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American Journal of Physics

Fiber-optic analogies with mechanics

Richard J. Black, Adrian Ankiewicz

Published Jun 1, 1985• 34 citations
Publisher

Abstract

We provide a new unified formalism for the analogy between fiber-optics and mechanics. It is based on the scalar theory of light for an optical fiber with length-independent refractive index, and the mechanics of a particle in a time-independent potential. Illustrative examples such as tunneling and precession are given. We also mention the conceptual analogies that arise when we include (i) polarization and (ii) length or time-dependent variations. This should prove useful both in teaching and as a bridge between research areas.

DOI
10.1119/1.14237

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We provide a new unified formalism for the analogy between fiber-optics and mechanics. It is based on the scalar theory of light for an optical fiber with length-independent refractive index, and the mechanics of a particle in a time-independent potential. Illustrative examples such as tunneling and precession are given. We also mention the conceptual analogies that arise when we include (i) polarization and (ii) length or time-dependent variations. This should prove useful both in teaching and as a bridge between research areas.
In this chapter, we shall expose several analogies between oscillatory phenomena in mecahanics and optics.The main subject will be the analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials.The basis of this analogy is the fact that both wave equations for electromagnetic monoenergetic waves (i.e. with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions -like behaviour at infinity and boundary conditions -are imposed, they generate similar eigenvalues problems, with similar solutions.The benefit of such analogies is twofold.First, it could help a researcher, specialized in a specific field, to better understand a new one.For instance, they might efficiently explain the fiber-optics properties to people already familiar with quantum mechanics.Also, even if such researchers work frequently with the quantum mechanical wave function, the electromagnetic modal field may provide an interesting vizualization of quantum probability density field [1].Second, it provides the opportunity of cross-fertilization between (for instance) electromagnetism and optoelectronics, through the development of ballistic electron optics in two dimensional (2D) electron systems (2DESs); transferring concepts, models of devices and experiments from one field to another stimulate the progress in both domains.Even if in modern times the analogies are not credited as most creative approaches in physics, in the early days of developement of science the perception was quite different."Men's labour ... should be turned to the investigation and observation of the resemblances and analogies of things... for these it is which detect the unity of nature, and lay the fundation for the constitution of the sciences.",considers Francis Bacon, quoted by [1].Some two centuries later, Goethe was looking for the "ultimate fact" -the Urphänomenon -specific to every scientific discipline, from botanics to optics [2], and, in this investigation, attributed to analogies a central role.However, if analogies cannot be considered anymore as central for the scientific investigation, thay could still be pedagogically useful and also inspiring for active scientific research.Let us describe now shortly the structure of this chapter.The next two sections are devoted to a general description of the two fields to be hereafter investigated: metallic and dielectric waveguides (Section 2) and 2DESs (Section 3).The importance of these topics for the development of optical fibers, integrated optics, optoelectronics, transport Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts
The optical fiber is a revolutionary technology of the past century. It enables us to manipulate single modes in nonlinear interactions with precision at the quantum level without involved setups. This setting is useful in the field of analogue gravity (AG), where gravitational phenomena are investigated in accessible analogue lab setups. These lecture notes provide an account of this AG framework and applications. Although light in nonlinear dielectrics is discussed in textbooks, the involved modelling often includes many assumptions that are directed at optical communications, some of which are rarely detailed. Here, we provide a self-contained and sufficiently detailed description of the propagation of light in fibers, with a minimal set of assumptions, which is relevant in the context of AG. Starting with the structure of a step-index fiber, we derive linear-optics propagating modes and show that the transverse electric field of the fundamental mode is well approximated as linearly polarized and of a Gaussian profile. We then incorporate a cubic nonlinearity and derive a general wave envelope propagation equation. With further simplifying assumptions, we arrive at the famous nonlinear Schrödinger equation, which governs fundamental effects in nonlinear fibers, such as solitons. As a first application in AG, we show how intense light in the medium creates an effective background spacetime for probe light akin to the propagation of a scalar field in a black hole spacetime. We introduce optical horizons and particle production in this effective spacetime, giving rise to the optical Hawking effect. Furthermore, we discuss two related light emission mechanisms. Finally, we present a second optical analogue model for the oscillations of black holes, the quasinormal modes, which are important in the program of black hole spectroscopy.
Review of the paraxial approximation to the problem of light beams propagating in optical fibers is presented. The analogy of time-dependent integrals of motion for the quantum system and space-dependent invariants for the radiation in inhomogeneous optical waveguides is elucidated. A specific tomographic probability distribution associated with the light-pulse profile is considered analogously to tomographic probabilities in quantum mechanics. The new inequality is presented for the tomographic probability.
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.