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Optics Express

Stochastic ray tracing for Fresnel diffraction

Keita Seto

Published Mar 15, 2024• 2 citations
Publisher

Abstract

We propose stochastic ray tracing for laser beam propagation in Fresnel diffraction to find the duality between wave and ray representations. We transform from the Maxwell equations to the Schrödinger equation for a monochromatic laser beam in the slowly varying envelope approximation. The stochastic ray tracing method interprets this Schrödinger equation as a stochastic process, of an analogy of Nelson's stochastic mechanics. It can illustrate the stochastic paths and the wavefront of an optical beam. This ray tracing method includes Fresnel diffraction effects naturally. We show its general theoretical construction and numerical tests for a Gaussian laser beam with diffraction, that stochasticity realizes the beam waist around the Rayleigh range.

DOI
10.1364/oe.521317

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We propose stochastic ray tracing for laser beam propagation in Fresnel diffraction to find the duality between wave and ray representations. We transform from the Maxwell equations to the Schrödinger equation for a monochromatic laser beam in the slowly varying envelope approximation. The stochastic ray tracing method interprets this Schrödinger equation as a stochastic process, of an analogy of Nelson's stochastic mechanics. It can illustrate the stochastic paths and the wavefront of an optical beam. This ray tracing method includes Fresnel diffraction effects naturally. We show its general theoretical construction and numerical tests for a Gaussian laser beam with diffraction, that stochasticity realizes the beam waist around the Rayleigh range.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
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.
We present a theoretical model for the propagation of non-self-interacting atom laser beams. We start from a general propagation integral equation and we use the same approximations as in photon optics to derive tools to calculate the atom-laser-beam propagation. We discuss the approximations that allow one to reduce the general equation whether to a Fresnel-Kirchhoff integral calculated by using the stationary phase method, or to the eikonal. Within the paraxial approximation, we also introduce the ABCDABCDABCD matrices formalism and the beam quality factor. As an example, we apply these tools to analyze the recent experiment by Riou et al. [Phys. Rev. Lett. 96, 070404 (2006)].
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.
This paper draws a comparison between atom optics and light optics, based on the observation that one can derive a Schrödinger equation for the slowly varying amplitude of the electromagnetic field in the paraxial approximation, the `optical Schrödinger equation'. A correction scheme between (monochromatic) light waves and matter waves is established and applied to cases of interest in atom or fibre optics. First the propagation in a periodic -potential is investigated, where well known effects, such as Bragg scattering or channelling are re-examined from a `matter wave versus light wave' point of view. In a second application the general theory is extended to axisymmetrical materials such as step-index fibres or graded-index fibres. In particular, a matter wave analogue of the so-called non-diffractive beams of electrodynamics is discussed.