Third-order Nonlinearity Research Articles

ADEQUATE SOLUTIONS OF JERK OSCILLATORS CONTAINING VELOCITY TIMES ACCELERATION-SQUARED: HAQUE’S APPROACH WITH MICKENS’ ITERATION METHOD

Haque’s Approach with Mickens’ Iteration Method is used to find the exact analytic solution of the nonlinear equation involving velocity times acceleration squared. A truncated Fourier series is used in different rhythms with different repetition steps. Our results are very close to the exact results and our results are comparatively closer to the exact results than others. Our solution method is obtained around the second-order angular frequency using Newton’s method. For some third-order (jerk) differential equations with cubic nonlinearities and nonlinear second-order differential equations; Mickens’ iteration method is used to determine the exact analytical approximate periodic solution. A numerical experiment of general differential equations with third-order, one-dimensional, autonomous, quadratic, and cubic nonlinearity has uncovered several algebraically simple equations involving the shaking of time-dependent acceleration that contain chaotic solutions.

Investigation on Bi-induced changes on linear and non-linear optical parameters of As45-Se (55-x)-Bix chalcogenide glasses for photonic application

The Bi-incorporated As45-Se(55-x)-Bix (x = 0, 2, 4, 6 at.%) chalcogenide glasses have been synthesized by deploying the conventional melt quenching method. The non-appearance of the peak in the X-ray diffraction pattern confirms the amorphous characteristics of these materials. The alteration in non-linear and linear optical properties has been investigated from UV–Vis spectroscopy data and associated empirical formulae. Tauc's plot show that the values of optical band gap energies decrease from 1.82 eV to 1.59 eV, while the assessed Urbach energies increase from 0.048 eV to 0.056 eV with Bi addition. The reflection and transmission data have been analyzed within the range of 300–800 nm. Various important optical parameters like refractive index, absorption coefficient, extinction coefficient, surface and volume energy loss functions, optical density, skin depth, third-order nonlinearity, thermal emissivity, etc., have been investigated. The Wimple Didomenico model has been used to analyze several dispersion parameters. The Chemical bond approach model has been used to estimate the cohesive energy, which is found to decrease from 2.761 eV to 2.747 eV. The increase in non-linear refractive index and third-order non-linear optical susceptibility indicates the possibility of these materials being used in optical switching devices as well as photonic devices.

Fine-tuning of plasmonics by Au@AuY/Au core–shell nanoparticle monolayer for enhancement of third-order nonlinearity

The manipulation of plasmonics on noble metal nanoparticles (NPs) is of great interest in developing nonlinear photonic devices, such as all-optical switches and frequency combs. An Au@AuY-core/Au-shell nanoparticle (Au@AuY/Au NP) monolayer is proposed for the fine-tuning of plasmonics and enhanced third-order nonlinearity. Based on the different thermodynamic mechanisms of Au and Y ions, the compact Au@AuY/Au core–shell architectures are designed and surface-modified in fused silica (SiO2) with enhanced free electron density, mobility, and quantum size effect. The flexible modulation of plasmonics is realized, resulting in significant absorption enhancement (165% for interband absorption and 38% for free electron absorption, respectively) and fine-tuning of the localized surface plasma resonance (LSPR) band. In addition, the physical mechanism is investigated by density functional theory (DFT) and Mie theory, which reveals a transition from size-independence to size-dependence of LSPR owing to the synergistic effect of multiple physical factors such as free electron density and mobility. With the above advantages, the third-order nonlinearity is enhanced by 4.4 times compared with traditional Au NPs. It indicates the significant potential of Au@AuY/Au core–shell NP monolayer in the performance improvement of nonlinear photonic devices.

Tunable Low-Threshold Optical Bistability in Optical Tamm Plasmon Superlattices

We propose a scheme to obtain tunable low-threshold optical bistability of reflected beams in optical Tamm plasmon superlattices (TPS). The low-threshold optical bistability is triggered due to the strong third-order non-linearity of graphene and the local field enhancement in the TPS. Our results show that the optical Tamm plasmon superlattices have the ability to lower the bistable threshold even further than the single optical Tamm state. The results show that the hysteresis behavior and optical bistability threshold can be continuously adjusted by changing the applied voltage and the number of graphene layers (N ≤ 4). In particular, the optical bistability in the TPS is affected by the incident angle. Our results introduce a new possible route for low threshold optical bistability in the THz range and provide a new method in the field of all-optical switching applications.

Modulation of photoluminescence and optical limiting properties of spray-coated tin oxide thin film through Eu doping

This work reports the effect of Eu doping on the structural, morphological, linear, and nonlinear optical properties of Sn1−xEuxO2 (x = 0 to 0.10) thin films prepared using spray pyrolysis. The films crystallized in tetragonal structure and the crystallite size reduced due to Eu incorporation. The film morphology consisted of evenly distributed dense grains with trapezohedron or bipyramidal shape. The Eu doping reduced the transmittance and energy band gap of the deposited films. The photoluminescence (PL) emission spectra comprised the contributions from both the transitions related to the Eu3+ ions and the defects, such as oxygen vacancies and tin interstitials in the host lattice. The films also showed a near white light emission due to the different defect-related PL emissions. The Z-scan technique revealed the third-order optical nonlinearity shown by the films. Amongst the doped films, Sn0.92Eu0.08O2 films exhibited least optical limiting threshold.Graphical abstract

Displacement current sources as nonlinear interactions in optical nanocircuits

Optical lumped circuit elements are the building blocks in the metatronics paradigm, whose goal is to extend the rules of RF circuit design into the field of nanophotonics by providing the advantages of lumpedness and modularity. In this paper, we aim at modeling within this framework nonlinear optical processes, based on the concept of optical lumped circuit elements. Displacement current sources are added to the previously introduced optical lumped elements in order to endow metatronics with nonlinear functionalities. This model not only simplifies the analysis of the nonlinear processes, but paves the road to develop nonlinear optical components and devices in this paradigm. Second- and third-order nonlinearities are investigated analytically in the case of a nanosphere. A step by step example of modeling a LiNbO3 nanorod, which is close to a practically realizable structure, is also presented. The results are compared with those of a full-wave simulation and the significance of the proposed model is discussed.

Microstructured All-Optical Switching Based on Two-Dimensional Material

Microstructured all-optical switching, possessing the unique function of light controlling light, is an important part of the on-chip ultra-fast optical connectivity network and integrated logic computing chip. Microstructured all-optical switching has attracted extensive research interest, the latest great developments of which have also yielded progress in nanophotonics, nonlinear optics, optical communications, and integrated optics, etc. The emergence of two-dimensional materials with good third-order optical nonlinearity provides an important driving force for the improvement of all-optical switches. This paper reviews the implementation principles, novel configurations, improved performance indexes, and research progress based on different two-dimensional materials for micro/nano all-optical switching. Not only is a systematic discussion of the current state provided, but also, a brief outlook is afforded on the remaining challenges in the pursuit of the application of practical on-chip microstructured all-optical switching that is based on two-dimensional materials.

The Nonlinear Eigenvalue Problem of Electromagnetic Wave Propagation in a Dielectric Layer Covered with Graphene

The paper focuses on the problem of a monochromatic terahertz TE-polarized wave propagation in a plane dielectric layer filled with a homogeneous isotropic medium; one of the boundaries of the waveguide is covered with a layer of graphene. In fact, the paper aims to find the eigenwaves of the described waveguiding structure. On the one hand, in the study, energy losses both in the dielectric layer and in the graphene layer are neglected; the latter assumption is reasonable in the terahertz range of electromagnetic radiation (on which the paper focuses), where graphene has a strong plasmonic response and much less loss. On the other hand, this study takes into account the significant third-order nonlinearity resulting from the interaction of the electromagnetic wave with the charge carriers in the graphene layer. The paper aims to study the guiding properties of the above structure using primarily an analytical approach. The wave propagation problem is reduced to an eigenvalue problem, where one of the boundary conditions is nonlinear with respect to the sought-for function. The main result of the paper is a dispersion equation allowing for a waveguide of a given thickness to determine a set of its propagation constants and, consequently, a set of its eigenwaves. It is worth noting that the dispersion equation being written in an explicit form can be used to obtain deep qualitative results related to the solvability of the problem and the properties of its solutions. For example, in the paper, the existence of several propagation constants (and, consequently, the eigenwaves) of the studied waveguiding structure is proved under some conditions. Besides studying the problem analytically, the paper presents some numerical results as well. In particular, the presented figures demonstrate how the nonlinearity in graphene affects the propagation constants and eigenwaves, providing the dispersion curves and eigenwaves for nonlinear graphene as well as for the linear one.

Electrically controlled liquid crystal nonlinear optical devices prepared by multi-layer composite structures

In this study, a multi-layer composite structure is used to prepare nonlinear optical devices from multi-walled carbon nanotubes (MWCNTs)/liquid crystal (LC) composite systems. Their third-order optical nonlinearity is investigated. The MWCNTs were coated and modified to improve their dispersion in the solvent. The coated and modified MWCNTs were characterized by IR absorption spectroscopy and transmission electron microscopy (TEM). The nonlinear optical (NLO) properties of the devices prepared by the multi-layer composite structures with varying design parameters were studied. The conventional devices were also prepared by mixed doping method for comparison with the proposed devices. Both devices underwent Z-scan testing. The third-order nonlinear polarizability of the conventional hybrid-doped devices was calculated with a value of 5.39 × 10-10 esu. The third-order nonlinear polarizability of the proposed devices prepared by the multi-layer composite structures has a value of 2.97 × 10-9 esu, which is about five times the corresponding value for the hybrid-doped devices. So, the devices prepared by the multi-layer composite structures have better NLO properties. Moreover, the third-order nonlinear polarizability of the proposed devices was further increased up to 3.88 × 10-9 esu, which is an increase of almost 31% after applying voltage to these devices prepared by the multi-layer composite structures. The limiting threshold of S-CTAB/MWCNTs@BHR33400 is 0.245 J/cm2 after voltage application. Applying voltage plays a role in promoting the NLO properties of the devices. Using the multi-layer composite structure to prepare the liquid crystal nonlinear optical devices can significantly improve the NLO properties of the devices. Such prepared devices can be electrically tuned, providing new avenues for preparing nonlinear optical devices.

π-Extended Octupolar Cyclized Indole Tetramer and Trimer Derivatives with Second- and Third-Order Nonlinear Optical Properties.

Several aromatic-fused octupolar tetraindole (TTI) and triindole (TI) derivatives have been synthesized via POCl3-promoted cyclopolymerization of the aromatic-fused oxindoles. As a result, the terminal phenyls of TTI and TI are fused with the phenyl group, indenyl group, and benzothienyl group separately in the π-extended target compounds TTIp and TIp, TTIid and TIid, and TTIbt and TIbt (including pairs of syn- and anti-isomers). Single-crystal X-ray diffraction disclosed that the tetramer syn-TTIbt keeps an S4-symmetric architecture and crystallizes in the P42/n space group. With the extension of the π-conjugation, the absorption and emission bands of the above tetramers and trimers are remarkably red-shifted. The second-order nonlinear optical (NLO) coefficients (β value) of syn-TTIbt and anti-TTIbt have been calculated to be 1.5 and 2.1 times that of their parent compound TTI, respectively. In the domain of the third-order NLO, the two-photon absorption sections (δ value) of syn-TIbt and anti-TIbt have been measured to be 37 and 102 times that of TI, respectively. These results indicate that the π-extension strategy here is successful, and the anti-isomers have better NLO properties and better π-conjugation than their corresponding syn-isomers.

Third-order nonlinearity with subradiance dark-state in ultra-strong excitons and surface plasmon coupling using self-antiaggregation organic dye

A strong coupling regime with dressed states is formed when a propagating surface plasmon (PSP) mode coherently exchanges energy with an ensemble of excitons at a rate faster than the system’s losses. These states are superpositions of superradiance excitons and PSP modes, accompanied by remaining subradiance or ‘dark’ exciton states. Dark-states are ubiquitous, especially in disordered systems, and they rise in number as the number of excitons increases. Here, the ultra-strong coupling regime was experimentally observed with the coupling strength to bare energy as high as g/∼ 0.23 using a self-antiaggregation organic dye, BOBzBT2 in an Otto-SPR configuration. We show that the hybrid system of excitons in a nonlinear organic dye layer and a PSP mode can be described by employing dark-state in a theory of nonlinear third-order sum-frequency generation (TSFG). Close agreement between the theory and the experiment has been demonstrated. The study opens up a new perspective for establishing a relationship between the optical properties of a third-order nonlinear material and the extent of strong coupling.

Non-Gaussian Statistics of Mechanically Generated Unidirectional Irregular Waves

An experimental study is presented on the non-Gaussian statistics of random unidirectional laboratory wave fields described by JONSWAP spectra. Relationships between statistical parameters indicative of the occurrence of large-amplitude waves are discussed in the context of the initial steepness of the waves combined with the effect of spectral peakedness. The spatial evolution of the relevant statistical and spectral parameters and features is also considered. It is demonstrated that over the distance the spectra exhibit features typical for developing nonlinear instabilities, such as spectral broadening and downshift of the peak, along with lowering of the high-frequency tail and decrease of the peak magnitude. The wave fields clearly show an increase of third-order nonlinearity with the distance, which can be significant, depending on the input wave environment. The steeper initial conditions, however, while favouring the occurrence of extremely large waves, also increase the chances of wave breaking and loss of energy due to dissipation, which results in lower extreme crests and wave heights. The applied Miche-Stokes-type criteria do confirm that some of the wave extremes exceed the limiting individual steepness. Eventually, this result agrees with the observation that the largest number of abnormal waves is recorded in sea states with moderate steepness.

The Z3RO Family of Precoders Cancelling Nonlinear Power Amplification Distortion in Large Array Systems

Large array systems use a massive number of antenna elements and clever precoder designs to achieve an array gain at the user location. These precoders require linear front-ends, and more specifically linear power amplifiers (PAs), to avoid distortion. This reduces the energy efficiency since PAs are most efficient close to saturation, where they generate most nonlinear distortion. Moreover, the use of conventional precoders can induce a coherent combining of distortion at the user locations, degrading the signal quality. In this work, novel linear precoders, simple to compute and to implement, are proposed that allow working close to saturation, while cancelling the third-order nonlinearity of the PA without prior knowledge of the signal statistics and PA model. Their design consists in saturating a single or a few antennas on purpose together with an negative gain with respect to all other antennas to compensate for the overall nonlinear distortion at the user location. The performance gains of the designs are significant for PAs working close to saturation, as compared to maximum ratio transmission (MRT) precoding and perfect per-antenna digital pre-distortion (DPD) compensation.

Hyperparametric Oscillation via Bound States in the Continuum.

Optical hyperparametric oscillation based on the third-order nonlinearity is one of the most significant mechanisms to generate coherent electromagnetic radiation and produce quantum states of light. Advances in dispersion-engineered high-Q microresonators allow for generating signal waves far from the pump and decrease the oscillation power threshold to submilliwatt levels. However, the pump-to-signal conversion efficiency and absolute signal power are low, fundamentally limited by parasitic mode competition and attainable cavity intrinsic Q to coupling Q ratio, i.e., Q_{i}/Q_{c}. Here, we use Friedrich-Wintgen bound states in the continuum (BICs) to overcome the physical challenges in an integrated microresonator-waveguide system. As a result, on-chip coherent hyperparametric oscillation is generated in BICs with unprecedented conversion efficiency and absolute signal power. This work not only opens a path to generate high-power and efficient continuous-wave electromagnetic radiation in Kerr nonlinear media but also enhances the understanding of a microresonator-waveguide system-an elementary unit of modern photonics.

Third-order optical nonlinearity and passively Q-switching operation with BiOCl nanosheets at 1.3 µm.

In this contribution, we measured the third-order nonlinear optical response of bismuth oxychloride (BiOCl) nanosheets with the open-aperture (OA) and the closed-aperture (CA) Z-scan techniques with a variable excitation intensity at 1.34 µm. The effective nonlinear absorption coefficient βeff and the nonlinear refractive index n2 of the prepared BiOCl nanosheets with abundant oxygen vacancies were obtained under the excitation intensity. The third-order nonlinear optical susceptibility |χ(3)| was 1.64 × 10-9 esu. The nonlinear optical features of BiOCl enabled it as a superb saturable absorber for pulse laser generation. As a consequence, we demonstrated the first passively Q-switched Nd:GdVO4 laser with the BiOCl saturable absorber, producing a shortest pulse duration of 543 ns and a highest repetition rate of 227 kHz, leading to a maximum pulse energy of 74 nJ. Our findings show that BiOCl nanosheets with oxygen vacancies have large nonlinear optical sensitivities and can be exploited to generate optical pulses.

Robust Pulse-Pumped Quadratic Soliton Assisted by Third-Order Nonlinearity

The generation of a quadratic soliton in a pulse-pumped microresonator has attracted significant interest in recent years. The strong second-order nonlinearity and high peak power of pumps offer a straightforward way to increase efficiency. In this case, the influence of the third-order nonlinearity effect becomes significant and cannot be ignored. In this paper, we study the quadratic soliton in a degenerate optical parametric oscillator driven synchronously by the pulse pump with third-order nonlinearity. Our simulations verify that the robustness of quadratic soliton generation is enhanced when the system experiences a perturbation from pump power, cavity detuning, and pump pulse width. These results represent a new way of manipulating frequency comb in resonant microphotonic structures.

Broadband optical wavelength conversion through four-wave mixing in W3-type AlGaAs photonic crystal waveguides

We realized a wide-band wavelength conversion method through four-wave mixing in W3-type AlGaAs photonic crystal waveguides. AlGaAs exhibits a large third-order nonlinearity. Furthermore, because of its large bandgap, two-photon absorption can be avoided in the 1550 nm range. A four-wave mixing efficiency of −7 dB was obtained for a pump peak power of 7 W. Furthermore, by utilizing the two even guided bands of the W3-type photonic crystal waveguide, a conversion bandwidth greater than 38 nm was achieved with a conversion efficiency of −22 dB.

Third-Order Optical Nonlinearities of 2D Materials at Telecommunications Wavelengths

All-optical signal processing based on nonlinear optical devices is promising for ultrafast information processing in optical communication systems. Recent advances in two-dimensional (2D) layered materials with unique structures and distinctive properties have opened up new avenues for nonlinear optics and the fabrication of related devices with high performance. This paper reviews the recent advances in research on third-order optical nonlinearities of 2D materials, focusing on all-optical processing applications in the optical telecommunications band near 1550 nm. First, we provide an overview of the material properties of different 2D materials. Next, we review different methods for characterizing the third-order optical nonlinearities of 2D materials, including the Z-scan technique, third-harmonic generation (THG) measurement, and hybrid device characterization, together with a summary of the measured n2 values in the telecommunications band. Finally, the current challenges and future perspectives are discussed.

Coherent terahertz radiation from indium tin oxide film via third-order optical nonlinearity

Coherent terahertz (THz) wave emission from indium tin oxide (ITO) in epsilon-near-zero region had been reported recently as a second-order nonlinear process. However, THz radiation from ITO through third-order nonlinear process has not been reported yet. Here, we demonstrate coherent THz radiation from an ITO film via four-wave mixing, a third-order nonlinear process, under the excitation of two-color pulsed laser fields. Experimentally, an in-line phase compensator is utilized to synthesize the asymmetric optical fields to induce photocurrent in an ITO film and consequently emit THz waves. Our experimental results show that THz radiation from ITO films can be coherently controlled by the relative phase between the two-color laser fields. Such a THz wave generation process can be attributed to the fast intra-band nonlinearity in the conduction band. Our demonstration reveals that the fast nonlinearity in ITO can be used to generate THz waves and promotes the investigation of ITO in ultrafast photonics.

Tailored Dispersion of Spectro-Temporal Dynamics in Hot-Carrier Plasmonics.

Ultrafast optical switching in plasmonic platforms relies on the third-order Kerr nonlinearity, which is tightly linked to the dynamics of hot carriers in nanostructured metals. Although extensively utilized, a fundamental understanding on the dependence of the switching dynamics upon optical resonances has often been overlooked. Here, all-optical control of resonance bands in a hybrid photonic-plasmonic crystal is employed as an empowering technique for probing the resonance-dependent switching dynamics upon hot carrier formation. Differential optical transmission measurements reveal an enhanced switching performance near the anti-crossing point arising from strong coupling between local and nonlocal resonance modes. Furthermore, entangled with hot-carrier dynamics, the nonlinear correspondence between optical resonances and refractive index change results in tailorable dispersion of recovery speeds which can notably deviate from the characteristic lifetime of hot carriers. The comprehensive understanding provides new protocols for optically characterizing hot-carrier dynamics and optimizing resonance-based all-optical switches for operations across the visible spectrum.