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1. Experimental demonstration of exceptional points of degeneracy in linear time periodic systems and exceptional sensitivity

   https://doi.org/10.1063/5.0084849  

   Kazemi, Hamidreza ; Nada, Mohamed Y. ; Nikzamir, Alireza ; Maddaleno, Franco ; Capolino, Filippo ( April 2022 , Journal of Applied Physics)

   We present the experimental demonstration of the occurrence of exceptional points of degeneracy (EPDs) in a single resonator by introducing a linear time-periodic variation of one of its components. This is in contrast with the requirement of two coupled resonators with parity time-symmetric systems with precise values of gain and loss. In the proposed scheme, only the tuning of the modulation frequency is required, which is easily achieved in electronic systems. The EPD is a point in a system parameters’ space at which two or more eigenstates coalesce, and this leads to unique properties not occurring at other non-degenerate operating points. We show theoretically and experimentally the existence of a second-order EPD in a time-varying single resonator. Furthermore, we measure the sensitivity of the proposed system to a small structural perturbation and show that the two shifted system’s eigenfrequencies are well detected even for relative perturbations of [Formula: see text], with distinguished peaks well above the noise floor. We show that the regime of operation of the system at an EPD leads to a unique square-root-like sensitivity, which can devise new exceptionally sensitive sensors based on a single resonator by simply applying time modulation. 

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2. High-sensitivity in various gyrator-based circuits with exceptional points of degeneracy

   https://doi.org/10.1051/epjam/2022005  

   Rouhi, Kasra ; Nikzamir, Alireza ; Figotin, Alexander ; Capolino, Filippo ( January 2022 , EPJ Applied Metamaterials)

   Mann, Sander ; Vellucci, Stefano (Ed.)

   Exceptional points of degeneracy (EPD) can enhance the sensitivity of circuits by orders of magnitude. We show various configurations of coupled LC resonators via a gyrator that support EPDs of second and third-order. Each resonator includes a capacitor and inductor with a positive or negative value, and the corresponding EPD frequency could be real or imaginary. When a perturbation occurs in the second-order EPD gyrator-based circuit, we show that there are two real-valued frequencies shifted from the EPD one, following a square root law. This is contrary to what happens in a Parity-Time (PT) symmetric circuits where the two perturbed resonances are complex valued. We show how to get a stable EPD by coupling two unstable resonators, how to get an unstable EPD with an imaginary frequency, and how to get an EPD with a real frequency using an asymmetric gyrator. The relevant Puiseux fractional power series expansion shows the EPD occurrence and the circuit's sensitivity to perturbations. Our findings pave the way for new types of high-sensitive devices that can be used to sense physical, chemical, or biological changes. 

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3. Linear Time-Periodic Systems with Exceptional Points of Degeneracy

   https://doi.org/10.1109/ICEAA.2019.8879088  

   Kazemi, Hamidreza ; Nada, Mohamed Y. ; Mealy, Tarek ; Abdelshafy, Ahmed F. ; Capolino, Filippo ( September 2019 , 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA))

   We show how exceptional points of degeneracy (EPDs), which a coalescence of multiple eigenmodes, emerge in a linear time-periodic (LTP) systems. We establish the necessary conditions that yield an EPD in a single LTP LC resonator, however, the presented theory can be generalized to any kind of resonator with a time varying element. Furthermore, we propose an application of the EPD in a LTP LC resonator as a sensing device and show the ultra-sensitivity of such system to external perturbations. 

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4. 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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5. Microwave Circuits with Exceptional Points and Applications in Oscillators and Sensors

   https://doi.org/10.1109/MMS.2018.8611828  

   Nada, Mohamed Y. ; Kazemi, Hamidreza ; Abdelshafy, Ahmed F. ; Yazdi, Farshad ; Oshmarin, Dmitry ; Mealy, Tarek ; Figotin, Alex ; Capolino, Filippo ( October 2018 , 2018 18th Mediterranean Microwave Symposium (MMS), 31 Oct.-2 Nov. 2018, Istanbul, Turkey)

   We investigate wave properties in coupled transmission lines (CTLs) under a special condition known as the exceptional point of degeneracy (EPD) at which two or more of the supported eigenmodes of the system coalesce. At an EPD, not only the eigenvalues (resonances or wavenumbers) of the system (a resonator or a waveguide) coalesce but also the eigenvectors (polarization states) coalesce, and the number of coalescing eigenmodes defines the order of the degeneracy. We investigate different structures, either periodic or uniform CTLs, that are capable of exhibiting EPDs in their dispersion diagram. Secondly, we show an experimental verification of the existence of EPDs through measuring the dispersion of microstrip-based CTLs in the microwave spectrum. For antenna array configurations, we discuss the effect of CTLs radiative and dissipative losses on EPDs and how introducing gain to the CTLs compensate for such losses restoring the EPD in a fully radiating array, in what we define as the gain and distributed-radiation balance regime. Therefore, we show how to obtain large linear and planar arrays that efficiently generate microwave oscillations, and by spatial combination they are able to generate collimated beams with large radiation intensity. Finally, we show other promising applications based on the concept of EPDs in ultra-sensitive sensors or reconfigurable antennas. 

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