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arXiv 2009.01307physics.optics
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光子共振的理论与数值模拟:准正规模展开--在电磁学中的应用

Theory and numerical modeling of photonic resonances: Quasinormal Modal Expansion -- Applications in Electromagnetics

Minh Duy Truong

摘要:

电磁学中的模式展开的思想源于对电磁谐振器的研究,而电磁谐振器在纳米光子学的发展中起着至关重要的作用。所有的电磁谐振器都有一个共同的特性:它们具有一组离散的特殊频率,这些频率在散射光谱中显示为峰值,称为谐振模式。这些谐振模式很快被认为决定了电磁谐振器和光之间的相互作用。这导致了一个假设,即谐振器的光学响应是系统中每个物理谐振状态的激发的合成:在外部脉冲的激发下,这些谐振模式最初被加载,然后释放它们的能量,这有助于谐振器的总光学响应。这些具有复频率的谐振模式在文献中被称为准简正模式(QNM)。在数学上,这些QNM对应于无源麦克斯韦方程组的本征值问题的解决方案。在谐振器的光学结构是无界的并且介质是色散的(并且可能是各向异性的和非互易的)的情况下,这需要解决非线性(在频率上)和非厄米特本征值问题。因此,整个问题归结为电磁麦克斯韦算符的谱理论的研究。因此,模态展开形式主义最近在光子学中受到了很多关注,因为它们能够在所考虑的系统的自然共振态基础上模拟物理特性,从而对数值结果进行透明的解释。这份手稿的目的是扩大QNM展开形式主义的研究,特别是,和非线性谱理论,一般。同时,给出了几个数值模拟的例子,说明模态展开法在计算中的应用。

英文摘要:

The idea of the modal expansion in electromagnetics is derived from the research on electromagnetic resonators, which play an essential role in developments in nanophotonics. All of the electromagnetic resonators share a common property: they possess a discrete set of special frequencies that show up as peaks in scattering spectra and are called resonant modes. These resonant modes are soon recognized to dictate the interaction between electromagnetic resonators and light. This leads to a hypothesis that the optical response of resonators is the synthesis of the excitation of each physical-resonance-state in the system: Under the excitation of external pulses, these resonant modes are initially loaded, then release their energy which contributes to the total optical responses of the resonators. These resonant modes with complex frequencies are known in the literature as the Quasi-Normal Mode (QNM). Mathematically, these QNMs correspond to solutions of the eigenvalue problem of source-free Maxwell's equations. In the case where the optical structure of resonators is unbounded and the media are dispersive (and possibly anisotropic and non-reciprocal) this requires solving non-linear (in frequency) and non-Hermitian eigenvalue problems. Thus, the whole problem boils down to the study of the spectral theory for electromagnetic Maxwell operators. As a result, modal expansion formalisms have recently received a lot of attention in photonics because of their capabilities to model the physical properties in the natural resonance-state basis of the considered system, leading to a transparent interpretation of the numerical results. This manuscript is intended to extend the study of QNM expansion formalism, in particular, and nonlinear spectral theory, in general. At the same time, several numerical modelings are provided as examples for the application of modal expansion in computations.

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