We introduce a new method for generating an array of programmable optical tweezers based on the principle of the Shack-Hartmann wave front sensor. In this approach, a lenslet array divides a laser beam into multiple point sources that are subsequently imaged onto the sample plane of an inverted microscope. This results in a matrix of tightly focused beams used for local confinement and manipulation of micron-sized dielectric particles in an aqueous solution. Using a spatial light-modulating device, the phase profile of the laser beam is computer-encoded providing for controlled spatial deflections of the trapping beams.
©2003 Optical Society of America
Optical tweezers  and the Shack-Hartmann wave front sensor  (SHWS) are individually noteworthy for their roles in several scientific advances. As a tool for micromanipulation of particles, optical tweezers has been applied in force measurements associated with cells and subcellular structures, elasticity studies of DNA molecules, and in the assembly and control of functional microfluidic devices, among others [3–5]. The SHWS, on the other hand, has progressed as a standard apparatus for a variety of wave front sensing-based applications in astronomy, optical diagnostics, and many areas of adaptive optics [6–8]. Like optical tweezers, the SHWS has also extended its practical use to the field of biomedical research. The implementation of the SHWS principle in quantifying human eye aberrations is a major breakthrough in modern ophthalmology .
In optical micromanipulation, the demand for simultaneous control of a collection of particles has appreciably increased. This has paved the way for the development of various multiple-beam trapping methods. Simultaneous generation of multiple optical traps has been achieved by means of computer-generated holograms , generalized phase contrast encoding [11, 12], and vertical-cavity surface emitting laser (VCSEL) arrays as light sources . Depending on the desired application, one method for creating multiple traps has advantages over the other approaches. Various multiple-beam optical tweezers are typically characterized in terms of trapping stability, beam steering and shaping functionalities, computational cost, and setup compactness. Because different types of optical trapping applications impose different requirements, the design of new and advanced multiple-beam tweezers still remains a challenge.
Here, we integrate the concept of light-induced confinement of microscopic particles with the principle of the Shack-Hartmann wave front sensing to derive a new dynamic multiple-beam optical tweezing methodology. We describe a method for producing two-dimensional matrices of optical tweezers by spatially dividing a beam using a lenslet array. A simple “proof-of-principle” experimental setup is used to demonstrate parallel trapping of a plurality of microscopic particles in three dimensions. The system offers fine positioning control within a localized region for each of the optically trapped particles. The principle and potential applications of the new multiple-beam optical tweezers are discussed.
2. The Shack-Hartmann optical tweezing methodology
The system architecture is conceptually based on the conventional SHWS configuration. As shown in Fig. 1, three main components are required for implementing the proposed tweezer configuration: a light source, a spatial phase-only modulating device, and a lenslet array. The wave front encoded on the incident light is spatially sampled by the lenslet array to create multiple adjacent point sources that stretch out along the transverse xy-plane. These beam spots are then imaged into the observation plane of a microscope producing an array of high-intensity optical traps used for manipulation of fluid-borne microscopic particles. In contrast to the standard SHWS where one deals with measurements of unknown wave front aberrations, this tweezing procedure requires a deliberate distortion of the input wave front by a simple tip-tilt or two-axis tilt encoding  at the phase modulator positioned between the light source and the lenslet array.
The input light source is either a collimated beam from a solitary laser or an array of independently addressable sources such as VCSEL arrays. With the latter type of laser source, temporal control of on-off states of each trapping beam in the array can be realized electronically . The wave front modulation is established either in transmission or reflection mode using liquid-crystal (LC) based spatial light modulators (SLM), deformable membrane mirrors (DMM) or MOEMS-based mirror arrays among others. Widely used in adaptive optics, these devices modulate the optical path either by refractive index control or by introducing geometrical path-length adjustments. By encoding the appropriate modulation patterns on the wave front modulator, corresponding beam deflections along the transverse directions are obtained and, as a consequence, positional fine-tuning of trapped particles can be realized. The principle can easily be extended to deflection encoding along the optical z-axis, as well, by using dynamic LC-based micro-lenses with tunable focal lengths  or simply by encoding additional quadratic phase (lens) functionality at each subaperture of the phase modulator.
The Shack-Hartmann-based optical tweezers can be considered as an N×N matrix of programmable phase gratings. Each phase grating has its corresponding micro-lens in front performing a spatial Fourier transform of each grating function. Hence, the micro-lens array illuminated by the phase-modulated beam projects a total of N2 diffraction patterns onto a common Fourier plane.
3. Experimental setup
The schematic of a simple experimental setup for implementing the Shack-Hartmann optical tweezers is shown in Fig. 2. An expanded and collimated beam emitted from a 135 mW diode laser, operating at a wavelength λ=830 nm, undergoes a phase modulation from a reflection-type phase-only SLM (Hamamatsu Photonics) with dynamic range of [0, 3π] phase depth. Note that in Fig. 2, the 480×480 pixels SLM is drawn in transmission mode for simplicity. The size of each SLM pixel is approximately 41 µm.
Different gray-scale patterns on the computer monitor (gray level approximately linear with phase depth) encode different phase functions on the SLM. Spatial Fourier transformation of the SLM-encoded phase gratings is performed by the lenslet array with lenslet diameter d=3 mm and focal length fl=40 mm (Control Optics, CA). The lenslets are adjacently packed in a rectangular lattice with interspacing that is not transparent to the incident laser beam. A convex lens (fc=75 mm) and a microscope objective (oil-immersion, NA=1.25) combine to rescale the set of lenslet array focal spots by a magnification factor of ~1/40 at the tweezer plane. The trapped particles are monitored by bright-field microscopy with the same objective lens and a CCD camera. Due to practical issues regarding the effective laser power obtained at each trapping location and the limited field of view of the CCD camera, we have only considered the case where N2=4 (number of traps) for all the experimental demonstrations reported in this paper.
4. Three-dimensional trapping capability of the optical tweezer array
When illuminated by focused laser light, a dielectric particle immersed in liquid of lower refractive index experiences two counteracting forces: the scattering force and the gradient force. The scattering force arises from the momentum transferred to the particle when a fraction of light is reflected by the liquid-particle interface. The gradient force is the momentum change due to refraction of light causing the attraction of the particle towards the region of highest intensity. To ensure three-dimensional confinement of the particle, the magnitudes of the oppositely directed forces should be equal at a position along the optical axis (slightly displaced from the beam focus). This is easily achieved with a high NA objective lens that can tightly focus the beam to a diffraction-limited spot generating high intensity gradients around the focal region.
As described in the previous section, the 2×2 beam spots at the Fourier plane of the lenslet array are imaged by an optical relay composed of a convex lens and a 1.25 NA objective lens. The convex lens collimates the four point sources and directs each of them to the back-aperture of the objective lens. Arriving at different angles, each of the four wave fronts effectively illuminates the back-aperture of the objective. Thus, for the case of a plane wave input (unperturbed by the SLM), four strongly focused trapping beams are established in a square lattice (lattice spacing ~76 µm) at the tweezer plane.
The three-dimensional trapping stability of the 2×2 optical tweezers with ~5.5 mW of effective power per trap is shown in Fig. 3. Four polystyrene microspheres (Bangs Laboratories, Fishers, IN), all 5 µm in diameter, are simultaneously trapped along a plane few microns above the bottom surface of the sample chamber. Both transverse and axial displacements of the sample stage have been introduced without detachment of the particles from their individual traps.
5. Wave front phase modulation for position tuning of optical traps
If the phase component of a plane wave incident on the lenslet array is spatially distorted, the location of a corresponding focal spot at the lenslet Fourier plane (and consequently the associated trapping beam at the tweezer plane) can be deflected away from its geometrical focus (reference position). The relation between local wave front slope and focal spot dislocation forms the basis of wave front reconstruction algorithms in the SHWS principle. Mathematically, this relation is expressed as
where (Δr, θ) denotes the transverse dislocation, in polar coordinates, of the focal spots at the lenslet Fourier plane from respective reference points. Subscripts i and j indicate the row and the column of the lenslet array. The function ϕ(x, y) defines the incident wave front phase distribution in radians and fl is the lenslet focal length.
In principle, a phase-only SLM can introduce a local linear tip-tilt in the wave front at each subaperture and a temporally varying linear phase function can result in dynamic and independent steering of each trapping beam within a certain radial distance from its reference point. Using Eq. (1) and the fact that our SLM can encode a maximum local tilt of ~πrad/mm over a single 3 mm-wide lenslet subaperture, we find that the maximum deflection of the beam in the tweezer plane is only ~λ/2 (415 nm). This has prompted us to encode a 2×2 array of blazed phase gratings on the SLM instead of an array of linear phase functions. With this blazing technique  (commonly applied in diffractive optical element designs), a linear phase function with a steeper slope can be equivalently represented by a periodic-ramp phase function as shown graphically in Fig. 4. Hence, effective slopes much greater than π rad/mm are achieved and the SLM only needs to be encoded with a phase depth in the range of [0, 2π].
With the blazed grating approach, the average wave front slope ∂ϕ/∂r at a subaperture denoted by indices i and j is 2π per grating period Tij. Hence, we can rewrite Eq. 1 as
Note that Eq. 3 is in agreement with the Fourier treatment of scalar diffraction theory where intensity patterns at the lenslet Fourier plane are regarded as the spatial frequency spectra of the sub-grating functions. Transverse dislocation (Δr, θ) is thus equivalent to the position of the main diffracted order in the lenslet Fourier plane. The magnitude of beam deflection varies inversely with the spatial periodicity Tij of each blazed grating and each angle θij follows the orientation angle of the associated grating about the optical axis. Fig. 5 shows an example of a 2×2 array of phase gratings, represented as gray-scale patterns, encoded on the SLM.
It is important to note that the SLM only provides a quantized version of each blazed grating. Moreover, the number of quantization levels for representing the 0 to 2π variation of the grating is proportional to period Tij. To avoid the creation of spurious diffraction orders and to maintain optimal diffraction efficiency for each main order, we consider an 8-level blazed grating as the coarsest quantization for our SLM. This sets the lower limit for the grating period. Using Eq. 3 and keeping the scaling factor of 1/40, the calculated maximum beam deflection magnitude at the tweezer plane is increased to ~2.53 µm.
Experimental results shown in Fig. 6 verify the functionality of the SLM-encoded phase gratings to independently introduce small positional deflections to the trapped particles during simultaneous confinement in their respective optical traps. In the movie linked to Fig. 6, the first sequence shows the synchronized deflection of the four microspheres following a diamond-like path and the next sequence shows distinct deflections in the horizontal and vertical directions. In these experiments, particle deflections are performed in a discrete manner. But it must be stressed that quasi-continuous motions of trapped particles can also be attained with a sequence of phase gratings encoded with sufficiently smooth transition in periodicity and orientation.
The availability of different types of optical tweezers, particularly those capable of generating multiple trapping beams, will allow users to choose a system with features that are appropriately matched for their intended application. For biological applications, the Shack-Hartmann multiple-beam optical tweezers can be used to fix an array of growth points for in vitro cell culture studies or to trap a matrix of beads attached to cell membranes whose elastic properties are to be determined . In materials research, this set of tweezers can be used to efficiently pick up colloidal particles and bring them to a substrate to form patterned templates for guided assembly of crystals . The selective deflection of trapping beams is of potential use for introducing structural asymmetry or defects in patterned templates. The proposed technique can also be implemented in array trapping of neutral atoms for applications in atom optics and quantum computation .
In order to generate a larger array of optical traps (N≥3), a lenslet array with a smaller subaperture diameter, d, can be used. However, to maintain the effective filling of the back-aperture (diameter ~ 6 mm) of the objective lens, the focal length of the lenslet fl and/or that of the convex lens fc must be properly compensated. For example, choosing the following parameters for the lenslet array, d=1.5 mm, fl=20 mm, and maintaining the same combination of the convex lens and the microscope objective, a 3×3 array of traps can be formed at the tweezer plane with a lattice spacing of ~38 µm. Note that with the same laser source, the effective power in each trap decreases as the number of traps is increased.
Alternatively, our current setup can also be improved with the use of a DMM as wave front modulator replacing the LC-SLM. Such continuous deformable mirrors are capable of producing optical surface deformations of up to 20 µm . This will significantly increase the magnitude of deflection of the trapping beams and eliminate undesirable diffraction effects that are inherent in phase quantization. We also propose the utilization of advanced light sources like VCSEL arrays to enhance the flexibility of the optical tweezers. Independently switched sources for each lenslet will allow the formation of different arrays of strongly focused optical traps and will potentially impart more applications for the proposed tweezing methodology.
A new tweezer system with multiple trapping beams has been proposed and demonstrated. The backbone of the optical setup is derived from that of the Shack-Hartmann wave front sensing principle. A two-dimensional matrix of optical tweezers, produced from the splitting of a single beam by a lenslet array, is used to trap multiple microscopic particles in unison. All traps are highly stable in both the axial and the transverse directions. Phase perturbations of the input beam in the form of blazed grating functions are introduced by a computer-controlled LC-SLM. The resulting array of diffracted spots defines the magnitude and direction of transverse deflections of the corresponding optical traps. Thus, independent positional fine-tuning of trapped particles has been achieved without any mechanical displacement in the setup.
We thank the Danish Technical Scientific Research Council for supporting this research and T. Hara and Y. Kobayashi of Hamamatsu Photonics for useful discussions on the operation of the SLM.
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