IV: Fractional transformations in optics

This chapter discusses the fractional transformations in optics. The chapter presents the fundamentals of the fractional Fourier transformation (FRT). A complete signal description, simultaneously displaying space and frequency information, can be achieved by the space-frequency Wigner distribution function (WDF). An application of the FRT is related to signal multiplexing. Due to the ability of the FRT to rotate the Wigner chart, the Wigner distribution of a signal may be arranged in a more efficient manner. The chapter explores the fractionalization procedure of Zernike's phase contrast microscope. This method is a spatial filtering process similar to the optical Hilbert transformation. The chapter discusses the harmonic real transformations, and provides the applications of several fractional transformations. Some aspects of the fractionalization techniques are also discussed in the chapter. These applications are “filtering processes”. The chapter also describes FRT and some other optical transformations based on the framework of group theory.