scholarly journals A Fiber Bundle Space Theory of Nonlocal Metamaterials

2020 ◽  
Author(s):  
Said Mikki
Keyword(s):  
Vector Bundle ◽  
Topological Structure ◽  
Electromagnetic Theory ◽  
Fiber Bundles ◽  
Dispersive Media ◽  
Proper Function ◽  
Space Theory ◽  
Infinite Dimensional ◽  
Spatially Dispersive ◽  
Special Case

It is proposed that spacetime is not the most proper space to describe metamaterials with nonlocality. Instead, we show that the most general and suitable configuration space for doing electromagnetic theory in nonlocal domains is a proper function-space infinite-dimensional (Sobolev) vector bundle, a special case of the general topological structure known as fiber bundles. It appears that this generalized space explains why nonlocal metamaterials cannot have unique EM boundary conditions at interfaces involving spatially dispersive media.

scholarly journals A Fiber Bundle Space Theory of Nonlocal Metamaterials

2020 ◽  
Author(s):  
Said Mikki
Keyword(s):  
Vector Bundle ◽  
Topological Structure ◽  
Electromagnetic Theory ◽  
Fiber Bundles ◽  
Dispersive Media ◽  
Proper Function ◽  
Space Theory ◽  
Infinite Dimensional ◽  
Spatially Dispersive ◽  
Special Case

It is proposed that spacetime is not the most proper space to describe metamaterials with nonlocality. Instead, we show that the most general and suitable configuration space for doing electromagnetic theory in nonlocal domains is a proper function-space infinite-dimensional (Sobolev) vector bundle, a special case of the general topological structure known as fiber bundles. It appears that this generalized space explains why nonlocal metamaterials cannot have unique EM boundary conditions at interfaces involving spatially dispersive media.

Cherenkov radiation in spatially dispersive media

1981 ◽  
Vol 374 (1759) ◽  
pp. 531-541 ◽  
Keyword(s):  
Rest Frame ◽  
Fourier Integral ◽  
Dispersive Media ◽  
Relativistic Velocity ◽  
Isotropic Plasma ◽  
Exciton Transition ◽  
Circular Field ◽  
Longitudinal Plasma ◽  
Spatially Dispersive ◽  
Special Case

The Cherenkov fields of a proton, and a neutron, moving with a relativistic velocity in a spatially dispersive medium are studied in the rest frame of the particle. The model of the medium used is typical of the behaviour of a dielectric near an exciton transition, and includes as a special case a screening medium like an isotropic plasma. The Fourier integral for the field of a proton is shown to split up into three integrals, each of which is identical to that in an ordinary medium but for a weight factor dependent on the frequency of the Fourier component. Each of these integrals is associated with one mode of Cherenkov emission, with its own threshold. The motion of the charge gives rise to three coaxial diffuse circular field cones with an azimuthally symmetric intensity distribution. The output of photons in each mode is evaluated. The field and output of a relativistic neutron are also evaluated for different orientations of the magnetic moment of the neutron relative to the direction of motion. It is shown that there are only two cones in this case, consistent with the fact that magnetic sources cannot excite the longitudinal plasma mode in a medium which is spatially dispersive only in its electrical properties.

Pulse propagation in spatially dispersive media

1983 ◽  
Vol 27 (2) ◽  
pp. 1044-1052 ◽  
Author(s):  
Ashok Puri ◽  
Joseph L. Birman
Keyword(s):  
Pulse Propagation ◽  
Dispersive Media ◽  
Spatially Dispersive

The structure of norm Clifford algebras

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Hans Keller ◽  
Herminia Ochsenius
Keyword(s):  
Clifford Algebra ◽  
Classical Case ◽  
Canonical Representation ◽  
Inner Product ◽  
Infinite Products ◽  
Infinite Dimensional ◽  
Significant Step ◽  
Group Of Isometries ◽  
Inner Automorphisms ◽  
Special Case

AbstractOrthomodular Hilbertian spaces are infinite-dimensional inner product spaces (E, 〈·, ·〉) with the rare property that to every orthogonally closed subspace U ⊆ E there is an orthogonal projection from E onto U. These spaces, discovered about 30 years ago, are constructed over certain non-Archimedeanly valued, complete fields and are endowed with a non-Archimedean norm derived from the inner product. In a previous work [KELLER, H. A.—OCHSENIUS, H.: On the Clifford algebra of orthomodular spaces over Krull valued fields. In: Contemp. Math. 508, Amer. Math. Soc., Providence, RI, 2010, pp. 73–87] we described the construction of a new object, called the norm Clifford algebra C̃(E) associated to E. It can be considered a counterpart of the well-established Clifford algebra of a finite dimensional quadratic space. In contrast to the classical case, C̃(E) allows to represent infinite products of reflections by inner automorphisms. It is a significant step towards a better understanding of the group of isometries, which in infinite dimension is complex and hard to grasp.In the present paper we are concerned with the inner structure of these new algebras. We first give a canonical representation of the elements, and we prove that C̃ is always central. Then we focus on an outstanding special case in which C̃ is shown to be a division ring. Moreover, in that special case we completely describe the ideals of the corresponding valuation ring $$\mathcal{A}$$. It turns out, rather unexpectedly, that every left-ideal and every right-ideal of $$\mathcal{A}$$ is in fact bilateral.

scholarly journals Collapsing Cavities and Converging Shocks in Non-Ideal Materials

2019 ◽  
Vol 72 (4) ◽  
pp. 501-520 ◽  
Author(s):  
Zachary M Boyd ◽  
Emma M Schmidt ◽  
Scott D Ramsey ◽  
Roy S Baty
Keyword(s):  
Gas Dynamics ◽  
Nonlinear Eigenvalue Problem ◽  
Ideal Gas ◽  
Test Problems ◽  
Infinite Dimensional ◽  
Cavity Collapse ◽  
Scaling Solution ◽  
Metal Deformation ◽  
Special Case ◽  
The Ideal

Summary As modern hydrodynamic codes increase in sophistication, the availability of realistic test problems becomes increasingly important. In gas dynamics, one common unrealistic aspect of most test problems is the ideal gas assumption, which is unsuited to many real applications, especially those involving high pressure and speed metal deformation. Our work considers the collapsing cavity and converging shock test problems, showing to what extent the ideal gas assumption can be removed from their specification. It is found that while most materials simply do not admit simple (that is scaling) solutions in this context, there are infinite-dimensional families of materials which do admit such solutions. We characterize such materials, derive the appropriate ordinary differential equations and analyze the associated nonlinear eigenvalue problem. It is shown that there is an inherent tension between boundedness of the solution, boundedness of its derivatives and the entropy condition. The special case of a constant-speed cavity collapse is considered and found to be heuristically possible, contrary to common intuition. Finally, we give an example of a concrete non-ideal collapsing cavity scaling solution based on a recently proposed pseudo-Mie–Gruneisen equation of state.

Energy Band Models for Spatially Dispersive Dielectric Media

1975 ◽  
Vol 53 (19) ◽  
pp. 2095-2122 ◽  
Author(s):  
J. E. Sipe ◽  
J. Van Kranendonk
Keyword(s):  
Optical Properties ◽  
Degrees Of Freedom ◽  
Energy Levels ◽  
Equations Of Motion ◽  
Dispersive Media ◽  
Energy Bands ◽  
Macroscopic Theory ◽  
Mechanical Coupling ◽  
Radiation Theory ◽  
Spatially Dispersive

The effects of spatial dispersion on the optical properties of dielectric crystals, arising from the broadening of the molecular energy levels into energy bands by the intermolecular interaction, are discussed both in the microscopic and the macroscopic theory. The microscopic equations of motion for the internal degrees of freedom describing the molecular excitations are derived using semiclassical radiation theory, and the conditions are given under which a description in terms of only the dipole moment is possible. The corresponding macroscopic equations are derived and the nature of the boundary conditions and integral relations appearing in the theory are discussed. The characterization of spatially dispersive media as nonlocal is shown to be based on a misinterpretation of the meaning of the integral kernels relating to infinite media. The breakdown of the macroscopic theory due to the previously predicted onset of an antiresonant response is explicitly demonstrated for slab geometries for which rigorous solutions are given of both the macroscopic and the microscopic equations. Finally, we introduce a mechanical coupling varying exponentially with the intermolecular separation, which provides a two parameter model for the exciton bands and which prevents the proliferation of microscopic refractive indices occurring in other models. The exp model is shown to be useful to study the dependence of the optical properties for example on the effective mass and the width of an exciton band.

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