![]() However, it applies to all sorts of quantum beams, including neutron and electron waves at atomic distances if there are a large number of atoms, as well as visible light with artificial periodic microscale lattices. Such law had initially been formulated for X-rays upon crystals. It encompasses the superposition of wave fronts scattered by lattice planes, leading to a strict relation between wavelength and scattering angle, or else to the wavevector transfer with respect to the crystal lattice. In physics and chemistry, Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a large crystal lattice. 2.3c.Physical law regarding scattering angles of radiation through a medium While this derivation assumes a constant or at least piecewise constant index of refraction, the HE is still an excellent description even for \(n \rightarrow n(\), and plot W as a function of 1/ M on double-logarithmic scale, see Fig. Further, the field intensity has to be sufficiently small to neglect the non-linear response of matter. The HE in the homogeneous form of ( 2.1) is derived from Maxwell’s equations for stationary fields in media, which are homogeneous, isotropic, non-magnetic, non-conductive, and do not contain free charges. Propagation of stationary X-ray fields in matter can be described by the well-known Helmholtz equation 1.1 Scalar Diffraction Theory and Wave Equations Finally, we present finite difference equations as a more general tool to treat X-ray propagation in matter and objects which cannot be approximated as thin. Next, we address the projection approximation which is ubiquitous in X-ray imaging to describe the complex transmission function of an optically thin object. Then we show how to compute the wavefield at a distance d along the optical axis z with respect to a known field distribution in a plane at \(z=0\), assuming free space between planes \(z=0\) and \(z=d\). We first justify the use of scalar wave theory and approximations of paraxial (parabolic) wave equations. Here we present an overview of fundamental principles of X-ray imaging and field propagation, with references to relevant literature. The chapter closes with a brief generalization of two dimensional coherent imaging to three dimensional imaging by tomography.Ĭoherent X-ray imaging is based on wave-optical propagation of electromagnetic waves, including free-space propagation and the interaction of short wavelength light with matter. We then present single-step and iterative fixed-point techniques based on alternating projections onto constraint sets as tools to decode the measured intensities (phase retrieval). The recorded intensities are inline holograms created by self-interference behind the object. This provides the basic tools to consider the mechanisms of coherent image formation in a lensless X-ray microscope. We start by an introduction to scalar wave propagation, first in free space, followed by propagation of short wavelength radiation within matter. It is meant as primer and tutorial which should help to understand later chapters of this book devoted to X-ray imaging, phase contrast methods, and optical inverse problems. More specifically, we consider lensless X-ray imaging based on free-space propagation. This chapter briefly summarizes some main concepts of coherent X-ray imaging.
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