Fermat's Principle Research Articles

Light in moving media

We embark on a journey through optics in dielectric media, inspired and guided by Fermat's Principle. We show how to modify the principle when dielectrics are moving non-uniformly and explain how the propagation of light in fluids is connected to phenomena in magnetism and in general relativity. Dielectric media may even serve to build artificial astronomical objects in an earthly laboratory.

I. Geometrical optics of variable-frequency light rays:Theoretical basis

A variable-frequency light ray (VF ray) is a new concept in optics; it is a ray that is monochromatic at every point of the path (no frequency spread) but its frequency changes along the path due to interactions occurring along that path. For example, the frequency of the light ray can be affected by gravitational time dilation, reflection from a moving mirror, or coherent Raman scattering. Since the Fermat principle of least time (PLT) implicitly assumes that the frequency of the light ray is an irrelevant constant, the PLT-based geometrical optics is insufficient to handle propagation of VF rays. In this paper, we derive a new extremum principle (NEP) that is sensitive to changes of frequency along the path of the light ray and constitutes a basis for geometrical optics of VF rays. The NEP is derived directly from the principle of least action applied to the quantized electromagnetic field associated with the light ray propagating in a vacuum or in a transparent material medium in the presence of the gravitational field. The relationship between the NEP, the classical version of the Fermat principle, and the general relativistic generalization of the Fermat principle (the extremum principle for null geodesics) is discussed in detail. Applications of the NEP are suggested in the nonrelativistic, special relativistic, and general relativistic regimes. PACS Nos.: 42.15-i, 12.20-m, 04.20-q

A Morse theory for massive particles and photons in general relativity

In this paper we develop a Morse theory for time-like geodesics parameterized by a constant multiple of proper time. The results are obtained using an extension to the time-like case of the relativistic Fermat principle, and techniques from Global Analysis on infinite dimensional manifolds. In the second part of the paper we discuss a limit process that allows to obtain also a Morse theory for light rays.

The general relativity dynamics in the Eisenhart geometry

It has been proposed a long time ago by Wheeler and deWitt to look at the evolution of three-metrics as a geodesic flow on the superspace. Since then a lot of attention has been paid towards better understanding the geometric structure of the superspace. In particular it has been appreciated that the minisuperspace can in a natural way be equipped with the Jacobi metric. However the Jacobi metric is degenerated on certain codimension one hypersurfaces (boundary sets) leading to severe difficulties. In this contribution we propose to use the so-called Eisenhart's principle as an alternative geometrical construction on minisuperspace. Then the dynamics of general relativity, represented by a constrain Hamiltonian system, is mapped onto a geodesic flow on a smooth manifold without boundary. Hence Eisenhart's proposal seems to be the right way to desingularization of motion in Jacobi metric (e.g. the dynamics of homogeneous cosmological models near the initial singularity) which is nontractable in the Jacobi picture. Different methods of desingularizing of the Jacobi metric through the isometric embedding into a flat space with the Lorentzian signature will also be presented. The extension of the Fermat principle for the case of timelike trajectories is also given.

Time-Distance Inversion Methods and Results - (Invited Review)

The current interpretations of the travel-time measurements in quiet and active regions on the Sun are discussed. These interpretations are based on various approximations to the 3-D wave equation such as the Fermat principle for acoustic rays and the Born approximation. The ray approximation and its modifications have provided the first view of the 3-D structures and flows in the solar interior. However, more accurate and computationally efficient approximations describing the relation between the wave travel times and the internal properties are required to study the structures and flows in detail. Inversion of the large three-dimensional datasets is efficiently carried out by regularized iterative methods. Some results of time-distance inversions for emerging active regions, sunspots, meridional flows and supergranulation are presented. An active region which emerged on the solar disk in January 1998, was studied from SOHO/MDI for eight days, both before and after its emergence at the surface. The results show a complicated structure of the emerging region in the interior, and suggest that the emerging flux ropes travel very quickly through the depth range of our observations. The estimated speed of emergence is about 1.3 km s-1. Tomographic images of a large sunspot reveal sunspot ‘fingers’ — long narrow structures at a depth of about 4 Mm, which connect the sunspot with surrounding pores of the same polarity.

Frequency-domain Green's function for a planar periodic semi-infinite phased array. II. Diffracted wave phenomenology

For pt.I see ibid., vol.48, no.1, p.67-74, 2000. This second part of a two-paper sequence deals with the physical interpretation of the rigorously derived high-frequency asymptotic wave-field solution in part I, pertaining to a semi-infinite phased array of parallel dipole radiators. The asymptotic solution contains two parts that represent contributions due to truncated Floquet waves (FWs) and to the corresponding edge diffractions, respectively. The phenomenology of the FW-excited diffracted fields is discussed in detail. All possible combinations of propagating (radiating) and evanescent (nonradiating) FW and diffracted contributions are considered. The format is a generalization of the conventional geometrical theory of diffraction (GTD) for smooth truncated aperture distributions to the truncated periodicity-induced FW distributions with their corresponding FW-modulated edge diffractions. Ray paths for propagating diffracted waves are defined according to a generalized Fermat principle, which is also valid by analytic continuation for evanescent diffracted ray fields. The mechanism of uniform compensation for the FW-field discontinuities (across their truncation shadow boundaries) by the diffracted waves is explored for propagating ad evanescent FWs, including the cutoff transition from the propagating to the evanescent regime for both the FW and diffracted constituents. Illustrative examples demonstrate: (1) the accuracy and efficiency of the high-frequency algorithm under conditions that involve the various wave processes outlined above and (2) the cogent interpretation of the results in terms of the uniform FW-modulated GTD.

Application-Oriented Relativistic Electrodynamics (2) - Abstract *

This article is a revised and upgraded edition of a previous one published in this journal, hence the label (2), see the General Remarks section below. Relativistic Electrodynamics, for many years a purely academic subject from the point of view of the applied physicist and electromagnetic radiation engineer, is nowadays recognized as pertinent to many practical applications. We therefore need to define a syllabus and explore the best methods for educating future generations of such users. Such an attempt is presented here, and is of course biased by personal preferences. What emerges as general guidelines are the facts that Relativistic Electrodynamics should be presented axiomatically, without trying to "explain the physical meaning" of Special Relativity, that four-vectors and their mathematical properties should be emphasized, and that the field tensors, an elegant formalism, albeit of limited practical use, should be avoided. Use of four-fold Fourier transforms not only greatly simplifies the relevant manipulations, it is also of paramount importance for discussion of dispersive media. This approach yields many concepts as mathematical results, e.g., the Relativistic Doppler effect, which therefore do not require a long phenomenological discussion with many "explanations". Introducing this approach as early as possible opens new vistas for the student and the educator, indeed some of the new results here do not appear in textbooks on Special Relativity. One of the main results shown here is the fact that the generalized Fermat principle states that the ray will propagate in such a manner that the proper time will be minimized (or extremized, in general). It also strips the mystique of this principle, showing that it is in fact equivalent to a modest mathematical condition on the smoothness of the phase function. The presentation is constructed in a way that allows the student to gradually overcome difficulties in assimilating new concepts and applying them. In that too it is different from many conventional presentations.

On the motion of rigid bodies in incompressible inviscid fluids of inhomogeneous density

The motion of a rigid body in an inviscid incompressible fluid of inhomogeneous density is considered. The size of the body is taken small with respect to the length scale of the density variations; its shape is otherwise arbitrary. The force and the torque acting on the body in an arbitrary motion are derived from Hamilton's principle of least action, thus offering a variational derivation of Kirchhoff's equations, supplemented by the terms due to the density gradient. The force and the torque due to a density gradient are proportional to the gradient and quadratic in the velocity and the angular velocity. The same coefficients are shown to govern both the inertial behaviour of the body, i.e. the response to accelerations, and the effects of density gradients. The free motion of spheres and two-dimensional circular cylinders is shown to obey a condition akin to the Fermat principle in optics.

Johann Bernoulli's brachistochrone solution using Fermat's principle of least time

Johann Bernoulli's brachistochrone problem is now three hundred years old. Bernoulli's solution to the problem he had proposed used the optical analogy of Fermat's least-time principle. In this analogy a light ray travels between two points in a vertical plane in a medium of continuously varying index of refraction. This solution and connected material are explored in this paper.

Gravitational lenses: odd or even images?

In this paper we first recall the Morse relations for light rays in vacuum. They can be obtained, under quite general assumptions, by the relativistic Fermat principle stated by Kovner and correctly proved by Perlick. They can be used for a global mathematical description of the gravitational lens effect. Afterwards we consider the well known thin lens model and we observe that the light rays having a geometrical index equal to 2 have a strong constraint about the surface density where they pass the deflector. Finally, we note that in situations similar, for example, to the Einstein cross (whose models predict the existence of one light ray having its geometrical index equal to 2), the constraint on the surface density suggests that the absorption may be so high that the related image is quite difficult to detect.

On a Fermat principle in general relativity. A Ljusternik-Schnirelmann theory for light rays

In this paper existence and multiplicity results for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions. Such results are obtained using an extension to Lorentzian Geometry of the classical Fermat principle in optics. The results are proved using critical point theory on infinite dimensional manifolds. An application to the gravitational lens effect is presented.

Optical path in birefringent media and Fermat's principle

In this paper we propose a definition for the optical path in birefringent crystals. We apply this definition to light refraction in a crystal-crystal interface by means of Fermat's principle. The result obtained is confirmed by applying Snell's law generalized for crystals and the relation obtained from Maxwell's equations between the direction normal to the wavefront and the ray.

A General-Relativistic Fermat Principle for Extended Light Sources and Extended Receivers

In an arbitrary Lorentzian manifold, we fix a spacelike submanifold P and a timelike submanifold Γ. We interpret P as (the surface of) a light source at a particular instant of time, and we interpret Γ as the history of (the surface of) a receiver. We prove the following version of Fermat's principle. Among all lightlike curves from P to Γ, the lightlike geodesics which are perpendicular to P and spatially perpendicular to Γ are characterized by stationary arrival time. Here, the arrival time is defined with the help of an arbitrary time function on Γ. Moreover, we show that the second variation of the arrival time at a stationary point is characterized by a Morse index theorem.

An investigation of pulse-timing techniques for broadband ultrasonic velocity determination in cancellous bone: a simulation study

Berlage wavelets are used to simulate ultrasonic pulses in an unbounded, homogeneous, isotropic and absorptive medium. Intrinsic absorption of the medium is properly described by Kolsky's attenuation, which considers velocity dispersion to meet the causality condition. Several current time-domain velocity measurement techniques have been investigated using numerically simulated pulses for three normalized BUA values: 20, 40 and 60 dB , which mimic experimentally determined values for cancellous bone. The velocities, calculated using first motion transit times, are used as references supported by the Fermat principle of least time. The simulated results for fixed sample thickness indicate that pulse-broadening increases with the transit time of the reference point and the intrinsic absorption of the medium. Comparison shows that the first zero-crossing method yields 3-6% errors in velocity results, better than the cross-correlation method. However, the zero-crossing method gives inconsistent velocity measurement for a medium of 40 dB absorption and three different thicknesses: 0.2, 0.4 and 0.6 cm. A novel technique for velocity measurement is presented using the peak of the envelope of a signal as a reference point to measure transit time difference. The envelope of a signal represents the instantaneous amplitude of the associated analytic signal. The velocities derived using this method differ from the true velocities by only 1.2-2.4%, more accurate than those obtained by the first zero-crossing method. The envelope peak has the additional merits of easy detection and robustness. Most importantly, the envelope technique may be used to yield accurate velocity measurement in cases where an accurate determination of the first motion transit time is sometimes prohibited due to the presence of noise.

Spatio-temporal Analysis of Electroconvection in Nematics

Abstract We investigate electroconvection in nematic liquid crystals by means of optical microscopy. Time resolved optical images are used to study the director dynamics. For the first time we present instant images in the dielectric regime. A numerical simulation of the optical transmission patterns is performed on the basis of Fermat's principle. In the instant images of dielectric rolls, the periodicity of the observed optical pattern is equal to the wavelength λ0 of the convection rolls. The well known low contrast 'stationary' optical texture observed in conventional experiments results from time averaging of these instant images; its wavelength is λ0/2.

A timelike extension of Fermat's principle in general relativity and applications

We define a variational problem based on the arrival time functional for timelike curves on a Lorentzian manifold ${\cal M}$ parameterized by a fixed constant multiple of their proper time. Under a causality assumption for the manifold ${\cal M}$ , we prove that the stationary points of our problem are geodesics, obtaining an extension of the Fermat's Principle for light rays proven in [14] (see also [2]). Moreover, we study the compactness pr operties of the arrival time functional by global variational techniques. Under intrinsic assumptions on the metric of ${\cal M}$ we get results of existence and multiplicity for geodesics with a given energy between an event and an observer of ${\cal M}$ .

Shortening null geodesics in Lorentzian manifolds. Applications to closed light rays

We show the existence of one spatially closed lightlike geodesic on a regular, conformally stationary Lorentzian manifold M , having a non-contractible, light-convex, timelike cylinder C. The result is obtained by using an extension of the classical Fermat's Principle in optics, proven in [2], and a shortening argument similar to that used in [21] for studying the existence of closed geodesics on Riemannian manifolds with boundary.

Fermat's principle, Huygens' principle, Hamilton's optics and sailing strategy

Fermat's principle asserts that light takes the path of minimum (actually extremal) time. Sailors often wish to find paths of minimum time. Thus by thinking of a sailboat as a light ray, a sailor can use Fermat's principle to describe the optimum sailing strategy. Huygens' principle and Hamiltonian optics follow from Fermat's principle, so a sailor can use Huygens' principle to visualize least-time paths, and Hamilton's optics provides a mathematical description of these paths. In especially simple cases, the optics-based formalism can be used to describe and quantify the basic tactics of sailboat racing.

Circular reasoning

The author discusses two interesting relationships between circles and lines. There should be a computer graphics application that can use the first topic to run faster or better. The second topic is Ptolemy's Theorem, which is a generalization of the triangle inequality. The author shows how it can be used to derive the angle addition formulas. He considers how Ptolemy's Theorem can be used to prove that Snell's Law and Fermat's Principle of Least Time both lead to the same geometry of refraction.

A timelike extension of fermat's principle in general relativity. Comparisons with the lightlike case

A timelike extension of fermat's principle in general relativity. Comparisons with the lightlike case