Fermat's Principle Research Articles

From Fermat principle to wave equation—quantization of “particle mechanics of light”

The Fermat principle states that light chooses the temporally shortest path. The action for this “motion” is the observed time, and it has no Lorentz invariance. In this Letter we show how this action can be obtained from a relativistic action of massive particle, and how the classical wave equation of light can be obtained from this action.

Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle

It is shown that the problem of designing a two- reflector system transforming a plane wave front with given in- tensity into an output plane front with prescribed output in- tensity can be formulated and solved as the Monge-Kantorovich mass transfer problem.

High-order pulse front tilt caused by high-order angular dispersion

We have found general expressions relating the high-order pulse front tilt and the high-order angular dispersion in an ultrashort pulse, for the first time to our knowledge. The general formulae based on Fermat's principle are applicable for any ultrashort pulse with angular dispersion in the limit of geometrical optics. By virtue of these formulae, we can calculate the high-order pulse front tilt in the sub-20-fs UV pulse generated in a novel scheme of sum-frequency mixing in a nonlinear crystal accompanied by angular dispersion. It is also demonstrated how the high-order angular dispersion can be eliminated in the calculation.

Huygens' Principle: A case against optimality

I will present evidence that nature does not optimize in the sense of Fermat's principle of least time, contrary to what Schoemaker's unintentionally ambiguous exposition might suggest. First, Huygens' principle, an alternative nonteleological account of Snell's law, is outlined. Second, I confront Fermat's principle with a substantive conceptual problem.

A Fast Method to Simulate Travelling Waves in Nonhomogeneous Chemical or Biological Media

Waves in excitable media can be treated by a simple geometric theory. The propagation velocity is assumed known and evolution of wave fronts is determined by elementary physical principles (Fermat's principle, Huygens' principle). Based on this geometric theory a fast computational method is developed. By this method the distorting effect of the spatial grid is avoided. The method is applied to the cases when a circular obstacle is surrounded by a homogeneous and heterogeneous medium, respectively. The numerical simulations show that the method is convenient, fast and reliable.

Fermat's principle for seismic rays in elastic media

We prove that rays in linearly elastic anisotropic nonuniform media obey Fermat's principle of stationary traveltime. First, we formulate the concept of rays, which emerges from the Hamilton equations. Then, we show that these rays are solutions of the variational problem stated by Fermat's principle. This proof is valid for all rays except the ones associated with infection points on the phase-slowness surface.

Holodiagrams in birefringent media.

The modifications to the holodiagram concept to describe free propagation (the extraordinary ray) inside birefringent materials are described. Holodiagrams are graphs showing the loci where the sum or the difference in the optical path from a generic point to two foci is the same. The holodiagrams obtained in this way give the shape of the surfaces that satisfy Fermat's principle, conjugate by reflection of one focus into the other, and represent the interference fringes obtained if both points are coherent sources. The reflection law in birefringent media is investigated in relation to this diagram. One direction for the optical axis is considered: parallel to the line joining the source and the observation point. Quartz-type and calcite-type crystals are studied.

New insight into optimization and variational problems in the 17th century

First, a synopsis of the major changes of natural science, mathematics and philosophy within the 17th century shall highlight the birth of the new age of science and technology. Based on Fermat's principle of the shortest light‐way and Galilei's first attempt of an approximative solution of the so‐called Brachistochrone problem using a quarter of the circle, Johann Bernoulli published a competition for this problem in 1696, and six solutions were submitted by the most famous scientists of the time and published in 1697, even though the variational calculus was only published in 1744 by Euler for the first time. Especially the analytical solution of Jakob Bernoulli contains already the main idea of Euler's variational calculus, i.e. to vary only one function value at a time using a finite difference method and proceeding to the infinitesimal limit. Also Leibniz' geometric solution is very remarkable, realizing a direct discrete variational method geometrically which was invented numerically much later in the 19th century by Ritz and Galerkin and generalized to the finite element method by introducing test and trial functions in finite subspaces. A new finite element solution of the non‐linear Brachistochrone problem concludes the paper. It is important to recognize that besides the roots of variational calculus also the first formulations of conservation laws in mechanics and their applications originated in the 17th century.

Fermat principle for a nonstationary medium.

One possible formulation of a variational principle of the Fermat type for systems with time-dependent parameters is suggested. In a stationary case, it reduces to the Mopertui-Lagrange least-action principle. A class of Hamiltonians (dispersion relations) is indicated, for which the variational principle reduces to the Fermat principle in a general nonstationary case. Hamiltonians that are homogeneous functions of momenta are in this category. For the important case of nondispersive waves (corresponding to Hamiltonians being homogeneous function of momenta order 1) the Fermat principle fully determines the geometry of the rays. Equations relating the variation of signal frequency with the rate of change of propagation time are established.

CDP mapping in tilted transversely isotropic (TTI) media. Part II: Velocity analysis by combining CDP mapping with a genetic algorithm

ABSTRACTThis paper presents a method for velocity analysis in tilted transversely isotropic (TTI) media by combining CDP mapping with a genetic algorithm. CDP mapping is a velocity analysis method for determining anisotropic velocity but has difficulties due to the following factors: (i) it involves a non‐linear and multimodal objective function; (ii) it is prohibitively expensive in the evaluation of candidate solutions, which often involves the calculation of images in the depth domain; (iii) there is often a very large parameter space. Recognizing the global and multimodal nature of the problem, a genetic algorithm is employed to search for the optimal velocity model. The efficiency of the method contributes to two critical processes: rapid model evaluation, achieved by generating CDP mapping only in the neighbourhood of specific reflectors, and fast computation, based on Fermat's principle, of the CDP points and traveltimes in TTI media. The method produces subsurface structure images in the depth domain, and can also solve for Thomsen's anisotropic parameters (ɛ and δ), the vertical velocity and the dip of the symmetry axis in the model space, simultaneously.

Gravitational lensing in metric theories of gravity

Gravitational lensing in metric theories of gravity is discussed. I introduce a generalized approximate metric element, inclusive of both post-post-Newtonian contributions and a gravitomagnetic field. Following Fermat's principle and standard hypotheses, I derive the time delay function and deflection angle caused by an isolated mass distribution. Several astrophysical systems are considered. In most of the cases, the gravitomagnetic correction offers the best perspectives for an observational detection. Actual measurements distinguish only marginally different metric theories from each other.

VSP traveltime inversion for anisotropy in a buried layer

ABSTRACTWe present a method for calculating the anisotropy parameter of a buried layer by inverting the total traveltimes of direct arrivals travelling from a surface source to a well‐bore receiver in a vertical seismic profiling (VSP) geometry. The method assumes two‐dimensional media. The medium above the layer of interest (and separated from it by a horizontal interface) can exhibit both anisotropy and inhomogeneity. Both the depth of the interface as well as the velocity field of the overburden are assumed to be known. We assume the layer of interest to be homogeneous and elliptically anisotropic, with the anisotropy described by a single parameter χ. We solve the function describing the traveltime between source and receiver explicitly for χ. The solution is expressed in terms of known quantities, such as the source and receiver locations, and in terms of quantities expressed as functions of the single argument xr, which is the horizontal coordinate of the refraction point on the interface. In view of Fermat's principle, the measured traveltime T possesses a stationary value or, considering direct arrivals, a minimum value, . This gives rise to a key result ‐‐ the condition that the actual anisotropy parameter . Owing to the explicit expression , this result allows a direct calculation of in the layer of interest. We perform an error analysis and show this inverse method to be stable. In particular, for horizontally layered media, a traveltime error of one millisecond results in a typical error of about 20% in the anisotropy parameter. This is almost one order of magnitude less than the error inherent in the slowness method, which uses a similar set of experimental data. We conclude by detailing possible extensions to non‐elliptical anisotropy and a non‐planar interface.

Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge–Ampere type. We show that this problem can be solved as a variational problem within the framework of Monge–Kantorovich mass transfer problem. We develop the techniques used by the authors in their work “Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle” [Preprint, 2003], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.

Refraction holodiagrams and Snell's law

Summary The refraction holodiagram RHD is analyzed here with respect to the law of refraction. Particularly, we study the surface that exactly conjugates by refraction a virtual point source with a real image or conversely. By using the total optical path as a parameter we build a diagram that consists in a family of Descartes ovals of the apple type that contains the Pascal's limacon as a particular extreme case and the spherical surface with the Weierstrass points as another. These representations permit the straightforward application of Fermat's principle in the case of arbitrary refracting surfaces and show the shape of generalized Fresnel's zones in the intersections with any surface. Snell's law is applied to rays incident on the apple type surfaces to find out the conditions for exact conjugation. Sensitivity to optical path variations is also discussed. The RHD curves family can be represented in a Cartesian way where the ovals appear as equally spaced straight lines.

Formulating Fermat's principle for light traveling in negative refraction materials

The formulation of Fermat's principle for electromagnetic waves traveling in materials with a negative refractive index is refined. It is shown that a formulation in terms of the minimum (or extremum) of wave travel time between two points is not correct in general. The correct formulation involves the extremum of the total optical length, with the optical length for the wave propagation through left-handed materials taken to be negative.

Gravitational lensing by spinning and radially moving lenses

The effect of currents of mass on bending of light rays is considered in the weak field regime. Following Fermat's principle and the standard theory of gravitational lensing, we derive the gravito-magnetic correction to time delay function and deflection angle caused by a geometrically-thin lens. The cases of both rotating and shifting deflectors are discussed.

Derivation of seismic cone interval velocities utilizing forward modeling and the downhill simplex method

The seismic cone penetration test (SCPT) has proven to be a very valuable geotechnical tool in facilitating the determination of low strain (<10–4%) in situ compression (P) and shear (S) wave velocities. The P- and S-wave velocities are directly related to the soil elastic constants of Poisson's ratio, shear modulus, bulk modulus, and Young's modulus. The accurate determination of P- and S-wave velocities from the recorded seismic cone time series is of paramount importance to the evaluation of reliable elastic constants. Furthermore, since the shear and compression wave velocities are squared in deriving the elastic constants, small variations in the estimated velocities can cause appreciable errors. The standard techniques implemented in deriving SCPT interval velocities rely upon obtaining reference P- and S-wave arrival times as the probe is advanced into the soil profile. By assuming a straight ray travel path from the source to the SCPT seismic receiver and calculating the relative reference arrival time differences, interval SCPT velocities are obtained. The forward modeling – downhill simplex method (FMDSM) outlined in this paper offers distinct advantages over conventional SCPT velocity profile estimation methods. Some of these advantages consist of the allowance of ray path refraction, greater sophistication in interval velocity determination, incorporation of measurement weights, and meaningful interval velocity accuracy estimators.Key words: seismic cone penetration testing (SCPT), downhill simplex method (DSM), forward modeling, Fermat's principle, weighted least squares (l2 norm), cost function.

Three-dimensional Travel-time Calculation Based on Fermat's Principle

We present a new travel-time calculation method based on Fermat’s principle. In the method, travel times are recursively calculated on horizontal planes of increasing depth. For typical configurations of exploration geophysics, the distance between the planes is on the order of 100 m. The travel times on the first plane are calculated by connecting straight ray segments from the source point to the grid points on the plane and integrating the slowness along each segment. The times on the other planes are calculated by finding the minimum of the combination of the times on the plane above plus the additional time along segments connecting grid points on the two planes. The travel-time calculation method is designed for calculating either the first arrival times, or the time of the shortest travel path arrival. The method is extended to handle vertically transverse isotropic (VTI) media by an approach which increases the computing time only slightly. The algorithm is tested against synthetic examples for isotropic and VTI wave propagation.

Fermat potentials for nonperturbative gravitational lensing

The images of many distant galaxies are displaced, distorted and often multiplied by the presence of foreground massive galaxies near the line of sight; the foreground galaxies act as gravitational lenses. Commonly, the lens equation, which relates the placement and distortion of the images to the real source position in the thin-lens scenario, is obtained by extremizing the time of arrival among all the null paths from the source to the observer (Fermat's principle). We show that the construction of envelopes of certain families of null surfaces consitutes an alternative variational principle or version of Fermat's principle that leads naturally to a lens equation in a generic spacetime with any given metric. We illustrate the construction by deriving the lens equation for static asymptotically flat thin lens spacetimes. As an application of the approach, we find the bending angle for moving thin lenses in terms of the bending angle for the same deflector at rest. Finally we apply this construction to cosmological spacetimes (FRW) by using the fact they are all conformally related to Minkowski space.

Correction of geometric and refractive image distortions in optical coherence tomography applying Fermat�??s principle

We describe a methodology for quantitative image correction in OCT which includes procedures for correction of nonlinear axial scanning and non-telecentric scan patterns, as well as a novel approach for refraction correction in layered media based on Fermat's principle. The residual spatial error obtained in layered media with a fan-beam hand-held probe was reduced from several hundred micrometers to near the diffraction and coherence-length limits.