Holographic microscopy of holographically trapped three-dimensional structures
Abstract.
Holographic optical trapping uses forces exerted by computer-generated holograms to organize microscopic materials into three-dimensional structures. In a complementary manner, holographic video microscopy uses real-time recordings of in-line holograms to create time-resolved volumetric images of three-dimensional microstructures. The combination is exceptionally effective for organizing, inspecting and analyzing soft-matter systems.
Holographic optical trapping (1); (2) uses computer-generated holograms to trap and organize micrometer-scale objects into arbitrary three-dimensional configurations. No complementary method has been available for examining optically trapped structures except conventional two-dimensional microscopy. Three-dimensional imaging would be useful for verifying the structure of holographically organized systems before fixing them in place (1); (3); (4); (5). It also would be useful for interactively manipulating and inspecting three-dimensionally structured objects such as biological specimens. Integrating three-dimensional imaging with holographic trapping might seem straightforward because both techniques can make use of the same objective lens to collect and project laser light, respectively. The problem is that most three-dimensional imaging methods, such as confocal microscopy, involve mechanically translating the focal plane through the sample. Holographic traps, however, are positioned relative to the focal plane, and would move as well. Although the trapping pattern could be translated to compensate for the microscope's mechanical motion, the added complexity, reduced imaging speed, and potential for disrupting the sample are clear drawbacks.
Digital holographic microscopy addresses all of these technical concerns, providing real-time three-dimensional imaging data without requiring mechanical motion (7); (8); (9); (10); (6). A particularly compatible variant of in-line holographic microscopy simply replaces the conventional illuminator in a bright-field microscope with a collimated laser (6). Light scattered out of the laser beam by the object interferes with the remainder of the incident illumination to produce a heterodyne scattering pattern that is magnified by the objective lens and recorded with a video camera. Provided that this interference pattern is not obscured by multiple light scattering, it contains comprehensive information on the scatterers' three-dimensional configuration. Each two-dimensional snapshot in the resulting video stream encodes time-resolved volumetric information that can be analyzed directly, or decoded numerically into three-dimensional representations. This Article demonstrates digital holographic microscopy in a holographic optical manipulation system, and uses the combined capabilities to directly assess both techniques' accuracy and limitations.

Figure 1 shows a schematic representation
of the integrated system, which is based on an inverted
optical microscope (Zeiss Axiovert S100-TV) outfitted with
a NA 1.4 oil immersion objective lens.
This lens is used both to project holographic optical traps,
and also to collect in-line holographic images of trapped
objects.
Holographic traps are powered by a frequency-doubled diode-pumped
solid state laser (Coherent Verdi) operating at
a wavelength of
.
A liquid crystal spatial light modulator (Hamamatsu PAL-SLM X7550)
imprints the beam's wavefronts with
phase-only holograms encoding the desired trapping pattern
(11).
The modified beam then is relayed to the input pupil of the
objective lens and is focused into optical traps.
The trapping beam is relayed to the objective lens with
a dichroic mirror tuned to the trapping laser's wavelength.
Other wavelengths pass through the dichroic mirror and form
images on a CCD camera (NEC TI-324AII).
We replaced the standard combination of incandescent illuminator
and condenser lens with
a helium-neon laser providing 5 mW collimated beam
of coherent light at a
wavelength of
in air.








Figure 2 demonstrates holographic imaging of
colloidal spheres holographically trapped in a three-dimensional
pattern.
These 1.53 diameter silica spheres (Bangs Labs Lot No. L011031B)
are dispersed in a 50
thick layer of water confined
within a slit pore formed by sealing
the edges of a #1 cover slip to the surface of a clean
glass microscope slide.
Each sphere is trapped in a separate
point-like optical tweezer (12),
and the individual optical traps are
positioned independently in three dimensions
(1); (13); (14); (15); (11).
Figure 2(a) shows a conventional bright-field
image of the particles arranged in the focal plane.
Projecting a sequence of holograms with the trapping positions
slightly displaced enables us to rotate the entire pattern in
three dimensions, as shown in Fig. 2(b).
As particles move away from the focal plane, their images
blur, as can be seen in Fig. 1(c).
Indeed, it is difficult to determine from this image whether the
most distant particles are present at all.
Figure 2(d) shows the same field of view, but with laser illumination. Each particle appears in this image as the coherent superposition of the laser light it scatters with the undiffracted portion of the laser beam. Other features in the image result from reflections, refraction and scattering by surfaces in the optical train. These have been minimized by subtracting off a reference image obtained with no particles in the field of view.
Enough information is encoded in two-dimensional
real-valued images such as Fig. 2
to at least approximately reconstruct the
three-dimensional complex-valued light field.
The image in Fig. 2(e) is an example
showing a numerically reconstructed
vertical cross-section through the array of particles.
This demonstrates the feasibility of
combining holographic microscopy with holographic
optical trapping.
The reconstruction is consistent with the anticipated
45 inclination of the array, and with the calibrated
5.4
separation between the particles.
Intended particle coordinates are shown as circles superimposed
on the image.
This quantitative comparison
demonstrates the utility of holographic microscopy
for verifying holographic assemblies.
Because holographic images such as Fig. 2(d)
can be obtained at the full frame rate of the video camera,
holographic microscopy offers the benefit of real-time data
acquisition over confocal
and deconvolution microscopies.






Previous implementations of in-line holographic digital video microscopy (6) have at least implicitly invoked the Fresnel far-field approximation to analyze digital holograms. Holograms such as Fig. 2(d), however, form at ranges comparable to the wavelength of light. More accurate results can be expected, therefore, from the Rayleigh-Sommerfeld formalism (16).
The field scattered by an object at height
above the microscope's focal plane propagates to the focal plane,
where it interferes with the
reference field,
, comprised of the undiffracted portion
of the laser illumination.
Here,
is the position in the focal plane.
The Rayleigh-Sommerfeld propagator describing the object field's
propagation along the optical axis is (16)
![]() |
(1) |
where and
is the
light's wavenumber in a medium of refractive index
.
The field in the focal plane is the convolution
.
The observed interference pattern, therefore, is
![]() |
(2) |
The first term in Eq. (2) can be approximated by measuring the intensity when no objects are in the field of view. Figure 2(d) was obtained by subtracting such a reference image from the measured interference pattern. If we further assume that the scattered field is much dimmer than the reference field, the second term in Eq. (2) dominates the third. In that case,
![]() |
![]() |
(3) | ||
![]() |
(4) |
provides a reasonable basis for
reconstructing .
The final approximation in Eq. (3) requires
gradients in the illuminating field's phase to be more gradual
than any phase gradients of interest.
The three-dimensional intensity field
is most easily reconstructed from
using the Fourier convolution theorem,
according to which
![]() |
![]() |
(5) | ||
![]() |
(6) |
where is the Fourier transform of
and
![]() |
(7) |
is the Fourier transform of the Rayleigh-Sommerfeld propagator (17); (9); (16).
The estimate for the Fourier transform of the
object field at height above the focal plane is
obtained by applying the appropriate Rayleigh-Sommerfeld propagator
to translate the effective focal plane:
![]() |
(8) |
The first term in Eq. (8)
is the reconstructed field, which comes into best
focus when .
The second is an artifact that is increasingly blurred as
increases.
Unfortunately, this term creates a mirror image around the plane
with the result that objects below the focal plane cannot
be distinguished from objects above.
This ghosting is apparent in Fig. 2(e).
Our final estimate for the complex light field at height above the
focal plane is
![]() |
![]() |
(9) | ||
![]() |
(10) |
Equation (9) can reconstruct
a volumetric representation of the
instantaneous light field in the sample
from a single holographic snapshot, .
The image in Fig. 2(e) is a cross-section through
the reconstructed intensity distribution,
.







Each sphere in Fig. 2(e) appears as a bright axial streak centered on the object's three-dimensional position. Circles superimposed on Fig. 2(e) indicate the coordinates used to compute (11); (18) the trap-forming hologram that arranged the spheres. The very good agreement between the optical traps' design and features in the resulting reconstructed field attests to the accuracy of both the projection and imaging methods.
Contrary to previous reports (6), images such as
those in Fig. 3 suggest that the axial resolution
of our holographic reconstruction approaches
the diffraction-limited in-plane resolution.
Figure 3(a) shows a hologram obtained
for a sphere held by an optical tweezer at height
above the focal plane.
Figure 3(b) is an axial section through
the real part of field reconstructed from (a),
.
This representation has the benefit of most closely resembling the
scattering field observed in conventional three-dimensional
bright-field microscopy (19).
The sphere, in this case, is centered at the crossover between
bright and dark regions.
The effective axial resolution can be assessed by scanning
the sphere past the focal plane and stacking the resulting images
to create a volumetric data set.
Figure 3(c) is a hologram of the same sphere
from Fig. 3(a) at .
Compiling a sequence of such images in axial steps
of
yields the axial section in
Fig. 3(d).
The
tilt in the scanned image reflects the inclination
of the trapping system's axis relative to the imaging train.
Figure 3(e) was obtained by scanning the sphere
with conventional incoherent illumination. It features the
same tilt seen in Fig. 3(d), but has a much
shallower depth of focus.
Figure 3(f) shows axial intensity profiles obtained
from the images in Figs. 3(b) and (d).
The very close agreement between these two traces demonstrates
that the holographic reconstruction approaches diffraction-limited
resolution.
The zero crossing in either case can be resolved to within
20 ,
which is comparable to the instrumentally limited in-plane
tracking resolution (20).
Structure in the spheres' images along the axial direction can be
analyzed to track the spheres in , as well as in
and
.
For the micrometer-scale particles studied here, for example, the
centroid is located in the null plane between the downstream intensity maximum
and the upstream intensity minimum along the scattering pattern's axis.
Holographic microscopy of colloidal particles therefore can be
used to extract three-dimensional trajectories more accurately than
is possible with conventional two-dimensional imaging
(20); (21) and far more rapidly than with scanned
three-dimensional imaging techniques (22).
In particular, in-plane tracking can make use of
conventional techniques (20), and tracking in depth
requires additional computation but no additional calibration.
Analyzing images becomes far more challenging when objects occlude
each other along the optical axis, as Fig. 4
demonstrates.
Here, the same pattern of spheres from Fig. 2
has been rotated by , so that four of the spheres
are aligned along the optical axis.
Figure 4(a) is a detail from the resulting
hologram and Figs. 4(b) and (c) are vertical sections through
the amplitude and imaginary part of the
reconstructed field, respectively. The latter has been
squared to mimic the contrast of a conventional
intensity representation.
Each spheres in Fig. 4 is centered on
a local maximum in
.
These maxima, in turn,
correspond to the points of inflection in
that were used to establish resolution
limits in Fig. 3.
The central observation from Fig. 4 is that all
four spheres are resolved, even though they directly
occlude each other.
An axial trace through along the
spheres' centers, plotted in Fig. 4(d),
clearly shows the three downstream spheres.
The conventionally in-focus sphere at
is suppressed in this
representation, but can be seen in the amplitude representation
in Fig. 4(b).
A fifth sphere, not directly occluded by the others was included
as a reference, and is visible to the right of the others in
Figs. 4(a), (b) and (c).
The lower spheres in Fig. 4 appear progressively brighter than the spheres they occlude because they act as lenses, gathering light scattered from above and focusing it onto the optical axis. Equation (9) does not take such multiple light scattering into account when reconstructing the light field. The resulting uncertainty in interpreting such results can be mitigated by acquiring images from multiple focal planes, or by illuminating the sample from multiple angles, rather than directly in-line (23). Results also would be improved by more accurate recordings. Each pixel in our holographic images contains roughly 6 bits of usable information, and no effort was made to linearize the camera's response. The camera was set to 1/2000 s shutter speed, which nonetheless allows for some particle motion during each exposure. A wider dynamic range, calibrated intensity response and faster shutter all would provide sharper, more accurate holograms, and thus clearer three-dimensional reconstructions.
With these caveats, the traces in Fig. 4(d)
highlight the potential importance of holographic imaging
for three-dimensional holographic manipulation.
The most distant particle appears to be very slightly
displaced along the optical axis relative to the reference particle
even though both were localized in optical tweezers projected
to the same height.
Three-dimensional visualizations confirm
the structure of the projected trapping field
(24).
The apparent axial displacement was not evident for
inclinations less than roughly .
It therefore reflects either a three-dimensional
imaging artifact or, more likely, a real displacement
of the particles from their designed configuration.
This is reasonable because light from the traps
projected closer to the focal plane exerts forces
on particles trapped deeper into the sample.
Inter-trap interactions are exacerbated by particles trapped
closer to the focal plane, which deflect light
onto more distant particles, altering their
effective potential energy wells.
This effect has been exploited for in-line
optical binding of particles trapped along
thread-like Bessel beams (25); (26).
Holographic imaging provides a means for measuring
such distortions, and thus a basis for correcting
them.
Adaptive structural optimization
can be critically important for processes
such as the holographic assembly of photonic
heterostructures, which rely on accurate placement
of microscopic-scale objects (4); (5).
This work was supported by the National Science Foundation through Grant Number DBI-0629584 and Grant Number DMR-0606415. SHL acknowledges support of a Kessler Family Foundation Fellowship.
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