## Optical micromanipulation using supercontinuum Laguerre-Gaussian and Gaussian beams

Optics Express, Vol. 16, Issue 14, pp. 10117-10129 (2008)

http://dx.doi.org/10.1364/OE.16.010117

Acrobat PDF (1353 KB)

### Abstract

We characterize a single beam supercontinuum “white light” trap and determine the trap stiffness in the transverse trapping plane. We realize a holographic white light trapping system using a spatial light modulator, and explore the generation of a dual beam trap and characterize its performance. We also demonstrate optical trapping and rotation of particles using a supercontinuum vortex beam. It is shown that orbital angular momentum can be transferred to spheres trapped in a supercontinuum vortex. Quantified rotation rates are demonstrated.

© 2008 Optical Society of America

## 1. Introduction

## 2. Theoretical considerations

### 2.1 Laguerre-Gaussian beams

*l*. The most general LG beams posses a number (

*p*+1) of rings, where

*p*is the radial mode index. Here, we only consider single ringed LG beams (

*p*=0) with a vortex line on the

*z*-axis defined by the complex solution of the paraxial equation [14]:

*z*=

_{r}*n*

_{0}

*k*

_{0}

*w*

^{2}

_{0}/2 is the Rayleigh range,

*w*the waist of the Gaussian envelope,

_{0}*k*the vacuum wavevector,

_{0}*n*the index of refraction of the host media and

_{0}*u*the amplitude. The first coefficient gives the longitudinal shape and decay of the beam including the LG Guoy phase shift, while the second coefficient describes the vortex part of the LG beam.

_{0}15. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. **66**, 2800–2802 (1989). [CrossRef]

**A**=(

*u*(

^{l}_{p=0}*x,y,z*)exp(

*-in*

_{0}

*k*

_{0}

*z*),0,0), by:

*x*-axis, and

*Z*is the vacuum impedance. This fully vectorial first order description of the LG beam can further be improved by introducing an additional vector potential for the magnetic field, which renders the relationship between the electric field

_{0}**E**and magnetic field

**H**more symmetric. Integrating the optical period averaged Poynting vector <

**S**>=Re(

**E**×

**H***)/2 (the asterisk denotes the complex conjugate and the chevron denotes a time average) over the azimuthal (transversal) plane gives the total power

*P*of the beam, which can be used to calculate the amplitude coefficient [

_{0}*u*in Eq. (1)]

_{0}*Z*is the wave impedance and

*s*=1/(

*n*

_{0}

*k*

_{0}

*w*

_{0}) is the Gaussian beam order parameter. This amplitude coefficient includes helicity changes and corrects for relatively tightly focussed LG beams as a function of the order parameter

*s*.

*iωt*), where

*ω*is the optical frequency and is given by the dispersion relation:

*ω*=

*ck*

_{0}where

*c*is the speed of light. However, in the case of the supercontinuum we need to sum over all the spectral components when defining the vector potential,

*A*(

*ω*) is the spectral amplitude of the supercontinuum and where the parenthesis contain the three vector components of the vector field.

### 2.2 Optical force and torque in the supercontinuum beam

*f*=

_{i}*-∂*-

_{t}g_{i}*∂*over the volume of the particle and average over the pulse duration. The subscripts i, j and k varying from 1 to 3, correspond to the three Cartesian coordinates. We further assume summation over repeating indices in products. The two parts in the force density correspond to the force arising from the variation of the electromagnetic momentum density,

_{j}T_{ij}*g*=

_{i}*ε*, and to the influx of momentum given by the divergence of Maxwell’s momentum-stress tensor [16,17

_{ijk}D_{j}B^{*}_{k}17. R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. **79**, 1197–1216 (2007). [CrossRef]

*E*,

_{i}*D*,

_{i}*H*and

_{i}*B*denote the electric field, the electric displacement, the magnetic field and magnetic flux respectively, and where we consider linear constitutive relations in SI units such as

_{i}*D*=

_{i}*ε*

_{r}ε_{0}

*E*and

_{i}*B*=

_{i}*µ*

_{r}µ_{0}

*H*. In these constitutive relations

_{i}*ε*,

_{r}*ε*

_{0}and

*µ*

_{0}refer to the relative dielectric constant, the vacuum permittivity and permeability.

*t*, is:

*g*, cancels out in the averaging process. The surface integral is evaluated on a closed surface surrounding the particle with

_{i}*ds*, which denotes the normal surface vector pointing outwards. The pulse can be seen as the superposition of many monochromatic waves [Eq. (4)] and the total average force becomes:

_{j}*ε*is the Levi-Civita antisymmetric tensor.

_{ijk}### 2.3 Numerical application

18. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A: Pure Appl. Opt. **9**, S196–S203 (2007). [CrossRef]

*n*=1.59) as a function of the wavelength (Fig. 1), and then used Eq. (7) to sum the spectral contributions of the SC beam. Here, we consider a linearly polarized Gaussian beam having a waist of

*w*=1.75µm. The 1µm polymer sphere is free to move in the focal plane of the beam and we assume the spheres to have a constant refractive index across the wavelength range of interest. We determined the global, transversal trap stiffness of 1.19pN/µm in the polarization direction and 1.14pN/µm in the perpendicular direction.

_{0}## 3. Experiments

### 3.1 Experimental setup

## 4. Results

### 4.1 Single beam trap stiffness

11. P. Li, K. Shi, and Z. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Lett. **30**, 156–158 (2005). [CrossRef] [PubMed]

*γ*

_{0}=6

*πηa*links the velocity to the force through the relation

*F*=

*-γ*

_{0}

*ν*. Here,

*a*and

*ν*are respectively the radius and velocity of the particle, and

*η*is the viscosity of the liquid (8.9×10

^{-4}sPa for water at 298K) [20

20. J. Kestin, J. V. Sengers, B. Kamgar-Parsi, and J. M. H. Levelt Sengers, “Thermophysical properties of fluid H2O,” J. Phys. Chem. Ref. Data **13**, 175–183 (1984). [CrossRef]

*f*is a random particle force having a Gaussian distribution with the moments <

_{b}(t)*f*(

_{b}*t*)>=0 and <

*f*(

_{b}*t*)

*f*(

_{b}*t*′)>=2

*γ*

_{0}

*k*. Here

_{B}T*k*and

_{B}*T*are respectively Boltzmann’s constant and the absolute temperature [21

21. P. Bartlett, S. I. Henderson, and S. J. Mitchell, “Measurement of the hydrodynamic forces between two polymer-coated spheres,” Philos. Trans. R. Soc. London, Ser. A **359**, 883–895 (2001). [CrossRef]

*k*is the optical spring constant. The motion of the particle is overdamped leading to the inertial term, the first term in the Langevin equation, to cancel out. The stochastic nature of this equation makes it possible to treat the solution of this equation, thus providing the particle position probability [22

22. W. P. Wong and K. Halvorsen, “The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur,” Opt. Express **14**, 12517 (2006). [CrossRef] [PubMed]

*U(x)*is the optical trapping potential.

*x*and

*y*directions,

*k*

_{x}=1.1pN/µm and

*k*

_{y}=1.3pN/µm respectively. The precise supercontinuum beam profile in the trapping plane is unknown due to uncertainty of the distance between the top coverslip and the focal plane of the Gaussian beam. However, a Gaussian beam with a spot radius of

*w*=1.75µm gives the correct trap stiffness within 10%.

_{z}## 5. Holographic supercontinuum trapping

### 5.1 Introduction

23. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. **24**, 608–610 (1999). [CrossRef]

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

25. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. **185**, 77–82 (2000). [CrossRef]

26. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*l*ħ per photon regardless of wavelength, where

*l*is the (integer) azimuthal index. A number of studies have seen the transfer of OAM from a monochromatic vortex to trapped particles by absorption [6

6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

8. V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

27. K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B **4**, S82–S89 (2002). [CrossRef]

### 5.2 Experiments

#### 5.2.1 Holographic experimental setup

#### 5.2.2 Dual beam supercontinuum trapping

*w*=1.2µm.

_{0}*n*=1.59) immersed in water, where trapping occurred up against the top coverslip in the sample chamber. As for the single beam trap, we estimated the individual trap stiffness for each beam by using their respective particle position histograms shown in Fig. 8.

*w*=1.1µm. Using Fig. 5(b) to estimate the power ratio between the two traps, we find an average trap stiffness of 16.5pN/µm for the upper trap and 15pN/µm for the lower trap.

_{0}#### 5.2.3 Supercontinuum vortex trapping

28. H. I. Sztul, V. Kartazayev, and R. R. Alfano, “Laguerre-Gaussian supercontinuum,” Opt. Lett. **31**, 2725–2727 (2006). [CrossRef] [PubMed]

*l*=3,

*p*=0) was generated using the SLM as shown in Fig. 5(c). All SC vortices generated were analyzed [28

28. H. I. Sztul, V. Kartazayev, and R. R. Alfano, “Laguerre-Gaussian supercontinuum,” Opt. Lett. **31**, 2725–2727 (2006). [CrossRef] [PubMed]

29. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. **45**, 1231–1237 (1998). [CrossRef]

*l*=3,

*p*=0,1,2,…) where more than 50% of the power is in the

*p*=0 mode. We used the measured beam profile to decompose the observed beam into the higher

*p*modes.

*n*=1.59). The spheres were trapped against the top coverslip of the sample and the optical confinement occurred in 2D. Rotation rates for the case of three trapped spheres, (inset Fig. 9), were determined using a fast camera (Basler pioneer plA640-210gm). The videos were then broken up into their individual frames and analyzed using a Mathematica tracking program to determine the rotation rates at different powers (Fig. 9). Each point on the graph is averaged over at least 50 rotations with a 10% error.

*p*=0 LG mode we find a rotation rate efficiency of 0.39Hz/mW. Some data point error bars lie outside the linear fit, which could be due to slight differences in the surface friction between measurements.

*l*=2,3,4 and 5 were generated using the SLM. The beam spot radius was determined by measuring the annular profile of the LG beam, see Fig. 10(a). We ensured that each spectral component of the temporally incoherent beam had an azimuthal phase step of appropriate order [30

30. P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, “Propagation and diffraction of Optical Vortices,” Physica C **468**, 514–517 (2008). [CrossRef]

*l*-values. Figure 10(b) shows the rotation rates of three spheres trapped within the LG beam as a function of

*l*. Again, at least 50 rotations were taken for each rotation rate at a power of 15mW measured in the trapping plane; the error was taken as 10% of the rotation rate.

*p*=0.

## 6. Conclusions

32. J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. **82**, 2211–2214 (1999). [CrossRef]

## Acknowledgments

## References and links

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

2. | K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. |

3. | E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. |

4. | K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett. |

5. | M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science |

6. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

7. | M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature |

8. | V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A |

9. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. |

10. | V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. |

11. | P. Li, K. Shi, and Z. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Lett. |

12. | P. Fischer, A. E. Carruthers, K. Volke-Sepulveda, E. M. Wright, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Enhanced optical guiding of colloidal particles using a supercontinuum light source,” Opt. Express |

13. | M. Guillon, K. Dholakia, and D. McGloin, “Optical trapping and spectral analysis of aerosols with a supercontinuum laser source,” Opt. Express |

14. | A. E. Siegman, |

15. | J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. |

16. | J. D. Jackson, |

17. | R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. |

18. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A: Pure Appl. Opt. |

19. | J. W. Goodman, |

20. | J. Kestin, J. V. Sengers, B. Kamgar-Parsi, and J. M. H. Levelt Sengers, “Thermophysical properties of fluid H2O,” J. Phys. Chem. Ref. Data |

21. | P. Bartlett, S. I. Henderson, and S. J. Mitchell, “Measurement of the hydrodynamic forces between two polymer-coated spheres,” Philos. Trans. R. Soc. London, Ser. A |

22. | W. P. Wong and K. Halvorsen, “The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur,” Opt. Express |

23. | M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. |

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

25. | J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. |

26. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

27. | K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B |

28. | H. I. Sztul, V. Kartazayev, and R. R. Alfano, “Laguerre-Gaussian supercontinuum,” Opt. Lett. |

29. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. |

30. | P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, “Propagation and diffraction of Optical Vortices,” Physica C |

31. | A. V. Filippov, “Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number,” J. Colloid Interface Sci. |

32. | J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. 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:**

Optical Trapping and Manipulation

**History**

Original Manuscript: April 8, 2008

Revised Manuscript: May 27, 2008

Manuscript Accepted: June 11, 2008

Published: June 23, 2008

**Virtual Issues**

Vol. 3, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer, and K. Dholakia, "Optical micromanipulation using supercontinuum Laguerre-Gaussian and
Gaussian beams," Opt. Express **16**, 10117-10129 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10117

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### References

- A. Ashkin, J. M. Dzeidzic, 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]
- K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008). [CrossRef] [PubMed]
- E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001). [CrossRef]
- K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Pattern formation and flow control of fine particles by laser-scanning micromanipulation," Opt. Lett. 16, 1463-1465 (1991). [CrossRef] [PubMed]
- M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002). [CrossRef] [PubMed]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995). [CrossRef] [PubMed]
- M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-350 (1998). [CrossRef]
- V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]
- A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002). [CrossRef] [PubMed]
- V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003). [CrossRef] [PubMed]
- P. Li, K. Shi, and Z. Liu, "Manipulation and spectroscopy of a single particle by use of white-light optical tweezers," Opt. Lett. 30, 156-158 (2005). [CrossRef] [PubMed]
- P. Fischer, A. E. Carruthers, K. Volke-Sepulveda, E. M. Wright, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Enhanced optical guiding of colloidal particles using a supercontinuum light source," Opt. Express 14, 5792-5802 (2006). [CrossRef] [PubMed]
- M. Guillon, K. Dholakia, and D. McGloin, "Optical trapping and spectral analysis of aerosols with a supercontinuum laser source," Opt. Express 16, 7655-7664 (2008). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986).
- J. P. Barton, and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999), Third Ed., p. 261.
- R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007). [CrossRef]
- T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical tweezers computational toolbox," J. Opt. A: Pure Appl. Opt. 9, S196-S203 (2007). [CrossRef]
- J. W. Goodman, Statistical Optics (Wiley-Interscience 1985), First Ed., p. 224.
- J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984). [CrossRef]
- P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001). [CrossRef]
- W. P. Wong, and K. Halvorsen, "The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur," Opt. Express 14, 12517 (2006). [CrossRef] [PubMed]
- M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999). [CrossRef]
- J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002). [CrossRef]
- J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
- K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002). [CrossRef]
- H. I. Sztul, V. Kartazayev, and R. R. Alfano, "Laguerre-Gaussian supercontinuum," Opt. Lett. 31, 2725-2727 (2006). [CrossRef] [PubMed]
- J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998). [CrossRef]
- P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008). [CrossRef]
- A. V. Filippov, "Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number," J. Colloid Interface Sci. 229, 184-195 (2000). [CrossRef] [PubMed]
- J. C. Meiners, and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999). [CrossRef]

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