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 fact, each ray from the slit will have another to interfere destructively, and a minimum in intensity will occur at this angle.In July 1816, the civil engineer Augustin-Jean Fresnel published his preliminary results confirming the wave theory of light. A ray from slightly above the center and one from slightly above the bottom will also cancel one another. Thus a ray from the center travels a distance \(\lambda / 2\) farther than the one on the left, arrives out of phase, and interferes destructively. In Figure 2b, the ray from the bottom travels a distance of one wavelength \(\lambda\) farther than the ray from the top. However, when rays travel at an angle \(\theta\) relative to the original direction of the beam, each travels a different distance to a common location, and they can arrive in or out of phase. When they travel straight ahead, as in Figure 2a, they remain in phase, and a central maximum is obtained. (Each ray is perpendicular to the wavefront of a wavelet.) Assuming the screen is very far away compared with the size of the slit, rays heading toward a common destination are nearly parallel. These are like rays that start out in phase and head in all directions. According to Huygens’s principle, every part of the wavefront in the slit emits wavelets. Here we consider light coming from different parts of the same slit. The analysis of single slit diffraction is illustrated in Figure 2. (b) The drawing shows the bright central maximum and dimmer and thinner maxima on either side. The central maximum is six times higher than shown. Monochromatic light passing through a single slit has a central maximum and many smaller and dimmer maxima on either side.
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