The synthesis, measurement, and manipulation of micrometer-sized objects are of great importance in many fields, with examples being catalysis, cell biology, quantum dots, colloidal chemistry, and paint design. Optical trapping combined with microfluidics [1, 2, 3] has been used, e.g., to size sort dielectric particles [4]. The evanescent edging field of a single guided optical mode has been used to propel particles over short distances $\left(\sim 0.1\text{\hspace{0.17em} mm}\right)$ on planar wave guides [5, 6] and in Si slot waveguides [7]. This approach has the disadvantage that the transverse optical field decays exponentially from the surface, making stable optical trapping difficult. Furthermore, the particles are guided very close to the waveguide surface, resulting in asymmetric drag forces. Guided mode radiation pressure has also been used to propel particles along hollow-core photonic crystal fibers (HC-PCFs) [8, 9] and along HC-PCFs in which only the hollow core is filled with liquid [10]. This last example results in highly multimode guidance by total internal reflection, yielding transverse intensity patterns that are difficult to control and in general axially varying.

Here we report the trapping and propulsion of single micrometer-sized particles in a single-mode photonic bandgap HC-PCF in which all the channels (hollow core and cladding holes) are filled with ${\text{D}}_2\text{O}$. Since the refractive index of ${\text{D}}_2\text{O}$ is less than that of silica glass, guidance is only possible by the photonic bandgap effect. This system allows single particles to be trapped transversely by optical forces close to the center of the guided mode and to be held stationary against a fluidic counterflow by radiation pressure. The precise measurement of the viscous forces acting on a single particle in a narrow microfluidic channel becomes possible. The low optical attenuation, in combination with the absence of diffraction, means that particles can be moved to and fro over extended distances (tens of centimeters) by ramping the laser power up and down. The mechanical flexibility of optical fiber, together with low bend loss, allows guidance of particles along reconfigurable curved paths, which is impossible using free-space beams.

The HC-PCF used had a core diameter of $17\mu \text{m}$ (see Fig. 1 ) and was designed, following known scaling laws [11, 12], for single-mode guidance at 1064 nm when filled with ${\text{D}}_2\text{O}$. The low absorption of ${\text{D}}_2\text{O}$ $\left(0.04\text{\hspace{0.17em} dB}{\text{cm}}^{-1}\right)$ at the trapping wavelength of 1064 nm minimizes the effects of laser heating. Launch efficiencies of $\sim 89\%$ into the fundamental core mode [see Fig. 1c] were achieved using an objective lens ($4\times$, 0.1 NA) whose NA matches that of the liquid-filled fiber. Robust single-mode guidance was obtained over the wavelength range from 790 to 1140 nm. Using a cut-back technique, the loss was measured to be $0.05\text{\hspace{0.17em} dB}{\text{cm}}^{-1}$ at 1064 nm—only slightly higher than the absorption of ${\text{D}}_2\text{O}$. The 11 cm length of liquid-filled fiber was placed horizontally on a glass plate, with its input face oriented parallel to a vertical glass window ($100\mu \text{m}$ thick) and immersed in a ${\text{D}}_2\text{O}$ droplet [Fig. 1d]. The end face of the fiber was enclosed in a pressure cell and imaged through an optical window using camera CCD4 [Fig. 1e]. The light from a continuous wave Nd:YAG laser (1064 nm) was divided at a beam splitter into guidance and loading beams [Fig. 1d]. The loading beam was focused by a long working-distance ($100\times$, 1.1 NA) water immersion objective, forming a conventional single-beam optical tweezers trap [13]. Cameras CCD1 and CCD2, monitoring the input face from orthogonal directions, allowed three-dimensional control of the particle position.

A small amount of dilute silica sol was added to the ${\text{D}}_2\text{O}$ droplet at the fiber input face. A single particle was selected from those in the droplet, trapped by the loading beam, and moved to the entrance of the fiber core [see Figs. 2a, 2b, 2c ]. The loading beam was blocked, and the horizontal guidance beam was used to push it into the core. The image from CCD2 in Fig. 2d shows the particle trapped just outside the core entrance by a combination of fluid counterflow and radiation force. Upon increasing the optical power, the particle is pushed into the core, after which the transmitted power drops by $\sim 40\%$. Once securely trapped inside the fiber, the particle could be moved to and fro by adjusting the laser power and the fluidic counterflow.

The laminar flow in the core is well described by the Hagen–Poiseuille theory for an incompressible fluid. However, the viscous drag force on a particle being pushed through a constrained counterflow is complicated to calculate, requiring numerical methods [14]. Two limiting regimes can be identified. The first arises when the flow is zero and the particle proceeds at a constant speed under the action of the optical force, and the second arises when the particle is held stationary against the flow by the optical force. In the general case, the net drag force is the sum of these two and can be written as $F_{net}=-6\pi \eta a\left(V_p{K}_1\left(\zeta \right)-V_{\text{max}}{K}_2\left(\zeta \right)\right)$, where $V_{\text{max}}=\left(R^2/4\eta \right)\text{d}p/\text{d}z$ is the fluid velocity in the center of the core, $R=8.5\mu \text{m}$ is the core radius, $V_p$ is the particle velocity, $\eta =0.00125\text{\hspace{0.17em} N s}{\text{m}}^{-2}$ is the viscosity, and p is the pressure at point z along the fiber. The numerically evaluated correction factors $K_1$ and $K_2$ were obtained from Al Quddus et al. [14].

The optical forces are more difficult to estimate in a waveguide geometry, where it is not clear whether the assumptions of the standard ray-optics approach [15] are valid. Nevertheless, to provide a basis for comparison, we carried out an analysis where the light guided in the core is represented by a bundle of rays traveling parallel to the axis, with intensities following the $J_0^2\left(j_{01}r/R\right)$ shape expected for the fundamental mode ($j_{01}$ is the first zero of the $J_0$ Bessel function). The momentum transferred to the particle is calculated for each ray and then integrated over all rays [15]. The result is that the propulsive force on a particle with radius a (in micrometers) sitting in the center of the guided mode (core radius of $8.5\mu \text{m}$) can be approximated by a polynomial: $F_p=0.214-1.65a+13.4a^2-1.64a^3\text{\hspace{0.17em} pN}{\text{W}}^{-1}$. The refractive indices of silica and ${\text{D}}_2\text{O}$ were taken to be 1.45 and 1.33.

Measurements were made for two different drag regimes. In the first, the particle velocity in the absence of any flow was measured via side scattering using camera CCD3. A sequence of typical photographs, taken at 1 s intervals, is shown in Figs. 2e, 2f, 2g, 2h. The velocities are plotted against the optical power in Fig. 3a , showing that the power-dependent optical transport velocity $\text{d}{V}_p/\text{d}{P}_{\text{opt}}$ lies in the range of $0.5–1\text{\hspace{0.17em} mm}{\text{s}}^{-1}{\text{W}}^{-1}$ for particle radii between 1 and $3\mu \text{m}$. In the second experiment, the reservoir was positioned so as to create a continuous flow against the direction of the light, and the optical power was adjusted so that the particle remained stationary in the laboratory frame. This was repeated for a range of different pressure gradients and for two different sphere sizes. The results show a linear relationship (with slopes ${\text{d}}^2{P}_H/\text{d}z\text{d}{P}_{\text{opt}}$ from 0.7 to $1\text{\hspace{0.17em} kPa}{\text{cm}}^{-1}{\text{W}}^{-1}$, depending on the particle size) between the pressure gradient and the optical power needed to keep the particle stationary [Fig. 3b].

Comparisons with the predictions of theory are shown in Table 1 . At zero flow, theory consistently overestimates, by a factor of $\sim 3$ for the larger particles and $\sim 1.8$ for the smaller ones, the power needed to reach a given particle velocity. The disagreement is slightly larger for the pressure gradient required to make the particles stationary; in this case theory overestimates the power required by factors of $\sim 2.5$ and $\sim 4$ for small and large particles. We suggest that this disagreement may be due to the waveguide geometry, which restricts the free propagation of rays escaping from the particle. A full explanation of this must, however, await the results of an ongoing analysis of the complex scattering behavior of a particle in HC-PCF.

The system offers fresh possibilities for studying the forces acting on particles in microfluidic channels. For example, if a trapped particle is pushed sideways using a laterally focused laser beam (which can be delivered through the cladding [16]), the imbalance of viscous drag on opposite sides will cause it to spin, enhancing chemical reactions at the particle surface. Such spinning has already been observed while the particle is being launched into the fiber [see Fig. 2d]. By loading the flowing liquid with chemicals in sequence (and perhaps activating them photolytically by side illumination), an optically trapped particle could be coated with multiple layers of different materials in a highly controlled manner, with the reaction being monitored using in- or through-fiber spectroscopy. In biomedical research, minute amounts of drugs (perhaps photoactivated) could be applied to a cell optically held against a counterflow. This would allow studying the effectiveness of chemical therapy at the single cell level. Since the refractive index of cancer cells is higher than that of healthy ones (1.37), it may even be possible to distinguish them by their larger velocities under optical propulsion through the fluid. Finally, the system could be used as a flexible optofluidic interconnect for transporting particles or cells between microfluidic circuits; we have readily achieved optical transport of microspheres over distances of up to 40 cm.

We gratefully thank Michael Scharrer, Amir Abdolvand, Johannes Nold, and Silke Rammler for help in designing and fabricating the fiber. The work was partly funded by the Koerber Foundation.

Table 1. Comparison of Theory and Experiment^a

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Fig. 1 (a) Scanning electron micrograph of the HC-PCF cross section with interhole spacing of $\Lambda =4.7\pm 0.1\mu \text{m}$ and core diameter of $17.1\pm 0.3\mu \text{m}$; (b) detailed view of the core structure; (c) mode profile measured at the output of an 11 cm ${\text{D}}_2\text{O}$-filled fiber at 1064 nm [same scale as (b)]. (d) Arrangement for loading and launching particles into the fiber and monitoring them while inside; the fiber end face is immersed in a drop of liquid at the vertex of two glass slides. (e) Pressure cell at output end of fiber.

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Fig. 2 Loading, launching, and guidance of a particle (diameter of $6\mu \text{m}$). (a)–(c) Tweezering a particle up to the entrance to the core; (d) bottom view (CCD2) of the particle held at the entrance to the core by optical forces balanced against the counterflow of liquid from the core. The particle is optically trapped slightly to one side of the core center. In this position the particle could be seen to revolve under the action of imbalanced viscous forces; (e)–(h) side-scattering patterns imaged through the cladding of the fiber, photographed at 1 s intervals.

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Fig. 3 Experimental data for the limiting cases of stationary fluid and stationary particle. (a) Particle velocity $V_p$ versus launched optical power $P_{\text{opt}}$ for three particle sizes (zero liquid flow). The relationship is approximately linear. At low powers the transverse trapping strength is weak, causing gravity to pull the particle closer to the wall, away from the center of the optical mode and thus lowering $V_p$. (b) Optical power needed to hold a silica sphere (for five different radii) stationary against the fluid flow driven by the pressure gradient $\text{d}{P}_H/\text{d}z$. The right-hand axis shows the velocity $V_{\text{max}}$ in the center of the flow. Once again the relationship is linear. The initial switch-on power, upon which the particle is lifted into the liquid by the light, could be used to study adhesion forces between particle and core wall.

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