One-dimensional optical thermal ratchets

Sang-Hyuk Lee and David G. Grier
Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003

Date: August 16, 2005

Abstract:

The ability to rectify Brownian forces with spatially extended time-varying light fields creates new opportunities for studying the statistical properties of thermal ratchet models and to exploit these models' interesting and useful properties for practical applications. This article describes experimental studies of one-dimensional thermal ratchets implemented with the holographic optical trapping technique applied to fluid-borne colloidal spheres. These studies demonstrate the complementary roles of global spatiotemporal symmetry and local dynamics in establishing the direction of ratchet-induced motion and also highlight avenues for future advances in higher-dimensional systems.

Thermal ratchets employ time-varying potential energy landscapes to break the spatiotemporal symmetry of thermally equilibrated systems (1). The resulting departure from equilibrium takes the form of a directed flux of energy or materials, which can be harnessed for natural and practical applications. Unlike conventional macroscopic machines whose efficiency is reduced by random fluctuations, thermal ratchets actually require noise to operate. They achieve their peak efficiency when their spatial and temporal evolution is appropriately matched to the scale of fluctuations in the heat bath.

Most thermal ratchet models involve locally asymmetric space-filling potential energy landscapes, and almost all are designed to operate in one dimension. Most practical implementations have exploited microfabricated structures such as interdigitated electrode arrays (2,3), quantum dot arrays (4), periodic surface textures (5,6), or microfabricated pores for hydrodynamic drift ratchets (7,8). Previous optical implementations have used a rapidly scanned optical tweezer to create an asymmetric one-dimensional potential energy landscape in a time-averaged sense (9,10), or a time-varying dual-well potential with two conventional optical traps (11,13,12).

This article describe a broad class of optical thermal ratchets that exploit the holographic optical tweezer technique (14,17,18,19,20,16,15) to create large-scale dynamic potential energy landscapes. This approach permits detailed studies of the interplay of global spatiotemporal symmetry and local dynamics in establishing both the magnitude and direction of ratchet-induced fluxes. It also provides a basis for possible practical applications.

Holographic optical tweezers use computer-generated holograms to project large arrays of single-beam optical traps. Our implementation (15), shown schematically in Fig. 1, uses a liquid crystal spatial light modulator (SLM) (Hamamatsu X7550 PAL-SLM) to imprint phase-only holograms on the wavefronts of a laser beam from a frequency-doubled diode-pumped solid state laser operating at 532 nm (Coherent Verdi). This SLM can vary the local phase, $\varphi(\vec{r})$, between 0 and $2 \pi~\unit{radians}$ at each position $\vec{r}$ in a $480 \times 480$ grid spanning the beam's wavefront. The modulated beam is relayed to the input pupil of a $100 \times$ NA 1.4 SPlan Apo oil immersion objective lens mounted in an inverted optical microscope (Zeiss S-100TV). The objective focuses the light into a pattern of optical traps that can be updated in real time by transmitting a new phase pattern to the SLM.

Figure 1: Schematic representation of the holographic thermal ratchet implementation.

The left-most photograph in Fig. 1 shows the focused light, $I(\vec{r})$, from a typical pattern of holographic optical traps, which is imaged by placing a front-surface mirror on the sample stage and collecting the reflected light with the objective lens. Each focused spot of light in this $20 \times 5$ array constitutes a discrete optical tweezer (21), which acts as a spatially symmetric three-dimensional potential energy well for a micrometer-scale object. The central image in Fig. 1 shows an aqueous dispersion of 1.53  $\unit{\mu m}$ diameter colloidal silica spheres (Bangs Laboratories, lot number 5328) interacting with this pattern of traps at a projected laser power of 2.5 mW/trap.

Each potential well may be described as a rotationally symmetric Gaussian potential well (22). Arranging the traps in closely spaced manifolds separated by a distance $L$ creates a pseudo-one-dimensional potential energy landscape, $V(x)$, which can be modeled as

$$V(x) = - V_0 \, \sum_{n = -N}^N \exp\left( - \frac{(x-nL)^2}{2\sigma^2} \right).$$ (1)

The well depth, $V_0$, approaches the thermal energy scale, $\beta^{-1}$, when each optical tweezer is powered with somewhat less than 1 mW of light. The holographically projected traps' strengths are uniform to within ten percent (15). Their widths, $\sigma$ are comparable to the spheres' radii (22,15). With the traps powered by 3 mW, diffusing particles are rapidly localized by the first optical tweezer they encounter, as can be seen from the center photograph in Fig. 1.

The potential energy landscape created by a holographic optical tweezer array differs from most ratchet potentials in two principal respects. In the first place, the empty spaces between manifolds comprise large force-free regions. This contrasts with most models, which employ space-filling landscapes. The landscape can induce motion only if random thermal fluctuations enable particles to diffuse across force-free regions. Secondly, the landscape is spatially symmetric, both globally and locally. Breaking spatiotemporal symmetry to induce a flux rests, therefore, with the landscape's time evolution. Details of the protocol determine the nature of the induced motion.