Elsevier

Physics Letters A

Volume 379, Issues 47–48, 18 December 2015, Pages 3059-3068
Physics Letters A

The dynamics of diffracted rays in foams

https://doi.org/10.1016/j.physleta.2015.10.011Get rights and content

Highlights

  • We obtained halos scattering light in foams.

  • We model the light scattering in foams using the geometrical theory of diffraction.

  • We examine the difference between rays and the diffracted rays.

  • We developed optical devices for diffracted rays.

Abstract

We have studied some aspects of the optics of the light scattering in foams. This paper describes the difference between rays and diffracted rays from the point of view of geometrical theory of diffraction. We have represented some bifurcations of light rays using dynamical systems. Based on our observations of foams, we created a solid optical device. The interference patterns of light scattering in foams forming Airy fringes were explored observing the pattern named as the eye of Horus. In the cases we examine, these Airy fringes are associated with light scattering in curved surfaces, while the halo formation is related to the law of edge diffraction. We are proposing a Pohl interferometer using a three-sided bubble/Plateau border system.

Introduction

The sight of a halo in the sky hardly ever fails to evoke a sense of wonder. Similar patterns are seen in different contexts, for example, the parlaseric circle in light scattering in foams [1]. A possible connection between these halo patterns of the atmospheric optics and light scattering in soap films could be based in the concept of diffracted rays, introduced by the geometrical theory of diffraction [2], [3]. From the point of view of this theory, the diffracted rays are produced by incident rays which hit edges, corners, or vertices of boundary surfaces, with the effects of ray splitting. Ray splitting is in fact a general phenomenon of wave propagation in optics at interfaces with sudden change in the refractive index, such as the Plateau border, which is the intersection of three thin films of soap bubbles. The pattern presented in Fig. 1 is a three-dimensional scattering of a laser beam hitting obliquely a Plateau border. The Plateau borders and ice crystals share this property, along with geometrical symmetry, and when these systems interact with light, halos and light spots can be formed, with the coexistence of light rays and light diffracted rays. However, it raises some questions: What are the conditions in which light behaves as a diffracted wave and a ray at same time? What is the difference between light rays and diffracted rays?

One possible way to answer these questions is doing experiments and comparing the observed effects with some patterns reported in the literature. For example, the physical phenomena explicitly associated with thin film interference colors have been observed since antiquity, in some Babylonians cuneiform texts [4], [5] four millennia ago, and people have used optical elements to change the direction of propagation of light such as lenses, prisms, mirrors, beam splitters, filters, diffusers, polarizers, diffracting gratings, halo generators, systems in optofluidics, and so on. In the context of halo generators, some authors have reported the observation of patterns such as the Keller's cone in laser/razor blade demonstrations [6], or witnessed halos related to the geometrical theory of diffraction in a hotel room [7]. According to Keller [8], the geometrical theory of diffraction is an extension of geometrical optics, and it earned a remarkable success mainly for engineering problems, by applying Fermat's principle, and by observing that high-frequency diffraction is a local phenomenon, reducing the solution for the scattering of electromagnetic waves from complex objects, such antennas or airplanes, to the superposition of simple canonical problems, such as wave scattering in an edge and a vertex, or creeping waves in cylinders. This theory was improved with its uniform versions, and some authors claim that the concept of diffracted rays is closely related to Young's edge wave picture [9], and these ideas have been pursued in quantum chaos [10], acoustics, and elasticity [11].

The observation of such patterns is also important in ophthalmology, because halos are symptoms described by patients, for example, medical students are taught that patients developing glaucoma report seeing colored halos, or the existence of epithelial corneal edema caused by poorly fitted contact lenses [12], and even Isaac Newton wrote about the observation of white, dark, and colored circles, while pressing his own eyes [13]. The role of colored circles fit into many of Newton's interests, for example, he discovered the colorful patterns when a convex lens is placed on a flat glass plate, and light rays are reflected by the plate and by the lower surface of the lens. The two groups of reflected rays interfere with each other to produce Newton's rings [9].

The aim of this Letter is thus to explore some patterns obtained by light scattering in detergent films and to investigate some dynamical systems related to these patterns. In order to accomplish this goal, this paper examines some aspects of the geometrical theory of diffraction in elements of foams and suggests new kind of optical element based on foams, and presents new patterns observed when light is scattered in foams. In the next section we describe our experimental apparatus. After that we present some results of light scattering in a straight surface Plateau border in Section 3. In Section 4, we discuss some aspects of dynamical systems in the formation of the parlaseric circle, including maps and bifurcations. We present a new pattern observed in laser beam scattered in three-sided bubbles in Section 5, named the eye of Horus, with a possible application to interferometry, and we close this paper in Section 6 with our conclusions.

Section snippets

Experimental apparatus

For a light wave, a foam is a complex system of arbitrary shaped interfaces, at which light is reflected, refracted, and diffracted multiple times. Because of this, we focused our attention in three specific film shapes shown in Fig. 2(a): a liquid bridge with surface Plateau borders, the Plateau border, and in the three-sided bubble, known as the tetrahedron bubble. The liquid bridge is observed between two Plexiglas plates separated by 2.0 cm, with these plates forming a box. In order to

Light scattering in a straight surface Plateau border

Before getting into more detail, let us establish the difference between rays and diffracted rays in optics. A ray is an idealized model associated with two statements: (a) a ray is a line drawn in space corresponding to the direction of flow of radiant energy, and (b) the trajectory of a ray can be computed by using the law of reflection or Snell's law of refraction. For the case of the concept of a diffracted ray in optics, the first statement is valid, while the statement (b) is not valid

Bifurcations of the light scattering in the Plateau border

The conical diffraction discussed previously is still valid for a groove in a polished surface, for example, the light pattern shown in Fig. 10, and some authors reported similar behavior in the light scattering in a cylindrical rod used as hollow conic beam generator [19], or the light scattering in ferrofluids [20]. The difference in the detail of each pattern is in the formation of the new reflected rays. Using the solid optical device described in Fig. 3, the number of laser dogs increased

The eye of Horus pattern

In this section we will present the effects of light scattering in curved surfaces. The pattern observed with diffracted rays in curved surfaces is presented in Fig. 14, in which we can see the image that we call “The eye of Horus”.

The optical element used for this case is constructed using a three-sided bubble shown in Fig. 15(a), with Plateau borders of two distinct natures (Plateau borders and surface Plateau borders), with thin films forming a vertex structure. The vertex structure is shown

Conclusions

This Letter reports the observation of halo formation and the dynamics of the diffracted rays in foams. Initially, we have explored the difference between rays and diffracted rays of light using some aspects of the geometrical theory of diffraction, and we have developed a connection between the wave theory and the diffracted rays, and for different values of the light wavelength. More specifically, the behavior of light scattering in some elements of foams, such as Plateau borders, is similar

Acknowledgements

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), FAPESP/CNPq #573560/2008-0. We also thank to Prof. César Antunes de Freitas from Faculty of Dentistry of Bauru (FOB-USP), Prof. Arthur Alves Fiocchi from Faculty of Engineering of Bauru (UNESP), and Luis Carlos de Castro of Trially Technology for manufacturing

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