Dynamic Holographic Optical Tweezers
Abstract.
Optical trapping is an increasingly important technique for controlling and probing the structure of matter at length scales ranging from nanometers to millimeters. This Article describes methods for creating large numbers of high-quality optical traps in arbitrary three-dimensional configurations and dynamically reconfiguring them under computer control. The resulting traps' unique capacity for reorganizing mesoscopically textured matter promises exciting new opportunities for research, engineering, diagnostics, and manufacturing.
An optical tweezer uses forces exerted by intensity gradients in a strongly focused beam of light to trap and move a microscopic volume of matter (1). Optical tweezers' unique ability to manipulate matter at mesoscopic scales has led to widespread applications in biology (2); (3), and the physical sciences (4). This Article describes how computer-generated holograms can dynamically reconfigure a single optical tweezer into hundreds of independent optical traps, each with individually specified characteristics, arrayed in arbitrary three-dimensional configurations. The enhanced capabilities of such dynamic holographic optical trapping systems offer new opportunities for research and engineering, as well as new applications in biotechnology, nanotechnology, and manufacturing.
Holographic optical tweezers (HOT) use a computer-designed diffractive optical element (DOE) to split a single collimated laser beam into several separate beams, each of which is focused into an optical tweezer by a strongly converging lens (5); (6); (7). Originally demonstrated with microfabricated DOEs (8), holographic optical tweezers have since been implemented with computer-addressed liquid crystal spatial light modulators (9); (10). Projecting a sequence of computer-designed holograms dynamically reconfigures the resulting pattern of traps. Unfortunately, calculating the phase hologram for a desired pattern of traps is not straightforward, and the lack of appropriate algorithms has prevented dynamic holographic optical tweezers from achieving their potential. This Article introduces new methods for computing phase holograms for optical trapping and demonstrates their use in a practical dynamic holographic optical trapping system.








Holographic optical tweezers take advantage of the same physical principles as do conventional optical gradient traps (1). A dielectric particle approaching a focused beam of light is polarized by the light's electric field and then drawn up intensity gradients toward the focal point. Radiation pressure competes with this optical gradient force and tends to displace the trapped particle along the beam's axis. For this reason, optical tweezers usually are designed around microscope objective lenses whose large numerical apertures and minimal aberrations optimize axial intensity gradients.
An optical trap can be placed anywhere within the objective lens' focal volume by appropriately selecting the input beam's propagation direction and degree of collimation. For example, a collimated beam passing straight into an infinity-corrected objective lens comes to a focus in the center of the lens' focal plane, while another beam entering at an angle comes to a focus proportionately off-center. A slightly diverging beam focuses downstream of the focal plane while a converging beam focuses upstream. By the same token, multiple beams entering the lens' input pupil simultaneously each form optical traps in the focal volume, each at a location determined by its degree of collimation angle of incidence. This is the principle behind holographic optical tweezers.
Our implementation, shown schematically in Fig. 1,
uses a Hamamatsu X7550 parallel-aligned nematic spatial light modulator (SLM)
(11) to reshape the beam from a frequency-doubled
Nd:YVO laser (Coherent Verdi) into a designated pattern of beams.
Each is transferred to the entrance
pupil of a 100
NA 1.4 oil immersion objective mounted
in a Zeiss Axiovert S100TV inverted optical microscope and then
focused into a trap.
A dichroic mirror reflects the laser light into the objective
while allowing images of the trapped particles to pass through to a
video camera.
When combined with a 1.6
widefield video eyepiece, this
optical train offers a
field of view.
The Hamamatsu SLM can impose selected phase shifts on the incident
beam's wavefront
at each 40 wide pixel in a
array.
The SLM's calibrated phase transfer function offers 150 distinct phase
shifts ranging from 0 to
at the operating wavelength
of
.
The phase shift imposed at each pixel is specified through a computer
interface with an effective refresh rate of 5 Hz for the entire
array.
Despite the SLM's inherently limited spatial bandwidth, quite
sophisticated trapping patterns still are possible.
The array of 400 functional optical traps shown in Fig. 1
is the largest created by any means.
Modulating only the phase and not the amplitude of the input beam is enough to establish any desired intensity pattern in the objective's focal volume and thus any pattern of traps (7). Such intensity-shaping phase gratings are often referred to as kinoforms. Previously reported algorithms for computing optical trapping kinoforms produced only two-dimensional distributions of traps (7); (9) or patterns on just two planes (10). Moreover, the resulting traps were suitable only for dielectric particles in low-dielectric media, and could not be adapted to handle metallic particles or samples made of absorbing, reflecting, or low-dielectric-constant materials. A more general approach relaxes all of these restrictions.
We begin by modeling the incident laser beam's
electric field
,
as having constant phase,
in the DOE plane,
and unit intensity:
.
Here
denotes a position in the DOE's aperture
,
is the real-valued amplitude profile of the
input beam, and we have omitted the field's polarization vector for clarity.
The DOE then imposes a phase modulation
onto the input beam's wavefront which ideally encodes the
desired pattern of outgoing beams.
The electric field at each
of the discrete traps is related to the electric field
in the plane of the DOE by
a generalized Fourier transform
![]() |
![]() |
|||
![]() |
(1) | |||
![]() |
(2) |
where is the effective focal length of the optical train, including
the relay optics and objective lens.
The kernel
can be used to transform the
-th trap from a
conventional tweezer into another type of trap, and
is its
inverse.
For conventional optical tweezers in the focal plane,
.
If the calculated amplitude, ,
were identical to the laser
beam's profile,
,
then
would be
the kinoform encoding the desired array of traps.
Unfortunately, this is rarely the case.
More generally, the spatially varying
discrepancies between
and
direct light away from the desired traps and into
ghosts and other undesirable artifacts.
Despite these shortcomings, combining kinoforms with Eq. (1)
is expedient and can produce useful trapping patterns (10).
Still better and more general results can be obtained by using
Eqs. (1) and (2)
as the basis for an iterative search for
the ideal kinoform.
Following the approach pioneered by Gerchberg and Saxton (GS)
(12), we treat the phase
calculated with Eqs. (1) and (2)
as an estimate,
,
for the desired kinoform and use this to calculate the fields at
the trap positions given the laser's actual profile
:
![]() |
(3) |
The index refers to the
-th iterative approximation to
.
The classic GS algorithm replaces the amplitude
in this estimate with the desired amplitude
, leaving
the corresponding phase
unchanged, and solves for the next
estimate
using
Eqs. (1) and (2).
The fraction
of the incident power actually delivered to the traps by the
-th
approximation is useful for tracking
the algorithm's convergence.
For the present application, the simple GS substitution leads to slow and non-monotonic convergence. We find that an alternate replacement scheme
![]() |
(4) |
leads to rapid monotonic convergence for .
The resulting estimate for
then can be
discretized and transmitted to the SLM to establish a trapping
pattern.
In cases where the SLM offers only a few distinct
phase levels, discretization
can be incorporated into each iteration to minimize
the associated error.
In all of the examples discussed below, this algorithm yields kinoforms
with theoretical efficiencies exceeding 80% in two or three iterations
starting from a random choice for the traps' initial phases
and often converges to solutions with better than 90% efficiency.
Iterative optimization with Eqs. (2) and (3)
is computationally efficient
because discrete transforms are calculated only
at the actual trap locations.
Figure 2(a) shows 26 colloidal silica spheres 0.99 in diameter suspended
in water and trapped in a planar pentagonal pattern
of optical tweezers created with Eqs. (2-3).
Replacing this kinoform with another whose traps are
slightly displaced moves the spheres into the new configuration.
Projecting a sequence of trapping patterns deterministically
translates the spheres into an entirely new configuration.
Figure 2(b) shows the same spheres after 16 such hops,
and Fig. 2(c) after 38.
Powering each trap with 1 mW of light
maintains the particles stably trapped against thermal
forces.
Increasing the trapping power to 10 mW and updating the
trapping pattern in
steps allows us to
translate particles at up to
.
Comparable planar motions also could be implemented by rapidly scanning a single tweezer through a sequence of discrete locations, thereby creating a time-shared trapping pattern (13). The continuous illumination of holographic optical traps offer several advantages, however. HOT patterns can be more extensive both spatially and in number of traps than time-shared arrays which must periodically release and retrieve each trapped particle. Additionally, the lower peak intensities required for continuously illuminated traps are less damaging to sensitive samples (14).
Similar rearrangements also would be possible with previous dynamic HOT implementations (10). These studies used fast Fourier transforms to optimize the projected intensity over the entire trapping plane, and routinely achieved theoretical efficiencies exceeding 95% (7). However, the discrete transforms adopted here allow us to encode more general patterns of traps.












Dynamic holographic optical tweezers need not be limited to planar configurations. If the laser beam illuminating the SLM were slightly diverging, then the entire pattern of traps would come to a focus downstream of the focal plane. Such divergence can be introduced with a Fresnel lens, encoded as a phase grating with
![]() |
(5) |
where is the desired displacement of the optical traps relative
to the focal plane in an optical train with effective focal length
.
Rather than placing a separate Fresnel lens into the input beam,
the same functionality can be obtained by adding the lens'
phase modulation to the existing kinoform:
.
Figure 3(a) shows a typical array of optical tweezers
collectively displaced out of the plane in this manner.
The accessible range of out-of-plane motion
in our system is approximately
.
Instead of being applied to the entire trapping pattern, separate lens functions can be applied to each trap individually with kernels
![]() |
(6) |
in Eqs. (1) and (3). Fig. 3(b) shows spheres being moved independently through multiple planes in this way.
Other phase modifications implement additional functionality. For example, the phase profile
![]() |
(7) |
converts an ordinary Gaussian laser beam
into a Laguerre-Gaussian mode (15),
and its corresponding optical
tweezer into a so-called optical vortex (16); (17); (18).
Here is the polar coordinate in the DOE plane
(See Fig. 1) and the integer
is the beam's topological charge (15).
Because all phases are present along the circumference of a Laguerre-Gaussian beam, destructive interference cancels the beam's intensity along its axis, all the way to the focus. Optical vortices thus appear as bright rings surrounding dark centers. Such dark traps have been demonstrated to be useful for trapping reflecting, absorbing (19) or low-dielectric particles (18) not otherwise compatible with conventional optical tweezers.
Adding to a kinoform encoding
an array of optical tweezers yields an array of identical optical
vortices, as shown in Fig. 4(a).
Here, the light from the array of traps is
reflected by a mirror placed in the microscope's
focal plane.
The vortex-forming phase function also can be applied to
individual traps through
![]() |
(8) |
as demonstrated in the mixed array of optical tweezers and optical vortices shown in Fig. 4(b).
Previously reports of optical vortex trapping have considered
Laguerre-Gaussian modes with relatively small topological charges,
.
The
examples in Fig. 4(b) are
thus the most highly charged optical vortices so far reported,
and traps with
are easily created with the present
system.
Fig. 4(c) shows multiple colloidal particles trapped
on the bright circumferences of a array of
vortices.
Because Laguerre-Gaussian modes have helical wavefronts,
particles trapped
on optical vortices experience tangential forces (19).
Optical vortices are useful, therefore, for driving motion at
small length scales, for example in microelectromechanical systems (MEMS).
Multiple particles trapped in
arrays of optical vortices have remarkable cooperative
behavior which will be discussed elsewhere.
The vortex-forming kernel can be combined with
to produce
three-dimensional arrays of vortices.
Such heterogeneous trapping patterns are useful for organizing
disparate materials into hierarchical
three-dimensional structures and for exerting controlled
forces and torques on extended dynamical systems.
While the present study has demonstrated how a single
Gaussian laser beam can be modified to create three-dimensional
arrays of optical tweezers and optical vortices,
other generalizations follow naturally, with virtually
any mode of light having potential applications.
For example, the axicon phase profile
creates an approximation of a Bessel mode
which focuses to an axial line trap whose length is controlled
by
(20).
Still better axial line traps
result from the more accurate mode-former
, where
is the effective aperture radius in the SLM plane.
These and other generalized trapping modes will be discussed elsewhere.
Linear combinations of optical vortices
and conventional tweezers have been shown to operate
as optical bottles (21) and controlled rotators
(22).
All such trapping modalities can be combined
dynamically using the techniques described above.
Trapping patterns' complexity is limited in practice
by the need to maintain three-dimensional intensity gradients for each trap,
and by the maximum information content
which can be encoded accurately in the SLM.
For example, the former consideration precludes forming a
three-dimensional cubic optical tweezer array with
a lattice constant much smaller than 10 , while the latter
limits our optical vortices to
.
Within these bounds, dynamic holographic optical tweezers are highly reconfigurable, operate noninvasively in both open and sealed environments, and can be coupled with computer vision technology to create fully automated systems. A single apparatus thus can be adapted to a wide range of applications without modification. Dynamic holographic optical tweezers should have a plethora of biotechnological applications including massively parallel high throughput screening, sub-cellular engineering, and macromolecular sorting. In materials science, the ability to organize materials into hierarchical three-dimensional structures constitutes an entirely new category of fabrication techniques. As research tools, dynamic holographic optical tweezers combine the demonstrated utility of optical tweezers with unprecedented flexibility and adaptability.
This work was funded by a sponsored research grant from Arryx, Inc. using equipment purchased under grant number 991705 from the W. M. Keck Foundation. The spatial light modulator used in this study was made available by Hamamatsu, Inc. as a loan to The University of Chicago. Additional funding was provided by the National Science Foundation through Grant Number DMR-9730189, and by the MRSEC program of the National Science Foundation through Grant Number DMR-980595.
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).
-
(2)
K. Svoboda, P. P. Mitra and S. M. Block.“Fluctuation analysis of motor protein movement and single enzyme-kinetics.”Proc. Nat. Acad. Sci.91, 11782–11786 (1994).
-
(3)
A. Ashkin.“History of optical trapping and manipulation of small-neutral particle, atoms, and molecules.”IEEE J. Sel. Top. Quantum Elec.6, 841–856 (2000).
-
(4)
D. G. Grier.“Optical tweezers in colloid and interface science.”Cur. Opin. Colloid Interface Sci.2, 264–270 (1997).
-
(5)
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).
-
(6)
D. G. Grier and E. R. Dufresne.“Apparatus for applying optical gradient forces.”U. S. Patent 6,055,106, The University of Chicago (2000).
-
(7)
E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier.“Computer-generated holographic optical tweezer arrays.”Rev. Sci. Instrum.72, 1810–1816 (2001).
-
(8)
D. G. Grier.“Colloids: A surprisingly attractive couple.”Nature393, 621–623 (1998).
-
(9)
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).
-
(10)
J. Liesener, M. Reicherter, T. Haist and H. J. Tiziani.“Multi-functional optical tweezers using computer-generated holograms.”Opt. Commun.185, 77–82 (2000).
-
(11)
Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi and T. Hara.“High efficiency electrically-addressable phase-only spatial light modulator.”Opt. Rev.6, 339–344 (1999).
-
(12)
R. W. Gerchberg and W. O. Saxton.“A practical algorithm for the determination of the phase from image and diffraction plane pictures.”Optik35, 237–246 (1972).
-
(13)
K. Sasaki, M. Koshio, H. Misawa, N. Kitamura and H. Masuhara.“Pattern formation and flow control of fine particles by laser-scanning micromanipulation.”Opt. Lett.16, 1463–1465 (1991).
-
(14)
K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman and S. M. Block.“Characterization of photodamage to Escherichia coli in optical traps.”Biophys. J.77, 2856–2863 (1999).
-
(15)
N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop and M. J. Wegener.“Laser beams with phase singularities.”Opt. Quantum Elect.24, S951–S962 (1992).
-
(16)
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).
-
(17)
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).
-
(18)
K. T. Gahagan and G. A. Swartzlander.“Optical vortex trapping of particles.”Opt. Lett.21, 827–829 (1996).
-
(19)
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).
-
(20)
J. Arlt, V. Garcés-Chávez, W. Sibbett and K. Dholakia.“Optical micromanipulation using a Bessel light beam.”Opt. Commun.197, 239–245 (2001).
-
(21)
J. Arlt and M. J. Padgett.“Generation of a beam with a dark focus surrounded by regions of higher intensity: The optical bottle beam.”Opt. Lett.25, 191–193 (2000).
-
(22)
L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant and K. Dholakia.``Controlled rotation of optically trapped microscopic particles.”Science292, 912–914 (2001).