Left-handed Nonlinear Transmission Line Research Articles

Impact of viscothermal loss on modulation instability and rogue waves in left-handed nonlinear diffractive acoustic transmission line metamaterials

In this study, the transmission line approach is used to describe the studied acoustic metamaterial model. Through Kirchoff’s pressure and volume-velocity laws and using multiple scales method, nonlinear coupled Schrödinger equations are obtained. Then, the amplitude disturbance method is applied to these equations to obtain and plot the modulational instability gain curves. Analytically, the impact of viscothermal loss on the modulational instability gain is studied. The similarity technique is used to derive integrable Manakov’s equations. First and second-order rational rogue wavelike solutions of coupled nonlinear Schrödinger are deduced. The results indicate that the modulational instability gain and Rogue wave intensities depend on the viscothermal parameter. This parameter can be considered in the design of nonlinear acoustic metamaterials to minimize the damage caused by the dynamics of freak waves.

Multi-soliton interactions, comb behavior, strange mixed breathers and other exotic rogue waves generated in a nonlinear left-handed transmission line

We establish a cubic–quintic-nonlinear Schrödinger equation (CQNLSE) including cubic- and quintic-nonlinearities, second-, third- and fourth-order dispersions in a left-handed nonlinear transmission line. We use the collective coordinates’ theory and the Gaussian Ansatz function with six coordinates to improve the comprehension of the system. Besides, the following innovations have been found when linear and nonlinear effects interact. (i) Some phenomena have emerged such as comb behavior, fragmentation of waves and wall of waves. (ii) Some solutions have also emerged such as the dark soliton, multi-peak waves, Kuznetsov–Ma breathers, periodical and non periodical mixed wave trains, Sasa–Satsuma waves, Peregrine waves, and crossed solitons. Otherwise, the internal perturbations leading to all aforementioned exotic rogue events are well studied in detail.

Peregrination, layers’ and multi-peaks’ generation induced by cubic-quintic-saturable nonlinearities and higher-order dispersive effects in a system of coupled nonlinear left-handed transmission lines

Peregrination, layers’ and multi-peaks’ generation induced by cubic-quintic-saturable nonlinearities and higher-order dispersive effects in a system of coupled nonlinear left-handed transmission lines

Spectral stability and dynamics of solitary waves in a coupled nonlinear left-handed transmission line

Spectral stability and dynamics of solitary waves in a coupled nonlinear left-handed transmission line

Backward- and forward-wave soliton coexistence due to second-neighbor coupling in a left-handed transmission line

We study analytically and numerically the impact of the second-neighbor interactions on the propagation of an initial wave-packet through a coupled nonlinear left-handed transmission line. We start with a detailed analysis of the dispersion curve and group velocity. We then use the quasi-discrete approximation with respect to long wavelength transverse perturbations to reduce the nonlinear discrete model of the lattice to a nonlinear Schrödinger equation. A detailed analysis of the product of the group velocity dispersion (P) and the nonlinear term (Q) is presented with respect to the relevant parameters of the system. Here we see the important role of the second-neighbor coupling, which makes the dispersion curve non-monotonic. Thus, for a given frequency, the group velocity can be both positive and negative, while for each choice the product, PQ, can be either positive or negative. Our numerical results confirm that for sufficiently strong second-neighbor coupling, bright backward-wave pulses and dark solitons can coexist at a common frequency, and these travel in opposite directions through the lattice.

Solitary waves and modulation instability with the influence of fractional derivative order in nonlinear left-handed transmission line

The resolution of the reduced fractional nonlinear Schrödinger equation obtained from the model describing the wave propagation in the left-handed nonlinear transmission line presented by Djidere et al recently, allowed us in this work through the Adomian decomposition method (ADM) to highlight the behavior and to study the propagation process of the dark and bright soliton solutions with the effect of the fractional derivative order as well as the Modulation Instability gain spectrum (MI) in the LHNLTL. By inserting fractional derivatives in the sense of Caputo and in order to structure the approximate soliton solutions of the fractional nonlinear Schrödinger equation reduced, ADM is used. The pipe is obtained from the bright and dark soliton by the fractional derivatives order. By the bias of MI gain spectrum the instability zones occur when the value of the fractional derivative order tends to 1. Furthermore, when the fractional derivative order takes small values, stability zones appear. These results could bring new perspectives in the study of solitary waves in left-handed metamaterials as the memory effect could have a better future for the propagation of modulated waves because in this paper the stabilization of zones of the dark and bright solitons which could be described by a fractional nonlinear Schrödinger equation with small values of fractional derivatives order has been revealed. In addition, the obtained significant results are new and could find applications in many research areas such as in the field of information and communication technologies.

Multi-Sasa Satsuma rogue events and multi-wave trains generation in a nonlinear left-handed transmission line

We demonstrate at the first time that the left-handed transmission line with varactor diodes can support higher-order effects. So, we establish at the first time, a nonlinear Schrodinger equation including second-, third- and fourth- order dispersions and self- and quintic-phase modulations. We use collective coordinate’s theory in order to give a good characterization of the light pulse. Moreover, we introduce a new Ansatz function called “type I Ansatz function” with seven coordinates compared to the conventional Ansatz function with six coordinates. The additional coordinate called “nonlinear phase” will give supplemental details on internal deformation which occurs during the propagation of the soliton light pulse. We find that the action of quintic-phase modulation on soliton light pulse leads to the generation of Peregrine solitons and several narrow wave trains. The introduction of third-order dispersion induces the appearance of multi-Sasa–Satsuma rogue events. Further, we introduce fourth-order dispersion and we assist to the generation of Parallel wave trains associated with strange wave’s field. Considering an increase in frequency combined to the action of all those effects, the precedent Parallel wave trains are changed into conventional Sasa–Satsuma rogue waves. Furthermore, we present physical conditions and theoretical frequencies leading to the generation of those specific extreme events. The new introduced Ansatz function improves the comprehension of rogue wave’s mechanism of generation.

Wave propagation of the perturbed nonlinear Schrödingers equation in the nonlinear left-handed transmission lines

This paper secures chirped dark and bright solitons of the perturbed nonlinear Schrödinger equation with parabolic law nonlinearity and self-steepening effect in the nonlinear left-handed metamaterials (NLHMs). We use the ansatz method, sine-cosine, csch function method and the auxiliary equation method to fall out the various soliton solutions. In view of the results obtained, rational, hyperbolic and trigonometric solutions emerge and there are new in CRLH TL. The existence criteria of these solutions are also discussed. Finally, we observed that the use of the auxiliary equation method leads to the abundant and efficient solutions than the other methods.

Exact Travelling Envelope Solitons and Kink-soliton Solutions for the Josephson Nonlinear Left-handed Transmission Line

Exact traveling soliton solutions for the Josephson nonlinear Left-handed transmission line (NL-JLHTL) based on the periodic structure of an array of Josephson junctions (JJs) are investigated. The nonlinearity of the Josephson left-handed transmission line (JLHTL) is due to the highly nonlinear nature of the JJs that provide the shunt inductances required to realize an LHTL. Applying the generalized Ricati methods, we analytically and successfully derive exact traveling kink solitons, bright and dark solitary wave solutions on this network. The left-handedness of the line is explicitly confirmed in numerical simulations with the existence of bright and dark soliton solutions in good agreement with analytical predictions.

Description and stability analysis of nonlinear transmission line type metamaterials using nonlinear dynamics theory

Nonlinear metamaterials offer a potential technology to realize applications at microwave, terahertz, and optical frequencies. However, due to the strong and controlled nonlinearity, the wave interactions can be quite complex. In the current article, a framework based on nonlinear dynamics theory is developed to describe such complex interactions. This is demonstrated for the case of a harmonically pumped nonlinear left handed transmission line through the use of bifurcation theory, stability analysis, and linearization about the limit cycle to calculate the autonomously generated frequencies and their spatial distributions. Higher order parametric interactions, which can be mediated by the strong nonlinearity, are automatically included in the model. It is demonstrated that autonomous components can be visualized in both the phase and the set of solution spaces. The framework is general in terms of the transmission line configuration, the nature and strength of the nonlinearity, and the number of stages. It also provides accurate results when the autonomous frequencies are in the vicinity of the Bragg frequency.

Broadband microwave frequency doubler based on left-handed nonlinear transmission lines**Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 61401373) and the Research Fund for the Doctoral Program of Southwest University, China (Grant No. SWU111030).

A bandwidth microwave second harmonic generator is successfully designed using composite right/left-handed nonlinear transmission lines (CRLH NLTLs) in a GaAs monolithic microwave integrated circuit (MMIC) technology. The structure parameters of CRLH NLTLs, e.g. host transmission line, rectangular spiral inductor, and nonlinear capacitor, have a great impact on the second harmonic performance enhancement in terms of second harmonic frequency, output power, and conversion efficiency. It has been experimentally demonstrated that the second harmonic frequency is determined by the anomalous dispersion of CRLH NLTLs and can be significantly improved by effectively adjusting these structure parameters. A good agreement between the measured and simulated second harmonic performances of Ka-band CRLH NLTLs frequency multipliers is successfully achieved, which further validates the design approach of frequency multipliers on CRLH NLTLs and indicates the potentials of CRLH NLTLs in terms of the generation of microwave and millimeter-wave signal source.

Wave solitons in a coupled left-handed nonlinear transmission line: Effect of the coupling parameter

In this work, we investigate the dynamics of modulated waves in a discrete coupled Left-Handed nonlinear transmission line, assuming a one-dimensional (1-D) propagation variation. A nonlinear Schrödinger equation (NLSE) is derived, analytical solitons are found and the instability region is presented for this model.

Envelope solitons in a left-handed nonlinear transmission line with Josephson junction

We consider a nonlinear left-handed transmission line that incorporates an array of Josephson junctions in its periodic lattice structure. We show that the system dynamics is described by a discrete sine-Gordon-like equation, where the left-handedness of the lattice manifests in the form of a non-standard second-time- derivative term. Since this modified discrete sine-Gordon equation has not yet been extensively studied in the literature, this paper opens up the possibility of additional mathematical analysis. It is also intriguing that by means of a semi-discrete approximation we can derive a nonlinear Schrödinger equation and thus show that the system supports both bright and dark envelope soliton solutions depending on the choice of carrier frequency. The left-handedness of the network is explicitly confirmed in numerical simulations which demonstrate the backward propagation of the bright and dark soliton, in good agreement with analytical predictions.

Wave propagation in two‐dimensional left‐handed non‐linear transmission line metamaterials

Recent research on harmonic generations in one-dimensional (1D) left-handed non-linear transmission line (LH NLTL) metamaterials has shown that the fundamental wave and its higher harmonics can realise quasi-phase matching, which in turn ensures the high conversion efficiency of the harmonics. In this study, the authors first present the electromagnetic (EM) theory to describe the generation and propagation of both second-harmonic (SH) and third-harmonic (TH) waves in a 2D non-linear transmission line metamaterial. Then, with the aid of the microwave circuit simulations, they fabricate a 2D LH NLTL metamaterial and demonstrate that an EM wave propagation in such medium has three states: left-handed (LH), right-handed (RH), and non-propagation. Furthermore, the response of the medium to applied field intensity shows that it can realise self-tuning between LH and RH states for the TH waves.

비선형 전송선로를 이용한 고조파 발생기 성능 비교

In this paper, we compared the performance of the Right Handed Nonlinear Transmission Line (RH-NLTL) and Left Handed Nonlinear Transmission Line (LH-NLTL) as a harmonic generator. For a performance comparison, we fabricated both a RH-NLTL and a LH-NLTL harmonic generator whose operational bandwidth is from 0.5 GHz to 1.5 GHz. Under the each condition for the RH-NLTL and the LH-NLTL to maximize second harmonic, the output power of the second harmonic was 9.33 dB lower than the input power for the RH-NLTL and 12.67 dB lower than the input power for the LH-NLTL. Under the each condition for the RH-NLTL and the LH-NLTL to maximize third harmonic, the output power of the third harmonic was 13.33 dB lower than the input power for the RH-NLTL and 14.83dB lower than the input power for the LH-NLTL. Also, we have observed that, generatlly, a RH-NLTL is useful in generating various multiple harmonics and a LH-NLTL is useful in generating a specific order of harmonic by adjusting a proper DC vias, input frequency and input power. These tendencies could be a good guideline to use NLTLs as a frequency multiplier.

A Suspended Stripline Frequency Tripler Using a Left-Handed Nonlinear Transmission Line

A suspended stripline frequency tripler using a left-handed nonlinear transmission line (LH NLTL) is presented. The proposed tripler using the LH NLTL is composed of a series of varactor diodes, shunt inductances, and a high-pass filter implemented with suspended stripline (SSL). An ultrawideband microstrip-to-suspended stripline transition is also utilized. The fabricated LH NLTL provides the minimum insertion loss of 1.7 dB and the maximum insertion loss of 4.7 dB for a wide frequency band from 2.6 to 18 GHz. As a tripler, the measured minimum third harmonic conversion loss is 15.3 dB at an input frequency of 2.4 GHz and typically 17 dB from 2 to 3.1 GHz.

Solitary wave solutions and modulational instability analysis of the nonlinear Schrödinger equation with higher-order nonlinear terms in the left-handed nonlinear transmission lines

We report the modulational instability (MI) analysis for the modulation equations governing the propagation of modulated waves in a practical left-handed nonlinear transmission lines with series of nonlinear capacitance. Considering the voltage in the spectral domain and the Taylor series around a certain modulation frequency, we show in the continuum limit, that the dynamics of localized signals is described by a nonlinear Schrödinger equation with a cubic–quintic nonlinear terms. The MI process is then examined and we derive the gain spectra of MI for the generation of solitonlike-object in the transmission line metamaterials. We emphasize on the effect of losses on the MI gain spectra. An exact kink-darklike solutions is derived through the auxiliary equation method. It comes out that the width of the darklike solution decreases as the attenuation constant increases. Our theoretical solution is in good agreement with our numerical observation.

Rogue waves generation in a left-handed nonlinear transmission line with series varactor diodes

We investigate the electromagnetic wave behavior and its characterization using collective variables technique. Second-order dispersion, first- and second-order nonlinearities, which strongly act in a left-handed nonlinear transmission line with series varactor diodes, are taken into account. Four frequency ranges have been found. The first one gives the so-called energetic soliton due to a perfect combination of second-order dispersion and first-order nonlinearity. The second frequency range presents a dispersive soliton leading to the collapse of the electromagnetic wave at the third frequency range. But the fourth one shows physical conditions which are able to provoke the appearance of wave trains generation with some particular waves, the rogue waves. Moreover, we demonstrate that the number of rogue waves increases with frequency. The soliton, thereafter, gains a relative stability when second-order nonlinearity comes into play with some specific values in the fourth frequency range. Furthermore, the stability conditions of the electromagnetic wave at high frequencies have been also discussed.

Analog study of near-field focusing and subwavelength imaging with nonlinear transmission-line metamaterial

In this paper, we report an analog study of realizing near-field focusing and near-perfect imaging at harmonic frequencies using a two-dimensional (2D) left-handed nonlinear transmission line (LH NLTL) metamaterial. Firstly, we give a detailed procedure of designing LH NLTL and right-handed (RH) transmission line media that match with each other at harmonics. Then, through microwave circuit simulations, we observe strong surface waves at the boundaries between RH and LH NLTL metamaterials, which are consistent with the theoretical analysis of an LH linear superlens. Finally, we show that the LH NLTL metamaterial can act as a flat nonlinear lens that can create, with a subwavelength resolution, an image of harmonic fields of the source which is opaque at the fundamental frequency.

Coupled backward- and forward-propagating solitons in a composite right- and left-handed transmission line

We study the coupling between backward- and forward-propagating wave modes, with the same group velocity, in a composite right- and left-handed nonlinear transmission line. Using an asymptotic multiscale expansion technique, we derive a system of two coupled nonlinear Schrödinger equations governing the evolution of the envelopes of these modes. We show that this system supports a variety of backward- and forward-propagating vector solitons of the bright-bright, bright-dark, and dark-bright type. Performing systematic numerical simulations in the framework of the original lattice that models the transmission line, we study the propagation properties of the derived vector soliton solutions. We show that all types of the predicted solitons exist, but differ on their robustness: Only bright-bright solitons propagate undistorted for long times, while the other types are less robust, featuring shorter lifetimes. In all cases, our analytical predictions are in very good agreement with the results of the simulations, at least up to times of the order of the solitons' lifetimes.