Optical tweezers: a light touch

Authors


Miles J. Padgett, School of Physics and Astronomy, SUPA, University of Glasgow G12 8QQ, United Kingdom. Tel: 01413305389; fax: 01413304464; E-mail: Miles.Padgett@glasgow.ac.uk

Summary

Optical tweezers use focused laser light to manipulate microscopic particles. We discuss the underlying physics of the technique in terms of a gradient force exerted by the light on the particles. The versatility of optical tweezers is highlighted, in particular, we explain how spatial light modulators and various imaging methods have greatly enhanced their range of applications.

The momentum carried by a light beam is typically very small, given by the power in the beam divided by its phase velocity. Lifting an apple against the force of gravity would require the momentum from a million or more laser pointers, at which point vaporization seems more likely than lifting! However, the simple cubic scaling between linear dimension and mass means that, for an object one-micron in diameter, even a few microwatts of optical power are sufficient to support the object.

From the 1970’s onwards, Arthur Ashkin pioneered the study of optical forces generated with laser beams and their interactions with microscopic objects. Easiest to understand is the scattering force where an incident laser beam exerts a radiation pressure on any object. Less obvious, but for our purposes more useful, is the gradient force which arises whenever a dielectric object is subject to a spatial gradient in the electromagnetic field. This gradient force can be derived from Maxwell’s equations but also has a simple ray-optical explanation. If a light beam is incident upon a transparent object, the scattering force is small. The transmitted light, however, is refracted and hence has changed its direction. This change in direction of the light and the associated momentum flow results in a reaction force on the object. Somewhat counterintuitively, a beam striking the edge of a spherical transparent object is refracted in a way such that the object experiences a force moving it towards the beam axis. If the beam is focused, a component of the gradient force pulls the object towards the focus. Ashkin’s seminal paper (Ashkin et al., 1986) described this approach as a ‘single beam gradient force trap for dielectric particles’, which we now call ‘optical tweezers’. It is usual that the samples to be studied comprise target particles suspended in water or some other fluid. This fluid gives both partial support of the particles against gravity and, more importantly, viscous drag force acting on the particles to ensure that the trap is dynamically stable. Target particles themselves can range from manufactured mircon-sized silica beads and metal nanoparticles to biological material such as diatoms and stem cells.

One appeal of optical tweezers is the simplicity of the required optical system. It needs both to focus the trapping laser to create a high field gradient and to view the trapped object with a sufficient magnification to see what’s happening. Both of these requirements are easily satisfied using a high-magnification optical microscope. In addition to the imaging optics it is usually straight forward to incorporate a beam splitter into the optical path, inserted between the infinity corrected objective lens and the microscope tube lens. A collimated laser beam introduced at this point is brought to a tight focus in the focal plane of the sample, creating an optical trap into which any nearby dielectric particle of higher refractive index than the surrounding fluid will be attracted. It is interesting to note that, if the particle has a lower refractive index than the surrounding fluid, then the optical forces act to repell it away from the beam focus (Fig. 1).

Figure 1.

Schematic of the optical gradient force in optical tweezers. A spherical particle is trapped by a tightly focused laser beam. If the target object moves away from the trap centre, the light passing through the object is refracted in a way such that the object experiences a force back towards the trap centre.

Trapping a micron-sized object at the centre of a microscope image is a beautiful demonstration of optical momentum, but how can we manipulate the object? Shifting the optical trap in a lateral direction requires a change in angle of the light going into the objective lens. Most simply, this can be achieved using a beam steering mirror that is telescopically imaged onto the plane of the objective lens. Automating this mirror or other angular beam steering technology gives the basis of a user interface. Creating two traps with the same microscope can be achieved by introducing a second laser or beam splitter and an additional mirror, but three or more traps would become complicated. Rather than use separate optical paths, early tweezers systems followed a time-shared approach where a rapid switching of the beam direction allows each trap to be created in turn. Providing that the return interval of the laser to the same trap is less than the diffusion time of the particle then multiple traps can be sustained.

In the late 1990’s, 10 years after the invention of optical tweezers, a number of groups began exploring alternative approaches based on diffractive optics. Consider the case of the beam steering mirror being replaced by a diffraction grating. Each of the diffraction orders has a different diffraction angle, therefore producing a linear array of optical traps in the sample plane. Alternative grating designs can be calculated to give different configurations of traps. We note that the diffractive optic is in the far-field of the trapping plane and hence these calculated optics are often called ‘computer generated holograms’ (Fig. 2).

Figure 2.

Simplified holographic optical tweezers. Here, a Laser beam is diffracted is a spatial light modulator, which enables splitting of the initial beam into many, each at an angle set by the periodicity of the diffraction grating. These beams are directed into an objective lens via telescopic imaging, where they are tightly focused. In addition to the apparatus shown here, white light is typical used to illuminate from above, and the sample imaged to a camera via a beamsplitter

Rather than using a pre-fabricated diffractive optic, the development of programmable spatial light modulators (SLMs) gives an interactive alternative. SLMs are pixellated liquid crystal devices where each pixel can introduce a phase change to the incident light between 0 and inline image. The SLMs are typically addressed using the video port output from a computer and allow rapid switching between diffractive patterns and hence configuration of optical traps. The computer control of the SLM also lends itself to the development of sophisticated interfaces where the multi-trap capability is reflected in the interface itself.

The relationship between the calculated diffraction pattern and the configuration of traps is a 2D Fourier transform. However, calculation of simple patterns can be performed more quickly. If all that is required is a single optical trap, the required pattern is simply that of a diffraction grating with a saw-tooth profile, blazed to concentrate all the diffracted light in the first order. Changing the period of the grating shifts the radial position of the trap and changing the angle shifts the angle of the trap. Unlike a simple mirror, an SLM can also be encoded with circular lines, acting as a Fresnel lens to change the wavefront curvature of the laser at the objective lens and shift the trap in an axial direction. Thus unlike a mirror, using an SLM programmed with a suitable hologram gives 3D control of trap position. For multiple traps one could switch rapidly between holograms; however, two or more holograms can be combined into a single design. Combining holograms in this way requires pixel by pixel addition, where each pixel is treated as a complex number (magnitude corresponding to the trap strength and argument equal to the phase of the individual hologram pixel). The required phase of each pixel in the composite hologram is the argument of the complex sum of the individual holograms. Thus the design of the holograms required to produce arbitrary configurations of optical traps requires nothing more than the addition of complex arrays of number, a task ideally suited to the use of graphics card processors, which can be connected directly to the SLM. Even consumer-grade components can calculate and display the required holograms at video frame rates. The use of spatial light modulators for optical trapping is referred to as ‘holographic optical tweezers’ (Grier, 2003) (Fig. 3).

Figure 3.

(a) Photograph of a holographic optical tweezer user interface. Here, a green 532 nm tweezing laser is incident on a Hamamatsu SLM seen at the bottom of the image, before being directed through a Zeiss microscope. Also visible is a high speed camera attached on the microscope’s left. (b) An image taken by the high speed camera whilst the tweezers are in operation. Four 2 inline imagem Silica spheres are seen, three of which are trapped in a multispot array determined by the user. (c) A typical diffraction pattern displayed on the SLM, with the grey scale representing a 0 to 2inline image phase change. Fork dislocations can be seen; this beam will exhibit angular momentum. The greyed out areas indicate there are two or more beams being diffracted. Curved lines imply the beam will be defocused.

The ability to configure complicated arrangements of optical traps, or shape the light field to form optical 2D and 3D landscapes allows a similar manipulation of colloidal structures. These optical structures can act as weak or strong templates around which larger inert or biological structures might form. Exchanging the surrounding water for a gel allows such structures to be made permanent, perhaps interesting as a biological scaffold?

The algorithms for designing the holograms are not restricted to creating simple spots. In addition to linear momentum, light also carries angular momentum, suggesting that not only should we be able to optically lift an object but that we should be able to spin it, too. Light's angular momentum has both spin and orbital components. The spin angular momentum is now associated with the spin of individual photons and is macroscopically manifest as circular polarization. Light's orbital angular momentum has only been truly appreciated since the early 1990’s and is manifest in light with helical phase fronts (Allen et al., 1992). Whereas the spin angular momentum has a maximum value of inline image per photon, the orbital angular momentum can be much larger, depending upon the rotation symmetry of the helical phase fronts. Phase fronts shaped like a screw thread, a DNA double helix or a pasta fuselli carry inline image, inline image and inline image per photon, respectively. Helically phased beams are easy to make. If a fork dislocation is embedded into the lines of a diffraction grating, then the first-order diffracted beam has helical phase fronts and an associated angular momentum. Using one of these beams, not only traps an object, but in most cases causes the particles to spin, or for larger beams, rotate around the beam axis. These “optical spanners” (Padgett & Bowman, 2011) are a natural choice for the optical drive of micro-machines and/or pumps.

Once the particle is trapped, its dynamics are described by those of a thermally driven, over-damped, oscillator. The residual Brownian motion of the trapped particle is described by a Lorentzian power spectrum with a cut-off frequency in the region of 100 Hz. Above this frequency the particle is effectively free with a dynamic unperturbed by the trap itself. The range of particle position is described by a Gaussian distribution with a standard deviation of tens of nanometres. Although these displacements are well below the optical wavelength, they can be measured based upon the image centroids. Similarly, the precision to which the trap can be positioned is not limited by the wavelength of light, but is rather a function of the stability of the optical system. To improve the stability, optical tweezers can be built on an air damped optical table and with design to minimize resonant vibrations. Nonetheless, instabilities still arise from sources such as thermal drift and beam pointing stability.

In most work to date, the nanometric displacements of the trapped particles were monitored using a quadrant photodiode to measure displacement of the particle image or, more sensitively, displacement of the transmitted laser. Best performance is achieved if the particles are near perfect spheres. Consequently, such beads are often used as handles to more complex objects. Whereas the difference signal between the quadrants of the photo-diode gives the lateral displacement, the sum signal can be related to the axial motion giving 3D position sensing at the nanometre level (Florin et al., 1998).

The ability to measure nanometric displacements within a well-characterized system in which the restoring force is accurately known has also been widely used to measure the pico-Newton forces. Following the invention of optical tweezers it was immediately recognized that the pico-Newton forces within an optical tweezers made them ideal for studies within life sciences in general, and the study of molecular motors in particular.

Moving from measuring a single particle to measuring many particles with a quadrant diode requires sophisticated optical multiplexing. As an alternative it is possible to use high-speed cameras to measure image centroids for many particles. Until recently such high-speed imaging would have been extremely expensive. However, the latest generation of CMOS cameras and consumer level interfaces can run a frame rate of 500 Hz for mega-pixel, 8-bit images. Selecting a reduced region of interest within these images allows even faster frame rates. The important parameter has been a frame-rate much higher than the position correlation time of the trap.

Typically, a high-magnification microscope image is over-sampled with respect to the optical wavelength. A typical magnification results in an image scale of 10 nm per pixel, that for a micron-sized bead can be interpolated to yield a precision in the position measurement of order 1nm. Comparing these centroid measurements to those made with a quadrant photodiode shows that both approaches yield a noise floor only limited by the thermally induced motion of the particle itself (Gibson et al., 2008).

One obvious drawback of position measurement based on images is that the axial motion of the particle is not immediately apparent. To overcome this limitation, an alternative approach has been demonstrated. Using white light LEDs, pig-tailed to large core fibres, the sample can be illuminated from two different directions. This dual illumination creates two images. A bi-prism inserted into the Fourier plane of the sample introduces an angular separation between the images that translates into a lateral separation in the plane of the camera, where the images are obtained side-by-side. Effectively, this approach gives us right and left images taken using different sub-sections of the numerical aperture of the objective lens. The convergence angle between the light sources is roughly 80inline image, giving strongly stereoscopic images from which multiple particles can be fully tracked in 3D to nanometric precision (Bowman et al., 2011).

Optical tweezers are a natural playground for 3D imaging, where one obtains nanometric co-ordinates that change dynamically. Various approaches to 3D imaging have been adopted including confocal imaging but perhaps most recently it is lensless holographic imaging which perhaps has the potential to give the most general approach to 3D centroid tracking, albeit with a large computational loading (Cheong et al., 2010).

In addition to optical tweezers being used to measure nanometre displacements and pico-Newton forces, they also are ideal probes of the local environment of the bio or micro-fluidic system. Rather than measuring the static displacement of a particle from the trap centre, it is the dynamics of the particle motion which reveals the mobility of the particles and hence the viscosity of the surrounding fluid. However, on these micron-length scales, the mobility of the particle is modified by neighbouring boundaries that need to be accounted for if the true viscosity is to be revealed. An interesting alternative is to measure the rotational dynamics or rotational drag for which the boundary effects have a much shorter range (Bishop et al., 2004).

The use of multiple traps allows these point measurements of flow or viscosity to be made over a network of positions, but rather than treating each particle independently it is also interesting to monitor their interactions. The ability to both trap and image multiple particles has provided the means to study hydrodynamic coupling between networks of particles. A hydrodynamic coupling occurs when the motion of one particle creates a fluid flow that then exerts a drag force on the particles that surround it. The particle networks should now be analysed in terms of their modes of collective motion, in general the mobility of the symmetric modes where particles move in the same direction is higher than the anti-symmetric modes. Even on this micron scale the effects are subtle, easily masked by the Brownian motion. Studies of such networks can be extended, or more accurately reduced, to 2D systems where the fluid flow decays more slowly making the coupling much stronger (Padgett & Di Leonardo, 2011). The effects of this coupling are complicated but seem to provide a mechanism where even a thermally driven system may become partially synchronized.

Since their inception, optical tweezers have caught the imagination of physicists recognizing that the science fiction of tractor beams is becoming, albeit on a microscopic scale, a reality. Beyond their immediate appeal, optical tweezers have been an enabling tool for studies of molecular motors, intercellular forces, colloidal systems and optical driven micro-machines and sensors. The early work of Ashkin, however, not only gave rise to what we now call optical tweezers but also to the optical cooling of atoms and molecules. The interaction of atoms with light is typically enhanced by tuning the laser to be near the resonance of the atomic or molecular system. The absence of a surrounding fluid means that extra consideration needs to be applied to a mechanism to dissipate the energy and, consequently, most of these techniques are beyond the capabilities of the simple optical tweezer set-up. However, recent work suggests that optical tweezers may still have more to offer. In fact, providing that the trap control is sufficiently fast, then the residual motion of a trapped particle can be sensed and fed back to the control, enabling us to suppress the motion of the particle. In the future it is likely that such active or resonant systems might allow optical cooling of macroscopic objects.

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