Electromagnetic Modelling of Fine Features in Photonic Applications

Straightforward application of numerical modelling approaches to Photonic Band-Gap structures demands the use of fine meshes, notably finer that the size of a single scattering element, in order that high accuracy is achieved. However simulations employing sufficiently fine meshing to correctly represent the geometry of the scatterers lead to very long computational run-times and huge memory consumption. Such direct numerical approaches to PBG characterisation are only suited to the analysis of single unit cells or for benchmarking. In practice, the computational overheads significantly increase when one deals with an array of cells or when modelling the global behaviour of a large number of devices integrated on a single substrate. Therefore in order to model the macroscopic response of PBG structures the possibility of employing meshes of much larger size than the individual scattering elements has been explored. A suitable approach, initially developed for Electromagnetic Compatibility predictions, has been modified to permit modelling of Photonic Crystal Waveguides. The method provides second order accuracy and leaves a designer with the flexibility to discretise the entire problem space with an arbitrary number of scatterers per mesh cell. Results are first presented for small clusters of scatterers and subsequently for complete photonic structures.     

Modeling comparison of second-harmonic generation in high-index-contrast devices

In this paper we model second-harmonic generation in a microphotonic structure using two different numerical methods. The proposed two-dimensional waveguide device is challenging as it incorporates a high-contrast grating, which instigates strong local enhancement but also radiation losses. The first simulation method extends the time-domain beam propagation method, whereas the second one builds upon frequency-domain eigenmode expansion. Good agreement between both tools is obtained for both the linear and nonlinear simulations.     

All-optical power limiting in waveguides with periodically distributed Kerr-like non-linearity

A slab waveguide with an embedded non-linear grating is considered as an all-optical power-limiting device. The limiting action is based on the power losses at boundaries between grating sections, where a mode-mismatch results from the self-focusing of the fundamental mode in each non-linear section. Time-harmonic and optical pulse propagation are both modelled numerically by finite-difference techniques to demonstrate the behaviour of the device. Received: 17 May 2001 / Revised version: 7 August 2001 / Published online: 23 October 2001     

Efficient and accurate models of buried spot-size convertors

The Free Space Radiation Mode method has been established as an accurate and extremely efficient technique for analysing buried optical waveguide structures. This work demonstrates its application to the simulation of spot-size convertors which are a crucially important component of many practical systems.     

Binary integer programs with two variables per inequality

Several recent papers have shown that some properties of the maximum weight stable set problem hold true in the more general setting of binary integer programs with two variables per inequality. In this paper, we show that in fact the two problems are equivalent by using the transitive closure of the binary integer program and (possibly) reducing the number of variables by fixing, complementing, or identifying them. We use this equivalence to prove two conjectures made by Johnson and Padberg regarding the perfection of bidirected graphs.     

On configuration-dependent loading

Summary A general investigation of the mathematical properties of configuration-dependent loading is described. This is loading which acts on the surface of an arbitrary body and which changes in a known way depending on the displacement of the neighbourhood of its point of application. Exact definitions of many possible kinds of such loading are given. Criteria for conservative loading are derived and applied. A virtual work principle for a sequence of perturbation problems is given. Suggestions are made for further research.     

Quantum fields on manifolds: PCT and gravitationally induced thermal states

We formulate an axiomatic scheme, designed to provide a framework for a general, rigorous theory of relativistic quantum fields on a class of manifolds, that includes Kruskal's extension of Schwarzschild space-time, as well as Minkowski space-time. The scheme is an adaptation of Wightman's to this class of manifolds. We infer from it that, given an arbitrary field (in general, interacting) on a manifold X, the restriction of the field to a certain open submanifold X⁽⁺⁾, whose boundaries are event horizons, satisfies the Kubo-Martin-Schwinger (KMS) thermal equilibrium conditions. This amounts to a rigorous, model-independent proof of a generalised Hawking-Unruh effect. Further, in cases where the field enjoys a certain PCT symmetry, the conjugation governing the KMS condition is just the PCT operator. The key to these results is an analogue, that we prove, of the Bisognano-Wichmann theorem. [J. Math. Phys.17 (1976), Theorem 1]. We also construct an alternative scheme by replacing a regularity condition at an event horizon by the assumption that the field in X⁽⁺⁾ is in a ground, rather than a thermal, state. We show that, in this case, the observables in X⁽⁺⁾ are uncorrelated to those in its causal complement, X^(?), and thus that the event horizons act as physical barriers. Finally, we argue that the choice between the two schemes must be dictated by the prevailing conditions governing the state of the field.