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Journal of Southwest University

The Huygens-Fresnel principle's fixed quantization

Luo Ling-xiao

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

A method of developing the half fixed quantity Huygens- Fresnel principle into a fixed quantity one under the old frame of the wave motion theory is introduced, then the principle is expressed into the same form as Rayleigh- Sommerfeld diffraction formula, and a form that fits probability amplitude of particle.

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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.

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

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Huygens' principle and the modelling of propagation

P. Enders

Huygens' principle (HP) is understood as a universal principle governing not only the propagation of light. According to Hadamard's rigorous definition, HP comprehends the principle of action-by-proximity (cf Faraday's field theory etc) and the superposition of secondary wavelets (Huygens' construction). Following Feynman and using appropriate field propagators (Green's functions) in the space - time domain and state space, respectively, HP is mathematically expressed through the Chapman - Kolmogorov equation, the equation of motion of Markov processes. This formulation leads to a unified formal representation of not only classical and Schrödinger (matter) waves as well as diffusive transport, but of virtually all propagation phenomena, which can be described through explicit differential and difference equations, respectively. It also overcomes certain problems of the physical interpretation of Kirchhoff's formula, leads to a relationship between the degrees of freedom of motion and the form of the equations of motion and simplifies the understanding of some features of the fractional Fourier transformation. Numerical algorithms which obey HP (in discrete form) provide a unified tool for the computation of propagation, transport, and other evolution processes; they are free of the known difficulties when applying Kirchhoff's formula to practical wave problems. Therefore, shedding new light on the 300 years old, but nevertheless actual HP yields not only new theoretical insights and practically useful results, but also conclusions for the methodology of the physics of propagation. For the physical understanding of these results, only undergraduate mathematics is required. Zusammenfassung. Das Huygenssche Prinzip (HP) wird als ein universelles Prinzip aufgefaßt, das nicht nur die Lichtfortpflanzung regiert. Gemäß Hadamards strenger Definition umfaßt es das Nahewirkungsprinzip (vgl. Faradays Feldtheorie u.s.w.) und die Überlagerung sekundärer Wellenfronten (Huygenssche Konstruktion). Verwendet man mit Feynman geeignete Feldpropagatoren (Greensche Funktionen) im Raum-Zeit-Bereich bzw. im Zustandsraum, so kann man es mathematisch in die Form der Chapman - Kolmogorov-Gleichung bringen, die Bewegungsgleichung der Markov-Prozesse. Diese Formulierung führt zu einer einheitlichen formalen Behandlung nict nur klassischer und Schrödinger(Materie)-Wellen sowie diffusiver Ausbreitung, sondern scheinbar aller Vorgänge, die durch explizite Differential- oder/und Differnzengleichungen beschrieben werden können. Sie überwindet zudem gewisse Probleme der physikalischen Interpretation der Kirchhoffschen Formel, führt zu einer Beziehung zwischen Freiheitsgraden der Bewegung und Bewegungsgleichungen, und sie erleichtert das Verständnis einiger Eigenschaften der gebrochen-rationalen Fourier-Transformation. Numerische Algorithmen, die dem HP (in diskreter Form) genügen, stellen ein einheitliches Werkzeug zur Berechnung derartiger Vorgänge bereit; sie sind frei von den bekannten Schwierigkeiten bei der Anwendung der Kirchhoffschen Formel auf praktische Wellenprobleme. Mithin liefert die Betrachtung des 300 Jahre alten, doch nach wie vor aktuellen HP unter neuen Blickwinkeln sowohl weitergehende theoretische Einsichten und praktisch nützliche Ergebnisse, als auch Schlußfolgerungen für die Methodik der Physik der Ausbreitungsvorgänge. Für das physikalische Verständnis dieser Zusammenhänge ist das mathematische Grundwissen des Physikers ausreichend.

European Journal of Physics•Jul 1, 1996•19 citations
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.
The theory of Fresnel diffraction is developed strictly by means of functional analytic method. Fresnel transforms and inverse Fresnel transforms, which give a basis for Fresnel diffraction, are formulated systematically in terms of Fresnel diffraction operator T( z ). Then, it is shown that the set { T( z )} is a one-parameter group of unitary and factor-type operators from the algebraic and topological properties of T( z ). Furthermore, from the representation with respect to the infinitesimal operator of T( z ), the differential formulation of Fresnel diffraction is obtained. This infinitesimal operator, which is -i k times the paraxial approximate expression of the quantized Hamiltonian operator of the variational problem which is given by Fermat's principle in optics, is closely related to the light ray slope.
In this paper it is shown the performing of an optical transform to state the scalar diffraction in the formulation of the wavelet transform and the 'wave equations'. From there, a bridge is build between equations of spherical waves presented in 1678 by Huygens and the continuous wavelet transform. For such a purpose, wavelets are introduced that meet the principles of waves and the properties of wavelets. The following equations are applied in solution to show a correspondence between the Huygens-Fresnel diffraction and the wavelet transform.
Diffraction mathematical theory of light can be derived from Quantum Mechanics and Relativity. Using the duality between position and momentum variables, we show that the momentum distribution of light coincides with the well-known angular spectrum distribution. The momentum distribution links quantum theory and relativity to classical diffraction theory. We also show that the Huygens Principle and the momentum distribution are conjugate expressions at the diffraction aperture. These considerations lead to the geometrical theory of diffraction.
Huygens' principle (HP) is understood as a universal principle governing not only the propagation of light. According to Hadamard's rigorous definition, HP comprehends the principle of action-by-proximity (cf Faraday's field theory etc) and the superposition of secondary wavelets (Huygens' construction). Following Feynman and using appropriate field propagators (Green's functions) in the space - time domain and state space, respectively, HP is mathematically expressed through the Chapman - Kolmogorov equation, the equation of motion of Markov processes. This formulation leads to a unified formal representation of not only classical and Schrödinger (matter) waves as well as diffusive transport, but of virtually all propagation phenomena, which can be described through explicit differential and difference equations, respectively. It also overcomes certain problems of the physical interpretation of Kirchhoff's formula, leads to a relationship between the degrees of freedom of motion and the form of the equations of motion and simplifies the understanding of some features of the fractional Fourier transformation. Numerical algorithms which obey HP (in discrete form) provide a unified tool for the computation of propagation, transport, and other evolution processes; they are free of the known difficulties when applying Kirchhoff's formula to practical wave problems. Therefore, shedding new light on the 300 years old, but nevertheless actual HP yields not only new theoretical insights and practically useful results, but also conclusions for the methodology of the physics of propagation. For the physical understanding of these results, only undergraduate mathematics is required. Zusammenfassung. Das Huygenssche Prinzip (HP) wird als ein universelles Prinzip aufgefaßt, das nicht nur die Lichtfortpflanzung regiert. Gemäß Hadamards strenger Definition umfaßt es das Nahewirkungsprinzip (vgl. Faradays Feldtheorie u.s.w.) und die Überlagerung sekundärer Wellenfronten (Huygenssche Konstruktion). Verwendet man mit Feynman geeignete Feldpropagatoren (Greensche Funktionen) im Raum-Zeit-Bereich bzw. im Zustandsraum, so kann man es mathematisch in die Form der Chapman - Kolmogorov-Gleichung bringen, die Bewegungsgleichung der Markov-Prozesse. Diese Formulierung führt zu einer einheitlichen formalen Behandlung nict nur klassischer und Schrödinger(Materie)-Wellen sowie diffusiver Ausbreitung, sondern scheinbar aller Vorgänge, die durch explizite Differential- oder/und Differnzengleichungen beschrieben werden können. Sie überwindet zudem gewisse Probleme der physikalischen Interpretation der Kirchhoffschen Formel, führt zu einer Beziehung zwischen Freiheitsgraden der Bewegung und Bewegungsgleichungen, und sie erleichtert das Verständnis einiger Eigenschaften der gebrochen-rationalen Fourier-Transformation. Numerische Algorithmen, die dem HP (in diskreter Form) genügen, stellen ein einheitliches Werkzeug zur Berechnung derartiger Vorgänge bereit; sie sind frei von den bekannten Schwierigkeiten bei der Anwendung der Kirchhoffschen Formel auf praktische Wellenprobleme. Mithin liefert die Betrachtung des 300 Jahre alten, doch nach wie vor aktuellen HP unter neuen Blickwinkeln sowohl weitergehende theoretische Einsichten und praktisch nützliche Ergebnisse, als auch Schlußfolgerungen für die Methodik der Physik der Ausbreitungsvorgänge. Für das physikalische Verständnis dieser Zusammenhänge ist das mathematische Grundwissen des Physikers ausreichend.