1. Introduction

A dual-beam optical trap using the radiation pressure of counter-propagating laser beams was first demonstrated by Ashkin in 1970 [1]. Since then this experimental configuration has become a powerful tool in fields as diverse as biological, biomedical sciences and physics [2]. In 1993, Constable reports on a dual-beam optical fiber trapping setup, which uses simple, inexpensive laboratory materials. In comparison to other trapping schemes, the dual-beam fiber-optic trap can be applied in optical stretcher [3, 4], optical rotator [5, 6] and optical binding [7]. Additionally, it’s quite convenient to integrate fiber-optic traps on low cost chips [8].

In many biological applications, the macromolecules are often biochemically linked to the microscopic dielectric beads, which serve as handles [9]. These beads are trapped by the optical forces and can be used to manipulate the molecules of interest. In this case, the action of the biological particles will be motored from the motion of the beads. Therefore, the sensitive detection of the bead position constitutes a key feature of quantitative optical trap. Lateral displacements of trapped particles can be measured with video-based position detection in most optical trap setup. The performance of this approach is restricted to video acquisition rates. Furthermore, it is not well suited to measure the relative position of an object with respect to the trap center [10]. Another convenient way of position detection uses a quadrant photodiode (QPD) placed in the back focal plane (BFP) of a condenser [11–13]. Interference pattern of the forward-scattered (FS) light and unscattered light impinging on the QPD is then converted to a normalized differential voltage, which is sensitive to relative position of the trapped particle [14, 15]. It’s convenient to utilize a single laser for both trapping and position detection, so this approach is generally used in many optical tweezers (OT) systems.

In a dual-beam fiber-optic trap system, however, the back-focal-plane position detection has been rarely explored. The light beam is introduced directly into the sample from the side by two single-mode optical fibers. Thus the forward-scattered light or the back-scattered (BS) light can only be collected by the fiber end instead of a condenser. Little of the scattered light can be coupled into the fiber core, since the NA of the fiber is small and the fiber separation is typically large. In consequence, FS light and BS light are inapplicable to the position detection in the dual-beam fiber-optic trap. Relatively, the side-scattered light hasn’t been fully utilized. In this paper we present a detection structure based on back-focal-plane position detection method using side-scattered light. This approach could develop a rapid and sensitive measurement in the dual-beam fiber-optic trap system.

In this paper we use side-scattered light as a signal for back-focal-plane position detection. The response of position sensitive detector is simulated by tracking scattering rays. Some parameters such as sphere radii, fiber separations and numerical apertures have influence on the sensitivity and linear region. Optimized parameters should be chosen to achieve high detection accuracy. Finally, we demonstrate how power instability brings in influences to the detection system.

2. Principle and experiment

We use ray optics to track the scattering of the polystyrene beads (refractive index n₂ = 1.59) trapped in water using two counter-propagating beams. The incident laser beam is decomposed into infinite optic rays. Consider first the scattered light due to a single ray of power P₀ hitting a dielectric sphere at an angle of incidence θ [Fig. 1]. The single ray is dispersed into infinite scattered rays containing reflected ray of power RP₀ and other refracted rays of power T²P₀, RT²P₀, R²T²P₀. The quantities R and T respectively represent Fresnel reflection and transmission coefficients. Some of these scattered rays are collected by the objective. The numerical aperture of this objective evaluates which scattered rays are available. Rays passing through the objective form scattered pattern in the back focal plane of the objective. This pattern is then imaged by a lens on the QPD. The intensity of the pattern is distributed across the four quadrants numbered 1 to 4. The axial displacement along the beam propagation direction is the focus of this study, and it can be obtained from the intensity signal by the following equation:

 

Fig. 1 Geometry for calculating the light power due to the scattering of a single incident ray by a microsphere.

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In the dual-beam fiber-optic trap, the microsphere is held at an equilibrium position. When the laser powers of two beams are equal, the bead is held at the center (point O) of the fiber interval as shown in Fig. 1. Then the scattered patterns from two beams are symmetrical in the back focal plane of the objective, i.e., I₊ = I_-. When the bead deviates from the trap center and moves to a new position (0, 0, z₀), there is a corresponding response from the QPD depended on z₀.

The correct response dependent on microsphere displacement is achieved by calculating the intensity signal summation caused by all incident rays, as shown in Fig. 2. Particle radius is 5 μm, fiber separation is 120 μm and NA is 0.54. The simulated results are denoted as red dots, and the blue solid curve is the fitting line. The results show that a linear range was located near the trap center. The normalized intensity difference I_signal decreases as z₀ increases.

 

Fig. 2 I_signal versus the axial offset of the particle.

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The schematic of the dual-beam fiber-optic trap is shown in Fig. 3, where S is the distance between two fiber ends. The microsphere is trapped by two counter-propagating beams emanating from single-mode fibers (Corning HI 1060, mode field diameter 5.9 ± 0.3 µm at 980 nm). An objective (Sigma, EPLE-50) is then placed perpendicular to the propagating beam to collect the scattered light. Following this, a quadrant photodiode detector (Thorlabs, PDQ-80A) is assembled in a plane conjugated to back focal plane of the objective. The QPD then collects side-scattered light from the optically trapped microsphere rather than the forward-scattered or back-scattered light.

 

Fig. 3 Schematic of the position detection system of dual-beam fiber-optic trap.

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A polystyrene microsphere (radius r = 5 μm)was suspended by the light force in water (refractive index n1 = 1.33). The laser power emitted from each fiber was P1 = P2 = 100 mW, and S = 169 µm. Both CCD and QPD were utilized to measure axial displacement [Fig. 4(a)], and the two curves matched well. Figure 4(b) shows experimental data from QPD and simulated results based on back-focal-plane displacement detection. The experimental and simulated results are denoted as circles and red solid curves, respectively. The blue line is the linear fitting curves of the experimental results. The simulated results and the measurements have the same tendency near the trap center. The difference of slope between simulations and measurements mainly includes the approximations of our simulation on signals, alignment of two fibers and deviation of equilibrium position induced by laser power. In the simulation, the trap center O is supposed to locate on the optical axis. However, we can hardly ensure no offset from axis in the experiments. Actually, the laser power from each fiber end will be affected by preparation processes of the dual-beam trap, such as peeling the fiber cladding. Those factors will deteriorate the ideal conditions of our previous simulation.

 

Fig. 4 (a) Displacement of the particle as a function of time detected by QPD and CCD. (b) I_signal versus the axial offset.

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3. Discussion

We have demonstrated that the position of the bead in a dual-beam fiber-optic trap can be determined using the side-scattered light by both experiment and simulation. To improve measurement sensitivity, the influences of sphere radii, fiber distances, and numerical aperture are discussed in this section. Laser power instability influence is also analyzed to optimize precision.

Figure 5(a) shows normalized intensity difference I_signal as a function of z₀ for varying microsphere radius r. Light power P emitted from each fiber is 100 mW, the beam waist radius ω₀ = 3µm, S = 100 µm and NA = 0.54. In each case of Fig. 1, I_signal reveals a linear response region in the vicinity of z₀ = 0. This I_signal decreases with z₀ until reaching a limit at which I_signal slope becomes 0. The slopes of r = 2 µm and r = 4 µm curves are negative, while the slope of r = 6 µm curve is positive. This means that a critical radius (r_c) with a fixed value between 4 µm and 6 µm will minimize sensitivity. Smaller r leads to higher sensitivity when r< r_c. Conversely, sensitivity increases with r when r> r_c. Figure 5(b) shows the slope of the curve for variable r. The slope near the trap center is expressed as follows:

 

Fig. 5 (a) I_signal versus the axial offset of the particle for varying r. (b) The slope of the curve versus r.

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The influence of fiber separation S is shown in Fig. 6(a), where r = 2 μm and NA = 0.54. The I_signal decreases with z₀ for different S, and slope increases as S decreases until S = 100 μm. When S decreases to 60 μm, the curve is much close to that of S = 140 μm. Figure 6(b) shows the slope of the curve for variable S. We can see that the maximum absolute values of the slope are located at 100 μm and 105 μm. Thus it would be better for S to be about 100 μm in this case.

 

Fig. 6 (a) I_signal versus the axial offset of the particle for varying S. (b) The slope of the curve versus S.

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Figure 7(a) shows the normalized intensity difference I_signal as a function of z₀ for varying NA, where r = 2 μm, S = 100 μm. The I_signal decreases versus z₀ when NA > 0.44. When NA < 0.44, slope becomes irregular and I_signal increases versus z₀ in the vicinity of z₀ = 0, such as when NA = 0.34 in Fig. 7(a). There is also a critical value of NA between 0.34 and 0.44 which leads to a null response. And the critical value is located at 0.355, approximately, as shown in Fig. 7(b). We can see that a broad linear response range can be achieved when the NA is larger than 0.44.

 

Fig. 7 (a) I_signal versus particle axial offset for varying NA. (b) The slope of the curve versus NA.

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Thus the small r and S may be benefit to the sensitivity detection when NA is fixed. However, a particle cannot be trapped when S decreases to a critical value in some cases. This is explained using a light force calculating model based on ray optics approximation [16].

The axial force that exerted on the trapped particle is calculated in the dual-beam fiber-optic trap. Figure 8 shows Q_axial as a function of a particle’s axial offset, where Q_axial is trapped efficiency and represents axial force [17]. In the case of r = ω₀ [Fig. 8(a)], we can see that when the fiber separation S = 14ω₀, the curve slope is negative. In this case, a particle escaping from the trap center will be dragged back to its initial position, thus stably trapping the particle. When S = 13ω₀, the curve slope becomes zero near the zero point, and the laser can hardly supply enough force to trap a particle. The critical value is S = 13ω₀, which is S_limit. A particle cannot be trapped when the fiber separation is smaller than S_limit, as shown by the S = 12ω₀ curve. The Q_axial as a function of axial offset dependent on bead radius is shown in Fig. 8(b). The r = 2μm curve slope is still negative even when S is small as 5ω₀, while the slopes of the other two curves are positive. This allows S to be as small as possible when r<ω₀. Thus, S_limit is decided based on the relationship between r and ω₀. There is no need to consider the fiber separation critical value only when choosing a particle with r<ω₀.

 

Fig. 8 (a) Q_axial as a function of axial offset of the particle for varying S when r = ω₀. (b) Q_axial as a function of axial offset of the particle for varying r when S = 5ω₀.

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Dual-beam optical traps use two counter-propagating beams to clamp particles, differing from single-beam optical tweezers. Thus, laser powers instability influences displacement detection. The intensity I₊ and I_- are both proportional to laser power. The laser powers can be expressed as P₁ = P₀ + δP₁ and P₂ = P₀ + δP₂, where δP₁ and δP₂ are power noises. When the noises changes, forces acting on the trapped microsphere vary, and thus holding the microsphere at a new equilibrium position. This will also cause microsphere displacement of Δz’. This situation is different from that a trapped sphere deviates from its initial position, thereby disturbing displacement detection. What follows is a discussion of power instability influence.

The axial forces of the two counter-propagating beams can be expressed as follows:

According to Sidick’s research, the trapping efficiency factor is proportional to the offset distance of a microsphere from trap position [17], thus:

The trapping position can be determined using Eq. (5) when the separation S is fixed. When δP₂/P₀≠0, the axial coordinate of the microsphere changes to z = Δz’, and a corresponding intensity shift can be calculated. Figure 9 shows the response δI_signal caused by power change δP₂/P₀ , which ranges from −1% to 1%. Other parameters include r = 2 μm, S = 120 μm, and NA = 0.54. This 1% power fluctuation results in a normalized intensity difference of 7 × 10⁻³. In the same case of Fig. 5, such a response is meant to be achieved by driving the trapped particle 500 nm away from the trap center. Thus, power noise greatly restricts detection precision. The two counter-propagating beams should emit from the same laser, so that δP₁ and δP₂ change simultaneously.

 

Fig. 9 δI_signal caused by single power change.

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4. Conclusion

We have demonstrated back-focal-plane displacement detection using side-scattered light in a dual-beam fiber-optic trap. The effects of main factors on the displacement detection sensitivity and their mechanism have been investigated and summarized in detail, based on the ray-optics model. The simulation showed that the displacement sensitivity can change sign as a function of the particle radius or the NA of objective. And a critical NA or radius will minimize the sensitivity. The simulation shows that a high detection precision and a broad linear range can be achieved with optimized parameters, such as $S\approx 100$ μm, r = 2 μm and$NA\ge 0.44$. The simulative and the experimental results show the same variation trends, although there is some deviation between them. It confirms that the displacements of held bead can be detected by side-scattered light. In addition, the power instability of two trapped laser will degenerate the actual resolution. The two trapped beams had better come from the same laser source.

We propose and demonstrate a novel measurement structure of the optically trapped particles displacements. That can be applied in those particular optical trap systems which the geometrical constraints may prevent access to the forward side or back side of the trap. Thus the technique has considerable significance in an integrated and microfluidic optical trap system. Moreover, this technique supplies a convenient way to detect the axial motion rather than measuring the lateral displacement, which is a major focus of most optical trapping work [10]. Combined with the traditional detection method, it shows the promise of achieving three-dimensional position detection of the particle in optical traps.