Narrow Results

The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in ℝ³ to describe the reflection of rays off a surface. This can be viewed as a parametric approach to geometric optics which has many possible applications. Recently, Jaffe and Scardicchio have developed a geometric optics approximation to the Casimir effect and the main purpose of this paper is to show that the quantities involved can be easily computed by this complex formalism. To illustrate this, we determine explicitly and in closed form the geometric optics approximation of the Casimir force between two non-parallel plates. By making one of the plates finite, we regularize the divergence that is caused by the intersection of the planes. In the parallel plate limit, we prove that our expression reduces to Casimir's original result.

We consider the stationary metrics that have both the black hole and the ergoregion. The class of such metric contains, in particular, the Kerr metric. We study the Cauchy problem with highly oscillatory initial data supported in a neighborhood inside the ergoregion with some initial energy $E_0$. We prove that when the time variable $x_0$ increases this solution splits into two parts: one with the negative energy $-E_1$ ending at the event horizon in a finite time, and the second part, with the energy $E_2=E_0+E_1>E_0$, escaping, under some conditions, to the infinity when $x_0\to +\infty$. Thus we get the superradiance phenomenon. In the case of the Kerr metric the superradiance phenomenon is “short-lived”, since both the solutions with positive and negative energies cross the outer event horizon in a finite time (modulo $O(\frac{1}{k})$) where $k$ is a large parameter. We show that these solutions end on the singularity ring in a finite time. We study also the case of naked singularity.

Considering plane gravitational waves propagating through flat spacetime, it is shown that curvatures experienced both in the starting point and during their arrival at the earth can cause a considerable shift in the frequencies as measured by earth and space-based detectors. In particular for the case of resonant bar detectors this shift can cause noise-filters to smother the signal.

In these expository notes we explain the role of geometric optics in wave propagation on domains or manifolds with corners or edges. Both the propagation of singularities, which describes where solutions of the wave equation may be singular, and the diffractive improvement under non-focusing hypotheses, which states that in certain places the diffracted wave is more regular than a priori expected, is described. In addition, the wave equation on differential forms with natural boundary conditions, which in particular includes a formulation of Maxwell's equations, is studied.

This article contains a review of the brachistochrone problem as initiated by Johann Bernoulli in 1696–1697. As is generally known, the cycloid forms the solutions to this problem. We follow Bernoulli's optical solution based on the Fermat principle of least time and later rephrase this in terms of Hamilton's 1828 paper. Deliberately an anachronistic style is maintained throughout. Hamilton's solution recovers the cycloid in a way that is reminiscent of how Newton's mathematical principles imply Kepler's laws.

The aim of this paper is to study the reflection-transmission of geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarisation. The framework is both that of Donnat and Williams since we consider dispersive media and profiles with hyperbolic (imaginary) phases and elliptic phases (complex with non-null real part). We first give hypothesis close to the Maxwell equation. Then we introduce a decomposition for both profile into boundary (tangential) and normal part and we solve the so-called "microscopic" equation of the small scales for each boundary frequency. Then we show that the non-linearities generate harmonics which interact at the boundary and generate new resonant profiles with harmonic tangential frequency. Lastly we make a WKB expansion at any order and give a precise description of the correctors.

This paper follows the work of Colin–Gallice–Laurioux⁶ in which a new model generalizing the Schrödinger (NLS) model of the diffractive optics is derived for the laser propagation in nonlinear media. In particular, it provides good approximate solutions of the Maxwell–Lorentz system for highly oscillating initial data with broad spectrum. In real situations one is given boundary data. We propose to derive a similar evolution model but in the variable associated to the direction of propagation. However, since the space directions for the Maxwell equations are not hyperbolic, the boundary problem is ill-posed and one needs to apply a cutoff defined in the Fourier space, selecting those frequencies for which the operator is hyperbolic. The model we obtain is nearly L² conservative on its domain of validity.

We then give a justification of the derivation. For this purpose we introduce a related well-posed initial boundary value problem. Finally, we perform numerical computations on the example of Maxwell with Kerr nonlinearity in some cases of short or spectrally chirped data where our model outperforms the Schrödinger one.

Recently (Int. J. Mod. Phys. D 27 (2018) 1847025) an interesting property of closed light rings in Kerr black holes has been noticed. We explain its origin and derive a slightly more general result.

We study, for times of order 1/h, solutions of Maxwell's equations in an modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite-dimensional kernel. The system structure requires many innovations.

In this paper, we introduce new frame and new transformation associated with combined Korteweg–de Vries–modified Korteweg–de Vries (KdV–mKdV) equation, the mKdV and the KdV equations with respect to the null Cartan frame in Minkowski 3-space. Direction of the state of polarization of monochromatic linear polarized light wave traveling in the optic fiber is given by the direction of electric field vector. We show that the electric field vector $\vec{E}$ can be expressed as a linear combination of null Cartan frame apparatus using this new transformation in Minkowski 3-space. Later, we study rotation of the polarization plane along optic fiber according to null Cartan frame and Frenet frame of pseudo-null curve in Minkowski 3-space. Finally, we obtain Lorentz force equations via null and pseudo null electromagnetic curves.

In this companion paper to our study of amplification of wavetrains J.-F. Coulombel, O. Guès and M. Williams, Semilinear geometric optics with boundary amplification, Anal. PDE7(3) (2014) 551–625, we study weakly stable semilinear hyperbolic boundary value problems with pulse data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency in the hyperbolic region. As a consequence of this degeneracy there is again an amplification phenomenon: outgoing pulses of amplitude O(ε²) and wavelength ε give rise to reflected pulses of amplitude O(ε), so the overall solution has amplitude O(ε). Moreover, the reflecting pulses emanate from a radiating pulse that propagates in the boundary along a characteristic of the Lopatinskii determinant. In the case of N × N systems considered here, a single outgoing pulse produces on reflection a family of incoming pulses traveling at different group velocities. Unlike wavetrains, pulses do not interact to produce resonances that affect the leading order profiles. However, pulse interactions do affect lower-order profiles and so these interactions have to be estimated carefully in the error analysis. Whereas the error analysis in the wavetrain case dealt with small divisor problems by approximating periodic profiles by trigonometric polynomials (which amounts to using a high frequency cutoff), in the pulse case we approximate decaying profiles with nonzero moments by profiles with zero moments (a low frequency cutoff). Unlike the wavetrain case, we are now able to obtain a rate of convergence in the limit describing convergence of approximate to exact solutions.

We prove energy estimates for exact solutions to a class of linear, weakly stable, first-order hyperbolic boundary problems with “large”, oscillatory, zeroth-order coefficients, that is, coefficients whose amplitude is large, $O(1)$, compared to the wavelength of the oscillations, $O(𝜖)$. The methods that have been used previously to prove useful energy estimates for weakly stable problems with oscillatory coefficients (e.g. simultaneous diagonalization of first-order and zeroth-order parts) all appear to fail in the presence of such large coefficients. We show that our estimates provide a way to “justify geometric optics”, that is, a way to decide whether or not approximate solutions, constructed for example by geometric optics, are close to the exact solutions on a time interval independent of $𝜖$. Systems of this general type arise in some classical problems of “strongly nonlinear geometric optics” coming from fluid mechanics. Special assumptions that we make here do not yet allow us to treat the latter problems, but we believe the present analysis will provide some guidance on how to attack more general cases.

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of Bérenger. We prove that the Bérenger and closely related layers define well-posed transmission problems in great generality. When the Bérenger method or one of its close relatives is well-posed, perfect matching is proved. The proofs use the energy method, Fourier–Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the Bérenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in Bérenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of Bérenger's algorithm for Maxwell's equations, our favorite choice leads to a strongly well-posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence. Open problems are indicated throughout.

Geometric optics approximation is sufficient to describe the effects in the near-Earth environment. In this framework Faraday rotation is purely a reference frame (gauge) effect. However, it cannot be simply dismissed. Establishing local reference frame with respect to some distant stars leads to the Faraday phase error between the ground station and the spacecraft of the order of 10⁻¹⁰ in the leading post-Newtonian expansion of the Earth’s gravitational field. While the Wigner phase of special relativity is of the order 10⁻⁴–10⁻⁵. Both types of errors can be simultaneously mitigated by simple encoding procedures. We also present briefly the covariant formulation of geometric optic correction up to the subleading order approximation, which is necessary for the propagation of electromagnetic/gravitational waves of large but finite frequencies. We use this formalism to obtain a closed form of the polarization dependent correction of the light ray trajectory in the leading order in a weak spherically symmetric gravitational field.