Evanescent wave optical trapping and transport of micro- and nanoparticles on tapered optical fibers

We investigate the manipulation of microscopic and nanoscopic particles using the evanescent optical field surrounding an optical fiber that is tapered to a micron-scale diameter, and propose that this scheme could be used to discriminate between, and thereby sort, metallic nanoparticles. First we show experimentally the concept of the transport of micron-sized spheres along a tapered fiber and measure the particle velocity. Having demonstrated the principle we then consider theoretically the application to the optical trapping and guiding of metallic nanoparticles, where the presence of a plasmon resonance is used to enhance optical forces. We show that the dynamics of the nanoparticles trapped by the evanescent field can be controlled by the state of polarization of the fiber mode, and by using more than one wavelength differently detuned from the nanoparticle plasmon resonance. Such a scheme could potentially be used for selectively trapping and transporting nanoor microscopic material from a polydisperse suspension. & 2012 Elsevier Ltd. Open access under CC BY license.


Introduction
It has long been known that a laser beam can exert significant optical forces on microscopic particles. Most commonly these forces are utilized in an optical tweezers [1], where a single laser beam is brought to a diffractionlimited spot by a microscope objective lens, capable of producing stable three-dimensional trapping for micron and sub-micron-sized particles. This realization of optical trapping has been applied to a variety of micro-and nanoscale materials including viruses [2], aerosols [3], microbubbles [4], semiconductor nanowires [5], carbon nanotubes [6] and graphene flakes [7]. Similarly, a microparticle exposed to an evanescent optical field experiences a radiative force, as was first demonstrated by the driving of water-suspended particles above the surface of a prism by a single laser beam, weakly focused and undergoing total internal reflection on the prism-to-water interface from below [8]. Later experiments used two counter-propagating laser beams which lead to stable trapping [9,10] and the formation of a variety of selfassembled particle structures dependent on the polarization state of the laser beams [11] or the number of bound particles [12], and enhanced the optical forces by enclosing the experiment in a resonant optical cavity [13].
An alternative scheme for generating an evanescent optical field suitable for trapping is to use an optical fiber that is tapered to a diameter of around 1 mm or smaller.
In such ultra-thin fibers a significant fraction of the optical power is contained in an evanescent field around the fiber core which penetrates an appreciable distance into the surrounding medium [14], and such evanescent field fiber traps have been used for trapping [15] and probing [16] laser-cooled atoms in addition to manipulations of microparticles [17]. Similar evanescent wave trapping and propulsion has also been achieved using waveguides with a rectangular geometry for both solid [18,19] and hollow particles [20].
In this work we demonstrate fabrication of tapered optical fibers using a simple and low-cost 'heat-and-pull' technique, and show that the biconical tapers produced in this way are suitable for evanescent-wave trapping and propulsion of microscopic spherical particles. Having experimentally demonstrated the principle of operation we then make calculations of the optical trapping and scattering forces on metallic nanoparticles, showing that such forces are enhanced for a trapping wavelength in the vicinity of the nanoparticle's plasmon resonance, and describe a scheme for controlling the nanoparticle trajectory along the taper using bichromatic laser fields and control of the fiber mode polarizations. We suggest that the sensitivity of the nanoparticle dynamics to the polarization and wavelength of the laser fields through the plasmon resonance could make such tapered fibers an effective tool for sorting and separating nanoparticles according to property-dependent criteria.

Fabrication of tapered optical fibers
Our tapered optical fibers are produced using a 'heatand-pull' method similar to that described in [21]. A length of standard single-mode optical fiber (SM-980-125-5.6 from Thorlabs Inc. NJ) has a short region of the protective polymer buffer jacket stripped away and is mounted on a pair of motorized translation stages. The heat source is a hand-held butane gas torch which, unlike the method described in [21,22], is kept static during the pulling process, limiting the length of the tapered region achievable. The fiber is tapered by driving the motorized stages apart a distance of 20-30 mm at speeds of between 0.5 and 1.0 mm s À 1 . After the tapering process is complete the flame is removed and the tapered fiber mounted on a custom-made microscope slide containing a 300 mm deep slot. A few tens of microliters of solution containing 2 mm diameter polystyrene microspheres suspended in deionized water (with 10% by volume Triton-X-100 to prevent particles sticking to the fiber) is added and sealed beneath a cover slip. Each end of the tapered fiber is cleaved and inserted into a fiber-to-fiber splice unit to couple in laser light. The slide-mounted tapered fiber is then viewed in an inverted microsope (Zeiss Axiovert 200).

Evanescent wave trapping and propulsion of microparticles
The laser source for these experiment is a single-mode Nd:YAG laser (wavelength l ¼ 1064 nm) with a maximum output power of 3 W. The output beam is coupled into a length of the same type of optical fiber that is used for making the fiber tapers using a NA ¼0.15 collimating lens. These fibers are then connected to the tapered fiber by means of the fiber-to-fiber splice. A diagram of the optical set-up is shown in Fig. 1.
The waist of the tapered fibers produced by this method is typically around 1 mm. When a single mode propagates unidirectionally in the fiber, microparticles are observed to be trapped against the fiber by the optical gradient force of the evanescent field, and propelled along it by the scattering force in the same direction as the mode propagates. The motion of the particle along the tapered optical fiber is tracked by video microscopy [23], from which we extract the trajectory of the propelled particle and measure its speed along the fiber.
A sequence of images is shown in Fig. 2(a)-(c) that shows a single 2 mm diameter polystyrene sphere pushed along the tapered region of the fiber by radiation pressure. An example of the particle trajectories extracted from a video recording is shown in Fig. 2(d), where the tracks of two particles which are pushed along the fiber are plotted. From a linear fit to the particle track we measure the speed to be v ¼ ð6:98 7 0:05Þ mm s À1 .

Distribution of electric field
The electric field distribution in the region of the tapered fiber waist is found by solving the wave equation for the cylindrical boundary conditions of the tapered fiber. A numerical solution of the fiber eigenvalue equation [24] yields the transverse component of the mode wavevector inside the glass fiber, h, which also satisfies where k 0 ¼ 2p=l is the vacuum wavevector, n 1 the refractive index of the fiber glass, and b the longitudinal component of the wavevector. In the region surrounding the waveguiding core with refractive index n 2 the transverse wavevector, Light from a Nd:YAG laser is coupled into a conventional optical fiber which is spliced to a short length of fiber, the central part of which has been drawn into a biconical taper by the pulling process. The tapered fiber is fixed to a slotted microscope slide and viewed in an inverted microscope. q satisfies and is imaginary, implying a rapidly decaying evanescent field. As an example we show in Fig. 3 the field distributions for a silica glass tapered fiber with parameters approximately those of the experiment in Section 2.2. The fiber is surrounded by water and has diameter a ¼ 1:0 mm, and is supporting a quasi-linearly (xÀ) polarized HE 11 mode. The normalized frequency of this tapered fiber, V ¼ ka ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 1 Àn 2 2 q ¼ 2:048, is below the cut-off frequency for single-mode operation, V C ¼ 2:405, thus the HE 11 modes are the only modes supported by this waveguide. As shown in [25] the strong confinement of the initially (quasi-) linearly polarized HE 11 fiber mode in the taper leads to significant components of electric field in the orthogonal directions.
For the HE 11 mode initially polarized in the x-direction the dominant component remains the x-polarized field, shown in Fig. 3(a) but the amplitudes of the electric fields in the y-and z-directions become significant for large refractive index differences between the silica fiber and its surroundings (large fiber numerical aperture), shown in Fig. 3(b) and (c). Similar cross-polarized components are also observed in the case of high numerical aperture focusing, e.g. by a microscope objective lens [26]. Furthermore, the conditions for continuity of electric field components normal and tangential to the boundary lead to an enhancement of the evanescent electric  field along the x-axis, resulting in a net field distribution with pronounced lobes along 7 x, shown in Fig. 3(d). The evanescent field of this mode penetrates a significant distance into the surroundings. For these exemplar parameters we calculate a penetration depth L ¼ 9q9 and approximately 26% of the mode power carried in the evanescent field.

Interaction with metallic nanoparticles
Metallic nanoparticles have unique useful optical properties due to the plasmon resonance in their scattering spectrum [27]. The enhancement of optical forces exerted on the nanoparticle that arises when the laser wavelength is tuned close to the plasmon resonance can be utilized to achieve stable optical trapping in single-beam optical tweezers [28,29] which might otherwise be difficult due to the volume scaling of optical gradient forces for such small particles [30], and recently good agreement between experimental measurement and theoretical calculation of trapping forces has been demonstrated for both single metallic spheres [31,32] and aggregates of nanoparticles [33]. Hybridization of metallic nanoparticles with objects such as whispering-gallery resonators or quantum dots and the resultant coupling between the plasmon and the narrow resonances has even been suggested as a novel method for achieving laser-cooling of these mesoscopic systems [34].
Evanescent wave manipulation of gold nanoparticles has been demonstrated using the field of a channel waveguide fabricated by an ion-exchange technique [35], and enhancement of optical forces using the surface plasmon of a thin gold film has also been applied to manipulation of both polystyrene [36] and gold particles [37].
Here we consider the manipulation of silver nanoparticles in the evanescent field around the tapered fiber. The motivation for using silver nanoparticles is that in addition to their potentially useful technological applications [38], the optical properties lend themselves to an additional degree of control over the optical forces. The relatively narrow plasmon resonance lineshape of silver when compared to, for example, gold leads to a region on the short-wavelength side of the resonance where the real part of the polarizability of the nanoparticle can be negative [39], and so the direction of the optical gradient force is reversed, which may be exploited to induce particle repulsion from a surface [42]. This situation is analogous to that used in the 'gravito-optical surface trap' for ultra-cold atoms [43].
First we consider the interaction between a nanoparticle and the evanescent field of a tapered optical fiber supporting a HE 11 mode of a wavelength tuned to the long-wavelength side of the plasmon resonance (red detuning). The optical properties of the silver nanoparticles are calculated using a Lorentz-Drude model [44] using the fitting parameters tabulated in [45] which produce a good fit to the experimental data of [46,47] in the range 0.125-6 eV (l ¼ 20021000 nm).
An example of the calculated optical properties is shown in Fig. 4 for a spherical silver nanoparticle of radius 20 nm. Part (a) of Fig. 4 shows the real part of the complex polarizibility, RefaðlÞg calculated in the range l 0 ¼ 3502700 nm. Of particular interest for the present work is the change in sign of Refag in the region of the plasmon resonance with Refag o 0 for blue detunings and Refag 4 0 for red detunings. Part (b) shows the imaginary part of the polarizibility, ImfaðlÞg with a clear resonance feature around l p ¼ 408 nm. This resonance leads to the peaks in the absorption, C abs , and scattering, C scat , crosssections shown in Fig. 4(c) and (d), respectively, which are related to the polarizibility as

Monochromatic trapping
We treat small (Rayleigh) particles as dipoles and account for depolarization effects at small but finite volume with the correction to the dipolar response of [48]. The interaction between a point dipolar particle and an optical beam is usually separated into two parts. Firstly a force arising from a spatial gradient in the dipole potential energy, that acts in the direction of a gradient of intensity in the beam, and secondly an optical scattering force, where C ext ¼ C abs þ C scat is the total extinction cross-section that acts in the direction of the time-averaged Poynting vector, S ¼ 1 2 RefE Â H n g. Recently a third component of optical force arising from the non-uniform distribution of the spin density of an optical field has been identified [49,50], is the time-averaged spin density. This contribution to the optical force (sometimes called the 'spin curl force') is the most significant for strongly localized fields such as in the focal volume of a high numerical aperture objective lens [51] and may be expected to play a role in the highly non-paraxial fields of a sub-micron tapered optical fiber. Consistent with previous work analyzing a similar system for trapping of nanoparticles [40], in calculating the optical forces we neglect the effects of multiple scattering between the nanoparticle and the tapered fiber. This approximation can be made provided the nanoparticle is not too close to the substrate (fiber), and the contrast in dielectric constant between the substrate and suspending medium is relatively low [41]. Let us take as an example the 40 nm diameter spherical silver nanoparticle considered above, and examine the interaction with the evanescent field of a tapered optical fiber with radius a ¼ 0:15 mm. We calculate the electric field distribution of an x-polarized HE 11 mode as above, and calculate the dipole potential energy for a number of different wavelengths taking into account also the dispersion of the silica glass of the optical fiber using tabulated Sellmeier coefficients [52]. The dipole interaction energies are then shown in Fig. 5(a)-(d), plotted in units of k B T (k B Boltzmann's constant) at temperature T ¼ 293 K, and normalized per unit power contained in the mode (i.e. the sum of the power carried in the fiber core and the power in the evanescent field).
The plasmon wavelength for the silver nanoparticle is found to be l p % 408 nm. Fig. 5(a) shows the interaction energy for a laser wavelength tuned slightly to the longwavelength side of the plasmon (red detuning), with l ¼ 425 nm. As expected there is a minimum of potential energy adjacent to the fiber, which is deepest along the predominant direction of polarization. For a laser wavelength detuned to the short wavelength side of the plasmon (blue detuning), with l ¼ 395 nm, the sign of the dipole potential is reversed, shown in Fig. 5(b), and the dipole potential repels the nanoparticle from the fiber. For unidirectional mode propagation in the tapered fiber, in addition to the gradient force that traps particles against the fiber, there is also a radiation pressure (scattering force and 'spin curl' force) directed parallel to the fiber that propels the particles along it, as seen in Fig. 2. For plasmonic nanoparticles the magnitude of the propelling force is maximum when the laser is tuned to the plasmon resonance, where we find for the cross-sections C abs ¼ 1:72 Â 10 À14 m 2 and C scat ¼ 6:58 Â 10 À15 m 2 , and a maximum propelling force adjacent to the fiber of F z ¼ 507 pN W À1 , decreasing to F z ¼ 84 pN W À1 at a distance from the fiber equal to the fiber radius (these forces are also normalized per unit power in the mode). At this distance approximately 90% of the force arises from the radiation pressure, and the remaining 10% from the curl of the spin density. The resulting propagation speed of the nanoparticle along the fiber is estimated from the Stokes drag on a sphere corrected for the presence of a cylindrical boundary [53] to be v z ¼ 224 mm s À1 W À1 .
Detuning to the red of the plasmon resonance such that gradient force trapping is achievable, we find at e.g. l ¼ 425 nm the cross-sections are reduced to C abs ¼ 8:87 Â 10 À15 m 2 and C scat ¼ 4:34 Â 10 À15 m 2 , and the propelling force at a distance d¼a from the fiber reduced to F z ¼ 50 pN W À1 . In this situation we estimate the nanoparticle velocity along the fiber to be v z ¼ 134 mm s À1 W À1 . These forces can be balanced by using counter-propagating modes with parallel polarizations [9] in the fiber, which will also modulate the intensity (and hence the optical dipole potential) along the length of the fiber providing an additional degree of localization [40].
Using a circularly polarized HE 11 mode changes all components of the forces acting on the nanoparticle. Firstly the optical dipole potential becomes symmetric around the fiber due to the rapidly rotating polarization direction. Secondly the Poynting vector acquires a component that acts in the azimuthal direction arising from the longitudinal component of the electric field, which contains an azimuthal phase variation similar to that which occurs in a free-space Laguerre-Gaussian laser mode. Thirdly the spin curl force also acquires an azimuthal component. For a high numerical aperture fiber there is a significant field component in the longitudinal direction, and for small diameter tapers a significant fraction of the optical power resides in the evanescent field outside the fiber. The net effect of these features is that a particle in the evanescent field may experience strong non-conservative forces arising from the optical scattering and spin curl contributions. Fig. 6(a) and (b) shows the distribution of the intensity of the transverse and longitudinal electric field components, respectively, for a circularly polarized HE 11 mode of a tapered optical fiber with the same diameter as above and at the plasmon wavelength of the 40 nm silver nanoparticle, l ¼ l p . As a result of the circular polarization there is now an azimuthal symmetry to the time-averaged electric field distribution. In parts (c) and (d) of Fig. 6 we show the spatial distributions of the longitudinal, S z , and azimuthal, S f , components of the Poynting vector, respectively. The presence of non-zero azimuthal components of Poynting vector and curl of the spin density gives rise to an additional component of the force transverse to the axis of the fiber, and it may therefore expected that a circularly polarized HE 11 mode will drive a nanoparticle in a helical trajectory along and around the taper. Fig. 7(a) and (b) shows the resulting electric dipole interaction energy at wavelengths detuned to the red and blue (respectively) of the plasmon resonance for the same 40 nm diameter silver nanoparticle in the vicinity of a ¼ 150 nm fiber. In this case the azimuthally directed component of the nonconservative (scattering þ spin curl) force, F f can be significant compared to F z , and we find for an exemplar case of red detuning from the plasmon resonance, l ¼ 425 nm an axial component of force (normalized) of F z ¼ 47 pN W À1 at a distance d¼a from the fiber, and an azimuthal component of F f ¼ 7 pN W À1 . In the absence of an azimuthally modulated gradient force we therefore expect the nanoparticles to be attracted to the fiber and pushed along it following a helical trajectory in a manner previously calculated for ultra-cold atoms [54]. From this we estimate (normalized) components of velocity of the nanoparticle of v z ¼ 126 mm s À1 W À1 in the axial direction and v f ¼ 17 mm s À1 W À1 in the azimuthal direction.

Bichromatic trapping
The ability to tune the interaction, and in particular produce either attractive or repulsive forces depending on the detuning from resonance naturally leads to the possibility of using two wavelengths simultaneously to obtain an extra degree of control over the optical trapping. The formation of a two-color or bichromatic trap for ultra-cold atoms in the evanescent field of a tapered fiber was considered in [55], and similar detuning-dependence of forces in an evanescent field have been considered for metallic nanoparticles and also for microparticles exhibiting whispering gallery modes [56]. As can be seen in Fig. 5 the penetration depths, L, of the evanescent fields depend on the optical wavelength, and due to the width of the plasmon resonance (some tens of nanometers for our silver nanoparticles) for two wavelengths detuned to the red and blue of l p these can be significantly different.
A plasmonic nanoparticle exposed to these evanescent fields may therefore experience competing gradient forces: one from the red-detuned beam that draws it toward the fiber, and one from the blue-detuned beam that provides a potential barrier preventing it from reaching the fiber surface. Fig. 8 shows some examples of the bichromatic potential experienced by the 40 nm silver nanoparticle around a fiber carrying two counter-propagating (parallel) linearly polarized HE 11 modes with different wavelengths.  The powers of the two modes are chosen such that the net potential energy is zero adjacent to the fiber. In part (a) the longer wavelength, l R ¼ 425 nm, and the shorter wavelength is l B ¼ 397 nm. For this combination the intensity of the blue-detuned mode is 1.37 times that of the red-detuned mode. Two potential minima can be seen a small distance from the fiber along the 7 x-axis. At this equilibrium trapping position the radiation pressures from the two beams are unequal. Due to the width of the plasmon resonance (some tens of nanometers) the typical difference in frequency between the red-and blue-detuned laser modes used to realize a bichromatic trap is sufficiently large that the effects of beating of the two frequencies may be neglected and the forces arising from the two modes simply added [57]. The net scattering force (radiation pressure þ spin curl forces) on the nanoparticle arising from this configuration is found to be 113 pN W À1 at the minimum of the dipole potential, which would drive the nanoparticle at an estimated speed of v z ¼ 460 mm s À1 . For a different combination of wavelengths the fraction of optical power in the evanescent field and the decay lengths are different. By way of example part (b) of Fig. 8 uses wavelengths l R ¼ 816:7 nm ¼ 2l p and l B ¼ 407 nm (very close to resonance). For these parameters the blue-detuned mode has an intensity 0.19 times that of the red-detuned mode. The potential minima can be seen to be broader, shallower, and displaced further from the fiber. At this equilibrium position the net radiation pressure from the counterpropagating beams is only 180 pN W À1 and a velocity of v z ¼ 178 mm s À1 .
When considering the utility of this technique for particle sorting and separation it is instructive to compare with a number of other methods that have previously appeared in the literature. A 'near-field' sorting technique was demonstrated for micron-sized particles in [58], where the evanescent field was created using a TIR objective lens. In this geometry the extent of the interaction region is limited to the size of the laser beam spot (given as 13:4 mm), whereas a tapered fiber can potentially produce a useful evanescent field for many tens of microns along the tapered region. Similarly, the large separation between particle collection regions at opposite ends of the taper may be advantageous when compared to the scheme of [59] where the relative displacements for particles of different sizes are comparatively small. The method of [60] uses a dynamic (translating or flashing) optical lattice for very effective separation of particles according to size or composition. By contrast the tapered fiber method proposed for sorting particles would not require active modulation of parameters during an experiment as the optical forces depend on the detuning from the plasmon resonance, and are thus strongly dependent on nanoparticle size and composition. The initial choice of laser wavelengths therefore controls the selectivity of the technique.

Conclusions
In conclusion, we have experimentally demonstrated that microscopic particles can be attracted to and propelled along sub-micron scale glass fibers by the evanescent field of a laser beam. We have made calculations of the optical gradient and scattering forces of the evanescent field for metallic (silver) nanoparticles and shown how control of the fiber mode polarization and wavelength can be used to control motion of nanoparticles along the fiber. We anticipate that the sensitivity of the interaction to nanoparticle plasmonic properties and fiber mode distribution could potentially be a useful tool for selectively trapping, sorting and trafficking micro-and nanoparticles drawn from a polydisperse distribution.