Electromagnetic equivalent model for phase conjugate mirror based on the utilization of left-handed material

An electromagnetic equivalent model for the phase conjugate mirror (PCM) is proposed in this paper. The model is based on the unique property of the isotropic left-handed material (LHM) - the ability of LHM to reverse the phase factors of propagative waves. We show that a PCM interface can be substituted with a LHM-RHM (right-handed material) interface and associated image sources and objects in the LHM. This equivalent model is fully equivalent in the treatment of propagative wave components. However, we note that the presence of evanescent wave components can lead to undesirably surface resonance at the LHM-RHM interface. This artefact can be kept well bounded by introducing a small refractive index mismatch between the LHM and RHM. We demonstrate the usefulness of this model by modelling several representative scenarios of light patterns interacting with a PCM. The simulations were performed by applying the equivalent model to a commercial finite element method (FEM) software. This equivalent model also points to the intriguing possibility of realizing some unique LHM based systems in the optical domain by substituting a PCM in place of a LHM-RHM interface

Instability of semi-Riemannian closed geodesics

A celebrated result due to Poincar\'e affirms that a closed non-degenerate minimizing geodesic $\gamma$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented manifold. A key role is played by the spectral index, a new topological invariant that we define through the spectral flow (being the Morse index truly infinite) of a path of selfadjoint Fredholm operators. A major step in the proof of this result is a em new spectral flow formula. Bott's iteration formula, introduced by author in 1956, relates in a clear way the Morse index of an iterated closed Riemannian geodesic and the so-called $\omega$-Morse indices. Our second result is a semi-Riemannian generalization of the famous Bott-type iteration formula in the case of closed (resp. timelike closed) Riemannian (resp. Lorentzian) geodesics. Our last result is a strong instability result obtained by controlling the Morse index of the geodesic and of all of its iterations.Comment: 33 pages, 2 figures. Fixed some typos and updated references. arXiv admin note: text overlap with arXiv:1705.0917