Optical solenoid beams
Abstract.
We introduce optical solenoid beams, diffractionless solutions of the Helmholtz equation whose diffractionlimited inplane 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.
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 singlebeam optical traps known as optical tweezers (1), which capture microscopic objects at the focus of a strongly converging beam of light. Stable threedimensional trapping results when axial intensity gradients are steep enough that the intensitygradient 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). 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); (3), their radial intensity profiles remaining invariant in the spiraling frame of reference (4).
Intensity gradients in a solenoid beam tend to draw small objects such as colloidal particles toward the onedimensional spiral of maximum intensity. Radiation pressure directed by the beam's phase gradients (5) then can drive the particle around the spiral. Under appropriate circumstances, the combination of intensitygradient localization and phasegradient 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 propagating along the direction may be written as
(1) 
where is the wave number of the light, is its polarization vector and measures the twodimensional displacement from the beam's axis. We derive the threedimensional optical solenoid field by considering the twodimensional field in the plane, . Because the light propagating to must first pass through the plane , the field in this plane completely specifies the beam. Moreover, a featureless beam imprinted with the complex field in the plane will propagate into the far field as . In this sense, may be considered the hologram encoding the desired beam.
Quite generally, may be obtained from by formally backpropagating the threedimensional field to . This can be accomplished in scalar diffraction theory with the RayleighSommerfeld formula (6),
(2)  
(3) 
is the RayleighSommerfeld propagator, and where the convolution is given by
(4) 
This formalism can be useful even if the desired field, , 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, , associated with can be obtained by propagating forward, again using the RayleighSommerfeld propagator,
(5) 
Those solutions for which , is independent of are said to be nondiffracting (2); (3).
We now apply this formalism to designing beams of light whose intensity maxima trace out specified onedimensional curves in three dimensions, with arbitrary amplitude and phase profiles along these curves. Such beams may be represented as
(6) 
Here, is the position of the beam's maximum at axial position , is its amplitude, and is its phase. This representation does not describe a physically realizable beam of light because it neither incorporates selfdiffraction nor locally conserves energy or momentum. Equations (2) through (5) nevertheless yield a physically realizable beam that has the desired properties along , provided that selfdiffraction may be neglected.
Equation (4) is most easily computed with the Fourier convolution theorem. In that case, the twodimensional Fourier transform of is
(7) 
An inverse Fourier transform then provides , and Eq. (5) yields the associated beam of light. This result extends to three dimensions our previous descriptions of holographic line traps and holographic ring traps in the plane (7); (8).
As a step toward deriving the solenoid beam, we first consider the case of an infinite line of light propagating along the optical axis, , with uniform amplitude, , but with a specified axial phase gradient, . For , Eq. (7) has solutions
(8) 
and , which is the zerothorder Bessel beam (9); (10); (11). Whereas we specified an infinitesimally finely resolved thread of light, formal backpropagation with Eq. (7) implicitly accounts for the beam's selfdiffraction. The limit corresponds to a plane wave propagating along . Smaller values of yield more finely resolved beams that carry less momentum along .
To create a solenoidal beam, we set and , where is the azimuthal angle around the optical axis in a spiral of radius and pitch . In addition to establishing a spiral structure for the beam's principal intensity maximum, we also impose a helical phase profile in the plane, , where the helical pitch, , is independent of . This helical phase profile will enable us to exert tunable phasegradient forces (5) along the solenoid.
As for the Bessel beam, we seek a nondiffracting solution of Eq. (7), and so integrate over all to obtain
(9) 
where and where is the integer part of . The solenoid beam thus is a particular superposition of th order Bessel beams. Superposition of nondiffracting modes previously has been used to synthesize multilobed spiral (12); (13); (14) and localized (15); (16) modes. More generally, Eq. (9) is a particular example of a rotating scaleinvariant electromagnetic field (4).
Figure 1(a) shows the threedimensional intensity distribution computed according to Eq. (9) for , and . As intended, the locus of maximum intensity spirals around the optical axis.
Quite clearly, the intensity distribution of a solenoid depends on , and so is not strictly invariant under propagation. Nonetheless, the inplane intensity distribution remains invariant, merely rotating about the optical axis. Such a generalization of the notion of nondiffracting propagation previously was introduced in the context of spiral waves (12). Solenoid beams therefore may be considered to be nondiffracting in this more general sense.
Distinct solenoid beams satisfy the orthogonality condition
(10) 
except if is an integer that falls in the range . This additional condition defines classes of congruent solenoid beams whose members are not mutually orthogonal and results from the solenoid modes' nontrivial periodicity along the optical axis.
Figure 2 shows the effect of changing the helicity of a solenoid beam with a fixed spiral pitch, . When , as in Fig. 2(a), the wave vector is directed along the solenoid. A particle confined to the spiral by intensitygradient forces therefore is driven downstream by this component of the radiation pressure. Changing does not alter , but changes the wavefronts' pitch relative to . At , 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 intensitygradient forces. Setting tilts the wavefronts in the retrograde direction, as shown in Fig. 2(c). The resulting reversesense phasegradient force can move the particle upstream along the spiral in the negative direction.
We experimentally projected solenoid beams using methods developed for holographic optical trapping (17); (18); (19). In this system, a phaseonly liquid crystal spatial light modulator (SLM) (Hamamatsu X769016 PPM) is used to imprint the hologram associated with onto the wavefronts of a linearly polarized laser beam with a vacuum wavelength (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 phaseonly SLM using the shapephase holography algorithm (7). The resulting beam includes the intended solenoid mode superposed with higher diffraction orders (20).
To visualize the projected beam, we mount a frontsurface mirror on the microscope's stage. The reflected light is collected by the objective lens, and relayed to a CCD camera (NEC TI324AII). Images acquired at a sequence of focal depths then are combined to create a volumetric rendering of the threedimensional intensity field (21). The example in Fig. 1(b) (Media 1) shows the serpentine structure of a holographically projected solenoid beam with .
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 in diameter dispersed in water. The sample was contained in the 50 thick gap between a glass microscope slide and a glass no. 1 cover slip, and was mounted on the microscope's stage. Brightfield 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 intervals. The sphere's appearance changes as it moves in in a manner that can be calibrated (22) to measure the particle's axial position. Combining this with simultaneous measurements of the particle's inplane position (22) yields the threedimensional trajectory data that are plotted in Fig. 3 (Media 2). The grayscale 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 .
The data plotted in Fig. 3 (Media 2) were obtained by alternately setting and without changing any other properties of the solenoid beam. The three blue traces show trajectories obtained with 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 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 intensitygradient 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 in our demonstration will require improved methods for projecting solenoid modes.
The foregoing results introduce solenoidal beams of light whose nondiffracting transverse intensity profiles spiral periodically around the optical axis and whose wavefronts can be independently inclined through specified azimuthal phase profiles. We have demonstrated that solenoid beams can trap microscopic objects in three dimensions and that phasegradient forces can be used to transport trapped objects not only down the optical axis but also up. The ability to balance radiation pressure with phasegradient forces in solenoidal beams opens a previously unexplored avenue for singlebeam control of microscopic objects. In principle, solenoid beams can transport objects over large distances, much as do Bessel beams (11); (23) and related nondiffracting modes (24), without the need for highnumericalaperture optics. Solenoid beams, moreover, offer the additional benefit of bidirectional transport along the optical axis.
This work was supported by the National Science Foundation through Grant Number DMR0855741 and by the W. M. Keck Foundation. S.H.L. acknowledges support from the Kessler Family Foundation.
References

(1)
A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a singlebeam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).

(2)
J. Durnin, “Exactsolutions for nondiffracting beams. 1. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).

(3)
J. Durnin, “Diffractionfree beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

(4)
J. Tervo and J. Turunen, “Rotating scaleinvariant electromagnetic fields,” Opt. Express 9, 9–15 (2001).

(5)
Y. Roichman, B. Sun, Y. Roichman, J. AmatoGrill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).

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

(7)
Y. Roichman and D. G. Grier, “Projecting extended optical traps with shapephase holography,” Opt. Lett. 31, 1675–1677 (2006).

(8)
Y. Roichman and D. G. Grier, “Threedimensional holographic ring traps,” Proc. SPIE 6483, 64830F (2007).

(9)
A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computergenerated holograms,” J. Opt. Soc. Am. A 6(11), 1748–1754 (1989).

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

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

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

(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).

(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).

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

(16)
J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, “Iterative algorithms for holographic shaping of nondiffracting and selfimaging light beams,” Opt. Express 14, 2108–2116 (2006).

(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).

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

(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).

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

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

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

(23)
V. GarcésChávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a selfreconstructing light beam,” Nature 419, 145–147 (2002).

(24)
T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Submicron particle organization by selfimaging of nondiffracting beams,” New J. Phys. 8, 43 (2006).