Wigner algebra as a tool for the design of achromatic optical processing systems

Achromatic optical processing systems can perform a variety of operations with temporally incoherent (polychromatic) light, without color blurring. The system design is a complicated task, since usually the scale at the output depends on the wavelength. The design goal is to eliminate this scale dependence as well as two other wavelength-dependent defects. Such a goal is generally achieved by modifying lens design procedures. Here we do it in a different manner. Specifically, we resort to matrix algebra, applied to the Wigner distribution function. The resulting Wigner matrix includes elements that characterize wavelength-dependent parameters of the optical systems. Such a characterization provides a clear insight into what is needed to reduce the wavelength dependence, and indeed achieve the achromatization of the systems. This design approach is valid with either wave optics or geometrical optics. The basic principles and specific design examples of achromatic optical Fourier transformers and Fourier processing systems with low chromatic aberrations over the entire visible spectrum are presented.