1. Introduction

Small dielectric objects can be trapped and moved using light alone. Part of the linear momentum carried by photons is transferred to microscopic samples when light gets reflected and refracted at their boundaries. Under the right conditions [1], the resulting balance of forces leads to stable confinement of the samples in a small region of space: the optical trap. Optical traps or tweezers are rapidly becoming an established technique for the manipulation of microscopic samples in several fields [2]. Accuracy, harmlessness and the possibility of calibration to measure the forces applied are salient features of the technique. There is also an excellent literature on how to build the required optical setups [3–6] and commercial systems are currently available from several vendors [3].

Typically, an optical tweezer setup consists of a high power laser and an inverted microscope modified with additional optics for beam shaping and steering [6]. A microscope objective with a high numerical aperture focuses the laser beam down to a diffraction-limited spot, creating the conditions for stable trapping.

A major improvement in optical tweezers has been brought about by the introduction of holograms displayed onto spatial light modulators (SLMs) [7–9]. With these in the setup, the laser wavefront can be spatially modified prior to the focusing step, resulting in a completely programmable intensity landscape over the sample plane. Large arrays of optical traps, position three-dimensional control or traps with exotic properties are among the new possibilities of holographic optical tweezers [8, 9].

Unfortunately, liquid crystal SLMs are notoriously unable to modulate the whole unit circle in complex space [10], that is, they are incapable of modifying both phase and amplitude of the incoming wavefront on an independent basis. They are constrained to modulate along one-dimensional manifolds through the complex plane, coupling phase and amplitude [11]. Most frequently the spatial light modulator is set to a phase-only configuration [12–14]. Although there is no control over the amplitude, the phase of the light beam can be changed at will.

The limited modulation capabilities of SLMs lead to problems in hologram generation that prevent holographic optical tweezers from reaching their full potential. Although, given a desired array of traps, the required hologram can be easily obtained by computing an inverse Fourier transform, the result is, in general, a full complex object that cannot be accommodated on the display. Algorithms have been developed that provide solutions by constraining the hologram to be a pure phase function but still generate usable traps, but they are time consuming. Computational load makes user interaction with the sample difficult in real time so there is a clear need for faster algorithms [15–17] and better displays.

There are alternatives but we believe these lack the simplicity and universality of the holographic approach. For example, acousto-optic scanners can produce arbitrary arrays of light spots at high speed [18], through time-sharing, but only in two dimensions. The generalized phase contrast method [19] allows an instant conversion of phase patterns into intensity patterns by optical means and is therefore extremely fast. However, it needs a specially fabricated phase plate and, probably, careful alignment. Also, its generalization to three dimensions seems rather elaborate [20].

The present paper introduces a new algorithm for producing holographic optical traps. It is based on the random mask encoding method [21] for multiplexing phase-only filters. The result is a very fast algorithm that gives the desired hologram with just a few simple computations. The algorithm can also be used to modify or multiplex any existing hologram very quickly and has an added advantage in that it does not produce the ghost traps or replicas that frequently plague other methods [16].

2. Experimental setup

Our experimental setup is sketched in Fig. 1. A continuous-wave, frequency-doubled Nd:YVO₄ laser beam (Viasho Technology, λ=532 nm, 1W) is expanded by a spatial filter, collimated by lens L1 and linearly polarized by a high quality polarizer. It illuminates a twisted-nematic liquid-crystal spatial light modulator (Holoeye Photonics, LC-R 2500) sandwiched between a half-wave plate and an analyzer with the proper orientations to achieve phase-only modulation [12–14]. Light then enters an inverted microscope (Motic AE-31) through the fluorescence port and is reflected upwards by a dichroic mirror to an oil-immersion, high numerical aperture, objective (Motic Plan achromatic 100x, 1.25 NA). Lenses L2 and L3, image the SLM onto the exit pupil of the microscope objective to prevent vignetting of high frequency Fourier components [8, 9]. They are arranged to form a telescope so as to still provide parallel illumination to the infinity-corrected objective. Finally, a CCD camera (Qimaging QICAM 1394) allows observation and recording of the experiments.

Since the spatial light modulator is illuminated by collimated light and the diffracted beams are observed at the focal plane of the objective lens (focal length, f’), the relation between the complex reflectance, R(u,v), of the modulator and the electric field at the observation plane, E(x,y), is, except for irrelevant phase terms [22], that of a Fourier transform:

 

Fig. 1. Optical setup for generating holographic optical tweezers.

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3. Algorithm

Given Eq. (1) above, when the spatial light modulator displays the hologram:

a set of N off-axis traps will appear at positions (x_k,y_k) on the sample plane, according to:

Hologram R(u,v) is the superposition of N linear phase functions with slopes (x_k,y_k). Unfortunately, R(u,v) is not a pure phase function and cannot be directly displayed on a modulator working in a phase-only configuration. Therefore, this problem needs to be solved if optical tweezers arrays by means of holographic optical elements on spatial light modulators are to be generated. The algorithms [7–9, 15–17] try to find a hologram that, being a phase function, does not deviate significantly from the expected goal, that of producing the desired trap array. Such algorithms are usually iterative and computationally expensive.

Our solution is non-iterative. It is an adaptation of the random-mask encoding technique [21] to this particular problem and consists of the multiplication of the linear phase functions in Eq. (2) by spatially disjoint binary masks, i.e.:

where

with

That is, the method involves dividing the spatial light modulator into as many subdomains, I_k, as traps are required so that these subdomains do not overlap and jointly cover the whole modulator area. Then, each linear phase function is displayed only on the pixels of a given I_k. (see Fig. 2).

 

Fig. 2. Encoding two linear phases by complementary random binary masks.

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Fig. 3. (a) Binary mask, 256×256 pixels. (b) Magnitude squared of its Fourier transform in logarithmic scale.

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With this arrangement, R(u,v) is trivially a pure phase function with no further modification.

Applying the convolution theorem [22] and Eq. (3), the field at the sample plane is:

where H_k(x,y) is the Fourier Transform of h_k(u,v). Thus, function H_k(x,y) appears centered at position (x_k,y_k). If the binary masks are selected such that their Fourier transforms H_k(x,y) consist of a single peak with flat sidelobes, then E(x,y) will be a good approximation to the desired array of optical traps.

Random masks, as proposed in Ref. [21], give good results in this respect. For example, Fig. 3(a) shows a random binary mask with 50% of its pixels set to one and the remaining 50% to zero. Figure 3(b) shows the magnitude squared of its Fourier transform, a sharp peak on a small random background. The scale on the Z axis is logarithmic so as better to show small intensity features, since the background is five orders of magnitude lower than the central peak.

Figure 4(a) shows a hologram that encodes a 2×2 array of optical traps using four disjoint binary masks, each of them having active the same number of pixels, one fourth of the total.

 

Fig. 4. (a) Hologram encoding an array of 4 optical traps and b) resulting traps.

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Fig. 4(b) shows the resulting traps obtained by computing the Fourier transform of the hologram. Notice the total absence of ghost traps since off-trap energy tends to scatter over the whole sample plane, instead of concentrating at specific locations (giving undesired trapping sites).

It should be finally mentioned that the algorithm can be extended to three dimensions quite straightforwardly by encoding a combination of linear and quadratic phase functions (see for example [9]) with no loss of functionality.

4. Additional useful properties

This procedure shows some other useful features that we comment on below.

4.1. Intensity control

The intensity of optical traps generated by the algorithm shows a remarkable uniformity for a small number of traps. For example, in our experiments we have found maximum variations in intensity of less than 4% for arrays of 2×2 optical traps (512×512 pixel holograms). However, for larger arrays (6×6) the intensity variations may increase up to 25%. When this is a problem or if the optical traps have to be of different intensity, a slightly more elaborate algorithm needs to be used [21]. Masks corresponding to traps that are required to be brighter are selected with a somewhat larger pixel count at the expense of other masks (those corresponding to traps need to be weaker).

4.2. Incremental updating and hologram multiplexing

Contrary to other algorithms, all information is very well localized within the binary masks so addition of new trapping sites can be done without recomputing the whole hologram. Specifically, for a hologram of N pixels that encode m traps, N/[m(m+1)] pixels from each binary mask are randomly discarded. Then, the resulting N/(m+1) pixels are used to codify the new linear phase. Only these latter pixels need to be updated.

Interestingly, this can be done over a hologram computed with any other algorithm, in which the information is distributed: discard a number of pixels and use them to produce a new trapping site with the random mask encoding technique. None of the existing traps is more affected than the others, the net effect is a lower-energy set of existing traps and a new trapping site at the desired location. This may be used to temporarily add a new trap to a preexisting, higher-quality hologram, for example, for loading an array of optical traps with microscopic samples. Finally, the loading trap can be removed by restoring the original pixels.

Figure 5 shows the result of adding a new trapping site to a hologram computed by the Gerchberg-Saxton algorithm [9, 15, 17] to produce an array of 2×2 optical traps. One fifth of its pixels were used to encode the new linear phase function. The figure also illustrates the main drawback of our method. The additional trap is significantly less energetic than equivalent traps computed by the other algorithm.

 

Fig. 5. New trapping site added to a Gerchberg-Saxton hologram.

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Fig. 6. (2.21 MB) Real-time, interactive manipulation of two yeast cells by means of tweezers generated with the algorithm.

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We are further studying the origin of this low efficiency and a possible solution based on reducing the randomness of the binary masks.

Finally, two or more holograms can be multiplexed by multiplication of binary disjoint random masks to merge their individual properties into a single hologram.

4.3. Speed

Once the random masks are selected, the hologram can be directly written onto the spatial light modulator without performing any further computation. Thus, the procedure is very fast and can be easily carried out at near video-rates, therefore enabling real-time interaction with the user. We have developed an interactive holographic optical manipulation system based on this algorithm, as shown in the accompanying video (Fig. 6). The control software is implemented in Java and is capable of displaying holograms (512×512 pixels) at an average rate of 10-12 Hz (including aberration correction of the Holoeye SLM and compensation of the operating curve nonlinearities), using a Pentium IV HT, 3.2 Ghz, computer.

5. Conclusion

We propose a new procedure for the generation of holographic optical tweezers based on the random mask encoding technique. The result is a direct, non-iterative algorithm that has a number of positive features. Specifically, the algorithm is very fast and video-rate generation is easy to achieve. Moreover, the algorithm does not produce ghost traps and can be used to add further trapping sites to existing holograms, even those generated by other algorithms, without the need to re-compute them. Finally, the main limitation of this procedure seems to be a reduced efficiency, being suitable only to generate a small number of traps.