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1. High-Q Substrate-Integrated-Waveguide Resonator with Degenerate Band Edge

   https://doi.org/10.1109/ICEAA.2019.8879299  

   Zheng, Tianyu; Casaletti, Massimiliano; Ren, Zhuoxiang; Abdelshafy, Ahmed F.; Capolino, Filippo; Valerio, Guido (September 2019, Published in: 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA))

   The novel concept of degenerate band edge (DBE) has been recently proposed by Figotin et al., in the framework of their study of slow wave propagation in photonic crystals [1]. The degenerate band edge is a special dispersion condition near the edge of the Brillouin zone ( kd=π/p, p is the spatial period) where four degenerate Bloch modes coalesce at a same frequency fd (two propagating modes and two evanescent modes). The dispersion relation is locally described by a quartic curve f−fd=−α(k−kd) where fd is the DBE frequency and α is a positive constant that depends on the parameters of the structure. 

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2. Experimental Demonstration of a New Oscillator Concept Based on Degenerate Band Edge in Microstrip Circuit

   Dmitry Oshmarin, Ahmed F. (September 2021, arXivorg)

   The first experimental demonstration of an oscillator based on a periodic, resonant microstrip circuit with a degenerate band edge (DBE) is presented. The DBE is a fourth-order exceptional degeneracy of the eigenmodes in a lossless periodic waveguide that is visible in the wavenumber-frequency dispersion diagram, and the periodic microstrip behaves as a frequency selective medium. The presence of the DBE condition and the associated DBE resonance allow for a stable, single-mode oscillation as well as stability with varying the load resistance. 

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3. General Conditions to Realize Exceptional Points of Degeneracy in Two Uniform Coupled Transmission Lines

   https://doi.org/10.1109/TMTT.2020.2999498  

   Mealy, Tarek; Capolino, Filippo (June 2020, IEEE Transactions on Microwave Theory and Techniques)

   We present the general conditions to realize a fourth-order exceptional point of degeneracy (EPD) in two uniform (i.e., invariant along z) lossless and gainless coupled transmission lines (CTLs), namely, a degenerate band edge (DBE). Until now the DBE has been shown only in periodic structures. In contrast, the CTLs considered here are uniform and subdivided into four cases where the two TLs support combinations of forward propagation, backward propagation, and evanescent modes (when neglecting the mutual coupling). We demonstrate, for the first time, that a DBE is supported in uniform CTLs when there is proper coupling between: 1) propagating modes and evanescent modes, 2) forward and backward propagating modes, or 3) four evanescent modes (two in each direction). We also show that the loaded quality factor of uniform CTLs exhibiting a fourth-order EPD at k=0 is robust to series losses due to the fact that the degenerate modes do not advance in phase. We also provide a microstrip possible implementation of a uniform CTL exhibiting a DBE using periodic series capacitors with very subwavelength unit-cell length. Finally, we show an experimental verification of the existence DBE for a microstrip implementation of a CTL supporting coupled propagating and evanescent modes. 

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4. Pulse Generation Based on a Microstrip Circuit with Fourth Order Degenerate Band Edge

   Dmitry Oshmarin, Alireza Nikzamir (June 2022, arXivorg)

   A pulse generation scheme is proposed based on a structured resonance in a cavity where the non-conventional energy distribution is concentrated in its middle part. The cavity is first used as an oscillator during the energy charging step and when a switch is activated the signal is extracted from its center. The key component of the proposed scheme is the periodic microstrip waveguide with a fourth-order degenerate band edge (DBE) of its wavenumber-frequency dispersion diagram. The DBE is an exceptional point degeneracy condition that is responsible for the energy to be localized at the cavity center and that can also have the quality factor easily destroyed by a perturbation. The waveguide is designed to have a DBE frequency of 2.86 GHz and produces pulses of approximately 0.1 V peak and 1.1 ns width. 

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5. Exceptional point of sixth-order degeneracy in a modified coupled-resonator optical waveguide

   https://doi.org/10.1364/JOSAB.385198  

   Nada, Mohamed_Y; Capolino, Filippo (July 2020, Journal of the Optical Society of America B)

   We demonstrate for the first time, to our knowledge, the occurrence of asixth-order exceptional point of degeneracy (EPD) in a realistic multimode optical photonic structure by using a modified periodic coupled-resonator optical waveguide (CROW) at the optical wavelength ${\mathrm{\lambda <\#comment/>}}_e=1550\text{nm}$. The sixth-order EPD is obtained in a CROW without the need of loss or gain, and such an EPD corresponds to a very special band edge of the periodic photonic structure where six eigenmodes coalesce, so we refer to it as the sixth-order degenerate band edge (6DBE). Moreover, we report a new scaling law of the quality factor $Q$of an optical cavity made of such a periodic 6DBE-CROW with cavity length as $Q\propto <\#comment/>N^7$, when operating near the 6DBE with $N$being the number of unit cells in the periodic finite-length CROW. Furthermore, we elaborate on the application of the 6DBE to ultralow-threshold lasers. We present a novel scaling law of the lasing threshold that scales as $N^{-<\#comment/>7}$when operating near the 6DBE. Also, we show the superiority of the threshold scaling of the 6DBE-CROW to the scaling of another CROW with the same size operating near a fourth-order EPD that is often referred to as the degenerate band edge (DBE). The lasing threshold scaling of the DBE-CROW laser is shown here for the first time to our knowledge. We also discuss the high sensitivity of the proposed 6DBE-CROW to perturbations, which may find applications in sensors, modulators, optical switches, nonlinear devices, and $Q$-switching cavities. 

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