## Escape trajectories of single-beam optically trapped micro-particles in a transverse fluid flow

Optics Express, Vol. 14, Issue 4, pp. 1685-1699 (2006)

http://dx.doi.org/10.1364/OE.14.001685

Acrobat PDF (266 KB)

### Abstract

We have studied the transverse and axial equilibrium positions of dielectric micro-spheres trapped in a single-beam gradient optical trap and exposed to an increasing fluid flow transverse to the trapping beam axis. It is demonstrated that the axial equilibrium position of a trapped micro-sphere is a function of its transverse position in the trapping beam. Moreover, although the applied drag-force acts perpendicularly to the beam axis, reaching a certain distance *r*_{0} from the beam axis (*r*_{0}/*a* ≃ 0.6, *a* being the sphere radius) the particle escapes the trap due to a breaking axial equilibrium before the actual maximum transverse trapping force is reached. The comparison between a theoretical model and the measurements shows that neglecting these axial equilibrium considerations leads to a theoretical overestimation in the maximal optical transverse trapping forces of up to 50%.

© 2006 Optical Society of America

## 1. Introduction

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

2. M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, “Optical manipulation of objects and biological cells in microfluidic devices,” Biomed. Microdevices **5**, 61–67 (2003). [CrossRef]

3. J. Enger, M. Goksor, K. Ramser, P. Hagberg, and D. Hanstorp, “Optical tweezers applied to a microfluidic system,” Lab. Chip **4**, 196–200 (2004). [CrossRef] [PubMed]

4. J. Glückstad, “Microfluidics: Sorting particles with light,” Nat. Mater. **3**, 9–10 (2004). [CrossRef] [PubMed]

5. M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol. **23**, 83–87 (2005). [CrossRef]

6. S. L. Neale, M. P. Macdonald, K. Dholakia, and T. F. Krauss, “All-optical control of microfluidic components using form birefringence,” Nat. Mater. **4**, 530–533 (2005). [CrossRef] [PubMed]

7. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. **75**, 2787–2809 (2004). [CrossRef]

*et al*. [8

8. 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]

*transverse*trapping force. Actual theoretical predictions for the maximal transverse trapping forces of optical tweezers - that rely on ray-optics [9

9. G. Roosen, “La lévitation optique de sphères,” Can. J. Phys. **57**, 1260–1279 (1979). [CrossRef]

10. 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]

11. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**, 4594–4602 (1989). [CrossRef]

12. K. F. Ren, G. Greha, and G. Gouesbet, “Radiation Pressure Forces Exerted on a Particle Arbitrarily Located in a Gaussian-Beam by Using the Generalized Lorenz-Mie Theory, and Associated Resonance Effects,” Opt. Commun. **108**, 343–354 (1994). [CrossRef]

13. W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric Study of the Forces on Microspheres Held by Optical Tweezers,” Appl. Opt. **33**, 1735–1748 (1994). [CrossRef] [PubMed]

14. A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. **41**, 2494–2507 (2002). [CrossRef] [PubMed]

15. D. Ganic, X. S. Gan, and M. Gu, “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express **12**, 2670–2675 (2004). [CrossRef] [PubMed]

16. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

*et al*. [17

17. S. Sato, M. Ishigure, and H. Inaba, “Optical Trapping and Rotational Manipulation of Microscopic Particles and Biological Cells Using Higher-Order Mode Nd-Yag Laser-Beams,” Electron. Lett. **27**, 1831–1832 (1991). [CrossRef]

10. 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]

*et al*. [18

18. A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. **459**, 3021–3041 (2003). [CrossRef]

*both*in the radial and the axial directions, before it finally escapes the trap.

13. W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric Study of the Forces on Microspheres Held by Optical Tweezers,” Appl. Opt. **33**, 1735–1748 (1994). [CrossRef] [PubMed]

19. H. Felgner, O. Muller, and M. Schliwa, “Calibration of Light Forces in Optical Tweezers,” Appl. Opt. **34**, 977–982 (1995). [CrossRef] [PubMed]

20. N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial
trapping efficiency,” J. Mod. Opt. **45**, 1943–1949 (1998). [CrossRef]

21. A. T. O’Neill and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre-Gaussian modes in inverted optical tweezers,” Opt. Commun. **193**, 45–50 (2001). [CrossRef]

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

*axial*equilibrium, before the actual maximum transverse force is reached, and that this has to be taken into account for a reliable mathematical prediction of the maximal transverse trapping forces.

## 2. Theory

*et al*. [23

23. P. Torok, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation,” J. Opt. Soc. Am. A **12**, 1605–1605 (1995). [CrossRef]

24. P. Torok, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation -errata,” J. Opt. Soc. Am. A **12**, 1605–1605 (1995). [CrossRef]

9. G. Roosen, “La lévitation optique de sphères,” Can. J. Phys. **57**, 1260–1279 (1979). [CrossRef]

### 2.1. EM-field calculations

*h*=

*f sin*ϕ

_{1}, where

*f*is the focal length in oil (of refractive index

*n*

_{1}). The origin O of the system of coordinates is positioned at the focus, as if the beam were focused into the index-matching oil. The position vector

**r**

_{P}, pointing from

*O*to

*P*, is given in spherical coordinates by

*z*= -

*d*. The angle of incidence at the interface is denoted by ϕ

_{1}and the angle of refraction by ϕ

_{2}. The time-independent components of the electric field

**e**=

*e*

_{x},

*e*

_{y},

*e*

_{z}) at any point

*P*to the right of the interface (Fig. 1) are expressed as a linear combination of three integral terms

*n*= 0,1,2

*et al*. [23

23. P. Torok, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation,” J. Opt. Soc. Am. A **12**, 1605–1605 (1995). [CrossRef]

*exp*(-γ

^{2}

*sin*

^{2}ϕ

_{1}) where γ =

*f*/ω, ω being the half Gaussian width of the incident collimated beam. α is the semi-aperture angle of the lens (in media

*n*

_{1},

*NA*=

*n*

_{1}

*sin*α), τ

_{p}and τ

_{s}are the Fresnel transmission coefficients at the glass-water interface, and

*J*

_{n}are the Bessel functions of the first kind, order

*n*. The wavenumbers read

*k*

_{0}=

*k*

_{1}/

*n*

_{1}=

*k*

_{2}/

*n*

_{2}= 2π/λ

_{0}and the focal length

*f*=

*n*

_{1}

*h*

_{max}/

*NA*, where

*h*

_{max}is the effective radius of the MO back-pupil. Ψ is the aberration function defined by

*e*

_{0}in (3) is the peak amplitude of the electric field incident onto the MO back-pupil, which is in contact with air (

*n*

_{0}= 1). Similar expressions as (2-4) exist for the magnetic-field vectors [23

23. P. Torok, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation,” J. Opt. Soc. Am. A **12**, 1605–1605 (1995). [CrossRef]

*I*, which is necessary for the computation of the optical force on dielectric particles (next section 2.2).

25. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in Confocal Fluorescence Microscopy Induced by Mismatches in Refractive-Index,” J. Microsc.-Oxford **169**, 391–405 (1993). [CrossRef]

26. K. C. Neuman, E. A. Abbondanzieri, and S. M. Block, “Measurement of the effective focal shift in an optical trap,” Opt. Lett. **30**, 1318–1320 (2005). [CrossRef] [PubMed]

*n*

_{2}shifts fairly linearly with the position of the interface

*k*

_{f}is the linearity constant for this focus shift. In the paraxial limit

*k*

_{f}= 1-

*n*

_{2}/

*n*

_{1}, but for high-NA systems a more precise estimation is to be done numerically.

*I*by

*f*/ω with

*f*being defined by

*f*=

*n*

_{2}

*h*

_{max}/

*NA*neglecting thus the effect of the mismatched coverglass-water interface.

### 2.2. Optical Force on a dielectric sphere

*dA*of the sphere is first obtained in a system of coordinates attached to the plane of incidence, and then projected on the optical system of coordinates, following Roosen [9

9. G. Roosen, “La lévitation optique de sphères,” Can. J. Phys. **57**, 1260–1279 (1979). [CrossRef]

*z*be the optical axis and

*S*= (0,

*r*

_{0},

*z*

_{0}) the center of the dielectric sphere of radius

*a*.

*M*is the impact point of a particular ray and

*C*is the center of curvature corresponding to the wavefront impinging on the sphere surface at

*M*. The plane of incidence is then defined by

*CSM*, whose orthogonal system of coordinates is (

*x*

_{1},

*y*

_{1},

*z*

_{1}), and has its center in

*S*(

*z*

_{1}being the unity vector in the direction

*CS̅*, and

*y*

_{1}lying in the plane of incidence and being orthogonal to

*z*

_{1}). In this second system of coordinates, a ray of intensity

*I*impinging in

*M*at an angle of incidence α

_{i}on a surface element

*dA*of the sphere induces an elementary force

*d*

**F**

_{1}

*n*

_{2}is the index of refraction of the media surrounding the sphere, α

_{r}is the angle of refraction of the ray entering the sphere,

*R*and

*T*are the intensity reflection and transmission coefficients, and

*c*is the speed of light in vacuum. Polarization effects are neglected by averaging

*R*and

*T*over the two polarization states.

*F*

_{z}and

*F*

_{y}on the whole sphere are obtained by projecting the elementary forces

*dF*

_{z1}and

*dF*

_{y1}along the

*y*and

*z*axis and by integration over the sphere surface, but only rays characterized by α

_{i}< π/2 (thus entering the sphere) are to be considered. For symmetry reasons, and since polarization effects are neglected, there is no difference between a displacement along the x-axis and the y-axis. In the following sections the displacements of the sphere center from the focus position will simply be given by its transverse (radial) and axial positions

*r*

_{0}and

*z*

_{0}.

*I*and the incident angles α

_{i}in eq. (9) using the ray-optics (RO) beam focusing model described by eq. (8) or by combining (9) to the vectorial diffraction (VD) focusing model, the local light intensity

*I*and incident angle α

_{i}being in this case found using (6).

*z*

_{0}→

*z*

_{0}-

*k*

_{f}

*d*is necessary when using the VD model, according to (7).

### 2.3. Special considerations for the drag-force technique

*r*

_{0}and

*z*

_{0}are the radial and axial coordinates of the sphere center with respect to the focus, and

*m*, ρ,

*V*are respectively the sphere mass, density and the immersed volume.

*v*

_{r}is the speed of the fluid surrounding the sphere, and β is the proportionality factor for the viscous drag according to Faxen’s law [27], describing the laminar viscous drag for a spherical particle of radius a which is displacing parallel to a very close planar surface (the microscope coverglass)

*a*is the sphere radius,

*b*is the distance between the sphere center and the planar surface (the trapping depth) and η is the viscosity of the fluid.

*v*

_{r}of the fluid surrounding the sphere increases linearly with time (

*a*

_{r}being the constant acceleration) and provided that the acceleration terms

*mr̈*

_{0},

*mz̈*

_{0}and the gravitational force are small compared to the other terms in (10) the time-dependent equilibrium of the trapped particle is described by the solutions of

*r*

_{0}.

### 2.4. Trapping efficiency

*c*being the speed of light in vacuum,

*n*

_{2}the index of refraction of the media surrounding the particle and

*P*the beam power at the focus [10

10. 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]

## 3. Experimental

### 3.1. Set-Up

^{**}) emitting at 974

*nm*and having a maximal output power of 100

*mW*. The output beam is collimated to a Gaussian diameter of 3.2

*mm*, and coupled into a high-NA oil-immersion objective via the fluorescence port of an inverted microscope (Leica DMIL). The focusing high-NA lens is a Leica C-PLAN 100

*X*/1.25 oil-immersion microscope-objective (MO). Its effective back-pupil radius and transmission were determined using two identical confocused objectives.

*μm*diameter polystyrene micro-beads to be trapped (Dynospheres, Dyno Particles AS) are diluted in water and the sample-cell, whose bottom consists in a microscope coverglass, is mounted on a programmable x-y motorized stage (Märzhäauser) just above the MO. The beads are observed through the microscope dicroïc mirror (reflective for infrared and transparent for visible wavelengths) using a CCD camera connected to a video acquisition board. All relevant experimental parameters are listed in Tab.1.

### 3.2. 3D position measurement

25. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in Confocal Fluorescence Microscopy Induced by Mismatches in Refractive-Index,” J. Microsc.-Oxford **169**, 391–405 (1993). [CrossRef]

26. K. C. Neuman, E. A. Abbondanzieri, and S. M. Block, “Measurement of the effective focal shift in an optical trap,” Opt. Lett. **30**, 1318–1320 (2005). [CrossRef] [PubMed]

*z*of the MO can be regarded as a shift of the coverglass-water interface in Fig. 1, thus the displacement Δ

*z*' of the effective focus position in water is given by

*k*

_{f}is the focus-shift parameter defined in (7).We calculated a ratio of Δ

*z'*/Δ

*z*≃ 0.78 for our experimental parameters using the vectorial diffraction focusing model (according to the value of

*k*

_{f}given in Tab. 1).

*μm*) the accuracy of the transverse position measurement with respect to the beam axis was estimated to be better than 0.1

*μm*. Concerning the axial position measurement, we were not able to ensure an accurate identical position of the laser focusing-plane and the plane imaged on the microscope CCD camera. The position of the laser focusing-plane depending on its precise collimation at the microscope objective back-pupil, and the plane imaged on the CCD relating to the accurate position of the CCD with respect to the focal plane of the microscope field-lens, the absolute axial position of the beads with respect to the laser focus could not be determined with an absolute precision greater than 1

*μm*. However, the

*relative*precision of the axial position measurements is better than 0.3

*μm*.

### 3.3. Drag force experiment

*v*

_{r}of a trapped bead was increased with a well-defined constant acceleration

*a*

_{r}with the help of the computer-controlled motorized

*x*-

*y*stage, thus the instantaneous drag-force -β

*a*

_{r}(

*t*-

*t*

_{0}) applied to the particle was known at any time. From the recorded videos the

*x*-

*y*positions and the area

*A*of the bright spot could be recorded as a function of time. The experiment was repeated 5 times in each direction (+

*x*,-

*x*,+

*y*,-

*y*) using 5 and 7

*μm*beads, and the data for each direction was averaged. We decided for a trapping depth of

*b*≃ 10

*μm*, which could be ensured with an accuracy of ±2

*μm*.

*Hz*. At each time step the bead transverse and axial positions

*r*

_{0}(

*t*) and

*z*

_{0}(

*t*) are retrieved by image processing.

## 4. Results

### 4.1. Numerical results

*r*

_{0}, is presented in Fig. 6(a) (solid line) for the parameters given in Tab. 1 and in the pure ray-optics (RO) approximation. Note that the

*r*

_{0}and

*z*

_{0}positions of the sphere with respect to the beam focus are normalized by the sphere radius

*a*. The model predicts that, as the particle is displaced from the optical axis, its axial equilibrium position shifts in the positive (beam propagation) direction. The curve breaks at

*r*

_{0}/

*a*= 0.74 because beyond this limit there exists no stable axial equilibrium position.

*Q*

_{r}=

*cF*

_{r}/(

*n*

_{2}

*P*) - calculated along the equilibrium trajectory

*r*

_{0}). It has to be emphasized that the the maximal transverse trapping efficiency, predicted to be

*z*

_{0}= 0 for all

*r*

_{0}as depicted by the dashed line in Fig. 6(a)) the resulting transverse force profile - shown by the dashed line of Fig. 6(b) - predicts a greater maximal transverse force corresponding to

*r*

_{0}/

*a*= 0.97 (escape positions and force shown by diamonds ◇).

*μm*diameter polystyrene bead at two different trapping depths (

*d*= 5

*μm*and

*d*=15

*μm*). Figure 6(d) shows the corresponding transverse forces calculated using the VD model. As for the RO model, the forces computed along

*z*

_{0}= 0 (dashed lines) are compared to the transverse forces along the equilibrium trajectories

*r*

_{0}) (solid lines).

*r*

_{0}it moves back in the positive z-direction.

*r*

_{0}= [0,1.2

*a*] and

*z*

_{0}= [0,

*a*] to reduce the computation time. Intermediate values were linearly interpolated.

### 4.2. Experimental results

*μm*, is compared to the measurements. Although the overall theoretical escape trajectory is not in fair agreement with the observed trajectories, the theoretical escape transverse position derived from this calculation is in agreement with the experimental mean value of

*r*

_{0}/

*a*= 0.56±0.5 (standard deviation over the 20 drag-force experiments).

*μm*beads can be found in Fig. 7(c) trajectories) and (d) (transverse displacement-force curves). The measured axial equilibrium position dependence on the transverse position is qualitatively different from that of the 5

*μm*beads. A smaller absolute axial displacement is observed. Moreover, for small transverse displacements the axial equilibrium position first shifts in the negative direction, then at larger transverse positions moves back in the positive axial direction. The measured mean escape transverse position of

*r*

_{0}/

*a*= 0.59±0.6 is correctly predicted by the VD model. The computed axial equilibrium position dependence on the transverse position reproduces the main observed characteristics, including an oscillating behaviour as the distance from the optical axis is increased. Nevertheless, an offset between the measurements and the theoretical curve remains.

## 5. Discussion

*r*

_{0}/

*a*≃ 0.8-0.9. However, no

*axial*equilibrium is theoretically possible at that distance from the optical axis. The appropriate way for predicting the transverse forces - only been followed by Mazolli et al. [18

18. A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. **459**, 3021–3041 (2003). [CrossRef]

*r*

_{0}). Consequently, the maximal transverse force corresponds to the transverse force at the furthermost transverse position where an axial equilibrium exists.

*r*

_{0}/

*a*≃ 0.6, in far better agreement with the transverse escape positions predicted by the broken axial equilibrium approach (

*r*

_{0}/

*a*≃ 0.5-0.65) than with the escape positions predicted by the absolute maximum transverse force within the laser focusing plane (

*r*

_{0}/

*a*≃ 0.8- 0.9). Furthermore, the experimental maximal transverse trapping efficiencies of

*a*>> λ), it does not take into account spherical aberrations, and it is scale-invariant. For the 5 and 7

*μm*we used, the escape distance and the maximal transverse trapping forces are overestimated, despite considering the axial trapping failure.

*μm*diameter.

**61**, 569–582 (1992). [CrossRef] [PubMed]

*πa*/

*λ*greater than 10 to 20. This corresponds to a sphere diameter of more than 3 to 6 μm for a 1μm laser in water. Thus the model is used at the limit sphere-sizes for its validity, which may explain why measurements on 7μm beads are better reproduced by this theoretical model. Particle sizes of 5 and 7μm were chosen because using larger particles the inertial terms in the motion equations (10) may become significant. For smaller beads, on the other hand, the axial-position measurement technique we used was not applicable anymore, in addition to the fact that the ray-optics model for the force calculation may not be justified any longer.

*r*

_{0}is not monotonically increasing but rather shows oscillations. This effect may in part be attributed to the spherical aberrations created by the coverglass-water interface, as predicted by the VD model (Fig. 6(c)), and we observed that this effect was more pronounced at larger trapping depths. However, these oscillations are also very likely to be a signature of geometrical resonance [12

12. K. F. Ren, G. Greha, and G. Gouesbet, “Radiation Pressure Forces Exerted on a Particle Arbitrarily Located in a Gaussian-Beam by Using the Generalized Lorenz-Mie Theory, and Associated Resonance Effects,” Opt. Commun. **108**, 343–354 (1994). [CrossRef]

16. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

18. A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. **459**, 3021–3041 (2003). [CrossRef]

29. P. C. Ke and M. Gu, “Characterization of trapping force in the presence of spherical aberration,” J. Mod. Opt. **45**, 2159–2168 (1998) [CrossRef]

30. E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. **236**, 145 (2004). [CrossRef]

20. N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial
trapping efficiency,” J. Mod. Opt. **45**, 1943–1949 (1998). [CrossRef]

21. A. T. O’Neill and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre-Gaussian modes in inverted optical tweezers,” Opt. Commun. **193**, 45–50 (2001). [CrossRef]

17. S. Sato, M. Ishigure, and H. Inaba, “Optical Trapping and Rotational Manipulation of Microscopic Particles and Biological Cells Using Higher-Order Mode Nd-Yag Laser-Beams,” Electron. Lett. **27**, 1831–1832 (1991). [CrossRef]

*on*-

*axis*does not imply that the axial trapping efficiency

*off*-

*axis*is increased, or that the transverse position area presenting an axial equilibrium position is extended. Nevertheless, any precise mathematical comparison between the maximal transverse forces obtained with focused Gaussian beams and those obtained with other types of beams has to take these axial equilibrium conditions into account.

## 6. Conclusion

*r*

_{0}/

*a*≃ 0.6, indicating that the escape is due to a failure in the axial trapping at that distance from the optical axis, and not because the maximal transverse force is reached. We have proven by comparing a mathematical model to experimental measurements that any reliable mathematical prediction of the maximal transverse forces has to take these essential axial equilibrium issues into account.

## References and links

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

2. | M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, “Optical manipulation of objects and biological cells in microfluidic devices,” Biomed. Microdevices |

3. | J. Enger, M. Goksor, K. Ramser, P. Hagberg, and D. Hanstorp, “Optical tweezers applied to a microfluidic system,” Lab. Chip |

4. | J. Glückstad, “Microfluidics: Sorting particles with light,” Nat. Mater. |

5. | M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol. |

6. | S. L. Neale, M. P. Macdonald, K. Dholakia, and T. F. Krauss, “All-optical control of microfluidic components using form birefringence,” Nat. Mater. |

7. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

8. | 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. |

9. | G. Roosen, “La lévitation optique de sphères,” Can. J. Phys. |

10. | A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. |

11. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. |

12. | K. F. Ren, G. Greha, and G. Gouesbet, “Radiation Pressure Forces Exerted on a Particle Arbitrarily Located in a Gaussian-Beam by Using the Generalized Lorenz-Mie Theory, and Associated Resonance Effects,” Opt. Commun. |

13. | W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric Study of the Forces on Microspheres Held by Optical Tweezers,” Appl. Opt. |

14. | A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. |

15. | D. Ganic, X. S. Gan, and M. Gu, “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express |

16. | O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B |

17. | S. Sato, M. Ishigure, and H. Inaba, “Optical Trapping and Rotational Manipulation of Microscopic Particles and Biological Cells Using Higher-Order Mode Nd-Yag Laser-Beams,” Electron. Lett. |

18. | A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. |

19. | H. Felgner, O. Muller, and M. Schliwa, “Calibration of Light Forces in Optical Tweezers,” Appl. Opt. |

20. | N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial
trapping efficiency,” J. Mod. Opt. |

21. | A. T. O’Neill and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre-Gaussian modes in inverted optical tweezers,” Opt. Commun. |

22. | N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurements of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. |

23. | P. Torok, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation,” J. Opt. Soc. Am. A |

24. | P. Torok, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation -errata,” J. Opt. Soc. Am. A |

25. | S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in Confocal Fluorescence Microscopy Induced by Mismatches in Refractive-Index,” J. Microsc.-Oxford |

26. | K. C. Neuman, E. A. Abbondanzieri, and S. M. Block, “Measurement of the effective focal shift in an optical trap,” Opt. Lett. |

27. | J. Happel and H. Brenner, eds., |

28. | H. C. van de Hulst, “Light Scattering by Small Particles,” pp. 114–227 (Dover Press, New York, 1981). |

29. | P. C. Ke and M. Gu, “Characterization of trapping force in the presence of spherical aberration,” J. Mod. Opt. |

30. | E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. |

31. | Y. Roichman, A. Waldron, E. Gardel, and D. G. Grier, “Performance of optical traps with geometric aberrations,” Appl. Opt. , in press (2005). |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

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

**ToC Category:**

Trapping

**History**

Original Manuscript: January 4, 2006

Revised Manuscript: February 3, 2006

Manuscript Accepted: February 10, 2006

Published: February 20, 2006

**Virtual Issues**

Vol. 1, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Fabrice Merenda, Gerben Boer, Johann Rohner, Guy Delacrétaz, and René-Paul Salathé, "Escape trajectories of single-beam optically trapped micro-particles in a transverse fluid flow," Opt. Express **14**, 1685-1699 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1685

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### References

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- D. Ganic, X. S. Gan, and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory," Opt. Express 12, 2670-2675 (2004). [CrossRef] [PubMed]
- O. Moine and B. Stout, "Optical force calculations in arbitrary beams by use of the vector addition theorem," J. Opt. Soc. Am. B 22, 1620-1631 (2005). [CrossRef]
- S. Sato, M. Ishigure, and H. Inaba, "Optical Trapping and Rotational Manipulation of Microscopic Particles and Biological Cells Using Higher-Order Mode Nd-Yag Laser-Beams," Electron. Lett. 27, 1831-1832 (1991). [CrossRef]
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- N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, "Optical tweezers with increased axial trapping efficiency," J. Mod. Opt. 45, 1943-1949 (1998). [CrossRef]
- A. T. O’Neill and M. J. Padgett, "Axial and lateral trapping efficiency of Laguerre-Gaussian modes in inverted optical tweezers," Opt. Commun. 193, 45-50 (2001). [CrossRef]
- N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, "Measurements of trapping efficiency and stiffness in optical tweezers," Opt. Commun. 214, 15-24 (2002). [CrossRef]
- P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation," J. Opt. Soc. Am. A 12, 1605-1605 (1995). [CrossRef]
- P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic Diffraction of Light Focused through a Planar Interface between Materials of Mismatched Refractive-Indexes - an Integral-Representation -errata," J. Opt. Soc. Am. A 12, 1605-1605 (1995). [CrossRef]
- S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in Confocal Fluorescence Microscopy Induced by Mismatches in Refractive-Index," J. Microsc.-Oxford 169, 391-405 (1993). [CrossRef]
- K. C. Neuman, E. A. Abbondanzieri, and S. M. Block, "Measurement of the effective focal shift in an optical trap," Opt. Lett. 30, 1318-1320 (2005). [CrossRef] [PubMed]
- J. Happel and H. Brenner, eds., Low Reynolds Number Hydrodynamics, 2nd ed. (Kluwer Academic, Dordecht, the Netherlands, 1991).
- H. C. van de Hulst, "Light Scattering by Small Particles," pp. 114-227 (Dover Press, New York, 1981).
- P. C. Ke and M. Gu, "Characterization of trapping force in the presence of spherical aberration," J. Mod. Opt. 45, 2159-2168 (1998) [CrossRef]
- E. Theofanidou, L. Wilson,W. J. Hossack and J. Arlt, "Spherical aberration correction for optical tweezers," Opt. Commun. 236, 145 (2004). [CrossRef]
- Y. Roichman, A. Waldron, E. Gardel and D. G. Grier, "Performance of optical traps with geometric aberrations," Appl. Opt., in press (2005).

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