Universal coupled-mode theory formulation of quasi-normal modes in a 1D photonic crystal
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Universal coupled‑mode theory formulation of quasi‑normal modes in a 1D photonic crystal Yann G. Boucher1,2 · Lamis Al Sheikh3 Received: 24 June 2020 / Accepted: 30 September 2020 / Published online: 6 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In terms of coupled-mode theory, only two real-valued normalized parameters are needed for describing the spectral properties of any 1D photonic crystal of finite size made of lossless materials: detuning (δL) with respect to the Bragg condition and coupling strength (κL). As far as its quasi-normal mode (QNM) spectrum is concerned, the usual “complex frequency” concept can be replaced by an effective “complex detuning” (δL + i αL), thus establishing a formal connection with the modal cartography, in the (κL, αL) plane, of purely index-coupled distributed-feedback lasers of normalized gain αL. Keywords Quasi-normal mode (QNM) · Coupled-mode theory (CMT) · Distributed Bragg reflector (DBR) · Distributed feedback (DFB) · Photonic band gap (PBG)
1 Introduction Quasi-normal modes (QNM) have been the subject of increasing interest in the last few decades, in many branches of physics such as optics (Ghatak 1985; Lai et al. 1990; Ching et al. 1996), gravitation (Kokkotas and Schmidt 1999), or acoustics (Pan et al. 2015), as well as in mathematics, in connection with so-called scattering resonances in spectral analysis (Zworski 2017). They provide an insightful way of describing open (leaky) resonators, since they represent their natural modes. In some contexts, a QNM is defined as an eigenstate solution to a wave equation, in Fourier space, with no source and suitable outgoing boundary conditions. Because the operator is non-Hermitian, eigenvalues become complex: the real part stands for the oscillation frequency, and the imaginary part for the leakage-induced broadening. Another consequence
* Yann G. Boucher [email protected] Lamis Al Sheikh lamis.al‑sheikh@u‑bourgogne.fr 1
CNRS, Institut Foton, UMR 6082, CS 80518, 22305 Lannion Cedex, France
2
École Nationale d’Ingénieurs de Brest (ENIB), CS 73862, 29238 Brest Cedex 3, France
3
CNRS, Institut de Mathématiques de Bourgogne, UMR 5584, BP 47870, 21078 Dijon Cedex, France
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of non-Hermiticity is the fact that natural modes are no longer orthogonal; for active optical resonators, the case of transverse modes has been addressed by Siegman (1989a, b), the longitudinal modes by Hamel and Woerdman (1989). In the specific case of electromagnetism, extensive work has been undertaken by Lalanne et al. (2018) notably in the important review where a solid theoretical background can be found, rich with analytical as well as numerical considerations. From 1 to 3D, realistic examples are provided of how the light scattered by a resonator can be decomposed into a few number of dominant QNMs, each characterized by a quality factor (Q) and a mode volume (V). For the sake of illustration, as a model system of an op
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