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| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 4 — May. 4, 2011
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Extended and knotted optical traps in three dimensions

Elisabeth R. Shanblatt and David G. Grier  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 5833-5838 (2011)
http://dx.doi.org/10.1364/OE.19.005833


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Abstract

We describe a method for projecting holographic optical traps that are extended along arbitrary curves in three dimensions, and whose amplitude and phase profiles are specified independently. This approach can be used to create bright optical traps with knotted optical force fields.

© 2011 OSA

Extended optical traps are structured light fields whose intensity and phase gradients exert forces that confine microscopic objects to one-dimensional curves in three dimensions [1

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

4

4. S.-H. Lee, Y. Roichman, and D. G. Grier“Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010). [CrossRef] [PubMed]

]. Intensity-gradient forces typically are responsible for trapping [5

5. 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]

] in the two transverse directions, while radiation pressure directed by phase gradients can move particles along the third [3

3. 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]

, 4

4. S.-H. Lee, Y. Roichman, and D. G. Grier“Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010). [CrossRef] [PubMed]

]. This combination of trapping and driving has been demonstrated dramatically in optical vortexes [6

6. 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]

10

10. K. Ladavac and D. G. Grier“Colloidal hydrodynamic coupling in concentric optical vortices,” Europhys. Lett. 70, 548–554 (2005). [CrossRef]

], ring-like optical traps that are created by focusing helical modes of light. Intensity gradients draw illuminated objects toward the ring, and phase gradients then drive them around [3

3. 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]

, 7

7. 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]

, 8

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

, 11

11. 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]

]. More recently, holographic methods have been introduced to design and project more general optical traps that are extended along lines [1

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

], rings [2

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

] and helices [4

4. S.-H. Lee, Y. Roichman, and D. G. Grier“Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010). [CrossRef] [PubMed]

], with intensity and phase profiles independently specified along their lengths. Unlike optical vortexes, these traps feature nearly ideal axial intensity gradients because they are specifically designed to achieve diffraction-limited focusing [1

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

4

4. S.-H. Lee, Y. Roichman, and D. G. Grier“Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010). [CrossRef] [PubMed]

].

Here, we describe a method for designing and projecting optical traps whose intensity maxima trace out more general curves in three dimensions with independently specified phase and amplitude profiles. Within limitations set by Maxwell’s equations, these three-dimensional light fields can be used to trap and move microscopic objects. We demonstrate the technique by projecting diffraction-limited holographic ring traps with arbitrary orientations in three dimensions.

More specifically, our goal is to project a beam of light that comes to a focus along a curve R⃗ 0(s) = (x 0(s), y 0(s), z 0(s)), parametrized by its arc length s, along which the amplitude a 0(s) and phase φ 0(s) also are specified. The three-dimensional light field u(r⃗,z) embodying this extended optical trap is projected by a lens of focal length f, and so passes through the lens’ focal plane where its value is u f (r⃗,0) = u f (r⃗). Associated with u f (r⃗) is the conjugate field u h(r⃗) in the back focal plane of the lens, over which we have control. A hologram that imprints this field onto the wavefronts of an otherwise featureless laser beam will project the desired trapping pattern u(r⃗,z) downstream of the lens, as illustrated in Fig. 1.

Fig. 1. Projecting three-dimensionally extended holographic traps. A laser beam is imprinted with a hologram in the input pupil of an objective lens. The hologram is projected through the objective’s focal plane, and comes to a focus along a three-dimensional curve parameterized by its arc length, s.

For extended traps that lie entirely within the focal plane, the projected field may be approximated by an infinitessimally fine thread of light u f(r⃗) ≈ a 0(s) exp( 0(s)) δ (r⃗r⃗ 0(s)), where r⃗ 0(s) = (x 0(s),y 0(s)) [1

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

3

3. 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]

]. Equation (2) then yields a hologram associated with this idealized design. The projected field comes to a focus of finite width because Eq. (2) naturally incorporates contributions from self-diffraction.

As an application of Eq. (5), we create a uniformly bright ring trap [2

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

] of radius R, rotated by angle β about the ŷ axis. The trap’s focus follows the curve
R0(s)=R(cos(sR)cosβ,sin(sR),cos(sR)sinβ),
(6)
with the arc length ranging from s = 0 to s = 2πR. Equations (2) and (5) then yield the hologram
uh(ρ)=J0(A(ρ))
(7)
up to arbitrary phase factors, where J 0(.) is the Bessel function of the first kind of order zero and A(ρ⃗) satisfies
(2fA(ρ)kR)2=ξ2+[ηcosβ+(4f2+ρ2)1/2sinβ]2.
(8)
This reduces to
uh(ρ)=J0(kR2fρ)
(9)
when β = 0, which is the previously reported result for a holographic ring trap aligned with the focal plane [2

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

].

Figure 2(a) is a volumetric reconstruction [18

18. Y. Roichman, I. Cholis, and D. G. Grier“Volumetric imaging of holographic optical traps,” Opt. Express 14, 10907–10912 (2006). [CrossRef] [PubMed]

] of the three-dimensional intensity distribution projected by the hologram in Eq. (7) of a ring trap of radius R = 9 μm tilted at β = π/4 rad. This complex-valued hologram was approximated with a phase-only hologram using the shape-phase algorithm [1

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

] so that it could be projected with a conventional holographic optical trapping system [19

19. 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]

]. The light from a diode-pumped solid-state laser (Coherent Verdi, λ = 532 nm) was imprinted with the computed hologram by a phase-only liquid-crystal spatial light modulator (Hamamatsu X8267-16 PPM) and relayed to the input pupil of an oil-immersion objective lens (Nikon Plan-Apo, 100×, numerical aperture 1.4) mounted in a conventional light microscope (Nikon TE-2000U). Transverse slices of the projected intensity distribution, such as the example in Fig. 2(b), were obtained by mounting a front-surface mirror on the microscope’s sample stage and moving the trap with respect to the focal plane. Light reflected by the mirror was collected by the same objective lens and relayed to a video camera (NEC TI-324A II) for recording. A sequence of such slices obtained in axial steps of Δz = 0.1 μm was stacked to create a volumetric map of the trap’s intensity. The surface in Fig. 2 (a) encompasses the brightest 70 percent of the pixels in each slice. The axial sections through the volume in Figs. 2(c) and 2(d) confirm the inclination of the ring.

Fig. 2. (a) Volumetric reconstruction of a tilted ring trap. (b) Horizontal section near the midplane showing tilt around the ŷ axis. (c) Vertical section along the axis. (d) Vertical section along the ŷ axis.

Figure 3 shows a typical bright-field image of 5.17 μm diameter silica spheres dispersed in water and trapped on the inclined ring. A sphere’s appearance varies from bright to dark depending on its axial distance from the microscope’s focal plane. This dependence can be calibrated to estimate the sphere’s axial position [20

20. J. C. Crocker and D. G. Grier“Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]

]. The image in Fig. 3 is consistent with the designed inclination of the ring trap and therefore demonstrates the efficacy of Eqs. (2) and (5) for designing three-dimensionally extended optical traps.

Fig. 3. (a) Bright-field microscope image of 5.17 μm diameter colloidal silica spheres trapped on an inclined ring. (b) Schematic representation of the geometry in (a). Media 1 shows particles circulating around the inclined ring under the influence of phase-gradient forces.

In addition to extending an optical trap’s intensity along a three-dimensional curve, Eq. (5) also can be used to specify the extended trap’s phase profile. Imposing a uniform phase gradient, φ (s) = (s/R), redirects the light’s momentum flux to create a uniform phase-gradient force [3

3. 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]

] directed along the trap. In the particular case of a ring trap, this additional tangential force may be ascribed to orbital angular momentum in the beam [21

21. 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]

] that is independent of the light’s state of polarization [22

22. 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]

] and makes trapped particles circulate around the ring [2

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

, 3

3. 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]

, 11

11. 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 principle can be applied also to inclined ring traps.

Unlike a horizontal ring trap [2

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

] at β = 0, the inclined ring inevitably has phase variations along its circumference, φ(s) = kz(s), whose azimuthal gradient s φ(s) = −ksin(s/R)sin β tends to drive trapped objects to the downstream end of the ring [3

3. 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]

]. Adding φ (s) to this intrinsic phase profile creates a tilted ring trap described by the hologram
uh(ρ)=J(A(ρ))exp(2πisR)
(10)
whose additional circumferential phase gradient that tends to drive particles around the ring. Here, J (.) is the Bessel function of the first kind of order . A particle should circulate continuously if this tangential gradient is large enough to overcome the overall downstream gradient, which occurs for > n m (R/λ)sinβ. Diffraction limits the maximum value of [2

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

, 23

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

, 24

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

] that can be imposed on a ring trap of radius R. This, in turn, limits the maximum tilt angle β for which free circulation is possible.

The images in Fig. 3 were obtained with no imposed phase gradient, = 0. Increasing to 20 directs enough of the beam’s radiation pressure in the tangential direction to drive the spheres around the ring at roughly 0.2 Hz. Colloidal spheres can be seen circulating around an inclined ring trap in the media file (Media 1) associated with Fig. 3.

Fig. 4. (a) Intensity in the focal plane of the microscope of two tilted ring traps projected simultaneously with opposite inclination, β = ±π/8. (b) Colloidal silica spheres trapped in three dimensions within the focused Hopf link. (c) Schematic representation of the three-dimensional intensity distribution responsible for the image in (a).

Acknowledgments

This work was supported by the MRSEC Program of the National Science Foundation through Grant Number DMR-0820341.

References and links

1.

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

2.

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

3.

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]

4.

S.-H. Lee, Y. Roichman, and D. G. Grier“Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010). [CrossRef] [PubMed]

5.

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]

6.

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]

7.

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]

8.

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

9.

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

10.

K. Ladavac and D. G. Grier“Colloidal hydrodynamic coupling in concentric optical vortices,” Europhys. Lett. 70, 548–554 (2005). [CrossRef]

11.

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]

12.

J. W. GoodmanIntroduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

13.

G. C. Sherman“Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967). [CrossRef] [PubMed]

14.

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]

15.

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]

16.

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

17.

J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. Laczik, and M. Padgett“Interactive approach to optical tweezers control,” Appl. Opt. 45, 897–903 (2006). [CrossRef] [PubMed]

18.

Y. Roichman, I. Cholis, and D. G. Grier“Volumetric imaging of holographic optical traps,” Opt. Express 14, 10907–10912 (2006). [CrossRef] [PubMed]

19.

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]

20.

J. C. Crocker and D. G. Grier“Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]

21.

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]

22.

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]

23.

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

24.

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

25.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett“Laser beams: knotted threads of darkness,” Nature 432, 165 (2004). [CrossRef] [PubMed]

26.

M. R. Dennis, R. P. King, J. Barry, K. O’Holleran, and M. J. Padgett“Isolated optical vortex knots,” Nat. Phys. 6, 53–56 (2010). [CrossRef]

27.

W. T. M. Irvine and D. Bouwmeester“Linked and knotted beams of light,” Nat. Phys. 4, 716–720 (2008). [CrossRef]

28.

L. Faddeev and A. J. Niemi“Stable knot-like structures in classical field theory,” Nature 387, 58–61 (1997). [CrossRef]

OCIS Codes
(090.1760) Holography : Computer holography
(140.7010) Lasers and laser optics : Laser trapping

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: December 23, 2010
Revised Manuscript: March 1, 2011
Manuscript Accepted: March 1, 2011
Published: March 15, 2011

Virtual Issues
Vol. 6, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Elisabeth R. Shanblatt and David G. Grier, "Extended and knotted optical traps in three dimensions," Opt. Express 19, 5833-5838 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-7-5833


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References

  1. Y. Roichman and D. G. Grier, “Projecting extended optical traps with shape-phase holography,” Opt. Lett. 31, 1675–1677 (2006). [CrossRef] [PubMed]
  2. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE 6483, 64830F (2007). [CrossRef]
  3. 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]
  4. S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010). [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. 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]
  8. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef] [PubMed]
  9. K. Ladavac and D. G. Grier, “Microoptomechanical pump assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (2004). [CrossRef] [PubMed]
  10. K. Ladavac and D. G. Grier, “Colloidal hydrodynamic coupling in concentric optical vortices,” Europhys. Lett. 70, 548–554 (2005). [CrossRef]
  11. 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]
  12. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).
  13. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967). [CrossRef] [PubMed]
  14. 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]
  15. 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]
  16. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]
  17. J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. Laczik, and M. Padgett, “Interactive approach to optical tweezers control,” Appl. Opt. 45, 897–903 (2006). [CrossRef] [PubMed]
  18. Y. Roichman, I. Cholis, and D. G. Grier, “Volumetric imaging of holographic optical traps,” Opt. Express 14, 10907–10912 (2006). [CrossRef] [PubMed]
  19. 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]
  20. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]
  21. 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]
  22. 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]
  23. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003). [CrossRef] [PubMed]
  24. S. Sundbeck, I. Gruzberg, and D. G. Grier, “Structure and scaling of helical modes of light,” Opt. Lett. 30, 477–479 (2005). [CrossRef] [PubMed]
  25. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004). [CrossRef] [PubMed]
  26. M. R. Dennis, R. P. King, J. Barry, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 53–56 (2010). [CrossRef]
  27. W. T. M. Irvine and D. Bouwmeester, “Linked and knotted beams of light,” Nat. Phys. 4, 716–720 (2008). [CrossRef]
  28. L. Faddeev and A. J. Niemi, “Stable knot-like structures in classical field theory,” Nature 387, 58–61 (1997). [CrossRef]

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