Fourier treatment of near-field scalar diffraction theory

A general treatment of scaler diffraction theory is presented and some interesting concepts are discussed which yield new insight into the phenomena of diffraction throughout the whole space in which it occurs. The direct application of Fourier transform theory to the diffraction process results in two equivalent descriptions of the diffracted wave field: one describes the wave field as a superposition of plane-wave components and corresponds to the transfer function approach in linear systems theory, and the other describes the wave field as a superposition of hemispherical-wave components (or Huygen wavelets) and corresponds to the impulse response approach in linear systems theory. The convolution of the initial disturbance with the impulse response results in the well-known Rayleigh-Sommerfeld formula for near-field diffraction. This formula is then rewritten in the form of the Fourier transform integral of a generalized pupil function which includes phase variations in the diffracting aperture. Any departures of the actual diffracted wave field from that predicted by the Fourier transform of the aperture function are shown to have the same functional form as the conventional wave-front aberrations of imaging systems. These aberrations are precisely the effects ignored when making the usual Fresnel and Fraunhofer approximations.