Wave Propagation Lab

Explore near- and far-field coherent X-ray propagation interactively.

Forward model

Modelling the object

We consider an object made from a single material, entirely described by its thickness \(d(x,y)\). At a fixed photon energy, the material is characterized by the complex refractive index

\[ n = 1-\delta+i\beta. \]

The refractive-index decrement \(\delta\) controls the phase shift, while \(\beta\) describes absorption. For hard X-rays, \(\delta\) is typically larger than \(\beta\). Relative to propagation through vacuum, the projected refractive object is

\[ [n-1]d(x,y) = [-\delta+i\beta]d(x,y). \]

For wavelength \(\lambda\) and wavenumber \(k=2\pi/\lambda\), the complex transmission function is

\[ O(x,y) = \exp\!\left\{ik[n-1]d(x,y)\right\} = \exp[-k\beta d(x,y)] \exp[-ik\delta d(x,y)]. \]

The first exponential describes attenuation and the second describes the phase shift. With unit plane-wave illumination, the field directly behind the object is

\[ \psi(x,y;0)=O(x,y). \]

After propagation over a distance \(z\), the detector records the intensity

\[ I(x,y;z)=|\psi(x,y;z)|^2. \]

Fresnel near-field propagation

In the near field, propagation over a distance \(z\) is described by the Fresnel transfer function:

\[ \psi(x,y;z) = \mathcal F^{-1} \left[ \mathcal F\!\left\{O\right\} \exp\!\left( -i\pi\lambda z(f_x^2+f_y^2) \right) \right], \]

Here, \((f_x,f_y)\) are spatial frequencies.

Fraunhofer far field

In the far field, the diffraction pattern is the Fourier intensity of the transmitted field:

\[ I_\mathrm{FF}(x,y;z) \propto \left| \mathcal F\!\left\{O\right\} \left( \frac{x}{\lambda z}, \frac{y}{\lambda z} \right) \right|^2. \]