1. Introduction

The fact that light transfers momentum to objects was demonstrated more than a hundred years ago [1] and later, lasers allowed for much stronger optical forces leading to particle trapping and manipulation at will [2–6]. Advances in optical forces to increase our control over light-matter interaction to better manipulate particles have revolutionized numerous fields of science and engineering [7–9]. The most basic yet powerful tool which implements optical forces for particle manipulation is the single-beam gradient force optical trap, also known as optical tweezers [3,10]. Even though the optical tweezer has a vast number of applications in particle manipulation and collective dynamics [11–16], it suffers from fundamental limitations. In order to create a stiff 3D trap, the single laser beam has to be tightly focused which requires expensive objectives with NAs exceeding one (making the use of immersion fluid unavoidable). This inevitably results in extreme local intensities creating undesired thermal effects, high aberration sensitivity, reduced available working distance (to a few hundred microns), and narrow field of view. In many applications, a less constrained optical trapping system is desirable for trapping and maneuvering micro and nanostructures while using low beam intensities at the focus [17–20]. To avoid the aforementioned limitations, a more practical alternative to optical tweezers is the counter-propagating (CP) optical traps [2,21–23], where the radiation pressure (scattering force) in the direction of propagation of each beam is balanced by its CP beam. Here, low-NA optics are used allowing longer working distances, widening our latitude for optical trapping and manipulation. The most commonly used CP geometries are shown in Fig. 1.

 

Fig. 1. Optical trapping via two counter-propagating: (a) diverging beams, (b) with overlapping foci, (c) converging beams, and (d) tube-like Gaussian beams.

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The two diverging beams shown in Fig. 1(a), create a stable CP trap [2], as long as the two foci are not far apart. Once the foci depart further, the low axial intensity gradient of the far-field beams leads to low axial stiffness and slow axial motion. This prevents any long-range particle manipulation. When the foci overlap as seen in Fig. 1(b) the transverse stiffness improves. However, this arrangement creates intensity hotspots, especially if high-NA optics are used, which can cause undesired thermal effects. Two converging beams as shown in Fig. 1(c) also create unstable traps longitudinally [24], since if due to Brownian motion the particle moves slightly off-center, it will get pushed out of the trap and drift away. Another possibility is the tube-like Gaussian beams shown in Fig. 1(d), which maintain an optimal transverse trap over a very long operating distance. Yet, due to the cancellation of axial forces at all points along the beams, this arrangement leads to an overall unstable 3D trap. The optical conveyor belt [23,25] which is based on a standing wave created by CP beams, has shown to be a superior technique to strongly confine nanometer-range particles and transport them successfully for several hundred microns. However, for larger particles, unstable trapping was reported [23]. Overall, the most stable CP trap for micron-sized particles so far demonstrated is the two diverging beams, realized first by Ashkin [2]. This trapping CP beams can be achieved by opposing objective lenses, fibers [21], placing a mirror behind the sample [22,26–28], optical phase conjugation [29] and holographic counter propagating traps [25]. In most studies, the counter propagating beams are identical and there are two main methods implemented to move a trapped particle along the beams. One is by varying the power of each beam and the other is by changing the distance between the foci. The latter can be achieved either mechanically or via phase modulation. Since the stability of the trap created by the two symmetric CP beams is a function of the foci separation [22,30] increasing the distance will significantly reduce the trapping stability. So, if the foci remain close enough (maximum 200µm), varying the powers will only move the particle axially by tens of microns at most while remaining in the trap.

So how can we modify the CP beams configuration to significantly extend particle (axial) mobility while increasing trap stability? Is there a way we could make particle confinement less sensitive to foci separation? In other words, how can we shape the light fields in order to engineer the optical forces to significantly expand particle manipulation while maintaining the specimen’s confinement? The answer to this question is entangled in the symmetry of the CP beams geometries that have been so far proposed. Here, we demonstrate the use of Asymmetrical Counter-Propagating (ACP) beams where the entire system works as a millimeter-long conveyor belt (without the use of standing waves) while the particle remains confined at all points. Instead of using two identical objectives to focus the CP beams, we break the symmetry by using only one objective in combination with a lens, engineering optical forces experienced by the trapped particle, leading to a long yet stable transporting tunnel. By doing so, the trap generated demonstrates larger axial stiffness values of at least one order of magnitude with respect to the symmetric counter-propagating beams previously reported [22,28,31]. This system allows for flexible positioning of any size particle ranging from tens of microns to sub-micron with only a few to tens of milliwatts of power, respectively.

2. Formation of optical tunnels

Here we present the idea behind the formation of optical tunnels via ACP beams as shown in Fig. 2(a) utilizing different optics on opposite sides, to focus the beams. The reason for this arrangement is as follows. While the objective allows for a tighter focus, the long focal length lens creates a tube-like Gaussian beam within its Rayleigh-range z_r, producing a constant axial radiation pressure along its path. Because of the low-NA optics used, the gradient forces in the axial direction of each beam are quite small so axial trapping is only due to the balancing of the scattering forces of the counter propagating beams. The tube-like beam is designed to balance axial forces of the first beam within its z_r, making 3D trapping possible at all points between the two foci. In the case of lateral trapping however, the gradient forces of the tighter focused beam are large enough making it responsible for transverse confinement near its focal point. So, by moving the tighter focus forward and back, we can move the 3D trap accordingly (Fig. 2.a) without needing to adjust laser power. While each beam by itself is incapable of confining particles and will push them away, it is only through the synergy of such CP asymmetric beams that long-range trapping and manipulation are possible forming an optical tunnel.

 

Fig. 2. The principal idea of ACP beams capable of particle trapping and transportation forming an optical tunnel. This is achieved by the configuration shown in (a) using a lens (on the left side here) to create a loose focus with a long Rayleigh range (z_r) and a microscope objective (on the right) to allow for a tighter focus on the right. B₁ and B₂ are the two ACP beams. The graph in (b) The red curve shows the sum of axial optical forces due to the ACP beams on a particle anywhere between the two foci which are 500μm apart. The z_r(lens)=500μm and z_r(obj)=10μm forming a stable trap. Insert graph shows axial forces of CP beams generated by two identical objectives (NA=0.25) when the foci are 500μm apart. (c) Is the same calculation as (b) but the foci are overlapped. In b, c the blue and yellow curves represent the axial forces from the lens and objective respectively.

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The speed of a particle in this tunnel is a function of its size and is affected by the power ratios of the ACP beams but is mainly controlled by the rate of displacement of the tighter focus. The latter was achieved in two different ways; by moving the objective using a motorized stage (with controllable speeds) or via SLM. Both methods will be further discussed in the following sections. The extent of this optical tunnel strongly depends on the focal length of the lens, and we were easily able to exceed a millimeter range. This is very different from the symmetric CP beams demonstrated so far where stable trapping and manipulation is limited to a few microns up to tens of microns in the region between the two foci of the beams [2,21,31].

Graphs (b,c) in Fig. 2 illustrate how the asymmetry introduced in the CP traps can extend the effectivity of axial optical forces. The forces calculated here are based on Rayleigh-regime approximation, assuming the scattering forces are directly proportional to the intensity on the beam axis [22] and ignoring interference. In these calculations, we used a 532nm laser beam that has been focused to a spot size of 9μm with a Rayleigh length of 490μm by the lens, and for the beam with a tighter focus, we used a spot size of 1.3μm with a Rayleigh length of 10μm with a 3 to 1 power ratio, respectively. In Fig. 2. (b) we present the axial forces of the ACP beams with their foci 500μm apart. This graph shows that there are two locations where trapping can occur. Those are where the sum of axial forces cancels out (F_axial=0), marked with X. However, only the blue X location forms a stable trapping position, for the following reason. If a particle is trapped at the blue X location and moves slightly to the left or right (due to Brownian motion), the axial optical forces of the system will push it right or left respectively, and back into the trap. However, for the black X position, if the particle slightly moves, the optical forces will completely push it away from the trap. Another result from Fig. 2(b) is that even at large foci separations for ACP beams, the particle can still be trapped. This result is in contrast with the conventional CP beams where trapping stiffness is a strong function of foci separation [22,30] so consequently trapping would have been impossible for such large foci distance. For the symmetric CP beams, if the foci separation becomes too large then there is a vanishing overlap such that axial forces become too small at the center. This is observed in the insert of Fig. 2(b) where only the same lateral trapping could be achieved near each focus. On the other hand, if the separation of the symmetric CP beams becomes zero (i.e. the foci overlap) then the scattering forces in the axial direction cancel out entirely, resulting in no trapping forces (red line in the insert of Fig. 2. c). This is contrary to ACP beams as seen in Fig. 2(c) where overlapping foci do not cancel out and are still able to create a stable trap.

While our proposed setup shares the simplicity and flexibility of single-beam traps, it allows us to use objectives with much lower NAs, which in turn results in a large field of view and working distance. Additionally, if one needs to have both front and side views, it can easily be achieved, both with a wide field of view. This is demonstrated below, in the setup segment.

3. ACP beams trap setup

Our setup for the proposed ACP beams is shown in Fig. 3. A 532nm laser beam (model Genesis from Coherent) is expanded and split into two perpendicularly polarized beams via a polarizing beam splitter with tunable power ratios using a half-wave plate. The power ratio is set at the beginning in such a way that particle transportation and 3D trapping could be achieved at any point inside the optical tunnel. The two beams are focused into the sample chamber from opposite directions using different optics as shown. On one side, a low NA (Nikon 10X, NA=0.25, f=1.8cm) movable microscope objective (MO₁) is used to focus the beam while on the other side a lens (L₁) with a relatively long focal length (12.5cm) is used. In the setup shown in Fig. 3, MO₁ is mounted on a motorized stage so that its focus can be moved with controllable speeds. This is the main method of controlling particle speed accurately when moving in the tunnel. But how well the particle keeps up with MO₁ motion is a function of total laser power utilized and the ratio of power distribution between the two ACP beams. A 4X microscope objective marked as MO₂ is used for side-view imaging of particle transportation along the optical tunnel. In the case of smaller particles, we use higher magnifications for MO₂. High-speed CMOS camera (model SC1 from edgertronic) is used in the setup with 2,000 fps to track particle motion and measure the trap stiffness. When measuring the trap stiffness, we used a 50X microscope objective as MO₂. Additionally, MO₁ can be utilized for frontal view and tracking of the particle inside the tunnel if needed. Examples of frontal imaging are presented in Fig. 4. (f-h).

 

Fig. 3. ACP setup: L₁ and L₂: two lenses with f₁=12.5 cm, f₂=50 cm, L₁ creates the long tunnel while MO₁(10X microscope objective) is the 3D trapping objective and allows for frontal view if needed. L₂ is used for front imaging. MO₂: 4X objective for side-view, MS: motorized stage with controllable speeds, PBS: polarizing beam splitter, HWP: half waveplate, WL: white light, NF: 532 nm notch filter, DM: dichroic mirror, M: mirror, S: the sample is inside a cuvette with 1 cm outer and 1 mm inner length, BE: beam expander. Here, L₁ and MO₁ demonstrate 7% and 5.5% losses, respectively.

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Fig. 4. Experimental demonstration of particle transportation in water along a millimeter-long optical tunnel by the ACP beams proposed in Fig. 3. (a-e): side-view of roundtrip propagation recorded via MO₂. (f-h): Front view of 1 mm transportation where there are three particles involved; two are on opposite chamber walls (1 mm apart), while the central one (indicated by red arrow) is confined inside the tunnel and is moved along it. (f) The central particle is at the right end of this tunnel, in focus with one of the particles (shown with green arrow). (g) The central particle has propagated 500 µm along the tunnel and is now the only particle in focus. (h) The central particle is transported another 500 µm, all the way to the left wall and in focus with the other particle (specified by the blue arrow).

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4. Results

Particle transportation along the optical tunnel via ACP beams is demonstrated in Fig. 4 (a-e) in a sequence. A 6µm polystyrene bead suspended in water is conveyed from right to left (a-c) for a total of 1mm distance and then moved back (c-e) to its original position using powers as low as 7mW in total. This total power was divided with a 3 to 1 ratio between the ACP beams where the higher portion entered the lens. Video clips of different size particles translated with the ACP beams are provided in the supplementary (Visualization 1, Visualization 2, Visualization 3, Visualization 4). In Fig. 4 (f-h) we demonstrate a frontal view of the same 6µm particle moving 1mm in one direction. In these images we see three identical particles, two are located at the two ends, 1mm away axially and about 25µm laterally, marked with green and blue arrows. The third is located in the middle of the two (literally speaking) and marked with a red arrow, which is confined by the ACP beams and can be moved along them. Figure 4(f) shows the middle particle at the right end, hence in focus with green marked particle while completely out of focus with the other. Figure 4(g) shows the middle particle which has propagated half the tunnel (500µm) and 3D trapped is now axially located between the two particles (both out of focus). Figure 4(h) shows successful transportation of the particle all the way to the other end so that it stands in the same axial plane as the previously out-of-focus particle marked with the blue arrow.

For the 6µm particle, the trapping stiffness was measured at three locations in the 1mm tunnel using the PSD (power-spectrum density) method. A total of 50mW power with the same 3 to 1 ratio was used, for better trap stiffness. These three locations are points 50 µm, 600 µm and 950 µm in the 1mm tunnel and κ_z (axial trap stiffness) values were measured to be 9.1, 10.6, 8.9 pN/µm respectively. These three data refer to foci separation of -450, 100, 450 µm respectively. Our measured κ_z values are at least an order of magnitude greater than reported previously when symmetric CP beams, with comparable experimental values were used to trap similar size polystyrene beads [22,28,31,32]. We also measured the lateral stiffness values to be about an order of magnitude smaller than the axial values. More work is required to further improve both lateral and axial trap stiffnesses for the ACP beams system.

Similar trapping and transportation ranges were achieved for either single or multi-particles (next to each other) along the tunnel with different sizes, ranging from tens of nanometer to larger than 20µm. Different size polystyrene beads and biological samples such as red blood cells and yeast particles were successfully trapped and translated. An image of yeast cell translation along the ACP beams has been provided in the supplementary documents, where a total of 20mW power with a 3 to 1 ratio (entering L₁ and MO₁ respectively) was applied. Particles smaller than 2µm demonstrated weaker trap stiffnesses and for smaller than 1µm we had to change MO₁ to a 40X objective with NA=0.5 (ELWD Nikon) for particle trapping and manipulation to be possible. A video clip of a 100nm polystyrene bead translated for 1mm is available in the supplementary (Visualization 4). In this clip, we also demonstrate the particle coming to a full stop (3D trapping) in the middle of its path. For this translation, a total of 40mW power was used with the same ratio as before.

As mentioned earlier, particle traveling can be achieved in both directions without moving MO₁, if the SLM is used. Figure 5 shows our second setup with the SLM (Model Pluto from HOLOEYE Photonics) added in the second beam path, to move particles in the tunnel in a similar manner but keeping MO₁ stationary. We have also added a 2X beam expander before the SLM and two lenses L₃ and L₄ with focal lengths f₃ = 40cm and f₄ = 20cm after the SLM. This lens combination would both image the SLM plane onto the back focal plane of the microscope objective and reduce beam size to match the back aperture of the objective MO₁. Here the focus of MO₁ moves due to the phase shift of the beam introduced by the varying phase mask on the SLM. The phase mask used were diffractive Fresnel lenses with varying focal lengths. We calibrated the grayscale level of the applied masks such that they mapped a phase shift from 0 to 2π. We used the SLM in an off-axis configuration so that once we add a grating to the phase pattern, we could separate the first-order diffracted beam from the zeroth-order (the reflected one without phase information) which would have otherwise been overlapped. Even though our masks and beam polarization have been optimized for maximum efficiency, we still have an overall loss of more than 65%. Nevertheless, due to the small powers needed, particle trapping and manipulation could be easily achieved. Using the SLM allowed us to control particle location at will and without moving any of the components, which results in higher precision. Also, our stiffness measurements showed comparable κ_z values to the previous case without the SLM. However, this method has a few disadvantages. By keeping MO₁ fixed we lose the front-view capability and for studies that require frontal view, it could be an issue. Another setback is the need for power modulation as the phase shift becomes larger for longer particle transportation. This limited the maximum translation length of the trap to less than half (∼450μm) of what could be achieved using the first setup, where we moved MO₁.

 

Fig. 5. Experimental realization of the optical tunnel by ACP beams using the SLM (spatial light modulator) to move the trapped particle without any moving parts in the setup. L₃ and L₄ with f₃=40 cm, f₄=20 cm are added to reduce beam size and image the SLM onto the back focal plane of MO₁. BE: 2X beam expander.

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In summary, we have demonstrated how by breaking the symmetry of the well-known counter-propagating optical traps, we can modify the overall optical forces allowing for long-range particle manipulation and trapping, exceeding that of the symmetric CP beams. The ACP beams work together to form an optical tunnel capable of transporting matter for more than a millimeter, with the ability to bring them to a full stop at any point, using inexpensive low-NA optics. Our trap stiffness measurements for ACP traps reveal at least an order of magnitude larger κ_z values compared to the symmetric CP beams with comparable experimental parameters (laser power, objective’s NA, and particle size). To fully understand all the parameters that could affect the ACP beams trap stiffness and improve it, further investigation is required. Our system is very simple but quite practical and versatile. It can confine and manipulate a large range of particle sizes, varying from tens of microns to tens of nm, using a few to tens of milliwatts of power, respectively. Due to the small intensities used, our method can avoid undesired thermal effects (such as photodamage) which can be inevitable in the case of single beam high-NA optical tweezers. This is specifically important when handling biological samples or absorbing particles, where high intensities can cause sample damage. Also, such transportation lengths are impossible to achieve with high-NA optical tweezers, due to their short working distances. It is worth mentioning that the alignment of the proposed asymmetric counter propagating system here, where only one low-NA objective is used, is very straightforward and significantly easier than when two objectives are utilized. A slight misalignment in the ACP beam system is quite forgiving and does not result in a significant reduction of the trap stiffness. Also, by avoiding high-NA objectives which are mostly oil immersion and have very short working distances, we can avoid high aberration sensitivity in the proposed system, while reducing the cost at the same time. Finally, the combination of “low-NA objective and a lens” utilized in our system to generate the ACP beams, has considerably reduced the sensitivity of the trap stiffness on foci separation.