1. Introduction

Optical trapping of small particles has established an indispensable basis for a wealth of scientific applications, ranging from fundamental physical research to commercialized manipulation systems for the life sciences.

The force exerted by a laser beam on a particle is in general classified into contributions from gradient forces that pull particles with higher index of refraction than the surrounding medium into regions with larger intensity, and scattering forces which stem from the transfer of momentum by absorption or scattering of photons and therefore push particles along the direction of the laser beam. In many cases scattering forces counteract the trapping of particles with light. Therefore several approaches have been devised to find a compensation for the often adverse effects of the scattering forces for stable trapping. One of the earliest is also one of the most successful solution, namely to use a single, tightly focused laser beam to trap transparent, high refractive index particles [1]. Probably its great popularity stems from the simplicity, which eases the practical implementation of this method. The tight focusing that is needed to outmatch the scattering force versus the gradient force in the axial direction typically requires microscope objectives of immersion type with large numerical aperture (NA>1). As a consequence this involves a small working distance and a narrow field of view, which often conflicts with the needs of practical applications. Another approach, already realized in the original work by Ashkin [2], is to use two counter-propagating laser beams to balance the scattering forces. If both beams are diverging, a stable axial trapping is possible. This relieves the burden of using large NA objectives with all its implications at the expense of a more involved optical setup. The realization of fiber-optical traps [3, 4, 5] that employ (at least) two optical fibers to create the trapping beams helped to considerably reduce the complexity of the optical setup, especially since it decouples the imaging from the trapping. They also offer the possibility to trap large cells, useful for Raman micro-spectroscopy [6]. However, it lacks the flexibility of conventional single or dual beam setups, especially in combination with modern wavefront shaping methods that use spatial light modulators (SLM) and allow to dynamically realize a variety of different trap configurations [7, 8].

In this work we present the experimental investigation of another approach reminiscent of a dual beam trap, which we will call optical mirror trap in the following: Two collinear beams are shaped by an SLM to form a predefined focus each, one before a planar mirror and one after reflection off the mirror (see inset of Figure 1 – note that the reflection of the beam forming focus 1 and the incoming beam later forming focus 2 do not significantly contribute to trapping, because they are both spread out at trapping positions with sufficiently large distances from the mirror). For focusing the trapping beams we use a microscope objective with a low numerical aperture of NA=0.2 and a long working distance of 7.4mm. In the axial direction the gradient forces are negligibly small and axial trapping is due to the balancing of the scattering forces in the region between the two foci of the trapping beams, whereas in the lateral direction the gradient forces are responsible for trapping. This method shares the simplicity and flexibility of single-beam traps, but allows to use objectives with low NA, which in turn results in a large field of view and a large working distance. Furthermore, our approach allows to have independent optical setups for trapping and imaging. We will present results that indicate this method is quite robust against misalignments or optical aberrations, which makes it easy to realize.

The trapping principle and trap performance are similar to the flexible, counter-propagating beam traps realized with the generalized phase-contrast method [9]. However, the way how the traps are created is different. Recently a very similar optical setup to ours has independently been proposed and analyzed theoretically in the work by Zwick et. al. [10, 11]. There the authors suggest an optical tweezers scheme with the same experimental configuration, but relying on trapping by gradient forces after complete cancellation of the scattering forces in the axial direction. They have successfully demonstrated trapping with NA=0.7, which is already smaller than the typical values for single-beam traps. The authors state that their method is quite sensitive to correct alignment, because the approach requires exact cancellation of axial scattering forces, and that optical aberrations have to be taken into account to achieve stable trapping.

2. Experimental setup

In this section we describe in more detail the experimental setup that we use for realizing optical mirror traps (see Figure 1). As a light source for the trapping beams we use an Ytterbium doped fiber laser (model PYL-10-1064-LP from IPG Photonics) at a wavelength of 1064nm and a FWHM linewidth of about 0.5nm and a corresponding coherence length of a few mm. The maximum output power is 10 W, but typically we use only between 0.1 W and 1 W. The collimated Gaussian beam with a 1/e ²-diameter of 5.1mm directly illuminates the SLM. We use the model HEO 1080 P from HOLOEYE Photonics. It has a resolution of 1920×1080 pixels with a pixelsize of 8 µm. We do not expect thermal effects that would affect the properties of the SLM for laser powers below 1 W with the above mentioned beam size [12], therefore we used a power limit of 1 W. The SLM shifts only the phase of the incoming light. We calibrated the adjustable dependency of the phase shift on the applied gray level in such a way that the gray levels are linearly mapped to a phase shift from 0 to 2π. We adjusted the linear polarization to maximize the diffraction efficiency. The surface of the SLM is slightly bent along the long dimension, which leads to a small astigmatism of the reflected beam, equivalent to a cylindrical lens with a focal length of about 24m. This was measured by the method described in [13]. To compensate for this aberration we incorporate a correspondingly opposite cylindrical lens term into the phase patterns that are used to create the trap beams. We use the SLM in an off-axis configuration, i.e., by adding a grating to the phase pattern we separate the (first order) diffracted beams from the zero order reflected beam which is blocked by a beam dump. Two lenses with focal length f ₁=300mm and f ₂=150mm image the SLM plane onto the back focal plane of the microscope objective and reduce the beam size by a factor of two to match the image size of the SLM to the aperture of the objective.

A dichroic mirror is used to couple the light into the optical path of a Zeiss Axiovert 135 microscope. For focusing the trap beams we use a Zeiss Ultrafluar 10 objective with a numerical aperture of 0.2 and a focal length of 16mm. The beam parameters of the focused trap beams (in air) are described by the waist diameter 2w ₀=8.6 µm and the Rayleigh length z _R=55 µm. Since we use water as a solvent in the probe chamber the Rayleigh length is increased to a value of z _R=74 µm. The beam convergence angle, corresponding to an NA=0. 08, is well below the limit given by the microscope objective (underfilled aperture) and therefore the Gaussian beam is not clipped. Actually trapping experiments should be possible even with a microscope objective with NA=0. 1, if its aperture is fully used.

 

Fig. 1. Experimental setup. A reflective SLM which is illuminated with a Gaussian laser beam creates multiple trapping beams. A telescope consisting of two lenses (f ₁=300mm, f ₂=150mm) decreases the beam size to match the back aperture size of the microscope objective. This objective (Zeiss Ultrafluar 10, NA=0.2) images two beams into the probe chamber creating the mirror trap which consists of 2 foci, one formed before the mirror and one after reflection. The inset shows a magnified view of the mirror trap light configuration. For imaging purposes we overlap broadband visible light on the trapping beams and observe the light passing through the dichroic mirror on top of the probe chamber.

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For the total light efficiency, given by the diffraction efficiency of the SLM and the transmission losses of the lenses, the dichroic mirror, and the microscope objective, we measure for a single beam an overall value of 21 %. The main losses arise from the fact that the SLM has a reflectivity of only 60%.

In order to measure the trapping properties of the mirror trap we use a test sample of spherical polystyrene beads (index of refraction n=1.58 at 1064nm) with diameters ranging from 1.4 µm to 45 µm. The beads are dispersed in water (index of refraction 1.320 at 1064nm). For larger beads (diameter >20µm) gravity starts to play an important role. In these cases we use a solution of 12% sucrose in water instead of pure water to match the specific density of 1.05 g/cm³ of the polystyrene beads. The refractive index is only slightly increased to a value of 1.34.

The probe chamber is made up of a dichroic mirror (thickness 3 mm), a plastic spacer with a thickness ranging from 300 µm to 2.1 mm and a cover glass (thickness 170 µm). Alternatively we use instead of the cover glass a glass slide (thickness 1 mm), which shows a better mechanical stability. The dichroic mirror (model FM03 from Thorlabs) reflects near infrared light (700 nm–1200 nm), which we use for trapping, and transmits light in the visual range, which we use for imaging. For imaging we employ another microscope objective (Olympus A10, with NA=0.25 or Olympus MSPlan50, NA=0.55) above the sample and a CCD camera. A small amount of infrared light leaks through the dichroic mirror and is optionally removed by additional dichroic mirrors. However, as shown later, observing the trap light passing through a trapped bead is useful for aligning the trap and for position measurements. The measurements of bead positions and trap parameters as described in the following sections are taken with this imaging setup, which is independent from the optical setup we use for trapping.

 

Fig. 2. Sketch of trap geometry. (a) If the central plane between both foci of a mirror trap (thick dashed line) lies on the surface of the mirror, then after reflection of one of the beams both foci coincide at a distance z _tr from the mirror. (b) Shifting the central plane away from the mirror surface by an amount d _s, the foci are separated by distance 2d _s, whereas the trap center still is at a distance z _tr from the mirror.

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3. Experimental procedures

In this section we will explain in more detail how to realize an optical mirror trap by holographic means. A sketch of the trap geometry is shown in Figure 2. For a single mirror trap we use the above described SLM to create two collinear beams which differ by their divergence angle. For this we essentially use two superposed diffractive Fresnel lenses with focal lengths f ₁ and f ₂. For convenience we choose a symmetric geometry with f ₂=-f ₁. Each lens is created by a phase pattern

where r is the distance from the center of the SLM, and k=2π/λ. To overlap both beams we use a random mask encoding technique, randomly choosing between one of the two phase patterns for each pixel of the SLM [14]. This method allows a quick calculation of the phase pattern and makes independent adjustments easy. In addition, it allows to easily control the power balance between both beams by changing the pixel ratio of the two shared lens phase patterns. Furthermore, as mentioned above, we add a blazed phase grating (period of 6 pixels) to separate the diffracted beams from the reflected light (zero diffraction order).

After focusing into the probe chamber the foci are axially separated by a distance ztr from the center plane (see Figure 2). The trap distance z _tr and the focal length f of the Fresnel lens are related by

where f ₀=16mm is the focal length of the microscope objective, m=2 the demagnification factor of our beam compression setup (see Figure 1), and n the refractive index of the medium in the trapping region (n=1.32 for water).

In water, for our optical setup z _tr=130µm corresponds to a focal length of f⋍10m. The limit for the shortest focal length that we can create with our SLM is reached when at the edge of the SLM the local grating period (see Eq. (1)) approaches 2 pixels. For our setup this is 65 mm, theoretically leading to a maximum trap distance of 10 mm. In our case this is unrealistically large, since this exceeds the working distance of our microscope objective.

 

Fig. 3. Theoretical calculations of axial forces of a mirror trap resulting from the intensity at the beam axis in dependence of the axial position, i.e., the distance from the mirror. The Rayleigh length of both beams is 74 µm, the bead diameter is assumed to be 1.4 µm. The three subplots show the results for different focus separations 2d_s: (a) 50 µm, (b) 190 µm and (c) 310 µm. The blue and cyan lines show the forces induced by the incoming and the retro-reflected beam, respectively, the former acting negatively, i.e., in direction towards the mirror, the latter positively, and the red line shows the sum of the two. The force near the equilibrium position at 400 µm distance from the mirror is approximated linearly by the black dashed line.

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To axially separate both foci we shift the central plane away from the surface by moving the microscope objective by a distance d_s. Then both foci are separated by 2d _s, whereas the trap center stays at the same position (see Figure 2). Being able to choose this focus separation gives us the possibility to easily tune the properties of our mirror trap, i.e., the stiffness at the center of the trap and the maximum force applied by the trap (escape force). If the separation is zero and the traps precisely overlap, then the scattering forces in the axial direction cancel out entirely along the axial beam range, and we have no trapping forces apart from weak residual gradient forces. If on the other hand the separation is much larger than the Rayleigh length, then there is a vanishing overlap, and again the axial forces are very small at the center. The optimum separations lie within 1–3 z _R depending on the property one wants to maximize.

In Figure 3 we present some calculations of the forces for different separations. In the underlying model we assume that the scattering forces are directly proportional to the intensity on the beam axis [2] and we ignore intensity variations due to interference. This approximation is valid only for beads significantly smaller than the beam diameter. The results show that for small separations the trap stiffness at the center is higher and the escape force lower, whereas for high separations it is the other way round. At a separation of around 190 µm the trap profile shows a maximally extended linear range at the center.

The above mentioned random mask encoding method to combine two phase patterns can straightforwardly be extended to a larger number of N traps. However, the diffraction efficiency per trap scales like 1/N ². Therefore to create several traps we use a slightly modified variant of the weighted Gerchberg-Saxton algorithm as described in [15]. We impose the additional constraint that both beams of an individual mirror trap are regarded as a single trap in order to have the same power balance between the two beams in each mirror trap. With this procedure the power per trap scales like 1/N with the number N of traps, and typically the total power of all traps together is about 90% of the power we measure if we create only a single beam. The coherence length of the trapping laser of a few millimeters is large enough that there is a standing wave interference between the counter-propagating trap beams.

 

Fig. 4. Demonstration of lateral and axial displacement of the mirror trap for a 4.5 µm bead. On the left we see the initial position with two beads caught in two traps, in the middle the trap on the right was moved laterally by several bead diameters, and on the right it was moved in axial direction, in both cases by changing the SLM pattern (Media1)(see movie road-movie.avi). Furthermore throughout the whole scene we repeatedly moved the microscope stage to demonstrate that the beads are trapped.

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This leads to a periodic intensity modulation along the axial direction with a periodicity of ≈500 nm. However the typical bead size is much larger than this and therefore the effect is averaged out. Indeed, in our measurements with bead sizes down to 1.4 µm and powers up to 50 mW we do not observe an effect that indicates that this interference pattern plays a significant role.

4. Experimental results

In this section we report the successful trapping of microspheres in optical mirror traps. Furthermore we present measurements that characterize the trapping properties in more detail. In Figure 4 we demonstrate the three-dimensional optical trapping of microspheres with a diameter of 4.5 µm. Figure 5 presents two images that show the simultaneous trapping of 20 beads. This result illustrates some of the most remarkable features of our implementation of optical mirror traps. First, due to the low (10×) magnification of the microscope objective the field of view is much larger than in single beam setups, which typically employ objectives with magnifications greater than 40. In our case the field of view of 500 µm is actually limited by the sensor size of the imaging camera and not by the object field of the microscope objective, which has a diameter of 2 mm. Furthermore, the use of holographic methods to create the trapping beams allows to simultaneously trap many beads in arbitrary and complex configurations.

The number of beads we can trap simultaneously is limited by the total available light power, which in our case is restricted by the maximum light intensity the SLM can withstand. We even observe stable trapping down to about 1–2 mW per trap, much lower than the 10 mW per trap we used for the data presented in Figure 5. However, with such low power levels the maximum velocity by which the microscope stage can be moved without losing the beads drops to less than 1 µm/s and manual loading of beads one after the other into the traps gets tedious.

The use of a microscope objective with low magnification not only provides a large field of view in the lateral direction, but also in the axial direction a large range is accessible. By controlling the focus positions (Eq. (2)) we explored trapping distances up to 2mm from the mirror. At such large distances the absorption of the trapping light by water (1/e-absorption length 10 mm at λ=1064nm) induces an imbalance of the power of both trapping beams which we compensate. In the same way we can use the power balance to displace the bead from the center position. However, changing the focus positions to axially move the bead offers much more flexibility due to the much larger accessible range while maintaining the trap profile.

 

Fig. 5. Simultaneous trapping of 20 microspheres (diameter is 4.5 µm) in a star-shaped pattern (indicated by red lines) with optical mirror traps. The field of view corresponds to 500 µm in x-direction, the trapping distance from the mirror is 150 µm. The power per trap is approximately 10 mW. Both images are taken from the beginning and the end of a movie during which the microscope stage was moved to show that the beads can indeed be transported collectively (Media2) (see movie star-movie.avi).

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If the mirror is not precisely aligned orthogonally to the incident beams, then the focus of the retro-reflected trap beam is not well aligned with the other focus at large trapping distances. As a consequence, the bead is not trapped in the center of the trap, but somewhat displaced laterally. We checked that even at a horizontal displacement of the trap beams equal to the waist radius (w ₀=4.3 µm) beads are still trapped, however, the escape velocity is roughly half the value of a well aligned trap. Adding a small angular deviation to one of the trapping beams allows to compensate for such a displacement. We want to stress that even at the edge of the field of view such lateral misalignments are not increased, as one might believe. Because of the telecentric design of standard microscope objectives the axis of the trapping beam is always parallel to the optical axis regardless of its lateral position.

To describe more quantitatively the properties of an optical mirror trap the crucial parameters are the maximum trapping force, the trap stiffness and the trapping range. In the following we want to present such measurements, both laterally and axially.

The strongly damped motion of a small spherical particle with radius r in a viscous medium (viscosity η=0.9·10^-3 Ns/m² for water), trapped in a harmonic trap is described by the differential equation κẋ+αx=0, where α denotes the trap stiffness and the friction coefficient κ is given by Stokes’ law κ=6πη r (note that even for bead sizes of 45 µm the inertial force is 3 orders of magnitude smaller than the friction force and therefore we neglect it). As a result a bead, which is initially displaced, approaches the equilibrium position in a way that $x\left(t\right)=x_0\exp (-\frac{\alpha }{\kappa }t)$. According to this approach we record the bead position after an initial displacement of the trap and fit an exponential from which we deduce a value for the trap stiffness. An advantage of this kind of force calibration is that it does not require a calibration of the underlying position measurements [16]. Additionally we directly calculate the position-dependent force acting on a bead from the relation F=-κẋ. This provides us with additional information about the range in which the force is linear with displacement and about the maximum force. To measure the bead position along the axial direction we employ the fact that the transmitted trap light shows a characteristic fringe pattern (see inset of Figure 6(a)) the temporal development of which is recorded by the camera. For sufficiently small movements of the bead (<100µm) the position of one fringe changes in good approximation linearly with the bead position.

 

Fig. 6. Measurement of trapping forces in axial direction. (a) The position of a bead (blue dots), measured from the fringe locations (see inset), approaches the final equilibrium position with an exponential time dependence (black line). (b) Calculation of the velocity yields the position-dependent trapping force. In the observed axial range of ±50µm around the equilibrium position the force scales linearly with the axial position. The slope gives an axial trap stiffness of 0.025 pN/µm. (c) With increasing axial separation of the foci the trap stiffness decreases. Experimental parameters: bead diameter 2.8 µm, trap distance z _tr=400µm, trap power 50mW.

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Figure 6 presents the results for the axial force measurements. Figure 6a shows the time evolution of the axial bead position approaching the equilibrium position which is well described by an exponential behavior. The plot of the position-dependent force (Figure 6b) indicates that the force is well approximated by a linear dependence within the observed range. This is in accordance with our simple model calculations (see Figure 3). Figure 6c depicts the dependence of the trap stiffness on the foci separation. Only for quite large separations it decreases. Theoretically we would expect a larger force for small focus separations than actually observed. A lateral misalignment, as probably present in this measurement, leads to a reduction of the trapping force, which is more pronounced for smaller focus separations, when the beam diameter at the bead position is small. This is supported by our theoretical estimate of the axial forces as derived from the intensity profile for slightly misaligned beams.

Figure 7 presents a measurement for the trap parameters in the lateral direction for several bead diameters. This is again done by laterally displacing the bead from the beam axis, and recording with the camera its exponentially damped approach to the equilibrium trapping position. We observe that the maximum trapping force increases with the bead diameter, together with the trap stiffness. This makes our method especially well suited for the trapping of larger particles. We even trapped beads with diameters as large as 45 µm. The trapping range where the force is linear with displacement also increases with the bead diameter. Similarly to the axial direction we observe that the trap stiffness and maximum trapping force in the lateral direction show only a weak dependence on the separation of the foci, e.g., for beads with 20 µm diameter the maximum force decreases only by 30% if the focus separation is increased from 200 µm to 600 µm. Absolute values of maximum forces are similar in the radial and axial directions, but because of the larger trapping range the stiffness in the axial direction is accordingly smaller. These results qualitatively agree with theoretical considerations in [17].

 

Fig. 7. Lateral force versus bead displacement for different bead sizes. The thick red line is a smoothing average of the raw data (blue dots) to remove the noise due to the Brownian motion. The settings for these measurements were: (a) bead diameter 2.8µm, trap power 50 mW, trap distance z _tr=425 µm, separation of foci 2d _s≈360 µm (b) bead diameter 4.5 µm, trap power 40 mW, trap distance z _tr=1200 µm, separation of foci 2d _s≈470 µm (c) bead diameter 20 µm, trap power 40 mW, trap distance z _tr=1200 µm, separation of foci 2d _s≈260 µm.

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Conclusion

To conclude, in this work we present the experimental demonstration of stable, three dimensional optical trapping of polystyrene microspheres with diameters ranging from 1.4µm to 45 µm in an optical mirror trap. We use an optical setup which employs a microscope objective with a low numerical aperture of NA=0.2 and offers a large field of view. Two collinear beams are shaped by a spatial light modulator to form a predefined focus each, one before the mirror and one after reflection off the mirror, which resembles the configuration in a conventional dual-beam trap. We demonstrated the simultaneous trapping of up to 20 beads, and our results show that with our method even more beads can be trapped. The trap configuration can be dynamically changed at video-rate by modifying the phase pattern displayed on the SLM. We explored a trapping range of 1 mm×1 mm×2 mm and characterized the mirror trap for typical experimental settings in terms of trap stiffness, maximum trapping force and capture range. We demonstrated that our setup is insensitive to misalignment of the counterpropagating beams and therefore robust trapping is easily achieved.

It is straightforward to implement our method in standard optical microscopy setups which leads to many possible applications. For use in biology or life sciences the large field of view opens up new perspectives, e.g., for trapping of large specimens. Comparably low intensities provide a better compatibility with living species. The large working distance allows to insert additional manipulation tools in the probe chamber. Our method can also be applied for complex assemblies in microfluidics. The necessary reflective surface can either be externally integrated into the setup by a mirror or internally by a reflective coating of the probe chamber.