Gain-loss Coefficient Research Articles

Integrability and trajectory confinement in -symmetric waveguide arrays

We consider -symmetric ring-like arrays of optical waveguides with purely nonlinear gain and loss. Regardless of the value of the gain–loss coefficient, these systems are protected from spontaneous -symmetry breaking. If the nonhermitian part of the array matrix has cross-compensating structure, the total power in such a system remains bounded—or even constant—at all times. We identify two-, three-, and four-waveguide arrays with cross-compensatory nonlinear gain and loss that constitute completely integrable Hamiltonian systems.

Enhanced spin Hall effect of light in the PT-symmetric trilayer structure containing epsilon-near-zero materials

We systematically study the spin Hall effect of light (SHEL) in the parity-time (PT)-symmetric trilayer structure containing epsilon-near-zero (ENZ) materials, and design a high sensitivity refractive index sensor with an adjustable sensing range. It is revealed that the SHEL shift in the PT-symmetric trilayer structure is clearly enhanced, which is two orders of magnitude larger than that in the conventional sandwich structure containing ENZ materials. The enhancement of the SHEL shift is attributed to the fact that the change of reflection coefficient induced by the quasi-bound states in the continuum (quasi-BIC) in the former structure is smoother than that induced by the bound states in the continuum in the latter structure. It is further found that when the refractive index of the interlayer dielectric in the PT-symmetric structure is fixed, the SHEL shift is significantly enhanced near the quasi-BIC resonance angle determined by the gain-loss coefficient. Meanwhile, the SHEL shift enhanced by excitation of quasi-BIC is very sensitive to the gain-loss coefficient and the refractive index of the interlayer dielectric. Finally, we design a high sensitivity refractive index sensor with an adjustable sensing range based on the quasi-BIC-enhanced SHEL shift. These studies provide a pathway to enhance the SHEL and may open avenues for the application of optical sensors.

Thermal assessment of a new planar thermal diode integrated collector storage solar water heater in different partial vacuums: An experimental study

Thermal diodes are novel mechanisms with high thermal resistance in one direction and low thermal resistance in the other. The current study introduces a new planar thermal diode integrated collector storage (ICSs) system and analyzes its transient dynamics and performance under field conditions in several scenarios, including operating at atmospheric pressure and partial vacuums: Ptotal = 0.7 atm and 0.5 atm. The acquired data illustrate that the collector has approximately the same transient dynamics during the collection period, while differences pronounce during the retention period. Utilizing partial vacuums is tricky and may cause to decrease in performance; however, the best retention and collection efficiencies were obtained at 16.97% and 29.33% when Ptotal = 0.7 atm was utilized. Increasing the partial vacuums to Ptotal = 0.5 atm decreases the retention efficiency significantly to nearly 3.26%. The heat gain (forward mode) coefficient was not affected significantly by partial vacuums. However, the heat loss (reverse mode) coefficient was reduced by nearly 17.63% at Ptotal = 0.7 atm compared to atmospheric pressure. The calculated heat gain and heat loss coefficients are 9.28 W/k and 19.65 W/k at Ptotal = 0.7 atm; hence, the constructed thermal diode successfully serves as a unidirectional heat transfer mechanism.

Gyrating solitons in a necklace of optical waveguides

We consider light pulses in a circular array of $2N$ coupled nonlinear optical waveguides. The waveguides are either hermitian or alternate gain and loss in a $\mathcal{PT}$-symmetric fashion. Simple patterns in the array include a ring of $2N$ pulses travelling abreast, and a breather -- a string of pulses where all even and all odd waveguides flash in turn. In addition, the structure displays solitons gyrating around the necklace by switching from one waveguide to the next. Some of the gyrating solitons are stable while other ones are weakly unstable and evolve into gyrating multiflash strings. By tuning the gain-loss coefficient, the gyration of solitons in a nonhermitian array may be reversed without changing the direction of their translational motion.

Topological split-ring resonator based metamaterials with PT symmetry relying on gain and loss

A one-dimensional metamaterial with parity-time (${\cal PT}$) symmetry that relies on balanced gain and loss is introduced, comprising of magnetically coupled split-ring resonators (SRRs). A particular topology that combines a non-trivial (topological) dimer configuration with a trivial (non-topological) dimer configuration which are separated by a central SRR with neither gain or loss, is investigated. By focusing on the dynamical aspects of such a topological ${\cal PT}$ metamaterial (PTMM), the existence of {\em topologically protected interface states} which are localized at the central SRR is demonstrated numerically. The solution of the corresponding {\em quadratic eigenvalue problem} reveals that the protected state is actually a robust eigenmode of the topological PTMM, whose eigenvalue is isolated in the middle of the gap (mid-gap state) of the two-band frequency spectrum. Direct numerical simulations have been further used to determine the robustness and dynamic stability of these states in the parameter space of the {\em dimerization strength} and the {\em gain-loss coefficient}.

Exceptional Points in Non-Hermitian Photonic Crystals Incorporated With a Defect

We explored exceptional points (EPs) in one dimensional non-Hermitian photonic crystals incorporated with a defect. The defect was asymmetric with respect to the center. Two EPs could be derived by modulating the normalized frequency and the gain-loss coefficient of defect. The reflection coefficient complex phase changed dramatically around EPs, and the change in complex phase was π at EPs. The electric field of EPs was mainly restricted to the defect, which can induce a giant Goos–Hänchen (GH) shift. Moreover, we found a coherent perfect absorption-laser point (CPA-LP) in the structure. A giant GH shift also existed around the CPA-LP. The study may have found applications in highly sensitive sensors.

Robust {bf{P}}{bf{T}} symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

The real spectrum of bound states produced by {bf{P}}{bf{T}}-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. The breakup essentially impedes the use of {bf{P}}{bf{T}}-symmetric systems for various applications. On the other hand, it is known that the {bf{P}}{bf{T}} symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) {bf{P}}{bf{T}} symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose {bf{P}}{bf{T}} symmetry remains unbroken for arbitrarily large values of the gain-loss coefficient. Further, we introduce an extended 2D model with the imaginary part of potential ~xy in the Cartesian coordinates. The latter model is not a {bf{P}}{bf{T}}-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the {bf{P}}{bf{T}} symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable {bf{P}}{bf{T}} symmetry, while generic soliton families are found in a numerical form. The {bf{P}}{bf{T}}-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with m = 1, 2, and 3. In the model with imaginary potential ~xy, families of single- and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.

Impact of near-𝒫𝒯 symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model.

We theoretically report the influence of a class of near-parity-time-(𝒫𝒯-) symmetric potentials on solitons in the complex Ginzburg-Landau (CGL) equation. Although the linear spectral problem with the potentials does not admit entirely-real spectra due to the existence of spectral filtering parameter α2 or nonlinear gain-loss coefficient β2, we do find stable exact solitons in the second quadrant of the (α2, β2) space including on the corresponding axes. Other fascinating properties associated with the solitons are also examined, such as the interactions and energy flux. Moreover, we study the excitations of nonlinear modes by considering adiabatic changes of parameters in a generalized CGL model. These results are useful for the related experimental designs and applications.

Integrable coupled Liénard-type systems with balanced loss and gain

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of co-ordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient. The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A few exact solutions for both the cases are presented for specific choices of the potential and space-dependent gain/loss co-efficients, which include periodic stable solutions. Quantization of the system is discussed with the construction of the integrals of motion for specific choices of the potential and gain–loss coefficients. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges are presented.

Families of fundamental and vortex solitons under competing cubic-quintic nonlinearity with complex potentials

We investigate the properties of two-dimensional(2D) fundamental and vortex solitons propagating in PT symmetric lattices with the competing cubic-quintic nonlinearity. We discuss the influence of the competing nonlinearity and the gain-loss coefficient on the existence and stability of both 2D fundamental solitons and vortex solitons, and obtain the existence and stability ranges of them. The stability of solitons under different competing CQ nonlinearity is analyzed by using the VK criterion or the anti-VK criterion. We demonstrate that the whole nonlinearity is determined by the coefficients of nonlinear terms and the propagation constants of solitons. In particular, the power curve of vortex soliton leap and move back near the Bloch band instead of bifurcating from it under the defocusing cubic and focusing quintic nonlinearity.

New design approach and implementation of solar water heaters: A case study in Michigan

The potential of evacuated tube solar water heaters is evaluated for the state of Michigan, which has regions with low solar resources, for year-round usage. Simple empirical equations for the initial sizing design stage are provided for the entire state for a given system. The analysis includes experimental results obtained for a setup at Michigan State University and model-based results for 26 locations dispersed throughout the state. These are obtained using the System Advisor Model (SAM) software package from National Renewable Energy Laboratory. The results show that evacuated tube collector solar water heaters are appropriate for typical household water usage for Michigan’s climate with payback periods as low as 8 years (variations are due to local weather conditions). In addition, a study on the optimum collector area for various hot water demands is performed; this led to identification of empirical equations useful for estimating the optimal area of a given system as a function of the hot water demand for locations throughout Michigan. A parametric study is carried out and it is found that the overall heat gain coefficient of an evacuated tube collector affects significantly its performance and it should be considered in order to select an appropriate system. The impact of additional parameters such as heat gain and heat loss coefficients on the performance of the system is also discussed.

Solitons in -symmetric ladders of optical waveguides

We consider a -symmetric ladder-shaped optical array consisting of a chain of waveguides with gain coupled to a parallel chain of waveguides with loss. All waveguides have the focusing Kerr nonlinearity. The array supports two co-existing solitons, an in-phase and an antiphase one, and each of these can be centred either on a lattice site or midway between two neighbouring sites. We show that both bond-centred (i.e. intersite) solitons are unstable regardless of their amplitudes and parameters of the chain. The site-centred in-phase soliton is stable when its amplitude lies below a threshold that depends on the coupling and gain–loss coefficient. The threshold is lowest when the gain-to-gain and loss-to-loss coupling constant in each chain is close to the interchain gain-to-loss coupling coefficient. The antiphase site-centred soliton in the strongly-coupled chain or in a chain close to the -symmetry breaking point, is stable when its amplitude lies above a critical value and unstable otherwise. The instability growth rate of solitons with small amplitude is exponentially small in this parameter regime; hence the small-amplitude solitons, though unstable, have exponentially long lifetimes. On the other hand, the antiphase soliton in the weakly or moderately coupled chain and away from the -symmetry breaking point, is unstable when its amplitude falls in one or two finite bands. All amplitudes outside those bands are stable.

Discrete multipole dark solitons in saturable nonlinearity media with parity-time symmetric lattices

We investigate numerically the existence and stability of discrete multipole dark solitons in saturable nonlinearity media with parity-time ( $\mathcal{PT}$ ) symmetric lattices. The dipole and triple solitons share the same existence domain associated with the propagation constant, gain-loss coefficient and the degree of saturable nonlinearity. However, the solitons stably propagate in a different area of the aforementioned three parameters. The beam power monotonously increases with the increase of propagation constant and the saturable degree, but gently decreases with the gain-loss coefficient. Moreover, the power of the triple solitons is higher than that of dipole solitons under the same conditions.

太阳能热水器热性能试验装置热性能测量不确定度分析

对研制开发的家用太阳能热水器热性能测试装置的日有用得热量和平均热损因数两个参数的测量不确定度进行了分析，分析结果表明，利用该装置测得的日有用得热量的测量不确定度受总辐射表的测量精度影响较大，平均热损因数的测量不确定度受温度测量传感器的不确定度影响较大，研究结果为提高太阳能热水器热性能测试装置的测量精度指明了方向。 The measurement uncertainty analysis of daily useful heat gain and average heat loss coefficient of the tank have been performed based on the test data from the solar water heater thermal performance test facility. The results show that the precision of pyranometer has dominated the measurement uncertainty of daily useful heat gain, and the temperature sensor measurement uncertainty has significant effect on the average heat loss coefficient of the tank. The work done in this paper gives the direction to improve the measure precision of solar water heater test facility.

Dimer with gain and loss: Integrability and -symmetry restoration

A -symmetric nonlinear Schrödinger dimer is a two-site discrete nonlinear Schrödinger equation with one site losing and the other one gaining energy at the same rate. In this paper, two four-parameter families of cubic -symmetric dimers are constructed as gain–loss extensions of their conservative, Hamiltonian, counterparts. We prove that all these damped-driven equations define completely integrable Hamiltonian systems. The second aim of our study is to identify nonlinearities that give rise to the spontaneous -symmetry restoration. When the symmetry of the underlying linear dimer is broken and an unstable small perturbation starts to grow, the nonlinear coupling of the required type will divert an increasingly large percentage of energy from the gaining to the losing site. As a result, the exponential growth will be saturated and all trajectories remain trapped in a finite part of the phase space regardless of the value of the gain–loss coefficient.

Gap solitons inPT-symmetric lattices with a lower refractive-index core

We address the existence and stability properties of families of gap solitons in a lower refractive-index core, sandwiched between two optical lattices with a parity-time ($\mathcal{PT}$) symmetry imprinted in a defocusing medium. The scale of flat-topped complex solitons can be controlled arbitrarily by varying the embedded index core. Multipeaked solitons are found to exhibit equal peak values in the region of the index core, and they have no analog in other lattice-modulated or bulk media. We demonstrate that, in sharp contrast to solitons in regular $\mathcal{PT}$ lattices, flat-topped and multipeaked solitons are either stable or suffer a negligibly weak instability, even when the gain-loss coefficient approaches the $\mathcal{PT}$-symmetry-breaking point. Our results, thus, build a bridge between the $\mathcal{PT}$ defect solitons in a narrow index core and the $\mathcal{PT}$ kink pairs in a broad index core. We also suggest an effective way for the observation of ``surface solitons'' in $\mathcal{PT}$-symmetric lattices.

Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity.

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain-loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multi-pole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain-loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain-loss coefficient.

An exactly solvable -symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

We show that a pair of coupled nonlinear oscillators, of which one oscillator has positive and the other one negative damping of equal rate, can form a Hamiltonian system. Small-amplitude oscillations in this system are governed by a -symmetric nonlinear Schrödinger dimer with linear and cubic coupling. The dimer also represents a Hamiltonian system and is found to be exactly solvable in elementary functions. We show that the nonlinearity softens the -symmetry breaking transition in the nonlinearly-coupled dimer: stable periodic and quasiperiodic states with large enough amplitudes persist for an arbitrarily large value of the gain–loss coefficient.

Two-dimensional solitons and clusters in dissipative lattices

We study the dynamics of two-dimensional spatial solitons in the structured optical medium modeled by the complex Ginzburg–Landau equation with cubic–quintic nonlinearity and a spatially periodic modulation of the local gain–loss coefficient [a dissipative lattice (DL)]. The analysis, following the variation of the DL’s amplitude and period, reveals several dynamical scenarios: stable or unstable propagation of a single dissipative soliton (the unstable propagation entails generation of an irregular multisoliton cluster), transformation of the input soliton into stable or unstable regular clusters patterned as the underlying DL, and decay of the input. Most results are obtained by means of systematic simulations, but the boundary of the single-soliton stability domain is explained analytically.

Stable long-distance propagation and on-off switching of colliding soliton sequences with dissipative interaction

We study propagation and on-off switching of two colliding soliton sequences in the presence of second-order dispersion, Kerr nonlinearity, linear loss, cubic gain, and quintic loss. Employing a Lotka-Volterra (LV) model for dynamics of soliton amplitudes along with simulations with two perturbed coupled nonlinear Schr\"odinger (NLS) equations, we show that stable long-distance propagation can be achieved for a wide range of the gain-loss coefficients, including values that are outside of the perturbative regime. Furthermore, we demonstrate robust on-off and off-on switching of one of the sequences by an abrupt change in the ratio of cubic gain and quintic loss coefficients, and extend the results to pulse sequences with periodically alternating phases. Our study significantly strengthens the recently found relation between collision dynamics of sequences of NLS solitons and population dynamics in LV models, and indicates that the relation might be further extended to solitary waves of the cubic-quintic Ginzburg-Landau equation.