Direct measurement of optical trapping force gradient on polystyrene microspheres using a carbon nanotube mechanical resonator

Optical tweezers based on optical radiation pressure are widely used to manipulate nanoscale to microscale particles. This study demonstrates direct measurement of the optical force gradient distribution acting on a polystyrene (PS) microsphere using a carbon nanotube (CNT) mechanical resonator, where a PS microsphere with 3 μm diameter is welded at the CNT tip using laser heating. With the CNT mechanical resonator with PS microsphere, we measured the distribution of optical force gradient with resolution near the thermal noise limit of 0.02 pN/μm in vacuum, in which condition enables us to high accuracy measurement using the CNT mechanical resonator because of reduced mechanical damping from surrounding fluid. The obtained force gradient and the force gradient distribution agree well with theoretical values calculated using Lorenz–Mie theory.

mechanical resonator to the measurement of optical radiation force, even with the mass sensitivity of the sub-ag range 25 . To elucidate the optical trapping force distribution acting on the microsphere, the position controlled attachment process of the microsphere on the very tip of the CNT cantilever is also necessary. In this study, we demonstrate direct measurements of the spatial distribution of optical radiation force acting on a PS microsphere using a CNT mechanical resonator in vacuum, for which we propose easy attachment process of an individual PS microsphere on the CNT mechanical cantilever.

Fabrication of CNT-PS mechanical resonator
A multiwall CNT examined in this study, synthesized using chemical vapor deposition (CVD), has a highly graphitized structure and a less amorphous carbon layer 28 because of its post-annealing treatment at temperatures higher than 1500 °C. The CNTs were aligned on the edge of the Au-coated Si chip. Figure 1a exhibits a scanning electron microscopy (SEM) image of the as-fabricated CNT resonator used for this study, where the CNT diameter and the length are, respectively, 80 nm and 18 μm. Note that the edge of the Si chip was inversely tapered to eliminate the background signal as schematically illustrated in Fig. 1b.
To investigate the optical force gradient, i.e., the optical spring constant k OPT acting on a PS microsphere, we attached a PS microsphere with diameter of 3 μm on the very tip of the CNT using a micromanipulator (Narishige, MHW-3) of an inverted microscope in air as follows (Fig. 1b). First, PS microspheres are deposited on a glass needle connected to the micromanipulator as schematically shown in Fig. 1c and d. One PS microsphere on the glass needle was mechanically contacted to the very tip of the CNT cantilever using the micromanipulator under the observation of an optical microscope. At this moment, a focused laser with power of 4 mW and wavelength of 633 nm was irradiated on a center of the CNT cantilever for 3-10 s through an objective lens (numerical aperture (NA) = 0.7, 60 × ) to weld the target PS microsphere to the very tip of the CNT cantilever. Under these circumstances, the temperature increase of the CNT induced by the laser irradiation was estimated as 110-210 °C from the thermal contact resistance of 0.15-0.35 K/μW between the CNT and the substrate described in an earlier report 26 , which is higher than the glass transition temperature of PS around 100 °C, and close to the melting point of 240 °C. Consequently, the PS microsphere was welded on the very tip of the CNT cantilever as presented in Fig. 1e with an attachment force larger than 650 pN (see estimation of attachment force and a movie for the welding process in Supplementary Information). Note that the PS microsphere was never attached without using the laser irradiation process.

Measurement setup for CNT mechanical resonator
The CNT-PS resonator fabricated was driven using a 0.1-mm-thick piezo-electric actuator (PbZr x Ti 1-x O 3 ceramic: PZT) with application of AC voltage of 100 mV pk in the vacuum chamber (10 −3 Pa) as shown in Fig. 2a. Note that the shear mode PZT actuator was used, so that the vibration direction of the CNT resonator is mainly perpendicular to the optical axis. The vibration direction before the attachment of PS sphere was confirmed by the SEM observation.
A laser (λ = 660 nm) was focused on the PS microsphere through an objective lens (NA = 0.7, 50×, WD = 2.2-3.0 mm) not only to probe the resonator oscillation but also to exert optical force on the PS microsphere. Scattered light from the PS was collected by the objective lens to a single-mode optical fiber with a mode field diameter of 3.6-5.3 μm, which acts as an aperture, as schematically shown in Fig. 2a. The scattered light intensity through the optical fiber was measured using a photomultiplier with high speed preamplifier. Initially, the scattered light at the equilibrium position of the PS was collected, where the position of the optical fiber entrance was adjusted to the half of the maximum signal of the scattered light along the oscillation direction as schematically illustrated in the inset of Fig. 2a. If the position of the CNT fluctuates, the signal intensity of the scattered light is changed depending on the position of the PS sphere.
To ascertain the temporal variation of resonance frequency shift Δf of the CNT-PS resonator, we used the phase-locked loop (PLL) technique, which induces self-oscillation at its resonance, where a commercially available high frequency lock-in-amplifier with PLL feedback loop (Zurich Instruments, HF2LI) was used. One can estimate the optical spring constant k OPT from the resonance frequency shift Δf of the CNT-PS resonator induced by the optical radiation pressure, which is given approximately as PS0 under the condition of Δf/f PS0 ≪ 1, where f PS0 and k CNT respectively denote the original resonance frequency of the CNT-PS resonator and the spring constant of the CNT cantilever (see Supplementary Information for derivation). Figure 2b portrays resonance curves before and after the attachment of the PS microsphere. A spring constant of the CNT cantilever, k CNT , was estimated as 194 pN/μm from the resonance frequency of 364 kHz before attachment of the PS sphere. The estimated spring constant k CNT is approximately two orders of magnitude smaller than that of the commercially available micro-fabricated cantilever for AFM. Although the resonance frequency after the attachment of the PS sphere was shifted from 364 kHz to 19.76 kHz, the resonance curve maintains the clear harmonic oscillation without marked degradation of the quality factor (Q-factor approx. 1000) and the nonlinear effect. The added mass of the PS sphere was estimated as 12.9 pg from the ratio of the resonance frequencies before (f 0 ) and after (f PS0 ) the attachment of PS sphere using the relation of where M and m respectively represent masses of the PS sphere and the CNT cantilever. The estimated added mass corresponds to the mass of PS sphere with diameter of 2.9 μm, which is close to the average diameter (3 μm) of PS spheres used in this experiment. These indicate that the spring constant of the CNT cantilever was maintained after the attachment of the PS sphere through the welding process in air and the attached PS sphere induces no significant mechanical dumping to the resonance properties.  The inaccurate position of the PS sphere to CNT axis might induce the significant error to the estimated force gradient. Now we consider the edge of PS sphere attaching to the CNT tip as the worst case as shown in Fig. S2. The diameter of the PS sphere and length of the CNT is 3 and 18 μm, respectively. The force acting on the PS sphere can be divided into two parts, perpendicular component f ⊥ or parallel component f // to the CNT axis under the presence of the position-error of the PS sphere. Note that the optical force gradient measured in this experiment is perturbed by f ⊥ . In the case of the focal point being in the center of the PS sphere, the measured f ⊥ would be f ⊥ cosθ, where θ is defined by θ = tan −1 (PS radius/CNT length). Using the diameter of PS sphere and the CNT length, the error is only 0.5%, which is much smaller than our experimental fluctuation. As a result, the effect of the position-error of the PS sphere is also negligible. Figure 3b presents the light intensity dependence of k OPT obtained from Δf, where the error bar was determined from the standard deviation of the five measurements. The optical spring constant k OPT is proportional to the light intensity. It is 0.18 pN/μm with applied laser power of 102 μW. Optical force F opt exerted by photon radiation pressure is described as where n represents the refractive index of the medium, c is the speed of light in vacuum, P stands for the input laser power, and Q opt is the dimensionless factor of the axial trapping efficiency described by electromagnetic theory. Consequently, the optical force and its spring constant are proportional to the light intensity, which is consistent with the experimentally obtained k OPT . The theoretical minimum detectable force gradient ′ F min based on the thermal noise 30 is estimated as 0.01 pN/μm in our experimental conditions using the relation given as

Results and Discussions
where A is the rms amplitude of the oscillation (30 nm), k B signifies the Boltzmann's constant, T stands for the temperature, and B represents the detection bandwidth (2-10 Hz). From the temporal fluctuation of the frequency shift, we estimate the minimum detectable k OPT of our experimental setup as roughly 0.01-0.02 pN/μm, which is close to the theoretical limit. The small spring constant of the CNT-PS resonator (194 pN/μm) enables us to measure the minute force variation, such as the optical gradient force. Further improvement of the sensitivity can be realized by reducing the pressure and lowering the temperature for the improvement of Q-factor and the reduction of thermal noise, respectively. To compare the experimentally obtained results with the theoretical predictions, we performed numerical calculations for the optical force based on the Lorenz-Mie theory using "Optical tweezers computational toolbox" software 31 . Here, we assumed that the incident laser light, from top to bottom shown in the inset of Fig. 3c, is the ideal focused Gaussian beam. Figure 3c and d respectively show the propagation (z) and transverse (x-y) axis position dependences of the F opt , where the PS sphere with 3 μm diameter was set at the focal position of NA = 0.7; 100 μW light intensity with 660 nm wavelength was used for the calculation. As observed in Fig. 3c, the optical force along the z-axis F opt,z is always positive, indicating no restoring force induced along z-axis because of the insufficient NA of the objective lens and shows no polarization dependence. The PS sphere is just pushed to + z direction. The weighted position should be shifted toward the propagation direction of the light. However, the optical force along the x-axis F opt,x changed from positive to negative depending on its position x, indicating that the sufficient optical intensity gradient is induced for the x-y plane, even under NA = 0.7. Additionally, F opt,x shows the polarization dependence of the incident light and the higher F opt,x was obtained at plane-polarized light along the x-axis indicated by [1 0].
From the slope of the position dependence of the optical force (force gradient), the optical spring constant k OPT is obtainable. In the experiment, the polarization of the laser was not conserved. Therefore, we only consider the average force induced under polarizations of [1 0] and [0 1]. The optical spring constant k OPT,x of 0.12 pN/μm for x-axis is much larger than k OPT,z of 0.02 pN/μm for the z-axis. It is consistent with the experimentally obtained results presented in Fig. 3b. As a result, we infer that the measured k OPT derives mainly from the optical force along the x-axis: F opt,x .
As presented schematically in Fig. 4a and b, we measured the position dependence of k OPT along the x-axis, where the relative position of the PS sphere to the focal point was scanned to x-axis as shown in Fig. 4b. Unlike the optical tweezers measurement, the spring constant of the CNT cantilever k CNT = 194 pN/μm is 3-4 orders of magnitude larger than that of k OPT , which results in accurate control of the relative position of the PS sphere. In the case of the optical tweezers measurement using single beam, the PS sphere tended to move to the focal point because of the optical tweezer effect, which might prevent accurate control of the relative position of the PS sphere to the focal point. Figure 4c shows the position dependence of k OPT perpendicular to the CNT axis at laser intensity of 100 μW, where the dot size in Fig. 4b represents the laser beam diameter. The measured k OPT are consistent with the calculated average curve of k OPT,x depicted in Fig. 4c. Measurement at the edge of the PS sphere is difficult because of the insufficient reflected light from the PS sphere for determining the accurate oscillation amplitude. Figure 5 shows the z-position dependence of the k OPT measured by changing the relative focal point to the PS sphere along the z-axis corresponding to the propagation direction of the laser light (100 μW) as illustrated schematically in the inset of Fig. 5, where the focal position for the radial direction was fixed at the PS microsphere center. The measured k OPT increases concomitantly with increasing position; it reaches its maximum value of 0.14 pN/μm at z = ± 1 μm. No abrupt change of k OPT (~0.14 pN/μm) was observed around the center of the PS sphere. Further increase of the z-position (z > 1 μm) decreases k OPT . The z-position dependence of numerically calculated k OPT,x induced along the x-axis for [1 0] and [0 1] polarization and their average are also shown. The spatial profile of the measured k OPT agrees well with the calculated k OPT,x . Therefore, we infer that the CNT-PS mechanical resonator is beneficial for analyzing the spatial profile of the optical force gradient.

Conclusions
We have demonstrated the direct measurement of optical force gradient k OPT acting on a PS microsphere connected to a CNT mechanical resonator in vacuum, in which condition enables high accuracy measurement using the CNT mechanical resonator because of reduced mechanical damping from surrounding fluid. In terms of the resonator fabrication, we attached the individual PS microsphere to the extreme tip of the CNT cantilever using a welding process with laser heating of the CNT. Using the thus-fabricated CNT-PS mechanical resonator, we measured k OPT with resolution near the thermal noise limit of 0.02 pN/μm, where the obtained values agreed well with theoretical results. The spatial profile of the k OPT acting on the PS sphere with the size of 3 μm was also demonstrated. We believe that this direct measurement of the optical force including spatial distribution can contribute to further elucidation of the photon radiation pressure acting on particles of nanometer scale.