1. Introduction

Optical tweezers (OT) are powerful tools able to trap and manipulate nano and micro-scaled particles without physical contact. After the first demonstrations [1], they have been successfully used to study biological systems also in combination with optical diagnostic techniques like two photon fluorescence, Raman scattering and coherent anti-Stokes Raman spectroscopy [2–5]. The standard OT scheme consists of a modified microscope with a high numerical aperture (NA) objective. An external laser is coupled to the objective and tightly focused inside the medium where the particles to be trapped are immersed. Despite considerable results have been obtained, this OT configuration presents some limitations due to its intrinsic structure, since the use of a microscope makes the device bulky and expensive. Moreover the need for a high NA objective to achieve the optical trap considerably limits the field of view of the device.

These limitations can be overcome by using optical fibers for the delivery and proper shaping of the trapping beam. Various approaches for the realization of single fiber OT are reported in literature, generally exploiting a hemispherical microlens on top of a standard “axial core” fiber. However the NA that can be obtained using this configuration is usually quite low preventing a complete three dimensional (3D) optical trapping [6]; on the other hand, in presence of an extremely tight focusing, the focal point is very close to the fiber tip, making difficult to obtain optical trapping without physical contact [7]. Recently we proposed and demonstrated a new approach to fiber tweezers development in which the high-NA light focusing is achieved by exploiting total-internal-reflection at the end-face of a microstructured probe constituted by a fiber bundle (BB-TOFT — bundle based total internal reflection fiber tweezers) [8]. BB-TOFT provides 3D optical trapping up to a distance of about 50 µm from the probe-end. In addition several functions like optical analysis and micromanipulation can be integrated in the same fiber probe, making the BB-TOFT a promising tool for biological studies.

The aim of this paper is to properly design the BB-TOFT structure in order to optimize its performances. Due to the peculiar geometry of the light intensity distribution, we propose a “modified ray-optics” description able to correctly evaluate the optical forces in the Mie regime [9] by taking into account the features of the beams emitted by the bundle. It is important to underline that this new method, described in the following, is significantly different with respect to the one used in [10]. The proposed approach takes properly into account the diffraction of the beams emitted from the fibers, hence giving the correct results for the electrical field calculation when the 4-fibers bundle structure is considered. Conversely the method proposed in [10], which was valid in the case of the annular core fiber tweezers, takes into account the diffraction-caused beam broadening only on the radial direction, while the azimuthal diffraction is not properly described, because of the considered annular geometry. Moreover, the present approach overcomes the limitations related to the ray-optics approximation in the Rayleigh range. The newly proposed method is described in Section 2, where we also introduce the figure of merit we used to assess BB-TOFT efficiency. In Section 3 we present the numerical results we have obtained considering different BB-TOFT configurations and we derive some simple rules for the design and optimization of the device.

2. Numerical analysis of the proposed fiber OT structure

The BB-TOFT structure is shown in Fig. 1. We consider the case of a probe composed by four fibers, see Fig. 1a, like that experimentally tested in [8]. With reference to the longitudinal cross-section depicted in Fig. 1(b), the end-faces of the fibers composing the bundle are properly cut at an angle θ in the region corresponding to the fiber cores so that the light propagating through the fibers experiences total internal reflection at the interface with the surrounding medium. Hence optical beams are first deflected into the cladding and then transmitted out of the fibers converging all in the same point, at an angle φ with respect to the fiber axis. It’s worth noticing that if the cut is not exactly perpendicular with respect to the y-z plane, the beams will not converge exactly in the same point. Anyway, as the fabrication error in the cut rotation is < 0.5°, the maximum offset of the beam centre from the “ideal” trapping position is less than 1 µm. This effect is not particularly critical, as the beam size in the trapping position is significantly larger than the maximum error in beam position.

All the used fibers are single-mode at the operating wavelength of 1070 nm, hence the optical mode can be described as a Gaussian beam characterized by a precise mode field diameter (MFD). The resulting structure provides the equivalent effect of a focused beam, with the advantage that the scattering force in the axial direction is highly suppressed, as in the case of Laguerre-Gauss beams in standard OT [11,12].

 

Fig. 1. Scheme of the OT under investigation. (a) Cross section of the four-fiber bundle. The cores are represented by dark gray spots lying on an ideal annulus with radius R. (b) Longitudinal section of the four- fiber bundle. Each beam propagating in the cores experiences total internal reflection at the cut surface and then refraction at the fiber-medium interface.

Download Full Size | PDF

The first step for the calculation of the forces is the evaluation of the intensity spatial distribution, given by the superposition of the Gaussian beams emitted by the fibers composing the bundle. In standard OT the optical forces in Mie regime are calculated by decomposing the strongly focused beam into individual rays. The power associated to each ray is determined by the corresponding amplitude in the far field, while the angular orientation, with respect to the propagation direction, is given by the gradient of the optical phase [10]. The total force is accomplished by summing up the force contributions coming from each ray. In such a representation the focal region is described as the crossing point of all the rays and the peculiar features of light propagation within the Rayleigh range (z_R) are completely neglected. However this decomposition works correctly in the case of standard OT, where the strong focusing leads to a focal depth much smaller than the particle radius (R_p). In such a case, as the momentum transfer between the photons and the particle is produced on the particle surface, which lies in the far field (being R_p ≫ z_R), the forces are reliably described.

Considering BB-TOFT, the propagation of the Gaussian beams emitted by each fiber is not adequately described by the usual ray decomposition. Indeed the assumption that the rays are originated from a single point in the centre of the optical fiber core cannot be accepted: the fiber MFD is in the range of few microns (typically 2~10) and the corresponding Rayleigh range can be much larger than the distance between the fiber-end and the trapping position. Hence, the particle surfaces cannot be considered as being always in the far-field, and the traditional ray optics description (reliable only when the distance between the fiber end-face and the particle surfaces is much larger than z_R) cannot be applied.

Here we adopt a different approach to the decomposition of the beam into rays [13], able to describe the optical field at distances both below and above z_R, provided that the considered beams are not tightly focused and that the propagation can be described through paraxial approximation. After each propagation step Δz’ along the fiber axis (z’ in Fig. 2a) a new set of optical rays is defined by calculating the amplitude and the wave-front curvature of the Gaussian beam: A(ρ, z’) and r(ρ, z’) respectively. The amplitude is used to assess the optical power associated to each ray, while the curvature radius is used to determine their propagation direction, which is perpendicular to the wave-fronts. It is worth noticing that the procedure to calculate the optical field is an approximation since the resulting rays’ directions depend on the propagation coordinate z. Anyway the obtained field profile introduces the proper correction for the finite beam dimension in the focal point.

 

Fig. 2. Scheme of the method exploited to calculate the optical field. At each step along the axis z’ we calculate (a) the amplitude and the curvature radius of the Gaussian beam and we exploit them to obtain respectively the power and the propagation direction, angle α, of each ray. The optical forces exerted on the particle are then calculated and by a roto-translation of the frame of reference (b) the total forces distribution in the BB-TOFT geometry is obtained.

Download Full Size | PDF

Once the optical field at each position is known, the optical force exerted on a particle is calculated as the sum of the scattering and gradient components of each ray [9,10]. The total optical force distribution is computed in the fiber bundle frame of reference (x, y, z) shown also in Fig. 1, where z is the probe axis, and the fiber cores lie on the x or y axis. As the propagation direction of the beams coming out from the fibers is tilted with respect to the bundle axis by a known angle φ, a simple change of the frame of reference is needed. For each possible position of the particle in the (x, y, z) space, we determine its position in the cylindrical frame of reference associated to each fiber (ρ, z’), we calculate the corresponding force and then we simply sum the contributions of all the four fibers.

A classical representation of the spatial force distribution is given by means of the dimensionless quantity Q, defined as a function of the total force (F_T), the total optical power (P), the medium refractive index (n_M), and the speed of light in vacuum (c): Q(x,y,z)=c F_T(x,y,z)/n_MP. Figure 3(a) shows the Q distribution obtained by illuminating a sphere whose radius and index of refraction are 5 µm and 1.59 respectively. The BB-TOFT has the structure shown in Fig. 1: the radius of the annulus along which the fibers’ cores are disposed is R=55 µm, the fibers have a MFD=8 µm and the cutting angle is θ=66°. It can be seen that the optical trap is formed at a distance of about 27 µm from the bundle end-face. The Q parameter is shown in the yz plane; due to the symmetric arrangement of the four fibers, identical results can be obtained in the xz plane.

A different parameter that can be used to quantify the efficacy of an optical trap is given by the “escape energy” (ε_esc), which is the minimum energy necessary to a particle in order to escape from the trap [10]. We note that this parameter can be used to easily compare the trapping efficiency of OT with different structures.

As the scattering force component is not conservative, a potential energy for the total force cannot be calculated; hence to define ε_esc a two-stage computation is performed. At first we calculate the work per unit power (ε_TP) that has to be done against the optical forces to move a particle along a straight-line connecting the centre of the trap to any possible target point in the surrounding space. As a second step, we evaluate, for any possible linear escape trajectory, the maximum value of ε_TP, thus identifying the energy required to move a particle along the corresponding path. The minimum among these values is defined as ε_esc and the corresponding trajectory is the most energetically favoured escape path. Hence it is easy to figure ε_esc as the “trap-depth” and it is also evident that the highest and lowest values of ε_TP can be found along the beam propagation directions. In particular, the most favoured escape path is along the propagation directions of the slanted rays coming from the fibers (ε_esc ≈ 2.53 fJ/W), whereas the less likely path is along the same direction, but in the opposite heading. It is worth noticing that the resulting ε_esc value is similar to those obtained considering a strongly focused Gaussian beam, as used in standard OT. Figure 3(b) shows a colormap of ε_TP obtained by integrating the forces of Fig. 3(a). The attached movie shows the three dimensional shape of the quantity ε_TP, where the optical trap clearly resembles a potential well.

 

Fig. 3. Example of a contour plot of the Q factor, (a), and of the work per unit power, (b) (Media 1), calculated for the fiber-bundle structure with geometrical parameter: bundle diameter R=55 µm, beam diameter w=8 µm and cutting angle θ=66°. Both parameters are shown in the yz plane: due to the BB-TOFT symmetry, identical results have been obtained in the xz plane

Download Full Size | PDF

3. BB-TOFT optimization

By using ε_esc as a figure of merit we can evaluate the BB-TOFT trapping efficiency in order to optimize its design. We consider three fundamental parameters: the radius (R) of the annulus along which the fibers’ cores are disposed, the fiber MFD and the cutting angle θ. The values of R and MFD affect essentially the conformation of the trap: as shown in Fig. 4(a), 4(b) for a fixed value of θ(θ=68°) the distance between the probe-end and the optical trap becomes larger, while ε_esc decreases by increasing R.

This behaviour can be explained considering that an increase in the radius results in a longer propagation of the beam outside the fiber before reaching the trapping point where all rays are converging. Hence the beam will be larger due to diffraction, this fact reducing the trapping efficiency. Such a trend is confirmed also by the several curves plotted in the same graphs, which report the results obtained for different values of MFD. Indeed for MFD values from 2 to 10 µm, ε_esc is larger for R < 50µm, since the propagation distance outside the fiber is short, showing a maximum value of 4.2 fJ/W for the case R=35 µm and MFD=8 µm. By further increasing the MFD (12 < MFD < 16 µm) we observe that the slope of the curves reported in Fig. 4(a) strongly decreases; this can be explained by considering that diffraction becomes less relevant as the beams have a larger MFD and a longer propagation doesn’t sensibly modify the trapping efficiency.

On the other hand, in these cases the maximum attainable value of ε_esc is lowered by increasing the MFD; this happens because the optical intensity is distributed on a larger region, and the beam’s intensity gradient is thus lower. The third fundamental parameter is the cutting angle θ that affects both the BB-TOFT’s NA and its trapping distance. Figure 4(c) and 4d show that, considering a fixed value of R (R=55 µm) ε_esc decreases when the angle θ increases, while the trapping distance increases almost linearly. Also in this case it is possible to explain the behaviour by geometrical consideration. Indeed an increase of θ corresponds to a decrease of the NA, thus thrusting the rays to propagate with a smaller angle φ with respect to the beam axis and increasing the scattering component of the forces. As in the previous case, the analysis has been performed for different values of MFD: as expected due to a lower impact of diffraction we find that an increase of MFD results in a higher trapping efficiency.

 

Fig. 4. Escape energy and trapping distance calculated in function of bundle radius R (a) and (b) respectively, and in function of the cutting angle θ (c) and (d). In all cases the analysis of the escape energy variation is evaluated for different values of the MFD.

Download Full Size | PDF

A further confirmation in this sense is given by the fact that ε_esc gets higher for MFD increasing from 2 to 10 µm, while a further increase of the size doesn’t result in any improvement (diffraction is no more limiting the trapping performance), and ε_esc is instead reduced because of the reduction in the beam intensity gradient. It is worth underlining that in the performances’ evaluation, reported in [10], the effect of diffraction and the actual behaviour of the beams emitted by the fibers at a distance lower than z_R from the fiber-end were not properly taken into account. Such a fact yielded to correct results in the case of OT realized with the annular-core fiber, but it caused an overestimation of the escape energy for the 4-fibers bundle structure. Indeed, at the same conditions (R=55 µm, MFD=6 µm and cut angle θ=70°) which yielded to an escape energy estimation of 1.75 fJ/W, a much lower value (0.31 fJ/W) is now obtained, thus confirming the need for an accurate description of the beam propagation features as they have a relevant impact on the trap formation.

In summary, the analysis clearly indicates that the optimization of the BB-TOFT performances needs proper design of the structure and an accurate choice of the fiber composing the bundle. In particular, a decrease of the cutting angle θ and of the bundle dimension R results in a more efficient optical trap, while the trapping position gets closer to the fiber end-facet. The experimental verification of the trapping distance, carried out as shown in [8] (Supplementary Information) for few different BB-TOFT structure, confirm the numerical simulation results. Moreover also the optimal value of fiber MFD, as a function of the trapping distance and of radius R, must be identified as a trade off between the impact of diffraction and the reduction of the intensity gradient produced using a larger MFD.

4. Conclusion

We performed a detailed numerical analysis of the trapping efficiency of a recently proposed fiber OT based on total internal reflection. The performance has been evaluated by assessing the minimum energy, per unit power of the optical beam, necessary for a particle to escape the trap ε_esc. Our study shows that in order to optimize the performance of the proposed fiber-optic tweezers the fiber MFD should range from 8 to 10 µm. Moreover we find out that once the beam size is chosen a further improvement is obtained by keeping the radius of the annulus and the cut angle as small as possible. It is worth noticing that the critical value of the angle for total reflection is θ_c=66.5°; anyway, if a metallic coating is deposited on the cut surfaces, the angle can be lowered down, increasing the value of ε_esc, while avoiding total internal reflection at the second interface between the fiber and the outer medium.