1. Introduction

Optical trapping and manipulation techniques [1–3] have piqued significant research interest in the fields of transport and positioning of nano and micro objects [4,5], as well as bioanalysis [6,7] and quantum information [8,9]. The conventional method is to use a far-field focus of single or multiple beams to trap and position particles along the optical axis [10–12]. In recent years, near-filed optical devices have attracted attention as a means of handling smaller particles by overcoming the diffraction limit, thermal issues and transporting tiny particles in flexible trajectories [13]. As a result, waveguides, such as nanosized optical fibers, photonic crystal resonators, nanoapertures [14,15] slot waveguides [16,17], and V groove waveguides [18] have become research targets. Yang analyzed the optical transport and trapping of gold and polystyrene nanoparticles smaller than 100 nm in silicon slot waveguides [16]. After analyzing the relationship between trapping stability and the consequent transport speed as a function of particle size, particle refractive index, and slot waveguide shape, they achieved trapping and transport for objects as small as 10 or 20 nm in diameter.

In recent years, more precise control, such as positioning, of small particles has been required. To trap a nanoobject along the waveguide propagating direction, Tong created a potential well by locally narrowing the slot to create a slit in the center [19]. As a result, they could increase higher intensity spots at the slit and create a significant gradient force, which restrains particles. However, because of the predesigned structure, it could not position particles at arbitrarily assigned places.

For more precise positioning and fine adjustment of nanoparticles, the ability to transport particles backward against the light propagating direction is also required. Wang achieved a waveguide-coupled optical conveyor using graded silver nanorods [20]. By switching the incident wavelengths, particles can be transported bidirectionally following the hot spots produced by resonance. The position of the particle can be restricted on a subwavelength scale. Along with precise control, the use of metals significantly increases the intensity loss and restricts the effective working length. In Schmidt’s work [21], microparticles could be transported over long distances and precisely positioned in a low-loss air-filled hollow-core photonic crystal fiber utilizing a coherent superposition of two co-propagating spatial modes balanced by a backward-propagating fundamental mode. Conversely, the counter-propagating beam requires a light source on both ends of the device, which increases the total size of the system and restricts its application fields.

We have proposed a single-input optical conveyor belt that can transport elliptical particles bidirectionally by adjusting incident wavelengths based on the absorption of Si and the scattering effect of the particles [22]. However, this method is only effective on elliptical particles with a major axis longer than a specific threshold. Besides, the fact that the pulling force is obtained by absorption and scattering significantly restricts the effective working length.

Based on the above discussion on existent optical conveyor belts and to integrate multiple functions, we propose a new concept called “optical burettes” that consists of dielectric bowtie cores [23,24] that can interact with nonluminous space, such as an intracellular atmosphere, by transporting or extracting nanoparticles based on its properties of single-end-source and bidirectional transmission induced by multimode excitation and interference effects. By recurrently changing the position of the incident beam waist, which can be realized by moving an objective lens against the incident surface of the bowtie burette, the position of intensity hot spots, known as an optical trap, can be moved continuously along fiber cores, and more importantly, with the fringe shape of the intensity distribution unchanged. Thus, by continuously moving or discretely switching the beam waist appropriately, the target nanoparticles can be bidirectionally transported along the capillary or positioned at arbitrarily assigned places. The distance between each optical trap can be adjusted by switching incident wavelengths, whereas the magnitude of the trapping force can be modulated by switching the incident angle.

The propagating modes in the bowtie fiber and their interference patterns along the bowtie center are introduced in the first part of this article. Then, we discuss and compare the effects of multiple parameters on the mode interference pattern, such as the position of the beam waist, input wavelength, beam waist width, and incident angle, and investigate how they affect the intensity peak distribution. The materials of bowtie cores and the effective working length are also discussed. The optical force and potential well are calculated to prove that the particles can be attracted into the capillary and transported bidirectionally.

2. Bowtie core capillary structure

The three-dimensional perspective and end face shape of the proposed bowtie core capillary are depicted in Fig. 1. x-polarized light has a high-intensity peak in the center of the core gap [23,24] due to the high-refractive-index contrast bowtie core structure. This structure is suitable for nanoparticle trapping because it produces a smaller, more intense light spot than the conventional rectangular core-slotted waveguide structure. In this study, we consider a structure, in which a bowtie core of Si₃N₄, which is typically used in photonic circuits, with a core diameter d_core and a gap distance d_gap of 30 nm, is enclosed by a thin SiO₂ capillary. The capillary is immersed in water (n_b = 1.33). The target nanoparticles are supposed to be polystyrene spheres (n_p = 1.6) to preserve a strong resemblance with biological particles with a diameter d_p of 20 nm dispersed in capillaries. A continuous, x-polarized beam with a Gaussian profile is launched into the fiber at z = 0 in our simulation. The width of the beam waist is denoted by w₀. The depth distance between the beam waist and the capillary incidence plane is denoted by d. The angle between the z-axis and the incident light is denoted by θ_x on the xz plane and θ_y on the yz plane. The incident wavelength used in the simulation shifts from 0.3 µm to 1.3 µm, and is denoted by λ. The power of the incident light was 1 mW. The parameters and their symbols are listed in Table 1.

 

Fig. 1. Schematic of (a) a three-dimensional view of bowtie core capillary trapping nanoparticles at hot spots. (b) xy cross-section of bowtie cores with a particle at the core center. (c) xy view of bowtie cores in SiO2 capillary. (d) xz view of incident scheme with different depths d along z. (e) xz view of incident scheme with different waist widths w₀. (f) xz (and yz) view of incident scheme with different incident angles to the z axis θ_x (and θ_y). At least one of θ_x and θ_y maintains 0. The components are not on an actual scale.

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Table 1. Symbols and values of the parameters

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3. Optical force

The total optical force applied on a particle is obtained by integrating the Maxwell stress tensor (MST) on an enclosed surface S surrounding it:

4. Simulation and discussion of transportation and positioning of particles

The longitudinal and lateral distributions of guided modes in a bowtie core capillary with d_core = 0.5 µm and d_gap = 0.03 µm at λ = 0.5 µm are illustrated in Fig. 2. The fundamental (LP₀₁) mode demonstrates that light is constrained to the center of the core gap, where particle capture is expected. In contrast, the LP₁₁ mode is weak and has little effect on particle capture at the core center. Both the fundamental and LP₀₂ modes exhibit a comparable confinement effect at the gap center; however, the light phase at the wings is reversed. If the fundamental mode is excited only in the fiber, the |E_x|² distribution along z remains stable, as depicted in Fig. 2(e). The scattering force plays the main role in pushing the particle in the + z direction. When the Gaussian beam with w₀ = 0.5 µm is launched under d = 0 and θ_x = θ_y = 0, high-order modes are also excited with the LP₀₁ mode. The mode interference forms a periodically distributed |E_x|², as illustrated in Fig. 2(d) and (e), because each mode has a unique group velocity. Force F_z along the |E_x|² gradient, shown in Fig. 2(f), is applied to particles trapped at the gap center, and a positive or negative |E_x|² slope causes particle transport in the + z or −z direction, respectively. A continuous transport happens if the periodic |E_x|² distribution shifts.

 

Fig. 2. Normalized E_x real part of (a) fundamental (LP₀₁) mode, (b) LP₁₁ mode, and (c) LP₀₂ mode in the bowtie core capillary with λ = 0.5 µm, w₀ = 0.5 µm, d = 0, θ_x(y) = 0, and d_core = 0.25 µm. (a), (b), and (c) share the same colormap. (d) Normalized |E_x|² distribution on xz plane at y = 0. (e) Normalized |E_x|² distribution along z at x = y = 0, the launch of Gaussian profile and LP₀₁ mode. (f) z component of optical force along z at x = y = 0, the launch of Gaussian profile and LP₀₁ mode.

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Because the beam waist can be treated as the first focal point of the mode interference, we start by thinking about changing the waist depth d of the beam to shift the distribution of |E_x|². The distributions of |E_x|² along the z-axis at λ = 0.75 and 0.5 µm are shown in Fig. 3(a) and (b). While the peak position is slightly altered, the period of interference fringes remains constant with respect to changes in d. Nevertheless, clear interference fringes are maintained at d within approximately 2 µm as |d| grows despite an increase in the optical coupling loss in the capillary. Figure 3(e) and (f) depict the F_z distributions on the particles produced by the |E_x|² gradient at λ = 0.75 and 0.5 µm, respectively. Here we focus on two particular cases of particle movement through d change.

 

Fig. 3. (a), (b) Normalized |E_x|² distribution at x = y = 0 along z axis with d = −5–5 µm at λ = 0.75 and 0.5 µm, respectively. (c) and (d) Normalized |E_x|² distribution at x = y = 0 along z when d = −2, 0, and 2 µm at λ = 0.75 and 0.5 µm, respectively. (e) and (f) z component of optical force F_z at x = y = 0 along z when d = −2, 0, and 2 µm at λ = 0.75 and 0.5 µm, respectively. (g) and (h) Displacement of |E_x|² peaks on z when d = −5–5 µm at λ = 0.75 and 0.5 µm, respectively. w₀ = 0.5 µm, d_core = 0.5 µm, d_gap = 0.03 µm, θ_x(y) = 0.

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We assume that a nanoparticle is trapped at the maximum |E_x|² point at z ≈ 10 µm when d = 2 µm in Fig. 3(c) and (e). When d is shifted to 0, the maximum |E_x|² point moves to z ≈ 10.5 µm in the + z direction, causing the particle to move under the + F_z in the same direction and be stuck at the peak position. Moreover, as d shifted to −2 µm, the fringe peak shifted to z ≈ 11 µm, and the particle was trapped there. The short-range nanoparticle transfer of 1 µm is now realized by shifting d from 2 to −2 µm. To transport the particle further along + z, we need to transmit the particle to the next adjacent peak in the + z direction. The next adjacent |E_x|² peak occurs when d = 2 µm. However, when d is shifted back to 2 µm, the particle is on a negative |E_x|² slope, and thus, −F_z pulls the particle backward to z ≈ 10.5 µm. Under λ = 0.75 µm and d = −2–2 µm, the particle will be constrained in one d period and will not be able to propagate through the entire capillary, as indicated by the arrows in Fig. 3(c). In Fig. 3(e), there exist + F_z bounce points (defined as the reversal point of force direction) at z ≈ 8.5, 11.25, and 13.75 µm, prohibiting the + z transmission of the particle. Likewise, the −z transmission is impeded by −F_z bounce points at z ≈ 9.75 and 12.5 µm.

Figure 3(d) and (f) show a successful example to transport a nanoparticle along the entire waveguide. We assume that a nanoparticle is trapped at z ≈ 7.2 µm for d = 2 µm at λ = 0.5 µm, w₀ = 0.5 µm, θ_x(y) = 0, and d_core = 0.25 µm, d_gap = 0.03 µm. The particle is subsequently transported to the |E_x|² peak positions, where z ≈ 7.6 and 8.2 µm, by + F_z when d shifts to 0 and then to −2 µm. If d changes back to 2 µm because the particle is still on the positive |E_x|² slope of the next adjacent peak, +F_z continues to propel the particle forward, beginning a new cycle, as indicated by the arrows in Fig. 3(d). In contrast, particle −z transmission can be realized by continuous −F_z when d shifts in the order of −2, 0, and 2 µm. In Fig. 3(g) and (f), when d changes within ±5 µm, the z positions of |E_x|² peaks shift within −0.76–0.56 µm at λ = 0.75 µm and within −0.48–0.52 µm at λ = 0.5 µm, respectively. This property indicates that the nanoparticles are moved continuously and smoothly with a d shift and can be trapped at any z position.

We establish a new function, namely, the transfer capacity T_c(d), as the |E_x|² peak shift range within ± d over the|E_x|² peak period to numerically evaluate the transfer continuity. The following condition is required to accomplish continuous transfer: T_c(d) > 0.5. In the successful example, T_c(2) ≈ 0.67, with a shift range of 1 µm and a peak period of 1.5 µm. In comparison, in the unsuccessful case, T_c(2) ≈ 0.4, with a shift range of 1 µm and a peak period of 2.5 µm. However, for the unsuccessful example, if |d| is increased to 5 µm, T_c(5) ≈ 0.528, which meets the continuous transfer condition. Thus, while designing an optical burette, one can increase |d| to obtain a continuous transfer; however, this must be balanced with F_z, because longer |d| contributes to a higher T_c(d) while decreasing F_z.

Modifying the incident wavelength λ is another way to shift |E_x|² distribution. The normalized |E_x|² distribution dependent on λ is shown in Fig. 4(a), with λ = 0.3–1.3 µm, d = 0, w₀ = 0.6 µm, and θ_x(y) = 0. The |E_x|² distribution is mainly divided into three bands. The first band is a “single-mode band” with λ ≈ 1.1–1.3 µm. In this single-mode band, w₀ is large compared with the bowtie core diameter, resulting in only the fundamental mode being coupled from the incident beam. The particles are mainly pushed in the + z direction by the scattering force. The second band is a “few-mode band” with λ ≈ 0.65–1.1 µm. In this band, w₀ becomes smaller than bowtie cores, exciting the high-order modes, LP₁₁ and LP₀₂ modes. The |E_x|² distribution appears periodic and unambiguous with longer λ producing a longer period. The particles will be trapped at the maximal |E_x|² points. The third band is a “multimode band” with λ ≈ 0.3–0.65 µm. In this band, w₀ is small compared with bowtie cores. As more high-order modes are excited in this band, the interference pattern becomes chaotic and erratic.

 

Fig. 4. (a) Normalized |E_x|² field distribution at x = y = 0 along the z-axis at λ = 0.3–1.3 µm, at w₀ = 0.6 µm, θ_x(y) = 0, d = 0, in the bowtie core capillary with d_core = 0.5 µm, d_gap = 0.03 µm. (b), (c) Normalized |E_x|² distribution and F_z distribution at x = y = 0 along z at λ = 0.5, 0.8, and 1.2 µm, respectively. (d) F_z distribution at x = y = 0 along z at λ = 0.7, 0.85 and 1 µm, respectively.

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To compare the transmission force in detail, we choose three typical wavelengths, λ = 0.5, 0.8, and 1.2 µm, as representations for the three bands. The distributions of |E_x|² and F_z along z at x = y = 0 are shown in Fig. 4(b) and (c). The distribution of |E_x|² is relatively smooth at λ = 1.2 µm because the fundamental mode dominates the intensity. The |E_x|² gradient-induced F_z has a smooth but low magnitude of approximately 0.01 pN/W. For λ = 0.8 µm in the “few-mode band,” the periodically changed |E_x|² field becomes more obvious, indicating that almost the same amount of LP₀₂ mode was excited, along with the LP₀₁ mode. The high contrast of |E_x|² also makes a higher optical force. For λ = 0.5 µm in the “multimode band,” the |E_x|² distribution has an irregular shape, making F_z unpredictable.

To transport particles along the entire capillary, continuous force pointing in one direction is required. However, Fig. 4(c) shows that there exist bounce points along the z-axis for both positive and negative F_z, which means that the continuous force cannot be formed by switching wavelengths between λ = 0.5, 0.8, and 1.2 µm.

Because the |E_x|² distribution in the “few-mode band” is reasonably regular and has high contrast, we choose λ = 0.7, 8.5, and 1 µm to investigate the transporting ability and illustrate the F_z distribution in Fig. 4(d). Although several bounce points still appear along z, preventing continuous transfer through the entire capillary, a short-range continuous transfer of −5.5 µm is realized.

The above analysis helps us choose the appropriate λ for different applications. For instance, a longer wavelength should be chosen for moving particles along the waveguide by the even-distributed intensity of the fundamental mode, and “few-mode band” wavelengths should be chosen for trapping and transporting particles in a short range. In addition, the trapping period can be adjusted by choosing different λ.

The |E_x|² distribution can also be changed by modifying the beam waist width w₀. We change w₀ from 0.25 to 1 µm at λ = 0.5 µm, θ_x(y) = 0, d = 0 with bowtie core parameters d_core = 0.25 µm and d_gap = 0.03 µm. The |E_x|² shown in Fig. 5(a) appears as a stable periodic distribution without a noticeable shift along z. Nevertheless, the maximal and minimal |E_x|² positions are inverse on each side of a transitional band of w₀ = 0.65–0.75 µm, which means that the peak position is discretely shifted by half period. To find a continuous |E_x|² shift, we magnify the |E_x|² distribution at w₀ = 0.65–0.75 µm to Fig. 5(b) and mark the boundary of the high-intensity area by solid lines at |E_x|² = 0.5. The |E_x|² peak shifts quickly around the threshold w₀ = 0.68 µm without a clear continuous shifting band. We then choose w₀ = 0.5, 0.68, and 1 µm as representatives from the three bands and show their |E_x|² and F_z distributions in Fig. 5(c) and (d). An antidirectional, regularly distributed F_z is produced by the high contrast, phase-opposite |E_x|² distribution at w₀ = 0.5 and 1 µm. This property can be used to switch the force direction without moving the optical trap. The |E_x|² distribution exhibits stability for w₀ = 0.68 µm, resulting in F_z with an amplitude of 0.005 pN/W, which is a comparatively modest value compared to F_z caused by other w₀. The one-directional F_z is always interrupted by F_z = 0, preventing continuous particle transport between |E_x|² peaks. Although the F_z created by w₀ = 0.68 µm can correct the continuity of the F_z produced by w₀ = 0.5 and 1 µm, it is not reasonable to believe that the low magnitude can aid in particle movement. In conclusion, changing w₀ alone is not an effective way to transport particles, but it still has value in force magnitude and direction adjustments when combined with other transport mechanisms.

 

Fig. 5. (a), (b) Normalized |E_x|² field distribution at x = y = 0 along the z-axis with w₀ = 0.25–1 µm (from 0.5λ to 2λ) at λ = 0.5 µm, θ = 0, d = 0, in bowtie capillary with d_core = 0.25 µm, d_gap = 0.03 µm. The solid lines indicate the normalized |E_x|² = 0.5. (c), (d) Normalized |E_x|² field distribution and F_z distribution at x = y = 0 along z at d = 0.5, 0.68, and 1 µm, respectively.

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Finally, we attempt to alter the distribution of |E_x|² by incident angle θ_x and θ_y. θ_x and θ_y are changed from 0° to 60° individually at λ = 0.5 µm, w₀ = 0.5 µm, d_core = 0.25 µm, and d_gap = 0.03 µm, and the corresponding normalized |E_x|² is displayed in Fig. 6(a) and (b) for θ_x and θ_y, respectively. The fringe form is unaffected by the incident angle in either of the figures, meaning that a change in the incident angle does not affect the interference pattern. The amplitude of |E_x|² reduces while θ_x(y) increases due to the growing coupling loss. The incident light is completely reflected at the incidence plane when θ_x(y) > 46°, causing the intensity of the bowtie core gap to disappear suddenly. The |E_x|² and F_z distribution at θ_x = 0, 25, and 43° are depicted in Fig. 6(c) and (d). Because of the approximately constant period and the peak position of the |E_x|² at the three angles, the direction of F_z along z is steady. Notably, |E_x|² at θ_x = 25° has a higher contrast between its maximal and minimal value, resulting in a greater optical force. Hence, altering the incident angle θ is an excellent way to change the magnitude of the force and contrast but not for particle transport.

 

Fig. 6. (a), (b) Normalized |E_x|² distribution at x = y = 0 along the z-axis at θ_x(y) = 0°–60°. (c), (d) Normalized |E_x|² distribution and F_z distribution at x = y = 0 along z at θ_x = 0, 25 and 43°. λ = 0.5 µm, w₀ = 0.5 µm, d = 0, θ_y = 0.

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In this section, we realized continuous nanoparticle transport by changing the depth of the incident beam waist. Modifying the incident wavelength can also transport nanoparticle in a short range and adjust the period of optical traps. Modifying the beam waist width and incident angle can effectively adjust the magnitude of the optical force.

5. Method of particle capture in the direction of the core center

The next issue is determining how to load particles into the bowtie gap. We considered the case where particles were trapped along the y direction. Figure 7(a–c) present the |E_x|² distribution on xy cross-sections at z = 2.85, 2.46, and 2.13 µm, which are the z positions of |E_x|² peak, half, and bottom values, respectively. The |E_x|² value at x = 0 was extracted from Fig. 7(d). It is obvious that |E_x|² near y = 0 is significantly higher than the surrounding area, which produces a gradient force near the bowtie center pointing to y = 0, acting as a trapping force, as depicted in Fig. 7(e). Although the amplitude of the trapping force F_y changes with different positions, the trapping center formed by the force remains unchanged, meaning that particles can be trapped in the bowtie center at any z position. For a simpler method, the fundamental mode can be used to form uniformly distributed intensity along z, attracting particles at any z position near the bowtie center with constant F_y. The fundamental mode can be produced by adjusting w₀ to a value greater than the bowtie size.

 

Fig. 7. Normalized |E_x|² distribution on the xy cross section at (a) |E_x|² peak (z = 2.85 µm), (b) |E_x|² half (z = 2.46 µm), and (c) |E_x|² bottom (z = 2.13 µm) at λ = 0.5 µm, w₀ = 0.5 µm, and d = 0, θ_x(y) = 0. (d) Normalized |E_x|² and (e) F_y along y axis at x = 0.

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6. Bowtie core material, loss effects and potential well

To explore the wider applicability of bowtie core capillaries, we simulated a high-refractive index (n = 3.03 at λ = 0.75 µm) semiconductor material AlAs, which is also extensively used in optical waveguides [26], to replace Si₃N₄ (n = 2.04 and 2.01 at λ = 0.5 and 0.75 µm, respectively) bowtie cores. The normalized |E_x|² in the bowtie center and F_z along z are depicted in Fig. 8(a) and (b), at λ = 0.75 µm, w₀ = 0.5 µm, θ_x(y) = 0, and d = −1, 0, and 1 µm, in the waveguide with d_core = 0.25 µm and d_gap = 0.03 µm. The F_z produced by the |E_x|² gradient always remains in the + z direction when d is shifted in the order 1, 0, and −1 µm recurrently. In contrast, continuous −F_z can be realized by changing d in the order −1, 0, and 1 µm. The simulation result indicates that continuous bidirectional transport can be achieved with shorter d in bowtie core capillaries with higher-refractive index materials.

 

Fig. 8. (a) Normalized |E_x|² distribution at x = y = 0 along z when d = −1, 0, and 1 µm at λ = 0.75 µm. (b) z component of optical force F_z at x = y = 0 along z when d = −1, 0, and 1 µm at λ = 0.75 µm, w₀ = 0.5 µm, d_core = 0.25 µm, d_gap = 0.03 µm, and θ = 0. (c) Normalized |E_x|² distribution at x = y = 0 along z with Si₃N₄ and AlAs bowtie cores. The parameters of incident light and shape of Si₃N₄ bowtie cores are the same as in Fig. 2(b) at d = 0.

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The propagation loss of bowtie cores is also considered, which decides the effective working length of the optical burette. We compare the normalized |E_x|² distribution at x = y = 0 along z of Si₃N₄ and AlAs bowtie cores with a capillary length of 100 µm, as shown in Fig. 8(c). The dashed lines indicate the peak values. According to Fig. 8(c), neither material exhibits significant intensity attenuation after being transported 100 µm, and the duration of the |E_x|² fringe remains constant, indicating that the optical force also maintains the same distribution. Therefore, it is reasonable to expect that the effective working length of the burette is significantly greater than 100 µm due to the negligible propagation loss of Si₃N₄ and AlAs bowtie cores.

In our proposal, the thermal effect is believed to be negligible. The optical trapping force is produced by the near-field enhancement which has a higher efficiency than using a focusing lens and reduces the energy aggregation and heat generation. Besides, the bowtie core is assigned as all dielectric materials without any metal or plasmonic effect in use, making the heat production by plasmonic absorption negligible [13,19]. In addition, the high thermal conductivity of Si₃N₄ [27] and AlAs [28], hundred times higher than water, also contributes to heat dissipation.

The capture stability of nanoparticles is represented by the potential well U [16,29]. The potential well corresponds to the amount of work necessary to move a particle from a specific position to infinity, which is calculated by integrating the force along the particle escaping direction. A particle can be stably trapped when |U| > 1 k_BT, where k_B is the Boltzmann constant (1.3805 × 10⁻²³ J/K), and T is the absolute temperature in Kelvin scale (T = 293.15 K). When |U| < 1 k_BT, the particle can easily hop away due to the Brownian motion. The potential well of trapping and transporting a nanoparticle with d_p = 20 nm by Si₃N₄ bowtie core capillary is shown in Fig. 9, which shows stable trapping ability in z and y direction. The weakest U_z locates at z ≈ 8.1 µm when d shifts from −2 to 2 µm as circled by dot line in Fig. 9(a). To satisfy the trapping condition |U_z| > 1 k_BT, the power threshold of the incident light can be extrapolated as P_in = 136 mW.

 

Fig. 9. Potential well of trapping a 20 nm particle by Si₃N₄ bowtie cores. (a) The potential well along z at x = y = 0 when d = −2, 0, and 2 µm. (b) The potential well along y at x = 0 and z positions of different |E_x|² magnitude. The other situations are the same as in Fig. 3(d).

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When transport particles forward by changing the waist depth, the depth should be shifted in the order of −|d|, 0, |d| and then back to −|d|. We also investigated the suitable switching time from −|d| to |d| to demonstrate the technical feasibility. We assume that d is shifted by moving the objective lens forward or backward at a constant velocity v_l in the z direction. Due to the ultrasmall size of the particle, the fluid viscosity resistance F_drag and Brownian force F_Br are considered, which can be calculated as [29]:

Here, we use the example when transport a particle forward as described in Fig. 3(d) and (f). We assume that the particle has already been trapped at z = 8.2 µm at d = −2 µm. To transmit the particle forward, the objective lens should be moved 4 µm backward to shift d from −2 to 2 µm. The instantaneous particle position shift Δz_p is shown in Fig. 10(a). When d shifts in a relatively large velocity v_l = −1000 µm/s, the particle shifts slightly in z direction with Δz_p = −0.04 µm. In this case, the particle can be captured by the next forward |E_x|² envelope. The speed threshold is obtained as v_l = −375 µm/s. The speed |v_l| lower than 375 µm/s causes the particle to be pulled back to the previous |E_x|² envelope, preventing the forward transmission. Thus, the faster |v_l| is, the more possible that the continuous transmission can be achieved. The v_l threshold at different P_in is shown as the dash line in Fig. 10(b). The slowest v_l allowing continuous forward transmission is 2.79 mm/s at P_in = 1 W, which is reasonably easy to achieve.

 

Fig. 10. (a) The instantaneous particle position shift Δz_p during d shifts from −2 to 2 µm with incident power P_in = 136 mW and objective lens moving speed v_l = −150, −375 and −1000 µm/s. (b) The particle position shift Δz_p at the moment when d shifts to 2 µm with a dependency on v_l and input power P_in. The initial status is assumed as that the particle is trapped at z = 8.2 µm at d = −2 µm. The other situations are the same as in Fig. 3(d).

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7. Conclusion

We proposed a tunable bowtie core optical burette capable of bidirectionally transporting nanoparticles with incident light from a single end. Because of the intensity of hot spots created by mode interference, nanoparticles are trapped and transported. Polystyrene spheres with diameters of 20 nm can be transported continuously along the + z direction at λ = 0.5 µm by recurrently switching the beam waist depth d in the order of 2, 0 and −2 µm. d shift from −2 back to 2 µm can be realized by moving objective lens backward with a velocity within mm/s level. Similarly, the −z transmission is achieved by switching d in the order of −2, 0 and 2 µm. The particles can be loaded into the fiber center from the y direction. The transporting distance is expected to be greater than hundreds of microns due to the low-loss material of bowtie cores. The transporting and positioning properties such as the magnitude of optical force, direction of particle transfer, and period of optical traps can be controlled by the incident angle θ_x(y), the beam waist width w₀, and incident wavelength λ.