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This tutorial is divided into two parts: the first examines the application of topology to problems in wave physics. The origins of the Chern number are reviewed, where it is shown that this counts the number of critical points of a complex tangent vector field on the surface. We then show that this quantity arises naturally when calculating the dispersion of modes in any linear system, and give examples of its ap plication to find one–way propagating interface modes in both continuous and periodic materials. The second part offers a physical interpretation for the Chern number, based on the idea that the critical points which it records can be understood as points where the refractive index vanishes. Using the theory of crystal optics, we show that when the refractive index vanishes in a complex valued direction, the wave is forced to circulate in only one sense, and this is the origin of the one–way propagation of topological interface states. We conclude by demonstrating that this idea of ‘zero refractive index in a complex direction’ can be used as a shortcut to find acoustic and electromagnetic materials supporting one–way interface states. [ABSTRACT FROM AUTHOR]

Electromagnetic waves at grazing incidence onto a planar medium are analogous to zero energy quantum particles incident onto a potential well. In this limit waves are typically completely reflected. Here we explore dielectric profiles supporting optical analogues of 'half–bound states', allowing for zero reflection at grazing incidence. To obtain these profiles we use two different theoretical approaches: supersymmetric quantum mechanics, and direct inversion of the Helmholtz equation, showing that discretized approximations to these profiles exhibit low reflectivity close to grazing incidence. [ABSTRACT FROM AUTHOR]

We report on the theory of a Luneburg lens for forward-volume magnetostatic spin waves and verify its operation via micromagnetic modelling. The lens converts a plane wave to a point source, and vice versa, by a designed graded refractive index, realized by modulating either the thickness or the saturation magnetization in a circular region. We find that the lens enhances the wave amplitude by about 5 times at the lens focus, and 47% of the incident energy arrives in the focal region. A lens with small deviations from the optimal profile can still result in good focusing if the index is graded smoothly. [ABSTRACT FROM AUTHOR]

Optical materials with a distribution of loss and gain can be used to manipulate waves in fascinating ways, seemingly impossible with ordinary lossless materials. Some recent results have shown that (for planar media) if the spatial distributions of the real and imaginary parts of the permittivity are related to one another by the Kramers-Kronig relations, then reflection can be eliminated. Moreover, if an additional "cancellation condition" is satisfied, then a material can be made invisible for incidence from one side. Here, we give a simple demonstration of these results that should be accessible to undergraduates. In addition, we show how this simple method can be used to prove results about the reflection from permittivity profiles, without ever requiring an exact solution of the Helmholtz equation. [ABSTRACT FROM AUTHOR]

When a planar dielectric medium has a permittivity profile that is an analytic function in the upper or lower half of the complex position plane x = x′ + ix″ then the real and imaginary parts of its permittivity are related by the spatial Kramers-Kronig relations. We find that such a medium will not reflect radiation incident from one side, whatever the angle of incidence. Using the spatial Kramers-Kronig relations, one can derive a real part of a permittivity profile from some given imaginary part (or vice versa) such that the reflection is guaranteed to be zero. This result is valid for both scalar and vector wave theories and may have relevance for designing materials that efficiently absorb radiation or for the creation of a new type of anti-reflection surface. [ABSTRACT FROM AUTHOR]

The advent of Transformation Optics established the link between geometry and material properties, and has resulted in a degree of control over electromagnetic fields that was previously impossible. For waves confined to a surface it is known that there is a simpler, but related, geometrical equivalence between the surface shape and the refractive index, and here we demonstrate that conventional devices possessing a singularity—that is, the requirement of an infinite refractive index—can be realised for waves confined to an appropriately sculpted surface. In particular, we redesign three singular omnidirectional devices: the Eaton lens, the generalized Maxwell Fish-Eye, and the invisible sphere. Our designs perfectly reproduce the behaviour of these singular devices, and can be achieved with simple isotropic media of low refractive index contrast. [ABSTRACT FROM AUTHOR]

A series of thin layers of alternating refractive index are known to make a good optical mirror over certain bands of frequency. Such a device, often termed the Bragg reflector, is usually introduced to students in isolation from other parts of the curriculum. Here, we show that the basic physics of wave propagation through a stratified medium can be used to illustrate some more modern developments in optics and quantum physics, from transfer matrix techniques to the optical properties of cold trapped atoms and optomechanical cooling. We also show a simple example of how such systems exhibit an appreciable level of optical nonreciprocity. [ABSTRACT FROM AUTHOR]

We find the action that describes the electromagnetic field in a spatially dispersive, homogeneous medium. This theory is quantized and the Hamiltonian is diagonalized in terms of a continuum of normal modes. It is found that the introduction of nonlocal response in the medium automatically regulates some previously divergent results, and we calculate a finite value for the intensity of the electromagnetic field at a fixed frequency within a homogeneous medium. To conclude we discuss the potential importance of spatial dispersion in taming the divergences that arise in calculations of Casimir-type effects. [ABSTRACT FROM AUTHOR]

Reciprocity is fundamental to light transport and is a concept that holds also in rather complex systems. Yet, reciprocity can be switched off even in linear, isotropic, and passive media by setting the material structure into motion. In highly dispersive multilayers this leads to a fairly large forward-backward asymmetry in the pulse transmission. Moreover, in multilevel systems, this transport phenomenon can be ail-optically enhanced. For atomic multilayer structures made of three-level cold 87Rb atoms, for instance, forward-backward transmission contrast around 95% can be obtained already at atomic speeds in the meter per second range. The scheme we illustrate may open up avenues for optical isolation that were not previously accessible. [ABSTRACT FROM AUTHOR]

We experimentally demonstrate a Purcell effect-based design technique for improved impedance matching, and thus enhanced the reflection coefficient from a small microwave emitter. Using an iterative process centred on comparing the phase of the radiated field of the emitter in air with that of the emitter in a dielectric environment, we optimise the structure of a dielectric hemisphere above a ground plane surrounding a small monopolar microwave emitter in order to maximise its radiation efficiency. The optimised system shows very strong coupling between the emitter and two omnidirectional radiation modes at 1.99 GHz and 2.84 GHz, yielding Purcell enhancement factors of 1762 and 411 times increase respectively, and near perfect radiation efficiency. [ABSTRACT FROM AUTHOR]

We analyse the evolution of the wavefunction of a quantum particle propagating on several compact manifolds, including the Klein bottle, Möbius strip and projective plane. We find analytically the stationary states and the energy spectrum and show that the wavefunction exhibits perfect revivals. Using the orbifold structure of the discussed manifolds, we establish the relation of wave evolution on the manifolds to Fresnel diffraction and consequently to the Talbot effect. This connection provides a novel method of optical simulation of the quantum motion on compact manifolds. We discuss some novel phenomena as well as the effects of topology on the properties of the waves on the manifolds. [ABSTRACT FROM AUTHOR]

We report an analytical theory of linear emission of exchange spin waves from a Bloch domain wall, excited by a uniform microwave magnetic field. The problem is reduced to a one-dimensional Schrödinger-like equation with a Pöschl-Teller potential and a driving term of the same profile. The emission of plane spin waves is observed at excitation frequencies above a threshold value, as a result of a linear process. The height-to-width aspect ratio of the Pöschl-Teller profile for a domain wall is found to correspond to a local maximum of the emission efficiency. Furthermore, for a tailored Pöschl-Teller potential with a variable aspect ratio, particular values of the latter can lead to enhanced or even completely suppressed emission. [ABSTRACT FROM AUTHOR]

The transmission of a wave through a randomly chosen "pile of plates" typically decreases exponentially with the number of plates, a phenomenon closely related to Anderson localization. In apparent contradiction, we construct disordered planar permittivity profiles which are complex valued (i.e., have reactive and dissipative properties) that appear to vary randomly with position, yet are one-way reflectionless for all angles of incidence and exhibit a transmission coefficient of unity. In addition to these complex-valued "random" planar permittivity profiles, we construct a family of real-valued, two-way reflectionless and perfectly transmitting disordered permittivity profiles that function only for a single angle of incidence and a narrow frequency range. [ABSTRACT FROM AUTHOR]

Ever since the inception of Transformation Optics (TO), new and exciting ideas have been proposed in the field of electromagnetics and the theory has been modified to work in such fields as acoustics and thermodynamics. The most well-known application of this theory is to cloaking, but another equally intriguing application of TO is the idea of an illusion device. Here, we propose a general method to transform electromagnetic waves between two arbitrary surfaces. This allows a flat surface to reproduce the scattering behaviour of a curved surface and vice versa, thereby giving rise to perfect optical illusion and cloaking devices, respectively. The performance of the proposed devices is simulated using thin effective media with engineered material properties. The scattering of the curved surface is shown to be reproduced by its flat analogue (for illusions) and vice versa for cloaks. [ABSTRACT FROM AUTHOR]

We propose a general method to arbitrarily manipulate an electromagnetic wave propagating in a two-dimensional medium, without introducing any scattering. This leads to a whole class of isotropic spatially varying permittivity and permeability profiles that are invisible while shaping the field magnitude and/or phase. In addition, we propose a metamaterial structure working in the infrared that demonstrates deep subwavelength control of the electric field amplitude and strong reduction of the scattering. This work offers an alternative strategy to achieve invisibility with isotropic materials and paves the way for tailoring the propagation of light at the nanoscale. [ABSTRACT FROM AUTHOR]

An exact solution is found for laminar fluid flow along the grooves of a family of surfaces whose shape is given by the Lambert W-function. This simple solution allows for the slip length in the direction parallel to the grooves to be calculated exactly. With this analytical model, we establish the regime of validity for a previously untested perturbation theory intended for calculating the surface mobility tensor of arbitrary periodic surfaces, finding that it compares well to the exact expression for nearly all choices of parameters of the conformal map. To test this perturbation theory further, the mobility tensor is evaluated for a simple sinusoidal surface for flow both parallel and perpendicular to the grooves, finding that the perturbation theory is less accurate in the latter of these two cases. [ABSTRACT FROM AUTHOR]

This Letter presents a method for making an uneven surface behave as a flat surface. This allows an object to be concealed (cloaked) under an uneven portion of the surface, without disturbing the wave propagation on the surface. The cloaks proposed in this Letter achieve perfect cloaking that only relies upon isotropic radially dependent refractive index profiles, contrary to those previously published. In addition, these cloaks are very thin, just a fraction of a wavelength in thickness, yet can conceal electrically large objects. While this paper focuses on cloaking electromagnetic surface waves, the theory is also valid for other types of surface waves. The performance of these cloaks is simulated using dielectric filled waveguide geometries, and the curvature of the surface is shown to be rendered invisible, hiding any object positioned underneath. Finally, a transformation of the required dielectric slab permittivity was performed for surface wave propagation, demonstrating the practical applicability of this technique. [ABSTRACT FROM AUTHOR]

By suitably patterning a metasurface, the phase velocity of surface waves may be manipulated. Here, a low-loss, thin (1/14th of the free-space wavelength), omnidirectional Luneburg lens, based upon a Sievenpiper "mushroom" array [ Sievenpiper et al. IEEE Trans. Microwave Theory Tech. 47 2059 (1999)], is fabricated and characterized at microwave frequencies. Surface waves excited using a near-field point source on the perimeter of the lens, exit the opposite side of the lens as planar wave fronts. The electric field of the surface wave is mapped out experimentally and compared to numerical simulations. [ABSTRACT FROM AUTHOR]

Graphene, despite its centrosymmetric structure, is predicted to have a substantial second order nonlinearity, arising from non-local effects. However, there is disagreement between several published theories and experimental data. Here we derive an expression for the second order conductivity of graphene in the non-local regime using perturbation theory, concentrating on the difference frequency mixing process, and compare our results with those already published. We find a second-order conductivity (σ(2) ≈ 10−17 AmV−2) that is approximately three orders of magnitude less than that estimated from recent experimental results. This indicates that nonlinear optical coupling to plasmons in graphene cannot be described perturbatively through the electronic nonlinearity, as previously thought. We also show that this discrepancy cannot be attributed to the bulk optical nonlinearity of the substrate. As a possible alternative, we present a simple theoretical model of how a non-linearity can arise from photothermal effects, which generates a field at least two orders of magnitude larger than that found from perturbation theory. [ABSTRACT FROM AUTHOR]