Inversion Formulas by the Use of Bessel Beams of Integer and Fractional Orders

A major application of the inversion methods is imaging all types of structural, physical, chemical and biological features of matter. The term vortex beam refers to a beam of electromagnetic radiation, electrons, photons or otherwise - whose phase changes in corkscrew-like manner along the direction of propagation. The paper is devoted to the use of scalar Bessel beams of integer and fractional modes for the reconstruction of scattering potential. In practical applications one naturally deals with Bessel beams truncated in the radial direction. The inversion formula for truncated Bessel beams is also obtained. Instead of the conventional Fourier diffraction theorem (Kak in Principles of computerized tomographic imaging, SIAM, New York, 2001), the relations connecting the scattering field and the scattering potential in the Fourier space are obtained in the explicit form.