A fundamental approach to a slowly varying amplitude formulation for nonlinear waves in metamaterials will be
established. The weakly nonlinear slowly varying amplitude approach will be critically examined and some
misunderstandings in the literature will be fully addressed. The extent to which negative phase behaviour has a
fundamental influence upon soliton behaviour will be exposed. The method will deploy nonlinear diffraction and a
special kind of diffraction-management. This is additional to a detailed modulation instability analysis. The examples
given involve waveguide coupling and a nonlinear interferometer. In addition, a strongly nonlinear approach will be
taken that seeks exact solutions to the nonlinear equations for a metamaterial. A boundary field amplitude approach will
be developed that leads to useful eigenvalue equations that expose, in a very clear manner, the possibility that new kinds
of waves can be generated.
There is now a global interest in the creation of creation of electromagnetic metamaterials. The substantive early work is
focused upon the GHz frequency range but almost immediately the desire to progress rapidly to the optical frequency
range gathered momentum. This is a natural desire because many applications operate at optical frequencies but the THz
range is also important for a range of medical applications as well. The concepts that underpin the need for
metamaterials, and their special properties, are explained in this article and why the creation of exotic, artificial,
molecules is required to produce material behaviour beyond any performance that could naturally be expected. It will be
shown that the major key lies in adding magnetic properties to special dielectric behaviour. This leads to composites that
have almost magical behaviour. This presentation will explore the current global experimental progress towards three-dimensional
metamaterials and will explain, in a straightforward manner, the concept of negative refraction that is
attracting such a lot of attention. The initial ideas, and even some of the early misconceptions, will be addressed and
clearly illustrated in a manner that enhances any understanding of the conceptual structure will be expressed. It will be
shown that even though negative refraction can be associated with both backward and forward waves, the novel
metamaterial concept is to associate backward wave phenomena with isotropic media, artificially endowed with negative
permittivity and permeability. The principle application shown here is to a nonlinear ring interferometer that is capable
of sustaining arbitrarily thin solitons or "optical needles" that can also be managed by an external magnetic field.
The fundamental approach to a slowly varying amplitude formulation for nonlinear waves in metamaterials will be
established. The weakly nonlinear slowly varying amplitude approach will be critically examined and some
misunderstandings in the literature will be fully addressed. The extent to which negative phase behaviour has a
fundamental influence upon soliton behaviour will be addressed and will include non-paraxiality, self-steepening and
nonlinear diffraction. A Lagrangian approach will be presented as a way of developing a clear picture of dynamical
behaviour. Exciting examples, involving waveguide and polarization coupling and interferometer systems will illustrate
the extent to which non-paraxiality, self-steepening and nonlinear diffraction will be required as part of the soliton
behaviour patterns, including coupler systems. In addition, a strongly nonlinear approach will be taken that seeks exact
solutions to the nonlinear equations for a metamaterial. The investigations will embrace "optical needles", or
autosolitons. A boundary field amplitude approach will be developed that leads to useful and elegant eigenvalue
equations that expose in a very clear manner the dependence of wave number upon the optical power density. All the
work will be beautifully illustrated with dramatic color-coded outcomes that will also embrace the soliton lens.
A discussion of gain control up to the THz frequency range is presented together with a study of diffraction-managed
solitons in metamaterials. Full dynamical simulations are used at every stage to illustrate the principles being evolved.
An interesting approach based upon the mapping of the complex frequency-plane onto the complex wave number-plane
is given as a way of demonstrating whether the offset of loss by using gain leads to a stable material. Nonlinear
behaviour is addressed through the possibility of soliton formation under the conditions for which nonlinear diffraction
can play a role whenever an attempt to manage the diffraction is made with a suitable combination of left-handed and
right-handed materials. All of this activity requires the modeling of curious artificial molecules but the computational
outcomes inspire confidence that is exemplified with exciting illustrations. The remit of control, exercised through
external influences such as an applied magnetic field, and cavity formation is briefly explored.
Negative phase velocity materials are engineered media that are currently enjoying a surge of interest due to their
interesting properties and potential applications, through negative refraction, to achieve cloaking that makes things
invisible. The literature is alive with papers devoted to the design of suitable metamaterials and there is a particular
desire to operate at THz frequencies and above. A full theory of gain control up to the THz frequency range is presented
together with a comprehensive study of diffraction-managed solitons. There are aspects of control that can be achieved
through externally imposed influences such as gyroelectromagnetic effects. Nonlinear behaviour is also intrinsic to the
Holy Grail quest for complete control, coupled to the possibility of beneficial competition between damping and gain.
The literature is alive with papers devoted to the design of metamaterials and there appears to be a particular desire to create photonic applications that will operate at THz frequencies and above. At one level the modelling of suitable artificial molecules is straightforward but nevertheless the approximations involved need to be able to inspire confidence for optical frequency operation. This presentation will set out a modelling activity that is known to be satisfactory only over certain frequency ranges. Split-ring and omega particles will be specifically investigated and the possibilities discovered will be related to the current experimental expertise. The detailed manner in which the constitutive relations can be controlled and the novel way in which an envelope equation emerges for even the most complex structure is exposed. The transmission and reflection properties of nano-structured materials will be discussed within a magneto-optic environment. Simulations of sub-wavelength transmission through holes in metallic and magneto-optic screens will be discussed using finite-difference time-domain (FDTD) methods. Modelling the interaction of light beams with metamaterials is developed, again using FDTD techniques, and it is shown that special care needs to be taken with structures that have sharp external edges. Finally, a summary of the problems surrounding efficient computations will be shown and some discussion of the role of genetic algorithms in metamaterial design will be featured.
Negative phase velocity metamaterials (NPM) are engineered media currently enjoying a surge of interest due to their interesting properties and potential applications. Their nonlinear behaviour will be intrinsic to the Holy Grail quest for power control. This is a hot topic that is only just being explored as evidenced by a rapidly increasing number of publications over the past few years. With the introduction of power comes the possibility of solitons and it is important to recognise that damping, arising from both the environment and the material, must be offset by the introduction of gain. In this context the investigation considers what are known as dissipative solitons, within a pumping, multi-stable configuration, designed as a ring or Fabry-Perot cavity. Several exciting scenarios will be presented and particular attention is devoted to the nonlinearity displayed by well-known 'artificial' molecules such as split rings and omega particles. The desire to create metamaterials that reach out to optical frequencies is acknowledged through a discussion of scalability. Detailed studies of the cavity stability regimes lead to some novel possibilities for cavity control. The presentation will be rounded off with a generalised theory of metamaterial behaviour in nonlinear environments that is based upon a novel approach using what is sometimes called the nonlinear Lorentz lemma. Extensive new numerical results will be used to illustrate the concepts outlined above.
A description of the fascinating coupling between gyrotropic media and negative refracting media will be presented. The article will address negative phase velocity media and particular types of dielectric-gyrotropic film-dielectric systems in which the applied magnetic field may result in a magneto-optic, or gyromagnetic, influence. The control features use a diverse family of dispersion
curves.
The evidence that double negative media, with an effective negative permittivity, and an effective negative permeability,
can be manufactured to operate at frequencies ranging from microwave to optical is ushering in a new era of
metamaterials. They are referred to here as 'left-handed', even though a variety of names is evident from the literature. In
anticipation of a demand for highly structured integrated practical waveguides, this paper addresses the impact of this
type of medium upon waveguides that can be also nonlinear. A planar guide is investigated first, in which the waveguide
is a slab consisting of a double negative medium, sandwiched between a substrate and cladding that are simple
dielectrics. The TE modes are addressed because they lend themselves to accurate analysis when the substrate and
cladding display a Kerr-type nonlinear response. Because of the nonlinear properties of the Kerr media, the power flow
direction can be controlled by the intensity of the electric field. The rest of the paper addresses a comprehensive finite difference-
time-domain analysis. It uses spatial soliton behaviour in the advanced example section. An interesting
soliton-lens arrangement is presented that deploys positive and negative slabs to create a novel cancellation effect.
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