Optics of Optical Tweezers

An optical tweezer traps particles with forces generated by optical intensity gradients. Dielectric particles polarized by the light's electric field are drawn up the gradients to the brightest point. Reflecting, absorbing and low-dielectric particles, by contrast, are driven by radiation pressure to the darkest point. Optically generated forces strong enough to form a three-dimensional trap can be obtained by bringing a laser beam with an appropriately shaped wavefront to a tight focus with a high numerical aperture lens. Microscope objective lenses offer an ideal combination of minimal aberration and large numerical aperture and often serve as the focusing element in practical implementations of optical tweezers [1] and variants such as the optical vortex [5,6].

Figure 1: Schematic representation of a typical holographic optical tweezer array. A collimated laser beam incident from the left is shaped by a diffractive optical element (DOE), transferred to an objective lens' back aperture (B) by lenses L1 and L2 and focused into a trapping array. OP$^\ast$ denotes the plane conjugate to the trapping plane. The point B$^\ast$ is conjugate to B. The phase pattern on the lower left (black regions shift the phase by $\pi$ radians) produced the traps shown in the lower right filled with 1 diameter silica spheres suspended in water.

The challenge in constructing an optical tweezer is to direct a laser beam into the objective lens' back aperture so that the beam fills the aperture and so that its axis coincides with the optical axis in the aperture's plane, at the point labelled B in Fig. 1. If the beam follows the optical axis, then it comes to a focus and forms a trap in the center of the lens' focal plane. If, on the other hand, it enters the back aperture at an angle, the resulting trap is offset from the center of the focal plane, as indicated schematically in Fig. 1.

Directing the beam into the objective with a dichroic mirror allows other wavelengths to pass through unimpeded and can be useful for imaging the trapped particles, as in Fig. 1. The problem remains, however, of aiming the beam.

The telescope formed by lenses L1 and L2 in Fig. 1 addresses this problem by creating a conjugate point, B$^\ast$, to the back aperture's center, B, at a convenient location. A beam of light passing through B$^\ast$ also passes through B and forms an optical trap. In our implementation, L1 and L2 are high quality plano-convex lenses with 250 mm focal lengths. Such long focal lengths help to minimize aberrations, particularly longitudinal spherical aberration, which would be detrimental to trapping [7,8,9]. More compact optical trains would require additional attention to minimizing wavefront distortions. References [10] and [11] offer more detailed discussions of this aspect of the optical design.

Multiple beams passing through B$^\ast$ all pass through B and thus all form optical tweezers. A diffractive optical element (DOE) at B$^\ast$, as shown in Fig. 1, can split a single collimated laser beam into any desired distribution of beams, each emanating from B$^\ast$ at a different angle, and thus each forming a separate trap [4]. Figure 1 shows the computer-generated pattern for a binary phase hologram together with a photomicrograph of colloidal particles trapped in the resulting array of optical tweezers. The remainder of this Article addresses the theory and practice of creating holograms such as the example in Fig. 1 suitable for projecting arbitrary arrangements of optical tweezers.