ABSTRACT

Em δ (r− rm), with (4)

Em = αm exp(iξm), (5)

where αm is the relative amplitude of the m-th trap, normalized by ∑Mm=1 |αm|2 = 1, and ξm is its (arbitrary) phase. Here δ (r) represents the amplitude profile of the focused beam of light, and may be approximated by a two-dimensional Dirac delta function. For simplicity, we may also approximate the input beam’s amplitude profile by a top-hat function with uj = 1 within the input pupil’s aperture, and u j = 0 elsewhere. In these approximations, the field at the m-th trap is [6]

Em = N

K−1j,m T j,m exp(iϕ j), (6)

with T j,m = Tj(rm). We introduce the inverse operator K−1j,m because the hologram ϕ j may modify the wavefronts of each of the diffracted beams it creates in addition to establishing its

direction of propagation. Such wavefront distortions are useful for creating three-dimensional arrays of multifunctional traps. However, they also distort the traps’ otherwise sharply peaked profiles in the focal plane, which were assumed in Eq. (4). The inverse operators correct for these distortions so that even generalized traps can be treated discretely.