Parity-time Symmetric Systems Research Articles

Design Considerations for Single-Mode Microring Lasers Using Parity-Time-Symmetry

Coupled microring arrangements with balanced gain and loss, also known as parity-time symmetric systems, are investigated both analytically and experimentally. In these configurations, stable single-mode lasing can be achieved at pump powers well above threshold. This self-adaptive mode management technique is broadband and robust to small fabrication imperfections. The results presented in this paper provide a new avenue in designing mode selective chip-scale in-plane semiconductor lasers by utilizing the complex dynamics of coupled gain/loss cavities.

Design Considerations for Single-Mode Microring Lasers Using Parity–Time Symmetry

Coupled microring arrangements with balanced gain and loss, also known as parity-time symmetric systems, are investigated both analytically and experimentally. In these configurations, stable single-mode lasing can be achieved at pump powers well above threshold. This self-adaptive mode management technique is broadband and robust to small fabrication imperfections. The results presented in this paper provide a new avenue in designing mode selective chip-scale in-plane semiconductor lasers by utilizing the complex dynamics of coupled gain/loss cavities.

Parity-Time Symmetry in Acoustics: Theory, Devices, and Potential Applications

Parity-time (PT) symmetric systems based on balanced distributions of gain and loss have attracted significant attention in wave physics, enabling the possibility of compensating material losses in photonic systems, and of inducing exceptional scattering phenomena, offering new possibilities for wave manipulation. A successful transition of these theoretical concepts to devices and practical applications requires a sufficient level of control over the frequency dispersion of the involved gain and loss elements, in order to engineer a PT-symmetric system that is stable and practical. Here, we show that such a degree of control can be achieved for acoustic waves, and obtain stable PT-symmetric acoustic systems that can manipulate sound in unprecedented ways. We describe the theory, design and potentials of PT-based negative refraction and phase compensation devices in free-space, induced by a pair of PT-symmetric acoustic metasurfaces engineered from active electro-acoustic resonators. Our study proves that acoustic systems represent an ideal platform to apply the unique scattering phenomena associated with PT-symmetric systems operated at exceptional points. We envision applications in acoustic imaging, furtive sensing, and noise control, and extension of these concepts to electromagnetics, optics, and nanophotonic systems.

Unidirectional Bragg Gratings Using Parity-Time Symmetry Breaking in Plasmonic Systems

Optical systems following concepts of parity-time (PT) symmetry have attracted significant attention because of their extraordinary behavior such as unidirectional reflectance or power oscillation. PT symmetric optical systems are realized by judiciously manipulating the complex refractive index to produce even- and odd-symmetric distributions for the real and imaginary indices, respectively. We propose two PT symmetric Bragg gratings based on step-in-width metal stripes and dielectric-loaded metal stripes operating with long-range surface plasmon polaritons. The gratings are designed to operate near 880 nm because optical gain can be conveniently provided by IR140-doped PMMA. Asymmetric reflectance is predicted in the proposed gratings based on modal and transfer matrix method computations. Moreover, we analyzed pulse reshaping and energy transport in generic gratings, and in the proposed plasmonic gratings, in terms of group and energy velocities. It is found that the group and energy velocities are dispersionless at the PT symmetry breaking point. Also, the group velocity dispersion can be inverted by changing the PT symmetric state from broken to unbroken or vice versa. Our designs are practical because a large left-right asymmetric reflectance contrast is produced for a wide range of physical dimensions. The proposed gratings are suitable as on-chip devices for optical processing providing new functionality such as switching and controlling the time delay of a data pulse without distortion.

Anomalous parity-time–symmetry transition away from an exceptional point

Parity-time (PT) symmetric systems have two distinguished phases, e.g., one with real energy eigenvalues and the other with complex conjugate eigenvalues. To enter one phase from the other, it is believed that the system must pass through an exceptional point, which is a non-Hermitian degenerate point with coalesced eigenvalues and eigenvectors. In this letter we reveal an anomalous PT transition that takes place away from an exceptional point in a nonlinear system: as the nonlinearity increases, the original linear system evolves along two distinct PT-symmetric trajectories, each of which can have an exceptional point. However, the two trajectories collide and vanish away from these exceptional points, after which the system is left with a PT-broken phase. We first illustrate this phenomenon using a coupled mode theory and then exemplify it using paraxial wave propagation in a transverse periodic potential.

Nonlinear waves inPT-symmetric systems

Recent progress on nonlinear properties of parity-time ($\cal PT$-) symmetric systems is comprehensively reviewed in this article. $\cal PT$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\cal PT$ symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a $\cal PT$-symmetric system. The natural inclusion of nonlinearity into these $\cal PT$ systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above $\cal PT$-symmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear $\cal PT$-symmetric systems arising from various physical disciplines are presented; nonlinear properties of these systems are thoroughly elucidated; and relevant experimental results are described. In addition, emerging applications of $\cal PT$ symmetry are pointed out.

Giant enhancement of optical high-order sideband generation and their control in a dimer of two cavities with gain and loss

Parity-time ($\mathcal{PT}$) symmetric systems, which rely on the balanced gain-loss condition and render the Hamiltonian non-Hermitian, have provided a new platform to engineer effective light-matter interactions in recent years. Here we explore the high-order sideband features of the output fields obtained from a $\mathcal{PT}$-symmetric optical system consisting of a passive nonlinear cavity coupled to an active linear cavity. By employing a perturbation technique, we derive analytic formulas used to determine the nonlinear transmission coefficient of optical second-order sideband in this structure. Using experimentally achievable parameters, it is clearly shown that the efficiency of the second-order sideband generation can be greatly enhanced in the $\mathcal{PT}$-symmetric dimer, extremely in the vicinity of the transition point from unbroken- to broken-$\mathcal{PT}$ regimes. Moreover, we further analyzed the influences of the system parameters, including the photon-tunneling rate between two cavities, Kerr nonlinearity strength, and optical detuning, on the second-order sideband generation. Subsequently we investigate the higher-order sideband output spectrum by numerical simulations, where the sideband amplitude also is largely enhanced in the $\mathcal{PT}$-symmetric arrangement, compared with the passive-passive double-cavity system. Our obtained results provide a new avenue for acquiring optical high-order sidebands and operating light, which may inspire further applications in chip-scale optical communications and optical frequency combs.

Exceptional Contours and Band Structure Design in Parity-Time Symmetric Photonic Crystals.

We investigate the properties of two-dimensional parity-time symmetric periodic systems whose non-Hermitian periodicity is an integer multiple of the underlying Hermitian system's periodicity. This creates a natural set of degeneracies that can undergo thresholdless PT transitions. We derive a k·p perturbation theory suited to the continuous eigenvalues of such systems in terms of the modes of the underlying Hermitian system. In photonic crystals, such thresholdless PT transitions are shown to yield significant control over the band structure of the system, and can result in all-angle supercollimation, a PT-superprism effect, and unidirectional behavior.

Controlling the exceptional points of parity-time symmetric acoustics

Parity-time (PT) symmetric systems experience phase transition between PT “exact” and “broken” phases at exceptional point. These PT phase transitions contribute significantly to the design of single mode lasers, coherent perfect absorbers, isolators, diodes, etc. However, such exceptional points are extremely difficult to access in practice because of the dispersive behavior of most loss and gain materials required in PT symmetric systems. Here, we introduce a method to systematically tame these exceptional points and control PT phases. Our experimental demonstration hinges on an active acoustic element that realizes a complex-valued potential and simultaneously controls the multiple interference in the structure. The manipulation of exceptional points offers new routes to broaden applications for PT symmetric physics in acoustics, optics, microwaves, and electronics, which are essential for sensing, communication, and imaging.

Nonlinear switching and solitons in PT‐symmetric photonic systems

AbstractOne of the challenges of the modern photonics is to develop all‐optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested Parity‐Time (PT) symmetric photonic systems with alternating regions of gain and loss can bring novel functionalities. In such systems, losses are as important as gain and, depending on the structural parameters, gain compensates losses. Generally, PT systems demonstrate nontrivial non‐conservative wave interactions and phase transitions, which can be employed for signal filtering and switching, opening new prospects for active control of light. In this review, we discuss a broad range of problems involving nonlinear PT‐symmetric photonic systems with an intensity‐dependent refractive index. Nonlinearity in such PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly‐induced PT‐symmetry breaking, and all‐optical switching. Nonlinear PT‐symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control. image

The asymmetric solitons in two-dimensional parity–time-symmetric potentials

We investigate the properties in two-dimensional (2D) special parity–time (PT) symmetric complex potentials. The linear case of this special 2D PT-symmetric complex potential and self-focusing nonlinear cases are discussed. For linear case, the eigenvalues and eigenfunction for different loss or gain level of the PT-symmetric complex potentials are obtained numerically. For nonlinear cases, the existence of asymmetric solitons and PT-symmetric solitons is studied in this PT symmetric system. The eigenvalue for linear case is equal to the critical propagation constant bc of existing PT-symmetric solitons. When the PT-symmetric soliton's propagation constant reaches a certain threshold bc1, a branch of asymmetric solitons can bifurcate out from the branch of PT-symmetric solitons.

Spawning rings of exceptional points out of Dirac cones.

The Dirac cone underlies many unique electronic properties of graphene and topological insulators, and its band structure--two conical bands touching at a single point--has also been realized for photons in waveguide arrays, atoms in optical lattices, and through accidental degeneracy. Deformation of the Dirac cone often reveals intriguing properties; an example is the quantum Hall effect, where a constant magnetic field breaks the Dirac cone into isolated Landau levels. A seemingly unrelated phenomenon is the exceptional point, also known as the parity-time symmetry breaking point, where two resonances coincide in both their positions and widths. Exceptional points lead to counter-intuitive phenomena such as loss-induced transparency, unidirectional transmission or reflection, and lasers with reversed pump dependence or single-mode operation. Dirac cones and exceptional points are connected: it was theoretically suggested that certain non-Hermitian perturbations can deform a Dirac cone and spawn a ring of exceptional points. Here we experimentally demonstrate such an 'exceptional ring' in a photonic crystal slab. Angle-resolved reflection measurements of the photonic crystal slab reveal that the peaks of reflectivity follow the conical band structure of a Dirac cone resulting from accidental degeneracy, whereas the complex eigenvalues of the system are deformed into a two-dimensional flat band enclosed by an exceptional ring. This deformation arises from the dissimilar radiation rates of dipole and quadrupole resonances, which play a role analogous to the loss and gain in parity-time symmetric systems. Our results indicate that the radiation existing in any open system can fundamentally alter its physical properties in ways previously expected only in the presence of material loss and gain.

Nonlinear parity-time symmetric closed-form optical quadrimer waveguides: attractor perspective

We report a study on a closed-form nonlinear parity-time symmetric optical quadrimer waveguides system with a specific coupling scheme. The system yields power saturation behavior in the modes, which may be attributed to the inherent attractor in the system. A detailed analysis has been provided to confirm the attractor aspect of the system. This work also addresses a crucial issue regarding choice of initial conditions while carrying out numerical simulation for such systems.

Analytical Study on Propagation Dynamics of Optical Beam in Parity-Time Symmetric Optical Couplers* *Supported by the National Natural Science Foundation of China under Grant No. 11465008, the Hunan Provincial Natural Science Foundation under Grant Nos. 2015JJ4020 and 2015JJ2114, the Scientific Research Fund of Hunan Provincial Education Department under Grant No. 14A118

We present exact analytical solutions to parity-time (PT) symmetric optical system describing light transport in PT-symmetric optical couplers. We show that light intensity oscillates periodically between two waveguides for unbroken PT-symmetric phase, whereas light always leaves the system from the waveguide experiencing gain when light is initially input at either waveguide experiencing gain or waveguide experiencing loss for broken PT-symmetric phase. These analytical results agree with the recent experimental observation reported by Rüter et al. [Nat. Phys. 6 (2010) 192]. Besides, we present a scheme for manipulating PT symmetry by applying a periodic modulation. Our results provide an efficient way to control light propagation in periodically modulated PT-symmetric system by tuning the modulation amplitude and frequency.

Negative refraction and planar focusing based on parity-time symmetric metasurfaces.

We introduce a new mechanism to realize negative refraction and planar focusing using a pair of parity-time symmetric metasurfaces. In contrast to existing solutions that achieve these effects with negative-index metamaterials or phase conjugating surfaces, the proposed parity-time symmetric lens enables loss-free, all-angle negative refraction and planar focusing in free space, without relying on bulk metamaterials or nonlinear effects. This concept may represent a pivotal step towards loss-free negative refraction and highly efficient planar focusing by exploiting the largely uncharted scattering properties of parity-time symmetric systems.

Multi-rogue wave and multi-breather solutions inPT-symmetric coupled waveguides

Abstract The coupled nonlinear Schrodinger equation in parity-time symmetric coupled waveguides is studied by means of the modified Darboux transformation method. The hierarchies of rational solutions and breather solutions are generated from the plane wave solution. Some basic properties of multi-rogue waves and multi-breathers including the superposed Kuznetsov–Ma solitons, Akhmediev breathers and their combined structures are discussed. Our results might provide useful information for potential applications of synthetic parity-time symmetric systems in nonlinear optics and condensed matter physics.

Nonautonomous solitons in parity-time symmetric potentials

Some analytical solutions of a (1+1)-dimensional nonlinear Schrödinger equation with inhomogeneous diffraction and nonlinearity in the presence of the parity-time symmetric potential are derived. All characteristic parameters such as amplitudes, speeds and widths of solutions are adapted both to the diffraction and to nonlinearity variations. The dynamical behaviors of two kinds of nonautonomous solitons with different parity-time symmetric potentials are investigated respectively both in the exponential diffraction decreasing waveguide and in the periodic modulated system. Moreover, phase changes for two kinds of nonautonomous solitons are illustrated in detail. These results may provide alternative methods in potential applications of synthetic parity-time symmetric systems.

Discrete vortex solitons and parity time symmetry

We study the effect of lifting the degeneracy of vortex modes with a parity time (PT) symmetric defect, using discrete vortices in a circular array of nonlinear waveguides as an example. When the defect is introduced, the degenerate linear vortex modes spontaneously break PT symmetry and acquire complex eigenvalues, but nonlinear propagating modes with real propagation constants can still exist. The stability of nonlinear modes depends on both the magnitude and the sign of the vortex charge; thus PT symmetric systems offer new mechanisms to control discrete vortices.

Dark solitons and vortices inPT-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinearPTphase transitions

We consider nonlinear analogues of Parity-Time (PT) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter $\varepsilon$ controlling the strength of the PT-symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as $\varepsilon$ is further increased, the ground state and first excited state, as well as branches of higher multi-soliton (multi-vortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear PT-phase transition ---thus termed the nonlinear PT-phase transition. Past this critical point, initialization of, e.g., the former ground state leads to spontaneously emerging solitons and vortices.

Nonlinear Modes in Finite-DimensionalPT-Symmetric Systems

By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schrödinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra does not imply similarity of the structure or stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. In addition, the stability is not directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use a graph representation of PT-symmetric networks allowing for a simple illustration of linearly equivalent networks and indicating their possible experimental design.