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Optics Express

Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15765–15776
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Theory of holographic optical trapping

Bo Sun, Yohai Roichman, and David G. Grier  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15765-15776 (2008)
http://dx.doi.org/10.1364/OE.16.015765


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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

The forces and torques exerted by an optical trap can be computed by first calculating the electromagnetic field surrounding an illuminated object and then integrating the Maxwell stress tensor over its surface. This approach has been used [8

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

, 9

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

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

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]

] to model the forces exerted by conventional optical tweezers [3

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]

]. More recently, finite-difference time-domain calculations have been used to compute the optical forces and torques on multiple spheres and cylinders arranged in holographically projected arrays of optical tweezers [12

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]

]. These studies use fully vectorial expansions of the light field, and so capture polarization-dependent effects as well as those due to intensity and phase gradients. Because they are based on particular models for the focused beam, however, they are not easily extended to the more general light fields that can be projected with the HOT technique.

This article presents an efficient method for computing the optical forces and torques exerted by holographic trapping patterns in practical HOT systems. Based on Debye-Wolf theory for light propagation through optical trains and the Lorenz-Mie theory of light scattering by small particles, this approach expresses the incident and scattered fields as solutions of the vector Helmholtz equation and so accurately describes the forces encountered in high-numerical-aperture trapping.

2. Optical forces and torques

The optical force and torque experienced by an illuminated object arises from the energy-momentum tensor 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]

]. In SI units,

μ0Tμν=FμρFρν+14FρσFρσδμν,
(1)

where the momentum flux in the electromagnetic field

Fμν=μAννAμ,
(2)

emerges from gradients in the light’s vector potential, A. The force on a particle is then obtained by integration over its surface ∑,

Fμ=ΣTμνdSν.
(3)

Similarly, the optical torque is obtained as

Mμ=12ΣεijμMijkdSk
(4)

where the angular momentum tensor corresponding to the symmetric stress tensor is

Mijk=riTjkrjTik
(5)

and where ε ijk is the Levi-Civita antisymmetric tensor.

In discussing the forces exerted by optical traps, it is conventional to define the trapping efficiency in the direction ν as [8

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

]

Qν=cnmPFν
(6)

where P is the laser power, n m is the refractive index of the medium, which we assume for simplicity to be homogeneous and isotropic, and c is the speed of light in vacuum. Similarly the torque efficiency for a sphere of radius a rotating about the ν direction is

τν=ωPMν.
(7)

This general formulation is common to all theoretical studies of optical trapping. Diverse implementations differ in how they treat the incident and scattered electromagnetic fields surrounding the particle, and in their strategy for performing the surface integrals in Eq. (3) and (4). The formalism described in the following sections computes the superposition of contributions from each pixel in a pixellated DOE to the overall force and torque exerted on an object in a holographic optical trapping system. Although our discussion focuses on optical trapping of isotropic homogeneous spheres, it can be generalized to particles of other shapes and compositions.

3. HOT light fields

3.1. The role of the relay lenses

Because the telescope forms an image in the far field, each pixel in the DOE may be treated as a magnetic dipole. The contribution to the vector potential from the 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]

]

Aj,μD(r)=aj2πcexp(iφj)exp(ikrrj)rrjPμ,
(8)

where k=2π/λ is the wavenumber of light of wavelength λ, and where a j and φ j are the phase and amplitude at the j-th pixel, respectively. The polarization of the ray propagating along

ŝ1=(sinθ1cosϕ1,sinθ1sinϕ1,cosθ1).
(9)

is given by

p=ŝ1×(ẑ×ε̂),
(10)

where ε̂ is the polarization of the incident light. The total vector potential at point r is then

AμD(r)=j=1NAj,μD(r)
(11)

for a DOE consisting of N pixels.

Fig. 1. Schematic representation of a holographic optical trapping system. A laser beam propagating from the left is imprinted with a computer-generated hologram by a diffractive optical element (DOE). The modified beam is relayed to an objective lens that focuses it onto the sample. The figure shows one of the plane waves projected from one of the DOE’s pixels at r j.

AμO(r2)=Ω2Bμ(ŝ2)exp(ikŝ2·r2)dΩ2,
(12)

where the complex amplitude of the plane wave propagating in direction ŝ 2 is

Bμ(ŝ2)=f22πcGμν(ŝ2)pνj=1Najexp(iφj)exp(ikŝ1·rj).
(13)

The geometric operator 𝔾(ŝ) accounts for rotation of the light’s polarization as it propagates through the telescope and may be represented as [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]

]

𝔾(ŝ2)=cosθ2cosθ11(ϕ1)𝕃(πθ2)𝕃(πθ1)(θ1),
(14)

in terms of the generalized Jones matrices

(ϕ)=(cosϕsinϕ0sinϕcosϕ0001)    and    𝕃(θ)=(cosθ0sinθ010sinθ0cosθ).
(15)

3.2. Focusing by an aberration-free objective

The light field formed by the telescope is focused into the sample by the objective lens. The plane wave propagating in the direction 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 m. The component propagating along ŝ with polarization µ may be written as

AμI(r,ŝ)=AμI(ŝ)exp(inmkŝ·r)
(16)

with a complex amplitude that also is described by a Debye-Wolf integral,

AμI(ŝ)=Ω2Hμν(ŝ,ŝ2)Bν(ŝ2)exp(inmkŝ·Δr)dΩ2,
(17)

where Δr=f sinθ 2 (cosϕ 2, sinϕ 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]

]

(ŝ,ŝ2)=iπŝ·ŝ2nm11(ŝ·ŝ2)2{(ŝ×ŝ2)(ŝ×ŝ2)+[ŝ2ŝ(ŝ·ŝ2)][ŝ+ŝ2(ŝ·ŝ2)]}.
(18)

Whereas r j is measured with respect to the center of the DOE, r 2 is centered on the objective lens’ input pupil and r is centered on the optical train’s focal point. These choices eliminate additional phase factors in the Debye-Wolf integrals.

Equations (12) through (18) describe the superposition of plane waves projected by the objective lens onto the sample arising from each pixel in the DOE. The total field includes the incident field from Eq. (17) and also the field scattered by the sample itself.

4. Light scattering by small objects

Although light scattering by small objects has been studied in great detail, analytic results are available only for a few specific cases. Lorenz-Mie theory provides exact expansions in vector spherical harmonics for the scattering of plane waves by spheres. More general systems can be treated with T-matrix theory [23

23. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

].

From Eq. (17), the object is illuminated by a superposition of plane waves. The component propagating along ŝ with polarization µ has the complex amplitude A I µ(ŝ). This component contributes A S µ (r, ŝ) to the scattered field at point r, where the subscript µ refers to the polarization of the incident light. The total scattered field at r is then

AS(r)=μ=12ΩAμS(r,ŝ)dΩ
(19)

and the total vector potential is

A(r)=AI(r)+AS(r).
(20)

This can be substituted into Eqs. (2) through (4) to obtain the optical force and torque on the object due to the particular hologram implemented with the DOE.

In many cases of practical interest, the object may be modeled as a smooth sphere made of a homogeneous isotropic material with complex refractive index n p. The imaginary part of n p, also known as the extinction coefficient, characterizes the material’s absorptivity. Although absorption plays a critical role in computations of optically-induced torque [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]

], its influence on the forces exerted by optical traps has received less attention [25

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

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

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]

].

Illuminating such a sphere with A I 1(r, ẑ), which describes a plane wave propagating along ẑ and linearly polarized along x̂, gives rise to a scattered field

AS(r,ẑ)=𝕃𝕄1(r)A1I(r,ẑ),
(21)

where the Lorenz-Mie scattering tensor for x̂-polarized light is given [28

28. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

, 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]

] as an expansion,

𝕃𝕄1(r)=n=1(ianNeln(3)(r)bnMoln(3)(r)),
(22)

in the vector spherical harmonics

Moln(3)(r)=cosϕsinθPn1(cosθ)jn(kr)θ̂sinϕdPn1(cosθ)dθjn(kr)ϕ̂
(23)

and

Neln(3)(r)=n(n+1)cosϕPn1(cosθ)jn(kr)krr̂
+cosϕdpn1(cosθ)dθ1krddr[rjn(kr)]θ̂
sinϕsinθPn1(cosθ)1krddr[rjn(kr)]ϕ̂.
(24)

Here, P 1 n (cosθ) 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

28. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

]

an=m2jn(mx)[xjn(x)]jn(x)[mxjn(mx)]m2jn(mx)[xhn(1)(x)]hn(1)(x)[mxjn(mx)],
(25)

where m=n p/n m is the particle’s relative refractive index relative, 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,

bn=jn(mx)[xjn(x)]jn(x)[mxjn(mx)]jn(mx)[xhn(1)(x)]hn(1)(x)[mxjn(mx)].
(26)

Once these scattering coefficients are known, equivalent results for other incident directions, ŝ, and polarizations, µ, can be obtained through coordinate rotations.

From the perspective of practical implementation, the sum in Eq. (22) converges after a number of terms, n c=x+4.05x 1/3+2, that depends on the particle’s size through x [23

23. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

, 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]

]. Computing additional terms does not improve accuracy because of the accumulation of round-off error. Making matters more difficult, the numerical implementations of the Bessel functions in at least some standard mathematical libraries suffer from large errors at large indexes 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]

] and [31

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

] and typically obtain convergences to within 10-5 over the entire range of sizes considered.

Fig. 2. (a) Magnitude Q max and (b) range r max of the axial trapping force for a sphere of radius a and relative refractive index n p/n m in an optical tweezer created from light of wavenumber k. The solid curve denotes marginal trapping conditions with Q max=0. The dashed curve in (b) indicates conditions for which the trap’s axial well extends 1 µm.

5. Superposition of Plane-Wave Contributions

The result analogous to Eq. (19) for scattering of light propagating along ẑ and polarized in the ŷ direction is obtained through a rotation of π/2 about ẑ: 𝕃𝕄2(r,θ,ϕ)=(π2)𝕃𝕄1(r,θ,ϕπ/2). The scattered wave due to a plane wave incident along ŝ=(θ s, ϕ s) has the same form in the coordinate system, r =(x , y , z ), that is rotated so that ŝ is aligned with ẑ. The necessary coordinate transformation can be performed with the Euler rotation tensor

𝔼(ŝ)=((cosθs1)cos2ϕs+1sinϕscosϕs(cosθs1)sinθscosϕssinϕscosϕs(cosϕs1)sin2ϕs(cosθs1)+1sinθssinϕssinθscosϕssin2ϕs(cosθs1)+1cosθs),
(27)

such that r =𝔼(ŝ) r. The general solution for the scattered wave therefore has the form

AS(r,ŝ)=𝔼1(ŝ)𝕃𝕄(𝔼(ŝ)r)𝔼(ŝ)AI(ŝ)
(28)

This can be combined with Eq. (17) for the incident field’s vector potential and substituted into Eqs. (1) through (4) to obtain the force and torque acting on the sphere.

6. Numerical Results

To illustrate the vectorial theory of holographic optical trapping outlined above, we have computed the force and torque fields associated with several types of optical traps acting on colloidal spheres.

6.1. Trapping by an optical tweezer

Following Ref. [32

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]

], we begin by computing the optical force experienced by a colloidal sphere in a conventional point-like optical tweezer as a function of the sphere’s radius, a, and its refractive index n p. The incident light field is obtained by setting φ j=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.

The results, summarized in Fig. 2, are similar in many respects to those reported in Ref. [32

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]

]. Most tellingly, Fig. 2(a) and Fig. 3 of [32

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]

] both demonstrate that particles with diameters larger than 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

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]

] are reduced to discrete islands of stability in Fig. 2 due in part to the influence of the relay optics on the light’s polarization, which was not considered in [32

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]

].

Fig. 3. (a) Computed intensity distribution in the focal plane, z=0 of an optical vortex of radius R . (b) In-plane force distribution, F(r), including six representative trajectories whose starting points are represented by dots. Hue indicates direction in the (x,y) plane according to the inset color wheel. Color saturation indicates relative magnitude. (c) In plane torque distribution. (d) Intensity distribution in the (x, z) plane along the optical axis, y=0. (e) Force distribution, with unbounded trajectories showing absence of stable axial trapping. (f) Axial torque distribution. Scale bars indicates 1 µm.

Both sets of results predict that high-index spheres can be trapped in a single-beam optical tweezer provided they are sufficiently small. This is an important observation for holographic assembly of photonic structures [33

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

], many of which rely on high-index materials for their interesting and useful optical properties.

Figure 3 of [32

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]

] suggests that spheres smaller than roughly 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

An optical tweezer may be transformed into a torque-exerting optical vortex [34

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

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

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

] by imposing the helical wavefront phase profile φ j=l θ j mod 2π 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 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

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

, 39

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]

]. The image in Fig. 3(a) shows the computed intensity in the focal plane, z=0, for half of such a ring. In this case, the topological charge =60 corresponds to the radius R =5 µm.

Figure 3(b) shows the in-plane component of the total optical force F(r) for a sphere with ka=5.9, n p=1.46+10-5 i, and n m=1.33. These values are appropriate for a 1 µm 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.

The six black dots in Fig. 3(b) show the starting points for the computed trajectories that are superimposed on the force field. These two-dimensional trajectories are calculated for particles constrained to move in the (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

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

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]

], where particles are pressed against a glass surface to prevented them from escaping along the axial direction. These representative trajectories show that particles are drawn by intensity-gradient forces to the bright ring and then are driven around the ring by phase-gradient forces [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]

, 42

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]

] This circulatory motion is a consequence of the helical beam’s orbital angular momentum, which amounts to h̄ 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

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]

]. The orbital angular momentum flux in a helical mode is independent of the photons’ spin, and thus is independent of the light’s polarization [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]

, 45

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]

].

Although the incident laser beam is assumed to be linearly polarized, the strongly focused light field has a far more complicated spatially varying polarization. Gradients in the intensity, phase and polarization of the light can exert torques as well as forces on illuminated objects, as the in-plane torque distribution in Fig. 3(c) demonstrates. The hue in Fig. 4(c) indicates direction of the torque in the (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]

]. The scale of the torque efficiency in Fig. 3(c) therefore is proportional to the imaginary part of n p. For the micrometer-diameter silica sphere in this calculation, 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.

The axial structure of an optical vortex, plotted in Figs. 3(d), (e) and (f), reveals its limitations as an optical trap. The axial intensity profile in Fig. 3(d) corresponds to a region around the principal focal ring, the dashed line indicating the position of the focal plane, 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

The data in Fig. 4 show the corresponding results for a micrometer-diameter silica sphere in a holographic ring trap [46

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

]. Unlike optical vortexes, these ring traps are designed to come to a diffraction-limited focus. The in-plane and axial intensity distributions in Figs. 4(a) and (d) demonstrate ring traps’ superior focusing characteristics. Holographic ring traps also may have arbitrary circumferential phase profiles. The example in Fig. 4 was created with a uniform azimuthal phase gradient corresponding to =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]

]

aj=J(kRrj2f)    and    φj=[θj+πH(J(kRrj2f))]mod2π,
(29)

where J (x) is the -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]

]. Whereas an optical vortex’s radius is coupled to its helicity, Eq. (29) shows that the radius R of a ring trap is independent of .

Fig. 4. (a) Intensity distribution of a holographic ring trap of radius R in the focal plane, z=0. (b) In-plane force distribution, with representative trajectories. Hue indicates direction in the (x,y) plane according to the inset color wheel. Saturation indicates relative magnitude. (c) In plane torque distribution. (d) Intensity distribution in the (x, z) plane along the optical axis, y=0. (e) Axial force distribution with representative trajectories showing strong axial trapping near the focal plane. (f) Axial torque distribution. Scale bars indicates 1 µm.

The ring trap’s in-plane characteristics, plotted in Figs. 4(b) and (c), qualitatively resemble those of the optical vortex. The radial force scale is substantially increased, however, by the stronger intensity gradients, with Q max=0.048. This also is reflected in the larger torque scale in Fig. 4(c), with τ max=2×10-5.

6.4. Holographic line trap

Figure 5 shows the intensity, force and torque distribution experienced by a micrometer-diameter silica sphere in a holographic line trap [20

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

]. This is another generalization of an optical tweezer whose focal point is extended along a line segment in the focal plane, as shown in Fig. 5(a). Holographic line traps also come to a diffraction-limited focus in the axial direction, as Fig. 5(d) shows, and therefore can trap objects stably in three dimensions, as shown in Fig. 5(e), with a trapping efficiency of 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]

], even when its intensity is uniform. The effect of a confining phase profile is demonstrated in Fig. 5(b), with superimposed trajectories converging on a region of mechanical equilibrium [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]

]. A uniform phase profile eliminates the inward force along the line, and would allow a particle to diffuse freely in the ŷ direction [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]

]. Switching the sign of the phase profile would drive particles to the ends of the line [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]

].

The computed torque distributions in Figs. 5(c) and (f) show that an illuminated particle again would tend to spin in the proximity of the strongly focused trap. The torque distribution for a holographic line trap is simpler than that for a holographic ring trap, with the particle’s rotation axis varying little along the line and its direction flipping as the particle crosses the line. The torque efficiencies are comparably large, however.

Fig. 5. (a) Intensity of a uniformly bright holographic line trap in the (x,y) plane. (b) Inplane force distribution, showing the influence of a parabolic phase profile. Superimposed trajectories converge at stable equilibrium point. Hue indicates direction of the force according to the color wheel, and saturation indicates magnitude. Full saturation corresponds to Q max=0.06. (c) In-plane component of the torque with τ max=2×10-5. (d) Intensity in the (x, z) plane showing diffraction-limited focusing. (e) Associated axial force distribution with superimposed trajectories demonstrating stable three-dimensional trapping. Q max=0.05 (f) Axial torque distribution. τ max=6×10-7.

7. Conclusion

We have used this vector theory to confirm the behavior of phase-gradient forces in extended traps that was predicted on the basis of scalar diffraction theory [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]

]. The spatially resolved images of the force and torque fields further reveal substantial consequences of polarization rotation in high-numerical-aperture optics. The transformation of the linearly polarized input beam into more general and spatially varying elliptical polarization can give rise to a highly structured torque field whose influence has yet to be observed experimentally. This suggests that polarization engineering can be viewed as an additional channel for control in holographically structured light fields. More generally, the vectorial theory of holographic optical trapping should provide a useful basis for designing and optimizing optical micromanipulation systems for particular applications.

This work was supported in part by the NSF through Grant Number DMR-0606415 and in part by a grant from the Keck Foundation. B.S. acknowledges support from a Kessler Family Foundation Fellowship.

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. 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]

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]

13.

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Royal Soc. London A 253, 349–357 (1959). [CrossRef]

14.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. 97, 133902 (2006). [CrossRef] [PubMed]

15.

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

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]

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]

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]

23.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

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]

28.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

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]

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]

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]

33.

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

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]

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]

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]

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]

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]

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]

46.

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

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]

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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References

  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]
  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]
  13. E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. Royal Soc. London A 253, 349-357 (1959). [CrossRef]
  14. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006). [CrossRef] [PubMed]
  15. R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004). [CrossRef]
  16. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computergenerated 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]
  21. P. R. T. Munro and P. Torok, "Calculation of the image of an arbitrary vectorial electromagnetic field," Opt. Express 15, 9293-9307 (2007). [CrossRef] [PubMed]
  22. S. S. Sherif, M. R. Foreman, and P. Torok, "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]
  23. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).
  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]
  28. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).
  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]
  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]
  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]
  33. Y. Roichman and D. G. Grier, "Holographic assembly of quasicrystalline photonic heterostructures," Opt. Express 13, 5434-5439 (2005). [CrossRef] [PubMed]
  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]
  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]
  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. D???avila Romero, "Orbital angular momentum exchange in the interaction of twisted light with molecules," Phys. Rev. Lett. 89, 143601 (2002). [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]
  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]
  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]
  46. Y. Roichman and D. G. Grier, "Three-dimensional holographic ring traps," Proc. SPIE 6483, 64830F (2007). [CrossRef]
  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]

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