Optical Tweezer Arrays

The optical forces generated by a milliwatt of visible light are more than enough to overwhelm the random thermal forces which drive the dynamics of microparticles. The goal in creating an optical tweezer is to direct the optical forces from a single laser beam to trap a particle in all three dimensions. While quite general formulations of this problem have been developed [23,24,25,26], a simplified discussion suffices to motivate the design of optical tweezer arrays. We will consider the forces exerted by monochromatic light of wavenumber k on a dielectric sphere of radius a in the Rayleigh limit, where $a \ll 2\pi/k$.The total optical force, $\vec{F}$, is the sum of two contributions [27]:  

$$\vec{F} = \vec{F}_{\nabla} + \vec{F}_{s},$$ (1)

the first of which arises from gradients in the light's intensity and the second of which is due to scattering of light by the particle. The gradient force on a particle of dielectric constant $\epsilon$ immersed in a medium of dielectric constant $\epsilon_{0}$ and subjected to an optical field with Poynting vector $\vec{S}$,

 

$$\vec{F}_{\nabla} = 2 \pi a^{3} \frac{\sqrt{\epsilon_{0}}}{c}... ...ilon+2\epsilon_{0}} \biggr) \vec{\nabla} \, \vert\vec{S}\vert,$$ (2)

tends to draw the particle toward the region of highest intensity. The scattering force,

 

$$\vec{F}_s = \frac{8}{3} \pi (ka)^{4} a^2 \frac{\sqrt{\epsilo... ...lon-\epsilon_{0}}{\epsilon+2\epsilon_{0}} \biggr)^{2} \vec{S},$$ (3)

drives the particle along the direction of propagation of the light.

An optical tweezer can be formed by focusing a laser beam to a diffraction limited spot with a high numerical aperture lens. The gradient force attracts the particle to the beam's focal point, while the scattering force drives the particle along the beam's axis. In order to form a full three dimensional trap, the axial intensity gradient must be large enough to overcome the scattering force.

In a typical experimental setup, a microscope objective lens focuses a Gaussian TEM₀₀ laser beam into a tweezer while simultaneously imaging the trapped particles. In order to maximize the axial intensity gradient, the incident beam should be expanded to fill the objective's back aperture. An optical tweezer can be translated across the microscope's focal plane by adjusting the beam's angle of incidence at the back aperture and can be displaced along the optical axis by changing the curvature of its wavefront.

Similarly, if several collimated beams pass through the back aperture at different angles, they form separate tweezers at different locations in the focal plane. For instance, dual optical tweezer systems have been created by splitting and recombining a single laser beam with beamsplitters and refractive optics[21,28]. This approach, however, becomes cumbersome for more than a few traps. An elegant alternative for creating two-dimensional arrays of traps involves scanning a single tweezer rapidly among a number of positions to create a time-averaged extended trapping pattern [21,29,30,31]. This approach has proved highly effective for shaping small colloidal assemblies.

Yet another alternative for creating two dimensional arrays of traps is to split and steer the light from a single beam with diffractive optical elements. As demonstrated in the next section, inexpensive commercially available diffractive pattern generators (MEMS Optical Inc., Huntsville AL) are ideally suited to this task. In addition, diffractive optics can be used to change the curvature of beam wavefronts, thereby facilitating the creation of three-dimensional arrays of traps. Additionally, diffractive optics can be used to modify beam profiles. For instance, computed holograms can convert a TEM₀₀ Gaussian beam into multiple Gauss-LaGuerre beams, otherwise known as optical vortices [32,33,34]. Tightly focused optical vortices can be used to trap low dielectric constant and reflective microparticles. Finally, the use of addressable liquid crystal phase shifting arrays allows for the dynamic reconfiguration of a tweezer array for active particle manipulation and assembly. Thus, diffractively generated optical fields can be used to configure arbitrary numbers of microscopic particles into useful and interesting arrangements. We will discuss some applications of these techniques after describing a practical demonstration of diffractively generated optical tweezer arrays.