Diffractive Optical Elements are optical elements that are used to introduce specific beam transformations in optical systems. These components, also known simply as DOEs, are typically thin optical windows that can be very versatile in the sense that almost any type of beam transformation can be accomplished.
A diffractive optical element consists of an array of small elements or pixels that apply a local phase delay to the corresponding area on the beam that is traversing the element. The transformation that is required from the diffractive optical element is written onto this array of modulating elements in a way that is not entirely intuitive. This means that if we were to look at the map of phase values, we will perhaps not be able to figure out the shape and distribution of the output beam. The reason is that the relation between the DOE plane and the output plane is given by a solution of the diffraction integral with some simplifications or approximations. One such approximation, that turns out to be of enormous practical interest, reduces the relation to a spatially scaled Fourier transform operation. Thus, starting from the desired shape of the final radiance distribution, to obtain the required array of phase delays at the DOE plane one can simply back propagate using an inverse Fourier transform calculation.
In practice, however, this inverse Fourier transform operation is far from sufficient. There are constraints having to do with the basic physics (i.e, it is often impossible to divert all energy to the desired transformation, some must be at undesired orders or non-diffracted), and there are almost infinite combinations of energy in various undesired diffracted orders that will give similar transformations. In addition, there are also restrictions that are imposed by manufacturing constraints. To start with, to achieve good efficiency, the DOE plane has to be completely transparent. Then, due to the photolithography techniques that are used to create the phase pattern, there is quantization of n the phase values that can be applied on each pixel.
Thus, in order to obtain an effective optical element, the DOE design has to reconcile all these constraints by using an optimization algorithm. One method that has proven its efficacy is the iterative Fourier transform algorithm, or IFTA. In this method there is a back-and-forth propagation process in which the manufacturing constraints and the desired output beam shape are imposed at each plane (input and output). This process ensures the convergence of the algorithm, and the result is an array of values that, upon propagation, yields the desired beam distribution.
There are other optimization algorithms for diffractive optical element design that are based on Monte Carlo methods or even in genetic algorithms, but often these achieve non-stable phase profiles that are very sensitive to tolerances.
