Vector Potential Electromagnetic Theory with Generalized Gauge for Inhomogeneous Anisotropic Media

Vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. The rapid development in quantum optics calls for electromagnetic solutions that straddle quantum physics as well as classical physics. The vector potential formulation is a good candidate to bridge these two regimes. Hence, there is a need to generalize this formulation to inhomogeneous media. A generalized gauge is suggested for solving electromagnetic problems in inhomogenous media which can be extended to the anistropic case. The advantages of the resulting equations are their absence of low-frequency catastrophe. Hence, usual differential equation solvers can be used to solve them over multi-scale and broad bandwidth. It is shown that the interface boundary conditions from the resulting equations reduce to those of classical Maxwell's equations. Also, classical Green's theorem can be extended to such a formulation, resulting in similar extinction theorem, and surface integral equation formulation for surface scatterers. The integral equations also do not exhibit low-frequency catastrophe as well as frequency imbalance as observed in the classical formulation using E-H fields. The matrix representation of the integral equation for a PEC scatterer is given.Comment: 16 pages, 2 figur

A brief historical perspective of the Wiener-Hopf technique

It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems

Open strings in relativistic ion traps

Electromagnetic plane waves provide examples of time-dependent open string backgrounds free of $\alpha'$ corrections. The solvable case of open strings in a quadrupolar wave front, analogous to pp-waves for closed strings, is discussed. In light-cone gauge, it leads to non-conformal boundary conditions similar to those induced by tachyon condensates. A maximum electric gradient is found, at which macroscopic strings with vanishing tension are pair-produced -- a non-relativistic analogue of the Born-Infeld critical electric field. Kinetic instabilities of quadrupolar electric fields are cured by standard atomic physics techniques, and do not interfere with the former dynamic instability. A new example of non-conformal open-closed duality is found. Propagation of open strings in time-dependent wave fronts is discussed.Comment: 43 pages, 11 figures, Latex2e, JHEP3.cls style; v2: one-loop amplitude corrected, open-closed duality proved, refs added, miscellaneous improvements, see historical note in fil

The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms

An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. At certain frequencies strongly directional dynamic anisotropy is observed, and the asymptotic theory is shown to capture the effect, giving highly accurate qualitative and quantitative results as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour

Scattering problems in elastodynamics

In electromagnetism, acoustics, and quantum mechanics, scattering problems can routinely be solved numerically by virtue of perfectly matched layers (PMLs) at simulation domain boundaries. Unfortunately, the same has not been possible for general elastodynamic wave problems in continuum mechanics. In this paper, we introduce a corresponding scattered-field formulation for the Navier equation. We derive PMLs based on complex-valued coordinate transformations leading to Cosserat elasticity-tensor distributions not obeying the minor symmetries. These layers are shown to work in two dimensions, for all polarizations, and all directions. By adaptative choice of the decay length, the deep subwavelength PMLs can be used all the way to the quasi-static regime. As demanding examples, we study the effectiveness of cylindrical elastodynamic cloaks of the Cosserat type and approximations thereof