Consuelo Bellver-Cebreros

Caustics and the Legendre transform

Abstract An application of the Legendre transform within the field of optics has been studied. The Legendre transform is used to obtain the caustics of light rays reflected off a surface profile and, reciprocally, to get the reflecting profile once the equation of caustic is given.

Amphoteric refraction at the isotropic–anisotropic biaxial media interface: an alternative treatment

Amphoteric refraction of light (refraction that can be positive or negative depending on the angle of incidence) at the interface between isotropic and anisotropic biaxial media is analysed by means of an alternative method previously developed by the authors. Unlike left-handed materials (LHMs), negative (or abnormal) refraction involving anisotropic media is only due to the intrinsic properties of the media and is only exhibited by non-collinear rays (rays that do not follow an isotropic behaviour). Moreover, the axes of wavevector ellipsoids must be rotated with respect to the normal to the incidence plane. In this paper, only planes of incidence lying normal to a principal axis of the dielectric tensor are considered, and consequently comparison between traditional and alternative methods is possible because both geometrical constructions are plane. First, a very simple procedure that allows us to find the direction of wavevector k from the knowledge of non-collinear (n.c.) ray direction is outlined together with the solution of the dual problem: to find the n.c. ray direction from the k one. Then, the abnormal refraction phenomenon is studied and the equivalence between traditional and alternative methods is clearly stated. Finally, two graphical constructions to find the critical angle of incidence and the maximum angle of negative refraction are also proposed.

Eikonal equation from continuum mechanics and analogy between equilibrium of a string and geometrical light rays

This work of didactic character on geometrical optics consists of two parts, whose main link is the use of methods associated with continuum physics. First, the formalism of continuum physics is applied in order to derive the eikonal equation from the study of propagation of discontinuities in electromagnetic fields in such a way that the eikonal equation is an exact derivation from Maxwell equations. The results obtained are well known from the works of Luneburg [Mathematical Theory of Optics (California U.P., Berkeley, 1964)], although the method used is new and efficient and provides a good occasion to use the continuum physics beyond its standard applications. Second, from the identity between the differential equation of light rays and the equilibrium equation of a flexible and inextensible string subjected to a conservative force, the analogy between both physical models is inferred. To illustrate this analogy, two applications in the realms of mechanics and optics are shown. The obtained results in...

Eikonal equation, alternative expression of Fresnel's equation and Möhr's construction in optical anisotropic media

Abstract A new approach to the study of propagation of plane light waves in anisotropic media, based on the local properties of second order symmetric tensors is described. First, the eikonal equation for anisotropic media is obtained following an alternative procedure based on the propagation of field discontinuities. It is emphasized that eikonal equation is an exact derivation from Maxwell equations. Also, an alternative expression for Fresnels equation is introduced. The main part of this paper deals with the use of Mohrs plane graphical construction in optics of anisotropic media. This construction, borrowed from mechanics, is proposed and justified in order to analyze (in the environment of a generic point) the propagation of plane light waves in both uniaxial and biaxial media. Mohrs construction is plotted on a plane and allows qualitative and quantitative analysis, in contrast with three-dimensional representations. This graphical method presents many advantages especially when dealing with uniaxial media or when one component of the unit wave vector vanishes in biaxial media. Likewise, the inverse problem of Fresnels equation is also regarded: if phase velocity is known, the directions of propagation to which this velocity belongs can also be found.

Obtention of meridian caustics and catacaustics by means of stigmatic approximating surfaces

We present an analysis of the problem of determination of meridian caustics and catacaustics by using the method of minimization of functionals. The well known result that a conic is an stigmatic profile for its foci is then reobtained for both refracting and refracting profiles. The original proposed method is to approximate each point P of the reflecting (or refracting) profile by an appropriated conic, whose focus is precisely the caustic point corresponding to P. The equations obtained for caustics and catacaustics go beyond some difficulties inherent to the classical method of envelopes and allow us to obtain caustics and catacaustics of non-analytical surfaces, even with irregularities and perceptible or imperceptible cracks.

Internal conical refraction in biaxial media and graphical plane constructions deduced from Möhr’s method

Abstract Following a recent alternative method based on the local properties of the permittivity tensor and on Mohr’s plane graphical construction, propagation of locally plane light waves in biaxial media are studied. The method is applied to re-obtain and synthesize in a very simple and compact manner some geometrical properties of a monochromatic locally plane wave in these media. Likewise, detailed studies for the case when the wave vector lies in the plane of optic axes or is close to these axes have been carried out. Then, phenomenon of internal conical refraction is easily explained and the aperture angle of internal conical refraction can be obtained graphically by means of simple procedures from the known eigenvalues of the dielectric tensor. Then, two graphical constructions to determine the ray direction are also proposed. The former is used when the electric field direction is known. The latter follows from the knowledge of the direction of propagation and uses an alternative tensor based on the relative dielectric permittivity one.

Analogies between the torque-free motion of a rigid body about a fixed point and light propagation in anisotropic media

An apparently unnoticed analogy between the torque-free motion of a rotating rigid body about a fixed point and the propagation of light in anisotropic media is stated. First, a new plane construction for visualizing this torque-free motion is proposed. This method uses an intrinsic representation alternative to angular momentum and independent of the modulus of angular velocity ω. The equivalence between this plane construction and the well-known Poinsots three-dimensional graphical procedure is also shown. From this equivalence, analogies have been found between the general plane wave equation (relation of dispersion) in anisotropic media and basic equations of torque-free motion of a rigid body about a fixed point. These analogies allow reciprocal transfer of results between optics and mechanics and, as an example, reinterpretation of the internal conical refraction phenomenon in biaxial media is carried out. This paper is intended as an interdisciplinary application of analogies for students and teachers in the context of intermediate physics courses at university level.

Geometrical connection between catacaustics and kinematics of planar motion of a rigid solid

Unnoticed and hidden optomechanical analogies between kinematics of planar motion of a rigid solid and catacaustics generated by mirror reflection on smooth profiles in geometrical optics are discussed. A concise and self-consistent theory is developed, which intends to explain and clarify many partial aspects covered by the literature.

The Eikonal Equation for Metamaterials Optics from a Moving Boundary Variational Principle

The eikonal equation for inhomogeneous anisotropic metamaterials with equal relative permittivity and permeability tensors ( ε̄(r) = μ̄(r)) is derived from a free boundary variational principle. An original approach is proposed considering the wavefront as a moving discontinuity surface in an extended continuous media described by the Lagrangian density of electromagnetic fields. The eikonal equation arises as natural (non prescribed) boundary conditions for variational problems.

On Inhomogeneous Metamaterials Media: a New Alternative Method for Analysis of Electromagnetic Fields Propagation

The analysis of waves propagation in homogeneous anisotropic media constitutes a classical topic in every field of science and has been preferentially discussed using locally plane waves. Specific physical quantities and their behaviour laws are really what make the difference. Although the use of Fourier transform enables an approach formally analogous to that of plane waves in linear evolution equations, its application to constitutive equations of inhomogeneous media involves cumbersome convolution products that mask the solution. This paper proposes a polar representation (amplitude and phase) of electromagnetic fields, that appears to be more suitable and provides two sets of equations that can be easily decoupled, reducing the problem to the superposition of two simpler ones. The procedure is based upon the following steps: a) The identification of dispersion equation with Hamilton-Jacobi equation yields the evolution laws of rays and/or wave-fronts. b) From the knowledge of tensor ¯ e(¯ r )a t any point ¯ r of the wave front (or the ray), the use of the intrinsic character (conjugation relations) of fields, introduced by the authors in a previous work, together with ray velocity or phase gradient (found in the first step) the remaining fields are immediately obtained.