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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 7 — Mar. 29, 2010
  • pp: 6988–6993
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Optical solenoid beams

Sang-Hyuk Lee, Yohai Roichman, and David G. Grier  »View Author Affiliations


Optics Express, Vol. 18, Issue 7, pp. 6988-6993 (2010)
http://dx.doi.org/10.1364/OE.18.006988


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Abstract

We introduce optical solenoid beams, diffractionless solutions of the Helmholtz equation whose diffraction-limited in-plane intensity peak spirals around the optical axis, and whose wavefronts carry an independent helical pitch. Unlike other collimated beams of light, appropriately designed solenoid beams have the noteworthy property of being able to exert forces on illuminated objects that are directed opposite to the direction of the light’s propagation. We demonstrate this through video microscopy observations of a colloidal sphere moving upstream along a holographically projected optical solenoid beam.

© 2010 Optical Society of America

Radiation pressure due to the momentum flux in a beam of light drives illuminated objects along the direction of the light’s wave vector. Additional forces arising from intensity gradients tend to draw small objects toward extrema of the intensity. These forces are exploited in single-beam optical traps known as optical tweezers [1

1. 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]

], which capture microscopic objects at the focus of a strongly converging beam of light. Stable three-dimensional trapping results when axial intensity gradients are steep enough that the intensity-gradient force overcomes radiation pressure downstream of the focus. The beam of light in a tightly focused optical tweezer therefore has the remarkable property of drawing particles upstream against radiation pressure, at least near its focal point [1

1. 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]

]. Collimated beams of light generally have no axial intensity gradients, and therefore ought not to be able to exert such retrograde forces.

In this Letter, we introduce optical solenoid beams whose principal intensity maximum spirals around the optical axis and whose wavefronts are characterized by an independent helical pitch. Figure 1 (Media 1) shows theoretical and experimentally realized examples. These beams are solutions of the Helmholtz equation, and thus propagate without diffraction [2

2. J. Durnin, “Exact-solutions for nondiffracting beams. 1. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

, 3

3. J. Durnin, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

], their radial intensity profiles remaining invariant in the spiraling frame of reference [4

4. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001). [CrossRef] [PubMed]

].

Intensity gradients in a solenoid beam tend to draw small objects such as colloidal particles toward the one-dimensional spiral of maximum intensity. Radiation pressure directed by the beam’s phase gradients [5

5. 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]

] then can drive the particle around the spiral. Under appropriate circumstances, the combination of intensity-gradient localization and phase-gradient driving can create a component of the total optical force directed opposite to the light’s direction of propagation, which can pull matter upstream along the beam’s entire length.

The vector potential for a beam of light at frequency ω propagating along the ẑ direction may be written as

A(r,z,t)=u(r,z)eiωtε̂,
(1)

where k = ω/c is the wave number of the light, ε̂ is its polarization vector and r measures the two-dimensional displacement from the beam’s axis. We derive the three-dimensional optical solenoid field u(r,z) by considering the two-dimensional field u 0(r) in the plane, z = 0. Because the light propagating to z > 0 must first pass through the plane z = 0, the field in this plane completely specifies the beam. Moreover, a featureless beam imprinted with the complex field u 0(r) in the plane z = 0 will propagate into the far field as u(r,z). In this sense, u 0(r) may be considered the hologram encoding the desired beam.

Quite generally, u 0(r) may be obtained from u(r,z) by formally back-propagating the three-dimensional field to z = 0. This can be accomplished in scalar diffraction theory with the Rayleigh-Sommerfeld formula [6

6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

],

Fig. 1. (a) Calculated three-dimensional intensity distribution of a solenoid beam propagating in the ẑ direction. (b) Volumetric rendering of the measured intensity in an experimental realization. Media 1 shows a rotating view of this static beam.
u0(r)=(uhz)(r)dz,where
(2)
hz(r)=12πz(eikr2+z2r2+z2)
(3)

is the Rayleigh-Sommerfeld propagator, and where the convolution is given by

(uhz)(r)=u(r,z)hz(rr)d2r.
(4)

This formalism can be useful even if the desired field, u(r,z), is not a solution of the Helmholtz equation, and so does not describe a physically realizable beam of light. In that case, the physical beam, up(r,z), associated with u(r,z) can be obtained by propagating u 0(r) forward, again using the Rayleigh-Sommerfeld propagator,

up(r,z)=(u0hz)(r).
(5)

Those solutions for which |up(r,z)|2, is independent of z are said to be non-diffracting [2

2. J. Durnin, “Exact-solutions for nondiffracting beams. 1. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

, 3

3. J. Durnin, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

].

We now apply this formalism to designing beams of light whose intensity maxima trace out specified one-dimensional curves in three dimensions, with arbitrary amplitude and phase profiles along these curves. Such beams may be represented as

u(r,z)={a(z)δ(rr0(z))e(z),z1zz20,otherwise.
(6)

Here, r 0(z) is the position of the beam’s maximum at axial position z, a(z) is its amplitude, and φ(z) is its phase. This representation does not describe a physically realizable beam of light because it neither incorporates self-diffraction nor locally conserves energy or momentum. Equations (2) through (5) nevertheless yield a physically realizable beam that has the desired properties along r 0(z), provided that self-diffraction may be neglected.

Equation (4) is most easily computed with the Fourier convolution theorem. In that case, the two-dimensional Fourier transform of u 0(r) is

u˜0(q)=z1z2a(z)e(z)eiq·r0(z)eiz(k2q2)1/2dz.
(7)

As a step toward deriving the solenoid beam, we first consider the case of an infinite line of light propagating along the optical axis, r 0(z) = 0, with uniform amplitude, a(z) = 1, but with a specified axial phase gradient, φ(z) = βz. For 0 ≤ βk, Eq. (7) has solutions

u0(r)=βJ0((k2β2)1/2r)
(8)

and up(r,z) = u 0(r) exp(iβz), which is the zeroth-order Bessel beam [9

9. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6(11), 1748–1754 (1989). [CrossRef] [PubMed]

, 10

10. P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992). [CrossRef] [PubMed]

, 11

11. J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001). [CrossRef]

]. Whereas we specified an infinitesimally finely resolved thread of light, formal back-propagation with Eq. (7) implicitly accounts for the beam’s self-diffraction. The limit β = k corresponds to a plane wave propagating along ẑ. Smaller values of β yield more finely resolved beams that carry less momentum along ẑ.

To create a solenoidal beam, we set a(z) = 1 and r 0(z) = Rcos(θ 0 (z)) x̂ + Rsin(θ 0 (z))ŷ, where θ 0(z) = z/γ is the azimuthal angle around the optical axis in a spiral of radius R and pitch γ. In addition to establishing a spiral structure for the beam’s principal intensity maximum, we also impose a helical phase profile in the plane, φ(z) = ℓθ 0(z), where the helical pitch, ℓ, is independent of γ. This helical phase profile will enable us to exert tunable phase-gradient forces [5

5. 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]

] along the solenoid.

As for the Bessel beam, we seek a non-diffracting solution of Eq. (7), and so integrate over all z to obtain

uγ,(r,z)=m=[γk][]mγ2Jm(qmR)ei(m)γzeimθJm(qmr),
(9)

where qm 2 = k 2 - (ℓ - m)2/γ 2 and where [x] is the integer part of x. The solenoid beam thus is a particular superposition of m-th order Bessel beams. Superposition of non-diffracting modes previously has been used to synthesize multi-lobed spiral [12

12. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996). [CrossRef]

, 13

13. V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: Rotating images,” J. Mod. Opt. 44, 1409–1416 (1997). [CrossRef]

, 14

14. P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 45, 2355–2369 (1998). [CrossRef]

] and localized [15

15. Z. Bouchal and J. Kyvalsky, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004). [CrossRef]

, 16

16. J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, “Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams,” Opt. Express 14, 2108–2116 (2006). [CrossRef] [PubMed]

] modes. More generally, Eq. (9) is a particular example of a rotating scale-invariant electromagnetic field [4

4. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001). [CrossRef] [PubMed]

].

Figure 1(a) shows the three-dimensional intensity distribution I γ,ℓ(r,z) = |u γ,ℓ(r,z)|2 computed according to Eq. (9) for kR= 10, θ = 30° and ℓ = 10. As intended, the locus of maximum intensity spirals around the optical axis.

Quite clearly, the intensity distribution of a solenoid depends on z, and so is not strictly invariant under propagation. Nonetheless, the in-plane intensity distribution remains invariant, merely rotating about the optical axis. Such a generalization of the notion of non-diffracting propagation previously was introduced in the context of spiral waves [12

12. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996). [CrossRef]

]. Solenoid beams therefore may be considered to be non-diffracting in this more general sense.

Distinct solenoid beams satisfy the orthogonality condition

uγ,*(r,z)uγ,(r,z)d2rdz=δℓℓδ(γγ),
(10)

except if m = (ℓγ - ℓ Γ)/(Γ - Γ) is an integer that falls in the range [max(ℓ - Γk, ℓ - Γ k)] ≤ m ≤ [min(ℓ,ℓ)]. This additional condition defines classes of m-congruent solenoid beams whose members are not mutually orthogonal and results from the solenoid modes’ non-trivial periodicity along the optical axis.

Figure 2 shows the effect of changing the helicity of a solenoid beam with a fixed spiral pitch, α = tan-1(Γk). When ℓ > 0, as in Fig. 2(a), the wave vector is directed along the solenoid. A particle confined to the spiral by intensity-gradient forces therefore is driven downstream by this component of the radiation pressure. Changing ℓ does not alter α, but changes the wavefronts’ pitch relative to ẑ. At ℓ = 0, the wavefronts are parallel to the solenoid’s pitch. as shown in Fig. 2(b). In this case, radiation pressure is directed normal to the spiral, and so can be balanced by intensity-gradient forces. Setting ℓ < 0 tilts the wavefronts in the retrograde direction, as shown in Fig. 2(c). The resulting reverse-sense phase-gradient force can move the particle upstream along the spiral in the negative ẑ direction.

Fig. 2. Retrograde forces in a helical solenoid beam. The local wave vector k is normal to the light’s wavefronts, with a component in the ẑ direction. (a) ℓ = +40: k is directed along the solenoid, resulting in a downstream phase-gradient force. (b) ℓ = 0: Wavefronts are parallel to the solenoid so that k is everywhere normal to the spiral. Particles trapped by intensity-gradient forces experience no net force. (c) ℓ= -40: A component of k is directed back down the spiral. A particle confined to the spiral therefore moves upstream.

We experimentally projected solenoid beams using methods developed for holographic optical trapping [17

17. 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]

, 18

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

, 19

19. M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005). [CrossRef] [PubMed]

]. In this system, a phase-only liquid crystal spatial light modulator (SLM) (Hamamatsu X7690-16 PPM) is used to imprint the hologram u 0(r) associated with u γ,ℓ(r) onto the wavefronts of a linearly polarized laser beam with a vacuum wavelength λ = 532 nm (Coherent Verdi). This hologram then is projected into the far field with a microscope objective lens (Nikon Plan Apo, 100×, oil immersion) mounted in a conventional inverted optical microscope (Nikon TE 2000U). The computed complex hologram is encoded on the phase-only SLM using the shape-phase holography algorithm [7

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

]. The resulting beam includes the intended solenoid mode superposed with higher diffraction orders [20

20. S.-H. Lee and D. G. Grier, “Robustness of holographic optical traps against phase scaling errors,” Opt. Express 13, 7458–7465 (2005). [CrossRef] [PubMed]

].

To visualize the projected beam, we mount a front-surface mirror on the microscope’s stage. The reflected light is collected by the objective lens, and relayed to a CCD camera (NEC TI-324AII). Images acquired at a sequence of focal depths then are combined to create a volumetric rendering of the three-dimensional intensity field [21

21. Y. Roichman, I. Cholis, and D. G. Grier, “Volumetric imaging of holographic optical traps,” Opt. Express 14, 10,907–10,912 (2006). [CrossRef]

]. The example in Fig. 1(b) (Media 1) shows the serpentine structure of a holographically projected solenoid beam with R = 5 μm.

To demonstrate the solenoid beam’s ability to exert retrograde forces on microscopic objects, we projected it into a sample of colloidal silica spheres 1.5 μm in diameter dispersed in water. The sample was contained in the 50 μm thick gap between a glass microscope slide and a glass no. 1 cover slip, and was mounted on the microscope’s stage. Bright-field images of individual spheres interacting with the solenoid beam were obtained with the same objective lens used to project the hologram, and were recorded by the video camera at 1/30 s intervals. The sphere’s appearance changes as it moves in z in a manner that can be calibrated [22

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

] to measure the particle’s axial position. Combining this with simultaneous measurements of the particle’s in-plane position [22

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

] yields the three-dimensional trajectory data that are plotted in Fig. 3 (Media 2). The gray-scale image in Fig. 3 was created by superimposing six snapshots of a single sphere that was trapped on a solenoid beam and moving along its length. This and the video sequence in Media 2 illustrate how the sphere’s image changes as it moves in z.

Fig. 3. Three-dimensional trajectory of a colloidal sphere moving along one turn of an optical solenoid beam together with a multiply-exposed image of the sphere at six points in its motion (Media 2). Alternating between ℓ = ±30 switches the direction of the particle’s motion relative to the propagation direction, ẑ. Arrows indicate the direction of motion for the downstream (blue) and retrograde upstream (red) trajectories.

The data plotted in Fig. 3 (Media 2) were obtained by alternately setting ℓ = +30 and ℓ = - 30 without changing any other properties of the solenoid beam. The three blue traces show trajectories obtained with ℓ = +30 in which the particle moved downstream along the curve of the solenoid, advancing in the direction of the light’s propagation. These alternate with two red traces obtained with ℓ= -30 in which the particle moves back upstream, opposite to the direction of the light’s propagation. These latter traces confirm that the combination of phase- and intensity-gradient forces in helical solenoid beams can exert retrograde forces on illuminated objects and transport them upstream over large distances.

Although the solenoid beam was designed to be uniformly bright, the particle does not move along it smoothly in practice. Interference between the holographically projected solenoid beam and higher diffraction orders creates unintended intensity variations along the solenoid that tend to localize the particle. Achieving retrograde motions over distances larger than the 8 μm in our demonstration will require improved methods for projecting solenoid modes.

This work was supported by the National Science Foundation through Grant Number DMR-0855741 and by the W. M. Keck Foundation. S.H.L. acknowledges support from the Kessler Family Foundation.

References and links

1.

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]

2.

J. Durnin, “Exact-solutions for nondiffracting beams. 1. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

3.

J. Durnin, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

4.

J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express 9, 9–15 (2001). [CrossRef] [PubMed]

5.

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]

6.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

7.

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

8.

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

9.

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6(11), 1748–1754 (1989). [CrossRef] [PubMed]

10.

P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992). [CrossRef] [PubMed]

11.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001). [CrossRef]

12.

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996). [CrossRef]

13.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: Rotating images,” J. Mod. Opt. 44, 1409–1416 (1997). [CrossRef]

14.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. 45, 2355–2369 (1998). [CrossRef]

15.

Z. Bouchal and J. Kyvalsky, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004). [CrossRef]

16.

J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, “Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams,” Opt. Express 14, 2108–2116 (2006). [CrossRef] [PubMed]

17.

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]

18.

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

19.

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005). [CrossRef] [PubMed]

20.

S.-H. Lee and D. G. Grier, “Robustness of holographic optical traps against phase scaling errors,” Opt. Express 13, 7458–7465 (2005). [CrossRef] [PubMed]

21.

Y. Roichman, I. Cholis, and D. G. Grier, “Volumetric imaging of holographic optical traps,” Opt. Express 14, 10,907–10,912 (2006). [CrossRef]

22.

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

23.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). [CrossRef] [PubMed]

24.

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

OCIS Codes
(090.1760) Holography : Computer holography
(260.1960) Physical optics : Diffraction theory
(350.5500) Other areas of optics : Propagation
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Physical Optics

History
Original Manuscript: February 1, 2010
Revised Manuscript: March 14, 2010
Manuscript Accepted: March 14, 2010
Published: March 19, 2010

Virtual Issues
Vol. 5, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Sang-Hyuk Lee, Yohai Roichman, and David G. Grier, "Optical solenoid beams," Opt. Express 18, 6988-6993 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6988


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References

  1. 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]
  2. J. Durnin, "Exact-solutions for nondiffracting beams. 1. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987). [CrossRef]
  3. J. Durnin, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
  4. J. Tervo and J. Turunen, "Rotating scale-invariant electromagnetic fields," Opt. Express 9, 9-15 (2001). [CrossRef] [PubMed]
  5. 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]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  7. Y. Roichman and D. G. Grier, "Projecting extended optical traps with shape-phase holography," Opt. Lett. 31, 1675-1677 (2006). [CrossRef] [PubMed]
  8. Y. Roichman and D. G. Grier, "Three-dimensional holographic ring traps," Proc. SPIE 6483, 64830F (2007). [CrossRef]
  9. A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6(11), 1748-1754 (1989). [CrossRef] [PubMed]
  10. P. L. Overfelt, "Scalar optical beams with helical symmetry," Phys. Rev. A 46, 3516-3522 (1992). [CrossRef] [PubMed]
  11. J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001). [CrossRef]
  12. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, "Nondiffracting beams: travelling, standing, rotating and spiral waves," Opt. Commun. 123, 225-233 (1996). [CrossRef]
  13. V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: Rotating images," J. Mod. Opt. 44, 1409-1416 (1997). [CrossRef]
  14. P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, "Rotating optical fields: experimental demonstration with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998). [CrossRef]
  15. Z. Bouchal and J. Kyvalsky, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004). [CrossRef]
  16. J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, "Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006). [CrossRef] [PubMed]
  17. 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]
  18. D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003). [CrossRef] [PubMed]
  19. M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, "Optimized holographic optical traps," Opt. Express 13(15), 5831-5845 (2005). [CrossRef] [PubMed]
  20. S.-H. Lee and D. G. Grier, "Robustness of holographic optical traps against phase scaling errors," Opt. Express 13, 7458-7465 (2005). [CrossRef] [PubMed]
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