Applied Digital Optics: From Micro-optics to Nanophotonics

Appendix C

FFTs and DFTs in Optics

The first attempts to describe linear transformation systems go back to the Babylonians and the Egyptians, likely via trigonometric sums. In 1669, Sir Isaac Newton referred to the spectrum of light, or light spectra (specter = ghost), but he had not yet derived the wave nature of light, and therefore he stuck to the corpuscular theory of light (for more insight into the duality of light as a corpuscule and a wave, see also Chapter 1).

During the 18th century, two outstanding problems would arise:

1. The orbits of the planets in the solar system: Joseph Lagrange, Leonhard Euler and Alexis Clairaut approximated observation data with a linear combination of periodic functions. Clairaut actually derived the first Discrete Fourier Transform (DFT) formula in 1754!
2. Vibrating strings: Euler described the motion of a vibrating string by sinusoids (the wave equation). But the consensus of his peers was that the sum of sinusoids only represented smooth curves.

Eventually, in 1807, Joseph Fourier presented his work on heat conduction, which introduced the concept of a linear transform. Fourier presented the diffusion equation as a series of (infinite) sines and cosines. Strong criticism at the time actually blocked publication: his work was finally published in 1822, in Théorie analytique de la chaleur (Analytic Theory of Heat).

C.1 The Fourier Transform in Optics Today

The Fourier Transform is a fast and efficient insight into any signal's building blocks. ...