Dynamics analysis of microsphere in a dual-beam fiber-optic trap with transverse offset

Xinlin Chen; Guangzong Xiao; Hui Luo; Wei Xiong; Kaiyong Yang

doi:10.1364/OE.24.007575

1. Introduction

Since first proposed by Arthur Ashkin in 1970 [1], Optical manipulation has been widely used in the fields of biology, fundamental physics and engineering [2–5 ]. Single-beam optical trap, which is commonly dubbed “optical tweezers”, has led to a vast number of applications owing to its easy realization and operation [6–8 ]. However, the required sharply focused beam and relatively short working distance may limit the applications of optical tweezers [9]. These restrictions could be overcome by dual-beam optical trap [10,11 ]. Additionally, compared with optical tweezers, dual-beam optical trap also has distinctive functions, such as optical stretcher [12], optical spanner [13,14 ] and optical binding [15].

The first dual-beam fiber-optic trap was reported by Constable et al. who utilized capillary tubes to hold and position two opposing optical fibers [16]. They pointed that the alignment of the two counterpropagating beams is critical for good trap operation. They classified beam misalignment into two possible types: positional misalignment (i.e., transverse offset) and rotational misalignment. Since then, dual-beam fiber-optic trap with rotational misalignment has been employed for many special applications. K. Taguchi et al. used plural optical fibers with rotational misalignment to levitate a microsphere against its gravity [17]. J. T. Blakely et al. showed that rotational misaligned fibers integrated with microfluidics would result in circulatory trajectories [18]. Relatively, dual-beam fiber-optic trap with transverse offset has been rarely explored until being proposed for spinning objects by B. J. Black in 2012 [14]. This spin technique is considered to be widely applicable because it is no longer limited by highly controlled beam profile or samples with special optical properties [19, 20 ]. Recently we demonstrated the orbital rotation of a trapped particle by a dual-fiber optical trap with transverse offset, which is another important motion type for the trapped particle apart from spin. The orbital rotation rate could be controlled by simultaneously varying the power of two counterpropagating beams [21].

In the dual-beam fiber-optic trap with transverse offset, the motion state of the trapped microparticle alters with the offset distance. Its motion types include capture, spiral motion, orbital rotation and escape. Dual-beam fiber-optic trap with transverse offset has offered a new optical manipulation technique with abundant and unique operation functions. This technique could be exploited in various fields like biology, physics and materials science [13,14,22 ].

In this paper, we explicitly identify all possible motion types of the microsphere confined in the dual-beam fiber-optic trap with transverse offset. The transformation process and mechanism of the four motion types is analyzed based on ray optics approximation. Dynamical simulations show that the existence of critical offset distances at which different motion types transform. The result is an important step toward explaining physical phenomena in a dual-beam fiber-optic trap with transverse offset. The theory presented will become a topic of general interest to many applications, such as microfluidic mixing, driven machines, biology imaging and cell manipulation [17,21 ].

2. Fundamentals

Figure 1 shows the schematic of a dual-beam fiber-optic trap with transverse offset. The laser beam emitted from the fiber has a Gaussian profile. The origin of coordinate system is set at the center of optical trap. In the following numerical simulation, we choose water (refractive index n ₁ = 1.33, viscosity coefficient η = 0.839 Pa·s) as surrounding medium and polystyrene microsphere (refractive index n ₂ = 1.59, radius r ₀ = 4.5 μm) as the captured particle. The other parameters are as follows: beam waist separation distance S = 150 μm, the light power from each fiber P ₁ = P ₂ = 100 mW, waist radius ω₀ = 3 μm, wavelength of the trapping laser λ ₀ = 1064 nm. The polarization directions of two trapping lasers are orthogonal to avoid the generation of coherent interference.

Fig. 1 Schematic of dual-beam fiber-optic trap with transverse offset. d: offset distance. S: beam waist separation distance.

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The size of the trapped microsphere is much greater than the wavelength of the optical fields such that the ray optics approximation can be applied [23]. In ray optics approximation, the light beam is decomposed into individual rays. The momentum of each ray changes when reflecting and refracting on the surface of the microsphere. The trapping force on the microsphere is equal to the momentum change of each ray per unit time. The total force can be calculated by summing the forces of all rays.

The total trapping force in the two-fiber configurations is the sum of the contributions of two single-mode optical fibers. Firstly, we calculate the trapping force generated by fiber 1. The force can be divided into the axial component F_z ₁ and the transverse component F_x ₁, which are given by [23]:

\begin{array}{l} F_{z 1} (z_{1}, d_{1}) = \frac{n_{1} P_{1}}{c} \times \frac{2 r_{0}^{2}}{π} \int_{0}^{π} d φ \int_{0}^{θ_{\max}} d φ \sin 2 θ \frac{\exp (- 2 r^{2} / ω^{2})}{ω^{2} R_{c}} \\ \times {q_{s} R_{z} + \frac{q_{g}}{\tan γ} \times [R_{z} - \frac{R_{c} (R_{z} + r_{0} \cos θ)}{a \cos γ}]}, \end{array}

\begin{array}{l} F_{x 1} (z_{1}, d_{1}) = \frac{n_{1} P_{1}}{c} \times \frac{2 r_{0}^{2}}{π} \int_{0}^{π} d φ \int_{0}^{θ_{\max}} d φ \sin 2 θ \frac{\exp (- 2 r^{2} / ω^{2})}{ω^{2} R_{c}} \\ \times {q_{s} (r_{0} \sin θ \cos φ - d_{1}) + \frac{q_{g}}{\tan γ} \times [r_{0} \sin θ \cos φ - d_{1} (1 - \frac{R_{c}}{a \cos γ})]}, \end{array}

where (z ₁, d ₁) denote the coordinates of the microsphere relative to fiber 1, θ and φ are the azimuthal angles. q_s and q_g are related to the fractions of the momentum transferred to the microsphere by the incident rays. They are given by [23, 24 ]:

\begin{array}{l} q_{s} = 1 + R \cos 2 α_{i} - T^{2} \frac{\cos (2 α_{i} - 2 α_{r}) + R \cos 2 α_{i}}{1 + R^{2} + 2 R \cos 2 α_{r}}, \\ q_{g} = - R \sin 2 α_{i} + T^{2} \frac{\sin (2 α_{i} - 2 α_{r}) + R \sin 2 α_{i}}{1 + R^{2} + 2 R \cos 2 α_{r}}, \end{array}

where α_i and α_r are the angles of incidence and refraction, R and T are the reflectance and transmittance at the surface of the microsphere. The parameters in Eqs. (1)-(3) are given by:

\begin{array}{l} R = \frac{1}{2} [\frac{\sin {(α_{i} - α_{r})}^{2}}{\sin {(α_{i} + α_{r})}^{2}} + \frac{\tan {(α_{i} - α_{r})}^{2}}{\tan {(α_{i} + α_{r})}^{2}}], T = 1 - R, \\ R_{c} = (z_{1} - r \cos θ) [1 + (^{\frac{π n_{1} ω_{0}^{2}}{λ_{0}}) 2}], R_{z} = (^{R_{c}^{2} - r^{2}) 1 / 2}, \\ α_{i} = \frac{1}{2 r_{0} R_{c}} {[^{d_{1}^{2} + (^{R_{z} + r_{0} \cos θ) 2}] 2} - r_{0}^{2} - R_{c}^{2}}, \sin α_{r} = \frac{n_{1}}{n_{2}} \sin α_{i}, \\ r^{2} = d_{1}^{2} + (^{r_{0} \sin θ) 2} - 2 d_{1} r_{0} \sin θ \cos φ, \\ a^{2} = d_{1}^{2} + (^{r_{0} \cos θ + R_{z}) 2}, θ_{\max} = \cos^{- 1} (\frac{r_{0}}{z_{1}}), \\ ω^{2} = ω_{0}^{2} [1 + (^{\frac{z_{p} λ_{0}}{π n_{1} ω_{0}^{2}}) 2}], γ = \sin^{- 1} (\frac{r_{0} \sin θ}{R_{c}}) . \end{array}

The axial component F_z ₂ and the transverse component F_x ₂ of the trapping force generated by fiber 2 can be calculated similarly. Thus, the total force can be given by:

\begin{array}{l} F_{z} = F_{z 1} (z_{1}, d_{1}) + F_{z 2} (z_{2}, d_{2}), \\ F_{x} = F_{x 1} (z_{1}, d_{1}) + F_{x 2} (z_{2}, d_{2}), \end{array}

in which (z ₂, d ₂) denote the coordinates of the microsphere relative to fiber 2.

When moving in the surrounding medium, the microsphere is also affected by the viscosity resistance F _v [25]:

F_{v} = - 6 π r_{0} v η,

where v is the velocity of the microsphere.

The dynamic trajectory of the trapped microsphere can be modeled by the Newton’s equation:

m \ddot{r} (t) = F_{x} (r) \hat{x} + F_{z} (r) \hat{z} + F_{v} (t),

where r, m represent the position and mass of the microsphere respectively. Gravity and buoyancy are ignored in the simulation.

The program designed to simulate the dynamic behavior of the microsphere is based on calculating its position according to Eq. (7) after a small increment of time. The next velocity and position of the microsphere is derived from its final velocity and position over the previous time interval Δt:

\begin{array}{l} v (t + Δ t) = v (t) + \ddot{r} (t) Δ t, \\ r (t + Δ t) = r (t) + v (t) Δ t . \end{array}

3. Numerical results

In this section, we simulate the dynamic trajectory of the microsphere in the dual-beam fiber optic trap with tansverse offset. The trajectory changes with offset distance, which is normalized as D = d/ω₀ on the basis of ω ₀. The motion type of the microsphere is transformed into another once the offset distance reaches a critical value. The transformation mechanism between each motion type is theoretically explained by the simulation.

3.1 Capture

Figure 2(a) shows the simulation result of the trapping forces exerted on the microsphere when the offset distance is D = 0. The colors and directions of the arrows respectively represent the magnitudes and directions of trapping forces. The red solid curve denotes the trajectory of microsphere calculated according to Eq. (7). When the counterpropagating beams are perfectly collinear, all trapping forces point to the origin of coordinate system. That will lead the microsphere to be fixedly captured in the center point between the fibers, as shown in Fig. 2(b).

Fig. 2 (a) Simulation results of the trapping forces exerted on the microsphere when D = 0. The colors and directions of arrows respectively represent the magnitudes and directions of trapping forces. The red solid curve denotes the dynamic trajectory of the microsphere. (b) Schematic showing the motion type of the microsphere when two fibers are perfectly collinear.

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3.2 Spiral motion

The force balance mentioned above is disrupted when introducing a small transverse offset between the two beams. Figure 3(a) shows the dynamic x-z trajectory of the microsphere for given offset distance of 2.58. Different from the situation of perfectly collinear, the trapping forces exerted on the microsphere are of spiral distribution when two fibers are slightly misaligned. As a result, the microsphere spirals inwards until reaching the trap center. In the ray optics approximation, the force from each refracted or reflected ray always acts through the center of the sphere. No torques are acted on the sphere [23]. At last, the sphere will be fixedly captured in the trap center, as shown in Fig. 3(b). When the captured particle is not a perfect sphere, the torques exerted on it will be nonzero and it will spin in the trap center. Such motion type has been experimentally observed by B. J. Black and S. K. Mohanty [14].

Fig. 3 (a)The dynamic trajectory of microsphere when D = 2.58. (b) Schematic of the spiral motion of the microsphere.

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3.3 Orbital rotation

The motion type of the microsphere changes when the offset distance is above a critical value D ₁. This can be explained by Fig. 4 , where axial trapping force F _z versus axial position z and transverse trapping force F _x versus transverse position x are calculated for varying D in the vicinity of trap center. The slope of F _z and F _x both decrease for increased D. Therefore the axial trap stiffness K _z and transverse trap stiffness K _x also decrease. We define the value of D in the condition of K _z = 0 and K _x = 0 as D _z and D _x severally. D _z and D _x are respectively calculated to be 4.5 [Fig. 4(a)] and 2.6 [Fig. 4(b)]. K _z and K _x are both positive in the condition that D<D _x. As a result, the center point is a stable equilibrium position. The microsphere will spiral inwards until reaching the trap center. When D >D _x, K _x turns to negative, which leads to an unstable trapping in transverse direction. As a result, the microsphere will escape from the trap center. D ₁ is verified equal to D _x.

Fig. 4 (a) Axial trapping force F _z versus axial position (b) Transverse trapping force F _x versus transverse position for varying D.

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For example, when the value of D is increased to 2.8 (D ₁ is calculated to be 2.6, D>D ₁), the dynamic trajectory of the microsphere is shown in Fig. 5(a) . The microsphere rotates along an approximate elliptic orbit in the $x z$ plane. The schematic of this motion type is shown in Fig. 5(b).

Fig. 5 (a) The dynamic trajectory of the microsphere when D = 2.8. (b) Schematic of the orbital rotation of the microsphere.

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Figures 6(a) and 6(b) show orbital rotation perimeter and rate of the microsphere versus transverse offset D for varying microsphere radius r ₀. The triangles represent the critical values D ₁. The squares represent the critical values D ₂ between dynamic behaviors of orbital rotation and escaping. The simulation results show an increasing of the orbital rotation perimeter from a minimum when D = D ₁ to a maximum when D = D ₂. The orbital rotation rate decrease from a maximum when D = D ₁ to a minimum when D = D ₂. The critical values D ₁ and D ₂ both increase with r ₀.

Fig. 6 (a) Orbital rotation perimeter and (b) orbital rotation rate versus transverse offset for varying microsphere radius r ₀. The triangles represent the critical values D ₁. The squares represent the critical values D ₂

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3.4 Escape

For larger fiber offset, the amplitude of orbital rotation increased. Once the dynamic trajectory is out of the feasible scope of optical trap, the microsphere will escape. This situation can be explained in Fig. 7 , where the transition of the motion type of the microsphere from orbital rotation to escape is analyzed with the increasing of D. The trajectories are illustrated as red solid curves. The stable and unstable transverse equilibrium points are donated as black solid lines with solid circles and hollow triangles, respectively. The transverse equilibrium points divide the trapping region into four intervals, with the symbol “+” and “-” indicating the direction of F _x in each interval. The microsphere moves along the transverse direction ( + x) in the “+” interval and the reversed direction (–x) in the “-” interval. Figures 7(a) and 7(b) show the motion trajectories of the microsphere when D = 3.0 and 3.194. The microsphere passes the stable transverse equilibrium points and moves alternately in “+” and “-” intervals in the center region of the optical trap. The orbital rotation amplitude increases with D until reaching the critical value D = D ₂ ( $D_{2}$ is simulated to be 3.194 in this example). When D>D ₂, as shown in Fig. 7(c), the microsphere passes the unstable equilibrium points and escapes.

Fig. 7 The transition of the motion type from orbital rotation to escape with the increasing of D. (a) D = 3.0 (b) D = 3.194 (c) D = 3.2. Red solid lines: motion trajectories of the microsphere. Black solid lines with solid circles: stable transverse equilibrium points. Black solid lines with hollow triangles: unstable transverse equilibrium points. The symbol “+” and “-” denote the direction of the transverse force F _x.

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Figure 8(a) shows the dynamic trajectory of the microsphere when D is increased to 3.2. The microsphere moves along a parallel trajectory and escapes from the trapping region. At last, it will be propelled to the opposite fiber surface, as shown in Fig. 8(b).

Fig. 8 (a) The dynamic trajectory of the microsphere when D = 3.2. (b) The microsphere escapes when D>D₂.

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4. Conclusion

The motion state of the microsphere captured in dual-beam fiber-optic trap with transverse offset has been systematically analyzed in this paper. As the offset distance D increases, the motion type of the microsphere starts with capture, then spiral inwards, then orbital rotation, and ends with escape. The two critical offset distances separating the motion types of spiral motion, orbital rotation and escape are defined as D ₁ and D ₂. When D = 0, the microsphere is fixedly captured in the center of optical trap. When 0<D<D ₁, the microsphere spirals into the trap center. When D ₁<D<D ₂, transverse trap stiffness turns to negative. The microsphere is broken away from the center position and begins to rotate along an approximate elliptic orbit. The orbital rotation amplitude increases from a minimum value when D = D ₁ to a maximum value when D = D ₂. When D>D ₂, the microsphere is pushed out of the trapping region and escapes.

In this paper we use transverse offset in dual-beam fiber-optic trap to control the motion trajectories and motion types of the trapped microsphere. We have successfully explained the physical principle of the transitions of these motion types theoretically. The results offered an exciting opportunity to bring about a deeper understanding of controlled motions of microspheres in dual-beam fiber-optic trap.

Acknowledgment

The project was supported by the Open Research Fund (SKLST201507) of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences.

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Dynamics analysis of microsphere in a dual-beam fiber-optic trap with transverse offset

Abstract

1. Introduction

2. Fundamentals

3. Numerical results

3.1 Capture

3.2 Spiral motion

3.3 Orbital rotation

3.4 Escape

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (8)

Equations (8)

Optics Express