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Abstract
Confinement in fiber traps with two optical fibers facing one another relies on balancing the optical forces originating from the interaction of a scattering micro-object with the light beams delivered through the fibers. Here we demonstrate a novel type of dual fiber trap that involves the use of nanobore fibers, having a nano-channel located in the center of their fiber cores. This nano-element leads to a profound redistribution of the optical intensity and to considerably higher field gradients, yielding a trapping potential with greatly improved tuning properties compared to standard step-index fiber types. We evaluate the trap performance as a function of the fiber separation and find substantially higher stiffness for the nanobore fiber trap, especially in the range of short inter-fiber separations, while intermediate distances exhibit axial stiffness below that of the standard fiber. The results are in agreement with theoretical predictions and reveal that the exploitation of nanobore fibers allows for combinations of transverse and axial stiffness that cannot be accessed with common step-index fibers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
After the first successful dual beam geometry realization in 1970 [1], optical traps gained substantial attention during recent times after the noble prize in physics was awarded [2]. Applications in various fields have been demonstrated, including the transfer to optical fibers [3] and trapping of micron-size biological specimen such as cells and bacteria [4–6], individual proteins [7,8] with Angstrom resolution [9], or single cooled atoms [10,11] and qubits [12]. Moreover, by for instance measuring Brownian kinetics and exerted sub-pN forces [8,13], information about viscoelastic properties of cells such as lipid membrane mechanics [5,14] has been revealed, or molecular motors and cell dynamics on the nanoscale [7,8,13] have been studied. Recent advances even showed in vivo optical trapping and optical manipulation inside living organelles [15].
In contrast to microscopic approaches that demand high NA-objectives, traps that rely on two optical fibers with facing ends – so-called dual fiber traps – provide a flexible, straightforward-to-implement and integrated platform for trapping and micro-manipulation of small objects [3,5,14]. Application of dual fiber traps include for instance rotating and stretching cells [5,14], nano-position detection and force measurements [16], as well as Raman micro-spectroscopy [17]. One restriction of current single-mode fiber traps is the Gaussian-type beam profile, which limits performance of dual fiber traps in terms of stiffness, confinement and tunability. For instance, changing the distance between the two fibers imposes both transverse and axial stiffness to monotonically drop, which for applications that demand precise and independent control of both parameters is undesired.
Here we introduce a novel dual fiber trap that relies on two nanobore optical fibers (NBFs), revealing greatly improved tuning capabilities compared to step-index single-mode fibers (SMFs). The key feature of the NBF is a central nano-channel incorporated into the fiber core, leading to a focus-type beam shape at the output [18], which represents a simple and cost-efficient way of shaping the fiber output compared to alternative means, in particular since the use of additional beam shaping elements before the fiber inputs is prevented [19–21]. Axially overlapping the two foci results in a dual fiber trap arrangement (Fig. 1(a)) that yields significantly different trap properties compared to SMFs, which is demonstrated here in the experiment on the example of silica microspheres. We evaluate trap performance by recording the displacement of a bead inside the trap, allowing us to determine optical forces and stiffness as a function of fiber separation, being supported by numerical simulations.
Fig. 1. The dual nanobore fiber optical trap. (a) Sketch of the trap consisting of two oppositely facing nanobore fibers. The colored distribution between the fibers resembles the intensity profile that results from overlapping both focus spots, leading to a maximum in the center of the trap. The gray object in the center is a 2 µm silica bead (fiber outer diameter not to scale). (b) Scanning electron micrograph (SEM) image of the cross-section of the used fiber (dark grey: silica, light grey: GeO$_2$-doped silica, black: air). (c) Intensity profile of the guided mode inside the nanobore fiber. (d) Selection of simulated circular cross-sections of the intensity profiles at different distances from the fiber end face ($z=3$ µm: center minimum increased to unity, $z=7$ µm: focus spot, $z=16$ µm: diverging Gaussian beam).
2. Nanobore fiber
The nanobore fiber consists of a GeO$_2$ doped core (5.3 mol%, $\Delta n=8\cdot 10^{-3}$, diameter 3 µm) and a silica cladding (diameter 180 µm). A circular nano-hole (nanobore, 0.7 µm diameter) located inside the core runs along the entire length of the fiber (Fig. 1(b)). This particular shape of the core region yields a donut-shape guided fundamental ring mode in the fiber (Fig. 1(c)), which at the output focuses at approximately 7 µm distance to the fiber end face and then diverges comparable to a Gaussian beam (Fig. 1(d)). The overall behavior can be qualitatively explained by assuming the overlap of the intensity profiles of two parallel propagating fundamental Gaussian beams that are transversely offset roughly by the diameter of the nano-channel (for details see [18]). Note that throughout this work the nano-channel is assumed to be filled with water, which due to the low refractive index imposes the field inside the channel to be evanescent thus overall enabling the focusing effect [18].
Dual fiber traps consist of two optical fibers facing each other that are aligned along their $z$-axes, separated by the inter-fiber distance $d$ (Fig. 1(a)) [3,5,14]. In the case of low-NA SMFs, the working principle mainly relies on compensation of scattering forces $F_{\textrm {scatt},\;z}(r=0,\;z)$ along $z$-direction [1] and balancing of gradient force $F_{\textrm {grad},\;r}(r,\;z_0)$ in radial direction. Trapping becomes possible where forces and intensities are equal, thus creating a stable equilibrium in the center of the trap at $(0,\;z_0=d/2)$.
To exemplify the difference between both fibers, Figs. 2(a) and 2(b) show simulation results comparing the intensities and their differences at $d=20$ µm in water at a vacuum wavelength of 635 nm. A clear increase of the intensity as well as a significantly different distribution at the trapping location is observed in the NBF case (Fig. 2(a)), suggesting an improved trap performance and an enhanced capability for tuning the trap properties via the fiber separation $d$.
Fig. 2. Comparison of simulated intensity profiles as well as their differences inside the dual fiber trap, emerging from the overlap of output beams at a separation of $d=20$ µm ((a) NBF, (b) SMF, $\lambda =635\,\mathrm{nm}$, medium between the fibers: water). Optical trapping is possible where the intensity difference vanishes (blue), i.e., forces are balanced. The insets show the evolution of intensity distributions along the radial direction of the left fiber at different locations inside the trap (indicated by the arrows).
3. Enhanced fiber trap performance and tuneability
3.1 Optical trapping, simulation issues
In this work, the used silica microbeads (Bangs Laboratories, Inc.) have mean diameters of $2R=2$ µm. At this stage, it is important to consider the variation of diameter of the bead ensemble used in the experiment. By analyzing microscopic images of trapped beads via image processing and by conducting an SEM analysis, we found that the standard deviation with respect to bead diameter is $\sigma =0.04$µm. Since the size is in the order of the used wavelength, simulation of the trap properties requires full numerical vector modeling. Specifically, an integration over the Maxwell stress tensor (MST) (known as generalized Lorentz-Mie scattering theory) is needed to fully characterize the trap behavior [13,22], where the optical force $\boldsymbol {F}_{\textrm {trap}}=\oint _S(\langle \hat {\boldsymbol {T}}_{\textrm {M}}\rangle \cdot \boldsymbol {n})\,\mathrm {d}S$ is calculated over a surface $S$ with normal $\boldsymbol {n}$ and averaged MST $\langle \hat {\boldsymbol {T}}_{\textrm {M}}\rangle$. Here we conduct finite element (FEM) simulations and calculate the trap parameters as a function of light power and fiber separation $d$.
Firstly, we performed FEM simulations in order to calculate electromagnetic field distributions on the output facets of both NB as well as SM fibers. The electromagnetic field was further propagated in water (refractive index $n_{\textrm {w}}=1.3316$) by means of the Fresnel diffraction integral [23] so that the fields could be evaluated on the surface of the silica sphere (refractive index $n_{\textrm {sph}}=1.457$). The optical force acting on such a particle was computed by the Barton’s formalism [24]. We performed force calculations for the sphere placed at various distances from the fiber output facet as well as from the optical axis. Since the beams coming from each of the two fibers in our counter-propagating geometry are not coherent, we obtained the dependence of the forces on fiber separation by summing the individual contributions of each beam.
Within the scope of this work, trap properties are quantified by trap stiffness $\boldsymbol {\kappa }(d,2R)$, which is calculated for small on-axis displacements $\boldsymbol {x}_d=(r,\;z)-\boldsymbol {x}_0$ around the equilibrium $\boldsymbol {x}_0=(0,\;z_0)$ through $\boldsymbol {F}_{\textrm {trap}}(\boldsymbol {x}_d,2R)=\boldsymbol {\kappa }(d,2R)\circ \boldsymbol {x}_d$ ($\circ$ denotes the elementwise Hadamard product [25]). Perpendicular to the fiber at $z_0=d/2$, the transverse trap stiffness is described by $\kappa _\perp$, whereas along $z$ at $r=0$ the axial stiffness is given by $\kappa _{||}$.
3.2 Tuning potential
For a better understanding of the trapping force $\boldsymbol {F}_{\textrm {trap}}(r,\;z)$, MST integration over a sphere of 2 µm diameter is carried out in 3D for various inter-fiber distances. Optical forces in all three directions are considered at this point, especially including axial gradient force $F_{\textrm {grad},\;z}(0,\;z)$ instead of the aforementioned heuristic pure scattering force $F_{\textrm {scatt},\;z}(0,\;z)$ only approach.
To reveal the difference between both types of fiber traps more clearly, we simulate $\kappa _\perp$ and $\kappa _{||}$ as a function of fiber separation (7.5 µm < d < 50 µm) and bead diameter (1.98 µm < 2R < 2.07 µm) (Figs. 3(a)–(d)), while the choice of the $y$-axis scale is justified by the small standard deviation of the bead diameter ($\sigma =0.04$ µm). Note that $R$ is a highly sensitive parameter for the stability of optical traps, which has been exploited in the past for optical sorting via optical conveyor belting [26,27].
Fig. 3. Simulated optical trap stiffness $\boldsymbol {\kappa }(d,2R)$ at the equilibrium point $\boldsymbol {x}_0=(0,\;d/2)$ as a function of fiber separation $d$ and particle diameter $2R$ ((a) NBF, $\kappa _\perp$; (b) SMF, $\kappa _\perp$; (c) NBF, $\kappa _{||}$; (d) SMF, $\kappa _{||}$). The color scale represents actual stiffness values given in pN/µm$\cdot$mW$^{-1}$.
Two features that solely arise from the unique intensity profile of the NBF geometry can be identified: First, higher stiffness as well as a maximum in transverse stiffness (Fig. 3(a)) are found in the NBF case, which is different to the monotonic increase of $\kappa _\perp$ for the SMF (Fig. 3(b)). Moreover, NBF separations between 10 µm and 20 µm lead to completely unstable axial stiffness, with even smaller values of $d$ imposing a fast increase of $\kappa _{||}$ (Fig. 3(c)). Both features are not present for the SMF, showing an almost constant value of $\kappa _{||}$ across the entire parameter domain (Fig. 3(d)).
4. Experimental setup and measurement procedure
The setup for trapping and analyzing microbeads in water consists of a light source, two fibers, a liquid chamber and a camera-based microscopic setup (Fig. 4(a)). Light from a continuous wave (CW) laser (output power $P_{\textrm {max}}=100\,\mathrm {mW}$, $\lambda =635\,\mathrm {nm}$) is split via a beam splitter and the resulting beams are individually coupled into two identical fibers via objectives (Olympus 20$\times$, 0.4 NA). The output power in each fiber is at max $P_{\textrm {out}}=30\,\mathrm {mW}$ and is correlated to the input power by a calibration measurement using an optical power meter prior to the trapping experiment. The optical fibers used throughout this work are a NBF with 3 µm core and 0.7 µm bore diameter (details see [18,28] and Fig. 1(b)) and a commercial Thorlabs SMF of comparable core size and numerical aperture ($\mathrm {NA}=0.13$, SM450) [29]. Both fibers used are single-mode at $\lambda =635\,\mathrm {nm}$ (single-mode cut-offs $\leq 600\,\mathrm {nm}$ [18,29,30]) and have lengths of 50 cm. The sidewise open liquid chamber is formed by two parallel glass slides (170 µm thick microscope coverslips) that are spaced $\sim 2\,\mathrm {mm}$ apart.
Fig. 4. Setup and experimental procedure used to analyze the properties of the dual fiber trap. (a) Sketch of the experimental setup consisting of a trapping laser, two fiber samples, liquid chamber and an imaging section with Köhler illumination including a fast camera. (b) Representative microscope image example of a 2 µm silica bead trapped with two NBFs at $d=30$ µm. Clearly visible is the water-filled bore (see Visualization 1 and Visualization 2 for sample videos). (c) Example of the tracked particle displacement ($d=10$ µm) plotted as a 2D histogram. The color scale is the bead’s density of occurrence at one specific spatial point over the entire trajectory length. Curves are projections of the data points on their respective axes, showing the corresponding one-dimensional distributions.
To create an aqueous sample solution, a stock solution including 2 µm silica beads is 2500-fold diluted, yielding a final particle number density of $4.7\cdot 10^6\,\mathrm {mL}^{-1}$. The fibers are sidewise introduced into the chamber before the solution ($\sim 150$ µL) is applied to avoid drift. Both fibers are aligned by focusing the imaging objective onto their water-filled bores from the side. Once a particle reaches the vicinity of the equilibrium position $x_0$ by free diffusion, it is optically trapped (example in Fig. 4(b)) if both fibers are well aligned, which is provided in the experiment by three-axis positioning stages with nm-precision. Via optimizing the fibers’ position, oscillations of the bead resulting from transverse misalignment and/or differences in tilt angles have been minimized. Its trajectory is time-resolved recorded using a high-speed CCD camera (Basler pilot piA640-210gm) and a home-built microscope (Olympus 40$\times$ dry objective, 0.65 NA + 250 mm tube lens for imaging) reaching a total magnification of $\approx 55\times$. The achieved frame rate and data acquisition time are 1000 fps and 30 s respectively, allowing for a sufficiently large number of frames required for statistical analysis. Illumination is provided via a multi-mode fiber (MMF) coupled LED (wavelength 617 nm) in standard Koehler geometry, while an additional notch filter is used to block undesired scattered trapping light to be recorded by the camera.
For investigation of the trap properties, all measurements have been taken with the same particle for a range of fiber separations $d$ from 50 µm to 10 µm. For every inter-fiber distance, various power levels are established by introducing variable optical density (OD) filters into the laser beamline before the beam splitter, which allows adjusting the output power within the range $P_{\textrm {out}}=30\,\mathrm {mW}\ldots 3\,\mathrm {mW}$ for each fiber. The input power is controlled for each measurement via an optical power meter. Special care was taken to ensure that the same and only one microsphere is kept trapped over the entire recording time. If a particle is lost, a second one is trapped, or a fiber has to be recoupled, the procedure is restarted.
After recording, videos are post-processed for tracking the motion of the bead. First, the static background is subtracted as the minimum value per pixel across all frames, followed by rescaling the remaining intensity back to full dynamic range, thus maximizing contrast. Second, each frame is spatially shifted using the Fourier shift theorem [31,32]
Figure 4(c) shows an example of a recorded 2D bead displacement histogram of one representative data set (NBF, $d=10$ µm, $P_{\textrm {out}}=30\,\mathrm {mW}$). Optical trapping is clearly visible from the Gaussian-shaped distribution, which is in-line with the probability $\rho _{\textrm {trap}}(\boldsymbol {x}_d(t))$ of finding a bead at position $\boldsymbol {x}_d(t)$, resulting from the underlying Boltzmann distribution that includes the thermal fluctuation energy $k_{\textrm {B}}T$ [13,33]
5. Direct comparison of fiber trap performance
The trap parameters have been extracted by the Ornstein-Uhlenbeck theory of Brownian motion via comparing experimental trajectories (Eqs. (1a) and (1b)) to an analytical model. The model results from the Langevin equation for a damped bead in a harmonic potential [34–36]. Dropping inertia terms for frequencies $f$ below the MHz regime and solving the differential equation via Fourier transformation leads to a power spectrum (PS) of the trajectory (example shown in Fig. 5(a)) [34–36]
Fig. 5. Evaluation and fitting of measured and post-processed data, shown for the example of $d=10$ µm. (a) Power spectrum $|\tilde {X}_{d,\perp }(f)|^2$ in transverse direction calculated from a tracked trajectory $x_{d,\perp }(t)$ for the NBF. Visible is a $1/f^2$-falloff at high frequencies as a result of free diffusion and a plateau for low frequencies due to bead confinement in the optical trap. Their respective tangents’ intersection yields the corner frequency $f_{\textrm {c}}$. The magenta curve refers to a close to Lorentzian shape maximum likelihood estimate (MLE, for details see main text). (b) Optical trap stiffness $\boldsymbol {\kappa }_{\textrm {trap}}(P_{\textrm {out}})$ as a function of output power $P_{\textrm {out}}$ ($d=10$ µm) (cyan: NBF, magenta: SMF; solid (dashed) lines: $\kappa _\perp$ ($\kappa _{||}$). Normalized power stiffness $\langle \boldsymbol {\kappa }_{\textrm {trap}}/P_{\textrm {out}}\rangle$ is obtained by linear fitting the data points (see Dataset 1 [39] for underlying data).
with $\tilde {X}_{d,\;i}(f)=\mathcal {FT}\{x_{d,\;i}(t)\}$. The two free parameters are the diffusion coefficient $D=k_{\textrm {B}}T/\gamma$ ($\gamma =6\pi \eta R$ friction coefficient, $\eta$: viscosity, $R$: particle radius) and the corner frequency $f_{\textrm {c}}=\kappa _i/(2\pi \gamma )$ [34–36]. Compared to common evaluation techniques such as, e.g. mean-square displacement (MSD), this method benefits from its computational speed by scaling with $N$ operations instead of $(N-1)N/2\sim N^2/2$ as compared to MSD-analysis and is used as the method of choice to recover the trap stiffness.
To reduce noise from the experimentally obtained PS and to correct for (i) the characteristic instrument response (e.g. motion blur from finite shutter time), and (ii) the finite sampling and limited data acquisition bandwidth (e.g., aliasing and spectral leakage) [33–35], we compress the PS by binning via Welch’s method [35–37] and fit Eq. (3) to the experimental data by minimizing the residuals $\chi _i^2$ from a maximum likelihood estimate (MLE) [34–36]
After obtaining the two fit parameters $D$ and $f_{\textrm {c}}$, the optical trap stiffness $\kappa _i=2\pi k_{\textrm {B}}Tf_{\textrm {c}}/D$ is determined for different output powers, showing a linear dependence (Fig. 5(b)) [33,38]. Linear regression is then used to obtain the power-normalized trap stiffness $\langle \boldsymbol {\kappa }_{\textrm {trap}}/P_{\textrm {out}}\rangle$, representing the parameter that allows to compare the performance of both fiber trap types (as well as reducing the influence of individual measurement errors on the overall error).
In addition to the discussed method, the power normalized trap stiffness is calculated by the EP theorem of Eq. (2) and the results of both methods for the two fiber types are presented aside in Figs. 6(a) and 6(b).
Fig. 6. Comparison of the power-normalized stiffness as a function of fiber separation for both types of fiber traps (cyan: NBF, magenta: SMF). The two left-handed columns refer to the two data processing methods ((a) power spectrum method, (b) equipartition theorem, while (c) shows the corresponding simulations). The top row refers to transverse stiffness $\kappa _\perp$, the bottom to axial stiffness $\kappa _{||}$. Error bars in (a) and (b) indicate tolerances obtained by the fits (see Dataset 1 [39]) for underlying data).
Both data evaluation methods are in excellent agreement (Figs. 6(a) and 6(b)), underlining the reliability of the obtained results. Originating from a low on-axis NA (focusing power on the optical axis), the axial trap stiffness $\kappa _{||}$ for both traps is about one order of magnitude smaller than $\kappa _\perp$ for all separations. Due to its focusing effect, the NBF shows significantly increased optical trap stiffness (e.g., $\sim 20\%$ in transverse direction) for $d<30$ µm and slightly higher values than the SMF trap at long distances.
Four key features that are qualitatively in-line with the simulations shown in Fig. 6(c) and the discussion within the simulation section can be identified (the bead diameter at which the simulated stiffness/fiber separation dependence was taken has been chosen by minimizing the deviation of the simulation results to the experimental data): (i) enhanced transverse stiffness in the NBF case for $d\leq 20$ µm, (ii) monotonic increase of $\kappa _\perp$ at all separations for the SMF situation, (iii) significantly reduced axial NBF-stiffness around $d\approx 20$ µm, and (iv) nearly constant SMF-$\kappa _{||}$ over the entire span of $d$. Compared to the ideal situation of perfect fiber alignment assumed in simulations (Fig. 6(c)), the measured stiffness values are slightly lower than those obtained from calculations, which we attribute to experimental uncertainties such as transverse misalignment, drift or vibrations of the fibers. Note that potential misalignments are likely to impact trap performance stronger for smaller inter-fiber distances, e.g. at $d=10$ µm compared to $d\geq 30$ µm.
6. Conclusion
Dual fiber traps represent a flexible and straightforward-to-use platform for the implementation of optical traps in optofluidic environments. Here we introduce a novel type of dual fiber trap that relies on two nanobore optical fibers, revealing greatly improved tuning properties compared to typically used step-index fiber-based traps. The axial overlap of two foci results in a dual fiber trap arrangement with significantly different properties compared to single-mode fibers, which was demonstrated here experimentally by trapping silica microspheres in water. We evaluate the trap performance by recording the displacement of micro-beads inside both traps, allowing us to determine optical forces and normalized stiffness as a function of fiber separation, fully supported by numerical simulations. Particular at small distances, substantially higher stiffness values can be achieved in the NBF case, while intermediate separations impose axial stiffness to drop below that of the single-mode fiber, which in all cases shows a simple monotonic behavior.
The results obtained clearly show that the nanobore fiber approach allows to access combinations of transverse and axial stiffness that are not in reach for its single-mode counterpart. This feature together with stiffness enhancement at short fiber distances can be of great importance for trapping objects and will allow for the implementation of traps that can be adjusted to the specific prerequisites of an anticipated application only by adjusting the inter-fiber distance.
Funding
Leibniz Science Campus InfectoOptics (Whole Blood Imaging SAS-2015); European Regional Development Fund (CZ.02.1.01/0.0/0.0/15_003/0000476).
Acknowledgments
The authors gratefully acknowledge Ondrej Stranik for help with fabrication of the sample cell and Ron Fatobene Ando for contribution to Fig. 1(a).
References
1. A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]
2. Nobel Media AB, “Arthur Ashkin – Nobel Lecture,” https://www.nobelprize.org/prizes/physics/2018/ashkin/lecture.
3. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18(21), 1867–1869 (1993). [CrossRef]
4. A. Ashkin and J. M. Dziedzic, “Optical Trapping and Manipulation of Viruses and Bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef]
5. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81(2), 767–784 (2001). [CrossRef]
6. T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6(11), 831–835 (2009). [CrossRef]
7. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365(6448), 721–727 (1993). [CrossRef]
8. M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with Optical Tweezers,” Biophys. J. 72(3), 1335–1346 (1997). [CrossRef]
9. E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by RNA polymerase,” Nature 438(7067), 460–465 (2005). [CrossRef]
10. A. Ashkin and J. P. Gordon, “Cooling and trapping of atoms by resonance radiation pressure,” Opt. Lett. 4(6), 161–163 (1979). [CrossRef]
11. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental Observation of Optically Trapped Atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986). [CrossRef]
12. J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P. A. Jones, G. Messin, A. Browaeys, and P. Grangier, “Two-dimensional transport and transfer of a single atomic qubit in optical tweezers,” Nat. Phys. 3(10), 696–699 (2007). [CrossRef]
13. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [CrossRef]
14. M. E. Solmaz, R. Biswas, S. Sankhagowit, J. R. Thompson, C. A. Mejia, N. Malmstadt, and M. L. Povinelli, “Optical stretching of giant unilamellar vesicles with an integrated dual-beam optical trap,” Biomed. Opt. Express 3(10), 2419–2427 (2012). [CrossRef]
15. P. L. Johansen, F. Fenaroli, L. Evensen, G. Griffiths, and G. Koster, “Optical micromanipulation of nanoparticles and cells inside living zebrafish,” Nat. Commun. 7(1), 10974 (2016). [CrossRef]
16. C. Ti, M.-T. Ho-Thanh, Q. Wen, and Y. Liu, “Objective-lens-free Fiber-based Position Detection with Nanometer Resolution in a Fiber Optical Trapping System,” Sci. Rep. 7(1), 13168 (2017). [CrossRef]
17. P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fibre trap for Raman microspectroscopy of single cells,” Opt. Express 14(12), 5779–5791 (2006). [CrossRef]
18. H. Schneidewind, M. Zeisberger, M. Plidschun, S. Weidlich, and M. A. Schmidt, “Photonic Candle – focusing light using nano-bore optical fibers,” Opt. Express 26(24), 31706–31716 (2018). [CrossRef]
19. A. Kuchmizhak, S. Gurbatov, A. Nepomniaschii, O. Vitrik, and Y. Kulchin, “High-quality fiber microaxicons fabricated by a modified chemical etching method for laser focusing and generation of Bessel-like beams,” Appl. Opt. 53(5), 937–943 (2014). [CrossRef]
20. A. Koshelev, G. Calafiore, C. Piña-Hernandez, F. I. Allen, S. Dhuey, S. Sassolini, E. Wong, P. Lum, K. Munechika, and S. Cabrini, “High refractive index Fresnel lens on a fiber fabricated by nanoimprint lithography for immersion applications,” Opt. Lett. 41(15), 3423–3426 (2016). [CrossRef]
21. M. Kreysing, D. Ott, M. J. Schmidberger, O. Otto, M. Schürmann, E. Martín-Badosa, G. Whyte, and J. Guck, “Dynamic operation of optical fibres beyond the single-mode regime facilitates the orientation of biological cell,” Nat. Commun. 5(1), 5481 (2014). [CrossRef]
22. W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63(6), 715–717 (1993). [CrossRef]
23. J. J. Stamnes, Waves in Focal Regions (Taylor & Francis Group, 1986).
24. J. P. Barton, D. R. Alexander, and S. H. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989). [CrossRef]
25. R. A. Horn, “The Hadamard Product,” in Matrix Theory and Applications, C. R. Johnson, ed. (American Mathematical Society, 1990), pp. 87–170.
26. T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006). [CrossRef]
27. D. B. Ruffner and D. G. Grier, “Optical Conveyors: A Class of Active Tractor Beams,” Phys. Rev. Lett. 109(16), 163903 (2012). [CrossRef]
28. S. Faez, Y. Lahini, S. Weidlich, R. F. Garmann, K. Wondraczek, M. Zeisberger, M. A. Schmidt, M. Orrit, and V. N. Manoharan, “Fast, Label-Free Tracking of Single Viruses and Weakly Scattering Nanoparticles in a Nanofluidic Optical Fiber,” ACS Nano 9(12), 12349–12357 (2015). [CrossRef]
29. Thorlabs Inc., “Single Mode Fiber,” https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=949.
30. K. Schaarschmidt, S. Weidlich, D. Reul, and M. A. Schmidt, “Bending losses and modal properties of nano-bore optical fibers,” Opt. Lett. 43(17), 4192–4195 (2018). [CrossRef]
31. M. Balci and H. Foroosh, “Subpixel estimation of shifts directly in the fourier domain,” IEEE Trans. on Image Process. 15(7), 1965–1972 (2006). [CrossRef]
32. I. T. Leite, S. Turtaev, X. Jiang, M. Šiler, A. Cuschieri, P. S. J. Russell, and T. Čižmár, “Three-dimensional holographic optical manipulation through a high-numerical-aperture soft-glass multimode fibre,” Nat. Photonics 12(1), 33–39 (2018). [CrossRef]
33. W. P. Wong and K. Halvorsen, “The effect of integration time on fluctuation measurements: calibrating and optical trap in the presence of motion blur,” Opt. Express 14(25), 12517–12531 (2006). [CrossRef]
34. B. M. Lansdorp and O. A. Saleh, “Power spectrum and Allan variance methods for calibrating single-molecule video-tracking instruments,” Rev. Sci. Instrum. 83(2), 025115 (2012). [CrossRef]
35. K. Berg-Sórensen and H. Flyvberg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004). [CrossRef]
36. S. F. Nórrelykke and H. Flyvberg, “Power spectrum analysis with least-squares fitting: Amplitude bias and its elimination, with application to optical tweezers and atomic force microscope cantilevers,” Rev. Sci. Instrum. 81(7), 075103 (2010). [CrossRef]
37. P. Welch, “The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms,” IEEE Trans. Audio Electroacoust. 15(2), 70–73 (1967). [CrossRef]
38. A. van der Horst and N. Forde, “Power spectral analysis for optical trap stiffness calibration from high-speed camera position detection with limited bandwidth,” Opt. Express 18(8), 7670–7677 (2010). [CrossRef]
39. M. Plidschun, S. Weidlich, M. Šiler, K. Weber, T. Čižmár, and M. A. Schmidt, “Analyzed trajectories of bead in dual fiber trap and fit results,” figshare (2019). [retrieved 8 October 2019] https://doi.org/10.6084/m9.figshare.11139881.
