11 - Kirchhoff Equation Solution for CRL, Pinhole, and Phase Contrast Imaging

Abstract

First, this chapter derives the Kirchhoff equation for 1-D biconcave, parabolic, or spherical compound refractive lens (CRL), which assumes wave amplitude and phase change, but not wave direction on passage through the thin sample. Next, a thick sample is assumed, where the wave can change direction, and the general Kirchhoff equation with 2-D biconcave parabolic or spherical CRL for X-rays or neutrons is derived, which includes gravity, and uses the thin-lens and paraxial approximations. From the CRL general Kirchhoff equation, the image amplitude and intensity distribution, point spread function (PSF), modulation transfer function, and field of view are derived. Without a CRL, the PSF of a pinhole camera is derived, followed by the phase contrast, image intensity distribution of an object illuminated with a distant X-ray or neutron point source. Finally, the stationary phase approach, which is applied to integral expressions of image amplitudes, is presented.