The optical Earnshaw theorem states that a small particle cannot be trapped solely by scattering forces. This limitation is overcome in a novel differential all-optical manipulator. It utilizes four collimated laser beams arranged along the axes of a tetrahedron to confine and move a microscopic sample in an aqueous medium. By adjusting the intensity of each beam individually the magnitude and direction of the optical forces acting on the sample, and via these its position, are controlled. Since only scattering forces are exploited the system is not confined to trapping near a geometrical focus, and therefore enables three-dimensional manipulation over ultra-long working distances. Latex beads 20μm in diameter can be positioned arbitrarily within a volume defined by the overlap of the four 100μm diameter beams. The sample is observed from four directions simultaneously, demonstrating the instrument’s potential as a universal manipulator in connection with high- and isotropic-resolution light microscopy.
© 2007 Optical Society of America
In classical optical microscopy the sample under investigation is spatially fixed between a glass slide and a cover slip. This mounting procedure is adequate for cell cultures in which the cells grow on cover slips and are already in contact with a glass interface. Contemporary biological experiments increasingly demand configurations in which the sample is kept under physiologically relevant conditions. This could e.g. be realized by embedding the sample in an appropriate scaffold or by having the sample freely floating in a liquid medium rather than stuck to a cover slip. Such systems are applicable to single cells, tissues, and whole embryos. There is still a requirement for a means to confine and translate the sample in a microscope in which the sample floats and is not mechanically fixed. It has been shown that the manipulation of small particles in solution can be performed gently by applying optical forces .
1.2 Optical tweezers
Since the first experimental demonstration of optical forces by Ashkin , the technique of optical trapping using single-beam optical tweezers  has become especially popular. In such experiments particles are confined in three dimensions (3D) by the steep intensity gradient, which occurs in the strongly focused electromagnetic fields produced by an objective lens of high numerical aperture (NA). This configuration is therefore also referred to as a single-beam gradient force trap.
Recently, the technique of shaping the trapping beam with a spatial light modulator (Holographic Optical Tweezers  and the Generalized Phase Contrast (GPC) technique ) in the entrance aperture of an objective lens has opened new possibilities for the application of optical traps. Multiple, dynamic traps can be created providing independent control over several particles simultaneously. Holographic techniques still rely on the use of a high NA lens to create an intensity gradient sufficient for trapping. This results in a small field of view and a short working distance, and thereby limits both the size of the particles that can be observed and the distances across which they can be moved . This limitation is overcome in the GPC technique, which has been shown to operate well with low NA objectives and long working distances .
1.3 Confinement by weakly focused beams
For long-range manipulation the short working distance inherent in existing single-beam trapping techniques is a strong limitation. This can be overcome by using a collimated (or weakly focused) beam instead of the highly-focused beam used in optical tweezers. However, without the axial intensity gradient, which provides a force that can balance the scattering force near the focus, an additional force must be present to allow axially stable trapping. In optical levitation , in which a weakly diverging laser beam is directed upwards, gravity provides this counter-balancing force axially, and lateral stability is achieved through a horizontal intensity gradient. This method has been demonstrated in a one-dimensional particle delivery system . Another technique for counteracting a single beam’s scattering force is the use of two diverging, counter-propagating, coaxial beams. The scattering forces generated by the beams are balanced between their foci, and lateral stability is obtained by a strong lateral gradient , as in levitation. One- and two-dimensional manipulation has been achieved by utilizing the diverging beams from optical fibers . The GPC method allows manipulation of multiple particles in a manifold of counter-propagating beam traps . However, a feature common to all of the above techniques is that they rely on lateral optical intensity gradients to provide stable traps.
1.4 Optical Earnshaw theorem
For a three-dimensional manipulator that relies only on scattering forces, the minimum requirement for stability is a set of four beams. A tetrahedral geometry has been proposed as a trap for atoms, but was soon declared to be unfeasible . According to the optical Earnshaw theorem , a small dielectric particle cannot be trapped using only the scattering force of optical radiation. Only for the one-dimensional case have alternating beams been demonstrated to create a temporal gradient sufficient to passively trap a particle at a fixed location . In this work we demonstrate that this restriction can be overcome and that microscopic objects can be confined and moved in a controlled fashion by introducing a feedback system that controls the powers of the beams.
2. Differential active optical manipulator
We have developed an all-optical manipulator that confines the sample using four collimated laser beams that overlap in a common volume. This Differential Active Optical Manipulator (DAOM) does not require high NA objectives: in fact, the intensities gradients are deliberately minimized so that there is no location at which passive stable trapping can occur. In such a system, varying the intensities of the four beams independently allows adjustment of the direction and magnitude of the optical scattering forces acting on the sample. This, coupled with a 3D position sensor that feeds back to control the beam intensities, allows arbitrary active positioning of the sample. In consequence, a particle can be moved in 3D over distances limited only by the beams’ diameters and the fields of view of the position sensors.
In the DAOM the four beams are arranged along the axes of a tetrahedron and are collimated in their volume of intersection. The optical forces generated by the beams add in their common volume (see Fig. 1) and the total force (magnitude and direction) acting on the sample is set by adjusting the intensities of the beams. This offers the possibility of moving the sample along arbitrary paths within a volume of about 0.5nL. Focusing and lateral translation of the microscopic sample, which are common tasks in three-dimensional light microscopy, are easily implemented.
2.1 Experimental setup
Our experimental setup of the DAOM consists of two units (Fig. 2). One unit (Fig. 2(a)) provides the four beams for sample manipulation. The output of a laser (Nd-YAG, wavelength 1064nm) is split into four beams by an arrangement of mirrors and beam splitter cubes. Each beam passes through an acousto-optic modulator (AOM), which provides fast, precise, and independent attenuation. Each beam is then coupled into a single-mode polarization-preserving optical fiber, which guides it to the second unit consisting of the optics required for the manipulator and the sample observation (Fig. 2(b)).
The tetrahedral configuration of the four beams is realized by a set of four identical microscope objectives (Zeiss, Achroplan 63×, N.A. 0.9, water dipping) arranged along the tetrahedron’s axes of symmetry . Figure 2(b) shows the arrangement in a simplified drawing, omitting the mechanical components that hold the objectives. The angle between any pair of objectives is arccos(-1/3) ≈109.5° . The distance between the objectives is chosen such that their focal planes intersect at a single point in the centre of the tetrahedron. With a working distance of 1.46mm these objective lenses provide sufficient space to inject the sample into the drop of water that fills the volume between the four lenses and is held in place by surface tension. The optical arrangement consists of four identical arms, only one of which is shown in Fig. 2(b). The light exiting the fiber is focused by a lens into the back focal plane of the objective lens. The resulting collimated beam passes through the volume between the four objective lenses and exits the sample region through the space between the opposing three objectives. The four beams, each 100μm wide (1/e2 criterion), intersect in the centre, defining the manipulation volume, which is currently about 0.5nL ≈ 4π (50μm)3/3. The sample is manually placed into the centre of the manipulation volume with a microinjection device. The laser beams are then switched on, and only optical forces are employed to further manipulate the sample.
2.2 Multi-view microscopy
In addition to manipulation, the objective lenses also allow high-resolution observation of the sample. A dichroic mirror (transparent to the NIR manipulation beam) reflects visible light from the sample to a tube lens, which forms an image of the object on the CCD of one of four cameras (Sony XCD-SX900, 1280×960 pixels). This allows one to follow the movement of the sample in three dimensions by analyzing the four simultaneously-recorded camera views. Moreover, the data acquired with the four cameras is complementary, since it yields multiple views of the sample. Such information provides better tracking accuracy (as shown in  with two orthogonal views) but more importantly these additional images can be used to render a high-resolution image of the sample. We have shown that the combination of such data sets by image processing results in significant resolution improvement (multi-imaging axis microscopy, MIAM, ). In this experiment each view represents a field of approximately 144μm by 108μm. The sample is illuminated with an external lamp (not shown) and is imaged by its scattered light. However, it should be noted the system is not restricted to scattered light: other microscopy techniques, e.g. fluorescence, can also be applied to visualize the sample .
2.3 Balance of forces in the DAOM
The sample in the DAOM is confined by a balance of optical scattering forces, buoyancy, friction, and thermal forces. If the sample is at rest, the sum of all forces vanishes,
The buoyancy F b is a gravitational effect and depends solely on the relative density of the sample and its volume. The friction F f is a function of the sample’s velocity v(t) relative to the immersion medium. F t(t) describes the thermal forces that cause the sample to exhibit Brownian motion. These forces are not under the direct control of the experimenter, and can cause the particle to be disturbed from its desired position. However, the optical forces F opt,i(x,t), can be controlled temporally via interactive feedback to keep Eq. (1) fulfilled.
In order to change the position of the sample, the optical forces are varied to provide a net force that pushes the sample in the desired direction. The target position and the intensity of the four trapping beams are controlled via a computer interface. The user is assisted by the real-time position information obtained from all four cameras. Once the target position is reached, the forces are again balanced and the DAOM ensures a zero net force. Only in the case of a desired displacement or an external perturbation is this balance briefly disrupted. It should be noted that Eq. (1) can be satisfied by an infinite set of optical forces. Theoretically, any combination of optical forces whose sum balances buoyancy and friction can be chosen. This gives us the freedom to reduce the total power sufficiently to exclude optical and thermal damage to the sample.
We demonstrate the DAOM by capturing and manipulating 20μm diameter latex beads within the field of view of the cameras. Figure 3 shows two selected frames from a video sequence taken with cameras CCD1 to CCD4, demonstrating the ability to move the sample into the centre of all fields of view. The bead, which was initially centered in the field of view of camera 1 (Fig. 3(a)) but happened to be too close to that lens, was moved towards the centre of the microscope by increasing the intensity of beam 1. Once the particle reached its destination, the beam powers were again balanced, bringing the bead to rest (Fig. 3(b)). The total distance the bead was moved from the edge of the field to its centre is 69μm (Fig. 3(c)). It should be noted that this was limited by the field of view of the cameras (144μm by 108μm). Trajectories in any direction covering distances on the order of the manipulation beam’s width (100μm) are feasible within the volume of beam intersection. The sample can be transported into the manipulation volume along the beams’ axes over even longer distances. Figure 3(d) shows the coordinates of the bead’s centre relative to the centre of the microscope, which have been extracted from the sequence by video analysis. The bead moved along -x and +z-direction while maintaining its position along the y-axis, consistent with a force applied in the direction of beam 1. The absolute speed was about 3μm/s. The sum of all four beam powers was kept constant (<50mW , i.e. <6μW/μm2) during the experiment: from frame a until frame b (phase II) beam 1 was more intense than the other three by a factor of 3.4, and in phases I and III the powers of all four beams were the same.
Figure 4 shows a situation where two beads are visible in the fields of view. One bead was selected (arrowhead) and by actively adjusting the intensities of the four manipulation beams it was moved into the centre of the microscope (Fig. 4(b)). Subsequently, this bead was moved back and forth across the fields of view (Fig. 4(c)–(d)). The second bead was ignored and quickly drifted out of the fields of view. This shows that the common functions of translation and focusing of a sample can be easily performed and any three-dimensional path can be achieved by controlling all beams simultaneously.
In summary, we present a new kind of all-optical manipulator, which actively controls the position of a sample by applying forces differentially along the four axes of a tetrahedron. By arranging four collimated laser beams in this symmetric fashion it is possible to build a manipulator that relies solely on optical scattering forces, thereby overcoming the optical Earnshaw theorem. In our current implementation, samples with diameters ranging from 1μm to more than 50μm (data shown for 20μm beads only) can be optically confined and moved across distances of ca. 100μm. The total distance that particles can be moved is limited by the field of view (i.e. the magnification of the system) and the size of the manipulation beams. The principle of the DAOM can be easily adjusted to other scales; beam diameters and magnification can be chosen as required. In our current implementation these parameters were chosen to fit samples that are of interest in the life sciences, ranging from single cells to small embryos.
By controlling the beam powers via the AOMs, we have developed a means of transporting samples in a system requiring no moving parts. This eliminates the vibrations, creeps, and hystereses that generally accompany positioning with a mechanical stage. The speeds achievable, several micrometers per second, are primarily limited by the viscous drag that the samples experience in the medium, and by the applied optical power.
In this work we have introduced the principle of the DAOM and experimentally demonstrated its validity. However, for this proof-of-principle we have employed a simple, relatively slow feedback system with a response time of ~1s. This limits the speed at which motion corrections can be performed, and accounts for the accuracy of a few microns seen in Fig. 3(d). However, we are currently implementing a much faster feedback system (response time <1ms), in which case we anticipate that the positioning accuracy will be limited by the resolution of the imaging optics, rather than the response of the feedback system.
The DAOM has great potential in many disciplines that require manipulation of relatively large samples, e.g. embryology and colloidal physics. Compared to optical tweezers, long-range 3D sample manipulation and stabilization is achieved, which gives a distinct advantage for many applications. The fact that the manipulator is set up inside a microscope that consists of four objective lenses makes it ideally suited for three-dimensional light microscopy, especially multi-lens arrangements that provide excellent resolution with low NA (and therefore long working distance) lenses in large samples . The ability to acquire views from four different directions while manipulating the sample without mechanical contact offers new perspectives in the fields of trapping and imaging.
This project, J. Huisken, and J. Swoger have been financially supported by the Bundesministerium für Bildung und Forschung (Germany).
References and links
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