Tiling

Figure 5: Tiling a hologram encoding a $3 \times 3$ array of tweezers scales the spacing between tweezers without sacrificing resolution. Marginal numbers indicate the number of copies tiled into each side.

Because fast Fourier transforms yield periodic functions, all holograms calculated with the AA algorithm can be tiled smoothly. That is, they can serve as the unit cell for new holograms without introducing phase discontinuities at the unit cell boundaries. The result of such tiling is to increase the spacing between tweezers by an amount proportional to the number of tilings along each dimension without reducing the resolution or trapping ability of the individual tweezers. Fig. 5 shows successive tilings of a hologram that generates a $3 \times 3$ square array of tweezers, each labelled by the number of unit cells tiled along each side of the hologram. The same number describes the relative spacings of the resulting tweezers.

We use this property to design holograms encoding tweezer arrays with large inter-tweezer spacings. Increasing an array's lattice constant requires smaller features in the input plane. In order to resolve these small features, the hologram's pixel size must be reduced. Since the width of the hologram is fixed, the number of pixels increases with the inverse square of the pixel size. Calculating such holograms can become computationally expensive. Instead of directly calculating the hologram for a desired lattice constant, therefore, we calculate the smaller hologram encoding the same pattern with a proportionately smaller lattice constant. This hologram can be tiled to create a hologram for the desired tweezer spacing. Tiling can be done either numerically or physically, via a step and repeat mask fabrication process.