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We study the conditions under which a particle, laser-guided in a vertically-oriented hollow-core photonic crystal fiber filled with liquid, can be kept stationary against a microfluidic counter-flow. An immobility parameter—the fluid flow rate required to immobilize a particle against the radiation force produced by unit guided optical power—is introduced to quantify the conditions under which this occurs, including radiation, viscous and gravity forces. Measurements show that this parameter depends strongly on the ratio of particle radius a to core radius R, peaking at an intermediate value of a/R. The results follow fairly well the theoretical estimates of the optical (calculated approximately using a ray optics approach) and numerically simulated drag forces. We suggest that the system has potential applications in, e.g., measurement of the diameter, refractive index and density of particles, synthesis and biomedical research.
©2011 Optical Society of America
1. Introduction
Following pioneering work by Kawata et al. [1], optical propulsion of particles in liquid waveguides has recently been used for particle separation in multimode hollow-core photonic crystal fiber (PCF) [2] and transport of bio-molecules along on-chip waveguides [3]. In a previous paper we reported how glass microspheres can be laser-trapped and propelled inside the hollow-core of a fluid-filled single-mode photonic crystal fiber (PCF) [4,5]. Radiation pressure from guided laser light was sufficient to hold a single micron-sized particle stably against a fluidic counter-flow. By ramping the laser power up and down, the particle could be moved reproducibly to and fro over tens of cm. In another experiment the particle speed was monitored remotely using a Doppler-based measurement scheme [6]. In this paper we provide a detailed analysis of the forces acting on guided particles in PCF, present new experimental data for a large range of particle sizes and introduce an immobility parameter to quantify more precisely the conditions under which a particle is stationary.
2. Experiments
A scanning electron micrograph (SEM) of the hollow-core photonic crystal fiber (HC-PCF) used in the experiments is shown in Fig. 1(a) . The cladding pitch is Λ = 4.7 µm and the hydraulic radius of the core is R = 8.7 µm [7]. The fiber was designed to guide a single-mode at 1064 nm when all core and cladding holes are filled with D2O (refractive index nfl = 1.33) [8,9]. D2O is chosen for its low absorption (4 dB.m−1) at 1064 nm. Launch efficiencies as high as 89% were achieved using an objective lens (4 × 0.1 NA) that matched the numerical aperture (NA) of the guided mode. Figure 1(b) shows the near-field intensity profile at the output face of a 1 m long piece of D2O-filled fiber at 1064 nm. Its shape agrees well with a J0 2 function that goes to its first zero at the core boundaries, showing that only the fundamental mode is excited. Indeed, robust single-mode guidance was obtained over the whole wavelength range 970 to 1140 nm. The loss spectrum was determined using the cut-back technique (Fig. 1(c)). At 1064 nm the loss was 5 dB.m−1, dominated not by intrinsic waveguide loss but by absorption in the D2O (dashed curve). Bend loss in the 970-1090 nm range was measured to be less than 0.2 dB per turn around a mandrel of radius of 7 mm. These characteristics make HC-PCFs highly suitable as low loss, flexible and reconfigurable single-mode liquid waveguides.
Fig. 1 (a) SEM of the HC-PCF structure, inter-hole spacing Λ = 4.7μm and hydraulic radius R = 8.7 μm; (b) Mode intensity profile measured at the output of a D2O-filled fiber at 1064 nm. Cross-sections along central horizontal and vertical axes are shown above and to the right. The measured profiles (symbols) are fitted to a J0 2(j 01 r/R) shape expected for the fundamental mode, where j 01 is the first zero of the J0 Bessel function (red curves); (c) Loss spectra for bulk D2O, the liquid-filled PCF and the excess loss introduced by the fiber. The shaded region indicates where light is guided in a low-loss single-lobed fundamental mode, the loss being dominated by D2O absorption.
Borosilicate glass spheres (Duke Scientific 9002, 9005, 9010) with refractive index n p = 1.55 and nominal radii a = 1, 2.5 and 5 µm were used. Figure 2(a-b) shows the set-up for launching and optically guiding the particles into HC-PCF (a detailed description is available in [4,5]). Once inside the core, the particle was propelled to a position 60 cm from the core entrance – far enough away to eliminate the disturbing effects of high loss higher order modes. In contrast to our previous experiments, the orientation of the fiber at the observation position was kept vertical so as to avoid gravity-driven radial particle displacements. The position of the particle along the fiber was determined to within 10 µm by side-scattering. A typical image recorded by CCD3 is shown in Fig. 2(c). As soon as the particle reached the observation region, the fluid counter-flow was increased until the fluidic and optical forces were balanced and the particle remained stationary.
Fig. 2 Experimental set-up. (a) A CW laser beam (Nd:YAG, 1064 nm) is divided between an optical tweezers trap and a particle guidance beam. The split ratio is controlled with a λ/2 plate and a polarizing beam splitter (PBS). A 100 × 1.1 NA water immersion objective forms a conventional optical trap [10] that is used to tweezer a selected particle up to the entrance of the HC-PCF core. Two cameras (CCD1 and CCD2) monitor the launching process in three dimensions. (b) Close-up of the open sample cell arrangement for loading and launching particles (to scale). The output facet is enclosed in a custom designed liquid cell (LC). The mode intensity profile is monitored by CCD4. The pressure head can be varied by moving the D2O reservoir up and down. (c) Image of an optically guided particle taken by CCD3. The vertical stripes represent the liquid-filled cladding holes that run along the fiber. P: polarizer; EOM: electro-optic modulator; CM: cold mirror.
The pressure head p H necessary to keep the particle stationary in the vertical fiber section divided by the fiber length L is plotted in Fig. 3 as a function of guided optical power P opt, estimated by measuring the power at the fiber output and extrapolating back to the particle position using the measured value of fiber loss (Fig. 1(c)). The results were reproducible to within 2% over a time periods of an hour or more and were repeated for 14 different particle radii from 1 µm to 6.25 µm. In all cases the value of p H needed to achieve immobility increased approximately linearly with optical power.
Fig. 3 Pressure head p H divided by fiber length L required for particle immobility, plotted against optical power for 4 different particle radii.
3. Microfluidic forces
The Reynolds number Re = 2ρ fl v m R/η (η = dynamic viscosity, v m = fluid velocity in core center, ρ fl = fluid density) is 0.0015 for the typical parameters in our experiments (v m ~100 µm.s−1, ρ fl = 1106 kg.m−3 and η = 0.00125 Pa.s at 293 K) so that the flow is laminar. Using Hagen-Poiseuille theory for an incompressible fluid, the local pressure gradient dp/dz is given by 4ηvm/R 2. In the absence of any particle-related constriction to the fluid flow (see next section) dp/dz = p H/L where L is the fiber length. Defining v p as the particle velocity in the laboratory frame (small random fluctuations due to Brownian motion are neglected), the viscous drag force acting on a particle can be generally written [11] as
where ζ = a/R and K 1 and K 2 are wall correction factors that are accurately approximated (for ζ < 0.9) by the polynomials:
Note that the values of K 1 and K 2, and thus the drag force, increase rapidly as ζ → 1, when the particle almost completely fills the channel.
4. Relating pressure head to flow rate
The fluid flow rate will in principle be constrained by the presence of the particle, particularly when ζ → 1. To estimate the particle sizes and fiber lengths at which this becomes significant, we modeled the liquid flow field in the vicinity of the particle numerically using a finite element approach (Comsol Multiphysics). Figures 4(a-c) show the results for a liquid-filled tube with length L = 5R and a fixed pressure difference p H between both ends. Note that we have normalized v m to the new parameter v m0 – the center-tube flow speed in the absence of a particle. It may be seen that in this very short fiber the effect on the flow is quite substantial.
Fig. 4 (a-c): Flow velocity field around spherical particles of different sizes, normalized to the on-axis velocity in an unobstructed tube, for L = 5R (images clipped) at a fixed pressure difference. (d) Normalized on-axis flow velocity v m/v m0 in the obstructed core, at an axial distance 2.5R from the particle position, plotted versus L/R. The numerical calculations (solid lines) are compared to the predictions of the analytical model in Eq. (4) (symbols).
The total pressure difference between both fiber ends p H may be approximated as:
where dp/dz is the pressure gradient far from the particle and Δp p is the pressure drop across it, taken to be proportional to the drag force divided by the core cross-sectional area, i.e., Δp p = C.F drag/πR 2 where C is a correction factor (to be determined). Equation (3) can be used to show that:
Comparing Eq. (4) with the results of finite element simulations for different particle sizes yields a best fit for C = 1.7, as shown in Fig. 4(d). The results show that if L > 1000 × R the reduction in v m is less than 5% for ζ < 0.75. For the experiments reported here L ~105 × R so that the flow speed is practically unaffected by the presence of a particle. For short fiber lengths L < 100R, or for ζ ~1, an optically propelled particle could even be used to drive a small fluid flow.
5. Optical forces
The optical forces acting on a spherical particle can be estimated using the ray-optics approach developed by Ashkin [12], assuming that the optical mode can be represented by a bundle of rays travelling parallel to the fiber axis. The momentum transferred to the particle is obtained via ray tracing, the contributions of all rays being integrated, taking into account the J0 2(j01 r / R) mode intensity profile for the fundamental mode, where j01 is the first zero of a J0 Bessel function and r the radial position. The resulting axial (q z) and radial (q r) forces per Watt of optical power acting on borosilicate glass particles with a = 1 and 2 µm, plotted versus radial displacement from the axis, are shown in Fig. 5 .
Fig. 5 Calculated axial (q z) and radial (q r) forces per W of guided power on particles of 1 µm (dashed curves) and 2 µm radius (solid curves) as a function of radial position. R = 8.7 µm, n p = 1. 55, n fl = 1.33.
Propulsive force
When it is on-axis the larger particle experiences a propulsive force of 84 pN/W, some 3.7 times larger than for the smaller particle and roughly proportional to the particle area intercepting the optical mode. For comparison, the value for a perfectly absorbing sphere of the same size as the particle, taking account of the J0 2 intensity profile, is 841 pN/W. Thus, for the 2 µm radius sphere only 10% of the incoming momentum is transferred to the particle. The axial propulsive force on a centrally-placed particle is plotted in Fig. 6(a) versus radius a for different values of particle index n p. The maximum value occurs at a particle radius of ~5 µm, almost independently of the value of n p. In Fig. 6(b) the wall correction factor K 2 is plotted versus a/R; note how strongly it increases as a/R → 1.
Fig. 6 Calculated axial optical force parameter q z versus a/R. The particle refractive index n p ranges from 1.45 to 1.60 while keeping the fluid index constant (n fl = 1.33). The points indicate the different borosilicate glass spheres studied experimentally. (b) Wall correction factor K2 as function of a/R for a stationary particle. As expected the drag force increases dramatically when a/R → 1. (c) Calculated trap stiffness at 1 W optical power for particles guided in a D2O-filled fiber. (d) Lateral displacement of the particles in a horizontally placed fiber under the influence of gravity. The density ρ p of the particles was taken to be 2500 kg/m3.
Trap stiffness
The radial restoring force increases linearly with radius for small displacements from core center. The trap stiffness κ (pN.µm−1W−1) at core center is plotted in Fig. 6(c) against a/R for a range of different particle refractive indices, keeping the fluid index constant at 1.33. The maximum trap stiffness is achieved for a/R ~0.86, i.e., when the particle fills ~75% of the core area. In case of a horizontal fiber, as in our previous work [4,5], gravity pulls particles away from core center. Figure 6(d) plots the particle displacement at which the trapping force equals the gravitational force, i.e., m'g.κ−1 (units of μm.W) where m' is the effective mass including buoyancy. It is less than 0.05 µm.W (inversely proportional to power) for particle radii in the range 0 < a/R < 0.74. For example, for 50 mW of guided power the displacement is less than 1 µm, and is accompanied by a ~4% reduction in axial force (Fig. 5(a)).
6. Immobility parameter
The forces acting on a particle in a vertically-oriented fiber are shown in Fig. 7 . We define the immobility parameter μ i as the center-channel flow velocity far from the particle needed to immobilize it (in the laboratory frame) when it is propelled by unit guided optical power, i.e.,
Fig. 7 Schematic illustrating the forces acting on a particle inside a vertically oriented fluid-filled HC-PCF. The Poynting vector distribution of the guided optical mode (propagating upwards) and the velocity distribution of the fluid flow (moving downwards) are shown. The effective mass of the particle, including buoyancy, is m'.
For a particle to be held stationary against a fluid flow in a vertically oriented fiber, the optical force must equal the sum of the microfluidic drag force and gravity:
where ρ p is the particle density. Assuming that the particle does not significantly constrain the flow, i.e., v m = v m0, the immobility parameter can be written:
As expected, μ i is zero when the propulsive optical force exactly balances gravity, i.e., when P opt = 4πa 3(ρ p−ρ fl)g/3q z. Note that in the limit of large optical powers, gravity can be ignored, leading to the simpler expression:
In this limit, experimental (symbols) and theoretical values (curves) for μi ∞ are plotted in Fig. 8 against particle radius for a range of different values of n p. Experimentally, μi ∞ is obtained by extrapolating the curves in Fig. 3 to 1 W of optical power and calculating the flow speed v m from the pressure gradient p H/L. In the experiment μi ∞ increases with particle radius, reaching a maximum value of 1.7 mm.s−1.W−1 at a = 2.8 µm and then decreasing again for larger values. The calculated immobility for n p = 1.55 (the borosilicate glass index) underestimates the experimental data slightly for smaller particles (a < 2 μm), the error being a factor of two in the worst case for larger particles. This we attribute to underestimation of the optical forces by the simplistic ray model that does not take into account the waveguide properties. Preliminary results of a separate theoretical study (in preparation) show that a rigorous scattering approach gives better agreement [13]. Reasonable agreement is obtained, however, if the optical force calculated from the ray-optics model is multiplied by a fixed correction factor of 1.7 (see dashed curve for n p = 1.55).
Fig. 8 Immobility parameter at high optical power, μi ∞, versus particle radius; theory (solid curves) and experimental data (circles). Reasonable agreement is found if the theoretical curve for n p = 1.55 is multiplied by 1.7 (dashed curve).
Figure 9 shows µ i against optical power for 3 different particle radii. At higher optical powers gravity can be ignored and μ i asymptotically approaches μ i ∞. At optical powers small enough so that gravity forces dominate, μ i goes negative, i.e., the propulsive optical force is insufficient to balance against gravity and the flow has to be reversed to achieve particle immobility. The theoretical curves were calculated using Eq. (7), using the correction factor (1.7) for the optical force. There is reasonable agreement with the experimental data.
Fig. 9 Measured (symbols) and calculated (curves) immobility parameter versus (low) optical power for three particle radii. A correction factor of 1.7 was used for the optical force (see Fig. 8). The estimated experimental errors (represented by the error bars) Δp H/p H = ±% and ΔP opt/P opt = ±0.1. The shaded regions show the spread of calculated µ i values, assuming uncertainties of Δn p/n p = ± 0.01 and Δa = ±0.3 µm in particle parameters.
7. Discussion and conclusions
Micron-sized dielectric particles trapped in the liquid-filled hollow core of a single-mode photonic crystal fiber can be held stably against a fluidic counterflow using radiation pressure. The trap stiffness is large enough to keep the particles close to the center of the guided optical mode even in the presence of gravity, resulting in highly reproducible radiation forces. An immobility parameter μ i gives the flow rate needed to keep a particle stationary for unit optical power level.
By fitting the theory (including a more accurate model for the optical forces [13]) to precise experimental measurements, it should be possible to determine the diameter, refractive index and density of an unknown particle. For small (a << R) and large (a/R close to 1) particles the immobility parameter varies strongly with particle radius, allowing accurate size measurements. In the intermediate region it depends strongly on the particle refractive index, while it is less sensitive to variations in particle size. For borosilicate spheres (n p = 1.55) it is 2.5 times larger than that of a silica sphere (n p = 1.45) of the same size.
Our system has three major advantages of over existing optical chromatography methods [14–16]. Firstly, the microfluidic drag depends strongly on the exact particle radius, due to confinement effects. Secondly, the approximately circular symmetry of the core causes particles to self-align in the center of both the flow profile and the optical mode. Thirdly, the immobility measurement does not require a precise measurement of particle position.
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 could be delivered through the cladding [17]), 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 [4,5].
In biomedical research, the system could be used to study optically induced deformations of cells [18]. In particular for large a/R values, small deformations would cause large changes in the immobility parameter. In addition, small concentrations of biochemicals could be applied to a cell optically held against a fluidic counterflow. Small changes in cell volume and refractive index could be observed through changes in the immobility. This would allow the study of the effectiveness of chemical therapy at the single cell level.
A useful extension to the existing setup would be to launch two counter-propagating beams into the fiber. In such a system the radial optical force can be changed independently of the axial force, as in Ashkin’s earliest optical trapping work [19]. By adjusting the intensity ratio between the two beams, particles could be moved to and fro in a stationary liquid.
Acknowledgments
We are grateful to Silke Rammler, Michael Scharrer, Amir Abdolvand, Johannes Nold and Jocelyn Chen for help in designing and fabricating the fiber, and thank Sarah Unterkofler for experimental assistance. The work was partly funded by the Koerber Foundation.
References and links
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