50 research outputs found
Speed-of-light limitations in passive linear media
We prove that well-known speed of light restrictions on electromagnetic
energy velocity can be extended to a new level of generality, encompassing even
nonlocal chiral media in periodic geometries, while at the same time weakening
the underlying assumptions to only passivity and linearity of the medium
(either with a transparency window or with dissipation). As was also shown by
other authors under more limiting assumptions, passivity alone is sufficient to
guarantee causality and positivity of the energy density (with no thermodynamic
assumptions). Our proof is general enough to include a very broad range of
material properties, including anisotropy, bianisotropy (chirality),
nonlocality, dispersion, periodicity, and even delta functions or similar
generalized functions. We also show that the "dynamical energy density" used by
some previous authors in dissipative media reduces to the standard Brillouin
formula for dispersive energy density in a transparency window. The results in
this paper are proved by exploiting deep results from linear-response theory,
harmonic analysis, and functional analysis that had previously not been brought
together in the context of electrodynamics.Comment: 19 pages, 1 figur
On the Spectral Theory of Linear Differential-Algebraic Equations with Periodic Coefficients
In this paper, we consider the spectral theory of linear
differential-algebraic equations (DAEs) for periodic DAEs in canonical form,
i.e., \begin{equation*}
J \frac{df}{dt}+Hf=\lambda Wf, \end{equation*} where is a constant
skew-Hermitian matrix that is not invertible, both and
are -periodic Hermitian -matrices with Lebesgue
measurable functions as entries, and is positive semidefinite and
invertible for a.e. (i.e., Lebesgue almost everywhere). Under
some additional hypotheses on and , called the local index-1 hypotheses,
we study the maximal and the minimal operators and , respectively,
associated with the differential-algebraic operator
, both treated as an unbounded operators
in a Hilbert space of weighted square-integrable
vector-valued functions. We prove the following: (i) the minimal operator
is a densely defined and closable operator; (ii) the maximal operator
is the closure of ; (iii) is a self-adjoint operator on
with no eigenvalues of finite multiplicity, but may have
eigenvalues of infinite multiplicity. As an important application, we show that
for 1D photonic crystals with passive lossless media, Maxwell's equations for
the electromagnetic fields become, under separation of variables, periodic DAEs
in canonical form satisfying our hypotheses so that our spectral theory applies
to them (a primary motivation for this paper).Comment: 47 page
Effective operators and their variational principles for discrete electrical network problems
Using a Hilbert space framework inspired by the methods of orthogonal
projections and Hodge decompositions, we study a general class of problems
(called Z-problems) that arise in effective media theory, especially within the
theory of composites, for defining the effective operator. A new and unified
approach is developed, based on block operator methods, for obtaining solutions
of the Z-problem, formulas for the effective operator in terms of the Schur
complement, and associated variational principles (e.g., the Dirichlet and
Thomson minimization principles) that lead to upper and lower bounds on the
effective operator. In the case of finite-dimensional Hilbert spaces, this
allows for a relaxation of the standard hypotheses on positivity and
invertibility for the classes of operators usually considered in such problems,
by replacing inverses with the Moore-Penrose pseudoinverse. As we develop the
theory, we show how it applies to the classical example from the theory of
composites on the effective conductivity in the periodic conductivity problem
in the continuum (2d and 3d) under the standard hypotheses. After that, we
consider the following three important and diverse examples of discrete
electrical network problems in which our theory applies under the relaxed
hypotheses. First, an operator-theoretic reformulation of the discrete
Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear
graph is given and used to relate the DtN map to the effective operator of an
associated Z-problem.\ Second, we show how the classical effective conductivity
of an electrical network on a finite linear graph is essentially the effective
operator of an associated Z-problem. Finally, we consider electrical networks
on periodic linear graphs and develop a discrete analog to classical example of
the periodic conductivity equation and effective conductivity in the continuum.Comment: 33 pages, 4 figure