We experimentally demonstrate optical rotation and manipulation of microscopic particles by use of optical vortex beams with fractional topological charges, namely fractional optical vortex beams, which are coupled in an optical tweezers system. Like the vortex beams with integer topological charges, the fractional optical vortex beams are also capable of rotating particles induced by the transfer of orbital angular momentum. However, the unique radial opening (low-intensity gap) in the intensity ring encompassing the dark core, due to the fractional nature of the beam, hinders the rotation significantly. The fractional vortex beam’s orbital angular momentum and radial opening are exploited to guide and transport microscopic particles.
©2005 Optical Society of America
Optical vortices embedded in a light beam are associated with doughnut-like intensity patterns and helical phase structures. It is well recognized that an optical vortex beam with a helical phase structure of exp(ilθ), where l represents the topological charge and θ the azimuthal angles, carries orbital angular momentum (OAM) . Numerous works have been done to implement the optical vortex beams for trapping and rotation of microscopic particles, for example, Laguerre-Gaussian (LG) beams  and high-order Bessel beams  were applied to induce rotation of particles due to the transfer of OAM from the light to the particles. However, it should be noted that all the reported optical vortex beams coupled in optical tweezers systems are with integer-order topological charges. In recent years, studies on the vortex beams with fractional topological charges have widely been reported, including fractional vortices imprinted in a Gaussian beam or a plane-wave beam [4–9] and fractional helical phase in Bessel beams [10–11]. Among the prior work, many interesting results have been obtained. Berry studied vortex structures with fractional phase steps and concluded that no fractional-strength vortices can propagate and their total vortex strength is the nearest integer ; Oemrawsingh et al found that the intrinsic OAM of a photon in an off-axis imprinted vortex can take on a continuous value of lħ (here l is either an integer or a fractional number) . Furthermore, reported simulations and experiments showed that an optical beam with fractional vortex structure is distinguished from an integer-order vortex beam in terms of the intensity pattern, which possesses a radial opening (low-intensity gap) in the annular intensity ring encompassing the dark core, and the studies predicted that such an optical beam with a fractional topological charge also carries OAM. To our best knowledge, however, no report has been found to experimentally demonstrate that an optical beam with fractional vortex charge has been used for trapping and rotating particles directly, and the problem how the unique radial opening in the annual intensity rings affects the rotating behavior of the trapped particles in particular has not been studied experimentally in optical trappings.
In this Paper, we demonstrate optical trapping experiments by using vortex beams with fractional topological charges, namely the fractional vortex beams. The transfer of OAM from fractional vortex beams to the trapped particles was verified experimentally. It was observed that the rotating fashion of the trapped particles induced by fractional vortex beams is different from that induced by the integer ones, owing to the influence of the fractional topological charges.
2. Fractional optical beams
On the basis of electromagnetic theory, the OAM density of an optical vortex beam can be calculated by the cross product of radius vector r and the Poynting vector S of the beam about the z axis. The OAM density can be written in terms of complex amplitude of the beam as below [1, 12],
where u is the complex amplitude of the beam, c is the speed of light in vacuum, ε 0 is the permittivity of free space, and w is the angular temporal frequency of the beam. The total OAM of the beam can be obtained by integrating the Eq. (1) within the cross-section of the beam. For the integer-order vortex beams, the angular momentum per photon is lħ, so the total angular momentum per second for linearly polarized light is :
where P is the laser power and v is the frequency of the light. However, for the fractional vortex beams, the angular momentum per photon is only proportional to the step height for integer and half-integer values . And the OAM per photon, which is expressed as the total OAM of the beam divided by the number of photons of the light field, can take on a fractional value of lħ [5–6].
We used a spatial light modulator (SLM) (Holoeye, LCR3000) to reconstruct a fractional vortex beam which was then directed into an inverted microscope (Carl-Zeiss Axiovert 25) to manipulate microscopic particles. In our experimental setup, as is shown in Fig. 1 schematically, firstly, a linearly polarized green light (wavelength of 532nm, with laser beam power adjustable) was emitted from a Nd: YVO4 laser (Coherent Verdi, 8 W), and then was steered by a mirror. Before the light projected onto the SLM screen, a half-wave retardation plate was inserted into the optical path to change the modulation type of the SLM as intensity-intensity or phase-intensity . We changed the polarization orientation of the linearly polarized beam incident to the SLM by rotating the half-wave retardation plate to make the SLM modulate the light’s phase only. Then the reconstructed light was reflected by the SLM screen and was condensed by two lenses (f1, focal length 100mm; f2, focal length 50mm) and then directed into the inverted microscope, in which a 100x oil-immersion objective lens (N.A. 1.25) was used to focus the light and a CCD camera was attached to observe and capture the images on the sample plane. Note that, to avoid the interference of the on-axis reconstructed beam with the DC component reflected from the SLM, we added a blazed grating to the phase-only hologram loaded on the SLM for an off-axis output, which was finally coupled into the microscope system for optical manipulations.
Patterns of the vortex beams with a negative charge of 3, 3.1–3.6, and 3.8 imaged onto the sample stage of the microscope are shown in Fig. 2, respectively.
In Fig. 2 the diameters of the split doughnut beams are estimated as around 10–15μm. It is seen that the radial openings in the circular intensity rings are becoming wider gradually when the fractional charges are increased from 3.1 to 3.5 and then narrower with the changes of the fractional charges from 3.6 to 3.8. This fashion complies with our simulation results. It is noted that the orientations of the low-intensity gaps in the fractional beams rotated a certain angle due to the Guoy phase shift of the beams in propagation.
3. Experimental optical manipulations
In the experiments, some latex sphere particles with a diameter of 3.1 μm were diluted in de-ionized water and used for trapping by the reconstructed laser beams at a power of 850 mW. We utilized different vortex beams similar to as shown in Fig. 2 to manipulate the particles on the sample stage and found that all the vortex beams, except the one with l = 3.5, were able to realize the annular rotation of particles, and when we reversed the sign of the vortex charge, the particles were found to rotate in the opposite direction. In Figs. 3(a) and (b) images representing the videos captured from the sample stage demonstrate the rotation of particles driven by respective vortex beams with a charge of 3.3 and 3.4. The white arcs with arrow in the figures show the orientation of the rotation. However, we can observe from the video represented by Fig. 3(b) that the reconstructed split doughnut for confining the particles has been distorted by the larger radial opening of the beam, compared with that in Fig. 3(a).
We observed that the particles’ rotating speed decreases with the increase of the radial opening width although the fractional vortex charge becomes greater. For example, the particle ring driven by the beam with a charge of 3.2 spent 26 seconds for a full rotating cycle but the one with a charge of 3.4 spent 36 seconds for a full cycle. A detailed plot based on the experimental results is shown in Fig. 4 to further reveal the relation between the rotating time for a full cycle and the topological charge of the vortex beams. It is known from Eq. (2) that the total OAM of a vortex beam is mainly determined by the power of the laser and the vortex charge. Practically, the rotation is also determined by factors such as viscosity of the fluid, refractive indices of the particles and the fluid, light absorption of the particles, the reconstructed beam quality, and the way that the particles are arranged along the beam. As a result, the rotating speed of a particle ring would vary even though the power of the laser and the charge of the beam are constant. However, we still can find out the trend for the rotations if these environmental factors are consistent. In Fig. 4 the measured rotating time is only considered and compared when the same laser power and the same particles are maintained in the experiments. We can observe from Fig. 4 that the variation of the rotating speed as a function of the fractional topological charges contradicts the experimental observations obtained with integer vortex beams. This can be understood as follows. The optical manipulations such as trapping and rotation of microscopic particles are dependent on the gradient and scattering forces of the light on the particles. For a particle whose refractive index is greater than that of the surrounding fluid would tend to be trapped in a more highly focused light field than a low- or zero- intensity one. Thus, when the microscopic particle is trapped in the high-intensity field, they need momentums to break the trapping and move to the low-intensity area, i.e. the radial opening in our case. Furthermore, although the phase of the fractional vortex beam is continuously rotating in propagation, the intensity of the ring encircling the zero-intensity core is discontinuous, and since the OAM can only be transferred by the change of photon’s momentum, this low-intensity gap would provide less or zero transfer of the OAM to the particles in the gap. As a result, the zero or lower intensity in the radial opening behaves as an obstacle hindering the rotation of the particles. The trapped particles would need more momentums gained from the high-intensity area to overcome the larger low-intensity barrier so as to sustain the rotation. Thus, it is not difficult for us to understand that, for the beams with smaller openings, trapped particles are easier to move across the openings during the rotation because of their inertia. However, as the fractional vortex beam with a half-integer charge has the biggest radial opening, particles trapped in the high-intensity region of the ring are most difficult to cross from the one end of the low-intensity gap to the other end. The hindrance resulted from the intensity variation of the tweezing beam has also been observed in the rotations with integer-order vortex beams [3, 15–16], and in our experiments we also observed that it is easier to see the rotation of more particles along the ring of a fractional vortex beam than a single particle.
For the beam with l =3.5 has the greatest radial opening width, in our experiments we failed to rotate the particles trapped along the intensity ring of the beam. However, to verify experimentally that the vortex beams with half-integer topological charge possess OAM and are able to rotate particles, alternatively, with the same setup we used the beam to trap a large volume of particles on the sample stage. We selected a bunch of 3.1 μm latex particles, which has been cohesively deposited by many particles in de-ionized water and covers almost the full vortex beam. We trapped the bunch of particles by using the vortex beam with a charge of -3.5 and observed the particles rotating intrinsically at a speed of 3.2 second/cycle. Fig. 5 shows the clockwise rotation of particles about the beam’s center, and the white arc arrow illustrates the direction of the rotation.
As an application, the fractional vortex beam possessing the OAM and radial opening was employed to present an example for particles guiding and/or sorting. In Fig. 6 we demonstrate that when the particles are placed initially near the vortex beam by moving the sample stage manually, they will then be aligned and transported to new positions caused by the OAM of the vortex beam of l = 3.5.
In Fig. 6(a) the white line arrow indicates the moving direction of the sample stage, the white arc arrow illustrates the intensity shape of the beam and the rotating direction of the particles and the blocking particle is annotated. Fig. 6(b) explains the mechanism of aligning and guiding with the vortex beam of l = 3.5. When particles are dragged near the arc of the beam with the slow platform movement, they would be trapped and aligned one by one along the arc. The first particle approaching to the gap would always be stopped by the low-intensity barrier and acts as a blocking particle. Since the following particles cannot overtake the blocking particle or overcome the barrier either, they would bypass the blocking particle and be transported to new positions by the movement of the sample stage. Alternatively, in practical applications, if the laser power or vortex charge is large enough, some kinds of particles can be sorted out and trapped at one end of the vortex beam arc first, then moved along, and exit finally at the other end.
In conclusion, we have verified experimentally that the fractional vortex beams possess OAM, which can trap and rotate particles intrinsically or extrinsically. Unlike the integer-order topologically charged vortex beams, the fractional vortex beams have a radial opening, which hinders or even stops the rotation of the particles. However, we can exploit the fractional vortex beams’ OAM and radial opening to guide and transport particles.
This work is supported by the Agency for Science, Technology and Research (A*STAR) of Singapore under A*STAR SERC Grant No. 032 101 0025.
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