Optimized holographic optical traps
Abstract.
Holographic optical traps use the forces exerted by computergenerated holograms to trap, move and otherwise transform mesoscopically textured materials. This article introduces methods for optimizing holographic optical traps' efficiency and accuracy, and an optimal statistical approach for characterizing their performance. This combination makes possible realtime adaptive optimization.
A single laser beam brought to a focus with a strongly converging lens forms a type of optical trap widely known as an optical tweezer (1). Multiple beams of light passing simultaneously through the lens' input pupil focus to multiple optical tweezers, each at a location determined by the associated beam's angle of incidence and degree of collimation as it enters the lens. Their intersection at the input pupil yields an interference pattern whose amplitude and phase corrugations characterize the downstream trapping pattern. Imposing the same modulations on a single incident beam at the input pupil would yield the same pattern of traps. Such wavefront modification can be performed by a computerdesigned diffractive optical element (DOE), or hologram.
Holographic optical trapping (HOT) uses computergenerated holograms (CGHs) to project arbitrary configurations of optical traps (2); (3); (4); (5); (6), and so provides exceptional control over microscopic materials dispersed in fluid media. Holographic micromanipulation provides the basis for a rapidly growing field of applications in the physical and biological sciences as well as in industry (7).
This article describes refinements to the HOT technique that help to optimize the traps' performance. It also introduces selfconsistent and statistically optimal methods for characterizing their performance. Section I describes modifications to the basic HOT optical train that compensate for practical limitations of dynamic holography. Section II discusses a direct search algorithm for HOT CGH computation that is both faster and more accurate than commonly used iterative refinement algorithms. Together, these modifications yield marked improvements in the holographic traps' performance that can be quantified rapidly using techniques introduced in Section III. These techniques are based on optimal statistical analysis of trapped colloidal spheres' thermallydriven motions, and lend themselves to simultaneous realtime characterization and optimization of entire arrays of traps through digital video microscopy. Such adaptive optimization is demonstrated experimentally in Section IV.
§ I. Improved optical train
Figure 1(a) depicts a conventional HOT implementation in which a collimated laser beam is modified by a computerdesigned DOE, and thereafter propagates as a superposition of independent beams, each with individually specified wavefront characteristics (4); (6). These beams are relayed to the input pupil of a highnumericalaperture lens, typically a microscope objective, which focuses them into optical traps. Although a transmissive DOE is shown in Fig. 1, comparable results are obtained with reflective DOEs. The same objective lens used to form the optical traps also can be used to create images of trapped objects. The associated illumination and imageforming optics are omitted from Fig. 1 for clarity.
Practical DOEs only diffract a portion of the incident light into the intended modes and directions. Some of the incident beam may not be diffracted at all, and the undiffracted portion typically forms an unwanted trap in the middle of the field of view (8). This “central spot” has been removed in previous implementations by spatially filtering the diffracted beam (8); (9). Practical DOEs also tend to project spurious “ghost” traps into symmetrydictated positions within the sample. Spatially filtering a large number of ghost traps generally is not practical, particularly in the case of dynamic holographic optical tweezers whose traps move freely in three dimensions. Projecting holographic traps in the offaxis Fresnel geometry automatically eliminates the central spot (10), but limits the number of traps that can be projected, and also does not address the formation of ghost traps.
Figure 1(b) shows a basic improvement that minimizes the central spot's influence and effectively eliminates ghost traps. Rather than illuminating the DOE with a collimated laser beam, a converging beam is used. This moves the undiffracted central spot upstream of the objective's normal focal plane. The intended traps can be moved back to the focal plane by incorporating wavefrontshaping phase functions into the hologram's design (6). Deliberately decollimating the input beam allows the central spot to be projected outside of the sample volume, thereby ensuring that the undiffracted beam lacks both the intensity and the intensity gradients needed to influence the sample's dynamics.
An additional consequence of the traps' displacement relative to the modified optical train's focal plane is that most ghost traps also are projected out of the sample volume. This is a substantial improvement for processes such as optical fractionation (11); (12) and optical ratchets (13); (14), which require defectfree intensity distributions.
Even though the undiffracted beam may not create an actual trap in this modified optical train, it still can exert radiation pressure on parts of the sample near the center of the field of view. This is a particular problem for large arrays of optical traps in that the central spot, which typically receives a fixed proportion of the input beam, can be brighter than the intended traps. Illuminating the DOE with a diverging beam (15) reduces the undiffracted beam's influence by projecting some of its light out of the optical train. In a thick sample, however, this has the deleterious effect of projecting both the weakened central spot and the undiminished ghost traps into the sample.
These problems all can be mitigated by placing a beam block as shown in Fig. 1(c) in the intermediate focal plane within the relay optics to spatially filter the undiffracted portion of the beam. The trapforming beams focus downstream of the beam block and therefore are only partially occluded, even if they pass directly along the optical axis. This has little effect on the performance of conventional optical tweezers and can be compensated by increasing the occluded traps' relative brightness.
§ II. Algorithms for HOT CGH Calculation
Holographic optical tweezers' efficacy is determined by the quality of the trapforming DOE, which in turn reflects the performance of the algorithms used in their computation. Previous studies have applied holograms calculated by simple linear superposition of the input fields (3), with best results being obtained with random relative phases (4); (6), or with variations (4); (5); (6) on the classic GerchbergSaxton and adaptiveadditive algorithms (16). Despite their generality, these algorithms yield traps whose relative intensities can differ greatly from their design values, and often project an unacceptably large fraction of the input power into ghost traps. These problems can become acute for complicated threedimensional trapping patterns, particularly when the same hologram also is used as a mode converter to project multifunctional arrays of optical traps (4); (6). This section describes faster and more effective algorithms for HOT DOE calculation based on direct search optimization.
The holograms used for holographic optical trapping typically operate only on the phase of the incident beam, and not its amplitude. Such phaseonly holograms, also known as kinoforms, are far more efficient than amplitudemodulating holograms, which necessarily divert light away from the traps. Quite general trapping patterns can be achieved with kinoforms because optical tweezers rely for their operation on intensity gradients and not on phase variations. The challenge is to find a phase pattern in the input plane of the objective lens that encodes the desired intensity pattern in the focal volume.
Most approaches to designing phaseonly holograms are based on scalar diffraction theory, in which the complex field , in the focal plane of a lens of focal length is related to the field, , in its input plane by a Fraunhofer transform,
(1) 
Here, and are the realvalued amplitude and phase, respectively, at position in the input pupil, and is the wavenumber of light of wavelength . If is taken to be the amplitude profile of the input laser beam, then is the kinoform encoding the intensity distribution . Finding the kinoform to project a particular pattern, , is nontrivial because the inherent nonlinearity of Eq. (1) defies straightforward inversion. Even so, kinoforms may be estimated through indirect search algorithms. The particular requirements of holographic trapping lend themselves to especially fast and effective computation.
Simply computing the superposition of input beams required to create a desired trapping pattern, disregarding the resulting amplitude variations, and retaining the phase as an estimate for turns out to be remarkably effective (3); (4); (6). Fidelity to design is particularly good if the input beams are assumed to have random relative phases. Not surprisingly, such randomly phased superpositions yield traps with widely varying intensities as well as a great many ghost traps.
The process of refining such an initial estimate begins by noting that most practical DOEs, including those projected with SLMs, consist of an array of discrete phase pixels, , centered at locations , each of which can impose any of possible discrete phase shifts on the incident beam. The field in the focal plane due to such an pixel DOE is, therefore,
(2) 
where the transfer function describing the light's propagation from input plane to output plane is
(3) 
Unlike more general holograms, the desired field in the output plane of a holographic optical trapping system consists of discrete bright spots located at :
(4)  
(5) 
where is the relative amplitude of the th trap, normalized by , and is its (arbitrary) phase. Here represents the amplitude profile of the focused beam of light, and may be approximated by a twodimensional Dirac delta function. For simplicity, we may also approximate the input beam's amplitude profile by a tophat function with within the input pupil's aperture, and elsewhere. In these approximations, the field at the th trap is (6)
(6) 
with . We introduce the inverse operator because the hologram may modify the wavefronts of each of the diffracted beams it creates in addition to establishing its direction of propagation. Such wavefront distortions are useful for creating threedimensional arrays of multifunctional traps. However, they also distort the traps' otherwise sharply peaked profiles in the focal plane, which were assumed in Eq. (4). The inverse operators correct for these distortions so that even generalized traps can be treated discretely.
For example, a trap can be displaced a distance away from the focal plane by curving the input beam's wavefronts into a parabolic profile
(7) 
The operator that displaces the th trap to is (4); (6)
(8) 
Its inverse, returns the th trap to best focus in the focal plane.
Similarly, a conventional TEM beam can be converted into a helical mode with the phase profile
(9) 
where is the azimuthal angle around the optical axis and is a winding number known as the topological charge. Such corkscrewlike beams focus to ringlike optical traps known as optical vortices, which can exert torques as well as forces (17); (18); (19); (20). The topologytransforming kernel (6)
(10) 
can be composed with in the same manner as the displacementinducing to convert the th trap into an optical vortex. A variety of analogous phasebased mode transformations have been described, each with applications to singlebeam optical trapping (7), all of which can be applied to each trap independently in this manner.
Calculating the fields only at the traps' positions greatly reduces the computational burden of HOT CGH refinement. It also eliminates the need to account for the beams' propagation through intermediate trapping planes when designing threedimensional patterns (4). Unlike more general FFTbased algorithms (5), this restricted approach does not directly optimize the field between the traps. If the converged amplitudes match the design values, however, no light is left over to create ghost traps.
Applying Eq. (6) directly in an iterative refinement algorithm (6) also has drawbacks. In particular, only the relative phases in Eq. (5) can be adjusted when inverting Eq. (6) to solve for . Having so few free parameters severely limits the improvement over simple superposition that can be obtained. Equation (6) suggests an alternative approach that not only is far more effective, but also is substantially more efficient.
The operator describes how light in the mode of the th trap propagates from the th phase pixel on the DOE to the trap's projected position . Changing the pixel's value by therefore changes each trap's field by
(11) 
If such a change were to improve the overall pattern, we would be inclined to retain it, and to seek other such improvements. This is the basis for direct search algorithms. The simplest involves selecting a pixel at random from a trial phase pattern, changing its value to any of the alternatives, and computing the effect on the projected pattern. Quite clearly, there is a considerable computational advantage in calculating changes only at the traps' positions, rather than over the entire focal plane. The updated trial amplitudes then are compared with their design values and the proposed change is accepted if the overall error is reduced. The process is repeated until the result converges to the design or the acceptance rate for proposed changes dwindles.
Effective and efficient refinement by the direct search algorithm depends on the choice of metric for quantifying convergence. The standard cost function, , assesses the meansquared deviations of the th trap's projected intensity from its design value , assuming an overall diffraction efficiency of . It requires an accurate estimate for and places no emphasis on uniformity in the projected traps' intensities. An alternative proposed by Meister and Winfield (21),
(12) 
avoids both shortcomings. Here, is the mean intensity at the traps and
(13) 
measures the deviation from uniform convergence to the design intensities. Selecting
(14) 
minimizes the total error and optimally accounts for nonideal diffraction efficiency (21). The weighting fraction sets the relative importance attached to diffraction efficiency versus uniformity, with providing a generally useful balance.
A direct binary search proceeds with any candidate change that reduces being accepted, and all others being rejected. In a worstcase implementation, the number of trials required for convergence should scale as , the product of the number of phase pixels and the number of possible phase values. In practice, this estimate is accurate if and are comparatively small and if the starting phase function is either uniform or purely random. Much faster convergence can be obtained by starting from the a randomly phased superposition of input beams. In this case, convergence typically is obtained within trials, even for fairly complex trapping patterns, and thus requires a computational effort comparable to the initial superposition.
As a practical demonstration, we have implemented a quasiperiodic array of optical traps, which is challenging because it has no translational symmetries. The traps are focused with a NA 1.4 SPlan Apo oil immersion objective lens mounted in a Nikon TE2000U inverted optical microscope. The traps are powered by a Coherent Verdi frequencydoubled diodepumped solid state laser operating at a wavelength of 532 . Computergenerated phase holograms are imprinted on the beam with a Hamamatsu X826716 parallelaligned nematic liquid crystal spatial light modulator (SLM). This SLM can impose phase shifts up to at each pixel in a array. The face of the SLM is imaged onto the objective's 5 diameter input pupil using relay optics designed to minimize aberrations. The beam is directed into the objective with a dichroic beamsplitter, which allows images to pass through to a lownoise chargecoupled device (CCD) camera (NEC TI324AII). The video stream is recorded as uncompressed digital video with a Pioneer 520H digital video recorder (DVR) for processing.
Figure 2(a) shows the intended planar arrangement of 119 holographic optical traps designed by the dual generalized method for generating quasiperiodic lattices (22). Even after adaptiveadditive refinement, the hologram resulting from simple superposition with random phases fares poorly for this aperiodic pattern. Figure 2(b) shows the intensity of light reflected by a frontsurface mirror placed in the sample plane. This image reveals extraneous ghost traps, an exceptionally bright central spot, and large variability in the intended traps' intensities. Imaging photometry on this and equivalent images produced with different random relative phases for the beams yields a typical rootmeansquare (RMS) variation of more than 50 percent in the projected traps' brightness. The image in Fig. 2(c) was produced using the modified optical train described in Sec. I and the direct search algorithm described in Sec. II, and suffers from none of these defects. Both the ghost traps and the central spot are suppressed, and the apparent relative brightness variations are smaller than 5 percent, a factor of ten improvement. Figure 2(d) shows 119 colloidal silica spheres, in diameter (Bangs Labs, lot 5238), dispersed in water at and trapped in the quasiperiodic array.
To place the benefits of the direct search algorithm on a more quantitative basis, we augment standard figures of merit with those introduced in Ref. (21). In particular, the DOE's theoretical diffraction efficiency is commonly defined as
(15) 
and its rootmeansquare (RMS) error as
(16) 
The resulting pattern's departure from uniformity is usefully gauged as (21)
(17) 
Figure 3 shows results for a HOT DOE encoding 51 traps, including 12 optical vortices of topological charge , arrayed in three planes relative to the focal plane. The excellent results in Fig. 3 were obtained with a single pass of directsearch refinement. The resulting traps, shown in the bottom three images, again vary from their planned relative intensities by less than 5 percent. In this case, the spatially extended vortices were made as bright as the pointlike optical tweezers by increasing their requested relative brightness by a factor of 15. This single hologram, therefore, demonstrates independent control over threedimensional position, wavefront topology, and brightness of all the traps. Performance metrics for the calculation are plotted in Fig. 3(b) as a function of the number of accepted singlepixel changes, with an overall acceptance rate of 16 percent. Direct search refinement achieves greatly improved fidelity to design over randomly phase superposition at the cost of a small fraction of the diffraction efficiency and roughly doubled computation time. The entire calculation can be completed in the refresh interval of a typical liquid crystal spatial light modulator.
§ III. Optimal characterization
Gauging a HOT system's performance numerically and by characterizing the projected intensity pattern does not provide a complete picture. The real test is in the projected traps' ability to localize particles. A variety of approaches have been developed for measuring the forces exerted by optical traps. The earliest involved estimating the hydrodynamic drag required to dislodge a trapped particle (23). This has several disadvantages, most notably that it identifies only the marginal escape force in a given direction and not the trap's actual threedimensional potential. Complementary information can be obtained by measuring a particle's thermally driven motions in the trap's potential well (24); (25); (26). For instance, the measured probability density for displacements is related to the trap's potential through the Boltzmann distribution
(18) 
where is the thermal energy scale at temperature . Similarly, the power spectrum of for a harmonically bound particle is a Lorentzian whose width is the viscous relaxation time of the particle in the well (24); (27).
Both of these approaches require amassing enough data to characterize the trapped particle's least probable displacements, and therefore oversample the trajectories. Oversampling is acceptable when data from a single optical trap can be collected rapidly, for example with a quadrant photodiode (24); (25); (26); (28). Tracking multiple particles in holographic optical traps, however, requires the area detection capabilities of digital video microscopy (29), which yields data much more slowly. Analyzing video data with optimal statistics (30) offers the benefits of thermal calibration by avoiding oversampling.
An optical trap is accurately modeled as a harmonic potential energy well (25); (26); (27); (28),
(19) 
with a different characteristic curvature along each axis. This form also is convenient because it is separable into onedimensional contributions. The trajectory of a colloidal particle localized in a viscous fluid by a harmonic well is described by the onedimensional Langevin equation (31)
(20) 
where the autocorrelation time , is set by the particle's viscous drag coefficient and by the curvature of the well, . The Gaussian random thermal force, , has zero mean, , and variance
(21) 
If the particle is at position at time , its trajectory at later times is given by
(22) 
Sampling such a trajectory at discrete times , yields
(23) 
where ,
(24) 
and where is a Gaussian random variable with zero mean and variance
(25) 
Because , Eq. (23) is an example of an autoregressive process (30), which is readily invertible.
In principle, the particle's trajectory can be analyzed to extract and , and thus the trap's stiffness, , and the particle's viscous drag coefficient . In practice, however, the experimentally measured particle positions differ from the actual positions by random errors , which we assume to be taken from a Gaussian distribution with zero mean and variance . The measurement then is described by the coupled equations
(26) 
where is independent of . We still can estimate and from a set of measurements by first constructing the joint probability
(27) 
The probability density for measuring the trajectory , is then the marginal (30)
(28)  
(29) 
where with transpose , is the identity matrix, and
(30) 
with the tridiagonal memory tensor
(31) 
Calculating the determinant, , and inverse, , of is greatly facilitated if we artificially impose time translation invariance by replacing with the matrix that identifies time step with time step 1. Physically, this involves imparting an impulse, , that translates the particle from its last position, , to its first, . Because diffusion in a potential well is a stationary process, the effect of this change is inversely proportional to the number of measurements, . With this approximation,
(32)  
and  
(33) 
so that the conditional probability for the measured trajectory, , is
(34) 
where is the th component of the discrete Fourier transform of . This can be inverted to obtain the likelihood function for , , and :
(35) 
Best estimates for the parameters are solutions of the coupled equations
(36) 
§ III.1. Case 1: No measurement errors ()
Equations (36) can be solved in closed form if . In this case,
(37) 
where
(38) 
is the barrel autocorrelation of at lag . The associated statistical uncertainties are
(39) 
In the absence of measurement errors, and constitute sufficient statistics for the time series (30) and thus embody all of the information that can be extracted.
§ III.2. Case 2: Small measurement errors ()
The analysis is less straightforward when because Eqs. (36) no longer are simply separable. The system of equations can be solved approximately if . In this case, the best estimates for the parameters can be expressed in terms of the errorfree estimates as
and  
(40) 
to first order in , with statistical uncertainties propagated in the conventional manner. Expansions to higher order in involve additional correlations, and the exact solution involves correlations at all lags . If measurement errors are small enough for Eq. (40) to apply, the computational savings relative to other approaches can be substantial, and the amount of data required to achieve a desired level of accuracy in the physically relevant quantities, and , can be reduced dramatically.
The errors in locating colloidal particles' centroids can be calculated from knowledge of the images' signal to noise ratio and the optical train's magnification (29). Centroid resolutions of 10 or better can be attained routinely for micrometerscale colloidal particles in water using conventional brightfield imaging. In practice, however, mechanical vibrations, video jitter and other processes may increase the measurement error. Quite often, the overall measurement error is most easily assessed by increasing the laser power to the optical traps to minimize the particles' thermally driven motions. In this case, , and can be estimated directly.
§ III.3. Trap characterization
The stiffness and viscous drag coefficient can be estimated simultaneously as
(41) 
with error estimates, and , given by
(42)  
(43) 
If the measurement interval, , is much longer than the viscous relaxation time , then vanishes and the error in the drag coefficient diverges. Conversely, if is much smaller than , then approaches unity and both errors diverge. Consequently, this approach does not benefit from excessively fast sampling. Rather, it relies on accurate particle tracking to minimize and . For trapparticle combinations with viscous relaxation times exceeding a few milliseconds and typical particle excursions of at least 10 , digital video microscopy provides the resolution needed to simultaneously characterize multiple optical traps (29).
In the event that measurement errors can be ignored (),
and  
(44) 
where
(45) 
These results are not reliable if , which arises when the sampling interval is much longer or much shorter than the viscous relaxation time, . Accurate estimates for and still may be obtained in this case by applying Eq. (40).
As a practical demonstration, we analyzed the thermally driven motions of a single silica sphere of diameter (Bangs Labs lot number 5328) dispersed in water and trapped in a conventional optical tweezer. With the trajectory resolved to within at 1/30 s intervals, 1 minute of data suffices to measure both and to within 1 percent error using Eqs. (41). The results plotted in Fig. 4(a) indicate trapping efficiencies of . Unlike , which depends principally on , also depends on , which is accurately measured only for . Over the range of laser powers for which this condition holds, we obtain the expected , as shown in Fig. 4(b). The viscous relaxation time becomes substantially shorter than our sampling time for higher powers, so that estimates for and its error both become unreliable, as expected.
§ IV. Adaptive Optimization
Optimal statistical analysis offers insights not only into the traps' properties, but also into the properties of the trapped particles and the surrounding medium. For example, if a spherical probe particle is immersed in a medium of viscosity far from any bounding surfaces, its hydrodynamic radius can be assessed from the measured drag coefficient using the Stokes result . The viscous drag coefficients, moreover, provide insights into the particles' coupling to each other and to their environment. The independently assessed values of the traps' stiffnesses then can serve as a selfcalibration in microrheological measurements and in measurements of colloidal manybody hydrodynamic coupling (32). In cases where the traps themselves must be calibrated accurately, knowledge of the probe particles' differing properties gauged from measurements of can be used to distinguish variations in the traps' intrinsic properties from variations due to differences among the probe particles.
These measurements, moreover, can be performed rapidly enough, even at conventional video sampling rates, to permit realtime adaptive optimization of the traps' properties. Each trap's stiffness is roughly proportional to its brightness. So, if the th trap in an array is intended to receive a fraction of the projected light, then instrumental deviations can be corrected by recalculating the CGH with modified amplitudes:
(46) 
Analogous results can be derived for optimization on the basis of other performance metrics. A quasiperiodic pattern similar to that in Fig. 2(c) was adaptively optimized for uniform brightness, with a single optimization cycle yielding better than 12 percent variance from the mean. Applying Eqs. (41) to data from images such as Fig. 2(d) allows us to correlate the traps' appearance to their actual performance.
With each trap powered by 3.4 mW, the mean viscous relaxation time is found to be . We expect reliable estimates for the viscous drag coefficient under these conditions, and the result with an overall measurement error of 0.01, is consistent with the manufacturer's rated 10 percent polydispersity in particle radius. Variations in the measured stiffnesses, and , can be ascribed to a combination of the particles' polydispersity and the traps' inherent brightness variations. This demonstrates that adaptive optimization based on the traps' measured intensities also optimizes their performance in trapping particles.
§ V. Summary
The quality and uniformity of the holographic optical traps projected with the methods described in the previous sections represent a substantial advance over previously reported results. We have demonstrated that optimized and adaptively optimized HOT arrays can be used to craft highly structured potential energy landscapes with excellent fidelity to design. These optimized landscapes have potentially wideranging applications in sorting mesoscopic fluidborne objects through optical fractionation (11); (12), in fundamental studies of transport (33); (9), dynamics (13); (14) and phase transitions in macromolecular systems, and also in precision holographic manufacturing.
We have benefitted from extensive discussion with Alan Sokal. This work was supported by the National Science Foundation under Grant Number DBI0233971 with additional support from Grant Number DMR0451589. S.L. acknowledges support from the Kessler Family Foundation.
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