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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 1 — Jan. 19, 2007
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Tapered fiber optical tweezers for microscopic particle trapping: fabrication and application

Zhihai Liu, Chengkai Guo, Jun Yang, and Libo Yuan  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 12510-12516 (2006)
http://dx.doi.org/10.1364/OE.14.012510


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Abstract

A novel single tapered fiber optical tweezers is proposed and fabricated by heating and drawing technology. The microscopic particle tapping performance of this special designed tapered fiber probe is demonstrated and investigated. The distribution of the optical field emerging from the tapered fiber tip is numerically calculated based on the beam propagation method. The trapping force FDTD analysis results, both axial and transverse, are also given.

© 2006 Optical Society of America

1. Introduction

Since first demonstrated by Ashkin in 1986 [1

1. 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, 288–290 (1986). [CrossRef] [PubMed]

], optical tweezers (a single-beam gradient force trap) has been widely used in biology, physics and chemistry. Applications now range from manipulation of cells to the assembly of microstructure. Especially in biology, optical tweezers has been applied in researches on cells, viruses, bacterias and DNA molecules.

Usually a high N.A. microscope objective is necessary to focus the laser beam in traditional optical tweezers, which is the disadvantage of this method for user is not convenient and easy. For this purpose, fiber optical tweezers has been developed since 1993 [2

2. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

]. Using optical fibers to carry the light to the microscopic object, in which it is to be trapped [3–4

3. E.R. Lyons and G.J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. 66, 1584–1586 (1995). [CrossRef]

], is much easier to handle, and much more suitable to practical use such as in trapping, levitating and rotating of microscopic particles [5–12

5. K. Taguchi, H. Ueno, and M. Ikeda, “Rotational manipulation of a yeast cell using optical fibers,” Electron. Lett. 33, 1249–1250 (1997). [CrossRef]

]. However, unlike traditional optical tweezers, these fiber optical tweezers can not form a 3-D optical trap by using just single one optical fiber due to the weakly focused laser beam emerging from the polished tapered fiber with spherical lens tip [4–6

4. K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fiber,” Electron. Lett. 33, 413–414 (1997). [CrossRef]

] result in gradient forces is not so strong. For this reason, etching method has been used by Taylor and Hnatovsky [13

13. R. S. Taylor and C. Hantovsky, “Particle trapping in 3-D using a single fiber probe with an annular light distribution,” Opt. Exp. 11, 2775–2782 (2003). [CrossRef]

] made a tapered, hollow tipped metalized fiber probe so that the optical confinement in 2-D was produced by the output annular light distribution, and the force produced by the light pressure on the particle in the direction of light propagation can be balanced by an electrostatic return force towards the highly tapered tip to produce particle trapping in 3-D.

In this paper a novel tapered single fiber optic tweezers is presented. Unlike the polishing and chemical etching manufacture techniques, a novel abruptly tapered fiber tip is fabricated by using heating and drawing method. The microscopic particle tapping performance of this special designed tapered fiber probe is demonstrated and investigated. The distribution of the optical field emerging from the tapered fiber tip is numerically calculated based on the beam propagation method. The ability of the optical forces to form a 3-D optical trap is simulated in detail.

2. Fabrication of tapered fiber optical tweezers

Figure 1 shows the tapered fiber probe used in our experiment. The probe, which is the most important device in our single tapered fiber optical tweezers system, was made from a single-mode optical fiber (with a core diameter of 9µ m). The tapered fiber optic probe has been manufactured by heating and drawing technology. The heating zone is about 2~3 mm along the fiber and the drawing speed is gradually changed from 0.03 mm/s to 0.32 mm/s as the diameter reduced from 125 µm to about 10 µm with length 600 µm. Then continue heating the waist zone of the tapered fiber and drawing at high speed of 1.6 mm/s until the fiber break at the waist point, thus the parabola-like profile fiber tip is formed due to the surface tension of the fused quartz material. It is convinced that the optical field emerging from this kind of fiber probe can form a 3-D trap for microscopic particles.

3. Yeast cells trapping experiment

In order to confirm the good performances of the 3-D fiber optical tweezers, a demonstration system has been conducted. The experimental setup of the single tapered fiber optical tweezers is shown in Fig. 2. The laser source is capable of producing maximum power outputs of 120mw, at the wavelength of 980nm. Through a SM coupler with a ratio of 95:5, the laser power was monitored by the 5 percent end, which connected to an optical power-meter. A manipulator was used to manipulate the fiber probe in 3 dimensions and also to rotate the probe to change the inserted angle. An inverted biological microscope with a CCD camera was equipped to observe the trapping particles, with the sample chamber on its moveable objective stage. Yeast cells were immersed in water taken as particles to be trapped. A filter was introduced to prevent the IR light to improve the quality of the images recorded.

Fig. 1. Image of the fiber probe manufactured by heating and drawing with a parabola-like profile fiber tip
Fig. 2. Experimental setup of single tapered fiber optical tweezers

The trapping of yeast cells in water was realized based on our single tapered fiber optical tweezers system mentioned above. Under the trapped status, we could freely move the trapped yeast cell to the forward and backward or right and left directions synchronized to the trapping fiber. The trapping force is also can be controlled by adjusting the optical power of the laser source, as shown in the movie given by Fig. 3.

The output optical power from the fiber tip versus the trapping force can be calibrated based on the viscous drag exerted by fluid flow according to Stokes’ law [14

14. O. Muller, M. Schliwa, and H. Felgner, “Calibration of light forces in optical tweezers,” Appl. Opt. 34, 977–980 (1995) [CrossRef] [PubMed]

],

F=6πξηau
(1)

here, u⃑ represents the moving velocity of the trapping particle. The diameter a of the trapping particle (yeast) used in our experiments is about 6.5µm which immersed in the water at room temperature 293 K and the friction coefficient of water at the room temperature is taken as η=1.002×10-3 (NS/M2), and ξ a coefficient which takes into account the fiber tip effects. The viscous force was balanced by the optical trap force up to the fluid velocity reaching a critical velocity at which the particle escaped from the potential well created by the laser light. Therefore the maximum trapping force could be derived from the critical velocity at which the trapping was no longer active [15

15. N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurement of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. , 214, 15–24 (2002). [CrossRef]

].

Fig. 3. (9.5Mb) The real-time movie showing that yeast cell was trapped by a single tapered fiber optical tweezers and moved in both axial and transverse directions.

Figure 4 shows the yeast trapping force toward to the fiber tip along the fiber direction is a function of the light source power and indicates that the relationship between the maximum trapping force and the laser power emerging from optical fiber tip is linearly.

Fig. 4. Measuring results between the trapping force and the light source power.

4. Simulation of the laser beam from the tapered fiber tip

It can be explained that a laser beam with a rather small waist size and large divergence angle (which can be concluded as high numerical aperture) can form a single beam gradient force trap, which is exactly the situation in our case. The distribution of the optical field emerging from the fiber probe used in our experiment was numerically calculated based on the BPM[16

16. C Youngchul and D Nadr, “An assessm ent of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1338 (1990). [CrossRef]

]. The wavelength of the laser is 0.98µ m, and the refractive index of the surrounding medium (water) is 1.33 and the refractive index of the fiber probe is 1.46. The laser beam with a Gaussian profile is considered as an input laser source [17

17. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (London, Chapman and Hall, 1983), chap.15, 337–339.

]. The simulation results are shown in Fig. 5.

As shown in Fig. 5(a), the laser beam emerging from the fiber tip is focused, at a place where the intensity of the laser beam is the highest, and the focal plane is 1µ m away from the fiber tip. The beam spot size was calculated to be 1µ m in diameter and the waist position is 1.2µ m away from the fiber tip. From Fig. 5(b), we can see that the output laser beam is with a large divergence angle in the far field. The laser beam emerging from the end of the fiber tip with a spot size of about 0.5µ m and large divergence angle is about 30°.

Fig. 5. The intensity of the optical field emerging from the fiber probe numerically calculated using BPM.

5. Trapping forces computation

The optical force exerted on the particles has been widely analyzed and many exciting results were achieved, but still it is too difficult to predict the optical force exactly, the reason of which lies in that the force is strongly relied on the intensity distribution of the optical field, the situation of surrounding medium and the characteristics of the particles. A ray-optics based method, which was firstly introduced by A. Ashkin [18

18. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992). [CrossRef] [PubMed]

], has been generally accepted as a satisfying theory. Though it is accurate merely for the particles with diameters far greater than the optical wavelength (d>10λ), it is easy to calculate and can give out many important results. It predicts that, in order to form a 3D single beam optical trap, the laser beam should be tightly focused so that the optical gradient force can be strong enough to resist the optical scattering force. It also points out that, take a Gaussian beam for example, the waist size of the beam is a critical parameter to form such a trap. If the waist size of a laser beam cannot be small enough (approximately less than 0.7µ m) the trap cannot be formed at all [19

19. W.H. Wright, G.J. Sonek, Y. Tadir, and M.W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990). [CrossRef]

]. This is just the reason why a tapered lensed fiber probe cannot trap microscopic particles in 3 dimensions [4

4. K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fiber,” Electron. Lett. 33, 413–414 (1997). [CrossRef]

, 6

6. K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176, 43–47 (2000). [CrossRef]

].

To reveal the trapping force characteristics of the tapered fiber tip, the 2-D Finite-Difference-Time-Domain (FDTD) algorithm [20

20. V. V. Kotlyar and A. G. Nalimov, “Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam,” Opt. Express 14, 6316–6321 (2006) [CrossRef] [PubMed]

, 21

21. R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Exp. 13, 3707–3718 (2005). [CrossRef]

] has been used to calculate the radiation pressure force in axial and radial directions for this parabolic-like fiber optic tip.

In the dielectric media, the radiation pressure force is given by [22

22. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, (Addison Wesley, Reading, Mass., 1968), chap.9, 242–243

]

f+gt=·T
(2)

It can be rewritten as

vfdV+ddtvgdV=v·TdV=sdS·T
(3)

where, g⃗ is the density of the electromagnetic field moment in vacuum and T⃡ represents the Maxwell stress tensor. T⃡ can be further represented as[20

20. V. V. Kotlyar and A. G. Nalimov, “Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam,” Opt. Express 14, 6316–6321 (2006) [CrossRef] [PubMed]

]

Tik=ε0ε1E2+μμ0H22δikε0ε1EiEkμμ0HiHk
(4)

Here, E⃑, H⃑ are the vectors of the electromagnetic field strength in free space, and foot notes i,k represents the direction x, y or z. ε 0 and ε 1 is the permittivity of vacuum and medium, respectively. μ is the permeability of medium, and µ 0 is the permeability of vacuum.

In the case of 2D TM mode continue electromagnetic field radiation, the non-zeroed electromagnetic field components are (Hy, Ex, Ez). Therefore, the axial (z direction) and the transverse (x direction) radiation pressure force for a microscopic sphere can be calculated as

Fz=12S[12(μμ0Hy2+ε0ε1Ex2ε0ε1Ez2)dx+μμ0Re(ExEz*)dz]
(5)
Fx=12S[12(μμ0Hy2+ε0ε1Ez2ε0ε1Ex2)dz+μμ0Re(ExEz*)dx]
(6)

here S is a contour that envelops the microscopic sphere cross-section in the XOZ-plane. The FDTD calculation coordinate is described as Fig. 6. In order to minimize the effects of reflections at the boundary of the mesh grid a perfectly matched layer (PML) is usually defined and encloses the entire computation domain [20

20. V. V. Kotlyar and A. G. Nalimov, “Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam,” Opt. Express 14, 6316–6321 (2006) [CrossRef] [PubMed]

].

Fig. 6. FDTD calculation coordinates with perfectly matched layer (PML) boundary condition.

In the FDTD calculation, a continue Gaussian beam come from the laser diode light source with wavelength 980nm propagating left to right along the parabolic-like fiber tip. The refractive index of the surrounding medium (water) is 1.33 and the refractive index of the fiber probe is 1.46. A dielectric spherical object of index of refraction 1.4, and radius r is shown centered on the point (dz, dx). The grid step is chosen as 0.05µm in the FDTD calculation.

Fig. 7. Calculation results of axial trapping force on a high refractive index dielectric sphere in water. The radius of the sphere has been chosen as 1.5 µm, 2µm and 2.5µm.
Fig. 8. Calculation results of transverse trapping force on a high refractive index dielectric sphere in water. The separate distance between fiber tip and the sphere center has been chosen from 5 to 9 µm.

6. Conclusion

A single tapered fiber optical tweezers system is demonstrated in this work. On the basis of a single tapered fiber probe heating and drawing fabrication technique, a very narrow parabola-like profile of tapered fiber tip is made and by using this new kind fiber probe, yeast cells in water can be trapped in 3 dimensions. The characteristics of the laser beam emerging from the fiber tip were analyzed and it is concluded that a laser beam with such a small waist size (approximately 0.5µ m) and a large divergence angle can form a 3D single beam optical trap, which has been proved by our experiments. The axial and transverse trapping forces are calculated by using FDTD method. The simulation result shown that 3D trapping force well is expected and stable. The single tapered fiber optical tweezers is convenient to handle, especially useful for build-up a multi-tweezers system for controlling and manipulating micro-scale particles.

Acknowledgments

This work was supported by the National Nature Science Foundation of China, under grant number 60577005, and The Teaching and Research Award Program for Outstanding Young Professors in Higher Education Institute MOE, P. R. C., to Harbin Engineering University.

References and links

1.

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, 288–290 (1986). [CrossRef] [PubMed]

2.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

3.

E.R. Lyons and G.J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. 66, 1584–1586 (1995). [CrossRef]

4.

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fiber,” Electron. Lett. 33, 413–414 (1997). [CrossRef]

5.

K. Taguchi, H. Ueno, and M. Ikeda, “Rotational manipulation of a yeast cell using optical fibers,” Electron. Lett. 33, 1249–1250 (1997). [CrossRef]

6.

K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176, 43–47 (2000). [CrossRef]

7.

W. Singer, M. Frick, T. Haller, P. Dietl, S. Bernet, and M. Ritsch-Marte, “Combined optical tweezers and optical stretcher in microscopy,” SPIE 4434, 227–232 (2001). [CrossRef]

8.

S.D. Collins, E. Sidick, A. Knoesen, and R.J. Baskin, “Micromachined optical trap for use as a microcytology workstation,” SPIE 2978, 69–74 (1997). [CrossRef]

9.

Z. Hu, J. Wang, and J. Liang, “Manipulation and arrangement of biological and dielectric particles by a lensed fiber probe,” Opt. Exp. 12, 4123–4128 (2004). [CrossRef]

10.

E. R Lyons and G.J. Sonek, “Demonstration and modeling of a tapered lensed optical fiber trap,” SPIE 2383, 186–198 (1995). [CrossRef]

11.

C. Jensen-McMullin, H. P. Lee, and E. R. Lyons, “Demonstration of trapping, motion control, sensing and fluorescence detection of polystyrene beads in a multi-fiber optical trap,” Opt. Exp. 13, 2634–2642 (2005). [CrossRef]

12.

M. Ikeda, K. Tanaka, M. Kittaka, M. Tanaka, and T. Shohata, “Rotational manipulation of a symmetrical plastic micro-object using fiber optic trapping,” Opt. Commun. 239, 103–108 (2004). [CrossRef]

13.

R. S. Taylor and C. Hantovsky, “Particle trapping in 3-D using a single fiber probe with an annular light distribution,” Opt. Exp. 11, 2775–2782 (2003). [CrossRef]

14.

O. Muller, M. Schliwa, and H. Felgner, “Calibration of light forces in optical tweezers,” Appl. Opt. 34, 977–980 (1995) [CrossRef] [PubMed]

15.

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurement of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. , 214, 15–24 (2002). [CrossRef]

16.

C Youngchul and D Nadr, “An assessm ent of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1338 (1990). [CrossRef]

17.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, (London, Chapman and Hall, 1983), chap.15, 337–339.

18.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992). [CrossRef] [PubMed]

19.

W.H. Wright, G.J. Sonek, Y. Tadir, and M.W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990). [CrossRef]

20.

V. V. Kotlyar and A. G. Nalimov, “Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam,” Opt. Express 14, 6316–6321 (2006) [CrossRef] [PubMed]

21.

R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Exp. 13, 3707–3718 (2005). [CrossRef]

22.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, (Addison Wesley, Reading, Mass., 1968), chap.9, 242–243

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.7010) Lasers and laser optics : Laser trapping

ToC Category:
Trapping

History
Original Manuscript: September 1, 2006
Revised Manuscript: November 8, 2006
Manuscript Accepted: November 8, 2006
Published: December 11, 2006

Virtual Issues
Vol. 2, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Zhihai Liu, Chengkai Guo, Jun Yang, and Libo Yuan, "Tapered fiber optical tweezers for microscopic particle trapping: fabrication and application," Opt. Express 14, 12510-12516 (2006)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-25-12510


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References

  1. A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). [CrossRef] [PubMed]
  2. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]
  3. E.R. Lyons, G.J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. 66, 1584–1586 (1995). [CrossRef]
  4. K. Taguchi, H. Ueno, T. Hiramatsu, M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fiber,” Electron. Lett. 33, 413–414 (1997). [CrossRef]
  5. K. Taguchi, H. Ueno, M. Ikeda, “Rotational manipulation of a yeast cell using optical fibers,” Electron. Lett. 33, 1249–1250 (1997). [CrossRef]
  6. K. Taguchi, K. Atsuta, T. Nakata, M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176, 43–47 (2000). [CrossRef]
  7. W. Singer, M. Frick, T. Haller, P. Dietl, S. Bernet, M. Ritsch-Marte, “Combined optical tweezers and optical stretcher in microscopy,” SPIE 4434, 227–232 (2001). [CrossRef]
  8. S.D. Collins, E. Sidick, A. Knoesen, R.J. Baskin, “Micromachined optical trap for use as a microcytology workstation,” SPIE 2978, 69–74 (1997). [CrossRef]
  9. Z. Hu, J. Wang, J. Liang, “Manipulation and arrangement of biological and dielectric particles by a lensed fiber probe,” Opt. Exp. 12, 4123–4128 (2004). [CrossRef]
  10. E. R Lyons, G.J. Sonek, “Demonstration and modeling of a tapered lensed optical fiber trap,” SPIE 2383, 186–198 (1995). [CrossRef]
  11. C. Jensen-McMullin, H. P. Lee, E. R. Lyons, “Demonstration of trapping, motion control, sensing and fluorescence detection of polystyrene beads in a multi-fiber optical trap,” Opt. Exp. 13, 2634–2642 (2005). [CrossRef]
  12. M. Ikeda, K. Tanaka, M. Kittaka, M. Tanaka, T. Shohata, “Rotational manipulation of a symmetrical plastic micro-object using fiber optic trapping,” Opt. Commun. 239, 103–108 (2004). [CrossRef]
  13. R. S. Taylor, C. Hantovsky, “Particle trapping in 3-D using a single fiber probe with an annular light distribution,” Opt. Exp. 11, 2775–2782 (2003). [CrossRef]
  14. O. Muller, M. Schliwa, H. Felgner, “Calibration of light forces in optical tweezers,” Appl. Opt. 34, 977–980 (1995) [CrossRef] [PubMed]
  15. N. Malagnino, G. Pesce, A. Sasso, E. Arimondo, “Measurement of trapping efficiency and stiffness in optical tweezers,” Opt. Commun., 214, 15–24 (2002). [CrossRef]
  16. C Youngchul, D Nadr, “An assessm ent of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1338 (1990). [CrossRef]
  17. A. W. Snyder, J. D. Love, Optical Waveguide Theory, (London, Chapman and Hall, 1983), chap.15, 337–339.
  18. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992). [CrossRef] [PubMed]
  19. W.H. Wright, G.J. Sonek, Y. Tadir, M.W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990). [CrossRef]
  20. V. V. Kotlyar, A. G. Nalimov, “Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam,” Opt. Express 14, 6316–6321 (2006) [CrossRef] [PubMed]
  21. R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Exp. 13, 3707–3718 (2005). [CrossRef]
  22. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, (Addison Wesley, Reading, Mass., 1968), chap.9, 242–243

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