## Transverse particle dynamics in a Bessel beam

Optics Express, Vol. 15, Issue 21, pp. 13972-13987 (2007)

http://dx.doi.org/10.1364/OE.15.013972

Acrobat PDF (1125 KB)

### Abstract

Spatially periodic optical fields can be used to sort dielectric microscopic particles as a function of size, shape or refractive index. In this paper we elucidate through both theory and experiment the behavior of silica microspheres moving under the influence of the periodic optical field provided by a Bessel beam. We compare two different computational models, one based on Mie scattering, the other on geometrical ray optics and find good qualitative agreement, with both models predicting the existence of distinct size-dependent phases of particle behavior. We verify these predictions by providing experimental observations of the individual behavioral phases.

© 2007 Optical Society of America

## 1. Introduction

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

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

03. C. Bustamante, Z. Bryan, and S. B. Smith, “Ten years of tension: single-molecule DNA mechanics,” Nature **421**, 423 (2003). [CrossRef] [PubMed]

04. K. Dholakia and P. Reece, “Optical micromanipulation takes hold,” Nano Today **1**, 18 (2006). [CrossRef]

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

*optical landscapes*) have been used to arrange and accumulate microparticles in pre-described arrays [6

06. P. T. Korda, C. Spalding, and D.G. Grier, “Evolution of a colloidal critical state in an optical pinning potential landscape,” Phys. Rev. B **66**, 024504 (2002). [CrossRef]

07. M.P. MacDonald, G.C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature **426**, 421 (2003). [CrossRef] [PubMed]

08. M. Pelton, K. Ladavac, and D. G. Grier, “Transport and fractionation in periodic potential-energy landscapes,” Phys. Rev. E **70**, 031108 (2004). [CrossRef]

09. I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “A modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. **88**, 121116 (2006). [CrossRef]

10. T. Cizmar, M. Siler, M. Sery, P. Zemanek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B **74**, 035105 (2006). [CrossRef]

11. S. Lee and D. G. Grier, “One-dimensional optical thermal ratchets,” J. Phys.: Condensed Matter **17**, S3685–S3695 (2006). [CrossRef]

13. P. Zemanek, A. Jonas, and M. Liska, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A **19**, 1025 (2002). [CrossRef]

15. T. Cizmar, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects”, Appl. Phys. Lett. **86**, 174101 (2005). [CrossRef]

19. K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multi-ringed light beams to spherical microparticles,” J. Opt. Soc. Am. B **21**, 1749 (2004). [CrossRef]

*ad hoc*tailored optical landscapes that may give rise to several potential applications, particularly with relation to optical sorting.

## 2. Optical forces and potentials: theoretical modeling

20. J. Durnin, “Exact solutions for diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651 (1987). [CrossRef]

21. D. McGloin and K. Dholakia, “Bessel Beams: Diffraction in a new light,” Contemp. Phys. **46**, 15 (2005). [CrossRef]

**k**of the plane waves that combine to form a Bessel beam as lying on the surface of a cone. The apex angle

*α*of this imaginary cone provides us with the expressions

*k*=

_{r}*k*sin(

*α*) and

*k*=

_{z}*k*cos(

*α*) for the radial and axial components of the wave vectors. The electric component of the optical field of an ideal zeroth-order Bessel beam is given in the paraxial approximation by

*E*is the amplitude of the electric field on the optical axis of the Bessel beam,

_{0B}*J*

_{0}is the first class Bessel function of zeroth-order and

*ω*is the angular frequency of the field. Experimentally, a reasonable and efficient approximation to an ideal Bessel beam can be generated within a finite region of space by illuminating a conical lens – also called an

*axicon*– with a well-collimated Gaussian beam. The distance over which this Bessel beam can be considered non-diffracting (Fig. 1(a)) is commonly known as the

*maximum propagation distance*and is denoted by

*z*. The corresponding intensity distribution can be approximated using the stationary phase method to yield [22

_{max}22. J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A **63**, 063602 (2001). [CrossRef]

*P*represents the power of the incident Gaussian beam, while

*w*is the beam’s half-width. The maximum propagation distance

_{c}*z*can be expressed as:

_{max}*α*,

*α*≈(

*n*-1)

_{ax}*γ*in air and

*z*≈

_{max}*w*/[(

_{c}*n*-1)

_{ax}*γ*].

*n*is the refractive index of the axicon and

_{ax}*γ*is its opening angle.

*J*

_{0}(

*k*) in Eq. (2) the characteristic transverse dimensions of the beam depend on

_{r}r*k*and consequently on the axicon parameter

_{r}*γ*. In practice a system of lenses can be added after the axicon to reduce the ring structure to a scale appropriate for optical micro-manipulation. We define the width of a particular ring as the distance between two consecutive radial intensity minima. Theoretically, the width of the

*n*-th Bessel beam ring is given by

*Δρ*= (

_{n}*Δx*/

_{n}*k*), where

_{r}*Δx*= (

_{n}*x*

_{n+1}-

*x*) represents the separation between two consecutive roots in the zero-order Bessel function. As

_{n}*n*→ ∞,

*Δx*quickly converges to

_{n}*π*and we can approximate

*λ*is the wavelength in the medium in which the Bessel beam is formed.

*z*plane at the position of maximum intensity along the beam propagation axis. From Eq. (2) this position is given by

*z*=

_{peak}*z*/2. It is important to note that because the Bessel beam does not change significantly over small displacements along the direction of propagation, slight movement of a microscopic particle along

_{max}*z*will not cause significant variations in the associated optical potential. This is not the case with conventional optical traps [1

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

12. L. McCann, M.I. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature **402**, 785 (1999). [CrossRef]

*R*, with respect to the wavelength

_{0}*λ*of the light in the medium. In the Rayleigh regime (

*R*≪

_{0}*λ*), the particles are implicitly very small and can be approximated as point dipoles. In this case the shape of the optical force and potential energy distributions as a function of the particle’s position turns out to be independent of the particle size or geometry, and depends only on the light distribution itself [24,25

25. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. **25**, 1085 (2000). [CrossRef]

*R*≈

_{0}*λ*or

*R*≫

_{0}*λ*forces can be calculated by a rigorous electromagnetic model based on Mie scattering theory [16

16. T. Cizmar, V. Kollarova, Z. Bouchal, and P. Zemanek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. **8**, 43 (2006). [CrossRef]

26. 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 (1989). [CrossRef]

19. K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multi-ringed light beams to spherical microparticles,” J. Opt. Soc. Am. B **21**, 1749 (2004). [CrossRef]

27. V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A **66**, 063402 (2002). [CrossRef]

28. R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B **9**, 1922 (1992). [CrossRef]

16. T. Cizmar, V. Kollarova, Z. Bouchal, and P. Zemanek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. **8**, 43 (2006). [CrossRef]

26. 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 (1989). [CrossRef]

**e**

*are the cartesian unit vectors and*

_{x,y,z}*ϕ*is the azimuthal angle in the beam-centred cylindrical system of coordinates (

*ϕ*=0 implies that the x-axis is aligned along the direction of the linear polarization of the beam incident on the axicon).

*E*is the electric field amplitude of this non-diffracting beam on the optical axis and

_{B0}*J*is the first class Bessel function of

_{m}*m*th order. The parameter

*β*= sin

*α*/( 1 + cos

*α*) is responsible for an asymmetry in this non-diffracting beam which becomes more significant as the angle

*α*increases. In the limit

*β*→ 0, we obtain a scalar electric field that has the same spatial transversal profile as the Bessel beam considered in Eq. (1).

*ξ*=

*2πR*/

_{0}*λ*,

*E*and

^{i}_{r}*B*are the radial components of the electric and magnetic fields respectively, incident on the surface of the spherical particle (expressed in spherical coordinates as (

^{i}_{r}*R*,

_{0}*φ*,

*θ*) with the origin at the sphere centre),

*Y*

^{*}

*are the complex-conjugated spherical harmonics and*

_{lm}*ψ*is the Ricatti-Bessel function. In this description the beam is not strictly azimuthally symmetrical and

*φ*is the azimuthal angle of the centre of the sphere in the cylindrical system of coordinates of the beam. The forces and torques acting on a spherical object can be calculated from the equations presented by Barton [26

_{0}26. 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 (1989). [CrossRef]

*A*and

_{tm}*B*and require only a single integration [16

_{tm}16. T. Cizmar, V. Kollarova, Z. Bouchal, and P. Zemanek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. **8**, 43 (2006). [CrossRef]

19. K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multi-ringed light beams to spherical microparticles,” J. Opt. Soc. Am. B **21**, 1749 (2004). [CrossRef]

27. V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A **66**, 063402 (2002). [CrossRef]

*z*). The basic equation for obtaining the radial optical force exerted on a dielectric sphere of radius R0 can be written as

*θ*∈[0,π],

*φ*∈[0,2π]; where

*θ*and

*φ*are the polar and azimuthal angles, respectively, in spherical coordinates. As we have assumed that all of the light rays arrive parallel to beam axis,

*θ*coincides with the incidence angle at each point on the sphere’s surface and the transmitted angle is denoted by

*θ*.

_{t}*R*and

*T*represent the reflectance and transmittance coefficients averaged over the two transverse polarization directions at each point and

*c*/

*n*represents the speed of light in the surrounding medium. We use Eq. (2) here to describe the intensity of the incident beam.

_{m}*r*is independent of the azimuthal coordinate and is defined as

## 3. Experimental set-up

*γ*=1°. A pair of lenses reduced the dimensions of the Bessel beam so that the typical transverse ring thickness was of the order of the diameter of the particles under examination. Great care was taken to ensure that the intensity distribution was azimuthally invariant. A dielectric mirror directed the beam into the sample chamber while allowing white light back illumination to enter the system. A half-wave plate and polarizing beam-splitting cube allowed precise control over the power of the beam without altering the operating characteristics of the laser. The sample plane was imaged and observed by means of a 100x oil-immersion objective and a Pulnix PE2015 CCD camera. A second dielectric mirror diverted most of the Bessel beam away from the camera, transmitting only a small portion of the beam through onto the camera surface. The video signal was fed into a computer via a National Instruments IMAQ PCI-1408 video capture card and stored for subsequent analysis. A single-beam optical trap was also incorporated into the set-up in order to allow precise and consistent positioning of individual particles at predetermined regions of interest within the Bessel beam profile. For this purpose, part of the output laser beam was diverted before the axicon and redirected through the back aperture of the imaging objective. A system of mirrors and lenses made the optical tweezers fully steerable in the sample plane. Following particle placement at a specific location within the Bessel beam, the tweezers beam was blocked, allowing the natural evolution of particle motion in the Bessel landscape to be monitored. Silica spheres with mean radii of R

_{0}=1.15±0.12μm, R

_{0}=2.5±0.21μm and R

_{0}=3.42±0.29μm (Bangs Laboratories) were suspended in deuterium oxide (D

_{2}O). D

_{2}O was used instead of H

_{2}O to minimize the effect of heating due to absorption of laser light at the wavelength of 1070nm. We did not consider spheres smaller than 2.3μm because their more pronounced Brownian motion and tendency to diffuse away from the substrate made them difficult to follow using video tracking. Dilute colloidal samples were used to avoid any unwanted colloid-colloid interactions. The silica particles were denser than the surrounding medium and settled quickly to the bottom of the sample cell. The Bessel beam illuminated the sample from below. Elevation can occur when the particles reach the core [18

18. L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A.C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. **87**, 123901 (2005). [CrossRef]

29. G. Milne, “St Andrews Tracker,” http://faculty.washington.edu/gmilne/tracker.htm.

## 4. Results and discussion

### 4.1 Theoretical results

*R*

_{0}=1.15μm (Fig. 3) and a Bessel beam with a characteristic ring width of

*Δρ*=3.1μm. Equilibrium points are represented in the Fig. 3. by circles and correspond closely to local intensity maxima (the dashed curve represents the transverse radial intensity profile.)

*untilted*, we still observe a washboard-like potential, with potential energy barriers lower on the inner side of each well (Fig. 3(b), inset). While we would expect small particles to remain radially localized at the potential energy minima, thermally activated escape from the well remains a possibility. Due to the lowering of the potential barriers on the inner side of each well, we expect to see particles move preferentially towards the beam core.

_{0}=2.15μm (Fig. 4(a, b)), the positive radial force regimes and corresponding potential energy barriers seen in Fig. 3 disappear. These particles are free to drift into the core of the Bessel beam without obstruction. This phenomenon is similar to the observed behavior of spheres in spatially periodic fields (standing waves) where spheres of a certain size do not feel the periodic field structure [9

09. I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “A modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. **88**, 121116 (2006). [CrossRef]

10. T. Cizmar, M. Siler, M. Sery, P. Zemanek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B **74**, 035105 (2006). [CrossRef]

13. P. Zemanek, A. Jonas, and M. Liska, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A **19**, 1025 (2002). [CrossRef]

15. T. Cizmar, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects”, Appl. Phys. Lett. **86**, 174101 (2005). [CrossRef]

_{0}=2.5μm. The potential energy wells associated with this phenomenon (Fig. 4(d), inset) are significantly shallower than those associated with the R

_{0}=1.15μm sphere (Fig. 3(b), inset). For example, at a radial position of r=10.4μm, we predict the R

_{0}=1.15μm sphere to experience a potential well with a depth of 166kT. The R

_{0}=2.5μm sphere, on the other hand, will experience a well at r=11.6μm with a depth of 34kT, almost five times shallower.

_{0}=3.42μm. In this case the core equilibrium position of the sphere is offset from the beam core by 1.16μm. This behavior is highlighted in Fig. 6 and in the associated movie. Similar effects have been observed both theoretically and experimentally in spatially periodic fields [9

09. I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “A modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. **88**, 121116 (2006). [CrossRef]

10. T. Cizmar, M. Siler, M. Sery, P. Zemanek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B **74**, 035105 (2006). [CrossRef]

13. P. Zemanek, A. Jonas, and M. Liska, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A **19**, 1025 (2002). [CrossRef]

15. T. Cizmar, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects”, Appl. Phys. Lett. **86**, 174101 (2005). [CrossRef]

### 4.2 Comparison with experiment

_{0}=1.15μm), positive radial force regimes exist across the profile. Stable equilibrium points exist at the roots of the force plot where the slope of the curve is negative. As expected, these positions correspond to local minima in the radial potential energy plot. For these minima, the height of the inner potential energy barrier (towards the beam core) is lower than the outer barrier. Under these circumstances, the particle behaves as it would in a tilted washboard potential [17

17. S. A. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. **91**, 038101 (2003). [CrossRef] [PubMed]

*hop*from one adjacent ring to another. We verified this behavior by recording the transitions of a particle between adjacent rings. A trajectory typical of a R

_{0}=1.15μm particle is illustrated in Fig. 7. The particles are confined in the radial direction, but are free to move azimuthally around the rings. This image shows two consecutive hops from the fifth ring to the third ring. The hopping events occur swiftly and are easily identifiable.

_{0}=1.15μm sphere at a precise point in the centre of the fifth ring. Blocking the tweezers beam released the particle and initiated a timer. Using the same sphere, 200 transition times were measured for four different Bessel beam power levels.

*τ*in. by fitting curves of the form (1 /

*τ*) exp(-

*t*/

*τ*) to the histograms. We see that the mean first passage time increases with increasing beam power, as we might expect.

_{0}=2.5μm, we observed that, as predicted, equilibrium positions exist in a configuration whereby the particle straddles two of the rings. At distances less than 8μm from the core our model predicts no stable equilibrium points and we observed experimentally that in this range the particle does indeed travel into the core of the beam uninhibited. This can be seen in Fig. 9(a), which represents a 21-second video sequence. The particle is briefly localized between the 2

^{nd}and 3

^{rd}rings (r≈8μm), which agrees well with our prediction of the presence of a shallow potential well at this location.

_{0}=1.15μm particle, the potential energy minima for R

_{0}=2.5μm correspond spatially to local intensity

*minima*of the beam profile. This scenario is similar to the case of spheres placed in a one-dimensional tilted standing wave, with all the consequences [13

**19**, 1025 (2002). [CrossRef]

14. P. Jakl, M. Sery, J. Jezek, A. Jonas, M. Liska, and P. Zemanek, “Behaviour of an optically trapped probe approaching a dielectric interface,” J. Mod. Opt. **50**, 1615 (2003). [CrossRef]

_{0}=1.15μm spheres at a similar radial distance and overall beam power.

_{0}=3.42μm, the core equilibrium position will be offset from the centre of the beam by 1.15μm (Fig. 5). The offset equilibrium position has been observed experimentally with excellent agreement and a typical particle trajectory can be seen in Fig. 10. The mean radial core separation distance for this stably trapped particle is ∼1μm.

### 4.3 Comparison with geometric ray optics model

_{0}< 5μm, we observe reasonable agreement between the radial force curves predicted by our two models (Fig. 11). Although there are some clear discrepancies in the absolute value of the force predicted by the two models for certain combinations of particle size and radial position (e.g. Fig. 11(b), r=2), for potential sorting applications the primary concern is the location of stable equilibrium points. For R

_{0}< 5μm, the equilibrium points overlap very closely for the two models.

### 4.4 Static Optical Sorting in a Bessel Beam

30. N. Chattrapiban, E.A. Rogers, D. Cofield, W.T. Hill, and R. Roy, “Generation of nondiffracting bessel beams by use of a spatial light modulator,” Opt. Lett. **28**, 2183 (2003). [CrossRef] [PubMed]

## 5. Conclusion

**21**, 1749 (2004). [CrossRef]

## Acknowledgments

^{th}Framework Progamme. We also thank the fp6 NEST ADVENTURE programme, through project ATOM3D (no. 508952), for supporting this work. Kishan Dholakia and Karen Volke-Sepulveda acknowledge the support of the Royal Society. Pavel Zemanek acknowledges the support of MEYS through the Centre of Modern Optics (LC06007) and MCT through the project FT-TA2/059. David McGloin is a Royal Society University Research Fellow.

## References and links

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

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

03. | C. Bustamante, Z. Bryan, and S. B. Smith, “Ten years of tension: single-molecule DNA mechanics,” Nature |

04. | K. Dholakia and P. Reece, “Optical micromanipulation takes hold,” Nano Today |

05. | MJ. Enger, M. Goksör, K. Ramser, P. Hagberg, and D. Hanstorp, “Optical tweezers applied to a microfluidic system,” Lab on a Chip |

06. | P. T. Korda, C. Spalding, and D.G. Grier, “Evolution of a colloidal critical state in an optical pinning potential landscape,” Phys. Rev. B |

07. | M.P. MacDonald, G.C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature |

08. | M. Pelton, K. Ladavac, and D. G. Grier, “Transport and fractionation in periodic potential-energy landscapes,” Phys. Rev. E |

09. | I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “A modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. |

10. | T. Cizmar, M. Siler, M. Sery, P. Zemanek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B |

11. | S. Lee and D. G. Grier, “One-dimensional optical thermal ratchets,” J. Phys.: Condensed Matter |

12. | L. McCann, M.I. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature |

13. | P. Zemanek, A. Jonas, and M. Liska, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A |

14. | P. Jakl, M. Sery, J. Jezek, A. Jonas, M. Liska, and P. Zemanek, “Behaviour of an optically trapped probe approaching a dielectric interface,” J. Mod. Opt. |

15. | T. Cizmar, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects”, Appl. Phys. Lett. |

16. | T. Cizmar, V. Kollarova, Z. Bouchal, and P. Zemanek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. |

17. | S. A. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. |

18. | L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A.C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. |

19. | K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multi-ringed light beams to spherical microparticles,” J. Opt. Soc. Am. B |

20. | J. Durnin, “Exact solutions for diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

21. | D. McGloin and K. Dholakia, “Bessel Beams: Diffraction in a new light,” Contemp. Phys. |

22. | J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A |

23. | D. McGloin, G.C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Comm. |

24. | K. Visscher and G.J. Brakenhoff A theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers. Optik |

25. | P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. |

26. | 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. |

27. | V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A |

28. | R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B |

29. | G. Milne, “St Andrews Tracker,” http://faculty.washington.edu/gmilne/tracker.htm. |

30. | N. Chattrapiban, E.A. Rogers, D. Cofield, W.T. Hill, and R. Roy, “Generation of nondiffracting bessel beams by use of a spatial light modulator,” Opt. Lett. |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(140.7010) Lasers and laser optics : Laser trapping

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Trapping

**History**

Original Manuscript: August 21, 2007

Revised Manuscript: October 2, 2007

Manuscript Accepted: October 3, 2007

Published: October 9, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Graham Milne, Kishan Dholakia, David McGloin, Karen Volke-Sepulveda, and Pavel Zemánek, "Transverse particle dynamics in a Bessel beam," Opt. Express **15**, 13972-13987 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-21-13972

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### References

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