Optical tweezers are the most efficient means of manipulating objects on the micrometer scale [1]. Typically, small particles floating in liquid with a refractive index lower than that of the particle are fixed in the focus. If the particle to be manipulated is much smaller than the wavelength, a spherical bead is fixed to the object to help the interaction with light [2]. The primary function of the tweezers is to fix the position of a particle in the focus. In many cases the orientation of the grabbed object is irrelevant, and in the case of bodies with spherical symmetry it is impossible to influence the orientation. However, the control of all motional degrees of freedom of the grabbed particle would greatly improve the utility of the system.

Orientation of the trapped object is possible on the basis of interaction of the focused laser beam with an anisotropic optical character of the object. Several possibilities have been described previously. For example, if the shape of the object is elongated (e.g., it is a rotational ellipsoid), it will be grabbed with the long axis pointing toward the optical axis [3–5]. Orientation around the optical axis of the trapping beam has been described for birefringent particles: if such particles are held in the focus formed by linearly polarized light, they can be oriented [5,6]. In addition, if birefringent particles are held by laser tweezers formed by circularly polarized light, they will rotate, and the rotation direction is determined by the direction of the circular polarization [5,6]. Several additional rotation mechanisms have also been reported [7–12].

It was observed previously that flat objects can be oriented in the optical tweezers if the trap is formed by an asymmetric beam. This can be produced by, e.g., placing a rectangular elongated aperture in the light path in front of the microscope objective [13].

Experience shows that there is another fairly simple way to orient particles around the optical axis. If the particle has a flat shape, it will be oriented in optical tweezers formed by linearly polarized light even if its material is not birefringent. The anisotropic character of the trap formed by linearly polarized light was investigated earlier [14]. It was calculated that the trapping force depends slightly upon the direction: the restoring force on a spherical bead with a size of approximately one wavelength in the lateral direction is ~10% larger in the direction of polarization. It was reported recently (while this manuscript was in preparation) that if small rod-shaped particles are held in optical tweezers close to the glass coverslip surface so that they are perpendicular to the laser beam direction, linearly polarized light can orient these particles [15].

In the present study we discuss the orienting torque of linearly polarized light acting upon a flat object. We also present microscopic objects that are designed and built to be conveniently manipulated by optical tweezers that utilize the orienting effect of linearly polarized light to enable the application of this phenomenon.

We used an experimental system based upon a Zeiss Axiovert 135 inverted microscope. The laser tweezers were a Cell Robotics Laser Tweezers 980–1000 unit. These tweezers consist of an infrared diode laser (SDL 5762 A6). The laser emits linearly polarized light with a wavelength of λ=994 nm. The light is focused in the sample area by a high-numericalaperture polarization-preserving (DIC) objective lens (Zeiss Plan-Apochromat oil immersion 100×/1.4). The power at the sample was ~20 mW.

We controlled the polarization state of the light by positioning optical elements in the light path before the objective of the microscope, corrected for infinity. A quarter-wave plate in a rotation stage was used to change the polarization state from linear to circular, when desired. In addition, a half-wave plate was also added to rotate the direction of the linear polarization. The half-wave plate could be rotated by an electric motor at arbitrary speeds, and the momentary position of the plate was known at all times during rotation.

The microscope image was either observed by a monochromatic video camera (COHU 4912) or projected onto a photomultiplier (Hamamatsu H5784) through a pinhole that cut out the image of the trapped object. The movement of the particle caused a fluctuation of the PMT signal, which was recorded by an oscilloscope (LeCroy 9310L). The signal was analyzed by computer.

Test objects of arbitrary shape were produced in the microscope by photopolymerization of light-curing resins [10,16]. To achieve high spatial resolution, two-photon excitation was used. The 514-nm light of an argon-ion laser was focused by the high-numerical-aperture objective, and in the focus the intensity was sufficient to polymerize the resin (Norland NOA 63) that needs excitation wavelengths shorter than 400 nm. The sample cuvette was positioned on an XYZ piezostage (Spindler & Hoyer P3D 20-100). When the cuvette is moved, the trajectory of the focus “draws” the object. By this method submicrometer spatial resolution could be easily achieved, and consequently particles of fairly complex shapes could be produced at a size of a few micrometers, optimal for the orientation experiments. Following photopolymerization, the unhardened material was dissolved with acetone. The rotation experiments were performed in the acetone solvent.

We found that linearly polarized light orients flat objects. The requirement is simply that, when grabbed, the particle should be flat as observed in the direction of the light propagation. Since it has been reported recently that small disks produced from nonbirefringent material are not oriented in linearly polarized light [5], we made careful tests to verify our contradicting results. Our material is not birefringent—we tested for the birefringence of the particles in the microscope with crossed polarizers. Moreover, we performed crucial orientation tests with circularly polarized light. In this case the polarization plane rotates with the frequency of light. This is too fast for a micrometer-sized trapped object to follow; consequently, if the origin of orientation is the flat shape, the particle will not rotate. On the other hand, if the origin of orientation is the birefringence of the particle, circularly polarized light will induce rotation, as demonstrated and characterized in detail earlier [5–7]. The flat particles produced by photopolymerization never rotated in circularly polarized light. We can reliably conclude that in our experiments the origin of orientation is the anisotropic scattering by the flat particles.

For a detailed quantitative analysis of the phenomenon, we have produced test particles with a well-defined shape. We have chosen a cross consisting of two perpendicular axes of different lengths as shown in Fig. 1. This particle is grabbed in the optical tweezers, preferably with the longest part pointing along the optical axis. The short axis is oriented by the polarized light: it points along the direction of polarization.

 

Fig. 1. Cross-shaped test object to study angular trapping in optical tweezers formed by linearly polarized light: a and c, drawing of the object from two different views; b and d, photographs of the produced objects viewed from respective directions. The scale bar is 3 µm long.

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We measured and characterized the torque exerted on the test object by the polarized light in rotation experiments. The plane of polarization is controlled by means of placing a halfwave plate in the microscope in front of the objective. When we rotate the half-wave plate with a rate ω, the plane of polarization rotates with 2ω. This rotation is followed by the test particle as seen in Fig. 2.

 

Fig. 2. Cross-shaped test object shown in Fig. 1 held in the optical tweezers formed by linearly polarized light. The cross is positioned with the longer rod along the optical axis: the subfigures show different angular trapped positions. The scale bar is 3 µm long. A corresponding video clip can be downloaded form the webpage of our laboratory [17].

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Owing to the viscous drag caused by the rotation, the particle follows the polarization with a phase delay, and this delay increases with increasing rate (results are shown in Fig. 3). The drag torque on the rotating cross can be estimated by use of the formula for the force acting on a cylinder moving in a direction perpendicular to its axis as in Ref. [7]. The viscous drag force on a cylinder of length l with a velocity v can be given by Eq. (1), where η is the viscosity, ρ is the density of the liquid medium, R is the diameter of the cylinder, and C is the Euler constant (C=0.577):

 

Fig. 3. Phase delay of the rotating particle to the rotating polarization as a function of the rotation rate.

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For the cross shape one can neglect the torque acting on the rod parallel to the rotation axis. The drag torque on the other cross rod (R=1 µm, l=2.6 µm) rotating in acetone (η=3.2×10^-4 kg/ms, ρ=790 kg/m³) can be calculated numerically. The result yields the torque at each rotation rate and, consequently, the torque exerted on the cross at different relative angles between the cross direction and the polarization plane (these values are presented in Fig. 4. as measured points).

 

Fig. 4. Torque exerted by the polarized light upon the trapped body as a function of the angle between the polarization plane and the long axis of the trapped flat object. ■, measured data; —, model calculation.

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As a major point of this study, the orienting torque of the polarized light was also calculated by means of estimating the interaction of light with the body. The model is based on the assumption that the origin of the torque exerted on the particle is the polarization dependence of the refraction and reflection of light on the surface of the particle, as described by the Fresnel formulas, Eqs. (2). These equations give the dependence of the reflected amplitudes of the electric field vector for polarization directions parallel (r _‖) and perpendicular (r _┴) to the plane of incidence as well as the respective transmittance coefficients t _‖ and t _┴. Here α is the angle of incidence and β is the angle of refraction:

We applied ray optics, using these correlations for numerically calculating the torque acting upon the test object. In our calculations we assumed that the light intensity is constant across the beam (as opposed to a Gaussian intensity profile). This is a good approximation, since in optical tweezers the laser beam is expanded before entering the objective lens to maximize the numerical aperture. We did not take into account the fact that the particle size is comparable to the wavelength of light—a most difficult task for objects with irregular shape [14,18].

When considering objects of several micrometers in size, this approximation is used with good results [3,4]. The simplifications can be reasonably justified. With the parameters of our optical system at a distance of more than 1 µm from the focus, the wave-front radius for a Gaussian beam agrees to within a few percent with that of the nondiffracting case. Consequently, at the positions where light refraction and reflection on the test object surface occur, the simplified ray optics yields appropriate momentum values. The approach used here has already proved successful in calculating the torque of nonpolarized light exerted upon objects with fairly complex shapes [11].

The laser beam was represented by 10,000 individual rays evenly distributed. The proper fraction of the total laser power was associated to each ray. The momentum current of each ray is given as Eq. (3) where S is the energy current and c is the speed of light:

The forces were calculated from the momentum current change upon reflection and refraction of the ray at the surface, taking into account the polarization dependence through the Fresnel formulas. The arising torque was given as the cross product of the position vector from the optical axis to the point of interest and the force as in Eq. (4):

These elementary torque values were then summed for all rays representing the laser beam. For reasons of symmetry, the resulting torque points along the optical axis.

The result of the calculations is shown in Fig. 4. The experimental and calculated values agree with each other extremely well: realistic parameters yield a perfect agreement between experiment and theory (the following parameters were used: 20-mW laser power, index of refraction of 1.36 for the medium and 1.56 for the particle with a length of 2.6 µm, and elliptical cross section with axes of 1 µm in the direction of the light propagation and 0.7 µm perpendicular to it). The calculation shows that the torque increases to a phase delay of ~60 deg. Beyond this value the system becomes unstable: further increase of the phase delay causes a decrease of the torque. In this regime the particle cannot follow the rotation of polarization; it slips. This property is also observed in the experiments. The agreement is a major justification of the applied treatment.

The results show that this orientation mechanism yields a torque of the order of 10^-19 Nm for reasonable system parameters. This value is in the range of effective torsional manipulation of biological objects such ATPase motors [19] or DNA molecules [20].

The angular trapping effect provides a means for precisely orienting objects within the optical tweezers. By arbitrarily changing the plane of polarization with the half-wave plate, one can orient the anisotropic object precisely, rotated in the focus. Since the torque exerted by the laser beam on the trapped object is known (see Fig. 4), this system can also be used to measure external torque. The presented cross-shaped microscopic objects make the application of the phenomenon practical. They are held in stable axial position in the focus; consequently, holding the particle in arbitrary location and rotation around the axis can be controlled separately. The introduced method offers a powerful additional possibility for manipulation as well: the originally linearly polarized light of the laser tweezers can be changed between linear and circular polarization by means of rotating a quarter-wave plate in the light path. As discussed above, circularly polarized light has no orienting or rotating effect on an asymmetric particle with no birefringence. It follows that by changing the light from linearly polarized to circularly polarized, the orienting effect can be switched on and off. This can be done by minimizing influencing the trapping force of the optical tweezers. The ease of realizing this property (simply by rotating quarter- and half-wave plates in the optical path of the microscope) with this mechanism, together with the ability to control and measure the torque, makes the presented method unique among the several other comparable possibilities for rotation and orientation by optical tweezers.

By attaching macromolecules to anisotropic objects as described here, the introduced method provides a powerful means to investigate the torsional properties of these molecules.

This research was supported by grants OTKA No. T29764, OM NKFP 3/064/2001, and OM NKFP 3/0005/2002.