Sang-Hyuk Lee [1], Yohai Roichman [2], and David G. Grier [2]
[1] Department of Molecular and Cell Biology,
Institute for Quantitative Biology, University of California - Berkeley
Berkeley, CA 94720-3220
[2] Department of Physics and Center for Soft Matter
Research, New York University, New York, NY 10003
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), 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). Collimated beams of light generally have no axial intensity gradients, and therefore ought not to be able to exert such retrograde forces.
In this article, 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 one-dimensional 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 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 propagating along the
direction may be written as
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Quite generally,
may be
obtained from
by formally back-propagating
the three-dimensional field to
.
This can be accomplished in scalar diffraction theory
with the Rayleigh-Sommerfeld formula (6),
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
Equation (4) is most easily computed with
the Fourier convolution theorem.
In that case, the two-dimensional Fourier transform of
is
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
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
phase-gradient forces (5) along the
solenoid.
As for the Bessel beam, we seek a non-diffracting solution of
Eq. (7), and so integrate over all to obtain
Figure 1(a) shows the three-dimensional
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 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).
Solenoid beams therefore may be considered to be non-diffracting
in this more general sense.
Distinct solenoid beams satisfy the orthogonality condition
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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
intensity-gradient 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 intensity-gradient forces.
Setting
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
direction.
We experimentally projected solenoid beams using
methods developed for holographic optical trapping
(17,19,18).
In this system, a phase-only liquid crystal spatial
light modulator (SLM) (Hamamatsu X7690-16 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 phase-only
SLM using the shape-phase 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 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).
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.
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
in a manner that can be calibrated (22)
to measure the particle's axial position.
Combining this with simultaneous measurements
of the particle's in-plane position (22)
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
.
![]() |
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 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
in our demonstration will require
improved methods for projecting solenoid modes.
The foregoing results introduce solenoidal beams of light whose non-diffracting 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 phase-gradient forces can be used to transport trapped objects not only down the optical axis but also up. The ability to balance radiation pressure with phase-gradient forces in solenoidal beams opens a previously unexplored avenue for single-beam control of microscopic objects. In principle, solenoid beams can transport objects over large distances, much as do Bessel beams (23,11) and related non-diffracting modes (24), without the need for high-numerical-aperture 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 DMR-0855741 and by the W. M. Keck Foundation. S.H.L. acknowledges support from the Kessler Family Foundation.