The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in free-space, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described h e by the UGO, fully agrees with the Airy beam solution. These observations explain that the "weak-diffraction" and the "self healing" properties are generated, in fact, by a continuum of sideways contributions to the field, and not by local self-curving dynamics. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in freespace, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the KirchhoffHuygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described here by the UGO, fully agrees with the Airy beam solution. These observations explain that the "weakdiffraction" and the "self healing" properties are generated, in fact, by a continuum of sideways contributions to the field. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

We investigate new sampling strategies for projection tomography, enabling one to employ fewer measurements than expected from classical sampling theory without significant loss of information. Inspired by compressed sensing, our approach is based on the understanding that many real objects are compressible in some known representation, implying that the number of degrees of freedom defining an object is often much smaller than the number of pixels/voxels. We propose a new approach based on quasi-random detector subsampling, whereas previous approaches only addressed subsampling with respect to source location (view angle). The performance of different sampling strategies is considered using object-independent figures of merit, and also based on reconstructions for specific objects, with synthetic and real data. The proposed approach can be implemented using a structured illumination of the interrogated object or the detector array by placing a coded aperture/mask at the source or detector side, respectively. Advantages of the proposed approach include (i) for structured illumination of the detector array, it leads to fewer detector pixels and allows one to integrate detectors for scattered radiation in the unused space; (ii) for structured illumination of the object, it leads to a reduced radiation dose for patients in medical scans; (iii) in the latter case, the blocking of rays reduces scattered radiation while keeping the same energy in the transmitted rays, resulting in a higher signal-to-noise ratio than that achieved by lowering exposure times or the energy of the source; (iv) compared to view-angle subsampling, it allows one to use fewer measurements for the same image quality, or leads to better image quality for the same number of measurements. The proposed approach can also be combined with view-angle subsampling.

The Airy beam (AiB) has attracted a lot of attention recently because of its intriguing features; the most distinctive ones are the propagation along curved trajectories in free space and the weak diffraction. We have previously shown that the AiB is, in fact, a caustic of the rays that radiate from the tail of the Airy function aperture distribution. Here we derive a class of ultra wideband Airy pulsed beams (AiPBs), which are the extension of the AiB into the time domain. We introduce a frequency scaling of the initial aperture field that renders the ray skeleton of the field, including the caustic, frequency independent, thus ensuring that all the frequency components propagate along the same curved trajectory and that the AiPB does not disperse. The resulting AiPB preserves the intriguing features of the time-harmonic AiB discussed above. An exact closed-form solution for the AiPB is derived using the spectral theory of transients. We also derive wavefront approximations for the field in the time window around the pulse arrival, which are valid uniformly in the vicinity of the caustic. These approximations are based on the so-called uniform geometrical optics, which is extended here to the time domain.

The three-dimensional Airy beam (AiB) is thoroughly explored from a wave-theory point of view. We utilize the exact spectral integral for the AiB to derive local ray-based solutions that do not suffer from the limitations of the conventional parabolic equation (PE) solution and are valid far beyond the paraxial zone and for longer ranges. The ray topology near the main lobe of the AiB delineates a hyperbolic umbilic catastrophe, consisting of a cusped double-layered caustic. In the far zone this caustic is deformed and the field loses its beam shape. The field in the vicinity of this caustic is described uniformly by a hyperbolic umbilic canonical integral, which is structured explicitly on the local geometry of the caustic. In order to accommodate the finite-energy AiB, we also modify the conventional canonical integral by adding a complex loss parameter. The canonical integral is calculated using a series expansion, and the results are used to identify the validity zone of the conventional PE solution. The analysis is performed within the framework of the nondispersive AiB where the aperture field is scaled with frequency such that the ray skeleton is frequency independent. This scaling enables an extension of the theory to the ultrawideband regime and ensures that the pulsed field propagates along the curved beam trajectory without dispersion, as will be demonstrated in a subsequent publication.