Optical metamaterials have redefined how we understand light in notable ways: from strong response to optical magnetic fields, negative refraction, fast and slow light propagation in zero index and trapping structures, to flat, thin and perfect lenses. Many rules of thumb regarding optics, such as μ = 1, now have an exception, and basic formulas, such as the Fresnel equations, have been expanded. The field of metamaterials has developed strongly over the past two decades. Leveraging structured materials systems to generate tailored response to a stimulus, it has grown to encompass research in optics, electromagnetics, acoustics and, increasingly, novel hybrid material responses. This roadmap is an effort to present emerging fronts in areas of optical metamaterials that could contribute and apply to other research communities. By anchoring each contribution in current work and prospectively discussing future potential and directions, the authors are translating the work of the field in selected areas to a wider community and offering an incentive for outside researchers to engage our community where solid links do not already exist.

Nonlinear metamaterial-driven waveguide geometries are discussed with an emphasis upon solitonic behaviour and the critical role of magnetooptics as a controlling influence that will drive not only applications but some future directions of metamaterial creation. The exotic family of excitations that emerge will include highly structured light in the form of optical vortices, which can be shaped to take advantage of a new, nonlinear, diffraction that dominates over nonparaxiality. Finally, a brief introduction to novel nonlinear energy concentrators is developed.

Metamaterial research is an extremely important global activity that promises to change our lives in many different ways, including making objects invisible and having a very dramatic impact upon the energy and medical sectors of society. Behind all of the applications, however, lies the design of metamaterials and this can be led by elegant routes that include nonlinearity, waveguide complexity and structured light. The associated optical device formats often involve coupling to soliton behavior. Vortex formation is going to be a critical feature for future applications focusing attention upon the role of angular momentum in special metamaterial-driven light beams. In this context nonlinear diffraction must be assessed and some discussion of a magnetooptical environment will be included. Solitonic behavior of light beams will be mentioned, including what have now become known as Peregrine solitons.

Controlling losses in metamaterials has now reached an advanced stage, so that building up a range of integrated waveguide devices based upon them is not only attractive from a fundamental point of view, but it is going to be possible to imagine a number of special down-stream applications. The degree of control obtained by manipulating planar metamaterial waveguides and interfaces is important and is popularly based upon solitons of various kinds. Complete control is assured by invoking transformation optics, as will be briefly demonstrated here.

Guided waves in metamaterials are attracting attention, even though, experimentally, there remain some substantial questions about the fabrication of wave guides. Nonlinear guided optical waves have always been attractive for their device potential, so the development of nonlinear waves in metamaterials is an important direction to take and this is the basis of the discussion put forward by this paper. Given an effective medium starting point, it is possible to highlight the metamaterial influences upon both exact nonlinear waves and the soliton behaviour that is characteristic of the weakly nonlinear regime. This paper progresses through a number of priorities that have been discussed in the literature. The outcomes are rapidly reviewed from the point of view of putting the field into the context of both strongly nonlinear waves and spatial solitons, since both scenarios emphasise the role of metamaterial control. Finally, the possibility of using magneto-optics as an external control to modify the metamaterial influences is briefly displayed.

A fascinating review of nonlinear waves in metamaterials is presented with an emphasis upon complex waveguides. Many opportunities exist for elegant control such as the capture of rainbows and an exciting deployment of magnetooptic environments.

The history of optical solitons is fascinating and any theory of these has a weakly guiding foundation. Vortex generation and propagation properties have also a beautiful history, and the possibility of generating them together with magnetooptic control in plasmonic metamaterials will be discussed in detail. An emphasis will be placed on the fact that spatial solitons have a lot of application possibilities, especially when placed into the context of materials being used in a light-controlling light environment that is suitable for optical chips of the future. In addition, temporal solitons will also be invoked. An initial emphasis will be placed upon narrow beams and extremely short pulses, but it will be pointed out very strongly that detailed control of light-packets can also be introduced by using plasmonic metamaterials in the optical frequency range. This feature requires an exact study of wave propagation in waveguides that are possibly tapered, or simply just power controlled. To any designs that are proposed can be added the advantage of using magnetooptics. The complicated structures that will be examined will include soliton-like channels near interfaces. Optically linear and nonlinear metamaterials will be discussed in this context. The applications of the outcomes should lead to a new range of optical switching.

The creation of electromagnetic metamaterials is an important activity. The latter should anticipate the kind of applications in which unique metamaterial behaviour can appear. This paper addresses nonlinear wave phenomena in both the strongly and the weakly nonlinear regimes. It inevitably involves novel nonlinear guided waves and solitonic beam activities. In this context, some magnetooptic control is introduced. In addition, the kind of structural complexity that can lead to trapped rainbows will be briefly examined. Finally, some aspects are made of vortex control in a diffraction-managed metamaterial is presented.

A fascinating review of nonlinear waves in metamaterials is presented. The usual weakly nonlinear approximation is dispensed with, and there is an emphasis upon complex waveguides. Many opportunities exist for elegant control using the deployment of magnetooptic environments.

The creation of electromagnetic metamaterials that will operate at THz frequencies, and into the visible frequency range, is an extremely important task that points to far-reaching medical, data storage, and processing applications. It is imperative, therefore, that these properties be associated with complex systems that can sustain both guided and surface waves in the nonlinear regime, and to offer the possibility of tunability through the addition of a gyromagnetic environment. In particular, a magneto-optic part of a metamaterial guiding structure will exert a dramatic influence because it can readily take advantage of the types of nanostructured geometries that are coming into existence. If the nonlinearity is strong, the shape of the modal fields of nonlinear guided waves changes significantly with power, as demonstrated a long time ago. The investigation of spatial and temporal solitons in double negative metamaterials is important to the future of integrated optical structures which rely upon specialized data manipulation. Some examples of strongly nonlinear waves will be given and the magnetooptic influences will be reserved for soliton management.

The first comprehensive exact theory of strongly nonlinear guided waves in a double-negative planar metamaterial waveguide is developed. The theoretical consequences are that novel surface and guided waves are predicted because of the special relationship of the boundary fields to each other. The analysis leads smoothly to tunability with power and direct access to group velocity control.

The nonlinear properties of metamaterials are going to be important for the control of new computing and sensor devices. In addition, an exciting dimension can be added through the inclusion of magnetooptical properties. Both temporal and spatial solitons will be considered for a range of metamaterials with an emphasis being placed upon bright and bright‐dark soliton interactions coupled to magnetic effects drawn from both the Voigt and the Faraday configurations. Strongly nonlinear waves will also be discussed in terms of their exciting ability to slow light and respond vigorously to both nonlinear and magnetooptic tuneability. A special emphasis will be placed upon the switching possibilities of solitons at an interface and complex waveguides, and shape effects will also be addressed.

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.

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 study of optical vortices solitons propagation under the influence of an inhomogeneous external magnetic field is presented. The external magnetic field is applied on the z-axis, which is also the direction of propagation, so that a Faraday configuration is created. This study of magnetooptic vortices in a bulk, nonlinear, gyrotropic media leads to an investigation of the coupling of the electric field components, Ex and Ey, in the (x,y) plane. An optical beam propagating in the bulk is modelled by coupled equations in which the nonlinear refractive index is Kerr-like bulk optical nonlinearity. A transformation to rotating coordinates enables circularly polarised waves to be selected and a peak in the magnetisation over the centre of the beam is used. An important spatial dependence of the magnetisation parameter, defined as Q(x), stimulates novel singular behaviour. To demonstrate this kind of gyrotropy experimentally the usual Kerr nonlinearity may be too weak for comfortable observations but semi-magnetic semiconductors and atomic gases are shown to be possible candidates for which Faraday rotations are impressive.

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. After an interesting historical overview and an exposure of some straightforward concepts, a planar guide is investigated, in which the waveguide is a slab consisting of a left-handed medium sandwiched between a substrate and cladding that are simple dielectrics. 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. A comprehensive finite-difference-time-domain (FDTD) analysis is presented that concentrates upon spatial soliton behaviour. An interesting soliton-lens arrangement is investigated that lends itself to a novel cancellation effect.

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.

This paper develops and applies a new quadrature method for TM waves that is capable of dealing with any kind of nonlinearity. Explicit solutions of the nonlinear equations are not required and power-effective index curves can be developed in terms of relatively straightforward integrations (quadrature) over one of the field components. The integration limits may be a function of frequency and/or magnetic field. The basic theory necessary for a study of nonlinear magneto-optic or magnetoplasma devices is created. Strongly nonlinear effects will require matched indices and a magnetic field can provide an extra tuning element that can enhance this matching. The theory has the potential to lead to novel coupler configurations.