The Fresnel diffraction arises now as a special case of the Kirchhoff diffraction. Gustav Kirchhoff readdressed the diffraction of scalar waves and put it on a rigorous mathematical foundation . The unifying electromagnetic theory of James Clerk Maxwell was the culmination of all previous developments on electromagnetism. 2.1 A brief historyįollowing the experimental demonstration of the wave nature of light by Thomas Young and the first theory on diffraction by Fresnel the nineteenth century was extremely successful in the investigation of wave phenomena, specially in optics. Moreover, we briefly review the concept of diffraction in time. Due to their different nature we distinguish in this discussion between light and matter waves.
In this section we first provide a historical overview of diffraction and then address the phenomenon of diffractive focusing. As an outlook we compare in “ Appendix C” the paraxial and non-paraxial results obtained for diffraction by slits and circular apertures. Indeed, “ Appendix A” contains the calculations associated with the Gaussian width of our wave packet and “ Appendix B” presents a detailed discussion of the Wigner function approach towards diffractive focusing. To keep our article self-contained while focusing on the central ideas we have included three appendices. Here we allude to the influence of the number of dimensions on the focusing and emphasize the importance of corrections to paraxial optics. 6 by summarizing our results and by providing an outlook. Moreover, we analyze the rays and envelopes of the Wigner function in more detail. In particular, we show that the phenomenon of focusing which reflects itself in a dominant maximum of the probability density on the optical axis follows from radial cuts through the initial Wigner function at different angles with respect to the momentum axis. Section 5 illuminates this focusing effect from quantum phase space using the Wigner function. Moreover, we make contact with the predictions of non-paraxial optics. We measure the intensity distributions of the light in the near-field of the slits and obtain the Gaussian width of the intensity field.
DIFFRACTION OF LIGHT BY A SINGLE SLIT PROGRAM FREE
Here we take advantage of the analogy between the paraxial approximation of the Helmholtz equation of classical optics and the time-dependent Schrödinger equation of a free particle. 4 we verify these predictions reporting on an experiment using laser light diffracted from a single slit. For this purpose we derive exact as well as approximate analytical expressions for the time-dependent probability amplitude and density. In particular, we show this effect manifests itself in the time-dependent probability density as well as the Gaussian width of the wave packet. 3 to the discussion of the focusing of a rectangular wave packet from the point of view of the time-dependent wave function. 2 we first give a brief history of the diffraction of waves, and then review several focusing effects especially those associated with the phenomenon of diffraction in time introduced in Moshinsky . Our article is organized as follows: in Sect. In the present article we illustrate this effect in Wigner phase space and verify it using classical light in real space. This phenomenon has been confirmed for light , water and surface plasmon waves . Indeed, we have recently found that a rectangular matter wave packet which undergoes free time evolution according to the Schrödinger equation focuses before it spreads. In the present article we discuss an effect related to the Poisson spot which is the one-dimensional analogue of the camera obscura . This phenomenon was experimentally confirmed by Francois Arago and led to the victory of the wave over the particle theory. It was on this occasion that Siméon Poisson predicted that an opaque disc illuminated by parallel light would create a bright spot in the center of a shadow. Three years later he participated with his Mémoire sur la Diffraction de la Lumière in the Grand Prix of the French Academy of Sciences . In July 1816, the civil engineer Augustin-Jean Fresnel published his preliminary results confirming the wave theory of light.
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