We report the guidance of dry micron-sized dielectric particles in hollow core photonic crystal fiber. The particles were levitated in air and then coupled to the air-core of the fiber using an Argon ion laser beam operating at a wavelength of 514 nm. The diameter of the hollow core of the fiber is 20 μm. A laser power of 80 mW was sufficient to levitate a 5 μm diameter polystyrene sphere and guide it through a ∼150 mm long hollow-core crystal photonic fiber. The speed of the guided particle was measured to be around 1 cm/s.
©2002 Optical Society of America
Ashkin  demonstrated that small particles could be propelled and suspended against gravity using only the force of radiation pressure. This levitation process results from a momentum change in the beam’s photons as they encounter the particle, which generates a force large enough to counterbalance the gravitational force. The force generated by this process can be divided into two components: the scattering force and the gradient force. The scattering force, which points in the propagation direction of the incident beam, is responsible for the motion of the particle in the direction of the beam. The gradient force is in general proportional to the gradient of the light intensity and for dielectric spheres points towards the high-intensity region, resulting in a transverse confinement of the particle.
The use of the radiation pressure principle provides a useful means for non-intrusive manipulation of microscopic objects e.g. biological objects, particles etc. Indeed, since its first demonstration it has been increasingly employed in different areas such as biology , chemistry , atomic physics  and engineering. The applications range over fields as varied as trapping and manipulation of viruses and bacteria , micromotor machining  and scanning force microscopy . However, these applications are intrinsically limited by the diffraction of the laser beam to micrometer length scales, as strong lateral confinement requires tight beam focusing. Overcoming this limitation is of particular interest in many areas where transportation of micro-sized objects over longer distances is required. For stable guidance, including cornering, one requires constant beam intensity focused to a small spot over many Rayleigh lengths, so that a hollow optical waveguide is a natural approach. Up until now, the only possibility was to use fiber capillaries as the waveguide. This was first demonstrated experimentally by Renn et al. who successfully transported Rb atoms , and then dielectric particles in water in a hollow-core fiber capillary . Unfortunately, two inherent light guiding properties of capillaries set a limit to the guidance length, so that for a laser power of 100 mW or less, the maximum guiding length one can hope for is a few millimeters. Those properties are (i) the guided modes in hollow capillaries are fundamentally leaky and (ii) the power loss coefficient increases with the third power of the bore radius of the waveguide . This severely limits the microparticle guidance by capillary fiber, especially in applications such as surface patterning  where sub-micron resolution is required. Also, effects such as optical absorption by the guided micro-objects set an upper limit for the laser power.
In hollow-core photonic crystal fiber (HC-PCF)  the above limitations vanish, opening new prospects for laser guidance of atoms and micro-objects. Unlike hollow fiber capillaries, HC-PCF guides light without leakage using a photonic band gap , enabling a much longer guidance length to be allied with stable strong transverse particle confinement by using a small hollow core.
Here, we report the observation of particle guidance in HC-PCF over a length of 150 mm with only 80 mW laser power. The transverse (gradient) force in our fibre would be sufficient to easily maintain the particle against the force of gravity (if the fibre were horizontal) or to steer the particle around sharp corners (if the fiber were bent). Such a strong gradient force with a comparable laser power would only be attainable over a distance of 0.6 mm using a focused beam in free space, or over 12 mm using a standard capillary fibre.
The experimental set-up is shown in figure 1. A collimated cw argon ion laser operating at 514.5 nm is focused by a low numerical aperture lens and directed vertically upwards into the HC-PCF. The particles are held on a glass plate located in the focused beam just below the fiber, and are inside a glass cell to minimize the air turbulence (see figure). A pair of PZT plates (resonant at ∼80 kHz) is glued to the glass plate in order to break the Van der Waals bonds between the particles and the plate. The HC-PCF is held vertically above the glass plate. The length of the HC-PCF used in our experiments varied from 100 mm to 200 mm. A 60 times objective at the fibre output enabled us to monitor the near field pattern of the laser beam exiting the HC-PCF. The two steps of the experiment, namely the levitation of the particle from the glass plate and the guidance in the air-core of the fiber are monitored using two CCD cameras equipped with telescopes and connected to a monitor, a PC and a video recorder.
Camera 1 (see Fig. 1) images the glass plate and the input end of the fiber, in order to view the particle being levitated in the focused laser beam. Camera 2 images an illuminated section of the fibre, located between 2 cm and 5 cm from the input end, to monitor particle movement within the fibre. A color filter which blocks the laser wavelength light is placed in front of camera 2 in order to eliminate the strong scattered light from both the fibre and the particle. The video signal from each camera was recorded by a video recorder and then transferred to a PC for data analysis.
The particles used in the experiment are standard dry polystyrene spheres provided by Duke Scientific Corporation, which, according to the specifications have an average diameter of 5 ± 0.3μm and a refractive index of 1.59 at 589 nm. A number of these particles were placed on a 0.5 mm thick glass slide using a pipette. The slide and the protective cell were mounted on a XYZ micropositioner, in order to locate individual particles within the laser beam and adjust their vertical location with respect to the input end of the fiber.
The fiber itself was fabricated using the capillary stacking technique  with a core formed by 7 missing capillaries. The resulting structure was drawn into ∼170μm outer-diameter fiber with 20μm core-diameter and a pitch of 3-4μm. The air filling factor was measured to be ∼75 %. Figure 2 shows the transmission spectrum of the fiber and the output face of a ∼5 cm long fiber viewed through an optical microscope when the input face is illuminated with white light. The yellow-green color of the guided modes corroborates the transmission spectrum, which shows three guiding bands, centered at ∼450 nm, ∼540 nm and ∼842 nm respectively. The loss at our laser wavelength (514 nm) was measured to be ∼5 dB/m. At longer wavelengths lower loss of <1dB/m was observed in the same fibers.
3. Theoretical background and results
Figure 3(a) illustrates the configuration of our experiment. A Gaussian laser beam is focused at the entrance of the fiber, represented by the point O in the schematic, which also coincides with the waist position of the beam. The center of a spherical particle is located at a vertical distance d from the point O. The sign of d is worked out as follows: d is negative when the particle is located in front of the beam waist (that is the converging side of the beam) and it is positive (in the absence of the fiber) when the particle is located behind the waist; the diverging side of the beam. Because, in our experiment, the mode profile of the laser beam changes from a Gaussian shaped beam to a Bessel shaped one as the laser beam is coupled to a fiber-guided mode we shall divide the particle motion analysis into two: the levitation, which concerns the motion before entering the fiber and the guidance for the case when the particle is in the fiber.
The theoretical study of laser levitation can be performed by use of either wave theory  or ray theory [15,16] when the particle size is much larger than the laser wavelength. In order to take into account the spatial profile of the laser beam and its overlap with the particle, we shall use here a numerical model based on ray optics and described in . The element of the axial force magnitude per unit area exerted by a laser beam on a dielectric particle can be written as follows:
where n, is the particle refractive index, c the velocity of light and P 0 the total incident power. The element of particle surface area, projected onto the plane perpendicular to the beam direction, is dA . I(r,z) is the normalised transverse profile of the laser beam, e.g., for a Gaussian beam, we have I(r,z) = (2/) exp[-2(r/ω 0)2]. The total force is deduced by integrating over the area where the laser beam and the particle overlap.
The model calculates the axial force for different particle locations with respect to the beam waist. This gives us an idea of the magnitude of the axial force acting upon the particle before it enters the fibre, and enables us to optimize the position of the glass plate holding the particles with respect to the input end of the fiber. The resultant force acting upon the particle in the vertical direction is, therefore, the sum of the scattering force mentioned above, the gravitational force and the drag force, which gives the following equation of motion:
where m = 4πa 3 ρ/ 3 where ρ is the density of the polystyrene (1.06′ 103 kg/m 3), a the particle radius and η air is the viscosity of air (1.8x10-5 N s m-2).
The solid curve in Fig. 3(b) shows the calculated magnitude of the resultant force, using equation (2), versus the distance d for a beam waist of 10 μm and a laser power of 80 mW. The points in the figure are experimental data deduced from a recorded video sequence of the particle motion (see Fig. 5). The movie was digitized and a sequence of equally time-spaced frames is obtained (a typical example of such a sequence is shown in Fig. 4). The position with time of the levitated particle is then deduced by calibrating the position using the known outer diameter of the fiber (170 μm). The time is deduced from the frame rate of the acquisition (30 frames/s). The measured force increases with d from ∼5 pN at -460 μm from the waist (i.e. the input end of the fiber) and reaches a maximum of ∼49 pN near the waist. The measured force magnitude and its evolution with d show a reasonable agreement with the theoretical predictions. The discrepancy between the experimental results and the theoretical predictions is due to the uncertainty in the measured distance d (i.e. location of the waist) and the laser power. Moreover, in the theoretical model, we assumed that the respective center of the laser beam and the particle coincide, which is not always the case with our experimental set-up. The typical measured speed of the levitated particles near the location of the input end of the fiber is 1.3 to 1.5 cm/s and agrees well with the theory.
3.2 Guidance in the HC-PCF
Once the particle is coupled to the HC-PCF, the assumed mode profile changes from a Gaussian to a Bessel beam. Experimental analysis of the supported propagation modes in the HC-PCF [18,19] show that the mode profiles of the propagation modes in HC-PCF can be well approximated as the modes of a perfect reflector . This can be understood intuitively by recognizing that while the cladding structure creates the band-gaps, i.e. the transmission spectrum of the fibre, the patterns of the guided modes are shaped by the defect. Consequently, the intensity expression within the HC-PCF can be modeled as:
where I 0 is the peak of the coupled intensity, rco the radius of the hollow-core, J 0 is the Bessel function of order zero and αloss is the attenuation coefficient.
Figure 6 shows the radial profile of the scattering and the gradient forces at the entrance of the fiber acting upon a 5 μm diameter polystyrene particle, assuming 80 mW incident optical power. The latter is deduced after substituting the expression of the laser intensity given by equation (3) in the expressions of the scattering and the gradient forces given in , which gives:
for the scattering force, and
for the gradient force where J 1 is a Bessel function of first order. It is worth mentioning that the use of equations (4) and (5) is for their functional form and for order-of-magnitude estimates only.
The magnitude of the scattering force agrees with the value found using the numerical model mentioned in the preceding section. For small displacements from the fiber center, the axial force is almost constant, while the gradient force is linearly proportional to the radial displacement and behaves like a restoring force of an harmonic oscillator that pushes back the particle to the high intensity region, i.e. Fgrad : - kr, k being the spring constant of the gradient force. From equation (5) one can easily deduce an expression for k:
In order to achieve a good transverse stability the magnitude of the gradient force must dominate over other possible transverse forces. If one neglects air turbulence inside the fiber core, the only transverse force present in a straight and vertically held fiber is the one induced by thermal noise. The potential energy of the gradient force for small displacements is ∼kr 2/2 (k : 8.4′ 10-6 N/m for our experimental conditions) and for a radius equal to r co/4 a quarter of the core radius (where the harmonic oscillator approximation is valid), the potential energy is : 10-17 J or 4 orders of magnitude larger than the potential energy of thermal noise kBT = 4′ 10-21 J. Moreover, the thermal noise acts as a perturbation to the transverse location of the particle, and hence sets a limit on the accuracy to which a particle can be placed precisely in a given location. This is estimated by equating the potential energy of the gradient force and the thermal noise, which gives . In the case of the fiber used here (i.e. rco =10 μm), racc is ∼0.15 μm. The value of r acc is crucial for any preliminary choice of the core radius of the fiber and the size of the particle to be transported. Also, it is of the highest importance in applications such as high-resolution surface patterning. For the latter, decreasing the core diameter of the HC-PCF (r acc scales with the square of the radius of the fiber core) by only a factor 3 would bring the resolution figure down by an order of magnitude.
Figure 7 shows a sequence of a guided polystyrene particle over a 1 mm section of 150 mm long HC-PCF localized at about 4 cm from the input end of the fiber. The real time video of the sequence is shown in Fig. 8. The sequence shows a well-centered particle and moving at a speed of 1 cm/s over the imaged section. The difference between the velocity of the particle inside and outside the fiber is due to two factors; (i) the drop in power due the coupling efficiency which was between 70 and 80%. (ii) The increased of the drag force due to the proximity of the fiber wall lowers the velocity compared to the velocity outside the fiber , For our experimental parameters (r co = 10μm and a = 2.5μm), the drag force coefficient is multiplied by 1.97.
The levitation height is taken as the location where the optical force is equal to the gravitational one, and can be written as z = z 0 ln(F 0/mg), F 0 being the scattering force at the entrance of the fiber and z 0 = 1/αloss is the exponential decay length. This gives a levitation height of 34 cm for a 5 μm diameter polystyrene particle, a laser power of 100 mW and a loss of 5 dB/m. For a loss of 1 dB/m (achievable at longer wavelengths in the fiber used here), one would expect a levitation height of ∼2 m. This is more than three orders of magnitude greater than the guidance length possible using a free space laser beam focused to the core size of the HC-PCF. Similarly, in the horizontal configuration (i.e., the fiber is held horizontally), the guidance is limited by the weight of the particle, which counteracts the confining effect of the gradient force and consequently pushes the particle towards the inner wall of the fiber core. Equating the maximum magnitude of the gradient force to the gravitational one, in the case of 20 μm fiber-core diameter with the parameters mentioned above, gives guidance lengths of 30 cm and 1.8 m for a loss figure of 5 dB/m and 1 dB/m respectively. It is noteworthy that the fiber bending has negligible effect on the guidance length. Indeed, for a fiber with a radius of curvature R greater than the critical radius of curvature of the fiber-which is of the order of 1 cm , and a particle velocity v of a few centimeters/s, the centrifugal force mv 2/R acting on the particle remains two orders of magnitude less than that of the gravitational force.
In conclusion, we have demonstrated levitation of dry polystyrene particles and their guidance in HC-PCF. Guidance over 150 mm at a speed of 1 cm/s with a laser power of 80 mW was observed. Moreover, the geometrical optics calculation agrees reasonably with the experimental results for the levitation experiment. With our actual loss figures, particle guidance length of 2 m is possible. The recent breakthrough improvement in hollow core losses (as low as 13 dB/km in a similar fiber ) would increase the possible guidance lengths to almost 130 m. This opens new prospects for all applications where the transportation of micro-objects is required.
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