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Physical Review A

Diffraction in time: Fraunhofer and Fresnel dispersion by a slit

Salvador Godoy

Published Apr 4, 2002• 39 citations
Publisher

Abstract

Moshinsky's work on diffraction in time is recognized as a particular case of monoenergetic, quantum dispersion process. Diffraction in time is extended to include new initial conditions, which under free evolution exhibit temporal quantum interference patterns in close analogy to spatial diffraction patterns found in optics. We show that free propagation of states, which initially are stationary states of infinite potential wells, diffract in time similarly as a plane wave by a double slit. We introduce the concepts of mass transport by transient Fraunhofer and Fresnel dispersion currents.

DOI
10.1103/physreva.65.042111

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 20, 2009•0 citations
Moshinsky's work on diffraction in time is recognized as a particular case of monoenergetic, quantum dispersion process. Diffraction in time is extended to include new initial conditions, which under free evolution exhibit temporal quantum interference patterns in close analogy to spatial diffraction patterns found in optics. We show that free propagation of states, which initially are stationary states of infinite potential wells, diffract in time similarly as a plane wave by a double slit. We introduce the concepts of mass transport by transient Fraunhofer and Fresnel dispersion currents.
In analogy to the well-known diffraction of waves at spatial structures, one can also define diffraction in time when a wave is temporarily modulated. In the present paper we investigate this phenomenon starting from a Green's-function approach. Diffraction in time appears for Schr\"odinger waves and not for light in vacuum. Specific cases of diffraction in time investigated here are diffraction at an ``edge in time'' and a Fresnel lens in time. We then investigate in detail the simultaneous diffraction both in space and in time representing a general solution. Then we analyze in detail diffraction at an edge, both in space and in time, of a single slit in space and in time, a double slit in space, and a single slit in time, a single slit in space and a double slit in time, and, finally, a double slit both in space and in time. In all cases we analyze the possibilities of various approximations which can be made, and show the limits and validity of the Fraunhofer approximation both in space and in time. We give explicit results for a gedanken experiment with very cold atoms.
A theoretical model of diffraction based on the concept of quantum measurement is presented. It provides a general expression of the state vector of a particle after its passage through an aperture of any shape in a plane screen (diaphragm). This model meets the double requirement of compatibility with the Huygens-Fresnel principle and with the kinematics of the particle-diaphragm interaction. This led to assume that the diaphragm is a device for measuring the three spatial coordinates of the particle and that the transition from the initial state (incident wave) to the final state (diffracted wave) consists of two specific projections which occur successively in a sequence involving a transitional state. In the case of the diffraction at infinity (Fraunhofer diffraction), the model predicts the intensity of the diffracted wave over the whole angular range of diffraction (0 ∘ - 90 ∘ ). The predictions of the quantum model and of the theories of Fresnel-Kirchhoff (FK) and Rayleigh-Sommerfeld (RS1 and RS2) are close at small diffraction angles but significantly different at large angles, a region for which specific experimental studies are lacking. A measurement of the particle flow intensity in this region should make it possible to test the FK and RS1-2 theories and the suggested quantum model.
This paper provides a complete physical model associated with mathematical analysis and formulation that is based on Schrödinger equation and Feynman path integral. This model answers to single Photon (electron) double slit experiment with high degree of accuracy. It also gives a new method of analysis and formulation for Fresnel and Fraunhofer diffractions that are completely according to the Schrödinger equation and Feynman path concept. The results of these methods are precisely in agreement with the near field and far field diffraction experiments. In other words the new formulation in all above mentioned phenomena i.e. single photon (electron) double slit experiment, Fresnel and Fraunhofer diffractions are completely confirmed by lab experiments(measurements). Lab experiments mean common intensity and pattern experiments resulting from Fresnel (near field) diffraction, Fraunhofer (far field) diffraction and patterns resulting in single photon (electron) lab experiment. The resultant formulas are sketched and calculated by matlab program, the resultant graphs and values are compared with lab experiments in each section. This analysis substitutes wave function and probability density function concepts instead of interference of light beams emitted from infinitesimal pinholes or classical interpretation of near field diffraction and far field diffraction phenomena. The ultimate goal of this paper is to give a very accurate answer to the single particle double slit experiment and also to unify diffraction concept of classical optics in to quantum mechanics, deleting the traditional concept of phases in classical optics and substituting the concept of deflection of Feynman paths. The above mentioned analysis may be a glad tiding to a reliable method to unify classical optics in to quantum mechanics.
This paper provides a complete physical model associated with mathematical analysis and formulation that is based on Schrödinger equation and Feynman path integral. This model answers to single Photon (electron) double slit experiment with high degree of accuracy. It also gives a new method of analysis and formulation for Fresnel and Fraunhofer diffractions that are completely according to the Schrödinger equation and Feynman path concept. The results of these methods are precisely in agreement with the near field and far field diffraction experiments. In other words the new formulation in all above mentioned phenomena i.e. single photon (electron) double slit experiment, Fresnel and Fraunhofer diffractions are completely confirmed by lab experiments(measurements). Lab experiments mean common intensity and pattern experiments resulting from Fresnel (near field) diffraction, Fraunhofer (far field) diffraction and patterns resulting in single photon (electron) lab experiment. The resultant formulas are sketched and calculated by matlab program, the resultant graphs and values are compared with lab experiments in each section. This analysis substitutes wave function and probability density function concepts instead of interference of light beams emitted from infinitesimal pinholes or classical interpretation of near field diffraction and far field diffraction phenomena. The ultimate goal of this paper is to give a very accurate answer to the single particle double slit experiment and also to unify diffraction concept of classical optics in to quantum mechanics, deleting the traditional concept of phases in classical optics and substituting the concept of deflection of Feynman paths. The above mentioned analysis may be a glad tiding to a reliable method to unify classical optics in to quantum mechanics.