## 2 Physical Foundations of X-ray Phase Contrast and Dark-Field Imaging
### Theoretical Coherent Case  
To introduce the first equations and phenomena, let us assume we are in a monochromatic case where the wave \(\psi_{\lambda}\) is emanating from a point-like source at an infinite distance and that the beam is parallel to the optical axis. As described in [11], this wave evolves when propagating through free space following the free space paraxial equation:  
\[(2 ik\frac{\partial}{\partial z}-\
abla_{\perp}^{2})\psi_{\lambda}(x,y,z)=0 \tag{1}\]  
\(k\) is the wavenumber and \(\lambda\) is the wavelength (\(\lambda=2\pi/k\)). Diffraction of the wave over distance \(z_{1}\) can then be written as:  
\[\psi_{\lambda}(x,y,z_{0}+z_{1})=D_{z_{1}}^{(F)}\ \psi_{\lambda}(x,y,z_{0}) \tag{2}\]  
where \(D_{z_{1}}^{(F)}\) is called the Fresnel propagator and is written as an operator:  
\[D_{z_{1}}^{(F)}=\exp(ikz_{1})\mathcal{F}^{-1}\exp(-\frac{iz_{1}}{2k}(k_{x}^{2 }+k_{y}^{2}))\mathcal{F} \tag{3}\]  
That was for free space propagation. Now, what happens when the wavefield encounters an object? In the case of thin samples that are sufficiently slowly varying in space, we can use the projection approximation, which allows us to consider that the X-ray beam follows a straight line within the sample. The projection approximation validity is discussed in [12; 13]. Under the approximation and using the complex refractive index introduced earlier, we describe two quantities: the phase shift (\(\Delta\phi\)) and attenuation (\(B\)) through the sample:  
\[\Delta\phi_{\lambda}(x,y)=-k\int_{z}\delta_{\lambda}(x,y,z)dzB_{\lambda}(x,y )=k\int_{z}\beta_{\lambda}(x,y,z)dz \tag{4}\]  
They are then combined as follows to compute the wave distortions through the sample:  
\[\psi_{\lambda}(x,y)=\psi_{0,\lambda}(x,y).\exp(i\Delta\phi_{\lambda}(x,y)-B_{ \lambda}(x,y)). \tag{5}\]where \(\psi_{0,\lambda}(x,y)\) is the scalar wave-field before the object. An important quantity that should be introduced at that point is the intensity of the wave-field: \(I_{\lambda}(x,y,z)=|\psi_{\lambda}(x,y,z)|^{2}\). This quantity is the one that will be ultimately measured by the detector in X-ray imaging. The relation between the intensity and the wave-field allows us to retrieve the well known Beer-Lambert Law, which gives the attenuation through an object:  
\[\begin{split} I_{\lambda}(x,y,z_{object})=&|\psi_{ \lambda}(x,y,z_{object})|^{2}\\ =& I_{\lambda}(x,y,z_{0})\exp(-2B_{\lambda}(x,y)) \\ =& I_{\lambda}(x,y,z_{0})\exp\left(-\int_{z}\mu_{ \lambda}(x,y,z)dz\right)\end{split} \tag{6}\]  
Under the assumption of local conservation of the optical energy, one can describe the evolution of the intensity upon propagation in free space with the transport of intensity equation (TIE) [14]:  
\[-\
abla_{\perp}.[I_{\lambda}(x,y,z)\
abla_{\perp}\phi_{\lambda}(x,y,z)]=k \frac{\partial I_{\lambda}(x,y,z)}{\partial z} \tag{7}\]  
This equation is fundamental in many phase contrast imaging methods.  
Now let us see what happens in more complex cases, when the beam is no longer parallel and is less coherent. We focus on the simple imaging case where we have a source, an optical element and a detector. It can easily be extended to several optical elements, as is the case in many phase contrast imaging set-ups. In this simple case, the wavefield propagates in free space from the source to the object over a distance \(z_{1}\), then gets distorted in the object and propagates again in free space to the detector over a distance \(z_{2}\). In the case of a divergent beam (or a convergent one, but we focus on the first case here), the wave distortion over distance can be calculated using the Fresnel scaling theorem. In this divergent case, we have a spherical wave and a magnification phenomenon. The magnification is simply calculated using: \(M=(z_{1}+z_{2})/z_{2}\). The Fresnel scaling theorem states that the Fresnel diffraction patterns due to a sample in a cone beam, observed after a distance \(z_{2}\), are equivalent to the one that would be observed at a distance \(z_{2}^{\prime}=z_{2}/M\) in a parallel beam.