89 research outputs found
From Whitney Forms to Metamaterials: a Rigorous Homogenization Theory
A rigorous homogenization theory of metamaterials -- artificial periodic
structures judiciously designed to control the propagation of electromagnetic
waves -- is developed. All coarse-grained fields are unambiguously defined and
effective parameters are then derived without any heuristic assumptions. The
theory is an amalgamation of two concepts: Smith & Pendry's physical insight
into field averaging and the mathematical framework of
Whitney-Nedelec-Bossavit-Kotiuga interpolation. All coarse-grained fields are
defined via Whitney forms and satisfy Maxwell's equations exactly. The new
approach is illustrated with several analytical and numerical examples and
agrees well with the established results (e.g. the Maxwell-Garnett formula and
the zero cell-size limit) within the range of applicability of the latter. The
sources of approximation error and the respective suitable error indicators are
clearly identified, along with systematic routes for improving the accuracy
further. The proposed approach should be applicable in areas beyond
metamaterials and electromagnetic waves -- e.g. in acoustics and elasticity.Comment: 23 pages, 10 figure
Nonasymptotic Homogenization of Periodic Electromagnetic Structures: Uncertainty Principles
We show that artificial magnetism of periodic dielectric or metal/dielectric
structures has limitations and is subject to at least two "uncertainty
principles". First, the stronger the magnetic response (the deviation of the
effective permeability tensor from identity), the less accurate ("certain") the
predictions of any homogeneous model. Second, if the magnetic response is
strong, then homogenization cannot accurately reproduce the transmission and
reflection parameters and, simultaneously, power dissipation in the material.
These principles are general and not confined to any particular method of
homogenization. Our theoretical analysis is supplemented with a numerical
example: a hexahedral lattice of cylindrical air holes in a dielectric host.
Even though this case is highly isotropic, which might be thought as conducive
to homogenization, the uncertainty principles remain valid.Comment: 11 pages, 5 figure
Can photonic crystals be homogenized in higher bands?
We consider conditions under which photonic crystals (PCs) can be homogenized
in the higher photonic bands and, in particular, near the -point. By
homogenization we mean introducing some effective local parameters
and that describe reflection, refraction
and propagation of electromagnetic waves in the PC adequately. The parameters
and can be associated with a hypothetical
homogeneous effective medium. In particular, if the PC is homogenizable, the
dispersion relations and isofrequency lines in the effective medium and in the
PC should coincide to some level of approximation. We can view this requirement
as a necessary condition of homogenizability. In the vicinity of a
-point, real isofrequency lines of two-dimensional PCs can be close to
mathematical circles, just like in the case of isotropic homogeneous materials.
Thus, one may be tempted to conclude that introduction of an effective medium
is possible and, at least, the necessary condition of homogenizability holds in
this case. We, however, show that this conclusion is incorrect: complex
dispersion points must be included into consideration even in the case of
strictly non-absorbing materials. By analyzing the complex dispersion relations
and the corresponding isofrequency lines, we have found that two-dimensional
PCs with and symmetries are not homogenizable in the higher
photonic bands. We also draw a distinction between spurious -point
frequencies that are due to Brillouin-zone folding of Bloch bands and "true"
-point frequencies that are due to multiple scattering. Understanding
of the physically different phenomena that lead to the appearance of spurious
and "true" -point frequencies is important for the theory of
homogenization.Comment: Accepted in this form to Phys. Rev. B. Small addition in Sec.V
(Discussion) relative to previous version. The title to appear in PRB has
been changed to "Applicability of effective medium description to photonic
crystals in higher bands: Theory and numerical analysis" per the journal
policy not to print titles in the form of question
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