On the Limits of Scalar Diffraction Theory for Conducting Gratings

Abstract

Many diffractive elements such as computer generated holograms and periodic gratings are designed by using scalar diffraction theory. The Lohmann hologram is an example of this [1]. The Lohmann method utilizes a two dimensional planar array of square cells each containing a rectangular aperture. Ideally by choosing the proper size and position of each aperture, any arbitrary wavefront can be created. The number of cells in the hologram determine how accurately it can produce the desired wavefront. So in order to reduce error, one would want as many cells in as small a space as necessary to achieve the desired accuracy. In the past, these holograms, as well as other diffractive elements, were limited by the manufacturing process which prevented very small cell sizes, or in more general terms, limited the space bandwidth product of the element. Many of the early computer generated holograms were plotted on paper or even laid out by hand, and then photo-reduced to the proper size. But today, thanks to advances in computer technology and microlithography devices such as electron beam machines, these diffractive elements can be made with micron or sub-micron feature sizes. Some devices made with very small features do not perform as expected. This is not due to fabrication constraints, but rather due to the design process itself. Scalar diffraction theories are used to describe the interaction of light with these diffractive elements [2]. But scalar diffraction theory begins to break down as feature sizes shrink to the order of a few wavelengths or less.