Optical tweezer arrays and optical substrates created with diffractive optics
Abstract.
We describe a simple method for creating multiple optical tweezers
from a single laser beam using diffractive optical elements.
As a demonstration of this technique, we have implemented a
square array of optical tweezers – the hexadeca-tweezer.
Not only will diffractively generated optical tweezers facilitate many
new experiments in pure and applied physics, but they also will be
useful for fabricating nanocomposite materials and devices, including
photonic bandgap materials and optical circuit elements.
§ I. Introduction
Since their introduction a decade ago (1); (2), optical tweezers have become indispensable tools for physical studies of macromolecular (3) and biological (4) systems. Formed by bringing a single laser beam to a tight focus, an optical tweezer exploits optical gradient forces to manipulate micrometer-sized particles. Optical tweezers have allowed scientists to probe the fantastically small forces which characterize the interactions of colloids (5); (6); (7); (8); (9), polymers (10); (11); (12); (13); (14); (15); (16) and membranes (17); (18); (19); (20), and to assemble small numbers of colloidal particles into mesoscopic structures (21); (22). These pioneering studies each required only one or two optical tweezers. Extending their techniques to larger and more complex systems will require larger and more complex arrays of optical tweezers.
We describe a simple and effective means to create multiple optical tweezers in arbitrary patterns from a single laser beam using diffractive optical elements. These tweezer arrays and their variants should have immediate applications for probing phenomena in biological systems and complex fluids and in organizing soft matter into mesoscopically textured composite materials.
§ II. Optical Tweezer Arrays
The optical forces generated by
a milliwatt of visible light are more than enough to overwhelm
the random thermal forces which drive the dynamics of microparticles. The
goal in creating an optical tweezer is to direct the optical forces
from a single laser beam to trap a particle in all three dimensions.
While quite general formulations of this problem have been developed
(23); (24); (25); (26), a simplified discussion suffices
to motivate the design of optical tweezer arrays.
We will consider the forces exerted by monochromatic light of wavenumber
on a
dielectric sphere of radius
in the Rayleigh limit,
where
.
The total optical force,
, is the sum of two
contributions (27):
![]() |
(1) |
the first of which arises from gradients in the light's intensity
and the second of which is due to scattering of light by the particle.
The gradient force on a particle of dielectric constant immersed
in a medium of dielectric constant
and subjected to an
optical field with Poynting vector
,
![]() |
(2) |
tends to draw the particle toward the region of highest intensity. The scattering force,
![]() |
(3) |
drives the particle along the direction of propagation of the light.
An optical tweezer can be formed by focusing a laser beam to a diffraction limited spot with a high numerical aperture lens. The gradient force attracts the particle to the beam's focal point, while the scattering force drives the particle along the beam's axis. In order to form a full three dimensional trap, the axial intensity gradient must be large enough to overcome the scattering force.
In a typical experimental setup, a microscope objective lens
focuses a Gaussian TEM laser beam into a tweezer while
simultaneously imaging the trapped particles.
In order to maximize the axial intensity gradient,
the incident beam should be expanded to fill the objective's back
aperture.
An optical tweezer can be translated across
the microscope's focal plane by adjusting the beam's angle
of incidence at the back aperture and can be displaced along the
optical axis by changing
the curvature of its wavefront.
Similarly, if several collimated beams pass through the back aperture at different angles, they form separate tweezers at different locations in the focal plane. For instance, dual optical tweezer systems have been created by splitting and recombining a single laser beam with beamsplitters and refractive optics(21); (28). This approach, however, becomes cumbersome for more than a few traps. An elegant alternative for creating two-dimensional arrays of traps involves scanning a single tweezer rapidly among a number of positions to create a time-averaged extended trapping pattern (21); (29); (30); (31). This approach has proved highly effective for shaping small colloidal assemblies.
Yet another alternative for creating two dimensional arrays
of traps is to split and steer the light from a single beam with diffractive
optical elements. As demonstrated in the next section, inexpensive
commercially available diffractive pattern
generators (MEMS Optical Inc., Huntsville AL) are ideally suited to this
task. In addition, diffractive optics can be
used to change the curvature of beam wavefronts, thereby facilitating the
creation of three-dimensional arrays of traps.
Additionally, diffractive
optics can be used to modify beam profiles.
For instance,
computed holograms can convert a TEM Gaussian beam into multiple
Gauss-Laguerre beams, otherwise known as optical vortices
(32); (33); (34).
Tightly focused optical vortices can be
used to trap low dielectric constant and reflective microparticles.
Finally, the use of addressable liquid crystal phase shifting arrays allows
for the dynamic reconfiguration of a tweezer array for active particle
manipulation and assembly.
Thus, diffractively
generated optical fields can be used to configure arbitrary numbers of
microscopic particles into useful and interesting arrangements.
We will discuss some applications of these techniques after
describing a practical demonstration of diffractively
generated optical tweezer arrays.
§ III. The Hexadeca-Tweezer

The schematic diagram in Fig. 1
shows one implementation
of diffractively generated optical tweezer arrays. This design was used to
create a square array of tweezers–the hexadeca tweezer.
The optical tweezer array is powered by
a 100 mW diode-pumped frequency-doubled
Nd:YAG laser operating at 532 nm.
Two Keplerian telescopes arranged in series produce
two planes conjugate to the back aperture of the microscope's
objective lens (100
, N.A. 1.40).
The laser beam passes through eyepoints in both planes to
create a conventional single optical tweezer.
Introducing a diffractive
square array
generator
(Edmund Scientific No. P53191,
angular divergence 25 mrad at
nm)
at the first eyepoint creates
the pattern of rays desired for the hexadeca tweezer.
A gimbal mounted
mirror centered at the second eyepoint allows us to translate the entire
tweezer array
in the microscope's field of view and to dynamically stiffen the
tweezers.
A spatial filter placed in the inner focal plane of either
telescope
removes spurious rays created by
imperfections in the low-cost diffractive optic.
Conventional white-light illumination is used to form an image
of the trapped particles.
The image passes through a dichroic beam splitter and is
captured with a video camera (NEC TI-324A) and recorded
with a VCR (NEC PC-VCR) for later analysis.
For our demonstration, we used the hexadeca-tweezer to trap silica
spheres (,
, Duke Scientific,
Palo Alto CA, Cat. No. 8100)
suspended in deionized water (
) at room temperature.
The suspension was sandwiched between a
microscope slide and cover glass separated by
.
The tweezer array was focused
above the lower glass
wall and left in place to acquire particles.




Fig. 2(a) shows all sixteen of the primary
optical traps filled with particles.
Since the focal plane is several sphere diameters from the nearest
wall, each sphere is trapped stably in three dimensions.
Residence times in excess of 100 sec, suggest a trapping potential
deeper than 6 per particle given the free spheres'
self-diffusion coefficient of
/sec.
Comparable trapping efficiency would be expected for a single
conventional optical tweezer operating at equivalent light intensity.
Two additional spheres are trapped less strongly at spurious peaks in
the diffraction pattern.
Fig. 2(b) shows the same field of
view sec after the laser was interrupted and
before the spheres have had time to diffuse away.
After 3.1 sec (Fig. 2(c)), the pattern has
completely dispersed with some of the spheres wandering out of
the imaging volume altogether.
Fig. 2(d) shows two-dimensional projections of the
particles' trajectories over this period (6).
Studying the dynamic relaxation of artificially structured
colloidal crystals is just one application we foresee for
manipulating soft matter with holographically patterned
light.
§ IV. Optical Substrates and Applications
Not all holographically generated optical fields will create arrays of traps. A large uniformly illuminated volume, for example, will not have the intensity gradients required for three-dimensional trapping. Such non-trapping patterns also have applications, however, and can be created in the same manner as the hexadeca-tweezer with diffractive optics. The resulting optical fields may be useful for forcing particles against a surface and into desired configurations.
Optically induced ordering has been demonstrated by intersecting several discrete beams (35); (36); (37); (38); (39); (40); (41); (42). The resulting periodic intensity patterns are observed to induce order in colloidal monolayers. In this respect, the pattern of light plays the role of a modulated substrate potential and affects the phase behavior of the illuminated two-dimensional system. This effect is believed to be closely analogous to the influence of atomic corrugation on the phase transitions of adsorbed atomic and molecular overlayers (43); (44). The optical substrate's symmetry, periodicity and depth of modulation all can be adjusted experimentally. This system therefore has great promise for exploring the mechanisms of surface phase transitions. Diffractively generated optical substrates, moreover, need not be periodic. Quasiperiodic and aperiodic patterns will be useful for studies of the effect of pinning on monolayer dynamics, a potentially powerful analog for dynamics in superconducting vortex lattices and sliding charge density waves.
Beyond these applications to studies in fundamental condensed matter physics, optical substrates should be useful for assembling composite systems textured on the micron scale. In this case, the patterned illumination acts as a template for depositing particles directly or assisting self-assembly. The resulting mesoscopically arranged structures could be gelled in place and combined to create extended structures and devices. The fabrication of optical circuit elements might be based on such an approach.
Both tweezing arrays and optical substrates can be used to direct the self-assembly of three-dimensionally ordered colloidal crystals. These crystals recently have been shown to have a promising future as photonic circuit elements (45); (46) and as quantitative chemical sensors (47); (48). Their full commercial exploitation will require the ability to fabricate large single crystals with desirable symmetry and lattice constants. Diffractive optical arrays can be used to organize the first layer of a growing colloidal crystal and register it with a desired substrate. Subsequent layers then grow epitaxially on the first, leading to much longer-ranged three-dimensional order than might be possible otherwise. This principle has been demonstrated with lithographically patterned substrates (49). The optical substrates can be adjusted in situ to achieve optical ordering.
Multilayer tweezer arrays could be used to directly form multilayer photonic crystals such as photonic bandgap materials (50), including optical waveguides, tuned defect states in the bandgap, and planned interstitials. The high-dielectric-constant materials required for realizing a full photonic bandgap most likely cannot be trapped with conventional tweezer arrays. Arrays of optical vortex tweezers, however, should be amenable to the task.
Finally, scanned and otherwise time-dependent optical substrates can provide the time-modulated spatially asymmetric potentials required for practical particle size fractionation through directed diffusion (51). The optical ratchet principle required for practical directed diffusion has been demonstrated with a single scanned tweezer (52); (53); (54). Extended optical substrates may turn this demonstration into a practical technique.
This work was supported primarily by the MRSEC Program of the National Science Foundation under Award Number DMR-9400379. Additional support was provided by the National Science Foundation under Grant Number DMR-9320378. Mr. Dufresne was supported by a GAANN Fellowship of the Department of Education under Award Number P200A-10052.
References
-
(1)
A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986).
-
(2)
A. Ashkin, Proc. National Academy Sci., USA 94, 4853 (1997).
-
(3)
D. G. Grier, Current Opinion Colloid Interface Sci. 2, 264 (1997).
-
(4)
K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomolecular Structures 23, 247 (1994).
-
(5)
J. C. Crocker and D. G. Grier, Phys. Rev. Lett. 73, 352 (1994).
-
(6)
J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298 (1996).
-
(7)
A. D. Dinsmore, A. G. Yodh, and D. J. Pine, Nature 383, 239 (1996).
-
(8)
A. E. Larsen and D. G. Grier, Nature 385, 230 (1997).
-
(9)
T. Sugimoto et al., Langmuir 13, 5528 (1997).
-
(10)
S. B. Smith, L. Finzi, and C. Bustamante, Science 258, 1122 (1992).
-
(11)
C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, Science 265, 1599 (1994).
-
(12)
H. Yin et al., Science 270, 1653 (1995).
-
(13)
T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, Science 268, 83 (1995).
-
(14)
S. B. Smith, Y. Cui, and C. Bustamante, Science 271, 795 (1996).
-
(15)
H. Felgner, R. Frank, and M. Schliwa, J. Cell Sci. 109, 509 (1996).
-
(16)
M. D. Wang et al., Biophys. J. 72, 1335 (1997).
-
(17)
R. Bar-Ziv, Phys. Rev. Lett. 73, 1392 (1994).
-
(18)
R. Bar-Ziv, R. Menes, E. Moses, and S. A. Safran, Phys. Rev. Lett. 75, 3356 (1995).
-
(19)
R. Bar-Ziv, T. Frisch, and E. Moses, Phys. Rev. Lett. 75, 3481 (1995).
-
(20)
J. D. Moroz, P. Nelson, R. Bar-Ziv, and E. Moses, Phys. Rev. Lett. 78, 386 (1997).
-
(21)
K. Sasaki et al., Opt. Lett. 16, 1463 (1991).
-
(22)
H. Misawa et al., Appl. Phys. Lett. 60, 310 (1992).
-
(23)
G. Gouesbet, B. Maheu, and G. Gréhan, J. Opt. Soc. Am. A 5, 1427 (1988).
-
(24)
A. Ashkin, Biophys. J. 61, 569 (1992).
-
(25)
K. F. Ren, G. Gréhan, and G. Gouesbet, Appl. Opt. 35, 2702 (1996).
-
(26)
Ø. Farsund and B. U. Felderhof, Physica A 227, 108 (1996).
-
(27)
Y. Harada and T. Asakura, Opt. Comm. 124, 529 (1996).
-
(28)
E. Fällman and O. Axner, Appl. Opt. 36, 2107 (1997).
-
(29)
L. P. Faucheux and A. J. Libchaber, Phys. Rev. E 49, 5158 (1994).
-
(30)
K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Selected Topics Quantum Elect. 2, 1066 (1996).
-
(31)
K. Sasaki et al., Jap. J. Appl. Phys. 30, L907 (1997).
-
(32)
N. R. Heckenberg et al., Opt. Quantum Elect. 24, S951 (1992).
-
(33)
H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
-
(34)
K. T. Gahagan and J. G. A. Swartzlander, Opt. Lett. 21, 827 (1996).
-
(35)
A. Chowdhury, B. J. Ackerson, and N. A. Clark, Phys. Rev. Lett. 55, 833 (1985).
-
(36)
B. J. Ackerson and A. H. Chowdhury, Faraday Discuss. Chem. Soc. 83, 1 (1987).
-
(37)
K. Loudiyi and B. J. Ackerson, Physica A 184, 1 (1992).
-
(38)
K. Loudiyi and B. J. Ackerson, Physica A 184, 26 (1992).
-
(39)
M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, Science 249, 749 (1990).
-
(40)
A. Chowdhury, B. J. Ackerson, and N. A. Clark, Science 252, 1049 (1991).
-
(41)
M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, Science 252, 1049 (1991).
-
(42)
A. E. Chiou et al., Opt. Comm. 133, 7 (1997).
-
(43)
J. Chakrabarti, H. R. Krishnamurthy, and A. K. Sood, Phys. Rev. Lett. 73, 2923 (1994).
-
(44)
J. Chakrabarti, H. R. Krishnamurthy, A. K. Sood, and S. Sengupta, Phys. Rev. Lett. 75, 2232 (1995).
-
(45)
G. Pan, R. Kesavamoorthy, and S. A. Asher, Phys. Rev. Lett. 78, 3860 (1997).
-
(46)
D. G. Grier, Phys. World 10, 24 (1997).
-
(47)
J. H. Holtz and S. A. Asher, Nature 389, 829 (1997).
-
(48)
D. G. Grier, Nature 389, 784 (1997).
-
(49)
A. van Blaaderen, R. Ruel, and P. Wiltzius, Nature 385, 321 (1997).
-
(50)
J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
-
(51)
M. Bier and R. D. Astumian, Phys. Rev. Lett. 76, 4277 (1996).
-
(52)
L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, Phys. Rev. Lett. 74, 1504 (1995).
-
(53)
L. P. Faucheux, G. Stolovitzky, and A. Libchaber, Phys. Rev. E 51, 5239 (1995).
-
(54)
R. C. Gauthier, Appl. Phys. Lett. 67, 2269 (1995).