Elisabeth R. Shanblatt and David G. Grier
Department of Physics and Center for Soft Matter
Research, New York University, New York, NY 10003
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 (2,4,1,3). Intensity-gradient forces typically are responsible for trapping (5) in the two transverse directions, while radiation pressure directed by phase gradients can move particles along the third (4,3). This combination of trapping and driving has been demonstrated dramatically in optical vortexes (7,8,10,6,9), 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 (11,7,8,3). More recently, holographic methods have been introduced to design and project more general optical traps that are extended along lines (1), rings (2) and helices (4), 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 (2,4,1,3).
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
,
parametrized by its arc length
, along which the amplitude
and phase
also are specified.
The three-dimensional light field
embodying this extended optical trap is projected by a lens of focal length
,
and so passes through the lens' focal plane
where its value is
.
Associated with
is the conjugate field
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
downstream of the lens,
as illustrated in Fig. 1.
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The ideal hologram,
,
is characterized by a real-valued amplitude
and
phase
both of which vary with position
in the back focal plane.
It is related to the projected field,
,
by a Fresnel transform (12)
For extended traps that lie entirely within the focal plane, the
projected field may be approximated by an infinitessimally
fine thread of light
,
where
(2,1,3).
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.
Extending this to more general three-dimensional curves
poses challenges that have been incompletely addressed in
previous studies (4).
The field
at height
above the focal plane is related to the field in the
focal plane
by the Rayleigh-Sommerfeld diffraction integral (12),
which is expressed conveniently with the Fourier convolution
theorem as
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As an application of Eq. (5), we
create a uniformly bright ring trap (2) of radius
,
rotated by angle
about the
axis.
The trap's focus follows the curve
Figure 2(a) is a volumetric reconstruction
(18)
of the three-dimensional intensity distribution projected by
the hologram in Eq. (7) of a ring trap
of radius
tilted
at
.
This complex-valued hologram was approximated with a
phase-only hologram using the shape-phase algorithm
(1)
so that it could be projected with a conventional holographic
optical trapping system (19).
The light from a diode-pumped solid-state laser
(Coherent Verdi,
)
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,
, 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
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 (d) confirm
the inclination of the ring.
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Figure 3 shows a typical bright-field
image of 5.17
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).
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.
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,
, redirects the light's momentum flux to create
a uniform phase-gradient force (3) 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) that is independent of the light's
state of polarization (22) and makes trapped
particles circulate around the ring (2,11,3).
This principle can be applied also to inclined ring traps.
Unlike a horizontal ring trap (2) at
,
the inclined ring inevitably has phase variations along its circumference,
, whose azimuthal gradient
tends to
drive trapped objects to the downstream end of the ring
(3).
Adding
to this intrinsic phase profile creates
a tilted ring trap described by the hologram
![]() |
(10) |
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The images in Fig. 3 were obtained with no imposed
phase gradient,
.
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 associated with Fig. 3.
The same formalism can be used to make more complicated three-dimensional
trapping fields. The image in Fig. 4(a) shows
the intensity in the focal plan of two tilted ring traps of radius
projected simultaneously but with opposite inclination,
.
Setting these rings' separation to
creates a pair of interlocking
bright rings in the form of a Hopf link.
These interpenetrating rings still act as three-dimensional optical traps,
and are shown in Fig. 4(b) organizing 3.01
diameter
colloidal silica spheres.
The trapped spheres pass freely past each other along the two rings.
Figure 4(c) offers a schematic view of the Hopf link
geometry.
This knotted light field differs from previously reported knotted vortex fields (26,25) in that its intensity is maximal along the knot, rather than minimal. In this respect, holographically projected Hopf links more closely resemble the Ranada-Hopf knotted fields that have been predicted (27) for strongly focused circularly polarized optical pulses. Knotted force fields arise in this latter case from knots in each pulse's magnetic field. Linked ring traps are not knotted in this sense. Nevertheless, the programmable combination of intensity-gradient and phase-gradient forces in linked ring traps can create a constant knotted force field for microscopic objects. Beyond their intrinsic interest, such knotted optical force fields have potential applications in plasma physics for inducing knotted current loops (27,28) and could serve as mixers for biological and soft-matter systems at the micrometer scale.
This work was supported by the MRSEC Program of the National Science Foundation through Grant Number DMR-0820341.