## Theory of holographic optical trapping

Optics Express, Vol. 16, Issue 20, pp. 15765-15776 (2008)

http://dx.doi.org/10.1364/OE.16.015765

Acrobat PDF (274 KB)

### Abstract

Optical traps use the forces exerted by structured beams of light to confine and manipulate microscopic objects in three dimensions. A popular implementation involves structuring the trap-forming beam with computer-generated holograms before focusing it into traps with a high-numerical-aperture optical train. Here, we present a fully vectorial theory for the forces and torques exerted by such systems.

© 2008 Optical Society of America

## 1. Introduction

1. E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. **69**, 1974–1977 (1998). [CrossRef]

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

3. 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]

4. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (1970). [CrossRef]

5. A. Ashkin, “Applications of laser radiation pressure,” Science **210**, 1081–1088 (1980). [CrossRef] [PubMed]

6. 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]

7. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**, 013602 (2008). [CrossRef] [PubMed]

8. P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. **50**, 702–708 (2000). [CrossRef]

9. A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optical trapped dielectric particles,” J. Appl. Phys. **91**, 5474–5488 (2002). [CrossRef]

10. 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]

11. A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. Royal Soc. London A **459**, 3021–3041 (2003). [CrossRef]

3. 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]

12. D. C. Benito, S. H. Simpson, and H. Simon, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express **16**, 2942–2957 (2008). [CrossRef] [PubMed]

## 2. Optical forces and torques

*T*

^{ν}*(*

_{µ}**r**) of the electromagnetic field. Among various formulations, the symmetric Maxwell tensor is found to simplify computation of forces and torques [15

15. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1134–1140 (2004). [CrossRef]

*A*. The force on a particle is then obtained by integration over its surface ∑,

*ε*

*is the Levi-Civita antisymmetric tensor.*

_{ijk}*ν*as [8

8. P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. **50**, 702–708 (2000). [CrossRef]

*P*is the laser power,

*n*

*is the refractive index of the medium, which we assume for simplicity to be homogeneous and isotropic, and*

_{m}*c*is the speed of light in vacuum. Similarly the torque efficiency for a sphere of radius

*a*rotating about the

*ν*direction is

## 3. HOT light fields

16. 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]

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

18. M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express **13**, 5831–5845 (2005). [CrossRef] [PubMed]

19. Y. Roichman, A. S. Waldron, E. Gardel, and D. G. Grier, “Performance of optical traps with geometric aberrations,” Appl. Opt. **45**, 3425–3429 (2006). [CrossRef] [PubMed]

20. Y. Roichman and D. G. Grier, “Projecting extended optical traps with shape-phase holography,” Opt. Lett. **31**, 1675–1677 (2006). [CrossRef] [PubMed]

## 3.1. The role of the relay lenses

9. A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optical trapped dielectric particles,” J. Appl. Phys. **91**, 5474–5488 (2002). [CrossRef]

10. 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]

21. P. R. T. Munro and P. Török, “Calculation of the image of an arbitrary vectorial electromagnetic field,” Opt. Express **15**, 9293–9307 (2007). [CrossRef] [PubMed]

*j*-th pixel thus has the form [21

21. P. R. T. Munro and P. Török, “Calculation of the image of an arbitrary vectorial electromagnetic field,” Opt. Express **15**, 9293–9307 (2007). [CrossRef] [PubMed]

*k*=2

*π*/

*λ*is the wavenumber of light of wavelength

*λ*, and where

*a*

*and*

_{j}*φ*

*are the phase and amplitude at the*

_{j}*j*-th pixel, respectively. The polarization of the ray propagating along

*ε*̂ is the polarization of the incident light. The total vector potential at point

*r*is then

*N*pixels.

**ŝ**_{2}is

*) accounts for rotation of the light’s polarization as it propagates through the telescope and may be represented as [21*

**ŝ**21. P. R. T. Munro and P. Török, “Calculation of the image of an arbitrary vectorial electromagnetic field,” Opt. Express **15**, 9293–9307 (2007). [CrossRef] [PubMed]

## 3.2. Focusing by an aberration-free objective

**ŝ**_{2}thus is transformed into a superposition of plane waves that are incident on the sample, which we assume to be immersed in a homogeneous isotropic medium of refractive index

*n*

*. The component propagating along*

_{m}*with polarization*

**ŝ***µ*may be written as

*=*

**r***f*sin

*θ*

_{2}(cos

*ϕ*

_{2}, sin

*ϕ*

_{2},0). Assuming further that the objective lens has well corrected aberrations, the geometric tensor describing the polarization’s rotation is [22

22. S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express **16**, 3397–3407 (2008). [CrossRef] [PubMed]

**r***is measured with respect to the center of the DOE,*

_{j}

*r*_{2}is centered on the objective lens’ input pupil and

**is centered on the optical train’s focal point. These choices eliminate additional phase factors in the Debye-Wolf integrals.**

*r*## 4. Light scattering by small objects

*with polarization*

**ŝ***µ*has the complex amplitude

*A*

^{I}

_{µ}(

*). This component contributes*

**ŝ**

*A*^{S}

_{µ}(

**,**

*r**) to the scattered field at point*

**ŝ****, where the subscript**

*r**µ*refers to the polarization of the incident light. The total scattered field at

*is then*

**r***n*

*. The imaginary part of*

_{p}*n*

*, also known as the extinction coefficient, characterizes the material’s absorptivity. Although absorption plays a critical role in computations of optically-induced torque [24*

_{p}24. P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly polarized electromagnetic wave,” Phys. Rev. A **30**, 2508–2516 (1984). [CrossRef]

25. K. F. Ren, G. Grehan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. **35**, 2702–2710 (1996). [CrossRef] [PubMed]

26. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Photonic force spectroscopy on metallic and absorbing nanoparticles,” Phys. Rev. B **71**, 045425 (2005). [CrossRef]

27. Y. Zhang, Y. Lie, J. Qi, G. Cui, H. Lui, J. Chen, L. Zhao, J. Xu, and Q. Sun, “Influence of absorption on optical trapping force of spherical particles in a focussed Gaussian beam,” J. Opt. A **10**, 085001 (2008). [CrossRef]

*A*

^{I}

_{1}(

*,*

**r***̂), which describes a plane wave propagating along*

**z****̂ and linearly polarized along**

*z**̂, gives rise to a scattered field*

**x***̂-polarized light is given [28, 29*

**x**29. P. W. Barber and S. C. Hill, *Light Scattering by Particles: Computational Methods*, vol. 2 of *Advanced Series in Applied Physics* (World Scientific, New Jersey, 1990). [CrossRef]

*P*

^{1}

_{n}(cos

*θ*) is the associated Legendre polynomial of the first kind, and

*j*

*(*

_{n}*kr*) is the spherical Bessel function of the first kind of order

*n*. The expansion coefficients in Eq. (22) are given by [28]

*m*=

*n*

*/*

_{p}*n*

*is the particle’s relative refractive index relative,*

_{m}*x*=

*ka*is its size parameter,

*h*

^{(1)}

*(*

_{n}*x*) is the spherical Hankel function of the first type of order

*n*, and where primes denote derivatives with respect to the argument. Similarly,

*, and polarizations,*

**ŝ***µ*, can be obtained through coordinate rotations.

*n*

*=*

_{c}*x*+4.05

*x*

^{1/3}+2, that depends on the particle’s size through

*x*[23, 29

29. P. W. Barber and S. C. Hill, *Light Scattering by Particles: Computational Methods*, vol. 2 of *Advanced Series in Applied Physics* (World Scientific, New Jersey, 1990). [CrossRef]

*n*, and either large or small arguments

*x*. The error can be assessed by computing the discontinuity in the total field just inside and just outside the particle’s surface. The implementations in recent versions of such commercial general purpose scientific computing packages as Mathematica, IDL and Matlab lead to relative discontinuities as large as 10 percent at the surface of a 100 nm diameter titania sphere in water. To avoid such errors, we employed the more robust numerical techniques described by [30

30. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. **15**, 668–671 (1976). [CrossRef] [PubMed]

31. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. **19**, 1505–1509 (1980). [CrossRef] [PubMed]

^{-5}over the entire range of sizes considered.

## 5. Superposition of Plane-Wave Contributions

*direction is obtained through a rotation of*

**ŷ***π*/2 about

*̂:*

**z***ŝ*=(

*θ*

_{s},

*ϕ*

*) has the same form in the coordinate system,*

_{s}

**r**^{′}=(

*x*

^{′},

*y*

^{′},

*z*

^{′}), that is rotated so that

*ŝ*is aligned with

*z*̂

^{′}. The necessary coordinate transformation can be performed with the Euler rotation tensor

**r**^{′}=𝔼(

**ŝ**)

**. The general solution for the scattered wave therefore has the form**

*r*## 6. Numerical Results

## 6.1. Trapping by an optical tweezer

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

*a*, and its refractive index

*n*

*. The incident light field is obtained by setting*

_{p}*φ*

*=0 for all of the pixels in the DOE. We focus our attention on the axial force profile, which tends to be weaker than the longitudinal force profile. Failure to achieve axial trapping is the principal failure mode of conventional optical tweezers.*

_{j}32. T. A. Nieminen, L. V. L. Y., A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A **9**, S196–S203 (2007). [CrossRef]

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

*λ*/2 can be trapped only if their relative refractive index is below roughly 1.4. The large domains of high-index trapping in [32

**9**, S196–S203 (2007). [CrossRef]

**9**, S196–S203 (2007). [CrossRef]

33. Y. Roichman and D. G. Grier, “Holographic assembly of quasicrystalline photonic heterostructures,” Opt. Express **13**, 5434–5439 (2005). [CrossRef] [PubMed]

**9**, S196–S203 (2007). [CrossRef]

*λ*/4 cannot be trapped at all, and that the condition for marginal trapping depends strongly on refractive index for small index mismatches. This differs qualitatively from Fig. 2, which shows stable trapping for very small spheres, even with modest relative refractive indexes. The difference in this case can be ascribed to the earlier study’s use of Matlab’s spherical Hankel functions, which are inaccurate for large indexes and small arguments. Consequently, Fig. 2 should be considered a more faithful guide for designing optical trapping experiments.

## 6.2. Forces and torques in an optical vortex

34. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. **42**, 217–223 (1995). [CrossRef]

35. N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. **43**, 2485–2491 (1996). [CrossRef]

36. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef] [PubMed]

*φ*

*=*

_{j}*l*

*θ*

*mod 2*

_{j}*π*with the DOE. The winding number ℓ controls the beam’s helicity and is referred to as the topological charge. The helical topology gives rise to destructive interference along the beam’s axis. Light therefore is redistributed to a ring of radius

*R*

*that is proportional to*

_{ℓ}*ℓ*in typical holographic implementations [37

37. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

38. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. **28**, 872–874 (2003). [CrossRef] [PubMed]

39. S. Sundbeck, I. Gruzberg, and D. G. Grier, “Structure and scaling of helical modes of light,” Opt. Lett. **30**, 477–479 (2005). [CrossRef] [PubMed]

*z*=0, for half of such a ring. In this case, the topological charge

*ℓ*=60 corresponds to the radius

*R*

*=5*

_{ℓ}*µ*m.

**(**

*F**) for a sphere with*

**r***ka*=5.9,

*n*

*=1.46+10-*

_{p}^{5}

*i*, and

*n*

*=1.33. These values are appropriate for a 1*

_{m}*µ*m diameter silica sphere dispersed in water and trapped at

*λ*=532 nm. Hues indicate the direction of the force in the (

*x*,

*y*) plane according to the inset color wheel. The saturation of the color corresponds to the magnitude of the force, with unsaturated white regions corresponding to weak forces, and brightly saturated regions corresponding to

*Q*

_{max}=0.007.

*x*,

*y*) plane, and do not account for the axial component of the force. They correspond to the motion typically described in experimental studies of colloidal particles in high-index optical vortexes [37

37. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

40. 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]

41. M. Babiker, C. R. Bennet, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. **89**, 143601 (2002). [CrossRef] [PubMed]

7. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**, 013602 (2008). [CrossRef] [PubMed]

42. 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]

*ℓ*

*h*̄ per photon [43

43. 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]

44. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**, 257901 (2002). [CrossRef] [PubMed]

43. 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]

45. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt. Lett. **22**, 52–54 (1997). [CrossRef] [PubMed]

*x*,

*y*) plane, and the saturation indicates the magnitude of the torque efficiency. A homogeneous isotropic sphere only experiences a torque if it absorbs light [24

24. P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly polarized electromagnetic wave,” Phys. Rev. A **30**, 2508–2516 (1984). [CrossRef]

*n*

*. For the micrometer-diameter silica sphere in this calculation,*

_{p}*τ*

_{max}=2×10

^{-6}. The maximum rotation frequency of 0.1 Hz/W would be challenging to observe experimentally, particularly on a background of vigorous brownian motion. Nevertheless, this demonstrates that optically-induced rotation can arise even in linearly polarized optical traps, and may become an important factor for materials such as polystyrene that absorb light more strongly than silica.

*z*=0. The associated force distribution in Fig. 3(e) shows that particles are driven downstream along the optical axis, and so are not axially trapped. Hues correspond to directions in the (

*x*,

*z*) plane corresponding to the color wheel, and maximum saturation corresponds to the force scale

*Q*

_{z}=0.006.

## 6.3. Trapping in a holographic ring trap

46. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE **6483**, 64830F (2007). [CrossRef]

*ℓ*=30. Such a ring trap can be created holographically with [46

46. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE **6483**, 64830F (2007). [CrossRef]

*J*

*(*

_{ℓ}*x*) is the

*ℓ*-th order Bessel function of the first kind and

*H*(

*x*) is the Heaviside step function. In practice, we have encoded this complex-valued hologram on a phase-only spatial light modulator using the shape-phase algorithm [46

46. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE **6483**, 64830F (2007). [CrossRef]

*R*of a ring trap is independent of

*ℓ*.

*Q*

_{max}=0.048. This also is reflected in the larger torque scale in Fig. 4(c), with

*τ*

_{max}=2×10

^{-5}.

*Q*

_{max}=0.042. Particles thus can be trapped in three dimensions as they circulate around the ring, without requiring additional external confinement. This should improve the performance of ring-based micro-optomechanical machines [47

47. K. Ladavac and D. G. Grier, “Microoptomechanical pump assembled and driven by holographic optical vortex arrays,” Opt. Express **12**, 1144–1149 (2004). [CrossRef] [PubMed]

## 6.4. Holographic line trap

20. Y. Roichman and D. G. Grier, “Projecting extended optical traps with shape-phase holography,” Opt. Lett. **31**, 1675–1677 (2006). [CrossRef] [PubMed]

*Q*

_{max}=0.05. The addition of an appropriate phase profile then facilitates creating a tailored force profile along the line’s length [7

7. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**, 013602 (2008). [CrossRef] [PubMed]

**100**, 013602 (2008). [CrossRef] [PubMed]

*direction [7*

**ŷ****100**, 013602 (2008). [CrossRef] [PubMed]

**100**, 013602 (2008). [CrossRef] [PubMed]

## 7. Conclusion

**100**, 013602 (2008). [CrossRef] [PubMed]

## References and links

1. | E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. |

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

3. | 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. |

4. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

5. | A. Ashkin, “Applications of laser radiation pressure,” Science |

6. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

7. | Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. |

8. | P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. |

9. | A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optical trapped dielectric particles,” J. Appl. Phys. |

10. | 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. |

11. | A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. Royal Soc. London A |

12. | D. C. Benito, S. H. Simpson, and H. Simon, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express |

13. | E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Royal Soc. London A |

14. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. |

15. | R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. |

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

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

18. | M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express |

19. | Y. Roichman, A. S. Waldron, E. Gardel, and D. G. Grier, “Performance of optical traps with geometric aberrations,” Appl. Opt. |

20. | Y. Roichman and D. G. Grier, “Projecting extended optical traps with shape-phase holography,” Opt. Lett. |

21. | P. R. T. Munro and P. Török, “Calculation of the image of an arbitrary vectorial electromagnetic field,” Opt. Express |

22. | S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express |

23. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

24. | P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly polarized electromagnetic wave,” Phys. Rev. A |

25. | K. F. Ren, G. Grehan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. |

26. | P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Photonic force spectroscopy on metallic and absorbing nanoparticles,” Phys. Rev. B |

27. | Y. Zhang, Y. Lie, J. Qi, G. Cui, H. Lui, J. Chen, L. Zhao, J. Xu, and Q. Sun, “Influence of absorption on optical trapping force of spherical particles in a focussed Gaussian beam,” J. Opt. A |

28. | C. F. Bohren and D. R. Huffman, |

29. | P. W. Barber and S. C. Hill, |

30. | W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. |

31. | W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. |

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

33. | Y. Roichman and D. G. Grier, “Holographic assembly of quasicrystalline photonic heterostructures,” Opt. Express |

34. | H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. |

35. | N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. |

36. | K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. |

37. | J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. |

38. | J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. |

39. | S. Sundbeck, I. Gruzberg, and D. G. Grier, “Structure and scaling of helical modes of light,” Opt. Lett. |

40. | 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. |

41. | M. Babiker, C. R. Bennet, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. |

42. | 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. |

43. | 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 |

44. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. |

45. | N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt. Lett. |

46. | Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE |

47. | K. Ladavac and D. G. Grier, “Microoptomechanical pump assembled and driven by holographic optical vortex arrays,” Opt. Express |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(090.1760) Holography : Computer holography

(140.7010) Lasers and laser optics : Laser trapping

(290.4020) Scattering : Mie theory

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: August 8, 2008

Revised Manuscript: September 9, 2008

Manuscript Accepted: September 15, 2008

Published: September 19, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Bo Sun, Yohai Roichman, and David G. Grier, "Theory of holographic optical trapping," Opt. Express **16**, 15765-15776 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15765

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