We demonstrate for the first time, to our knowledge, the occurrence of a sixth-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 λ e = 1550 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 ∝ 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 − 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.

We present an approach and a theoretical framework for generating high-order exceptional points of degeneracy (EPDs) in photonic structures based on periodic coupled resonator optical waveguides (CROWs). Such EPDs involve the coalescence of Floquet-Bloch eigenwaves in CROWs, without the presence of gain and loss, which contrasts with the parity-time symmetry required to develop exceptional points based on gain and loss balance. The EPDs arise here by introducing symmetry breaking in a conventional chain of coupled resonators through periodic coupling to an adjacent uniform optical waveguide, which leads to unique modal characteristics that cannot be realized in conventional CROWs. Such remarkable characteristics include high quality factors (Q factors) and strong field enhancement, even without any mirrors at the two ends of a cavity. We show for the first time the capability of CROWs to exhibit EPDs of various orders, including the degenerate band edge (DBE) and the stationary inflection point. The proposed CROW of finite length shows an enhanced quality factor when operating near the DBE, and the Q factor exhibits an unconventional scaling with the CROW's length. We develop the theory of EPDs in such unconventional CROW using coupled-wave equations, and we derive an analytical expression for the dispersion relation. The proposed unconventional CROW concepts have various potential applications including Q switching, nonlinear devices, lasers, and extremely sensitive sensors.

We propose a novel scheme for enhancing the quality factor (Q factor) of coupled resonators optical waveguides (CROWs) side coupled to a straight waveguide when operating near a degenerate band edge (DBE). The DBE is an exceptional point of degeneracy occurring without the need of loss or gain when four Bloch eigenwaves coalesce, providing a resonance condition with field enhancement and other unique properties associated with such frozen mode. We derive an analytic expression that describes the flatness of the dispersion diagram at the DBE and show how this is related to engineering the quality factor. We report an anomalous scaling law of the quality factor for a cavity made of the DBE-CROW with the fifth power of the number of rings when operating at the resonance frequency closest to the DBE frequency, even in the presence of losses and structural perturbations. Remarkably, the Q factor of the proposed DBE-CROW can be engineered to exceed that of a single ring resonator having a diameter equal to the total DBE-CROW length; hence, the DBE leads to an overall huge area reduction. We also demonstrate the robustness of the DBE-CROW's Q factor scaling near the DBE against perturbations and disorders that could arise from tolerances in fabrication. The findings reported in this paper are critical for enhancing field's amplitudes to giant levels and the Q factor of ring resonators, and are very beneficial for various applications including four-wave mixing, Q switching, lasers, and highly sensitive sensors.

We present a general theory of exceptional points of degeneracy (EPD) in periodically time-variant systems. We show that even a single resonator with a time-periodic component is able to develop EPDs, contrary to parity-time- (PT) symmetric systems that require two coupled resonators. An EPD is a special point in a system parameter space at which two or more eigenmodes coalesce in both their eigenvalues and eigenvectors into a single degenerate eigenmode. We demonstrate the conditions for EPDs to exist when they are directly induced by time-periodic variation of a system without loss and gain elements. We also show that a single resonator system with zero time-average loss-gain exhibits EPDs with purely real resonance frequencies, yet the resonator energy grows algebraically in time since energy is injected into the system from the time-variation mechanism. Although the introduced concept and formalism are general for any time-periodic system, here, we focus on the occurrence of EPDs in a single LC resonator with time-periodic modulation. These findings have significant importance in various electromagnetic and photonic systems and pave the way for many applications, such as sensors, amplifiers, and modulators. We show a potential application of this time-varying EPD as a highly sensitive sensor.