## 4 Two Examples of Application
### Library of Ice Crystal Objects for the Calibration of Interferometric Airborne Instruments  
We develop an airborne instrument to detect and characterize droplets and ice crystals in the atmosphere [36]. The technique used is interferometric particle imaging [37; 38; 39]. It has been chosen because fast algorithms can be developed that analyze directly a global image [36]. The instrument records on a CCD sensor the out-of-focus image of the scattering particle. In the case of droplets, the out-of-focus image shows regular interference fringes due to interference of both reflected and refracted beams. In the case of irregular rough particles such as ice crystals, the out-of-focus image shows a  
Figure 8: (**a**) Experimental reconstruction of inclusions in a droplet; (**b**) Simulated reconstruction of opaque and transparent inclusions in a droplet.  
Figure 7: Experimental set-up for the reconstruction of droplet’s inclusions.  
speckle-like pattern. The size of the droplet is deduced from the frequency of the fringes, while size and ellipticity of irregular particles are deduced from size and ellipticity of the speck of light in the speckle pattern [40]. In this last case, it can be demonstrated that the 2-Dimensional Fourier transform of the speckle pattern gives the 2-Dimensional autocorrelation of the global shape of the object [41]. The development of the airborne instrument requires its calibration in laboratory using a second well-known technique. Digital In-line Holography is perfectly appropriate. But the experimental set-up in laboratory requires experiments in cold chambers with different optics and windows. Our matrix-based numerical model allows to describe such environmental conditions by the appropriate definition of the correct optical transfer matrices. For future studies, we have developed a library of the most-common ice crystals that can be observed in the atmosphere. Some examples are presented in Figure 9: a dendritic crystal with sector-like ends (top left) a malformed crystal (top right), an ordinary dendritic crystal (bottom left) and a crystal with sectorlike branches (bottom right).  
Using our simulator, Figure 10 shows the digital in-line hologram generated by a transparent particle whose form is the one of a dendritic crystal with sector-like ends (left) and its optimal reconstruction using 2D-FRFT (right). As the object is not opaque, the exact form of the reconstructed can be affected by different effects as for example an intense central peak, or ambiguous limits in different parts of the reconstruction. We think that the models developed should allow a better interpretation of experiments and artifacts encountered in the reconstructions. With the definition of the object presented in the previous section, any kind of object shapes can be simulated. Objects can be opaque or phase objects. Any optical system can be considered by the choice of the appropriate transfer matrices.  
Figure 9: Some examples from our library of transparent phase objects describing ice crystals. A dendritic crystal with sector-like ends (**top left**); A malformed crystal (**top right**); An ordinary dendritic crystal (**bottom left**) and a crystal with sectorlike branches (**bottom right**).