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European Journal of Physics

Complementary Huygens principle for geometrical and nongeometrical optics

Alfredo Luis

Published Jan 8, 2007• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/28/2/008

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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.
While optics and mechanics are two distinct branches of physics, they are connected. It is well known that the geometrical/ray treatment of light has direct analogies to mechanical descriptions of particle motion. However, connections between coherence wave optics and classical mechanics are rarely reported. Here we report links of the two through a systematic quantitative analysis of polarization and entanglement, two optical coherence properties under the wave description of light pioneered by Huygens and Fresnel. A generic complementary identity relation is obtained for arbitrary light fields. More surprisingly, through the barycentric coordinate system, optical polarization, entanglement, and their identity relation are shown to be quantitatively associated with the mechanical concepts of center of mass and moment of inertia via the Huygens-Steiner theorem for rigid body rotation. The obtained result bridges coherence wave optics and classical mechanics through the two theories of Huygens.
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.
Few-photon systems are best described by their wave function rather than by the usual quantum field formalism. In this work, we develop a photon wave function (PWF) formalism suitable for analyzing a wide variety of quantum optical problems related to propagation, diffraction and imaging with quantum states of light. We establish a generalized Huygens-Fresnel (GH-F) principle that describes the propagation of any paraxial N-photon state. This tool is very helpful for predicting photo-detection correlations in space and time due to an initial N-particle entanglement, even in complex situation. The effect of lenses, beam splitters, filters ... on the photon paths can be easily taken into account. We apply the PWF formalism and the GH-F principle to three specific problems in quantum optics. First, we revisit the Hong-Ou-Mandel two-photon interference effect and analyze the effect of photon shape mismatch in space, time and polarization using the PWF formalism. Second, we show how to use the GH-F principle to analyze "ghost" imaging and diffraction experiments with entangled photon pairs such as those realized by Strekalov et al. [Phys. Rev. Lett. 74, 3600 (1995)] and Pittman et al. [Phys. Rev. A 52, R3429 (1995)] in the nineties. Finally, we use the GH-F principle to analyze the resolution enhancement in a recent quantum imaging proposal based on N incoherent single-photon sources [Phys. Rev. Lett. 99, 133603 (2007) and Phys. Rev. A 80, 013820 (2009)].
The paper presents a review of some applications of the Wigner distribution function to optics. The Wigner distribution function F(x,u) of a signal describes the signal in space x and spatial frequency u, simultaneously, and can be considered as a local spatial frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics. Some examples are given to show this resemblance. Properties of the Wigner distribution function are discussed, showing, for instance, its relation to Heisenberg’s uncertainty principle and to some well‐known radiometric quantities. The concept of the Wigner distribution function is not restricted to deterministic signals; it can easily be extended to stochastic signals. Some examples of stochastic signals are considered. The propagation of signals through linear systems can readily be expressed in terms of Wigner distribution functions. Again, the description of systems by Wigner distribution functions can be interpreted directly in terms of geometrical optics: for Luneburg’s first‐order system, for instance, such a description immediately yields the ray transformation matrix of the system, and the relation between the input and the output Wigner distributioin function reads Fo(x,u) = Fi(Ax+Bu,Cx+Du). Transport equations for the Wigner distribution function in a general linear medium are derived. In the Liouville approximation they take the form of a first‐order partial differential equation, which allows, once more, a geometric‐optical interpretation: along the path of a geometric‐optical light ray the Wigner distribution function has a constant value.
The premise of this submission is to identify the role of the Wigner function [1] in paraxial wave optics, where it can be naturally introduced on the basis of the analogy between the paraxial wave equation and the Schrodinger equation of nonrelativistic quantum mechanics. Also known as the Wolf function, it describes the cross section of a paraxial beam using a real-valued function of displacement from axis and transverse components of the wave vector. This can be perceived as a quasi-probability distribution over the system's phase space, following the dynamics of a state space density of an ensemble of free noninteracting particles. In a (2+1)–dimensional picture this corresponds to the propagation of straight linear rays where the Wolf function describes the local phase space intensity. There is no notion of relative phase but negative values can be present, providing an alternative basis for description of interference phenomena. This corresponds exactly to the concept of pencils of positive and negative rays, developed by E. C. G. Sudarshan in the late 70’s and 80’s [2, 3].