Exceptional Points Of Degeneracy Research Articles

Non-linear coherent perfect absorption in the proximity of exceptional points

Coherent perfect absorption is one of the possibilities to get high absorption but typically suffers from being a resonant phenomena, i.e., efficient absorption only in a local frequency range. Additionally, if applied in high power applications, the understanding of the interplay of non-linearities and coherent perfect absorption is crucial. Here we show experimentally and theoretically the formation of non-linear coherent perfect absorption in the proximity of exceptional point degeneracies of the zeros of the scattering function. Using a microwave platform, consisting of a lossy nonlinear resonator coupled to two interrogating antennas, we show that a coherent incident excitation can trigger a self-induced perfect absorption once its intensity exceeds a critical value. Note, that a (near) perfect absorption persists for a broad-band frequency range around the nonlinear coherent perfect absorption condition. Its origin is traced to a quartic behavior that the absorbance spectrum acquires in the proximity of the exceptional points of the nonlinear scattering operator.

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

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.

How to achieve exceptional points in coupled resonators using a gyrator or PT-symmetry, and in a time-modulated single resonator: high sensitivity to perturbations

We study the rise of exceptional points of degeneracy (EPD) in various distinct circuit configurations such as gyrator-based coupled resonators, coupled resonators with PT-symmetry, and in a single resonator with a time-varying component. In particular, we analyze their high sensitivity to changes in resistance, capacitance, and inductance and show the high sensitivity of the resonance frequency to perturbations. We also investigate stability and instability conditions for these configurations; for example, the effect of losses in the gyrator-based circuit leads to instability, and it may break the symmetry in the PT-symmetry-based circuit, also resulting in instabilities. Instability in the PT-symmetry circuit is also generated by breaking PT-symmetry when one element (e.g., a capacitor) is perturbed due to sensing. We have turned this instability “inconvenience” to an advantage, and we investigate the effect of nonlinear gain in the PT-symmetry coupled-resonator circuit and how this leads to an oscillator with oscillation frequency very sensitive to perturbation. The circuits studied in this paper have the potential to lead the way for a more efficient generation of high-sensitivity sensors that can detect very small changes in chemical, biological, or physical quantities.

Exceptional points of degeneracy in traveling wave tubes

The traveling wave tube (TWT) is a powerful vacuum electronic device used to amplify radio-frequency signals with numerous applications, including radar, television, and telephone satellite communications. TWT design in a nutshell comprises a pencil-like electron beam (e-beam) in vacuum interacting with a guiding slow-wave structure (SWS). In our studies here, the e-beam is represented by one-dimensional electron flow and SWS is represented by a transmission line (TL). The interaction between the e-beam and the TL is modeled by an analytic theory that generalizes the well-known Pierce model by taking into account the so-called space-charge effects, particularly electron-to-electron repulsion (debunching). Many important aspects of the analytic theory of TWTs have been already analyzed in our monograph on the subject. The focus of the studies here is on degeneracies of the TWT dispersion relations, particularly on exceptional points of degeneracy and their applications. The term exceptional point of degeneracy (EPD) refers to the property of the relevant matrix to have a nontrivial Jordan block structure. Using special parameterization, particularly suited to chosen EPD, we derive exact formulas for the relevant Jordan basis, including the eigenvectors and the so-called root vector associated with the Jordan block. Based on these studies, we develop a constructive approach to sensing of small signals.

Waveguides with Exceptional Points of Degeneracy of Order 2, 3, 4 and 6 without Loss and Gain

We show various examples of fully-planar two-way and three-way microstrip coupled waveguides that exhibit second, third, fourth and sixth order exceptional points of degeneracy (EPD). EPDs are shown in reciprocal waveguides that do not have gain or loss. The occurrence of EPDs is observed using three methods: (i) the mathematical description of modes and the similarity of the system matrix to a Jordan block; (ii) the dispersion diagram with prescribed flatness; (iii) the coalescence of eigenvectors that is observed analytically or experimentally. We have defined a coalescence parameter that is used in conjunction with experimental results to measure the separation of the coalescing eigenvectors (polarization states). The presented structures can serve various applications like leaky-wave antennas, distributed amplifiers, oscillators, delay lines, pulse generators, and sensors.

High-Power Backward-Wave Oscillator Using Folded Waveguide With Distributed Power Extraction Operating at an Exceptional Point

The concept of exceptional point of degeneracy (EPD) is used to conceive a degenerate synchronization regime that is able to enhance the level of output power and power conversion efficiency for backward wave oscillators (BWOs) operating at millimeter-wave and Terahertz frequencies. Standard BWOs operating at such high frequency ranges typically generate output power not exceeding tens of watts with very poor power conversion efficiency in the order of 1%. The novel concept of degenerate synchronization for the BWO based on a folded waveguide is implemented by engineering distributed gain and power extraction along the slow-wave waveguide. The distributed power extraction along the folded waveguide is useful to satisfy the necessary conditions to have an EPD at the synchronization point. Particle-in-cell (PIC) simulation results shows that BWO operating at an EPD regime is capable of generating output power exceeding 3 kwatts with conversion efficiency of exceeding 20% at frequency of 88.5 GHz.

Exceptional degeneracies in traveling wave tubes with dispersive slow-wave structure including space-charge effect

The interaction between a linear electron beam and a guided electromagnetic wave is studied in the context of exceptional points of degeneracy (EPD) supported by such an interactive system, focusing on a linear beam traveling wave tube (TWT) with a realistic helix waveguide slow-wave structure (SWS). The interaction is formulated by an analytical model that is a generalization of the Pierce model, assuming a one-dimensional electron flow along a dispersive single-mode guiding SWS and taking into account space-charge effects. The augmented model using phase velocity and characteristic impedance obtained via full-wave simulations is validated by calculating gain vs frequency and comparing it with that from more complex electron beam simulators. This comparison also shows the accuracy of our new model compared with respect to the nondispersive Pierce model. EPDs are then investigated using the augmented model, observing the coalescence of complex-valued wavenumbers and the system's eigenvectors. The point in the complex dispersion diagram at which the TWT-system starts/ceases to exhibit a convection instability, i.e., a mode starts/ceases to grow exponentially along the TWT, is the EPD. We also demonstrate the EPD existence by showing that the Puiseux fractional power series expansion well approximates the bifurcation of the dispersion diagram at the EPD. This latter concept also explains the “exceptional” sensitivity of the TWT-system to changes in the beam's electron velocity when operating near an EPD.

Scaling theory of absorption in the frozen mode regime.

A stationary inflection point (SIP) of the Bloch dispersion relation of a periodic system is a prominent example of an exceptional point degeneracy (EPD) where three Bloch eigenmodes coalesce. The scattering problem for a bounded photonic structure supporting a SIP features the frozen mode regime (FMR), where the incident wave is converted into the "frozen mode" with vanishing group velocity and diverging amplitude. We analyze the effect of losses and disorder on the FMR and develop a scaling formalism for the absorbance in the FMR that takes into consideration losses, disorder, and system size. The signatures of the EPD appear as an abrupt growth of absorbance for system sizes greater than a characteristic length that follows a parallel resistance law involving the absorption length and the Anderson localization length.

High-Power X -Band Relativistic Backward-Wave Oscillator with Exceptional Synchronous Regime Operating at an Exceptional Point

Abstract An exceptional point of degeneracy (EPD) is induced in a system made of an electron beam interacting with an electromagnetic (EM) guided mode. This enables a degenerate synchronous regime in backward wave oscillators (BWOs) where the electron beams provides distributed gain to the EM mode with distributed power extraction. Current particle-in-cell simulation results demonstrate that BWOs operating at an EPD have a starting-oscillation current that scales quadratically to a non-vanishing value for long interaction lengths and therefore have higher power conversion efficiency at arbitrarily higher level of power generation compared to standard BWOs.

Enhanced avionic sensing based on Wigner's cusp anomalies.

Typical sensors detect small perturbations by measuring their effects on a physical observable, using a linear response principle (LRP). It turns out that once LRP is abandoned, new opportunities emerge. A prominent example is resonant systems operating near Nth-order exceptional point degeneracies (EPDs) where a small perturbation ε ≪ 1 activates an inherent sublinear response [Formula: see text] in resonant splitting. Here, we propose an alternative sublinear optomechanical sensing scheme that is rooted in Wigner's cusp anomalies (WCAs), first discussed in the framework of nuclear reactions: a frequency-dependent square-root singularity of the differential scattering cross section around the energy threshold of a newly opened channel, which we use to amplify small perturbations. WCA hypersensitivity can be applied in a variety of sensing applications, besides optomechanical accelerometry discussed in this paper. Our WCA platforms are compact, do not require a judicious arrangement of active elements (unlike EPD platforms), and, if chosen, can be cavity free.

Exceptional degeneracy in a waveguide periodically loaded with discrete gain and radiation loss elements

We demonstrate that a periodic waveguide comprising of uniform lossless segments together with discrete gain and radiating elements supports exceptional points of degeneracy (EPDs). We provide analytical expressions for all possible conditions that guarantee the occurrence of an EPD, i.e., the coalescence of eigenvalues and eigenvectors. We show that EPDs are not only achieved using symmetric gain and radiation periodic loading, but they are also obtained using asymmetric gain and radiation loss conditions. We illustrate the characteristics of the degenerate electromagnetic modes, showing the dispersion diagram and discussing the tunability of the EPD frequency. We show a special condition, and we refer to it as a parity-time-glide symmetry, which leads to a degeneracy that is occurring at all frequencies of operation. The class of EPDs proposed in this work is very promising for many applications that incorporate discrete-distributed coherent sources and radiation loss elements; operating in the vicinity of such special degeneracy conditions leads to a potential performance enhancement in a variety of microwave and optical resonators, antennas, and devices and can be extended to a new class of active integrated antenna arrays and radiating laser arrays.

Enhanced energy harvesting near exceptional points in systems with (pseudo-)PT-symmetry

Exceptional point degeneracies, occurring in non-Hermitian systems, have challenged many well established concepts and led to the development of remarkable technologies. Here, we propose a family of autonomous motors whose operational principle relies on exceptional points via the opportune implementation of a (pseudo-)PT-symmetry and its spontaneous or explicit violation. These motors demonstrate a parameter domain of coexisting high efficiency and maximum work. In the photonic framework, they can be propelled by thermal radiation from the ambient thermal reservoirs and utilized as autonomous self-powered microrobots, or as micro-pumps for microfluidics in biological environments. The same designs can be also implemented with electromechanical elements for harvesting ambient mechanical (e.g., vibrational) noise for powering a variety of auxiliary systems. We expect that our proposal will contribute to the research agenda of energy harvesting by introducing concepts from mathematical and non-Hermitian wave physics.

Perturbations of circuit evolution matrices with Jordan blocks

In our prior studies, we synthesized special circuits with evolution matrices featuring degenerate eigenfrequencies and nontrivial Jordan blocks. The degeneracy of this type is sometimes referred to as exceptional point of degeneracy (EPD). Our focus here is on the simplest of our circuits featuring EPDs that are composed of only two LC-loops coupled by a gyrator. These circuits, when near an EPD state, can be used for enhanced sensitivity applications. With that in mind, we develop here a comprehensive perturbation theory for these simple circuits near an EPD. Using this theory, we propose an approach to sensing, allowing one to benefit from the proximity to an EPD on the one hand when providing for stable operation on the other hand. We also address a broader scope of problems related to perturbations of Jordan blocks and their numerical treatment that allow us to effectively detect proximity to Jordan blocks.

Observing exceptional point degeneracy of radiation with electrically pumped photonic crystal coupled-nanocavity lasers

Controlling gain and loss of coupled optical cavities can induce non-Hermitian degeneracies of eigenstates, called exceptional points (EPs). Various unconventional phenomena around EPs have been reported, and are expected to incorporate extra functionalities into photonic devices. The eigenmode exactly under EP degeneracy is also predicted to exhibit enhanced radiation. However, such responses have yet to be observed in on-chip lasers because of both the limited controllability of their gain and loss and the lifting of degeneracy by pump-induced cavity detuning. Here, we report, to the best of our knowledge, the first non-Hermitian nanophotonic platform based on two electrically pumped photonic crystal lasers and its spontaneous emission at EP degeneracy. Systematically tuned and independent current injection to our wavelength-scale active heterostructure cavities enables us to demonstrate the clear EP phase transition of their spontaneous emission, accompanied with the spectral coalescence of coupled modes and reversed pump dependence of the intensity. Furthermore, we find experimentally and confirm theoretically the peculiar squared Lorentzian emission spectrum very near the exact EP, which indicates a four-fold enhancement of the photonic local density of states induced purely by the degeneracy. Our results open a new pathway to engineer the light–matter interaction by non-Hermiticity and explore larger reconfigurable laser arrays for further non-Hermitian features and physics.

Frozen Mode in Three-Way Periodic Microstrip Coupled Waveguide

We demonstrate theoretically and experimentally that a simple periodic fully planar three-way microstrip coupled waveguide exhibits a stationary inflection point (SIP). The SIP is a third-order exceptional point of degeneracy where three eigenmodes of the guiding system coalesce to form a stationary frozen mode with zero group velocity. Here, the frozen mode is shown in a reciprocal three-way waveguide supporting three modes in each direction. We illustrate the occurrence of the frozen mode regime by observing the dispersion diagram and by the coalescence parameter that is a measure of the separation of the three coalescing eigenvectors. Results based on full-wave simulations and measurements demonstrate the coalescence of three modes and how this coalescence is not perfect due to the presence of losses in the microstrip structure at 2 GHz. The coalescence parameter is used to determine how close the three-way system is to the ideal frozen mode condition.

Ultra-Sensitive Radio Frequency Biosensor at an Exceptional Point of Degeneracy Induced by Time Modulation

We exploit the premises of exceptional points of degeneracy (EPDs) induced in linear time-periodic (LTP) systems to achieve extremely sensitive biosensors. The EPD is formed in a single LC resonator where the total capacitance is comprised of a time-varying capacitor in parallel to a biosensing capacitor. We use the time-periodic variation of a system parameter (e.g., capacitance) to achieve a second order EPD aiming at improving the sensitivity of liquid based radio frequency biosensors, leading to an intrinsic ultra sensitivity. We show the emergence of EPDs in such a system and the ultra sensitivity of the degenerate resonance frequency to perturbations compared to conventional RF sensors. Moreover, we investigate the capacitance and conductance variations of an interdigitated biosensing capacitor to the changes in the concentration of a biological material under test (MUT), leading to subsequent large changes in the resonance frequency of the LTP-LC resonator. A comparison with a standard LC resonator demonstrates the ultra-high sensitivity of the proposed LTP-LC based biosensor. In addition, we show the scalability of the biosensor sensitivity across different frequency ranges.

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

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: (i) propagating modes and evanescent modes, (ii) forward and backward propagating modes, or (iii) 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 sub-wavelength 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.

Exceptional Points of Degeneracy Directly Induced by Space–Time Modulation of a Single Transmission Line

We demonstrate how exceptional points of degeneracy (EPDs) are induced in a single transmission line (TL) directly by applying periodic space-time modulation to the per-unit-length distributed capacitance. In such space-time modulated (STM)-TL, two eigenmodes coalesce into a single degenerate one, in their eigenvalues (wavenumbers) and eigenvectors (voltage-current states) when the system approaches the EPD condition. The EPD condition is achieved by tuning a parameter in the space-time modulation, such as spatial or temporal modulation frequency, or the modulation depth. We unequivocally demonstrate the occurrence of the EPD by showing that the bifurcation of the wavenumber around the EPD is described by the Puiseux fractional power series expansion. We show that the first order expansion is sufficient to approximate well the dispersion diagram, and how this "exceptional" sensitivity of an STM-TL to tiny changes of any TL or modulation parameter enables a possible application as a highly sensitive TL sensor when operating at an EPD.

Exceptional Point of Degeneracy in a Backward-Wave Oscillator with Distributed Power Extraction

We show how an exceptional point of degeneracy (EPD) is formed in a system composed of an electron beam interacting with an electromagnetic mode guided in a slow wave structure (SWS) with distributed power extraction from the interaction zone. Based on this kind of EPD, a new regime of operation is devised for backward wave oscillators (BWOs) as a synchronous and degenerate regime between a backward electromagnetic mode and the charge wave modulating the electron beam. Degenerate synchronization under this EPD condition means that two complex modes of the interactive system do not share just the wavenumber, but they rather coalesce in both their wavenumbers and eigenvectors (polarization states). In principle this new condition guarantees full synchronization between the electromagnetic wave and the beam's charge wave for any amount of output power extracted from the beam, setting the threshold of this EPD-BWO to any arbitrary, desired, value. Indeed, we show that the presence of distributed radiation in the SWS results in having high-threshold electron-beam current to start oscillations which implies higher power generation. These findings have the potential to lead to highly efficient BWOs with very high output power and excellent spectral purity.

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

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.