2 - Fourier optics

Summary

The classical theory of diffraction originated in the work of the French physicist Augustin Jean Fresnel, in the first quarter of the nineteenth century. Fresnel's ideas were subsequently expanded and elaborated by, among others, William Rowan Hamilton, Gustav Kirchhoff, George Biddell Airy, John William Strutt (Lord Rayleigh), Ernst Abbe, and Arnold Sommerfeld, leading to a complete understanding of light in its wave aspects.

The Fourier-transform operation occurs naturally in any formulation of the theory of diffraction, giving rise to a body of literature that has come to be known as Fourier optics. The prominence of Fourier transforms in physical optics is rooted in the fact that any spatial distribution of the complex amplitude of light can be considered a superposition of plane waves. (Plane waves, of course, are eigenfunctions of Maxwell's equations for the propagation of electromagnetic fields through homogeneous media.)

Many students of Fourier optics are intimidated by the approximations involved in deriving its basic formulas, but it turns out that the majority of these approximations are in fact unnecessary: by starting from a plane-wave expansion of the light amplitude distribution, rather than the traditional Huygens' principle, one can readily arrive at the fundamental results of the classical theory either directly or after applying the stationary-phase approximation. (For a detailed discussion of the stationary-phase method see the appendix to this chapter.