As promised, we now return to the absorption contrast, where we will hear a bit more from Manuel, about how the recorded intensities relates to the absorption, experienced by the X-rays as they pass through the sample. We want to use absorption contrast. This is basically based on the Beer-Lambert law. The output intensity is related to the input intensity, through this exponential with the integral of the absorption coefficient, distributed in 3D. If we want to reconstruct this, then we have to get an equation that is linear on some of the line integrals, across this absorption coefficient. So, basically one has to take the output intensity divided by the input intensity, see if there is any structure in the input beam. Then, you have to take a negative natural logarithm. With that, we do linearize the problem after taking the logarithm of the intensity. You can then apply filtered back projection or any other technique, in order to get a reconstruction of the absorption coefficient inside the sample. With absorption contrast thus established, let us move on to phase contrast. In the end, I will return with a small discussion about the advantages of absorption contrast versus phase contrast imaging, but first let us hear more from Manuel about phase contrast. So, one way to get some indirect access to this phase information or the most basic way and probably one of the first ones that was used in X-ray community, is Fresnel face contrast or people sometimes call it also propagation based phase contrast. The basic idea is, if you have enough coherence in your beam, and this is also, for example, is applicable for synchrotrons, when you have a lot of coherence, but is also applicable to lab source. You can have enough coherence to do this type of imaging. So, if you have enough coherence in the input beam, then the boundary waves, even though the object is invisible, it still has a boundary where you creates this phase shift and this boundary waves propagate a little bit away, and they interact with the normal reference or the throughput wave that is going through, and this interference creates ripples that become visible. So, basically an object that would be invisible, it becomes visible through this propagation. So, just to show you an example, here I have actually put two disks. One you can see, because it's a zero propagation distance. This assimilation of two disks completely against the detector. So, you can also see the disk of absorption and the disk that creates a phase shift is invisible to the detector. But, if we propagate a little bit, even as short as 50 millimeters away, you can already see the boundary of the other phase disk, and as you propagate further and further this becomes more and more evident. So, this is a very quick way to get access to the phase information, and from this type of information you can also do tomography after a little bit more post-processing. I hope you recognize the analogy through the example I showed with the glasses for visible light, so this is the expected result for an X-ray experiment. As Manuel mentioned, it requires some more analysis to get access to the phase contrast in practice, and in most cases it also requires a highly coherent source of X-rays. This can be synchrotrons or special types of lab equipment, but it is not something that is easily accessible. Even so, it does come into play for certain applications, for example, organic materials, soft materials with low density there. It can be necessary in order to get sufficient contrast to do the actual imaging. But, this is an extra level of complexity, that we are not treating in this course. So, all the data that you will get experience with and see represented in the course material will be based on absorption contrast imaging.
