We experimentally investigate the size-selective trapping behavior of Laguerre-Gaussian beams (“doughnut-beams”) and “cogwheel”-shaped beams which are collinear superpositions of two doughnut beams of equal opposite helical index. Experimentally they are created by diffraction of a Gaussian laser beam at a high resolution refractive spatial light modulator (SLM). In the focus of an optical microscope such a beam looks similar to a “cogwheel”, i.e. the light intensity is periodically modulated around the circumference of a sphere with a precisely adjustable diameter. In an optical tweezers setup these modes can be used to trap particles or cells, provided their sizes exceed the ring diameter by a fixed amount. This promises a convenient method of constructing an optical tweezers system in microscopy which acts as a passive sorter for particles of differing sizes.
©2004 Optical Society of America
Optical tweezers have become versatile tools for various applications in microscopy, like manipulation of cells and cell components, measurement of interaction forces, or assembly of micro-structures (a detailed overview is given in ). Recently, there were many important developments in increasing the flexibility of these optical micro-tools. One very promising approach is to control the optical tweezers systems with commercially obtainable high resolution spatial light modulators (SLMs), as for instance small liquid crystal displays. These can be used to spatially modulate the wavefront of a laser beam before it enters the optical microscope. Using the established methods of diffractive optics allows to design arbitrary light intensity distributions in the object plane of the microscope, e.g. to create optical traps in various shapes [2, 3, 4], whole arrays of traps with individually steerable spot positions [5, 6, 7, 8], or three dimensional trapping structures [9, 10, 11]. In most cases such light structures can be dynamically changed thus allowing micro-manipulations at video rate, or faster . Control over the spatial phase distribution of a wavefront also enables an almost arbitrary transformation of laser beam modes into each other. Lasers used for optical tweezers systems usually emit a Gaussian (TEM00) mode, which can be shaped by diffractive optics into a variety of other modes which are useful for particular purposes [12, 13]. One interesting feature is to transform the Gaussian (TEM00) mode into a doughnut-shaped Laguerre-Gaussian mode of precisely controllable order [5, 14], by adding a helical pitch to the phase of the original TEM00 wavefront, corresponding to a phase shift which depends linearly on the polar angle θ around the beam axis, i.e. ~ exp(ilθ), where l is an integer corresponding to the so-called azimuthal mode index, also called sometimes the “topological charge”. It has been already demonstrated that such light modes can be used to trap objects with an even higher axial trapping efficiency than obtainable with “normal” TEM00 modes of the same intensity , and that each photon in such an “optical vortex” carries an orbital momentum lℏ̄ which can be transferred to an interacting particle causing it to rotate around the laser beam axis [16, 17].
One very interesting feature of such focused doughnut modes is that the light intensity is concentrated at the narrow circumference (with a width on the order of the light wavelength) of a sphere. In contrast to a direct projection of a ring-shaped intensity distribution, the center of such a doughnut beam has an intensity minimum along its complete longitudinal extension. The diameter D of the intensity minimum in the focus of an objective depends linearly on the helical pitch index l of the doughnut mode. It has been reported  that the proportionality constant depends on the light wavelength and (reciprocally) on the numerical aperture of the microscope objective. This provides the possibility of creating optical traps with a precisely defined size. It is plausible that such a ring-shaped optical intensity distribution used as optical tweezers will trap only particles with a diameter exceeding at least the ring diameter, since otherwise a particle in the center of the trap would not interact with any photons. This size selectivity still remains, if such a doughnut mode is collinearly superposed with a second doughnut mode of opposite helical pitch index-l. The resulting light intensity distribution is located on a ring with the same precisely adjustable diameter as the original doughnut modes, but now it resembles a “cogwheel”, since there is a periodic intensity modulation around the ring circumference, with a possibly large number of intensity maxima (i.e. the “cogs”) corresponding to 2l. In contrast to a doughnut mode, such a “cogwheel” mode carries no net orbital momentum, i.e. if it is used for optical trapping in microscopy, then no rotation of trapped particles is induced. We investigated both, the size selectivity of trapping with such “cogwheel” modes, and with doughnut modes, and we found the same behavior - however it turned out that it is experimentally easier to quantify the trapping behavior of the “cogwheel” modes, since there the measurements are not influenced by orbital angular momentum transfer. Thus, the topic of our present paper is to quantitatively investigate the trapping behavior of such “cogwheel” modes for particles of different sizes, with the main intention to use it for size selective trapping in microscopy.
2. Trapping with doughnut and “cogwheel” modes
As already mentioned, “cogwheel” modes are coherent superpositions of two doughnut modes with opposite equal helicity. Interference of an l=2 and an l=-2 mode used by  for 3D stacking of particles led to a precursor to our cogwheel modes, though with only a very
small number of “cogs”. There, rotation of the stacked particles was possible by inducing a relative frequency shift. In our case, however, the mode is computer-generated by displaying a corresponding modified phase pattern at the SLM. Thus it is possible to generate very pure cogwheel modes of a fixed helicity, and to induce rotation by directly changing the relative phase without need of changing the frequency. The phase front of a “cogwheel” mode resembles a “pie” divided into a number of slices corresponding to twice the mode index 2l. Between two adjacent slices there is always a phase jump of π, going up and down in turn (see Fig. 2). Our approach has the advantage to allow clean generation of modes of exactly one helicity, without admixtures from other modes even for large l, that can be changed dynamically on demand.
An image of such a “cogwheel” mode in the focal plane of our microscope is presented in Fig. 1, together with an image of a polystyrene micro-bead with a diameter of 5 µm for size comparison. The dark area of the “cogwheel” mode has a minimal diameter of 3.7 µm in focus, and cannot be focused to any smaller spot with the given oil immersion microscope objective (100×magnification, N.A.=1.3). The diameter of the light intensity maxima, and the distance between adjacent “cogs” is on the order of half of the light wavelength, depending on the numerical aperture of the objective. We observed that at a high laser power the “cogs” can act as independent traps for small particles, providing the ability to perform a controlled rotation of a whole number of trapped particles around the beam axis by rotating the optical “cogwheel” by changing the corresponding SLM holograms at video rate. However, at lower laser intensities it was only possible to trap particles which centrally fitted into the “cogwheel”, starting form a size which crucially depended on the helical mode index.
A sketch of our optical setup is displayed in Fig. 2. For creating the optical trap we use a continuous wave Ytterbium fiber laser which emits a linearly polarized pure TEM00 mode at a wavelength of 1064 nm out of a single mode optical fiber, with up to 5 W adjustable intensity. Holographic steering of the beam is performed by reflecting it at a spatial light modulator (SLM), i.e. a liquid crystal phase modulator (Holoeye, LCR3000) with a pixel size of 10× 10µm2 and a resolution of 1920×1200 pixels. At our wavelength, the maximal obtainable diffraction efficiency in the desired order is approximately 40 %. We avoid the appearance of undesired diffraction orders by illuminating the SLM with a strongly diverging light wave (lens L1). This beam divergence is compensated only in the desired first diffraction order after reflection of the beam at the SLM. Technically this is done by mathematically adding a strongly focussing lens phase term to the “conventionally” calculated hologram, i.e. by multiplication with exp([iπr 2)/fλ]modulo2π), where λ is the laser wavelength, r is the radial coordinate in the SLM plane (measured from the center of the SLM) and f is the desired focal length of the computer designed Fresnel lens. If the focal length is appropriately programmed, the diffracted first order beam is re-collimated and travels through another set of two lenses to the input of the optical microscope, whereas undiffracted light continues to propagate as a strongly diverging wave.
We used this setup to measure the lateral stiffness (within the object plane) of “cogwheel” tweezers with a method described in . It is based on a time-resolved position detection of a micro-bead while it is being drawn (by light forces) into the center of the optical trap. From the time-resolved position detection, the velocity of the bead at each position can be obtained as a derivative, and this velocity is directly proportional to the local optical force at the respective position. We first caught a bead in the optical tweezers, and then switched the SLM display in the shortest possible time (about 15 ms) such that the center of the corresponding optical trap jumped to an adjacent position, separated by about one bead radius. The originally captured bead was then drawn to its new equilibrium position in the center of the displaced laser mode in an exponentially damped movement. The corresponding trapping dynamics was tracked by projecting the shadow of the bead at a fast position sensitive detector (quadrant photo diode) and recorded by computer hardware. From the data, the lateral trap stiffness was determined. This measurement method gave results which were reproducible within an accuracy of 5 % for beads of different sizes, different light intensities and different doughnut modes.
Such measurements of the force constant were only performed for the “cogwheel” modes, and not for “normal” doughnut modes. The reason was that doughnut modes transfer orbital momentum to the beads, and therefore there is no straight path from a bead to the center of a doughnut mode, but instead a curved one. Since our hardware was not adapted for recording such curved pathes, we could use this kind of force calibration only with the “cogwheel” modes which did not transfer orbital momentum to the beads. Comparisons of the absolute trapping forces of doughnut and “cogwheel”-modes of the same sizes and intensities, however, were performed by measuring the critical flow velocities where trapped beads jumped out of the optical trap. These data showed that the trapping forces were comparable, such that our results obtained for “cogwheel” modes are also representative for the doughnut modes.
On the left of Figure 3 the results of measurements of the trap-stiffness for “cogwheel” modes of different azimuthal mode indices are displayed, measured for different sizes of micro-beads. All of the beads consisted of the same material (polystyrene) and trapping was performed in all cases with the same total light intensity. The data points were recorded for each bead size up to the maximal azimuthal mode index (proportional to the diameter of the “cogwheel” mode) where the force constant measurements could be performed. The data show that the trap stiffness at a constant light intensity mainly depends on the bead diameter, i.e. there is an approximately reciprocal relation between bead size and force constant. This behavior is in accordance with other measurements (i.e. ) and can be explained at least for the case of bead trapping in a point-like laser focus by a geometric argument: if a bead is optically trapped in such a point focus, then the only relevant length scale is given by the size of the bead. Therefore, if a bead of any size is displaced by a fixed fraction of its diameter (e.g. 10%) out of the focus, then the absolute restoring force will be the same. Consequently, the trap stiffness which corresponds to the absolute force divided by the absolute displacement will scale reciprocally with the bead size.
On the other hand, the data demonstrate that the stiffness is almost constant if the helical pitch index of a laser mode is increased for a bead of a fixed size (horizontal lines in the diagram). This behavior is clearer pronounced for larger beads (10 and 15 µm). This can be explained by the fact that the directional momentum distribution of the photons in a strongly focused laser mode depends mainly on the numerical aperture of the used objective, but less on the mode index. Thus, if the bead size is sufficiently larger than the “cogwheel” diameter, then the situation at the surface of the bead (i.e. the in and outgoing photons there) does not depend on the area of the destructive interference zone in the center of the beam, i.e. on the mode index. Therefore, the refraction of the incoming and outgoing beams at the bead surface is almost size independent, resulting in an approximately constant trap stiffness.
The situation changes when the size of the “cogwheel” ring is on the order of the bead diameter. Clearly, there can be no restoring force at a bead in the center of a doughnut mode, if the ring is larger than the bead diameter, since then the bead does not interact with any photons. Our experimental data show that for a given bead size there is a sharply defined mode index where trapping is not possible any more, even for considerably increased laser power. On the right side of Fig. 3 the maximal “cogwheel” diameter Dmax that could just trap a bead of a given size is plotted as a function of bead diameter B. We find a linear dependence with a slope of ΔDmax/ΔB=0.53, i.e. the “cogwheel diameter” has to be smaller than 53 % of the bead diameter in order to enable stable trapping, independent of the absolute bead size. We found the same dependence of the trapping condition on the ratio between bead- and laser mode diameters also for trapping in “normal” doughnut modes - where we, however, could not determine the trap stiffness with our experimental method, as explained above. The insert exemplifies an application to living cells: yeast cells growing in a solution typically span a range in cell diameters of several µm. By choosing a specific ring diameter, one can selectively trap the sufficiently large ones. The yeast cells thus demonstrate the same effect for a different refractive index regime (n lies between 1.36 and 1.40, lower than the refractive index of polystyrene) with comparably reduced radiation pressure force.
If these laser modes are used for size selective trapping, then the corresponding resolution is limited by the size resolution of the optical doughnut ring diameter. In our setup we find in agreement with  a linear increase in doughnut ring diameter D as a function of the helical mode index l (in a range up to l=200), i.e. D=(1.00+0.27l)µm. Particularly, the diameter of the dark area in the center of the “cogwheel” or doughnut modes is very sharply confined. Thus we can expect that particles with a diameter difference of about 0.25 µm can be selectively trapped. This is a very useful size range for biological applications, because this is close to the resolution limit of optical microscopes and gives a convenient handle to a size selective manipulation of biological samples (e.g. cells) which have sizes on the order of 10 µm and individual size differences of a few microns.
For practical applications it is important to note that size selective trapping with “cogwheel” or doughnut tweezers of different diameters gives only a minimal size threshold from where on no smaller particles can be trapped - similar to a fishing net with a particular mesh size. However, if a specific object size has to be selected, then there is the requirement for a second method to also reject objects of a size too large. One easy solution to this problem is to perform all trapping experiments in a liquid flow of adjustable velocity, e.g. either by diffractive steering of the optical trap through the liquid by dynamically changing the SLM holograms, or by driving a motorized object stage with constant speed. Since the friction force acting at trapped particles is proportional to their diameters, larger objects will experience larger forces and will be pulled off the trap at flow velocities where smaller objects are still stably trapped. A combination of a doughnut mode trap with such a controlled flow system will then result in a tailored trapping condition for a selected particle size. A recently developed setup working in the holographic Fresnel-regime  is currently investigated as a practical solution for dragging a trapped particle at a controlled speed through the liquid, just by computer-controlled movement of a hologram window across the SLM display.
Summarizing, we have presented quantitative investigations about the trapping behavior of doughnut and “cogwheel” modes, with the main intention to demonstrate their suitability for size selective trapping in microscopy. With our present setup it is possible to produce precisely controllable “cogwheel” and doughnut modes with mode indices up to l=200 corresponding to ring diameters between 1.5 and 55 µm, thus covering a large range of biologically interesting samples to be manipulated. A specific application that we have in mind is the search for un-frequently occurring metastatic tumor cells or micro-metastases in blood or spinal marrow, being significantly larger than normal blood cells. The set-up allows both, the possibility to passively “fish” for cells larger than a specific size without the necessity of permanent supervision, and the possibility to view the trapped particles through the microscope. Moreover, a highly desirable side-effect of size-selective trapping is the fact that this type of optical trap cannot get “clogged” by dirt and smaller cell debris floating around. The size selectivity of the optical trapping, together with the option of easy implementation in the diffractive tweezers systems that are nowadays widely used, thus lend these special laser modes a significant potential for applications in practical microscopy.
This work was supported by the Austrian Science Foundation (FWF), Project No. P14263MED.
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