Airy beam.

In real-life applications, optical transport of numerous particles results in a challenging problem. For instance, the capability of reconfigurable transport to delivery the particles/cells avoiding obstacles present in the host environment is required. In this work we introduce the concept of dynamic morphing conceived to smoothly transform the 3D shape of the curved laser trap with independent control of the propelling optical forces. This makes possible harnessing the potential of freestyle laser traps to achieve reconfigurable 3D optical transport and more sophisticated particle manipulation problems. Dynamic morphing can be exploited for programmable transport of particles but it also allows shaping the curved laser trap according with the outline shape (in general, a non-parametric curve) of the object (e.g., a cell or micro-device) to be manipulated. Specifically, the particle trajectory or the object outline are represented by using a set of linked Bézier curves that are easy and fast to compute. Note that Bézier curves are extensively used, for example, to describe object shapes in computer graphics and define motion planning in robotics. Here we experimentally validate the proposed dynamic morphing technique on the example of all-optical transport of colloidal dielectric micro-particles (silica spheres of 1 *μ*m), which are trapped up to 25 *μ*m deep within the sample avoiding perturbations due to proximal wall effects.

The article is organized as follows. Section 2 describes the design principles of the freestyle laser trap and the proposed dynamic morphing technique. In Section 3 we experimentally study the performance of dynamic morphing of different curved laser traps demonstrating reconfigurable 3D all-optical transport of multiple particles. The work ends with concluding remarks.

## 2. Design principles

#### 2.1. Generation of a 3D freestyle laser trap with local phase control

Let us first summarize the mathematical fundamentals of the curved laser beams considered here. The freestyle laser trap, in the form of a parametric 3D curve in cylindrical coordinates **c**(*t*) = (**R**(*t*), *u _{z}*(

*t*)) with

**R**(

*t*) = (

*R*(

*t*) cos

*t*,

*R*(

*t*) sin

*t*), is generated by using a direct (non-iterative) holographic beam-shaping technique. A hologram addressed into a spatial light modulator (SLM) encodes the polymorphic beam [8,10]

**r**

_{0}= (

*x*

_{0},

*y*

_{0}) are spatial coordinates in the transverse plane. The function

*g*(

*t*) is a complex valued weight (with dimension of electric field) of the spherical waves described in paraxial approximation comprising the beam. The parameter

*T*stands for the maximum value of the azimuth angle

*t*, where

*k*= 2

*π*/

*λ*with

*λ*being the light wavelength. To create the laser curve, the polymorphic beam

*E*(

**r**

_{0}) is focused by using a convergent lens −with a focal length corresponding to the normalization constant f of Eq. (1)− leading to the following expression for the proximity of the focal plane (corresponding to

*z*= 0):

**r**= (

*x*,

*y*). By introducing the expression Eq. (1) into Eq. (2) we derive, after straightforward calculations similar to the ones reported in [10], that the complex field amplitude is distributed in the 3D space along the target curve

**c**(

*t*) according with the expression

*Ẽ*(

**R**(

*t*),

*u*(

_{z}*t*)) =

*Ẽ*(

*t*) =

*γ*exp [i

*ku*(

_{z}*t*)]

*g*(

*t*) /|

**c′**(

*t*)|, where: $\left|{\mathbf{c}}^{\prime}(t)\right|=\sqrt{R{\left(t\right)}^{2}+{R}^{\prime}{(t)}^{2}+{u}_{z}^{\prime}{(t)}^{2}}$ with

**c′**(

*t*) = d

**c**/d

*t*, and

*γ*= −i

*λ*f/

*L*with $L={\int}_{0}^{T}\left|{\mathbf{c}}^{\prime}(\tau )\right|\text{d}\tau $ being the length of the 3D curve.

It is evident that *g*(*t*) = |*g*(*t*)| exp [iΦ(*t*)] plays a crucial role in the design of the curved laser trap. Its modulus controls the intensity distribution along the curve *I* = |*Ẽ*(*t*)|^{2} which is uniform when |*g*(*t*)| ∝ |**c′**(*t*)|. Note that the beam is strongly focused into a diffraction limited laser curve, see Eq. (3), exhibiting high intensity gradients in the normal directions to the curve, which are responsible for stable 3D particle confinement along the it [8,10]. While, the phase of *g*(*t*) is responsible for particle propulsion along the laser curve. In particular, it is the function Ψ(*t*) = Φ(*t*) + *ku _{z}*(

*t*) that controls the phase variation of

*Ẽ*(

*t*). This function is expressed for convenience as

*S*(

*t*) being an arbitrary real function. The phase distribution can be set uniform along the curve

**c**(

*t*) by using or be non-uniform if the particles need to be speed up or slow down at certain points along the trajectory [8, 10]. We underline that the parameter

*l*defines the phase accumulation along the entire curve. For closed curves the phase accumulation is 2

*πl*and

*l*can be understood as the vortex topological charge $l=\frac{1}{2\pi}{\oint}_{C}\nabla \mathrm{\Psi}\text{d}\mathbf{c}$ [13]. The direction of the phase gradient coincides with the curve tangent

**v**= d

**c**/|d

**c**| and then ∇Ψ =

**v**· dΨ/d

**c**, where dΨ/d

**c**= 2

*πlS′*(

*t*) /[

*S*(

*T*) |

**c′**(

*t*)|]. Since the particles experience a propelling force

**F**∝

_{v}*I*∇Ψ proportional to the phase gradient [4] the polymorphic beam allows easily controlling not only the trajectory but also the speed and direction of particle movement. Thus, an uniform phase distribution

*S*(

*t*) yields a constant propelling force

**F**∝

_{v}*I*·

*l*

**v**whose sign coincides with

*l*. The motion of the particle along the curve can be reversed by simply changing the sign of

*l*. While

*l*provides a global control of the phase-gradient propelling forces along the trajectory, the function

*S*(

*t*) can be used to vary them locally in order to speed up or slow down the particles at certain points of the curve.

To experimentally generate the trapping beam, we have used the same system reported in [8,10], which comprises an inverted microscope Fig. 1(a) and a programmable reflective SLM phase-only hologram (see Appendix for technical details).

#### 2.2. Curve design for optical transport

In optical transport applications an analytic expression for the curve **c** is often unknown and only a priori knowledge that the curve must pass through certain points **P**^{(n)} is available. Thus, the problem consists in finding a continuous curve passing through *m* points **P**^{(n)} given in order, *n* = 1, 2, ..., *m*. A similar problem has been solved in computer graphics by using a set of parametric Bézier splines **b*** _{n}*(

*τ*) (e.g., cubic polynomials) joined to each other in the so called knot points

**P**

^{(n)}, see for example [14]. Specifically, each Bézier spline

**b**

*(*

_{n}*τ*) is defined by four points: two knot points ${\mathbf{P}}_{s}^{(n)}$ and ${\mathbf{P}}_{e}^{(n)}$ and two associated control points ${\mathbf{T}}_{s}^{(n)}$ and ${\mathbf{T}}_{e}^{(n)}$, all given in Cartesian coordinates. Here, the sub-indices

*s*and

*e*stand for the start and end points of the spline, correspondingly. The parametric equation for the Bézier spline is given by

*τ*∈ [0, 1]. The coordinates of the control points can be manually specified or automatically determined [14]. Since the piecewise-defined curve

**c**= {

**b**

_{1}(

*τ*), ...,

**b**

*(*

_{m}*τ*)} has to be continuous (${\mathbf{P}}_{e}^{(n)}={\mathbf{P}}_{s}^{(n+1)}={\mathbf{P}}^{(n+1)}$) and differentiable then the points

**P**

^{(n+1)}, ${\mathbf{T}}_{e}^{(n)}$ and ${\mathbf{T}}_{s}^{(n+1)}$ belong to the same line tangential to the curve at the knot point

**P**

^{(n+1)}. The curve can be closed if ${\mathbf{P}}_{e}^{(m)}={\mathbf{P}}_{s}^{(1)}$ or open otherwise. Note that

**c**(

*t*) is simply obtained by representing {

**b**

_{1}(

*τ*), ...,

**b**

*(*

_{m}*τ*)} in cylindrical coordinates, which is further indicated as

**c**(

*t*) ← {

**b**

_{1}(

*τ*), ...,

**b**

*(*

_{m}*τ*)}.

In our case the Bézier splines are calculated using the method proposed in [14] that automatically gives the control points ${\mathbf{T}}_{s,e}^{(n)}$ of the spline **b*** _{n}*(

*τ*), see Appendix. This method provides fast generation of smooth (differentiable) curves (avoiding cusps) that are solved in linear time, we refer the reader to [14] for further information. A set of parametric Bézier splines allows representing an arbitrary trajectory as a piecewise-defined smooth curve

**c**= {

**b**

_{1}(

*τ*), ...,

**b**

*(*

_{m}*τ*)} used further for the generation of freestyle laser traps. We recall that the curve smoothness is required for phase prescription and enables easy particle circulation along the desired trajectory. For example, the closed trajectory

**c**

_{1}(

*t*) sketched in Fig. 1(b) illustrates how six knot points have been connected in order by the corresponding Bézier splines indicated with six different colors. Note that only the control points ${\mathbf{T}}_{s}^{(1)}$ and ${\mathbf{T}}_{e}^{(1)}$ of

**b**

_{1}(

*τ*) have been explicitly indicated in such a sketch for the sake of clarity, see also the Appendix.

Certain experiments could require adapting the curve **c**(*t*) to a target non-parametric object outline (e.g., a cell or micro-device), for example to optically manipulate the object or simply for selective illumination with the laser curve. In such a case the piecewise-defined function **c**(*t*) ← {**b**_{1}(*τ*), ..., **b*** _{m}*(

*τ*)} can be also created by the proposed method but using as many knot points as needed to approximate the object outline.

The complex field amplitude *E*(**r**_{0}) of the trapping beam is easy to compute by numerical calculation of Eq. (1). In our case sequential computation of *E*(**r**_{0}) has been completed in about 10 s (Matlab R2017b and Intel Xeon E5-1620v3 CPU). Nevertheless, since the trajectory is given as a piecewise function **c** = {**b**_{1}(*τ*), ..., **b*** _{m}*(

*τ*)} then parallel computation of

*E*(

**r**

_{0}) for each curve segment is possible. Thus, the computation time of the trapping beam can be performed almost in real time by harnessing the power of parallel computing architectures such as the current graphics processing unit (GPU, e.g. NVIDIA’s CUDA, a device often exploited for real-time calculation of holograms [15]). This is another practical advantage of the proposed curve design.

#### 2.3. Dynamic morphing of the curved laser beam

Optical transport, and other manipulation applications, could require continuous shape variation of **c**(*t*) motivated by several reasons such as particle delivering towards a moving target or transport of particles avoiding suddenly appearing obstacles. Moreover, the change in the trajectory shape has to be such to prevent the particles escaping in the trap transition. Therefore, a method for smooth transformation of a source trajectory into target ones is required. Here, we propose a technique for shape control of **c**(*t*) based on a dynamic morphing process that exploits the degrees of freedom of the composite Bézier curve **c**(*t*) ← {**b**_{1}(*τ*), ..., **b*** _{m}*(

*τ*)}. An easy mechanism to achieve dynamic morphing of the curved laser traps consists in parametric shifting of the knot points comprising the composite Bézier curve. For example, one or several knot points can be simultaneously shifted following a prescribed 3D or 2D path. This idea is illustrated in Fig. 1(b) where only the first knot point has been progressively shifted following a line of length ∼ 5

*μ*m contained in the transverse plane, while leaving the rest of knot points in the same positions. Specifically, in Fig. 1(b) it has been sketched the transformation from an initial 2D closed trajectory

**c**

_{1}(

*t*) to a target one

**c**

*(*

_{N}*t*) in

*N*steps by using six Bézier splines [indicated with six different colors in Fig. 1(b)] linked by corresponding knot points. The intensity and phase of the trapping beams corresponding to the initial

**c**

_{1}(

*t*) and final

**c**

*(*

_{N}*t*) trajectories (trap states) are displayed in Fig. 1(c). Both the created intensity and phase (for

*l*= 30) are uniform along the curves without any artifacts in spite of the curves are constructed by a piecewise function. Here we have considered the transition between closed curves as an example. The transition between closed and open trajectories is straightforward just by untying one knot point. More complex dynamic morphing configurations can be obtained by combining parametric shifting with curvature changes of the Bézier splines, see Appendix.

## 3. Optical manipulation with dynamic morphing of freestyle laser traps

Here, we evaluate the experimental performance of the developed dynamic morphing technique for freestyle laser traps. The characteristics of the optical trapping setup are provided in the Appendix. The experimental results displayed in Fig. 2(a), see also
Visualization 1, correspond to a 2D dynamic morphing process in the *xy*–plane with *N* = 22 states **c**_{1,...,N} (*t*) (described in the previous Section) to warrant continuous and smooth trap transitions. Once the final state **c*** _{N}*(

*t*) is reached the morphing is reversed to return the initial trap state

**c**

_{1}(

*t*) and then the process is continuously repeated. The particles are stably trapped and transported along each trajectory

**c**

_{1,...,N}(

*t*) confirming the success in dynamic morphing of freestyle laser traps. The mean speed of the particles is ∼ 5

*μ*m/s for the trap state

**c**

*(*

_{N}*t*) while it is halved for

**c**

_{1}(

*t*) due to its multiple turns. Let us underline that in the considered dynamic morphing example, see Visualization 1, the experiment starts with the trap state

**c**

*(*

_{N}*t*) corresponding to a curve longer than in the state

**c**

_{1}(

*t*). The trap

**c**

*(*

_{N}*t*) is full of trapped particles, then, when dynamic morphing is activated the length of the curve progressively decreases and therefore a few particles are lost because there is no space for all the previously trapped particles.

In this experiment we have considered two different values of the switching time (100 ms and 500 ms) between consecutive trap states **c**_{1,...,N} (*t*). A switching time as short as 100 ms provides rapid trap morphing as observed in
Visualization 1. The time lapse image displayed in Fig. 2(b) shows a flow of particles that spreads on a surface according with the programmed morphing shape Fig. 1(b). The particle tracking analysis, performed by using an open source tracking software [16], allows estimating the particle trajectories for the switching time of 100 ms and 500 ms indicated in Fig. 2(c) and 2(d) by a dashed trajectory, respectively. For a switching time of 100 ms the first knot point of the curve travels from **c*** _{N}*(

*t*) to

**c**

_{1}(

*t*) with a speed of 2.4

*μ*m/s, and viceversa. In this case the trajectory of a particle starting in the point A (that belongs to

**c**

*(*

_{N}*t*) state) results affected by the motion of the curve. The resulting trajectory (from point A to B) is given by the combination of two different motions in the transverse plane: one due to the phase gradient force (propelling the particle at a speed of ∼ 5

*μ*m/s) and another due to the motion of the laser curve itself. Let us recall that the intensity gradient force continuously confines the particles in the laser curves. For the switching time 500 ms the speed of the first knot point is 0.48

*μ*m/s, then the particle A stays in each curve for 500 ms traveling along it at ∼ 5

*μ*m/s due to the phase gradient force. Thus, in this case a particle at the point A reaches the point B describing a path slightly deviated from the curve

**c**

*(*

_{N}*t*) containing the point A, as depicted in Fig. 2(d). Note that in the fixed curved path connecting the knot point 2 and 6, the particles (e.g., C and D) are exclusively transported by the optical propelling force with a mean speed of ∼ 5

*μ*m/s.

Faster switching of the laser curves **c**_{1,...,N} (*t*) enables time-shared trapping. Specifically, when the return interval of the laser to the same trap state is less than the Brownian diffusion time of the particle (about 10 ms for a 1 *μ*m diameter sphere to diffuse 100 nm) [17] each curved trap is created before the particle can escape. As a result, a time-shared trapping pattern extended over a surface that contains all the curved laser traps **c**_{1,...,N} (*t*) can be created. This could be interesting for mimicking a laser trap surface in which the confined particles can be optically transported by the created landscape of phase gradient forces. Indeed, the time-lapse image displayed in Fig. 2(b) predicts the expected confining surface created by *time-shared* freestyle laser traps corresponding to this particular configuration of 2D dynamic morphing. In practice, this kind of time-shared trapping requires a programmable SLM able to display the corresponding holograms [encoding the beam *E*(**r**_{0}), Eq. (1)] at high frame rates of at least 200 Hz (e.g., Meadowlark Optics LCoS-SLM), which is much faster than our video rate SLM (30 Hz, Holoeye PLUTO). Let us underline that time-shared trapping with freestyle laser traps is different to the familiar time-shared optical tweezers (point-like laser trap) that are often obtained by deflecting a laser beam using moving mirrors or acousto-optic modulators (scan the beam from point to point with rates typically ranging from 50 Hz to 2000 Hz) [18–20]. In particular, *time-shared* freestyle laser traps can create a trapping pattern in form of a *virtual* surface in 3D with the advantage of optical transport provided by the phase gradient forces prescribed for each 3D curve.

Dynamic morphing in 3D can be easily achieved just by shifting in 3D the knot points of the trajectories **c**_{1,...,N} (*t*). For instance, Fig. 3(a) shows a 3D composite Bézier trajectory made by using four knot points, **c**(*t*) ← {**b**_{1}(*τ*), ..., **b**_{4}(*τ*)}. The slope at each point of the trajectory **c**(*t*) is shown in Fig. 3(b). An uniform phase prescribed along the 3D curve with a value *l* = −20 as shown in Fig. 3(b), has been used to exert a propelling force strong enough to drive the particle motion. The intensity and phase distributions of the trapping beam are displayed in Fig. 3(d) for three different axial positions *z*, confirming the correct shaping of the beam. As expected the particles move counterclockwise along the curved laser trap performing a repetitive downstream and upstream motion, see Fig. 3(e) and
Visualization 2. In this example the morphing process consists of transforming this 3D curve into a new one by 2D shifting (in the *xy*–plane) of the second and fourth knot points, obtaining *N* = 5 states as indicated in Fig. 4(a). The particles maintain stable counterclockwise motion along the curved laser traps during the whole transition, see Fig. 4(b) and
Visualization 2. A switching time of 250 ms allows for rapid morphing, however, a longer time of 2.5 s have been also considered here to help the visualization of the particle motion performed along each 3D curve. As in the previous experiment, Fig. 2, once the final state **c*** _{N}*(

*t*) is reached the morphing is reversed to return the initial trap state

**c**

_{1}(

*t*) and then the process is repeated. Nevertheless, in this new experiment not only the curve shape but also the topological charge

*l*has been progressively changed from

*l*

_{1}= −20 to

*l*

_{5}= −30 corresponding with the initial state

**c**

_{1}(

*t*) and final state

**c**

_{N=5}(

*t*), respectively. This illustrates the ability of dynamic morphing for programmable shape transformation of 3D trajectories with independent control of the phase gradient forces propelling the particles along them.

## 4. Conclusions

We have provided an introductory guide to construction and dynamic morphing of 3D freestyle laser traps. First, we have established the mathematical fundamentals of the laser beam shaping technique that allows creating arbitrary diffraction-limited 3D light curves with easily tailored phase distribution along it. This new income in the phase control of 3D freestyle traps is important for the design of optical propelling forces (arising from the prescribed phase gradient) that is crucial for particle transport along the curve (trajectory). Second, based on an approach developed in computer graphics to draw arbitrary curves [14], we have shown how to design and generate on-task adaptive trajectories. Third, we have proposed a simple and computationally inexpensive method for dynamic morphing of such trajectories. This provides a solution for easy and rapid transformation of such curved laser traps according to the considered application, with independent control of the optical propelling forces.

The proposed technique has been experimentally verified on the example of colloidal dielectric micro-particles (silica spheres of 1 *μ*m). Closed trajectories have been used in the experimental demonstrations to provide a continuous flow of particles. Nevertheless, open trajectories can be also created if needed. From the experimental results, we envision that dynamic morphing control of a freestyle laser trap can be further exploited to generate vibrating laser-trap *strings* that are also able to simultaneously confine and transport the particles in 3D. Moreover, we conclude that time-sharing trapping with freestyle laser traps could be useful for the creation of complex trapping patterns in form of a virtual 3D surface, with the advantage of optical transport of numerous particles provided by prescribed phase gradient forces. Dynamic morphing also allows shaping the curved laser trap according with a time-dependent outline shape of a 3D object (e.g., a cell and micro-device) to be manipulated. These facts and the described experimental results illustrate the potential of dynamic morphing of freestyle laser traps for sophisticated optical manipulation tasks yet to explore. Another potential application of the dynamic morphing of freestyle laser curves could be to perform optical manipulation tasks next to surfaces, including highly reflective ones (e.g. silver substrates) for plasmon-assisted optical trapping of nano-particles [9,27].

Further research is required to completely characterize the observed particle dynamics, speed and acceleration, that indeed can be affected by several factors such as: the curve geometry and phase gradients prescribed along it, the type and size of particle and its surrounding medium, hydrodynamic interactions between the particles, etc.

## Appendix

## Optical trapping setup

Our trapping setup comprises an inverted microscope, see Fig. 1(a), a programmable reflective SLM (Holoeye PLUTO, pixel size of 8 *μ*m, refresh rate of 60 Hz) and a high speed sCMOS camera (Hamamatsu, Orca Flash 4.0, 16-bit gray-level, pixel size of 6.5 *μ*m). To generate the trapping beam, a phase-only hologram addressed into the SLM modulates an input collimated laser beam (Laser Quantum, Ventus, *λ* = 532 nm, 1.5 W), which is then projected (with a 1× Keplerian telescope) into the back aperture of the microscope objective lens (Olympus UPLSAPO, 1.4 NA, 100×, oil immersion), as reported in [8, 10]. This lens highly focuses the latter beam (power of 170 mW at lens input) generating the 3D trap in a sample comprising 1 *μ*m silica particles (Bang Labs) dispersed in deionized water. To reach the ability of stable trapping deep within the sample, an immersion oil with higher refractive index (*n* = 1.56, Cargille Labs, Series A) than the standard one (*n* = 1.52) has been used [8, 21, 22]. Finally, let us recall that a polymorphic beam Eq. (1) allows for creating diffraction-limited laser curves of arbitrary geometry and size. The maximum volume in space in which the 3D laser curve can be extended significantly depends on the focusing lens. For example, in the case of a high NA objective lens, as the one (NA 1.4) used in this work, the 3D curve can be axially extended over 50 *μ*m preserving stable trapping of the particles. This is due to both the diffraction-limited condition of the trapping beam and the use of an immersion oil with higher refractive index (*n* = 1.56, Cargille Labs, Series A) than the standard one (*n* = 1.52), that has been experimentally demonstrated crucial to achieve trapping deep within the sample up to 50 *μ*m [8,21,22].

## Bézier spline with automatic calculation of its control points for dynamic morphing

In our case each Bézier spline Eq. (6) has been calculated using the method proposed in [14] that automatically gives the required control points ${\mathbf{T}}_{s,e}^{(n)}$. Here, we provide a short description of this method. The control points, see Fig. 5(a), have been found by using

*ν*and

_{s}*ν*are the tension values given at the knot point ${\mathbf{P}}_{s}^{(n)}$ and ${\mathbf{P}}_{e}^{(n)}$, respectively. The functions with

_{e}*α*=

*a*(sin

*θ*−

*b*sin

*φ*)(sin

*φ*−

*b*sin

*θ*)(cos

*θ*− cos

*φ*) guarantee splines that have approximate second-order continuity [14], with real numbers:

*a*= 1.597,

*b*= 0.07 and

*c*= 0.37. Note that $\theta =\text{arg}({w}_{s})-\text{arg}\left({\mathbf{P}}_{e}^{(n)}-{\mathbf{P}}_{s}^{(n)}\right)$ and $\phi =-\text{arg}({w}_{e})+\text{arg}\left({\mathbf{P}}_{e}^{(n)}-{\mathbf{P}}_{s}^{(n)}\right)$ are angles defined by the knot points. While

*w*= (

_{s}**P**

^{(n)}−

**P**

^{(n−2)}) /‖

**P**

^{(n)}−

**P**

^{(n−2)}‖ and

*w*= (

_{e}**P**

^{(n+1)}−

**P**

^{(n−1)}) /‖

**P**

^{(n+1)}−

**P**

^{(n−1)}‖ have been used in Eq. (7). This method provides fast generation of smooth (differentiable) curves (avoiding cusps) that are solved in linear time, we refer the reader to [14] for further information.

In the considered laser curves we have used Bézier splines with tension *ν*_{s,e} = *ν* = 1. Other tension values also allow obtaining smooth curves, for example Fig. 5(b) shows a comparison for three different values: *ν* = 1, 0.75 and 0.5. This illustrates the possibility of dynamic morphing with variable tensions that can be combined with parametric shifting of the knot points **P**^{(n)}, which in turn underlines the versatility of the proposed technique.

## Quantitative characterization of curved laser traps

The particles are confined along the curve due to the intensity-gradient trapping force while they are optically propelled in the tangential direction to the curve thanks to the prescribed phase gradient force. To quantitatively estimate the trapping force of the curved laser trap, the 3D positions of the particles have to be detected from the images recorded as a video by using particle tracking techniques. Here, we apply the well-known particle tracking algorithm reported in [16] for this purpose.

Let us first illustrate how to estimate the trap stiffness associated to the intensity-gradient optical force in the transverse direction. As an example we consider the case of the optical trap **c**_{16}(*t*) corresponding to Fig. 1(b) and 2(a), which has a shape of 2D ring and topological charge *l* = 30. In this ring trap **c**_{16}(*t*) the particles are confined in the radial direction while they circulate along the azimuthal direction. The particle position is affected by thermal fluctuations that provide information about the harmonic optical potential *V _{r}* in the radial direction. Specifically, the probability density of position in the radial direction follows the Boltzmann statistics:

*P*∝ exp (−

_{r}*V*/

_{r}*k*), where

_{B}T*k*is the Boltzmann constant and

_{B}*T*the sample’s temperature [1]. Therefore the strength of the harmonic restoring force

*F*=

_{r}*k*|

_{r}*r*−

*R*| experienced by the particle, with

*R*and

*k*being the radius of the ring and the radial stiffness of the trap, can be estimated from its distribution of radial positions. In particular, the radial stiffness is expressed as

_{r}*k*=

_{r}*k*/

_{B}T*δ*

^{2}where

*δ*is the standard deviation in the particle’s radial position, assuming thermal equipartition [1]. The value of

*δ*is extracted from the normal probability distribution that fits the histogram of positions Δ

*r*=

*r*−

*R*. In Fig. 6(a) it is displayed the measured positions when 15 particles are confined in the ring trap

**c**

_{16}(

*t*) with

*R*= 3.8

*μ*m, whereas Fig. 6(b) displays the corresponding histogram of positions Δ

*r*(red curve is the fitted normal distribution). In this case it has been obtained

*δ*= 32 nm and then the radial stiffness is

*k*= 3.9 pN/

_{r}*μ*m for a temperature of

*T*= 293 K, the measurement was performed at 30 frames per second. The radial stiffness estimated from the positions of only one particle confined in the ring trap is

*k*= 3.3 pN/

_{r}*μ*m, see Fig. 6(c) and 6(d). These results show that a single particle as well as multiple particles have been properly confined. Indeed, the estimated value of radial stiffness are compatible to the one obtained when using a conventional single-beam laser trap [21, 22], thus it confirms that multiple dielectric particles of 1

*μ*m can be well-confined in optical traps created 25

*μ*m deep in the sample.

The analysis of the measured positions, see Fig. 6(a) and 6(c), can be further exploited for characterizing the created potential landscape. Note that the measured positions provide a quantitative plot of the probability distribution of the particle in the curved trap. Specifically, we observe in Fig. 6(a) that when multiple particles are trapped together they push each other over two eventual soft potential barriers that are revealed in the case of one trapped particle Fig. 6(c). Note that the optical potential landscape of the experimental curved laser trap is not completely uniform due to small intensity variations and imperfections in the optical train yielding such eventual soft potential barriers. The capacity to overcome eventual potential barriers is explained by hydrodynamic interactions when using multiple particles, as pointed out in [23,24] where a similar toroidal trap has been characterized. In our case these two soft potential barriers, whose locations have been indicated by arrows in Fig. 6(c), did not significantly affect the motion of the particle along the ring trap. Nevertheless, if needed, the value of the topological charge can be increased to provide a propelling phase-gradient force strong enough to overcome eventual potential barriers along the tangential direction to the curve. An alternative method is to exploit the capability of the polymorphic beams Eq. (1) to locally increase the phase gradient force [10] to overcome such potential barriers, without altering the value of the topological charge.

The described approach is also suited for the rest of trap states **c*** _{N}*(

*t*) in which case the corresponding curve trajectory is used instead of the circle to estimate histogram of positions Δ

*r*. The axial stiffness

*k*for each point of the curve can be also determined by following the described method, however, it could be more difficult to accurately determine the axial positions. Let us underline that other methods for measuring the force acting on the particles and for characterizing different optical potential landscapes have been reported elsewhere [24–26].

_{z}## Funding

The Spanish *Ministerio de Economía y Competitividad* is acknowledged for the project TEC2014-57394-P.

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