Graded-index fiber tip optical tweezers: Numerical simulation and trapping experiment

Yuan Gong; Ai-Yan Ye; Yu Wu; Yun-Jiang Rao; Yao Yao; Song Xiao

doi:10.1364/OE.21.016181

1. Introduction

Optical tweezers has been extensively investigated since it was developed by Ashkin et al. [1] and widely used in both physics and biology [2, 3]. In order to strongly focus a Gaussian beam to form an optical trap, conventional optical tweezers employed high numerical aperture (NA) objective lens, making it bulky and expensive. Optical fiber tweezers (OFT) is one of the most important branches of optical tweezers and has the advantages of small size and low cost. Single beam OFT has been attracting much attention as it is easy to fabricate and manipulate [4–11]. In an optical stretcher [12], two fiber ends are aligned and an optical trap is formed by two counter-propagating divergent beams. This kind of optical tweezers is often used to investigate the elasticity of single living cells [12,13]. Optical tweezer array, which is of great importance to simultaneously trap a large number of microscopic objects, has been demonstrated by using an optical fiber bundle [14–17].

Single beam OFTs are often achieved using tapered fibers, which are fabricated by fusing and drawing [4–7] or chemical etching [8–10]. Most of them are fabricated on common single mode fibers (SMF) [4, 5, 7–11]. The SMF has small fiber core with a diameter of <10μm, leading to several disadvantages of SMF-OFTs. First, the small fiber core limited the diameter of the micro lens formed at the fiber tip, and further limited the trapping distance as the focus length is so small. On the other hand, the micro lens needs a special shape fabricated at the fiber tip to enhance the convergence ability [4] to increase the gradient force. This may make the fabrication process complex. Second, in case there are drawbacks or bad roughness around the fiber tip, the light constrained in the fiber core is easier to be scattered in the SMF than in the multimode fiber (MMF), as the diameter of the MMF core, 62.5μm or even larger, is several times larger than that of the SMF.

In order to reduce the requirements of fabrication process of OFTs and further enhance trapping performance, in this paper a novel optical tweezer based on a graded-index multimode fiber (GIMMF) is proposed. Based on a numerical model, the light distribution is simulated and the trapping force is calculated. The simulated results indicate that compared with the SMF-OFT, the GIMMF-OFT has longer trapping distance and larger gradient force, both of which are key characteristics for optical tweezers. The feasibility of the concept is proved by optically trapping yeast cells using a chemically-etched GIMMF tip experimentally.

2. Numerical simulations

2.1 Numerical model

The structure and index profile of the etched optical fiber taper are schematically shown in Fig. 1(a). The laser beam is launched into a section of single mode fiber and then coupled into the GIMMF. Both geometric and index parameters are given. The refractive index profile of the GIMMF used in our experiment can be expressed as

n (r) = {\begin{matrix} n_{1 m} \sqrt{1 - 2 Δ {(\frac{r}{a_{m}})}^{α}}, r < a_{m}, \\ n_{2 m}, r \geq a_{m} . \end{matrix}

where

n_{1 m}

is the maximum index at

r = 0

,

Δ = (n_{1 m}^{2} - n_{2 m}^{2}) / 2 n_{1 m}^{2}

is the index difference between the core and cladding materials and

a_{m}

is the radius of the fiber core.

α

is a factor that determines the index profile.

Fig. 1 (a) Schematic diagram of the etched GIMMF taper and (b) Optical intensity distribution near the GIMMF fiber taper

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By carefully measuring the geometric parameters of the fiber tip by an optical microscope, the cross-section of the fiber tip can be approximated by a hyperbola function, i.e., $z^{2} / a^{2} - x^{2} / b^{2} = 1$ . $a$ and $b$ are the major and minor semi-axes, respectively. The structure of the GIMMF tip is also shown as lines in Fig. 1(b). The angle between two asymptotes is the cone angle, $θ$ , which is about 32.9° in our experiment. The parameter values used in the numerical simulations are given in Table 1. The optical intensity distribution near the fiber tip is shown in Fig. 1(b) and will be discussed in detail in the next sub-section.

Table 1. Values of parameters used in the numerical simulations

View Table

2.2 Optical field distribution

The optical intensity distribution emerging from the GIMMF taper is calculated by the beam propagation method [18] and shown in Fig. 1(b). The length of GIMMF is selected to be $L_{m} = 446 μm$ so that the light beam emerging from the fiber tip is highly focused to obtain the maximum power density. The 1/e width of the light beam at the focus is about 1μm. The shape factor, $a$ , is fixed at 22.5μm. The normalized optical power along the propagation direction, z, is calculated by an integral of the light intensity within −0.5μm<x<0.5μm in the center of the fiber core and shown in Fig. 2(a1). The normalized maximum power is 0.891 and located at z_max = 461.7μm. The light intensity distribution along the transverse direction at z_max is shown in Fig. 2(b1). The working distance, $d_{w}$ , is defined as the spacing between the fiber tip and the focal point with maximum intensity. Here $d_{w}$ is about 5.4μm.

Fig. 2 Propagation properties along z axis and light intensity distribution along x axis at z = z_max.

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Similar to Figs. 2(a1) and 2(b1), the normalized optical power along the propagation direction and the light intensity distribution along the transverse direction are given by Figs. 2(a2) and 2(b2) for $L_{m} = 928 μm$ , and by Figs. 2(a3) and 2(b3) for $L_{m} = 1411 μm$ , respectively. As well-known, the light propagates, i.e., diverges and converges, periodically in the GIMMF, as shown in Fig. 2(a3), and follow sinusoidal paths if the index profile of the fiber core is parabolic. As the GIMMF length increases from 446μm to 928μm and 1411μm, the beam propagation properties near the fiber tip changes slightly, except for the period number increases. Both normalized maximum power in z direction, p_max, and the maximum intensity in x direction, I_max, decrease slowly due to the propagation loss. The working distance, $d_{w}$ , remains almost constant around 5.4μm.

In order to compare the focusing performance of the fiber tapers, we calculate the light intensity distribution of different types of fiber tapers. The light intensity distribution at z_max for GIMMF, SMF tapers are given by Figs. 3(a) and 3(b), respectively. We also calculate the light intensity distribution for the step-index multimode fiber (SIMMF) taper and the distribution indicates multimode style (simulated results not shown). Thus, the SIMMF taper is not proper to be used as optical tweezers. The peak of the light intensity using the GIMMF taper is ~0.88, while using the SMF taper it is ~0.58. The SMF taper has the same shape with that of the GIMMF taper. For SMF, the light intensity distribution near the fiber tip is not influenced by its length, but only influenced by the shape of it. The NA values for the SMF is 0.14. The simulated results indicate that both the GIMMF tapers and SMF tapers can be used as optical tweezers. However, the SMF taper has a lower maximum light intensity and is less convergent than that of the GIMMF. Moreover, the trapping performance of SMF optical tweezers is strongly influenced in the experiment, by the asymmetry and roughness of the SMF taper in the fiber core region. As the dimension of GIMMF core is much larger than that of SMF, the requirements on its structure symmetry and roughness will be lower.

Fig. 3 Light intensity distribution along the X axis when L_m = 446μm, for (a) GIMMF taper and (b) SMF taper.

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2.3 Influence of structural parameters

Different from the SMF tapers, the light emerging from GIMMF tapers is strongly dependent on the length of GIMMF, $L_{m}$ . The major semi-axis, $a$ , of the hyperbola-shape fiber taper is linearly proportional to the radius of curvature of the fiber tip. Thus, by adjusting the length and the shape of the GIMMF, the focusing performance and working distance can be optimized.

Figures 4(a) and 4(b) show the normalized maximum optical power, p_max, and working distance, $d_{w}$ , respectively, as a function of the length of GIMMF, $L_{m}$ . p_max denotes the optical power at the focal point outside the GIMMF tip, as shown in the inset of Fig. 1(b). $d_{w}$ is defined as the distance between the fiber tip to the focal point. As $L_{m}$ increases, the maximum power at the focal point outside the fiber tip changes periodically. Therefore, there are periodic optimal GIMMF lengths to obtain maximum gradient force in optical tweezers and the corresponding performance is similar. The light distribution along the GIMMF without taper is shown in Fig. 4(a) as a reference by the black curve to show clearly the period. Compared with the reference, the tapered GIMMF tip has greater power, which is important to enhance the gradient force in optical tweezers. The working distance also changes periodically with $L_{m}$ , as shown in Fig. 4(b). It is interesting that the peak of the optical power distribution in each period in the reference curve splits into two peaks. A valley in the optical power distribution is formed at $L_{m} = 500 μm$ between the two peaks. A large amount of light is reflected inside the GIMMF taper, because in the latter half-period near $L_{m} = 500 μm$ , the shape of the light distribution and the shape of the fiber taper is matched, that is, the incident angle of the focused light exceeds the critical angle at the surface of the fiber taper.

Fig. 4 (a) Normalized maximum optical power and (b) working distance as a function of GIMMF length. Curves in different colors correspond to different values of a (the semi-axis of the taper hyperbola), i.e. different shapes of the taper. The black curve in (a) is the power distribution along the GIMMF without a taper, shown as a reference of the period.

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The influence of the shape of the GIMMF taper on the maximum optical power and the working distance with L_m = 446μm is given in Fig. 5. As the cone angle of the fiber taper is fixed as 32.9°, according to the chemical etching experiment, the shape is mainly determined by the radius of curvature of the fiber tip, $ρ$ , which is linearly proportional to the major semi-axis of the hyperbola shape by $ρ = a \tan^{2} (θ / 2)$ . Considering the dimension of the fiber tip obtained by chemical etching, the range of a, from 1μm to 40μm, are selected so that the radius of curvature of the fiber tip, $ρ$ , is around 1μm. As the radius of curvature increases, the working distance increases while the maximum optical power fluctuates.

Fig. 5 Normalized maximum optical power and working distance as a function of the shape of GIMMF taper.

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It is worth noting that the optimal value of $L_{m}$ , with which maximum p_max is obtained, is not consistent with that the maximum working distance requires. Take a = 12.5μm for example, there are two peaks of the optical power in the first period, corresponding to $L_{m} = 446 μm$ and $L_{m} = 525 μm$ , respectively. When $L_{m} = 446 μm$ , better focusing performance with p_max = 0.94 while shorter working distance, $d_{w} = 3 μm$ , are obtained. When $L_{m} = 525 μm$ , relatively weaker focusing performance with p_max = 0.78 but much longer working distance, $d_{w} = 8 μm$ , are obtained. On the other hand, as a decreases, i.e., the fiber taper becomes shaper, p_max increases, indicating better convergent ability of the optical tweezers. However, the working distance decreases as a decreases. Therefore, in the experiment, one should compromise between the two factors to optimize the trapping performance.

2.4 Gradient force distribution

According to the calculated light distribution around the GIMMF tip, the gradient force of the GIMMF tip tweezers when trapping a yeast cell can be calculated by [10]

{\vec{F}}_{g r a d} (r) = 2 π ε_{0} n_{2}^{2} r_{0}^{3} (\frac{m^{2} - 1}{m^{2} + 2}) \nabla I (r, t) .

Here m denotes the relative refractive index, m = n₁/n₂. n₁ and n₂ are the refractive indices of the trapped cell and the aqueous solution, respectively.

d_{0}

is the diameter of the cell.

ε_{0}

is dielectric constant in vacuum.

The Gradient force distribution along the transverse direction is calculated for both GIMMF tip tweezers and SMF tip tweezers, as shown in Fig. 6. The length of GIMMF is selected to be $L_{m} = 446 μm$ . The gradient force of the GIMMF tip tweezers is about 4 times higher than that of the SMF tip tweezers with a same shape. Therefore, by selecting optimal GIMMF length, the GIMMF tip tweezers may achieve much better trapping performance than the SMF tip tweezers.

Fig. 6 Gradient force distribution along the transverse direction for (a) the GIMMF tip tweezers and (b) SMF tip tweezers.

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3. Fabrication of GIMMF taper

Chemically-etched optical fiber tips have been extensively investigated in 1990s for the application of near field optical microscopy [19]. In order to achieve a high scanning resolution, fiber tips with small cone angle [20] and apex with nanoscale dimensions [21] were needed. The requirement on the fiber tips when used as optical tweezers is different from that used as scanning fiber probes. Large cone angle is required to form large gradient force and relatively long working distance for in-depth manipunation. Cone angles of 60° and 90° were obtained for SMF optical tweezers [8, 9] by the so-called two-step etching technique [22]. However, when changing the etching solution in two-step etching technique [8], it was difficult to precisely control the relatively position of the fiber end and the surface between etching solution and protective layer. On the other hand, one-step chemical etching method is simple and more su, but cone angle of optical fiber tapers etched by this method is relatively small. Hoffmann et al. demonstrated the cone angle can be varied from 8° to 41° by the adequate choice of the organic protective layer on the etching solution [23]. Single mode optical fiber tapers with cone angle in this range have low trapping efficiency although submicrometric tapers with even smaller cone angles can be used for optical trapping [7]. Graded-index multimode fibers have the periodic focusing effect [24], which is employed in our experiment to enhance the trapping efficiency in the case of small cone angle.

The chemical etching procedure is simple. In order to estimate the etching performance, three kinds of optical fibers were simultaneously etched, including common single-mode fiber, high numerical aperture step-index multimode fiber and the GIMMF with a quasi-parabolic index profile. Optical fibers were cleaved and dipped into 40% HF acid containing a protective layer ∑[ isooctane at the top. The organic liquid layer reduces evaporation of the etching solution and protects the cladding side wall from being etched. The etching time was about 120 minutes. The etching process is self-terminating, so that replicable production of the fiber tip can be easily realized as long as the etching time exceeds certain minimum value. The ambient temperature is about 15°C. The photos of these fiber tapers are given in Fig. 7. Although the refractive index profiles of three kinds of optical fibers were largely different from each other, all the cone angles were nearly the same. In Fig. 7, Ti denotes the thickness of isooctane. By changing the thickness of isooctane from 1.8 μm to 2.6μm or even larger, the variation of cone angle can be ignored.

Fig. 7 Chemically etched optical fiber tapers.

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4. Optical trapping experiment

The experimental setup for the graded-index fiber tip optical tweezers is shown in Fig. 8. A 980nm laser with a maximum output power of 400mW is employed as the light source for the optical tweezers due to the relatively low transmission loss at this wavelength in water. A small fraction, 5%, of the light is used to monitor the laser power. A 5-dimensions manipulator is used to precisely adjust the position of the GIMMF taper. An inverted optical microscope is used to observe the optical trapping of polystyrene (PS) microspheres and yeast cells. The average diameter of the PS microspheres in aqueous suspension is 916 nm with standard deviation of 311 nm, measured by the manufacturer (Sphere Scientific, China). Optical trapping of a PS microsphere is shown in the inset of Fig. 8.

Fig. 8 Experimental setup for the graded-index fiber tip optical tweezers. The optical trapping of a polystyrene microsphere is shown in the inset.

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By the proposed GIMMF tip optical tweezers, optical trapping of yeast cells with diameter of ~5μm is observed. The length of the graded-index fiber tip is 369.6μm. In the chemical etching experiment, the GIMMF length is difficult to be precisely controlled. However, by other micromachining technologies like the focused ion beam (FIB) or laser micromachining technology, the GIMMF length can be controlled and further optimized. The trapping performance is shown in the movies given in Fig. 9. The laser power of the 95% branch is about 10.83mW when the yeast cell is trapped.

Fig. 9 Optical trapping of a yeast cell with a diameter of ~5μm with graded-index fiber tip optical tweezers. (Media 1) 5.89Mb.

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According to the Stokes' law, the trapping force can be determined by [10]

F = 6 π η r v .

Here

η

stands for the coefficient of viscosity of the medium, for water at 15°C,

η

= 1.1447mPa

\cdot

s. The radius,

r

, of the trapped yeast cell, is 2.5μm. When the trapping laser power is about 10.83mW, the trapping force is calculated to be about 4.56 pN according to the critical velocity at which the cell escaped from the optical trap. The trapping force can be easily changed by adjusting the laser power. The working distance of the optical tweezers can be further improved by optimizing the shape of the GIMMF tip.

5. Conclusion

In this paper, a novel optical tweezer based on the graded-index multimode fiber tip has been proposed and demonstrated. The optical field distribution near the fiber taper and its dependence on the structural parameters have been numerically simulated and analyzed in details. The gradient force distribution has also been calculated. The simulated results indicated that by selecting optimal GIMMF length, The gradient force of the GIMMF tip tweezers is about 4 times higher than that of the SMF tip tweezers with a same shape. As a proof of concept, the GIMMF taper has been fabricated by the chemical etching method and optical trapping of both polystyrene microspheres and yeast cells has been realized. The trapping efficiency can be further enhanced by optimizing the length of the graded-index multimode fiber and the cone angle of the fiber taper. It is anticipated that this kind of GIMMF tip optical tweezers can find a number of applications in the field of bio-photonics.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant (61107073, 61107072 and 61106045), Fundamental Research Funds for the Central Universities (ZYGX2011J002), Research Fund for the Doctoral Program of Higher Education of China (20110185120020), and the Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences (SKLST201005).

References and links

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Parameter	Value	Unit	Parameter	Value	Unit
$n_{1 m}$	1.4829	-	$2 a_{m}$	62.5	μm
$n_{2 m}$	1.4572	-	$2 a_{s}$	4	μm
$n_{1 s}$	1.4639	-	$d_{c l a d}$	125	μm
$n_{2 s}$	1.4572	-	$a$	2-40	μm
$N A_{s}$	0.14	-	$θ$	32.9	deg
$N A_{m}$	0.275	-	$L_{s}$	10	μm
$α$	1.926	-	$L_{m}$	446, 928, 1411	μm
$n_{w a t e r}$	1.33	-

Graded-index fiber tip optical tweezers: Numerical simulation and trapping experiment

Abstract

1. Introduction

2. Numerical simulations

2.1 Numerical model

2.2 Optical field distribution

2.3 Influence of structural parameters

2.4 Gradient force distribution

3. Fabrication of GIMMF taper

4. Optical trapping experiment

5. Conclusion

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (1)

Equations (3)

Optics Express