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Trajectory-Based Interpretation of Young's Experiment, the Arago–Fresnel Laws and the Poisson–Arago Spot for Photons and Massive Particles

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Trajectory-Based Interpretation of Young's Experiment, the Arago–Fresnel Laws and the Poisson–Arago Spot for Photons and Massive Particles Home Search Collections Journals About Contact us My IOPscience Trajectory-based interpretation of Young's experiment, the Arago–Fresnel laws and the Poisson–Arago spot for photons and massive particles This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 Phys. Scr. 2013 014015 (http://iopscience.iop.org/1402-4896/2013/T153/014015) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 147.91.1.43 This content was downloaded on 10/11/2016 at 16:25 Please note that terms and conditions apply. You may also be interested in: Photon field and energy flow lines behind a circular disc D Arsenovi, D Dimic and M Boži Electromagnetic energy flow lines as possible paths of photons M Davidovi, A S Sanz, D Arsenovi et al. On the influence of resonance photon scattering on atom interference M Boži, D Arsenovi, A S Sanz et al. Photon trajectories, anomalous velocities and weak measurements: a classical interpretation Konstantin Y Bliokh, Aleksandr Y Bekshaev, Abraham G Kofman et al. Visibility of interference in Feynman's atomic light microscope D Arsenovi and M Boži Theory and experimental verification of Kapitza–Dirac–Talbot–Lau interferometry Klaus Hornberger, Stefan Gerlich, Hendrik Ulbricht et al. Hydrodynamic view of electrodynamics: energy rays and electromagnetic effective stress Chia-Chun Chou and Robert E Wyatt IOP PUBLISHING PHYSICA SCRIPTA Phys. Scr. T153 (2013) 014015 (5pp) doi:10.1088/0031-8949/2013/T153/014015 Trajectory-based interpretation of Young’s experiment, the Arago–Fresnel laws and the Poisson–Arago spot for photons and massive particles Milena Davidovic´1, Angel´ S Sanz2, Mirjana Boziˇ c´3, Dusanˇ Arsenovic´3 and Dragan Dimic´4 1 Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade, Serbia 2 Instituto de F´ısica Fundamental (IFF-CSIC), Serrano 123, E-28006 Madrid, Spain 3 Institute of Physics, University of Belgrade, Pregrevica 118, Belgrade, Serbia 4 Faculty of Natural and Mathematical Sciences, University of Nis,ˇ Nis,ˇ Serbia E-mail: [email protected], [email protected], [email protected], [email protected] and [email protected] Received 14 September 2012 Accepted for publication 23 October 2012 Published 28 March 2013 Online at stacks.iop.org/PhysScr/T153/014015 Abstract We present a trajectory-based interpretation of Young’s experiment, the Arago–Fresnel laws and the Poisson–Arago spot. This approach is based on the equation of the trajectory associated with the quantum probability current density in the case of massive particles, and the Poynting vector for the electromagnetic field in the case of photons. Both the form and properties of the evaluated photon trajectories are in good agreement with the averaged trajectories of single photons observed recently in Young’s experiment by Steinberg’s group at the University of Toronto. In the case of the Arago–Fresnel laws for polarized light, the trajectory interpretation presented here differs from interpretations based on the concept of ‘which-way’ (or ‘which-slit’) information and quantum erasure. More specifically, the observer’s information about the slit that the photons went through is not relevant to the existence of interference; what is relevant is the form of the electromagnetic energy density and its evolution, which will model consequently the distribution of trajectories and their topology. Finally, we also show that the distributions of end points of a large number of evaluated photon trajectories are in agreement with the distributions measured at the screen behind a circular disc, clearly giving rise to the Poisson–Arago spot. PACS numbers: 03.50.De, 03.65.Ta, 42.25.Hz, 42.50. p − (Some figures may appear in color only in the online journal) 1. Introduction simultaneously wave and particle properties, and would move within an interferometer along trajectories determined by the Very refined and ingenious interferometers and detectors for guidance equation electrons [1], neutrons [2–4], atoms [5–7], molecules [7, 8] dr S(r, t) v E ∇ E , (1) and photons [9, 10] have been devised to demonstrate E = dt = m that the quantum interference pattern can be built up by where S(r, t) is the phase of the particle wave function E means of the accumulation of single detection events. Even iS(r,t)/h 9(r, t) 9(r, t) e E , (2) before the realization of these experiments, de Broglie [11] E = | E | ¯ and Bohm [12] argued that particles with mass possess which satisfies the time-dependent Schrodinger¨ equation. 0031-8949/13/014015+05$33.00 1 © 2013 The Royal Swedish Academy of Sciences Printed in the UK Phys. Scr. T153 (2013) 014015 M Davidovic´ et al Using the method proposed by De Broglie [11] and Recently, average trajectories of single photons in a Bohm [12], Philippidis et al [13] plotted the trajectories double-slit experiment were observed experimentally for the of massive particles in the double-slit experiment [13], first time by Kocsis et al [22]. Their result motivated us to Dewdney [14] showed trajectories for neutrons inside a apply the method of EME flow lines to numerically evaluate (neutron) interferometer and Sanz and Miret-Artes´ [15] photon trajectories behind the double-slit grating with the explained the Talbot effect for atoms by plotting their same parameters as in the Kocsis et al experiment. We show associated trajectories behind a diffraction grating. these results in section 2. In section 3, we study how polarizers Motivated by the works of Laukien [16], presented put behind the slits affect the photon trajectories, thus in [17], and Prosser [18], Davidovic´ et al [19] explained providing a trajectory interpretation for the Arago–Fresnel the emergence of interference patterns in experiments with laws. By adding orthogonal polarizers behind the slits, Kocsis photons by determining electromagnetic energy (EME) flow et al could observe average photon trajectories in the presence lines behind an interference grating. The equation of such of orthogonal polarizers and check directly this interpretation. EME flow lines reads as Section 4 is devoted to the trajectory-based interpretation of the Poisson–Arago spot. dr S (r) E E E , (3) ds = cU (r) E 2. Electromagnetic energy (EME) flow lines: average where s denotes a certain arc-length along the corresponding photon trajectories in a Young’s interferometer path, S(r) is the real part of the complex-valued Poynting E vector E Let us consider a monochromatic electromagnetic wave in S(r) 1 Re[E(r) H (r)] (4) vacuum incident onto a two-slit grating located on the E 2 E E ∗ E = E × E XY-plane, at z 0. In order to simplify the treatment, we will and U(r) is the time-averaged EME density = E assume that the electric and magnetic fields do not depend on 1 h i the y-coordinate. This assumption is justified when the slits U(r) ε0 E(r) E∗(r) + µ0 H(r) H ∗(r) . (5) E = 4 E E · E E E E · E E are parallel to the y-axis and their width along the y-axis is much larger than the width along the x-axis. In such a Here E(r) and H(r) are, respectively, the spatial parts of the E E case from Maxwell’s equations one obtains two independent electric andE magneticE field vectors, which satisfy Maxwell’s sets of equations: one involving the Hx and Hz components equations and have been assumed to be harmonic, i.e. of the magnetic field and the Ey component of the electric field (commonly referred to as E-polarization), and another iωt H (r, t) H (r) e− , E E H H ˜E E = E E involving x , z and y ( -polarization). As shown in [19], iωt the electric and magnetic fields behind the grating are given E (r, t) E (r) e− . (6) ˜E E = E E by iβ ∂9 iβ ∂9 Davidovic´ et al [19] pointed out that it is useful to write the E (r) ex + ez + α9ey, (9) equation of the Bohmian trajectories for massive particles (1) E E = − k ∂z E k ∂x E E in terms of the probability current density, iα ∂9 iα ∂9 kβeiϕ H (r) ex ez + 9ey, (10) h E E = ωµ0 ∂z E − ωµ0 ∂x E ωµ0 E J(r, t) ¯ 9 9∗ 9∗ 9 , (7) E E = 2im ∇ − ∇ where 9 is a scalar function that satisfies the Helmholtz because from the latter form, one can recast the guidance equation and the boundary conditions at the grating. The equation (1) as solution for 9 can be written as a Fresnel–Kirchhoff integral, dr J(r, t) r + E E E , (8) k Z ∞ 2 2 iπ/4 ikz + ik(x x0) /2z dt = 9(r, t) 9 (x, z) e− e ψ x0, 0 e − dx0, | E | = 2πz from which one may conclude that the equation for the EME −∞ (11) + flow lines and the equation of the Bohmian trajectories for where ψ(x0, 0 ) is the wave function just behind the grating. massive particles have the same form. In other words, the We consider a grating with two Gaussian slits [23], so that Poynting vector in the case of photons plays the same role as the wave function just behind the grating is given by the quantum probability current density in the case of particles + + + with a mass. ψ(x0, 0 ) ψ1(x0, 0 ) + ψ2(x0, 0 ), (12) This analogy is even more apparent in the cases where = the spatial parts of the magnetic and electric fields can be where expressed in terms of a scalar function that satisfies the 2 2 + 2 1/4 (x0 µ1) /4σ ψ1(x0, 0 ) (2σ π)− e− − 1 w(x0 µ1, a1), (13) Helmholtz equation [19, 20].
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