Diffractive optical elements with spiral dislocations

A typical set-up for a geometrical mapping using a HOE is described, and the relationship between the grating vector field of the HOE and the mapping is derived. A method is given for finding a mapping function that transforms one intensity distribution into a second distribution, using the principle of conservation of light energy. In general, the grating vector field required to implement this mapping with a HOE is not conservative. A method of replacing this non-conservative vector field with the sum of a conservative field and a set of spiral dislocations is described: a vector field which has the same value of curl as the required grating vector is calculated as a continuous distribution of spiral-dislocation functions over the field; this is then subtracted from the grating vector, and replaced by an approximation to it consisting of a discrete distribution of spiral dislocations. The resulting vector field can then be implemented in a quasi-continuous HOE, which produces a wavefront with a set of spiral dislocations. The method is demonstrated for a circle to square patch intensity mapping