Fig. 1. Illustration of four different plane wave configurations that form hexagonal lattice structures. Each configuration is described by its k-space representation (left column), by a transverse cut (center column) and an axial cut (right column) of the intensity distribution each taken through a lattice vector direction. (a) The simplest configuration of three waves in k-space, which results in a sinusoidal transverse intensity pattern and a uniform axial profile. (b) A 6-wave configuration giving sharper peaks with small sidelobes transversely and similarly a uniform axial profile. (c) A 12-wave configuration formed by adding the second k-space harmonic of the fundamental hexagon, which increases the peak intensity of the traps but the does not eliminate the sidelobes. Having a second value of k_z results in a sinusoidal modulation in the axial intensity. (d) Our experimental configuration with 18 waves containing three values of k_z , giving the sharpest peaks and virtually eliminating the transverse sidelobes. The three values of k_z , make the Talbot image revival more complicated and give a larger axial periodicity. All intensity plots have been scaled so that there is equal power in each configuration. The transverse periodicity in (a), (b), (c), and (d) is 8λ; the axial periodicity in (c) is 32λ, while the axial periodicity in (d) is 94λ.

Fig. 2. The interference patterns produced when (a) all three sets of beams are in phase (ΔΦ=0) and the image at the half Talbot distance is the phase contrast image, and (b) when the phase is set to give the half Talbot image. In the later instance, the hexagonal trapping pattern is recovered after propagation through a fraction of the Talbot distance.

Fig. 3. (a) Intensity distribution of the interfering beams in the image plane of the SLM where all the 18 beams are in phase. (b) A similar distribution when the phase of all 18 beams are set to be midway between Talbot images. It is this image (b), with the addition of a linear phase, that forms the multiplicative mask upon which the hologram design is derived.

Figure 4. Schematic of the optical tweezers 4f setup. The lens L1, L2, L3 and L4 have focal lengths of 100mm, 160mm 140mm and 80mm respectively.

Fig. 5. Multiple patterns of 2 µm diameter silica spheres trapped in the hexagonal interference pattern. (a) A 9-sphere array, (b) A 12-sphere arrow pattern array, and (c) a 19-sphere hexagonal array.

Fig. 6. A square array structure consisting of 15 2 µm diameter silica spheres trapped in a hexagonal interference pattern. (a) The structure is lifted axially by 8µm. (b) Structure is moved to the left by 46 µm. (c) Structure then moved in the positive y direction by a total of 25µm, (d)–(e). Array was then moved right by a total of 52 µm. (f) Structure then moved in the negative y direction by 27 µm,(g), before being lowered by 8 µm back down to its original position (h).