We report the measurements of charge density of tapered optical fibers using charged particles confined in a linear Paul trap at ambient pressure. A tapered optical fiber is placed across the trap axis at a right angle, and polystyrene microparticles are trapped along the trap axis. The distance between the equilibrium position of a positively charged particle and the tapered fiber is used to estimate the amount of charge per unit length of the fiber without knowing the amount of charge of the trapped particle. The charge per unit length of a tapered fiber with a diameter of 1.6 μm was measured to be C/m.
© 2016 Optical Society of America
Tapered optical fibers are a versatile tool in light-matter coupling [1–12] as well as in sensing applications [13, 14]. A variety of systems have been investigated for the interaction between evanescent field surrounding tapered fibers  and matter. Coupling between a waveguide and an optical microresonator has been studied for application in cavity quantum electrodynamics . Trapping of neutral atoms around a tapered fiber have been proposed  and demonstrated with laser-cooled cesium atoms [4–6]. Besides cold atoms, recently, solid-state systems such as quantum dots [7, 9, 12] and nanodiamonds containing nitrogen vacancy defect [10, 11, 16] have been investigated for realization of single photon generators. These novel light-matter-coupling systems will offer promising tools for quantum networks .
For effective coupling between a quantum particle and a tapered fiber, it is necessary that the quantum particle near a fiber be precisely positioned. Trapped atomic ions meet this purpose because of their long storage period and good controllability of their position with applied voltage values. A few groups including ours are currently interested in this type of system . One of the predicted issues in such a system is that the trapping potential will be modified if a tapered fiber accumulates charge. However, the charge density of a tapered optical fiber has not been studied so far in detail as far as we know.
The information of the charge density of a tapered optical fiber is also important for optical sensing techniques using tapered fibers, which have been developed in many of fields [13, 14]. Such techniques are based on the sensitivity of evanescent fields to change in therefractive index around tapered fibers. When using tapered fibers as sensors, any effect of fibers on the object to be measured should be as little as possible. Because optical fibers are made of dielectric material, it is possible that a fiber accumulates charge and affects the surrounding electric properties of the target medium. Such information is critically important for other hybrid quantum systems because the charge on the tapered fiber may cause extra decoherence.
In order to meet these demands, here we report charge measurements of a tapered optical fiber integrated in a linear Paul trap. We conducted the experiments at room temperature and ambient pressure using charged polystyrene particles. Experiments with a tapered fiber integrated in a linear Paul trap have been performed by Gregor et al., where they developed a soft-landing technique of preselected microparticles on a tapered fiber surface . Here, we employed an alternative approach based on our findings that there exists an equilibrium position for positively charged particles at which the force produced by trap electrodes and that by fiber charge are balanced. An advantage of this method is that the charge per unit length of a tapered fiber can be derived from the distance between the particle and the fiber without knowing the amount of microparticle charge. To the best of our knowledge, this is the first proposal and demonstration of charge measurements of a tapered optical fiber. This method and our result will provide information not only for a tapered fiber-integrated ion trap but also for many sensing applications and other integrated hybrid quantum systems using tapered optical fibers. Furthermore, our method extends the possibility of application of trapped microparticles or nanoparticles. Recently, trapped nanoparticles have been regarded as a promising device for sensitive force detection [20–22]. Here we show that trapped microparticles can be used for highly sensitive charge measurements.
2.1 Basic theory of charge density measurement using trapped microparticles
A linear Paul trap is a well-established technique for control of charged particle motion. An alternating current (AC) voltage produces radial confinement, whereas direct current (DC) voltages produce axial confinement along the z-axis as shown in Fig. 1(a) [23, 24]. Consider a tapered fiber is set in a horizontal plane and perpendicular to the z-axis. If the tapered fiberaccumulates charge, it can be treated as a line charge with an infinite length when the fiber length is much larger than the distance between the fiber and a trapped particle. If the line charge is assumed to be uniformly distributed and the induced charges by the fiber on the surface of the trap electrodes are neglected, the magnitude of electric field along the z-axis at a distance r from the fiber can be written as,Figure 1(b) shows a schematic diagram of the case of positive charge. It should be noted that, according to Eq. (2), it is not necessary to measure Q for derivation of λ. Thus, λ is obtained by the following equation,
2.2 Numerical calculation of electric field for axial confinement
As explained in the previous section, we need to know Eend to estimate the charge density λ. We derive Eend by using a commercially available solver (Ansys, Maxwell 3D), which is based on the finite element method. Because we need to calculate potential only around the trapping region, we use a simplified model of trap electrode consisting of four parallel rods as shown in Fig. 2 (a). The rod radius and the distance between a diagonal pair of rods correspond to the curvature of the electrode tip and 2R0, respectively. The dimensions of R0, and the length along the z direction of electrodes A, B, and C are set to be 0.7 mm, and 6.4 mm, 3 mm, and 6.4 mm, respectively. The spacing between the DC electrodes is set to be 0.1 mm. The typical mesh size is in the order of ten micrometers. Figure 2(a) shows a result of the calculated static potential on a plane including the z-axis, when voltages of 50 V, 0 V, and 20 V are applied to electrodes A, B, and C, respectively. Figures 2(b) and 2(c) show a calculated static potential and Eend, respectively. A positively charged particle will be trapped at the minimum potential shown in Fig. 2(b). As described in section 4, the calculation is verified by a comparison between the calculated minimum potential and actual particle position.
3. Experimental setup
We used a three-segmented linear Paul trap made of stainless blades as shown in Fig. 1(a). The dimensions of our trap were R0 = 0.7 mm and 2Z0 = 3.2 mm. An AC voltage was applied to a pair of non-segmented electrodes with a length of 16 mm. The three segmented DC electrodes of lengths 6.4 mm, 3 mm, and 6.4 mm were aligned with a spacing of 0.1 mm. A function generator and an amplifier (Matsusada Precision, HOPS-1.5B2-L2) were used to supply the AC voltages with various amplitudes (600 V-800 V) and frequencies (240 - 260 Hz). DC voltages up to ± 500 V were applied to the segmented electrodes. The trap was operated at ambient pressure, and placed in a box made of acrylic to prevent air fluctuation. We used polystyrene spheres with a diameter of 1 μm (Polyscience, Polybead Polystyrene Dry Form). The particles were charged up by being rubbed against a container which had a valve and a tube. They were brought to the trapping region from the top of the trap through the tube. A helium-neon laser at 633 nm was directed along the z-axis to observe the trapped particles via the scattered light. We recorded images of the trapped particles and the fiber using a commercially available camera, and measured the distance between them by using these images. The resolution of these images was 0.03 mm, which was restricted by the number of pixels of 1.8 × 107.
The tapered fibers were fabricated using standard single-mode optical fibers by simultaneously heating and pulling them by using a computer-controlled system equipped with a ceramic heater and motorized stage . The fiber was glued on a mount, and set across the trapping region perpendicular to the trap axis. This mount was placed on a linear translation stage, and hence it can be moved away from the center of the trapping region while loading the charged particles. The diameter of the tapered fiber was estimated using a scanning electron microscope image.
4. Results and analysis
4.1 Trapping charged particles
We trapped a particle in the linear Paul trap as shown in Fig. 3. The motion of the charged particle in the linear Paul trap obeys the Mathieu equation. The trapping parameter q = 2QVAC/mR02Ω2 characterizes stability of solution of the equation, where VAC is the AC amplitude, m is the mass of particle, and Ω is the frequency of the AC voltage. In ion trapping operated in a vacuum chamber, the condition q < 1 must be satisfied; however, this is largely modified under the ambient pressure, where the Mathieu equation includes a damping term [25, 26]. Using the experimental results performed with the linear Paul trap in  as a reference, a q value up to 3 is allowed for particle trapping. Taking this point into consideration, the upper bound of a charge-to-mass ratio Q/m in our experiments is estimated to be in the order of 10−3 C/kg.
We found that the microparticles were positively charged since the particle experienced repulsive force when applying a positive voltage to the static electrodes labeled A. After trapping, we moved the tapered optical fiber towards the trapped particle by moving the linear translation stage and obtained the value of 𝜆 by measuring the distance between the particle and the tapered fiber.
4.2 Electric field estimation
As described in section 2, the Eend(z) was estimated using numerical calculation. To verify validity of the calculation, we compared the calculated positions of the static minimum potential along the z-axis with the actual positions of the trapped particle. We applied several values of voltages to electrodes A, B, and C and measured the particle position (Table 1). The maximum difference between the measured and calculated values was found to be 0.11 mm. The added resolution of the image limited the precision of the measurements to the difference between the calculated and observed positions of the minimum potential, which was 0.03 mm. From these measurements, the difference between the calculated and actual position was estimated to be 0.14 mm. This difference δz will be taken into consideration when Eend(z) is derived, i. e., we consider that the Eend should range from Eend (z + δz) to Eend (z-δz) .
4.3 Line charge density
We measured 𝜆 using Eq. (3) for a tapered fiber with a diameter of 1.6 μm. To check whether our measurement method depends on the trap potential, we measured 𝜆 by changing two parameters, i. e., the AC and DC voltages. The AC frequency Ω/2π was fixed at 250 Hz. Figure 4(a) shows the obtained values when amplitude of the AC voltage was changed from 600 V to 800 V while the voltages of electrodes A, B, and C were set to be 15 V, 0 V, and 30 V, respectively. The typical distance between the charged particle and the tapered fiber was a few hundred micrometers. The averaged value of 𝜆 was C/m. The error in the measurements mainly is from the uncertainty of the Eend and particle position. The error in the Eend was estimated according to the results described in subsection 4.2. The total error was estimated to be C/m and C/m. From these results, we do not observe an obvious dependence of 𝜆 on the AC voltage.
Figure 4(b) shows the results when the voltage of electrode C was changed from 20 V to 40 V at the AC voltage of 800 V. The averaged value of 𝜆 was C/m. We estimated the error in the same manner as Fig. 4(a). Under the current precision of the measurements, we do not observe an obvious dependence of 𝜆 on the DC voltage. From the results shown in Figs. 4(a) and 4(b), we conclude that the averaged value of the line charge density of the tapered optical fiber we used was C/m. We did not observe time variation in the obtained value. We measured a few different tapered fibers of similar diameter and found that the results were in order-of-magnitude agreement with the above-mentioned value.
In order to improve the precision of the measurements, it is necessary to reduce the error in the Eend. To reduce the difference between the calculated and actual trap potential, improving machining accuracy of trap electrodes would be necessary. In addition, improving the resolution of the image is required for measurements that are more precise.
Another possible error may arise from the simplified model when Eq. (1) is derived. The induced charge on the electrode surface is not taken into consideration. Although this effect is considered to be sufficiently small because the induced charge will distribute only around the tip of the electrodes and not surround the z axis, it may cause an error in the obtained values.
For probing surface charge distribution, atomic force microscopes  provides atomic scale spatial resolution and high charge sensitivity [28, 29]. Using atomic force microscopes, measurements of localized charge distribution such as DNA on the order of 10−2 C/m2  or measurements of relative charge distribution  have been performed. On the other hand, the spatial resolution of our method is restricted by the dimension of particles and by the distance between the particle and the object to be measured. However, it can be applied for highly sensitive evaluation of line charge distribution on the order of 10−11 C/m for a large region, typically a millimeter-scale region.
We have studied the charge of the tapered optical fibers by trapping microparticles in a linear Paul trap. Our method enables to measure the line charge density without knowing the number of trapped particle charge. However, the magnitude of the electric field in a trapping region is necessary, which is estimated using numerical calculations. We performed measurements for several different trap conditions and found that the trap potential does not affect the measurements. We examined the tapered fiber with a diameter of 1.6 μm and obtained its charge density as C/m.
In this study, the typical distance between the charged particle and the tapered fiber was a few hundred micrometers. To implement a fiber-integrated ion trap, the distance must be in the range of sub-micrometers. As a result, the reduction of the surface charge density is important. It can be expected that the line charge density of the tapered fiber decreases as the fiber diameter (1.6 μm in this study) decreases. One could also try to use special coating, for example conductive polymer, on the tapered fiber. Such efforts to control the line charge density of the tapered fiber should be useful for many other applications of the tapered fibers, including sensing and hybrid quantum systems. For example, the coupling of nano fiber Bragg cavity  with trapped ions will realize the perfect coupling of photons from trapped ions into optical fiber networks and thus will be one of the most ideal interfaces for the distributed ion trap quantum information system .
The coupling of nano diamond particles with such surface-charge-free tapered optical fibers will also be useful for nano-scale sensing and mapping of single fundamental charges . The demonstrated method to quantitatively measure the charge density of a tapered optical fiber using trapped microparticles will be a fundamental technology for these novel photonic applications using tapered optical fibers.
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