Locating and steering entire ensembles of microscopic objects has become extremely practical with the emergence of holographic optical tweezers. Application of this technology to single molecule experiments requires great accuracy in the spatial positioning of optical traps. This paper calculates the theoretical position resolution of a single holographic beam, predicting that sub-nanometer resolution is easily achieved. Experimental corroboration of the spatial resolution’s inverse dependence on the hologram’s number of pixels and phase levels is presented. To at least a nanometer range position resolution, multiple optical tweezers created by complex superposition holograms also follow the theoretical predictions for a single beam.
©2005 Optical Society of America
Developments in the optical manipulation of atoms and microscopic objects continue to inspire new possibilities for those objects’ quantitative measurement. Optical gradient traps , also called optical tweezers (OT), localize mesoscopic objects and can accurately measure the picoNetwon forces that often arise in biological and colloidal arenas [2, 3]. Laser cooling techniques slow atoms in a step to form a Bose-Einstein condensate, which can then be transported with optical tweezers . The optical stretcher distorts soft objects such as cells to measure their viscoelastic properties . The optical Bessel beam transports microscopic objects in a diffractionless tube of light over large distances .
Many of these approaches have recently been realized using holographic optical tweezers (HOT) [7, 8, 9, 10, 11], or like HOT and the Generalized Phase Contrast technique , employ a spatial light modulator (SLM) to steer, shape, and produce multiple beams for optical manipulation [13, 14, 15]. An OT is a tightly-focused laser beam whose steep electric field gradients draw high dielectric microparticles into the beam’s focus where they are trapped . A HOT apparatus can uniquely locate a single to hundreds of OTs in a three-dimensional microscopic volume. Individual traps may be independently translated, eliminated, and new ones created. To accomplish these varied tasks, HOT applies phase-only wavefront engineering to a single incident laser beam at the aperture of the SLM. This beam then propagates and interferes with itself to produce multiple beams which are subsequently focused into OTs (see Fig. 1).
Several future applications of HOT and other SLM-based beam shaping and steering techniques including atomic physics  will require high-resolution beam placement and motion. One application, the optical tweezer force clamp, is often used to study single molecule mechanics . A force clamp maintains a constant force on a microsphere by the constant monitoring and rapid, precision adjustment of a microparticle’s position in the trap. Currently, the relatively slow nematic liquid-crystal SLMs limit this process; however the bandwidth achievable by ferroelectric SLMs may make such force clamps possible . In other applications such as multi-handled manipulation of molecules, the precise distance of traps with respect to one another will also be important.
Here the system limitations in positioning a single holographic optical trap are calculated. We demonstrate theoretically that sub-nanometer positioning is attainable and supply compelling experimental verification. The calculation predicts that HOT’s position resolution is proportional to Ng where N 2 is the SLM’s total number of pixels and g is the total number of phase levels. The compensation of g by N and vice versa is validated experimentally. We then briefly address whether the theoretical resolution is changed by the simultaneous positioning of several optical traps.
1. Experimental set-up
The essential component of the HOT technique is the SLM, a computer-addressable diffractive optical element whose liquid crystal display has N × N liquid crystal cells (pixels). The HOT beam-splitting element, called a phase mask or hologram, is created on the SLM face where each pixel is assigned one of g calibrated phase levels between 0 and 2π. In this paper, a HOT apparatus with a 512×512 reflective SLM (Boulder Nonlinear Systems, N = 512) imprints the desired phase profile onto the wavefront of a collimated TEM00 laser beam (Spectra-Physics J20-BL-106C, 5 W, λ = 1064 nm). The SLM produces traps with a measured diffraction efficiency of 60 %. Phase level calibration gives g = 130. A 4f-telescope with the focal length f 1 and f 2 adjusts the beam’s diameter to perfectly match the back aperture of a microscope objective that focuses the laser light into an optical trap. The telescope also locates the SLM and the high-NA objective (100×, NA = 1.45, oil immersion)  in conjugate planes (Fig. 1).
In some experiments, the number of SLM pixels is reduced by binning. Nevertheless, the area of the binned phase mask remains identical to the unbinned phase mask, and the transferred laser beam still perfectly fills the objectives’s aperture. To determine a trap’s position in the focal plane with sub-pixel resolution, the center of mass of the trap’s intensity is calculated and related to its position, using a method described by Wuite et al. . This yields a maximum spatial resolution of 10 nm for our system.
2. Spatial resolution of holographic beam steering
HOT uses a computer-generated hologram to create and position multiple beams. It is well-known that calculation of the hologram by fast Fourier transform (FFT) based schemes like the Gerchberg-Saxton algorithm  superficially limits beam steering to a grid with a lattice spacing of several hundred nanometers. Such discrete FFT-based hologram calculation algorithms start from an idealized image of the desired trap pattern, whose pixelation necessarily reflects that of the SLM. This restricts the placement of OTs to integer multiples of δf = λf/D in the objective’s focal plane, where λ is the wavelength of the laser, D is the area of back aperture and f the focal length of the objective. For our system with λ = 1064 nm, f = 1.65 mm and D = 4.4 mm, the resolution is δf = 400 nm, insufficient for high-precision experiments.
By avoiding the discrete FFT, the lateral spatial resolution can be greatly increased. However, there exist no theoretical estimations nor measurements of the minimum distance between two holographically-steered optical traps.
To study the position resolution of just one optical trap, a phase mask is easily produced without an iterative algorithm. A single beam can be located using a phase grating that has a phase profile equivalent to the phase modulation created by an optical prism. The phase grating for a single OT located at position in the focal plane is given by
where is the coordinate in the SLM plane and (f 2/f 1) the telescope’s magnification factor. The same solution would be found by discrete FFT hologram-calculating algorithms, and can be derived analytically.
An example of a full phase grating is shown in the inset of Fig. 2, as calculated from Eq. (1). The phase structure of three similar phase gratings before mod (2π) are also plotted, representing the value of the phase versus the horizontal axis, x. The solid lines represent ideal phase gratings. The horizontal bars illustrate the phase structure of pixelated holograms with N = 6 pixels and g = 5. The phase gratings (a) and (b) displace the laser beam by δf and 2δf in the x-direction respectively, as expected when using ρx = δf, 2δf in Eq. 1. Since the two displacements are multiples of δf, these gratings are included in the class of the pixel-restricted holograms obtainable by FFT-based algorithms.
The beam’s displacement is also related to the slope of the phase gratings. For the two above examples, the slope is given by 2πj/N,j = 1,2 and N = 6, and the displacement is ρx = j · δf. Phase gratings with slopes in between that of (a) and (b), and thus smaller displacements, are also obtainable by using Eq. (1). One example is the phase grating represented in Fig. 2 by (*), which positions the beam at a distance of ρx = 1-5δf.
We propose that a phase grating defined by Eq. (1) can position an OT with a precision up to
where kmax is the maximum achievable number of phase gratings that produce a displacement between jδf and (j+1)δf . Therefore, kmax must be related to the area enclosed by two adjacent phase gratings that displace the beam by an integer value of δf (the shaded area in Fig. 2), divided by the minimum incremental phase modulation of the hologram, 2π/g, which yields
Thus in principle, it is possible to achieve virtually continuous beam placement using a standard SLM with N = 512 pixels and g = 130 phase levels: the theoretical maximum resolution δf/kmax is more than sufficient to achieve sub-nanometer resolution. Figure 2b shows the trap position determined by video microscopy for 400 phase gratings that shift the trap a total distance of 2δf , each with a predicted step size of δf /200 = 2 nm (N = 512, g = 130). Direct measurement of δf yielded exactly 400 nm, in agreement with the predicted value. Here the number of holograms used to cover the distance δf, k = 200 < kmax determines the resolution rather than N and g. The theoretical position resolution is clearly an upper limit which will be decreased by various defects including dead pixels, non-linear phase level characteristics, and electronic noise. Nevertheless, the experimentally achievable spatial resolution, discussed in the next section, is impressive.
3. High-precision steering with reduced pixels (N) or phase levels (g)
To verify Eq. 2, the HOT position resolution is artificially limited to experimentally resolvable values by binning N and g. We also test the prediction that high precision beam placement should be feasible with low resolution in g when compensated with high resolution in N, and vice versa. To measure this compensation and the resolution, an OT’s position is monitored as it moved from 0 to δf in experimentally resolvable steps. Two cases are examined, one with fixed N = 12 and varying g; the other with fixed g = 2 and varying N. The root mean square deviation (rmsd) of the measured postions from the idealized positions obtainable with a perfect prism (N,g → ℞) is calculated and plotted in Fig. 3. One point corresponds to the rmsd of all steps from 0 to δf. This deviation is related to the maximum resolution of the beam placement.
Figure 3. shows that, in agreement with the theoretical prediction, equal products of Ng for both series give similar results. For values above Ng/2 = 200 the measurements are restricted to ~ 10 nm due to limitations in particle tracking. A very good fit with the theoretical prediction given by Eq. (2) implies that the same should hold true for sub-nanometer positioning.
Remarkably, these experiments demonstrate that sub-δf positioning can be achieved even when one SLM parameter is limited to very small values . This is very interesting for applications that require the faster refresh rate of the ferroelectric SLM, which has only two phase levels, g=2. It may also be advantageous for applications where fast hologram calculation is desirable and a restriction of the number of pixels is possible. However, this second possibility will be highly application dependent since restricting the number of pixels also limits the maximum displacement, which is given by Nδf .
Other deleterious effects occur when constricting g. A hologram’s efficiency is directly related to g . Binary holograms (g = 2) show diffraction efficiencies around 40 %. For g > 5 efficiencies ranging from 80 % up to more than 90 % can be achieved. Additionally, our measurements show that the beam’s shape is distorted for g < 8. A hologram with N = 512 and g = 2 or 8 produces a beam with an elliptical focus as shown in inset (a) of Fig. 3. For a similar Ng-value (N = 12 and g = 85), there is no distortion of the diffraction limited laser focus (inset b, Fig. 3). This effect probably depends on the degree of the unavoidable binning of the ideal phase levels, which is more prevalent for low g.
One might also consider the spatial resolution in the axial direction, perpendicular to the focal plane. For this, a similar analysis must be performed to obtain a complementary prediction, since the axial displacement is controlled with a lens phase mask rather than a phase grating .
4. Multiple optical traps
One of the advantages of working with a HOT apparatus is the possibility to simultaneously create more than one OT. It is therefore relevant to ask whether a more extensive theory is needed to predict the position resolution of multiple beams produced by the SLM, or whether the existing prediction is sufficient. This question is addressed experimentally by comparing the measured displacements of several simultaneous holographic optical traps with Eq. 2. A hologram that creates multiple traps with a spatial resolution higher than δf is most simply produced by the complex superposition of the phase gratings associated with each individual trap. A single trap has an amplitude and phase described by ψj =Ajexp(iϕj), where Aj is the amplitude, ϕj is the phase described by Eq. 1, and j is an index to distinguish it from other phase gratings. Therefore, the complex addition of j single traps gives ψ =A exp(iϕ) = ψ 1 + ψ 2 + ψ 3 + ⋯ + ψj. To extract a phase-only hologram that will create the multiple trap field, the amplitude A is discarded and the phase ϕ becomes the new hologram. Many iterative algorithms use this resultant phase as a starting input guess. Our recent work has shown that significant improvements in efficiency (> 2%) can only be achieved for patterns with both high symmetry and periodicity . Therefore, the complex superposition method is used here to experimentally examine the position resolution of multiple holographic beams.
Figure 4 shows the trajectories of three OTs generated by superposition holograms. By defining a unique k value for the phase gratings affiliated with each OT, the outer three particles should move with theoretical step sizes of δf /25, δf /50 and δf /100 (16 nm, 8 nm and 4 nm) respectively. The different k values result in different speeds and final total displacement. The predicted step sizes of 8 nm and 16 nm were verified using particle tracking, where Fig. 5 shows the displacement of the laser focus of trap 2. The fitted line shows trap positions smoothed by adjacent averaging (20 frames), and the horizontal bars represent the mean position of the trap during 100 frames of each of 6 consecutive holograms. The mean step size over the entire trajectory was measured to be 8nm in agreement with the prediction for a single trap.
Very high lateral position resolution of single traps can be achieved with HOT, as shown by our theoretical prediction and experimental measurements. The dependence of the resolution on the product (Ng)-1 presents the option to reduce hologram calculation time by limiting the number of effective pixels when appropriate. The influence of reduced phase levels g on the beam quality is identified and attributed to distorted phase gratings that arise with low g.
The prediction can also be reliably applied to multiple traps for displacements down to at least 8 nm. An open question remains concerning whether complex addition of single beam phase gratings reduces the resolution at the extremes of the minimum displacement. These results should be useful in planning and designing experiments with fine-tuned SLM beam placement, for example in single molecule optical tweezer experiments and the guiding and trapping of atoms.
This work was supported by the Alfried Krupp-Stiftung, the Alexander von Humboldt-Stiftung (J.C.), and the Fonds der chemischen Industrie (C.S.).
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24 . Due to total internal reflection of rays at the glass coverslip / water interface, the real NA of the objective (1.45) is reduced to 1.33.