Resonant-state Expansion for Periodic Open Optical Systems

共振态展开（RSE）是卡迪夫物理与天文学院最近提出的一种严格微扰方法。RSE已被开发并用于描述具有各种微扰的平面、圆柱和球形光学谐振腔等开放系统的相对论波动方程。该方法特别适用于计算这种开放系统中的高质量模式，并且在计算其微扰时效率高、精度高。研究结果表明，RSE的计算速度比ComSol或Lumerical等商业软件快几个数量级。该项目的主要目标之一是将RSE应用于非相对论波动方程，并发展量子力学系统中的共振态理论。特别是，RSE将被应用到薛定谔的波动方程描述电子/激子在各种纳米结构有效的一维，二维和/或三维的潜力和各种扰动。新方法的效率和质量将被调查和测试对现有的有限差分方法和从头计算。共振态的概念也将被用来研究量子态的短时间和长时间行为，并计算量子态在短（长）时间的生存概率的非二次（非指数）衰减。该项目还将考虑RSE在相对论问题中的一些进一步应用。这可以包括计算光学平面波导和光子晶体光纤中的本征态，其中永久波导模式有助于绿色函数的扩展。最后，将尝试对相对论狄拉克波动方程的RSE的推广与窄带半导体纳米结构的方法的潜在应用。

The resonant state expansion (RSE) is a rigorous perturbative method recently invented in Cardiff School of Physics and Astronomy. The RSE has been developed and used for relativistic wave equations describing such open systems as planar, cylindrical and spherical optical resonators with various perturbations. The method was demonstrated to be particularly suitable for calculating high-quality modes in such open system and efficient and accurate in calculating their perturbations. It has been shown that the RSE is a few orders of magnitude quicker computational tool as compared to existing commercial software packages like ComSol or Lumerical.One of the main aims of the project is to apply the RSE to non-relativistic wave equations and develop a theory of resonant states in quantum-mechanical systems. In particular, the RSE will be applied to Schrödinger's wave equation describing an electron/exciton in various nanostructures with effectively one-, two- and/or three-dimensional potentials and various perturbations. The efficiency and the quality of the new method will be investigated and tested against the existing finite difference methods and ab initio calculations. The concept of resonant states will also be used to investigate the short and long-time behaviour of quantum states and to calculate a non-quadratic (non-exponential) decay of the survival probability of quantum states at short (long) times.The project will also consider some further applications of the RSE in relativistic problems. This may include calculation of eigenstates in optical planar waveguides and photonic crystal fibres where everlasting waveguide modes contribute to expansions of the Green's function. Finally, there will be made attempts towards generalisation of the RSE for relativistic Dirac wave equation with a potential application of the method to narrow-band semiconductor nanostructures.