Exceptional Points Of Degeneracy Research Articles

Simple reciprocal electric circuit exhibiting exceptional point of degeneracy

Simple reciprocal electric circuit exhibiting exceptional point of degeneracy

Unidirectional amplification in the frozen mode regime enabled by a nonlinear defect.

A stationary inflection point (SIP) is a spectral singularity of the Bloch dispersion relation ω(k) of a periodic structure where the first and the second derivatives of ω with respect to k vanish. An SIP is associated with a third-order exceptional point degeneracy in the spectrum of the unit-cell transfer matrix, where there is a collapse of one propagating and two evanescent Bloch modes. At the SIP frequency, the incident wave can be efficiently converted into the frozen mode with greatly enhanced amplitude and vanishing group velocity. This can be very attractive for applications, including light amplification. Due to its non-resonant nature, the frozen mode regime (FMR) has fundamental advantages over common cavity resonances. Here, we propose, a novel, to the best of our knowledge, scheme for FMR-based unidirectional amplifiers by leveraging a tailored amplification/attenuation mechanism and a single nonlinear defect. The defect breaks the directional symmetry of the periodic structure and enables nonlinearity-related unidirectional amplification/attenuation in the vicinity of the SIP frequency. We demonstrate the robustness of the amplification mechanism to local impurities and parasitic nonlinearity.

Frequency-Domain Analysis of an Oscillator With an Exceptional Point of Degeneracy

Frequency-Domain Analysis of an Oscillator With an Exceptional Point of Degeneracy

Time modulation to manage and increase the power harvested from external vibrations

We investigate how a single resonator with a time-modulated component extracts power from an external ambient source. The collected power is largely dependent on the precise modulation signal frequency choice. We focus on the power absorbed from external vibration using a mechanical resonator and how time modulation of the damper can make a significant difference in the amount of harvested power, leading to more than 10 times enhancement compared to an analogous system without time modulation. We also find that a narrow band pair of peak and dip in the spectrum of the absorbed power occurs because of the presence of an exceptional point of degeneracy (EPD). In this narrow frequency range, the delay between the damper modulating signal and the external vibrating signal largely affects the collected power. The high frequency-selectivity of EPD-induced power management could potentially be used in sensing and spectrometer applications.

Gain-loss-induced non-Abelian Bloch braids

Onsite gain-loss-induced topological braiding principle of non-Hermitian energy bands is theoretically formulated in multiband lattice models with Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss parameter is tuned across exceptional point degeneracy. Laboratory realizable effective-Hamiltonians are proposed to realize braid groups B2 and B3 of two and three bands, respectively. While B2 is trivially Abelian, the group B3 features non-Abelian braiding and energy permutation originating from the collective behavior of multiple exceptional points. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesizing exceptional materials for applications in topological computation and information processing.

Noise resilient exceptional-point voltmeters enabled by oscillation quenching phenomena

Exceptional point degeneracies (EPD) of linear non-Hermitian systems have been recently utilized for hypersensitive sensing. This proposal exploits the sublinear response that the degenerate frequencies experience once the system is externally perturbed. The enhanced sensitivity, however, might be offset by excess (fundamental and/or technical) noise. Here, we developed a self-oscillating nonlinear platform that supports transitions between two distinct oscillation quenching mechanisms – one having a spatially symmetric steady-state, and the other with an asymmetric steady-state – and displays nonlinear EPDs (NLEPDs) that can be employed for noise-resilient sensing. The experimental setup incorporates a nonlinear electronic dimer with voltage-sensitive coupling and demonstrates two-orders signal-to-noise enhancement of voltage variation measurements near NLEPDs. Our results resolve a long-standing debate on the efficacy of EPD-sensing in active systems above self-oscillating threshold.

Design of a Modified Coupled Resonators Optical Waveguide Supporting a Frozen Mode

We design a three-way silicon optical waveguide with the Bloch dispersion relation supporting a stationary inflection point (SIP). The SIP is a third order exceptional point of degeneracy (EPD) where three Bloch modes coalesce forming the frozen mode with greatly enhanced amplitude. The proposed design consists of a coupled resonators optical waveguide (CROW) coupled to a parallel straight waveguide. At any given frequency, this structure supports three pairs of reciprocal Bloch eigenmodes, propagating and/or evanescent. In addition to full-wave simulations, we also employ a so-called “hybrid model” that uses transfer matrices obtained from full-wave simulations of sub-blocks of the unit cell. This allows us to account for radiation losses and enables a design procedure based on minimizing the eigenmodes’ coalescence parameter. The proposed finite-length CROW displays almost unitary transfer function at the SIP resonance, implying a nearly perfect conversion of the input light into the frozen mode. The group delay and the effective quality factor at the SIP resonance show an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> scaling, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> is the number of unit cells in the cavity. The frozen mode in the CROW can be utilized in various applications like sensors, lasers and optical delay lines.

Lasing at a stationary inflection point

The concept of lasers based on the frozen mode regime in active periodic optical waveguides with a 3rd-order exceptional point of degeneracy (EPD) is advanced. The frozen mode regime in a lossless and gainless waveguide is associated with a stationary inflection point (SIP) in the Bloch dispersion relation, where three Bloch eigenmodes coalesce forming the frozen mode. As a practical example, we consider an asymmetric serpentine optical waveguide (ASOW). An ASOW operating near the SIP frequency displays a large group delay of a non-resonant nature that scales as the cube of the waveguide length, leading to a strong gain enhancement when active material is included. Therefore, a laser operating in the close vicinity of an SIP has a gain threshold that scales as a negative cube of the waveguide length. We determine that this scaling law is maintained in the presence of small distributed losses, such as radiation associated with waveguide bends and roughness. In addition, we show that although gain causes a distortion in the modes coalescing at the SIP, the properties of the frozen mode are relatively resistant to such small perturbations and we still observe a large degree of exceptional degeneracy for gain values that bring the system above threshold. Finally, our study also reveals that lasing near an SIP is favored over lasing near a photonic band edge located in close proximity to the SIP. In particular, we observe that an SIP-induced lasing in an ASOW displays lower gain threshold compared to lasing near the photonic regular band edge (RBE), even though the SIP resonance has a lower quality factor than the RBE resonance.

Frozen mode regime in an optical waveguide with a distributed Bragg reflector

We introduce a glide symmetric optical waveguide exhibiting a stationary inflection point (SIP) in the Bloch wavenumber dispersion relation. An SIP is a third-order exceptional point of degeneracy (EPD) where three Bloch eigenmodes coalesce to form a so-called frozen mode with vanishing group velocity and diverging amplitude. We show that the incorporation of chirped distributed Bragg reflectors and distributed coupling between waveguides in the periodic structure facilitates the SIP formation and greatly enhances the characteristics of the frozen mode regime. We confirm the existence of an SIP in two ways: by observing the flatness of the dispersion diagram and also by using a coalescence parameter describing the separation of the three eigenvectors collapsing on each other. We find that, in the absence of losses, both the quality factor and the group delay at the SIP grow with the cubic power of the cavity length. The frozen mode regime can be very attractive for light amplification and lasing in optical delay lines, sensors, and modulators.

Reconfigurable enhancement of actuation forces by engineered losses in non-Hermitian metamaterials

While boosting signals with amplification mechanisms is a well-established approach, attenuation mechanisms are typically considered an anathema because they degrade the efficiency of the structures employed to perform useful operations on these signals. An emerging alternate viewpoint promotes losses as a novel design element by utilizing the notion of exceptional point degeneracies (EPDs)—points in parameter space where the eigenvalues of the underlying system and the associated eigenvectors simultaneously coalesce. Here, we demonstrate a direct consequence of such eigenbasis collapse in elastodynamics—an unusual enhancement of actuation force by a judiciously designed non-Hermitian metamaterial supporting an EPD that is coupled to an actuation source. Intriguingly, the EPD enables this enhancement while maintaining a constant signal quality. Our work constitutes a proof-of-principle design which can promote a new class of reconfigurable nano-indenters and robotic-actuators. Importantly, it reveals the ramifications of non-Hermiticity in boosting the Purcell emissivity enhancement factor beyond its expected value, which can guide the design of metamaterials with enhanced emission that does not deteriorate signal quality for mechanical, acoustic, optical, and photonic applications.

Exceptional points of degeneracy with indirect band gap induced by mixing forward and backward propagating waves

We demonstrate that exceptional points of degeneracy (EPDs) are obtained in two coupled waveguides without resorting to gain and loss. We show the general concept that modes resulting from a proper coupling of forward and backward waves exhibit EPDs of order two and that there the group velocity vanishes. We verify our insight by using coupled mode theory and also by fullwave numerical simulations of light in a dielectric slab coupled to a grating, when one supports a forward wave whereas the other (the grating) supports a backward wave. We also demonstrate how to realize photonic indirect bandgaps in guiding systems supporting a backward and a forward wave, show its relations to the occurrence of EPDs, and offer a design procedure.

Highly Sensitive Coupled Oscillator Based on an Exceptional Point of Degeneracy and Nonlinearity

We propose a scheme for obtaining highly-sensitive oscillators in a coupled-resonator system with an exceptional point of degeneracy (EPD) and a small instability. The oscillator with the exceptional degeneracy is realized by using two coupled resonators with an almost balanced small-signal gain and loss, that saturates due to nonlinear effects of the active component, resulting in an oscillation frequency that is very sensitive to a perturbation of the circuit. Two cases are investigated, with two parallel LC resonators with balanced small-signal gain and loss that are either coupled wirelessly by mutual inductance or coupled-wired by a capacitor. This paper demonstrates theoretically and experimentally the conditions to obtain a second-order EPD oscillator and analyzes the ultrasensitivity of the oscillation frequency to components' perturbation, including the case of asymmetric perturbation that breaks PT-symmetry. We discuss the effects of nonlinearity on the performance of the oscillator and how the proposed scheme improves the sensing's sensitivity of perturbations. In contrast to previous methods, our proposed degenerate oscillator can sense positive or negative changes of a circuit component. The degenerate oscillator circuit may find applications in various areas such as ultrasensitive sensors, tunable oscillators and modulators.

Non-resonant exceptional points as enablers of noise-resilient sensors

Exceptional point degeneracies (EPDs) in the resonant spectrum of non-Hermitian systems have been recently employed for sensing due to the sublinear response of the resonance splitting when a perturbant interacts with the sensor. The sublinear response provides high sensitivity to small perturbations and a large dynamic range. However, the resonant-based EPD sensing abides to the resolution limit imposed by the resonant quality factors and by the signal-to-noise ratio reduction due to gain-elements. Moreover, it is susceptible to local mechanical disturbances and imperfections. Here, we propose a passive non-resonant (NR) EPD-sensor that is resilient to losses, local cavity variations, and noise. The NR-EPD describes the coalescence of Bloch eigenmodes associated with the spectrum of transfer matrices of periodic structures. This coalescence enables scattering cross-section cusps with a sublinear response to small detunings away from an NR-EPD. We show that these cusps can be utilized for enhanced noise-resilient sensing.

Observation of Exceptional Point Degeneracy with Current-injected Photonic Crystal Lasers

Observation of Exceptional Point Degeneracy with Current-injected Photonic Crystal Lasers

High-Sensitive Parity-Time Symmetric Oscillator in Coupled Transmission Lines With Nonlinear Gain

A scheme for generating oscillations based on an exceptional point of degeneracy (EPD) is proposed in two-coupled resonators made of two coupled transmission lines terminated on balanced gain and loss, exhibiting a double pole. The EPD is a point in the parameter space of the system at which two or more eigenmodes coalesce in both their eigenvalues (here, resonance frequencies) and eigenvectors. We show that a finite-length single transmission line terminated with gain and loss possesses no degeneracy point, whereas second-order EPDs are enabled in two finite-length coupled transmission lines (CTLs) terminated with balanced gain and loss. We demonstrate the conditions for EPDs to exist for three different termination configurations with balanced gain and loss, and show the eigenfrequency bifurcation at the EPD following the fractional power expansion series related to the Puiseux series. We study the oscillatory regime of operation assuming the gain element is nonlinear, and the extreme sensitivity of the degenerate self-oscillation frequency to perturbations and how it compares with the sensitivity of the linear-gain case. Finally, we show that the sensitivity of the EPD-CTL resonator is much higher than the one of a single-TL resonator. The very sensitive EPD based oscillator can be used as sensors when very small variations in a system shall be detected.

Third order modal exceptional degeneracy in waveguides with glide-time symmetry

The dispersion of a three-way waveguide is engineered to exhibit exceptional modal characteristics. Two coupled waveguides with Parity-Time (PT) symmetry have been previously demonstrated to exhibit second order exceptional points of degeneracy (EPDs). In this work, we introduce and investigate a particular class of EPDs, applicable from radio frequency to optical wavelengths, whereby three coupled waveguides satisfy Glide-Time (GT) symmetry to exhibit a third order modal degeneracy with a real-valued wavenumber. GT symmetry involves glide symmetry of lossless/gainless components of the waveguide in addition to changing the sign of passive/active elements while applying a glide symmetry operation. This GT-symmetry condition allows three Floquet-Bloch eigenmodes of the structure to coalesce to a real-valued wavenumber at a single frequency, in addition of having one branch of the dispersion diagram with a purely real wavenumber. The proposed scheme may have applications including but not limited to distributed amplifiers, radiating arrays, and sensors, from radio frequency to optics.

Frozen Mode in an Asymmetric Serpentine Optical Waveguide

The existence of a frozen mode in a periodic serpentine waveguide with broken longitudinal symmetry is demonstrated numerically. The frozen mode is associated with a stationary inflection point (SIP) of the Bloch dispersion relation, where three Bloch eigenmodes collapse on each other, as it is an exceptional point of order three. The frozen mode regime is characterized by vanishing group velocity and enhanced field amplitude, which can be very attractive in various applications including dispersion engineering, lasers, and delay lines. Useful and simple design equations that lead to realization of the frozen mode by adjusting a few parameters are derived. The trend in group delay and quality factor with waveguide length that is peculiar of the frozen mode is shown. The symmetry conditions for the existence of exceptional points of degeneracy associated with the frozen mode are also discussed.

Triple Ladder Lumped Circuit With Sixth Order Modal Exceptional Degeneracy

We introduce a circuit topology based on a simple triple-ladder circuit realized with lumped reactive components that provides a sixth order degenerate band-edge (6DBE). The 6DBE is a special kind of sixth-order exceptional point of degeneracy in a lossless and gainless periodic ladder. This degeneracy provides a very flat band edge in the phase-frequency dispersion diagram. The proposed topology exhibits unique structured resonance features associated with a high loaded Q-factor. We investigate the Floquet-Bloch modes in an infinite-length periodic triple-ladder and their dispersion relation using the S parameter formalism. We also provide the approximate analytic expressions of the eigenmodes and dispersion relation around the degenerate point based on the Puiseux series expansion. We investigate the filtering characteristics of a finite-length structure terminated with loads to highlight the special properties of the 6DBE compared to ladders with regular band edge (RBE) and fourth order degenerate band edge (DBE). The circuit framework introduced here with a 6DBE can be exploited in designing novel high Q-factor oscillators, filters, sensors, and pulse shaping networks.

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

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 0.3%, 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.

Exceptional point in a degenerate system made of a gyrator and two unstable resonators

We demonstrate that a circuit comprising two unstable LC resonators coupled via a gyrator supports an exceptional point of degeneracy (EPD) with purely real eigenfrequency. Each of the two resonators includes either a capacitor or an inductor with a negative value, showing a purely imaginary resonance frequency when not coupled to the other via the gyrator. With external perturbation imposed on the system, we show analytically that the resonance frequency response of the circuit follows the square-root dependence on perturbation, leading to possible sensor applications. Furthermore, the effect of small losses in the resonators has been investigated, and we show that losses lead to instability. In addition, the EPD occurrence and sensitivity are demonstrated by showing that the relevant Puiseux fractional power series expansion describes the eigenfrequency bifurcation near the EPD. The EPD has the great potential to enhance the sensitivity of a sensing system by orders of magnitude.