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Latin American journal of physics education

Huygens' Principle as Universal Model of Propagation

P. Enders

Published Jan 1, 2009• 26 citations
OpenAlex

Abstract

Huygens' Principle (HP) contains both the principle of action-at-proximity and the superposition principle. Although\nthe propagation of sharp, non-spreading wave fronts is included in Huygens' original formulation, it can be left\nout without touching those principles. The formulation of HP by means of the Chapman-Kolmogorov equation\n(following Feynman ) comprises both versions and overcomes misunderstandings like "Huygens" principle is not\nexactly obeyed in Optics' and "HP is incompatible with Green's functions'. This way, HP applies not only to\nthe propagation of light, but also to heat and matter diffusion, Schrödinger matter waves, i. e., to virtually all\npropagation phenomena, which can be described through explicit linear differential and difference equations,\nrespectively. HP for Maxwell's equations is discussed in terms of the Helmholtz-decomposed fields and currents. The\nappearances of HP in mechanics and in fractional Fourier transformation being exploited in optics are also mentioned.

Related papers

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

Complementary Huygens principle for geometrical and nongeometrical optics

Alfredo Luis

We develop a fundamental principle depicting the generalized ray formulation of optics provided by the Wigner function. This principle is formally identical to the Huygens–Fresnel principle but in terms of opposite concepts, rays instead of waves, and incoherent superpositions instead of coherent ones. This ray picture naturally includes diffraction and interference, and provides a geometrical picture of the degree of coherence.

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Huygens Wave Equations in the Field of 2D-CWT

V. V. Vermehren, H. M. de Oliveira

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.

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Jacques Vanier, Cipriana Tomescu

The Lagrangian could be written in terms of kinetic and potential energy, functions of coordinates and velocity and the amplitude of the fields at that point. The essential condition is that the Lagrangian assumed for the field, when studied with Lagrange’s equation, gives Maxwell’s equations as solution and leads to the same wave equation. There are many other fields of physics and many mechanical systems to which the Lagrangian approach can be applied. The goal of the present text is to introduce the beginner to the least action principle and the Lagrangian approach in an elementary way. The chapter argues that the Lagrangian chosen for the fields involved must be an expression that leads to those fields and equations. In wave mechanics, particle properties are described by a so-called wave representing the classical matter field ψ used in Schrodinger’s approach of wave mechanics.

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Equation of Motion

P. Enders

Having got the quantum-mechanical equation of change of stationary states, it is rather straightforward to derive the quantum-mechanical equation of motion. As anticipated in the foregoing chapter, this is the time-dependent Schrödinger equation. Moreover, Huygens’ principle is shown to apply to QM, as obtained by Feynman in another way.

BENTHAM SCIENCE PUBLISHERS eBooks•Feb 12, 2019•0 citations

Huygens-Fresnel principle: Analyzing consistency at the photon level

Elkin A. Santos, Ferney Castro, Rafael Torres

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.

arXiv:1806.08464•Physical Review A•Apr 23, 2018•14 citations
Huygens' Principle (HP) contains both the principle of action-at-proximity and the superposition principle. Although\nthe propagation of sharp, non-spreading wave fronts is included in Huygens' original formulation, it can be left\nout without touching those principles. The formulation of HP by means of the Chapman-Kolmogorov equation\n(following Feynman ) comprises both versions and overcomes misunderstandings like "Huygens" principle is not\nexactly obeyed in Optics' and "HP is incompatible with Green's functions'. This way, HP applies not only to\nthe propagation of light, but also to heat and matter diffusion, Schrödinger matter waves, i. e., to virtually all\npropagation phenomena, which can be described through explicit linear differential and difference equations,\nrespectively. HP for Maxwell's equations is discussed in terms of the Helmholtz-decomposed fields and currents. The\nappearances of HP in mechanics and in fractional Fourier transformation being exploited in optics are also mentioned.
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.
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.
The Lagrangian could be written in terms of kinetic and potential energy, functions of coordinates and velocity and the amplitude of the fields at that point. The essential condition is that the Lagrangian assumed for the field, when studied with Lagrange’s equation, gives Maxwell’s equations as solution and leads to the same wave equation. There are many other fields of physics and many mechanical systems to which the Lagrangian approach can be applied. The goal of the present text is to introduce the beginner to the least action principle and the Lagrangian approach in an elementary way. The chapter argues that the Lagrangian chosen for the fields involved must be an expression that leads to those fields and equations. In wave mechanics, particle properties are described by a so-called wave representing the classical matter field ψ used in Schrodinger’s approach of wave mechanics.
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.