Introduction to the Fractional Fourier Transform and Its Applications

The concept of fractional Fourier transform and its applications is discussed. The fractional Fourier transforms has several applications in the area, analog optical information processing, or Fourier optics. Fourier optical systems can be analyzed using geometrical optics, Fresnel integrals (spherical wave expansions), and plane wave expansions. The class of Fourier optical systems (or first order optical systems) consist of arbitrary thin filters lies in between arbitrary quadratic-phase systems. Quadratic graded-index media have a natural and direct relationship with the fractional Fourier transform. Light is simply fractional Fourier transformed as it propagates through quadratic graded-index media. Quadratic graded-index media realize fractional Fourier transforms in their purest and simplest form. The fractional Fourier transform can describe all systems composed of an arbitrary number of lenses separated by arbitrary distances, whereas imaging and Fourier transforming systems are only special cases.