Volumetric imaging of holographic optical traps
Abstract.
The holographic optical trapping technique creates arbitrary threedimensional configurations of optical traps, each with individually specified characteristics. Holographic modification of the individual traps' wavefronts can transform conventional pointlike optical tweezers into traps with different structures and properties, and can position them independently in three dimensions. Here, we describe a technique for rapidly characterizing holographic optical traps' threedimensional intensity distributions. We create volumetric representations by by holographically translating the traps through the optical train's focal plane, acquiring a stack of twodimensional images in the process. We apply this technique to holographic line traps, which are used to create tailored onedimensional potential energy landscapes for mesoscopic objects. These measurements highlight problems that can arise when projecting extended traps with conventional optics and demonstrates the effectiveness of shapephase holography for creating nearly ideal line traps.
A singlebeam optical trap uses forces exerted by focused beam of light to confine a small object to a particular point in space. Commonly known as optical tweezers (1), these optical micromanipulators have revolutionized research in several branches of biology, chemistry and physics. Extended optical tweezers, which sometimes are called line traps, differ from pointlike optical tweezers by acting as onedimensional potential energy landscapes for mesoscopic objects. Line traps can be used to rapidly screen the interactions between colloidal particles and biological materials (2); (3); (4) and thus have potentially valuable applications in biological research, medical diagnostics and drug discovery. These applications, however, require methods for projecting line traps with precisely specified characteristics. Recently, we introduced a class of extended optical traps created with shapephase holography (5) whose intensity and phase profiles can be independently specified. In this article, we provide a detailed view of these traps' threedimensional intensity distributions and contrast their performance with other classes of extended optical traps.
Our method is based on the optimized holographic trapping technique (6); (7), shown schematically in Fig. 1. Here, a beam of light from a frequencydoubled solidstate laser (Coherent Verdi) operating at a wavelength of is directed to the input pupil of a highnumericalaperture objective lens (Nikon Plan Apo, NA 1.4, oil immersion) that focuses it into an optical trap. The laser beam is imprinted with a phaseonly hologram by a computeraddressed liquidcrystal spatial light modulator (SLM, Hamamatsu X8267 PPM) in a plane conjugate to the objective's input plane. As a result, the light field, , in the objective's focal plane is related to the field in the plane of the SLM by the Fraunhofer transform (8)
(1) 
where is the objective's focal length, where is the optical train's aperture, and where we have dropped irrelevant phase factors. Assuming that the laser illuminates the SLM with a radially symmetric amplitude profile, , and uniform phase, the field in the SLM's plane may be written as
(2) 
where is the realvalued phase profile imprinted on the beam by the SLM. The SLM in our system imposes phase shifts between 0 and at each pixel of a array. This twodimensional phase array can be used to project a computergenerated phaseonly hologram, , designed (7) to transform the single optical tweezer into any desired threedimensional configuration of optical traps, each with individually specified intensities and wavefront properties.
Ordinarily, the pattern of holographic optical traps would be put to use by projecting it into a fluidborne sample mounted in the objective's focal plane. To characterize the light field, we instead mount a frontsurface mirror in the sample plane. This mirror reflects the trapping light back into the objective lens, which transmits images of the traps through a partially reflecting mirror to a chargecoupled device (CCD) camera (NEC TI324AII). In our implementation, the objective, camera and camera eyepiece are mounted in a conventional optical microscope (Nikon TE2000U).
Threedimensional reconstructions of the optical traps' intensity distribution can be obtained by translating the mirror relative to the objective lens. Equivalently, the traps can be translated relative to the fixed mirror by superimposing the parabolic phase function
(3) 
onto the hologram encoding a particular pattern of traps. The combined hologram, , projects the same pattern of traps as but with each trap translated by along the optical axis. The resulting image obtained from the reflected light represents a crosssection of the original trapping intensity at distance from the objective's focal plane. Translating the traps under software control is particularly convenient because it minimizes changes in the optical train's properties due to mechanical motion and facilitates more accurate displacements along the optical axis. Images obtained at each value of are stacked up to yield a complete volumetric representation of the intensity distribution.
As shown schematically in Fig. 3, the objective captures essentially all of the reflected light for . For , however, the outermost rays of the converging trap are cut off by the objective's output pupil, and the contrast is reduced accordingly. This could be corrected by multiplying the measured intensity field by a factor proportional to for . The appropriate factor, however, is difficult to determine accurately, so we present only unaltered results.
Figure 2 shows a conventional optical tweezer reconstructed in this way and displayed as an isointensity surface at 5 percent peak intensity and in three crosssections. The former representation is useful for showing the overall structure of the converging light, and the crosssections provide an impression of the three dimensional light field that will confine an optically trapped object. The angle of convergence of in immersion oil obtained from these data is consistent with an overall numerical aperture of 1.4. The radius of sharpest focus, , is consistent with diffractionlimited focusing of the beam.
These results highlight two additional aspects of this reconstruction technique. The objective lens is designed to correct for spherical aberration when light passing through water is refracted by a glass coverslip. Without this additional refraction, the projected optical trap actually is degraded by roughly of spherical aberration, introduced by the lens. This reduces the apparent numerical aperture and also extends the trap's focus along the axis. The trap's effective numerical aperture in water would be roughly 1.2. The effect of spherical aberration can be approximately corrected by predistorting the beam with the additional phase profile
(4) 
the Zernike polynomial describing spherical aberration (9). The radius, , is measured as a fraction of the optical train's aperture, and the coefficient is measured in wavelengths of light. This procedure is used to correct for small amount of aberration present in practical optical trapping systems to optimize their performance (7); (10).
This correction was applied to the array of 35 optical tweezers shown as a threedimensional reconstruction in Fig. 4. These optical traps are arranged in a threedimensional bodycentered cubic (BCC) lattice with a 10.8 lattice constant. Without correcting for spherical aberration, these traps would blend into each other along the optical axis. With correction, their axial intensity gradients are clearly resolved. This accounts for holographic traps' ability to organize objects along the optical axis (11); (12).
Correcting for aberrations reduces the range of displacements, , that can be imaged. Combining with and increases gradients in , particularly for larger values of near the edges of the DOE. Diffraction efficiency falls off rapidly when exceeds , the maximum phase gradient that can be encoded on an SLM with pixel size . This problem is exacerbated when itself has large gradients. We therefore study more complex trapping patterns without aberration correction. In particular, we use uncorrected volumetric imaging to illustrate the comparative advantages of extended optical traps created by recently introduced holographic techniques.
Extended optical traps have been projected in a timeshared sense by rapidly scanning a conventional optical tweezer along the trap's intended contour (13); (14); (15); (2); (3); (4). A scanned trap has optical characteristics as good as a pointlike optical tweezer, and an effective potential energy well that can be tailored by adjusting the instantaneous scanning rate. Kinematic effects due to the trap's motion can be minimized by scanning rapidly enough (14). For some applications, however, continuous illumination or the simplicity of an optical train with no scanning capabilities can be desirable.
Continuously illuminated line traps have been created by expanding an optical tweezer along one direction. This can be achieved, for example, by introducing a cylindrical lens into the objective's input plane (16); (17). Equivalently, a cylindricallens line tweezer can be implemented by encoding the function on the SLM (18). The result, shown in Fig. 5 appears serviceable in the plane of best focus, , with the pointlike tweezer having been extended to a line with nearly parabolic intensity and a nearly Gaussian phase profile. The threedimensional reconstruction in Fig. 5, however, reveals that the cylindrical lens merely introduces a large amount of astigmatism into the beam, creating a second focal line perpendicular to the first. This is problematic because the astigmatic beam's axial intensity gradients are far weaker than a conventional optical tweezer's. Consequently, cylindricallens line traps typically cannot localize objects against radiation pressure along the optical axis.
Replacing the single cylindrical lens with a cylindrical Keplerian telescope (17) eliminates the astigmatism and thus creates a stable threedimensional optical trap. Similarly, using an objective lens to focus two interfering beams creates an interferometric optical trap capable of threedimensional trapping (19); (20). These approaches, however, offer little control over the extended traps' intensity profiles, and neither affords control over the phase profile.
Shapephase holography provides absolute control over both the amplitude and phase profiles of an extended optical trap at the expense of diffraction efficiency. It also yields traps with optimized axial intensity gradients, suitable for threedimensional trapping (5). If the line trap is characterized by an amplitude profile and a phase profile along the direction in the objective's focal plane, then the field in the SLM plane is given by Eq. (1) as
(5) 
where the phase is adjusted so that . Shapephase holography implements this onedimensional complex wavefront profile as a twodimensional phaseonly hologram (5)
(6) 
where the shape function allocates a number of pixels along the row proportional to . One particularly effective choice is for to select pixels randomly along each row in the appropriate relative numbers (5). The unassigned pixels then are given values that redirect the excess light away from the intended line. Typical results are presented in Fig. 6.
Unlike the cylindricallens trap, the holographic line trap focuses as a conical wedge to a single diffractionlimited line in the objective's focal plane. Consequently, its transverse angle of convergence is comparable to that of an optimized point trap. This means that the holographic line trap has comparably strong axial intensity gradients, which explains its ability to trap objects stably against radiation pressure in the direction.
The line trap's transverse convergence does not depend strongly on the choice of intensity profile along the line. Its threedimensional intensity distribution, however, is very sensitive to the phase profile along the line. Abrupt phase changes cause intensity fluctuations through Gibbs phenomenon. Smoother variations do not affect the intensity profile along the line, but can substantially restructure the beam. The line trap created by a cylindrical lens, for example, has a parabolic phase profile. Inserting this choice into Eq. (2) and calculating the associated shapephase hologram with Eqs. (1) and (6) yields the same cylindrical lens phase profile. This observation opens the door to applications in which the phase profile along a line can be tuned to create a desired threedimensional intensity distribution, or in which the measured threedimensional intensity distribution can be used to assess the phase profile along the line. These applications will be discussed elsewhere.
This work was supported by the National Science Foundation through grant number DMR0451589.
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