1. Introduction

Accurate positioning and transport of nano- and micro-scale objects by optical forces is of importance in many fields [1], examples being particle sorting [2], single cell analysis [3] and DNA transport [4]. In single beam optical tweezers, a dielectric particle is trapped near the focus of a tightly focused laser beam by radiation pressure [5]. The axial positioning range of single beam traps can be extended to a few times the Rayleigh length by use of two co-aligned interfering Bessel beams [6,7]. By scanning the trapping beam the particle can be positioned transversely within the field of view of the objective (typically tens of microns). While the translation range can be extended in all directions in the focal plane by moving the optical trapping system [8], all-optical methods would be preferred.

A much greater Rayleigh length, and thus axial trapping range, is achievable in dual-beam traps consisting of weakly focused, counter-propagating, incoherent laser beams [9]. However, a long Rayleigh length inherently increases the beam waist, resulting in strongly reduced transverse trapping stiffness (at fixed optical power). Much larger forces can be optically induced by making use of photophoresis, which although limited to absorbing particles can extend to meters the range over which such particles can be optically propelled [10]. In dual-beam traps, the axial trapping position is controlled by adjusting the relative power in the counter-propagating beams. Since, however, the axial trap stiffness decreases with increasing Rayleigh length, it is difficult to achieve accurate axial positioning in long-range trapping configurations.

Much higher axial positional accuracy can be attained by interfering two coherent counter-propagating beams so as to create a standing wave pattern along the beam axis. The result is a series of optical trapping sites spaced by half the wavelength. These traps can be moved in either direction by changing the phase difference between the two beams. Such ‘standing wave optical conveyor belts’ have been used, for instance, to accurately transport atoms [11] and nanoparticles [12]. A limitation of such counter-propagating standing wave conveyor belts is that the axial stiffness critically depends on particle size [12], making it difficult to reliably transport particles much larger than the fringe spacing. Standing wave optical conveyor concepts have recently been combined with waveguides to controllably transport atoms in the evanescent field of a tapered fiber [13].

Whereas free-space optical manipulation systems and substrate-based hollow waveguides [14–16] are typically limited to predefined, straight trajectories, hollow fiber capillaries permit propulsion of particles and atoms along a flexible path [17,18]. Since the loss in a capillary scales as the cube of the bore diameter, however, and small bore diameters are desirable for high trapping stiffness, the transport length is strongly limited by waveguide losses. Hollow-core photonic crystal fiber (HC-PCF) side-steps this limitation by providing low-loss light guidance in hollow cores with diameters in the 10 µm range [19]. These unique properties have recently allowed propulsion of microparticles along flexible paths of several meters in air-filled HC-PCF [20–22].

Axial traps can be created in HC-PCF using counter-propagating modes. A simple analysis indicates that the trapping position is given by [16]

Here we show how these limitations can be overcome, and an effective optical conveyor belt realized, by making use of the intermodal beat-pattern between a pair of co-propagating modes in HC-PCF. The coherent superposition of the two lowest order forward-propagating modes is combined with a counter-propagating fundamental mode. This creates a series of trapping positions along the fiber spaced by half the intermodal beat-length, which can be up to three orders of magnitude longer than the wavelength, allowing high axial trap stiffness for a wide range of particle sizes. The trap positions can be moved along the fiber simply by changing the phase between the two forward-propagating modes, allowing accurate positioning and long-range transport of microparticles.

2. Intermodal beating in fibers

In a first experiment, shown in Fig. 1(a), a silica sphere with a diameter of 6 µm was trapped and launched into the 12 µm diameter core of an air-filled HC-PCF using two counter-propagating LP₀₁ modes [22]. Once the backward propagating beam is blocked, the particle is propelled through the 70 cm long fiber, its speed being monitored via in-fiber Doppler velocimetry [23]. The measured velocity trace in Fig. 1(a) shows a periodic speed variation with a period of 120 µm. Similar fluctuations have been observed in an optofluidic waveguide [24], and are caused by unintended excitation of a two-lobed LP₁₁ mode. Since the LP₀₁ and LP₁₁ modes have different phase velocities, the intensity distribution across the core varies periodically along the fiber, modulating the particle velocity. In Fig. 1(a) 90% of the power is carried by the LP₀₁ and 10% by the LP₁₁ mode (see Fig. 1(b)). By approximating the HC-PCF as a hollow cylindrical core surrounded by a dielectric medium [25,26], the intermodal beat-length can be shown to take the form:

 

Fig. 1 Intermodal beating in HC-PCF. (a) Experimental speed variations of 6 µm diameter particle propelled along a HC-PCF (core diameter 12 µm). (b) Calculated intensity patterns of the fundamental and higher order modes. (c) Intensity distribution in the yz-plane of a 90% LP₀₁, 10% LP₁₁ mode mixture. The dashed line indicates the computed particle trajectory.

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The calculated intensity distribution along one beat-period in the yz-plane is shown in Fig. 1(c). The particle follows a zigzag trajectory indicated by the dotted line. The field intensity on the particle, and thus the axial component of the optical force, peaks at the off-center positions, resulting in a speed variation with a period of $L_{\text{B}}/2=150\text{µm}$, in reasonable agreement with the experimentally measured period of 120 µm.

3. Optical trapping

In order to halt the particle, a counter-propagating LP₀₁ mode of the same optical power is added (see Fig. 2(a)). While a narrowband fiber laser generated all the fiber modes, we have not observed any polarization-dependent standing wave effect [27], most likely since we used microparticles much larger than the fringe spacing given by half the wavelength. The counter-propagating modes thus create a series of stable trapping positions spaced by half the intermodal beat-length. Changing the relative phase Δφ between the two forward-propagating modes shifts the intermodal beating pattern, and thus the trap positions, along the fiber axis (Fig. 2(b)). A phase shift of $\Delta \phi =2\pi$ corresponds to one intermodal beat-length L_B. In contrast to standing wave dual-beam traps, which have beat-lengths of λ/2 and provide a displacement of λ for $\Delta \phi =2\pi$, this mode-based trap increases the axial movement (for a given phase shift) by a factor of $L_{\text{B}}/\lambda$, which works out at more than 200 in the current experiment. Not being limited to nanoparticles, this intermodal conveyor belt can be used to trap microparticles, which would be advantageous, e.g., for manipulating cells. For instance, red blood cells (< 10 µm) fit inside the hollow core, where they undergo deformation driven not only by optical forces but also by enhanced drag forces in the confined space of the hollow fiber core [28]. Moreover, fibers with larger cores are also available that provide few-mode guidance suitable for manipulating larger biological objects.

 

Fig. 2 Principle of mode-based conveyor belt. (a) Schematic of beam configuration. (b) The intermodal beat-pattern creates a series of trapping sites along the fiber, spaced by half the beat-length of the two forward-propagating modes. Shifting the relative phase between the modes results in a moving intensity pattern that carries the microparticle along the HC-PCF.

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The optical forces within the trap were estimated using a ray optics model [29]. Computed results are shown in Fig. 3 for an LP₁₁ mode content of 10% and 6 µm particle diameter. The density plot of the transverse optical force in Fig. 3(a) depicts the gradient force, the arrows showing the force direction. A silica microparticle ($n=1.45$) is pulled to positions of zero transverse force (the dark dashed line) and laterally trapped. The axial optical force in Fig. 3(b), also known as the scattering force, can push microparticles either forwards or backwards. The presence of the counter-propagating mode causes the scattering forces to cancel at certain positions, indicated by the solid lines. Intersections between the solid and dashed lines indicate possible trapping positions of the particle – note that they are slightly off-axis. The directions of the optical forces for small displacements determine whether these traps are stable or unstable.

 

Fig. 3 Optical force landscape created by the mode-based conveyor belt calculated using a ray-optics approach. (a) Transverse component of the optical force acting on a 6 µm microparticle; the arrows indicate the force direction. (b) Axial component of the optical force, the encircled intersections between the two zero-force lines indicating possible trapping positions. (c) Calculated transverse displacement of microparticle as a function of LP₁₁ content. (d) Effect of LP₁₁ content on axial stiffness of dual-beam trap. All data are normalized to 1 W of total (forward plus backward) optical power.

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In Fig. 3(c) the transverse displacement increases with LP₁₁ content until the particle is no longer axially trapped. The maximum radial displacement is ~2 µm, and depends only slightly on the particle diameter. If a trapped microparticle touches the silica wall of the air-filled fiber core, van der Waals interactions cause it to strongly adhere to the glass surface. We note that the van der Waals interaction is much weaker in liquid-filled PCFs, allowing particles to be optically lifted from the core interface [23]. In Fig. 3(d) the axial trap stiffness (restoring force per axial displacement from trapping site) depends both on LP₁₁ content and particle diameter. For increasing LP₁₁ content the axial stiffness first reaches a maximum and then returns to zero (no axial trapping) whereas the transverse stiffness (not shown) is one to two orders of magnitude greater than the axial stiffness and varies only weakly with LP₁₁ content. These results show that both the radial displacement and the axial trap stiffness can be tuned by changing the LP₁₁ content of the mode mixture.

4. Mode-based conveyor belt

To realize a mode-based optical conveyor belt, mode mixtures were excited in the HC-PCF using a spatial light modulator (SLM) beam-shaping technique [30] combined with an optimization algorithm for the optical field distribution [31]. The wavefront of a linearly polarized LP₀₁ mode was altered using a phase-only liquid-crystal-on-silicon SLM (Hamamatsu X10468-03), see Fig. 4(a). The first diffraction order was spatially filtered at a pinhole and coupled into the HC-PCF. A quadrant photodiode (QPD) monitored the near-field pattern of the counter-propagating LP₀₁ mode emerging from the fiber, camera 1 (Cam1) imaged the light side-scattered by the particle and camera 2 (Cam2) recorded the near-field intensity distribution of the mode mixture at the fiber endface. A scanning electron micrograph of the HC-PCF structure is shown in Fig. 4(b) and the loss spectrum of the LP₀₁ mode is given in Fig. 4(c).

 

Fig. 4 Optical set-up for optimized excitation of coherent superpositions of fiber modes. (a) Set-up for launching a superposition of forward-propagating LP₁₁ and LP₀₁ modes together with a counter-propagating LP₀₁ mode; SLM: spatial light modulator, QPD: quadrant photodiode. (b) Scanning electron micrograph of HC-PCF structure with a core diameter of 12 µm (the inset shows a schematic of a 6 µm diameter particle). (c) Cut-back measurement of LP₀₁ fiber loss. (d) SLM phase pattern and corresponding intensity pattern at fiber endface after optimization.

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4.1 Fiber mode optimization

Before trapping a microparticle the excitation of the LP₀₁ mode was optimized using a computer algorithm to control the phase pattern on the SLM in response to the near-field intensity pattern at the fiber endface (Cam2). A global optimization routine was combined with a local minimization algorithm for wavefront correction using the first six Zernike polynomials. The global optimization was based on differential evolution [32], in which the genetic algorithm proceeds through several generations of populations, only the fittest “individuals” (vectors containing the complex-valued Zernike coefficients) being allowed to survive in each generation. Starting from randomly generated individuals in the first generation, each individual was stochastically mutated and recombined with other individuals. The resulting individual was compared to the initial one, and the fitter one selected for the next generation. This stochastic process ensured an efficient search while avoiding convergence to local maxima. After a fixed number of generations, the algorithm stops, the fittest individual in the population being the best-found solution. This solution was then locally optimized using the Nelder-Mead simplex algorithm [33].

As fitness function for the LP₀₁ mode we used the power-weighted overlap integral between the actual and the ideal normalized intensity: $F_{0\text{1} }= P_{\text{Cam2}} {\displaystyle \int I_{\text{Cam2}} I_{0\text{1} }\text{d}A}$. In contrast, the fitness function used for the LP₁₁ mode was based on the negative logarithm of the least-squared deviation between the normalized intensities, i.e., $F_{11}=-\text{log}{\displaystyle \int {\left(I_{\text{Cam2}} – I_{\text{11} }\right)}^{\text{2}}\text{d}A}$. To optimize excitation of the LP₁₁ mode, the SLM was initially set up to mimic a half-wave plate covering one half of the beam cross-section. The position and orientation of this virtual phase plate was first globally and then locally optimized via the algorithms described above.

Figure 4(d) shows the resulting SLM patterns and corresponding mode profiles at the fiber endface. The sawtooth-shaped grating structure in the SLM patterns allows efficient generation of modes in the first diffraction order, while avoiding interference with the beam reflected at the front surface of the SLM. To generate the LP₁₁ mode, a phase shift of π was applied to one half of the SLM pattern. In addition, the contrast of the grating structure was varied spatially, allowing the phase-only SLM to be conveniently used to amplitude-shape the incoming beam [34,35]. For mixtures of LP₀₁ and LP₁₁ modes the computed phase distribution of the modal superposition was used instead of the π-phase step, and the amplitude shaping was modified in response to the mode mixture intensity distribution [36]. This enabled the generation of mode mixtures of arbitrary LP₁₁ content and relative phase shift Δφ. In addition the grating contrast was adjusted so as to keep the total power of the mode mixture (which varies with Δφ) constant. Figures 5(a) and 5(b) show how the endface intensity distributions vary with Δφ for a mode mixture with 10% LP₁₁ content. A comparison with Fig. 1(c) shows reasonable agreement.

 

Fig. 5 Experimental demonstration of mode-based conveyor belt. (a,b) Intensity profile of the beat-pattern of a mode mixture (10% LP₁₁) emerging from PCF, measured by increasing Δφ (Media 1). (c,d) Sequence of optical images of light side-scattered from the microparticle while Δφ is ramped up and down (Media 2). (e,f) QPD signals while increasing and decreasing Δφ.

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Once the optimization was completed, a silica microsphere with a diameter of 6 µm was trapped 30 cm away from the fiber end in the hollow fiber core using a total power of 50 mW. The transverse trap stiffness was 5 pN/µm, whereas the axial trap stiffness was 30 fN/µm. The trapping potential provides a barrier of 50 aJ, which is four orders of magnitude larger than the thermal energy of the particle, enabling stable trapping over hours. The position of the trapped particle could be controlled by slowly increasing or decreasing Δφ (see Figs. 5(c) and 5(d)). Ramping the phase from 0 to 12π over 7 seconds caused the particle to move by 1.4 mm. The beat-length can be increased if a larger core radius is used; for example, Eq. (2) predicts $L_{\text{B}}=1 \text{mm}$ for $a=11\text{µm}$. Since the SLM device provides a linear phase shift $\Delta \phi =2\pi$ via 216 gray-scale values, axial displacements as small as $L_{\text{B}}/216=1\text{µm}$ could be measured.

4.2 Particle dynamics

The dynamics of particle motion were further studied using the quadrant photodiode to monitor the intensity distribution of the counter-propagating light emerging from the fiber endface (Fig. 4(a)). Depending on its radial position, the particle scatters a portion of the backward LP₀₁ mode into other modes (mainly the LP₁₁). The resulting asymmetry is detected by the QPD. In Figs. 5(e) and 5(f) Δφ is varied from 0 to 20π, and then back to 0π, causing the particle to move forwards and backwards over 2 mm. The transmitted sum (SUM) and difference (ΔX and ΔY) signals vary periodically with Δφ and are mutually correlated. The QPD signals for increasing and decreasing Δφ are in reasonable agreement, showing that this mode-based optical conveyor belt can indeed be used for reliable transportation of microparticles. The deviations we attribute to a slightly asymmetric particle shape, which results in imbalanced viscous forces that alter the orientation of the particle when the transport direction reverses.

To better understand the QPD signals we now construct a simple analytical model for the scattering of the backward-going LP₀₁ mode at the particle (Fig. 6). The backward-going field distribution at the fiber endface ($z=0$) may be written:

 

Fig. 6 Model for mode conversion by the moving particle. Sketch of the particle-induced scattering of a backward-propagating LP₀₁ mode into guided and radiating modes. The asymmetry of the guided mode mixture is analyzed by the QPD.

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Using Eq. (5) to interpret the QPD signals, let us first consider the case where the particle moves along a straight line parallel to, but not necessarily on, the fiber axis, i.e., ${\dot{x}}_p+{\dot{y}}_p=0$. Under these circumstances the scattering coefficients remain constant, the difference signals fluctuate periodically with z_p, and the sum signal remains constant except for the effects of exponential decay (or growth for $v_{\text{p}}<0$) caused by fiber loss. This does not however agree with the experimental results in Figs. 5(e) and 5(f), which show all the signals varying periodically at two principal frequencies: $v_{\text{p}}/L_{\text{B}}$ and $2v_{\text{p}}/L_{\text{B}}$. To explain this, we postulate the likely existence of a small (unintended) amount of LP₁₁ mode in the counter-propagating field. This would create a stationary intermodal beat-pattern that periodically perturbs the particle sideways as it moves along the axis, i.e., (x_p, y_p) no longer remains constant. Since all the scattering coefficients depend strongly on x_p and y_p, both sum and difference signals will then fluctuate periodically as seen in the experiment.

5. Conclusions

We have successfully realized a mode-based optical conveyor belt in HC-PCF. A suitably chosen coherent superposition of LP₀₁ and LP₁₁ modes, together with an uncorrelated backward-propagating LP₀₁ mode, results in a stiffness-tunable three-dimensional optical trap. By varying the phase between the forward-propagating modes a particle can then be accurately positioned in both the axial and transverse directions. Higher order modal losses of ~1 dB/m will limit the potential travel range of the conveyor belt to a few meters, although this could be extended to many meters by adjusting the modal powers while the particle is moving. The modal interference pattern acts as an “optical ruler” that is present all along the HC-PCF, permitting particle placement at positions one beat-length apart. For fast transport over long distances it is practical to vary the power ratio between the counter-propagating beams; the progress of the particle can then be followed by counting the beats in the transmitted power. Fine adjustment of the particle position is possible by controlling the relative phase between the fiber modes. In the experiments particles were transported to a position 30 cm away from the fiber end facet, followed by phase-controlled transport over a travel range of six beat-lengths. The technique may in particular be of interest for particle delivery into vacuum systems and for on-chip optofluidic devices.