Simple wave solutions for the Maxwell equations in bianisotropic, nonlinear media, with application to oblique incidence

Sjöberg, Daniel

doi:10.1016/s0165-2125(00)00039-1

Cited by 6 publications

(13 citation statements)

References 32 publications

(27 reference statements)

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“…This is an eigenvalue problem with n (or 1/n) as the eigenvalue, and (E 0 , H 0 ) as the eigenvector [22]. More explicitly, the existence of plane waves requires the following determinant condition to hold:…”

Section: Applicationsmentioning

confidence: 99%

Low-Frequency Scattering Analysis and Homogenisation of Split-Ring Elements

Stén¹,

Sjöberg²

2011

PIER B

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Abstract-A key structure in so-called metamaterial mediums is the elementary split-ring resonator. We consider in this paper the lowfrequency electromagnetic scattering by a split-ring particle modelled as a perfectly conducting wire ring, furnished with a narrow gap, and derive analytical solutions for the electric and magnetic dipole moments for different kinds of incidence and polarisation in the quasistatic approximation. Through a vectorial homogenisation process, the expressions discovered for the dipole moments and the related polarisability dyadics are linked with the macroscopic constitutive equations for the medium. We further show that the condition for resonance of a medium consisting of simple split-rings cannot be achieved by means of the given quasi-static terms without violating the underlying assumptions of homogenisation. Nevertheless, the results are applicable for sparse medium of rings, and we derive numerical guidelines for the applicability with some examples of the effect of the considered split-ring medium on electromagnetic wave propagation.

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Section: Applicationsmentioning

confidence: 99%

Low-Frequency Scattering Analysis and Homogenisation of Split-Ring Elements

Stén¹,

Sjöberg²

2011

PIER B

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“…We use the scaled time t = c 0 t SI , where c 0 = 1/ √ 0 µ 0 is the speed of light in vacuum, and the constants 0 and µ 0 are the permittivity and permeability of free space, respectively. The sixvector notation from [28,8], i.e.,…”

Section: The Quasi-linear Maxwell Equationsmentioning

confidence: 99%

“…The source free version of this system has been extensively studied in [28], where it is shown that the equations in general support two waves, the ordinary and the extraordinary wave, each with its own refractive index. Due to the quasi-linearity, the system (6) may exhibit shock solutions, i.e., even if we give smooth data, the solution becomes discontinuous in finite time.…”

Section: Sjöbergmentioning

confidence: 99%

On Uniqueness and Continuity for the Quasi-Linear, Bianisotropic Maxwell Equations, Using an Entropy Condition

Sjöberg¹

2007

PIER

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Abstract-The quasi-linear Maxwell equations describing electromagnetic wave propagation in nonlinear media permit several weak solutions, which may be discontinuous (shock waves). It is often conjectured that the solutions are unique if they satisfy an additional entropy condition. The entropy condition states that the energy contained in the electromagnetic fields is irreversibly dissipated to other energy forms, which are not described by the Maxwell equations. We use the method employed by Kružkov to scalar conservation laws to analyze the implications of this additional condition in the electromagnetic case, i.e., systems of equations in three dimensions. It is shown that if a cubic term can be ignored, the solutions are unique and depend continuously on given data.

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“…We use the scaled time t = c 0 t SI , where is the speed of light in vacuum, and the constants ϵ 0 and μ 0 are the permittivity and permeability of free space, respectively. The six‐vector notation from the works of Sjöberg [2000] and Gustafsson [2000], i.e., enables us to write the source free Maxwell equations in the compact form In this paper we treat the six vectors as column vectors, i.e., we write the scalar product as e T d = Σ i = 1 6 e i d i . This is merely for notational convenience and does not capture the full mathematical structure, which is not needed here.…”

Section: The Maxwell Equations Constitutive Relations and Entropy Cmentioning

confidence: 99%

“…The initial value problem for the Maxwell equations with an instantaneously reacting constitutive model is and since d ′( e ) is positive definite and symmetric, this is by definition a quasi‐linear, symmetric, hyperbolic system of partial differential equations [ Taylor , 1996, p. 360]. This system has been extensively studied by Sjöberg [2000], where it is shown that the equations in general support two waves, the ordinary and the extraordinary wave, each with its own refractive index.…”

Section: The Maxwell Equations Constitutive Relations and Entropy Cmentioning

confidence: 99%

Shock structure for electromagnetic waves in bianisotropic, nonlinear materials

Sjöberg

2003

Radio Science

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Shock waves are discontinuous solutions to quasi‐linear partial differential equations and can be studied through a singular perturbation known as the vanishing viscosity technique. The vanishing viscosity method is a means of smoothing the shock, which we use to study the case of electromagnetic waves in bianisotropic materials. We derive the conditions arising from this smoothing procedure for a traveling wave, and the waves are classified as fast, slow, or intermediate shock waves.

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Simple wave solutions for the Maxwell equations in bianisotropic, nonlinear media, with application to oblique incidence

Cited by 6 publications

References 32 publications

Low-Frequency Scattering Analysis and Homogenisation of Split-Ring Elements

Low-Frequency Scattering Analysis and Homogenisation of Split-Ring Elements

On Uniqueness and Continuity for the Quasi-Linear, Bianisotropic Maxwell Equations, Using an Entropy Condition

Shock structure for electromagnetic waves in bianisotropic, nonlinear materials

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