Holographic microscopy of holographically trapped threedimensional structures
Abstract.
Holographic optical trapping uses forces exerted by computergenerated holograms to organize microscopic materials into threedimensional structures. In a complementary manner, holographic video microscopy uses realtime recordings of inline holograms to create timeresolved volumetric images of threedimensional microstructures. The combination is exceptionally effective for organizing, inspecting and analyzing softmatter systems.
Holographic optical trapping (1); (2) uses computergenerated holograms to trap and organize micrometerscale objects into arbitrary threedimensional configurations. No complementary method has been available for examining optically trapped structures except conventional twodimensional microscopy. Threedimensional 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 threedimensionally structured objects such as biological specimens. Integrating threedimensional 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 threedimensional 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 realtime threedimensional imaging data without requiring mechanical motion (7); (8); (9); (10); (6). A particularly compatible variant of inline holographic microscopy simply replaces the conventional illuminator in a brightfield 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' threedimensional configuration. Each twodimensional snapshot in the resulting video stream encodes timeresolved volumetric information that can be analyzed directly, or decoded numerically into threedimensional 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 S100TV) outfitted with a NA 1.4 oil immersion objective lens. This lens is used both to project holographic optical traps, and also to collect inline holographic images of trapped objects. Holographic traps are powered by a frequencydoubled diodepumped solid state laser (Coherent Verdi) operating at a wavelength of . A liquid crystal spatial light modulator (Hamamatsu PALSLM X7550) imprints the beam's wavefronts with phaseonly 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 TI324AII). We replaced the standard combination of incandescent illuminator and condenser lens with a heliumneon 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 threedimensional 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 pointlike 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 brightfield 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 twodimensional realvalued images such as Fig. 2 to at least approximately reconstruct the threedimensional complexvalued light field. The image in Fig. 2(e) is an example showing a numerically reconstructed vertical crosssection 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 realtime data acquisition over confocal and deconvolution microscopies.
Previous implementations of inline holographic digital video microscopy (6) have at least implicitly invoked the Fresnel farfield 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 RayleighSommerfeld 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 RayleighSommerfeld 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 threedimensional 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 RayleighSommerfeld 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 RayleighSommerfeld 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 crosssection through the reconstructed intensity distribution, .
Each sphere in Fig. 2(e) appears as a bright axial streak centered on the object's threedimensional position. Circles superimposed on Fig. 2(e) indicate the coordinates used to compute (11); (18) the trapforming 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 diffractionlimited inplane 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 threedimensional brightfield 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 diffractionlimited resolution. The zero crossing in either case can be resolved to within 20 , which is comparable to the instrumentally limited inplane 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 micrometerscale 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 threedimensional trajectories more accurately than is possible with conventional twodimensional imaging (20); (21) and far more rapidly than with scanned threedimensional imaging techniques (22). In particular, inplane 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 infocus 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 inline (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 threedimensional reconstructions.
With these caveats, the traces in Fig. 4(d) highlight the potential importance of holographic imaging for threedimensional 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. Threedimensional 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 threedimensional 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. Intertrap 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 inline optical binding of particles trapped along threadlike 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 microscopicscale objects (4); (5).
This work was supported by the National Science Foundation through Grant Number DBI0629584 and Grant Number DMR0606415. SHL acknowledges support of a Kessler Family Foundation Fellowship.
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