Fourier optics, filtering and spectrometry

Abstract

The fundamental work of E. Abbe on image formation of periodic objects in microscopy marked a decisive step towards a proper comprehension of wave phenomena which form the basis of optical information processing. Abbe investigated the problem of image formation of extended objects and showed that in coherent light the formation of an image is equivalent to a two-step diffraction process. It was probably at this occasion that the advantage of using Fourier transform techniques in optics became apparent for the first time. As mathematical tool for the study of periodic functions, the Fourier analysis appears very naturally in the representation of all wave phenomena in Physics and particularly in optics. A few examples of the application of Fourier analysis to optical phenomena are discussed. The diffracted amplitude in the Fraun-hofer region is given by the Fourier transform of the amplitude distribution on the diffracting aperture; a converging lens produces in its back-focal plane an amplitude distribution which is the two-dimensional Fourier transform of the amplitude in the pupil plane. The degree of spatial coherence of light vibrations at two points illuminated by a quasi-monochromatic, incoherent source is given by the Fourier transform of the intensity function of the source. The application of Fourier analysis to the problem of image formation in incoherent light brings out the important fact that an imaging system acts as a spatial frequency filter. In coherent optical imaging the spatial spectrum of an object and hence the output image can be modified by inserting a complex amplitude filter in the Fourier plane. These frequency filtering techniques are used, for example, to improve the quality of photographs and for character recognition. The spectral distribution, S(σ), of energy in a source is related to the Fourier transform of output flux, I(Δ). in an interferogram where Δ is the path difference between the two interfering beams. An arrangement is described that produces optically the Fourier transform of I(Δ) and thus directly gives the spectral distribution S(σ). This setup can be easily modified to generate the derivative, S(σ) or to correlate S(σ) with a reference spectral distribution. We believe that the possibilities offered by Fourier analysis in the field of optics and particularly in interference spectroscopy have not been exhausted.