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arXiv:1503.05820

Huygens Wave Equations in the Field of 2D-CWT

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

Published Mar 14, 2015• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1503.05820

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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.
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.
In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ 3 , wavelets in ℝ n ( n > 3), and wavelets on the 2-sphere S 2 . Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis. In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform. Three-dimensional wavelets Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].
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.
A theory of generalized Fresnel transformations, which is an important part of transformation of scalar-wave propagating between two quadratic surfaces, is presented. The generalized Fresnel transformation is formulated in terms of generalized Fresnel transform operator T( Z ). A set of { T( Z )} has group properties. The differential formulation of the generalized Fresnel transform is obtained by means of functional analytic method. In a special case of the generalized Fresnel transform, one may obtain the differential formula of Fresnel diffraction and Fourier transform.