Several techniques are available for achieving the necessary phase modulation, and some of the associated practical considerations are discussed in Section 6. For the purposes of the present discussion, we will refer to the phase modulating element as a hologram or a diffractive optical element and treat it as if it acts in transmission, as shown in Fig. 1.
After passing through a phase modulating hologram, the electric field in the input plane has a modified wavefront
In a typical application of holographic optical tweezer arrays, the undiffracted beam, , projects a single optical tweezer into the center of the focal plane with output wavefront , and the goal is to create displaced copies of this tweezer in the focal plane. One possible wavefront describing an array of optical tweezers at positions in the focal plane is a superposition of single (non-overlapping) tweezers
Equations (6) and (8) relate to the associated input wavefront:
The phases of the complex weights, , must be selected so that is a real-valued function. Unfortunately, the resulting system of equations has no analytic solution. Still greater difficulties are encountered in designing more general systems of optical traps, including tweezers which trap out of the focal plane or mixed arrays of conventional and vortex tweezers. Rather than deriving solutions for particular tweezer configurations, we have developed more general numerical methods which we apply in the following Sections to creating planar arrays optical tweezers.