In this paper, we present recent developments in our modal expansion technique for electromagnetic structures with highly dispersive media and its application for unbounded geometries. The expansion formula, based on a simple version of Keldys’s theorem, make use of Dispersive Quasi-Normal Modes (DQNMs), also known as natural modes of photonic structures, obtained by solving spectral problems associated to the Maxwell's equations. Such structures can be defined very generally by their geometry (bounded or unbounded), and the electromagnetic properties of various media (permeability and permittivity can be dispersive, anisotropic, and even possibly non reciprocal). As an example, a dispersive benchmark case, a diffraction grating, made of a periodic slit array etched in a free-standing silver membrane, is presented.
We present a new method for the direct computation of the resonances associated with electromagnetic structures including media with highly dispersive permittivities. The FEM discretization of this problem leads to a generalized polynomial eigenvalue problem (PEP). We have developped a very general method to compute such modes and to limit the order of the PEP when the structure involves several dispersive media. As the accurate description of the dispersive permittivity in the form of a rational function of pulsation is a critical point in our method, we also describe a very general procedure to obtain a causal fit of the permittivity of materials from experimental data with very few parameters. Unlike other closed forms proposed in the literature, the particularity of this approach lies in its independence towards the material or frequency range at stake.
Our approach consists in finding the eigenmodes and the complex eigenfrequencies of structures using
a finite element method (FEM), that allows us to study mono- or bi-periodic gratings with a maximum
versatility : complex shaped patterns, with anisotropic and graded index material, under oblique incidence
and arbitrary polarization. In order to validate our method, we illustrate an example of a four layer
dielectric slab, and compare the results with a specific method that we have called tetrachotomy, which
gives us numerically the poles of the reflection coefficient (which corresponds to the eigenfrequencies of
the structure). To illustrate our method, we show the eigenvalues of one- and two-dimensional gratings.
The purpose of this paper is to investigate the scattering by a nonlinear crystal whose depth is about the wavelength
of the impinging field. More precisely, an infinite nonlinear slab is illuminated by an incident field which is the sum
of three plane waves of the same frequency, but with different propagation vectors and amplitudes, in such a way
that the resulting incident field is periodic. Moreover, the height of the slab is of the same order of the wavelength,
and therefore the so-called slowly varying envelope approximation cannot be used. In our approach we take into
account some retroactions of the scattered fields between them (for instance, we do not use the nondepletion of the
pump beam). As a result, a system of coupled nonlinear partial differential equations has to be solved. To do this,
the finite element method (FEM) associated with perfectly matched layers is well suited. Nevertheless, when using
the FEM, the sources have to be located in the meshed area, which is of course impossible when dealing with plane
waves. To get round this difficulty, the real incident field is simulated by a virtual field emitted by an appropriate
antenna located in the meshed domain and lying above the obstacle (here the slab).
In recent years, transformation optics has become a very active new field. It has been popularized through the idea
of J.B. Pendry that an invisibility cloak can be designed by transforming space and considering the corresponding
equivalent material properties. Indeed, it is a deep property of the Maxwell's equations that they are purely
topological (when written in the proper formalism) and that all the metric aspects can be encapsulated in the
electromagnetic material properties. A direct consequence is that any continuous transformation of space can be
encoded in an equivalent permittivity and permeability. In this paper, we discuss the meaning of transform optics
to show how global quantities defined by geometric integrals are in fact left invariant by the transformation.
Extending this principle beyond continuous transformations allows to design exotic optical devices such as the
invisibility cloak. Another example of transformation optics devices are the superlenses : even if these devices were
proposed a few years before the rise of transform optics, they are nicely interpreted as corresponding to a folding of
the space on itself. It has been suggested that such devices allow a kind of "remote action" of the scatterers making
possible things such as immaterial waveguides called "invisible tunnels". In this paper, we investigate numerically
(using finite element modelling) the behaviour of invisibility cloaks and cylindrical superlenses to show some of
their amazing possibilities but also to define some of their limitations.
The low-frequency behavior of a set of wires with a very high conductivity is studied. The effective non-local
constitutive relation is derived for wires with a finite height. Some numerical examples are described.
We demonstrate the accuracy of the Finite Element Method (FEM) to characterize an arbitrarily shaped
crossed-grating in a multilayered stack illuminated by an arbitrarily polarized plane wave under oblique
incidence. To our knowledge, this is the first time that 3D diffraction efficiencies are calculated using the
FEM. The method has been validated using classical cases found in the literature. Finally, to illustrate
the independence of our method towards the shape of the diffractive object, we present the global energy
balance resulting of the diffraction of a plane wave by a lossy thin torus crossed-grating.
We present a new formulation of the finite element method (FEM) dedicated to the rigorous solution of Maxwell's equations and adapted to the calculation of the scalar diffracted field in optoelectronic subwavelength periodic structures [for both transverse electric (TE) and transverse magnetic (TM) polarization cases]. The advantage of this method is that its implementation remains independent of the number of layers in the structure, the number of diffractive patterns, the geometry of the diffractive object, and the properties of materials. The spectral response of large test photodiodes that can legitimately be represented in 2-D has been measured on a dedicated optical bench and compared to the theory. The validity of the model as well as the possibility of conceiving in this way simple processible diffractive spectral filters are discussed.
Microstructured optical fibers have much more degrees of freedom concerning the geometries and index contrasts than
step-index fibers. This richness opens totally new fields of application for fiber optics. The finite element method appears
as an extremely versatile tool to compute the propagation modes in such systems as it allows to take into account
arbitrary geometries of the cross section and also anisotropic and inhomogeneous (i.e. not only piecewise constant)
dielectric permittivities. In this paper, we review some more advanced features: how to compute leaky modes (crucial for
the understanding of such kind of fibers) by using perfectly matched layers, how to use helicoidal coordinate systems to
determine the influence of a twist on the modes via a
two-dimensional model (using equivalent materials), and how to
compute spatial solitons in fibers involving Kerr optical medium by taking into account the refractive index
inhomogeneities caused by the nonlinearity.
We present a new formulation of the Finite Element Method (FEM) dedicated to the 2D rigorous solving of
Maxwell equations adapted to the calculation of the diffracted field in optoelectronic subwavelength structures.
The advantage of this method is that its implementation remains independent of the number of layers in the
structure, of the number of diffractive patterns, of the geometry of the diffractive object and of the properties of
the materials.
The spectral response of large test photodiodes that can legitimately be represented in 2D has been measured
on a dedicated optical bench and confronted to the theory. The representativeness of the model as well as the
possibility of conceiving this way simply processable diffractive spectral filters are discussed.
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