scholarly journals DIFFRACTION OF BLOCH WAVE PACKETS FOR MAXWELL'S EQUATIONS

2013 ◽  
Vol 15 (06) ◽  
pp. 1350040 ◽  
Author(s):  
GRÉGOIRE ALLAIRE ◽  
MARIAPIA PALOMBARO ◽  
JEFFREY RAUCH
Keyword(s):  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Approximate Solutions ◽  
Bloch Wave ◽  
System Structure ◽  
Periodic Medium ◽  
Scalar Wave ◽  
Infinite Dimensional

We study, for times of order 1/h, solutions of Maxwell's equations in an [Formula: see text] 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.

scholarly journals Internal stabilization of Maxwell's equations in heterogeneous media

2005 ◽  
Vol 2005 (7) ◽  
pp. 791-811 ◽  
Author(s):  
Serge Nicaise ◽  
Cristina Pignotti
Keyword(s):  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Heterogeneous Media ◽  
Space Variable ◽  
Smooth Boundary ◽  
Variable Coefficients ◽  
Bounded Region ◽  
Scalar Wave ◽  
Internal Stabilization

We consider the internal stabilization of Maxwell's equations with Ohm's law with space variable coefficients in a bounded region with a smooth boundary. Our result is mainly based on an observability estimate, obtained in some particular cases by the multiplier method, a duality argument and a weakening of norm argument, and arguments used in internal stabilization of scalar wave equations.

scholarly journals Two basic systems of maxwell’s equations in a rotating frame: application in theory of ring laser gyro

2020 ◽  
pp. 64-73
Author(s):  
Evgen Bondarenko
Keyword(s):  
Ring Laser ◽  
Electromagnetic Waves ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Rotating Frame ◽  
Cylindrical Coordinates ◽  
Laser Gyro ◽  
Component Form ◽  
Velocity Approximation

In the paper, using a linear in angular velocity approximation, two basic well-known systems of Maxwell’s equations in a uniformly rotating frame of reference are considered. The first system of equations was first obtained in the work [L. I. Schiff, Proc. Natl. Acad. Sci. USA 25, 391 (1939)] on the base of use of the formalism of the theory of general relativity, and the second one – in the work [W. M. Irvine, Physica 30, 1160 (1964)] on the base of use of the method of orthonormal tetrad in this theory. In the paper, in the approximation of plane waves, these two vectorial systems of Maxwell’s equations are simplified and rewritten in cylindrical coordinates in scalar component form in order to find the lows of propagation of transversal components of electromagnetic waves in a circular resonator of ring laser gyro in the case of its rotation about sensitivity axis. On the base of these two simplified systems of Maxwell’s equations, the well-known wave equation and its analytical solutions for the named transversal components are obtained. As a result of substitution of these solutions into the first and second simplified systems of Maxwell’s equations, it is revealed that they satisfy only the second one.  On this basis, the conclusion is made that the second system of Maxwell’s equations is more suitable for application in the theory of ring laser gyro than the first one.

WAVE AND MAXWELL'S EQUATIONS IN CARNOT GROUPS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250032 ◽  
Author(s):  
BRUNO FRANCHI ◽  
MARIA CARLA TESI
Keyword(s):  
Vector Potential ◽  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Higher Order ◽  
Carnot Groups ◽  
New Class ◽  
Euclidean Setting

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

Static and propagating exponential solutions of Maxwell’s equations for crystals

1992 ◽  
Vol 438 (1904) ◽  
pp. 485-510
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Electric Permittivity ◽  
Complex Vector ◽  
Time Harmonic ◽  
Inhomogeneous Plane ◽  
Isotropic Media ◽  
Special Case ◽  
Inhomogeneous Plane Waves

Solutions of Maxwell’s equations are considered for anisotropic media for which the electric permittivity κ and magnetic permeability µ are assumed to be arbitrary real positive definite symmetric second order tensors. The propagation of time-harmonic electromagnetic inhomogeneous plane waves or ‘propagating exponential solutions’ in such media has been presented previously. These solutions were systematically obtained by prescribing an ellipse - the directional ellipse associated with a bivector (complex vector) C - and finding the corresponding slowness bivectors. Here, it is shown that for some prescribed directional ellipses, not only propagating exponential solutions (PES), but also static exponential solutions (SES), may be obtained. There are a variety of possibilities. For example, for one choice of directional ellipse it is found that two SES and one PES are possible, whereas, for some other choices, only one SES or only one PES is possible. By a systematic use of bivectors and their associated ellipses, all the possible SES and PES are classified. To complete the classification it is necessary to examine special elliptical sections of the ellipsoids associated with the tensors κ , µ , κ -1 , µ -1 . In particular, sections by planes orthogonal to special directions called ‘generalized optic axes’ and ‘generalized ray axes’ play a major role. These axes reduce to the standard optic axes and ray axes in the special case of magnetically isotropic media.

scholarly journals Extension of the electrodynamics in the presence of the axion and dark photon

2019 ◽  
Vol 34 (03n04) ◽  
pp. 1950012 ◽  
Author(s):  
Fa Peng Huang ◽  
Hye-Sung Lee
Keyword(s):  
Magnetic Field ◽  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Dark Photon

We present the extended electrodynamics in the presence of the axion and dark photon. We derive the extended versions of Maxwell’s equations and dark Maxwell’s equations (for both massive and massless dark photons) as well as the wave equations. We discuss the implications of this extended electrodynamics including the enhanced effects in the particle conversions under the external magnetic or dark magnetic field. We also discuss the recently reported anomaly in the redshifted 21 cm spectrum using the extended electrodynamics.

Exact solutions of the three-dimensional scalar wave equation and Maxwell's equations from the approximate solutions in the wave zone through the use of the Radon transform

1989 ◽  
Vol 422 (1863) ◽  
pp. 351-365 ◽  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Radon Transform ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Three Dimensional ◽  
Asymptotic Solutions ◽  
Scalar Wave ◽  
Wave Zone ◽  
One Dimensional

In our earlier paper we have shown that the solutions of both the three-dimensional scalar wave equation, which is also the three-dimensional acoustic equation, and Maxwell’s equations have forms in the wave zone, which, except for a factor 1/ r , represent one-dimensional wave motions along straight lines through the origin. We also showed that it is possible to reconstruct the exact solutions from the asymptotic forms. Thus we could prescribe the solutions in the wave zone and obtain the exact solutions that would lead to them. In the present paper we show how the exact solutions can be obtained from the asymptotic solutions and conversely, through the use of a refined Radon transform, which we introduced in a previous paper. We have thus obtained a way of obtaining the exact three-dimensional solutions from the essentially one-dimensional solutions of the asymp­totic form entirely in terms of transforms. This is an alternative way to obtaining exact solutions in terms of initial values through the use of Riemann functions. The exact solutions that we obtain through the use of the Radon transform are causal and therefore physical solutions. That is, these solutions for time t > 0 could have been obtained from the initial value problem by prescribing the solution and its time-derivative, in the acoustic case, and the electric and magnetic fields, in the case of Maxwell’s equations, at time t = 0. The role of time in the relation between the exact solutions and in the asymptotic solutions is made very explicit in the present paper.

Sign in / Sign up

Export Citation Format

Share Document