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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

On The Connection Between The Extended Huygens-Fresnel Principle And The Path-Integral Approximate Computation Based On Orthogonal Expansions

Valerian I. Tatarskii, V. V. Zavorotnyi

Published Nov 25, 1986• 7 citations
Publisher

Abstract

In the present paper a connection between the approximate Huygens-Fresnel formulae and the exact path-integral solutions is studied. We consider the approach which makes it possible to separate from the path integral the contribution that corresponds to the phase approximation of the extended Huygens-Fresnel principle. This approach is used to derive the corrections to the phase approximation.

DOI
10.1117/12.975509

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In the present paper a connection between the approximate Huygens-Fresnel formulae and the exact path-integral solutions is studied. We consider the approach which makes it possible to separate from the path integral the contribution that corresponds to the phase approximation of the extended Huygens-Fresnel principle. This approach is used to derive the corrections to the phase approximation.
Typically the use of the Rayleigh-Sommerfeld diffraction formula as a photon propagator is widely accepted due to the abundant experimental evidence that suggests that it works. However, a direct link between the propagation of the electromagnetic field in classical optics and the propagation of photons where the square of the probability amplitude describes the transverse probability of the photon detection is still an issue to be clarified. We develop a mathematical formulation for the photon propagation using the formalism of electromagnetic field quantization and the path-integral method, whose main feature is its similarity with a fractional Fourier transform (FRFT). Here we show that because of the close relation existing between the FRFT and the Fresnel diffraction integral, this propagator can be written as a Fresnel diffraction, which brings forward a discussion of the fundamental character of it at the photon level compared to the Huygens-Fresnel principle. Finally, we carry out an experiment of photon counting by a rectangular slit supporting the result that the diffraction phenomenon in the Fresnel approximation behaves as the actual classical limit.
We present a simple pedagogical discussion of the physics of path integrals in nonrelativistic quantum mechanics. Rather than show that the path integral gives the standard results of quantum mechanics, as is usually done in the literature, we instead discuss why it does so. Specifically, we discuss why the phase factor associated with each path is given by the classical action. We construct the path integral using Huygens’ principle. Finally, we show that the existence of many paths is a consequence of the spreading of the quantum-mechanical wave function.
An alternative picture for photon diffraction had been proposed describing diffraction by a distribution of photon paths determined through a Fourier analysis of a scattering lattice. The momentum exchange probabilities are defined at the location of scattering, not the point of detection. This contrasts with the picture from classical optical wave theory that describes diffraction in terms of the Huygens-Fresnel principle and sums the phased contributions of electromagnetic waves to determine probabilities at detection. This revised picture, termed “Momentum Exchange Theory,” can be derived through a momentum representation of the diffraction formulas of optical wave theory, replacing the concept of Huygens wavelets with photon scattering through momentum exchange with the lattice. Starting with the Rayleigh-Sommerfeld and Fresnel-Kirchoff formulas, this paper demonstrates that diffraction results from positive and negative photon dispersions through virtual particle exchange probabilities that depend on the lattice geometry and are constrained by the Heisenberg uncertainty principle. The positive and negative increments of momentum exchange exhibit harmonic probability distributions characteristic of a “random walk,” dependent on the distance of momentum exchange. The analysis produces a simplified prediction for the observed intensity profile for a collimated laser beam diffracted by a long, straight edge that lends conceptual support for this alternative picture.
This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n -fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.