## Dynamic micro-bead arrays using optical tweezers combined with intelligent control techniques

Optics Express, Vol. 17, Issue 26, pp. 24102-24111 (2009)

http://dx.doi.org/10.1364/OE.17.024102

Acrobat PDF (357 KB)

### Abstract

Dynamic micro-bead arrays offer great flexibility and potential as sensing tools in various scientific fields. Here we present a software-oriented approach for fully automated assembly of versatile dynamic micro-bead arrays using multi-beam optical tweezers combined with intelligent control techniques. Four typical examples, including the collision-free sorting of array elements by bead features, are demonstrated in real time. Control algorithms and experimental apparatus for these demonstrations are also described.

© 2009 OSA

## 1. Introduction

1. W.-H. Tan and S. Takeuchi, “A trap-and-release integrated microfluidic system for dynamic microarray applications,” Proc. Natl. Acad. Sci. U.S.A. **104**(4), 1146–1151 (2007). [CrossRef] [PubMed]

1. W.-H. Tan and S. Takeuchi, “A trap-and-release integrated microfluidic system for dynamic microarray applications,” Proc. Natl. Acad. Sci. U.S.A. **104**(4), 1146–1151 (2007). [CrossRef] [PubMed]

2. P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature **436**(7049), 370–372 (2005). [CrossRef] [PubMed]

3. H. Noda, Y. Kohara, K. Okano, and H. Kambara, “Automated bead alignment apparatus using a single bead capturing technique for fabrication of a miniaturized bead-based DNA probe array,” Anal. Chem. **75**(13), 3250–3255 (2003). [CrossRef] [PubMed]

4. C. D. Onal and M. Sitti, “Visual servoing-based autonomous 2-D manipulation of microparticles using a nanoprobe,” IEEE Trans. Contr. Syst. Technol. **15**(5), 842–852 (2007). [CrossRef]

5. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**(5), 288–290 (1986). [CrossRef] [PubMed]

6. D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef] [PubMed]

7. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**(1-6), 169–175 (2002). [CrossRef]

8. P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glueckstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express **10**(26), 1550–1556 (2002). [PubMed]

9. C. Mio and D. W. M. Marr, “Optical trapping for the manipulation of colloidal particles,” Adv. Mater. **12**(12), 917–920 (2000). [CrossRef]

10. J. M. Tam, I. Biran, and D. R. Walt; “An imaging fiber-based optical tweezer array for microparticle array assembly,” Appl. Phys. Lett. **84**(21), 4289–4291 (2004). [CrossRef]

## 2. Software-oriented approach for versatile dynamic arrays

16. S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express **14**(26), 13095–13100 (2006). [CrossRef] [PubMed]

### 2.1 Fully automated optical trapping

### 2.2 Simultaneous transportation

*M*×

*N*lattice array. Our control algorithm consists of two processes: destination assignment (process 1) and path generation (process 2).

^{d}

**p**

*=[*

_{k}^{d}x

*,*

_{k}^{d}y

*]*

_{k}^{T}to initial positions

^{i}

**p**

*=[*

_{k}^{i}x

*,*

_{k}^{i}y

*]*

_{k}^{T}, which can be represented by

*M*×

*N*matrices

^{d}

**P**and

^{i}

**P**, respectively. Let us denote the

*i*-th row of

^{d}

**P**by

^{d}

**P**

*=[*

_{i}^{d}

**p**

_{i}_{1},

^{d}

**p**

_{i}_{2}, …,

^{d}

**p**

*], and*

_{iN}^{i}

**P**by

^{i}

**P**

*=[*

_{i}^{i}

**p**

_{i}_{1},

^{i}

**p**

_{i}_{2}, …,

^{i}

**p**

*], in which*

_{iN}^{d}

**p**

*, namely, the element of the*

_{ij}^{d}

**P**at the

*i*-th row and

*j*-th column is the position vector of

*k*-th destination,

^{d}

**p**

*=[*

_{k}^{d}x

*,*

_{k}^{d}y

*]*

_{k}^{T}=

^{d}

**p**

*, of the lattice array, where*

_{ij}*k*=

*N*(

*i*-1)+

*j.*Since the destination is 2D

*M*×

*N*lattice, we can assume the following relations:

^{i}y

*, that is, the elements of*

_{k}^{i}

**P**satisfy the following relation:which means that the initial trapped beads at position from

^{i}y

_{N}_{(}

_{i}_{-1) + 1}to

^{i}y

_{N}_{(}

_{i}_{-1)+}

*are assigned*

_{N}^{d}

**P**

*corresponding to*

_{i}^{i}

**P**

*. For example, in the case of sixteen beads in Fig. 2 , each set of four beads stained the identical color is assigned one row of a 4×4 matrix. Second, for all beads which constitute*

_{i}^{i}

**P**

*, re-numbering is done on the basis of the identified*

_{i}^{i}x

*, that is, all the beads constituting the same row are renumbered in ascending order of the*

_{k}^{i}x

*. In Fig. 2, for example, respective elements of*

_{k}^{i}

**P**

*, that is, four beads stained the identical color are renumbered in ascending order of the*

_{i}^{i}x

*. After the renumbering, the elements of*

_{k}^{i}

**P**satisfy the following relations: Note that the inequality in Eq. (3) no longer holds after the renumbering. Thus, all rows of

^{i}

**P**corresponding to

^{d}

**P**are determined.

^{i}

**P**and

^{d}

**P**. First, since we have found that an optimal step size for smooth transportation is less than the radius of the bead, each step size

*δ*

**p**

*is determined by the following equations: where*

_{k}*r*is the radius of the

_{k}*k*-th bead and

*n*is the smallest integer satisfying the inequality (7). Second, new positions in the next time-step are gradually generated using the

*M*×

*N*matrix

*δ*

**P=[**

*δ*

**p**

*].*

_{k}**P**(

*t*

_{0})=

^{i}

**P**,

**P**(

*t*)=

_{n}^{d}

**P**. Each bead is transported directly towards its destination along the linear trajectory as illustrated in Fig. 2(a). Using the inequalities (2) and (4), Eq. (8) derivesthat is,with respect to the new x-positions of bead, which belonged to one row of

^{i}

**P**, at arbitrary time-step

*t*

_{s}. Similarly, the inequalityis derived by using the inequalities (1) and (5). The inequalities (10) and (11) are remarkable in that they state that all beads which are transported along the trajectories gradually generated by Eq. (8) retain the inequalities with respect to their initial positions. Therefore, each bead never overtakes or collides with others, if the beads are regarded as points geometrically. Indeed, in the case of a

*3*×

*3*array, fewer collisions occur, although the potential for collisions does exist.

^{m}

*δ*

**p**

*is the element of the modified step size matrix*

_{k}^{m}

*δ*

**P**=[

^{m}

*δ*

**p**

*] and*

_{k}*c*is the largest value in all collision counters

_{stop}**c**=[

*c*]. In order to avoid collisions, the modified control algorithm checks the potential collisions in the next time-step during Step 1. If the

_{k}*k*-th bead is projected to collide from behind or head-on with others, then it stops the update of its position, namely

^{m}

*δ*

**p**

*=[0,0]*

_{k}^{T}, one time and adds one to the

*k*-th collision counter

*c*. Under the modified algorithm, each bead is transported to its destination along the trajectory parallel to the grid, as illustrated by the dotted arrow in Fig. 2(b). Beads transported along the trajectory parallel to the x-axis never collide with others during Step 2, because the inequalities (2) and (10) are satisfied with respect to the x-positions. Therefore, no checks of the potential collisions are required after the Step 1. Note that the only physical limitations of the collisionless parallel transportation under the modified algorithm are grid size, bead size and the number of columns in the array. The sufficient condition for completing process 2 iswhere

_{k}^{i}

*L*is the initial grid size,

*D*

_{max}is the largest bead size, and

*N*is the number of columns in the array. After the process2, if necessary, we can shrink/expand the grid from

^{i}

*L*to arbitrary size

*L*. Thus the final grid size,

*L*, is independent of the number and size of beads. Therefore, the performance of our approach scales well with smaller/larger array sizes, namely grid widths, bead diameters and the number of beads. The advantage of our approach is that once transport paths have been generated for the destinations with

^{i}

*L*, the beads can be simultaneously transported without collisions and subsequent sorting can also be achieved using collision-free cyclic shifts described in next Section 2.3. Furthermore, the algorithm is faster and more reliable as compared to previously published work [16

16. S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express **14**(26), 13095–13100 (2006). [CrossRef] [PubMed]

### 2.3 Collision-free sorting of array elements

*3*×

*2*array in Fig. 3(a) bywhere G

*and B*

_{i}*are grid nodes numbered*

_{i}*i*and beads numbered

*i*, respectively. On the basis of group theory [12], the expression (15) implies a permutation of the grid nodes G

*to B*

_{i}*, and all of the 6!=720 permutations form a group S*

_{i}_{6}, called the permutation group. The operations of CS6 form a cyclic group

*C*

_{6}, and those of CS4 form a

*C*

_{4}. We can interchange two beads at arbitrary nodes of the

*3*×

*2*array using the combination of CS6 and CS4 [13], like solving the well-known Rubik’s Cube puzzle [14]. This fact implies that all of transpositions forming S

_{6}are generated by the combination of CS6 and CS4 — a mathematical proof will be provided in a future report. According to the proposition of group theory, any permutation can be expressed as a product of transpositions. Therefore, we can sort the six beads forming a

*3*×

*2*array in arbitrary order. A

*M*×

*N*array that is larger than a

*3*×

*3*array can be divided into

*3*×

*2*arrays, for example a

*4*×

*3*array is divided into two

*2*×

*3*arrays as shown in Fig. 3(e). Thus, we can complete the sorting for a

*M*×

*N*array using the combination of CS6 and CS4.

## 3. Experimental apparatus

15. Y. Tanaka, H. Kawada, K. Hirano, M. Ishikawa, and H. Kitajima, “Automated manipulation of non-spherical micro-objects using optical tweezers combined with image processing techniques,” Opt. Express **16**(19), 15115–15122 (2008). [CrossRef] [PubMed]

## 4. Demonstrations

### 4.1 Fully automated assembly of bead arrays

*4*×

*4*array. The sample was glass spheres (Duke Scientific, Borosilicate, 2.5μm). First, all positions of beads dispersed in the pepetted water on a cover glass were detected by the circular Hough transform, and then sixteen beads nearest to the center position, o, were simultaneously trapped at the initially detected positions using the T3S optical tweezers (Fig. 5(b)). The micro-beads in the droplet are diffused by Brownian motion while untrapped. Therefore, after image digitizing, we have to complete the recognition processes for initial traps within the allowable time in which the beads stay in the neighborhood of the identified positions. This allowable time depends on the size of the beads, the viscosity of droplets and the temperature, and can be estimated by the Langevin equation [16

16. S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express **14**(26), 13095–13100 (2006). [CrossRef] [PubMed]

*4*×

*4*array. These collisionless paths were gradually generated using Eqs. (12) and (13). Figure 5(c) shows the image after the procedure of step 1 in Eq. (13). Note that in Fig. 5(c), the sixteen beads are almost aligned in the assigned rows. However, bead number 3 could not reach its row because it would have collided with bead number 4 which reached the row before it; beads number 6 and 7 could also not reach their row because they would have collided head on. These bead situations after step 1 are illustrated by white circles in Fig. 5(a). Third, after the forming of the

*4*×

*4*array with initial grid size

^{i}

*L*=7.5μm (Fig. 5(d)), the grid size was shrunk to 70% of its initial size, to

*L*=5.2μm (Fig. 5(e)). Finally, we rotated the array counterclockwise in the XY-plane.

*6*×

*6*array using the same three-beam system as used in our previous paper [17

17. Y. Tanaka, K. Hirano, H. Nagata, and M. Ishikawa, “Real-time three-dimensional orientation control of non-spherical micro-objects using laser trapping,” Electron. Lett. **43**(7), 412–414 (2007). [CrossRef]

*6*×

*6*array for the chosen 36 beads were divided into three sets of destinations for

*2*×

*6*arrays. Next, under supervisory controls by a PC, the 36 beads were simultaneously transported along the collisionless paths based on the proposed algorithm to form the

*6*×

*6*array, where each divided

*2*×

*6*array was assembled with one set of the T3S system. After shrinking the grid size (Fig. 6(b)), subsequent operations such as Z-axis translation of the

*2*×

*6*arrays (Fig. 6(c)) were also demonstrated.

### 4.2 Fully automated sorting by bead features

*3*×

*2*array and subsequent sorting by identified colors. First, all bead positions in an image were identified by the circular Hough transform and their colors by the thresholding of RGB signals using the Discriminant Threshold Selection Method (DTSM) [18, 19], and then three pairs of beads with identical colors were simultaneously trapped using a T3S system (Fig. 7(a)). Second, they were transported to form a

*3*×

*2*array (Fig. 7(b)). Note that the

*3*×

*2*array is a primitive array for the cyclic shift operations, CS4 and CS6, to sort an arbitrary

*M*×

*N*array. Third, the successive operations of CS4 and CS6 were carried out to interchange the array elements under the collision-free manner (Figs. 7(b)-7(d)). Under the knowledge data based on group theory, this procedure continued until the sorting was completed to rearrange the colored

*3*×

*2*array like a traffic light, that is, in the order of red-yellow-blue (Fig. 7(e)). Finally, the array was translated and rotated in the XY-plane (Fig. 7(f)).

*3*×

*3*array consisting of nine beads with different sizes was automatically rearranged to sort its elements by size. The sample was glass spheres (Duke Scientific, Borosilicate, 2.5μm±0.5μm). First, all bead positions and their radii in an image were identified by the circular Hough transform, and then nine beads nearest to the center were simultaneously trapped using a T3S system (Fig. 8(a)). Second, under the control algorithm mentioned in Section 2.2, they were transported to form a

*3*×

*3*array. Note that the array was not sorted at that moment; therefore the bead sizes at grid nodes were random (Fig. 8(b)). Third, under the knowledge data based on group theory for

*3*×

*3*arrays, the successive operations of CS4 and CS6 were carried out to interchange the beads at specified nodes, like solving the Rubik’s Cube puzzle operations (Figs. 8(c)-8(k)). Note that a

*3*×

*3*array is divided into overlapped two

*3*×

*2*arrays; therefore we can complete the sorting only using the combination of CS6 and CS4. Finally, after the nine operations the beads at the grid nodes were sorted by size (Fig. 8(l)).

## 4. Conclusion

2. P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature **436**(7049), 370–372 (2005). [CrossRef] [PubMed]

20. G. S. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. J. Laczik, “Assembly of 3-dimensional structures using programmable holographic optical tweezers,” Opt. Express **12**(22), 5475–5480 (2004). [CrossRef] [PubMed]

## Acknowledgements

## References and Links

1. | W.-H. Tan and S. Takeuchi, “A trap-and-release integrated microfluidic system for dynamic microarray applications,” Proc. Natl. Acad. Sci. U.S.A. |

2. | P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature |

3. | H. Noda, Y. Kohara, K. Okano, and H. Kambara, “Automated bead alignment apparatus using a single bead capturing technique for fabrication of a miniaturized bead-based DNA probe array,” Anal. Chem. |

4. | C. D. Onal and M. Sitti, “Visual servoing-based autonomous 2-D manipulation of microparticles using a nanoprobe,” IEEE Trans. Contr. Syst. Technol. |

5. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

6. | D. G. Grier, “A revolution in optical manipulation,” Nature |

7. | J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

8. | P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glueckstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express |

9. | C. Mio and D. W. M. Marr, “Optical trapping for the manipulation of colloidal particles,” Adv. Mater. |

10. | J. M. Tam, I. Biran, and D. R. Walt; “An imaging fiber-based optical tweezer array for microparticle array assembly,” Appl. Phys. Lett. |

11. | D. H. Ballard, and C. M. Brown, |

12. | J. Chen, |

13. | Y. Tanaka, H. Kawada, K. Hirano, M. Ishikawa, and H. Kitajima, Japan patent 2008–101060 (April, 9, 2008). |

14. | |

15. | Y. Tanaka, H. Kawada, K. Hirano, M. Ishikawa, and H. Kitajima, “Automated manipulation of non-spherical micro-objects using optical tweezers combined with image processing techniques,” Opt. Express |

16. | S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express |

17. | Y. Tanaka, K. Hirano, H. Nagata, and M. Ishikawa, “Real-time three-dimensional orientation control of non-spherical micro-objects using laser trapping,” Electron. Lett. |

18. | N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Sys., Man, Cybernetics |

19. | A. Murakami, Y. Tanaka, and Y. Kinouch, “Laser manipulation system for automatic control of microscopic particles,” in |

20. | G. S. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. J. Laczik, “Assembly of 3-dimensional structures using programmable holographic optical tweezers,” Opt. Express |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(150.0150) Machine vision : Machine vision

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: October 5, 2009

Revised Manuscript: December 3, 2009

Manuscript Accepted: December 4, 2009

Published: December 17, 2009

**Virtual Issues**

Vol. 5, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Yoshio Tanaka, Hiroyuki Kawada, Shogo Tsutsui, Mitsuru Ishikawa, and Hiroyuki Kitajima, "Dynamic micro-bead arrays using optical tweezers combined with intelligent control techniques," Opt. Express **17**, 24102-24111 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-26-24102

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### References

- W.-H. Tan and S. Takeuchi, “A trap-and-release integrated microfluidic system for dynamic microarray applications,” Proc. Natl. Acad. Sci. U.S.A. 104(4), 1146–1151 (2007). [CrossRef] [PubMed]
- P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature 436(7049), 370–372 (2005). [CrossRef] [PubMed]
- H. Noda, Y. Kohara, K. Okano, and H. Kambara, “Automated bead alignment apparatus using a single bead capturing technique for fabrication of a miniaturized bead-based DNA probe array,” Anal. Chem. 75(13), 3250–3255 (2003). [CrossRef] [PubMed]
- C. D. Onal and M. Sitti, “Visual servoing-based autonomous 2-D manipulation of microparticles using a nanoprobe,” IEEE Trans. Contr. Syst. Technol. 15(5), 842–852 (2007). [CrossRef]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
- J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002). [CrossRef]
- P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glueckstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express 10(26), 1550–1556 (2002). [PubMed]
- C. Mio and D. W. M. Marr, “Optical trapping for the manipulation of colloidal particles,” Adv. Mater. 12(12), 917–920 (2000). [CrossRef]
- J. M. Tam, I. Biran, and D. R. Walt; “An imaging fiber-based optical tweezer array for microparticle array assembly,” Appl. Phys. Lett. 84(21), 4289–4291 (2004). [CrossRef]
- D. H. Ballard, and C. M. Brown, Computer Vision (Prentice-Hall, 1982), Chap. 3–4.
- J. Chen, Group Representation Theory for Physicists (World Scientific, 1989), Chap. 1.
- Y. Tanaka, H. Kawada, K. Hirano, M. Ishikawa, and H. Kitajima, Japan patent 2008–101060 (April, 9, 2008).
- http://www.rubiks.com/
- Y. Tanaka, H. Kawada, K. Hirano, M. Ishikawa, and H. Kitajima, “Automated manipulation of non-spherical micro-objects using optical tweezers combined with image processing techniques,” Opt. Express 16(19), 15115–15122 (2008). [CrossRef] [PubMed]
- S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express 14(26), 13095–13100 (2006). [CrossRef] [PubMed]
- Y. Tanaka, K. Hirano, H. Nagata, and M. Ishikawa, “Real-time three-dimensional orientation control of non-spherical micro-objects using laser trapping,” Electron. Lett. 43(7), 412–414 (2007). [CrossRef]
- N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Sys., Man, Cybernetics SMC-9, 62–66 (1979).
- A. Murakami, Y. Tanaka, and Y. Kinouch, “Laser manipulation system for automatic control of microscopic particles,” in Proceedings of the 4th Asian Control Conference, Singapore, 25–27 Sept. 2002, pp.414–419.
- G. S. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. J. Laczik, “Assembly of 3-dimensional structures using programmable holographic optical tweezers,” Opt. Express 12(22), 5475–5480 (2004). [CrossRef] [PubMed]

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