Higher-order Nonlinear Response Research Articles

Efficient High-Order Harmonic Generation from the van der Waals Layered Crystal Copper Indium Thiophosphate.

Layered metal thio- and selenophosphates (MTPs) are a family of van der Waals gapped materials that exhibit a multitude of functionalities in terms of magnetic, ferroelectric, and optical properties. Despite the recent progress in terms of understanding the material properties of these compounds, the potential of MTPs as a material class yet needs further scrutiny, especially in terms of nonlinear optical properties. Recent reports of efficient low-order harmonic generation and extremely high third-order nonlinear optical properties in MTPs suggest the potential application of these materials in integrated nanophotonics. In this article, we investigate the high-order nonlinear response of bulk and exfoliated thin-film crystals of copper indium thiophosphate (CIPS) to intense mid-infrared fields through experimental and computational studies of high-order harmonic generation (HHG). From a driving laser source with a 3.2 μm wavelength, we generate odd and even harmonics up to the 10th order, exceeding the bandgap of the material. We note conversion efficiencies as high as 10-7 measured for the fifth and seventh harmonics and observe that the harmonic intensities follow a power law scaling with the driving laser intensity, suggesting a perturbative nonlinear optical origin of the observed harmonics for both bulk and thin flakes. Furthermore, first-principles calculations suggest that the generation of the highest harmonic orders results from electron-electron interactions, suggesting a correlation-mediated enhancement of the high-order optical nonlinearity.

Spectroscopic fingerprints of gapped quantum spin liquids, both conventional and fractonic

We explain how gapped quantum spin liquids, both conventional and 'fractonic', may be unambiguously diagnosed experimentally using the technique of multidimensional coherent spectroscopy. 'Conventional' gapped quantum spin liquids (e.g. $Z_2$ spin liquid) do not have clear signatures in linear response, but do have clear fingerprints in non-linear response, accessible through the already existing experimental technique of two dimensional coherent spectroscopy. Type I fracton phases (e.g. X-cube) are (surprisingly) even easier to distinguish, with strongly suggestive features even in linear response, and unambiguous signatures in non-linear response. Type II fracton systems, like Haah's code, are most subtle, and may require consideration of high order non-linear response for unambiguous diagnosis.

Estimating multiple latencies in the auditory system from auditory steady-state responses on a single EEG channel

The latency of the auditory steady-state response (ASSR) may provide valuable information regarding the integrity of the auditory system, as it could potentially reveal the presence of multiple intracerebral sources. To estimate multiple latencies from high-order ASSRs, we propose a novel two-stage procedure that consists of a nonparametric estimation method, called apparent latency from phase coherence (ALPC), followed by a heuristic sequential forward selection algorithm (SFS). Compared with existing methods, ALPC-SFS requires few prior assumptions, and is straightforward to implement for higher-order nonlinear responses to multi-cosine sound complexes with their initial phases set to zero. It systematically evaluates the nonlinear components of the ASSRs by estimating multiple latencies, automatically identifies involved ASSR components, and reports a latency consistency index. To verify the proposed method, we performed simulations for several scenarios: two nonlinear subsystems with different or overlapping outputs. We compared the results from our method with predictions from existing, parametric methods. We also recorded the EEG from ten normal-hearing adults by bilaterally presenting superimposed tones with four frequencies that evoke a unique set of ASSRs. From these ASSRs, two major latencies were found to be stable across subjects on repeated measurement days. The two latencies are dominated by low-frequency (LF) (near 40 Hz, at around 41–52 ms) and high-frequency (HF) (> 80 Hz, at around 21–27 ms) ASSR components. The frontal-central brain region showed longer latencies on LF components, but shorter latencies on HF components, when compared with temporal-lobe regions. In conclusion, the proposed nonparametric ALPC-SFS method, applied to zero-phase, multi-cosine sound complexes is more suitable for evaluating embedded nonlinear systems underlying ASSRs than existing methods. It may therefore be a promising objective measure for hearing performance and auditory cortex (dys)function.

High-order nonlinear response correction method for infrared radiation detector

The infrared detector can generate nonlinear response error when the Fourier transform infrared spectrometer is used for implementing the radiometric calibration or observing the high temperature targets. Based on the relationship between the incident radiation intensity and the electron concentration in the optical conduction band, the mechanism of the nonlinear response error caused by the high incident photon flow is analyzed. According to Planck radiation law and interference principle, the effect of nonlinear error on spectrum is studied by simulating blackbody radiation data with nonlinear error. It is found that the nonlinear response with a different order has a different influence region, and the higher-order nonlinear response has a wider influence range and generates a larger nonlinear response error. By the general nonlinear response correction method the nonlinear response coefficient is obtained through constructing the nonlinear response model of the interference data and then the spectral distortion produced by the detector is corrected. According to the convolution iteration method, the polar orbit meteorological satellite CrIS constructs the convolution equation to correct the second-order nonlinear response by taking the low-wave number band of 50－500 cm<sup>–1</sup> as the characteristic region. The European Meteorological Agency’s Airborne Infrared Interferometer Evaluation System (ARIES) selected two feature areas, 50－500 cm<sup>–1</sup> and 2000－2500 cm<sup>–1</sup>, and iteratively corrected the second-order and third-order nonlinear response. The gradient descent method is often used to solve the optimization problems of unconstrained multivariate functions. Based on the gradient descent algorithm, an iterative method suitable for correcting the high-order nonlinear response errors is proposed in this paper. In this method, the information about the iteration point is obtained by constructing the nonlinear response function of the high-order detector and setting the appropriate iteration initialization. According to the initial value of the iteration and the information about the known iteration point, the gradient of the iteration variable is calculated to determine the iteration value of the next unknown variable, thus quickly searching for the global minimum point and determining the nonlinear response coefficient. We use Fourier transform infrared spectrometer to carry out radiometric calibration experiment and compare the effects of three correction methods: convolution, cross iteration and gradient descent method. The results show that the three correction methods can effectively reduce the nonlinear error, and improve the fitting extent by 0.15%, 0.29% and 0.39% respectively. The spectral data corrected by gradient descent method are more accurate.

Ultrasonic frequency mixing via local defect resonance for defect imaging in composites

A new approach to nonlinear frequency mixing based on local damage resonance is proposed, analysed and tested experimentally for flexural waves in composites. The method is free from stringent requirements on the mode types and frequencies for interacting waves. The resonance of damage enhances strongly its higher-order nonlinear response and boosts the efficiency of generation for numerous-order combination frequencies. The damage resonance combined with its strong nonlinearity also provides locality of nonlinear interaction even for continuous wave operation. The combination frequencies generated locally in the damaged area are the footprints of damage and used for its detection, location and visualization. A single C-scan yields a number of images of the defect corresponding to various nonlinearly generated frequencies. Various versions of the resonant frequency mixing are considered and applied to nonlinear imaging of defects in composite materials.

Stopping power of hot dense deuterium-tritium plasmas mixed with impurities to charged particles.

In this work, we studied the stopping power of deuterium-tritium (DT) plasmas mixed with impurities to the injected charged particles. Based on the Brown-Preston-Singleton model, the analytical expression for the change ratio of stopping power (denoted by η) induced by impurities in DT plasmas is developed, in which both classical short-distance collision part and quantum correction contribution are purely linear response to the impurity concentration ξ_{X}, while the classical long-range collision brings about higher-order nonlinear response to ξ_{X}. Furthermore, the expression for change ratio of deposition depth (denoted by χ) of charged particles induced by impurities in DT plasmas is also derived. As applications, we systemically investigated the energy loss of α particles deposited into a hot dense DT plasma mixed with impurity X(X=C, Si, Ge), where the temperature and density of DT are smaller than 10 keV and 500 g/cm^{3} and the concentration of Xξ_{X} is less than 5%. The numerical results suggest that (i) for the case of C mixed into DT, both change ratios of stopping power and deposition depth of α particles (i.e., η and χ) are linear response to the concentration of C ξ_{C}; (ii) for the case of Si mixed into DT, the second-order nonlinear response of η and χ to ξ_{Si} cannot be ignored when the densities of DT are larger than 200 g/cm^{3}; and (iii) for the case of Ge mixed into DT, the second- and third-order nonlinear response of η and χ to ξ_{Ge} are very remarkable because of the higher ionization degree and heavier atomic mass of Ge. The formulas and findings in this work may be helpful to the research of internal confinement fusion (ICF) related implosion physics and may provide useful theoretical guidance and data for the design of ICF target.

Nonlinear optical characterization of copper oxide nanoellipsoids

Recently, nonspherical nanoparticles took attention due to advanced properties of these structures. We report the study of the nonlinear optical properties of copper oxide nanoellipsoids using 800 nm and 400 nm, 60 fs pulses. The optical limiting effect of copper oxide nanoellipsoids is analyzed. The influence of band gap of copper nanoparticles and copper oxide nanoellipsoids on their nonlinear optical response was studied. For the first time, the low- and high-order nonlinear optical responses of copper nanoellipsoids were studied. The magnitudes of nonlinear optical parameters of the suspension of copper oxide nanoellipsoids were measured to be γ = 1.23 × 10−15 cm2 W−1, and β = 1.0 × 10−11 cm W−1 respectively. We observed the four-fold enhancement of the nonlinear optical refraction of copper oxide nanoellipsoids at the wavelength of 400 nm, 60 fs probe pulses compared to 800 nm radiation. We also analyzed the high-order nonlinear response of CuO nanoellipsoids through generation of high-order harmonics of 800 nm, 60 fs pulses in the plasmas produced during laser ablation of the nanoellipsoid-contained targets. We demonstrated the harmonics up to the 35th order (E = 50 eV) in case of single-color pump and 24th (30 eV) in case of two-color pump.

High-Order Nonlinearities of Gold Nanoparticles: The Influence of Size, Filling Factor, and Host

High-order nonlinearities of gold nanoparticles embedded in a nonlinear host were studied using the Maxwell-Garnett model and the degenerate electron gas model. From the nonlinear response, the dependence of the effective third-, fifth-, seventh-, and ninth-order nonlinear optical susceptibilities of gold nanoparticles on the filling factor, the nanoparticle size, and refraction index of the host was calculated. The predicted high-order nonlinear response arose from the enhanced local field factor and was attributed only to the cascaded contribution coming from the third-order nonlinear susceptibility of the nanoparticles. From the calculations, the reversal signal on the high-order nonlinear susceptibilities varying the host refractive index was observed. In addition, tuning the filling factor or the nanoparticle (NP) size, the cancelation of the real part of the third-order susceptibility of the composite material was proposed in order to obtain a refractive macroscopic nonlinear response due only to the real part of the fifth-, seventh-, and ninth-order nonlinearities. It shows a new way to manage the high-order nonlinear optical response of composite materials containing metallic nanoparticles.

Phantom investigation of phase-inversion-based dual-frequency excitation imaging for improved contrast display

Objective The goal of this work is to examine the effects of pulse-inversion (PI) technique in combination with dual-frequency (DF) excitation method to separate the high-order nonlinear responses from microbubble contrast agents for improvement of image contrast. DF excitation method has been previously developed to induce the low-frequency ultrasound nonlinear responses from bubbles by using the composition of two high-frequency sinusoids ( f 1 and f 2). Motivation Although the simple filtering was conventionally utilized to provide signal separation, the PI approach is better in the sense that it minimizes the mutual interferences among these high-order nonlinear responses in the presence of spectral overlap. The novelty of the work is that, in addition to the common PI summation, the PI subtraction was also applied in DF excitation method. Methods DF excitation pulses having an envelope frequency of 3 MHz (i.e., f 1 = 8.5 MHz and f 2 = 11.5 MHz) with pulse lengths of 3–10 μs and the pressure amplitudes from 0.5 to 1.5 MPa were used to interrogate the nonlinear responses of SonoVue™ microbubbles in the phantom experiments. The high-order nonlinear responses in the DF excitation were extracted for contrast imaging using PI summation for even-order nonlinear components or PI subtraction for odd-order nonlinear ones. Results Our results indicated that, as compared to the conventional filtering technique, the PI processing effectively increases the contrast-to-tissue ratio (CTR) of the third-order nonlinear response at 5.5 MHz and the fourth-order nonlinear response at 6 MHz by 2–5 dB. For these high-order nonlinear components, the CTR increase varies with the transmission pressures from 0.5 to 1.5 MPa due to the microbubbles’ displacement induced by the radiation force of DF excitation. Conclusions For DF excitation technique, the PI processing can help to extract either the odd-order or the even-order nonlinear components for higher CTR estimates.

Phase-dependent dual-frequency contrast imaging at sub-harmonic frequency

Sub-harmonic imaging techniques have been shown to provide a higher contrast-to-tissue ratio (CTR) at the cost of relatively low signal intensity from ultrasound contrast agents (UCAs). In this study, we propose a method of dual-frequency excitation to further enhance the CTR of subharmonic imaging. A dual-frequency excitation pulse is an amplitude-modulated waveform which consists of two sinusoids with frequencies of f₁ (e.g., 9 MHz) and f₂ (e.g., 6 MHz) and the resulting envelope component at (f₁ - f₂) (e.g., 3 MHz) can serve as a driving force to excite the nonlinear response of UCAs. In this study, the f₂, at twice of the resonance frequency of UCAs, is adopted to efficiently generate a sub-harmonic component at half of the f₂ frequency, and f₁ is included to enhance the high-order nonlinear response of UCAs at the sub-harmonic frequency. The second- and third-order nonlinear components resulting from the envelope component would spectrally overlap at the sub-harmonic frequency when f₁ and f₂ are properly selected. We further optimize the generation of the sub-harmonic component by tuning the phase terms between second- and third-order nonlinear components. The results show that, with dual-frequency excitation, the CTR at sub-harmonic frequency improves compared with the conventional tone-burst method. Moreover, the CTR changes periodically with the relative phase of the separate frequency component in the dual-frequency excitation, leading to a difference of as much as 9.1 dB between the maximal and minimal CTR at 300 kPa acoustic pressure. The echo produced from the envelope component appears to be specific for UCAs, and thus the proposed method has the potential to improve both SNR and CTR in sub-harmonic imaging. Nevertheless, the dual-frequency waveform may suffer from frequency-dependent attenuation that degrades the generation of the envelope component. The deviation of the microbubble's resonance characteristics from the selection of dual-frequency transmission may also decrease the CTR improvement.

Effective higher-order nonlinear coefficients of composites with weakly nonlinear media

The field equations, based on the third-order perturbation expansion of electrostatic potential, are derived, and our general formulae for higher-order effective nonlinear coefficients based on the energy definition, are presented and applied to dielectric composites consisting of dilute linear cylindrical inclusions randomly dispersed in a weakly nonlinear host media. The effective nonlinear coefficients are determined up to the ninth order. In addition, the results are also compared to those obtained using the average field method and likely to provide more accurate predictions of effective higher-order nonlinear responses.

Identification of material parameters for McGinty’s model using adaptive RBFs and optimization

In this paper, an optimization approach is adopted to obtain the 12 material parameters used in McGinty’s Model for AL6022 by minimizing the differences between simulation and experimental stress–strain curves. Since the differences between the two stress–strain curves are implicitly related to the change of material parameters, the metamodeling technique is utilized to create explicit, approximate functions of these relationships. Radial basis functions (RBFs), which are shown from previous studies to be effective for both low- and high-order nonlinear responses, are used for the metamodels that are adaptively updated in the optimization work. Two optimization formulation schemes are studied to address the issue of using inaccurate RBF models in optimization. The sampling, metamodeling, and optimization works are performed using the integrated optimization framework HiPPO.

Observation of a Microscopic Cascaded Contribution to the Fifth-Order Nonlinear Susceptibility

Typically, low-order nonlinearities are much stronger than high-order nonlinearities. In this Letter, we demonstrate a procedure by which strong high-order nonlinearities can be synthesized out of low-order nonlinearities. Our procedure involves the use of the previously largely overlooked process of microscopic cascading, which results from local-field effects. We have performed an experiment that allows us to distinguish the influence of microscopic cascading from the more-well-known process of macroscopic cascading, and we find conditions under which microscopic cascading can be the dominant effect. The ability to create a large high-order nonlinear response could prove useful for applications in quantum-information science that require the detection of the simultaneous presence of N entangled photons.

Global response approximation with radial basis functions

In this article, a study is performed on the accuracy of radial basis functions (RBFs) in creating global metamodels for both low- and high-order nonlinear responses. The response surface methodology (RSM), which typically uses linear or quadratic polynomials, is inappropriate for creating global models for highly nonlinear responses. The RBF, on the other hand, has been shown to be accurate for highly nonlinear responses when the sample size is large. However, for most complex engineering applications only limited numbers of samples can be afforded; it is desirable to know whether the RBF is appropriate in this situation, especially when the augmented RBF has to be used. Because the types of true responses are typically unknown a priori, it is essential for high-fidelity metamodeling to have an RBF or RBFs that are appropriate for linear, quadratic, and higher-order responses. To this end, this study compares a variety of existing basis functions in both non-augmented and augmented forms with various types of responses and limited numbers of samples. This article shows that the augmented RBF models created by Wu’s compactly supported functions are the most accurate for the various test functions used in this study.

Higher-order nonlinear responses in weakly nonlinear composite materials within the effective medium approximation

Based on the spectral representation theory and the perturbative expansion metho d, the general expressions for effective nonlinear responses (χe a nd ηe) are derived for any complicated micro structures. The higher_orde r nonlinear response of two_component composite is studied by using the effectiv e medium approximation. The effective dielectric constant of the composite is gi ven by ε～e=εe+χe|E0|2+ηe|E0|4. The effe ctive cubic order nonlinear response χe and the higher_order response ηe for the whole volume fraction p of the nonlinear component ar e investigated and the general expressions of χeand ηeare g iven in this paper. We also perform numerical simulations for χe and ηe in the case of complex dielectric constant of the nonlinear co mponent. The effect of the higher order nonlinear responses on the effective di electric constant of the composite is theoretically studied.

Powder second-harmonic generation study of (K2O)15(Nb2O5)15(TeO2)70 glass ceramic

A power second-harmonic generation (SHG) study has been carried out on (K2O)15(Nb2O5)15(TeO2)70 glass ceramic, which consists of centrosymmetric crystals embedded in a centrosymmetric glass matrix. The effective second-order nonlinear optical coefficient, χ(2), of this material at 532nm is found to be 1.3pm∕V or approximately ten times that of quartz, leading to a maximum second-harmonic intensity approximately 100 times that of quartz. This material is nonphase matchable, with a coherence length of 30μm. We conclude that the SHG in this material originates from the abrupt changes in optical susceptibility at the interfaces between the glass and crystal, together with the higher-order nonlinear response.

Spectral representation theory for higher-order nonlinear response in random composites

The higher-order nonlinear response of random two-phase composites is investigated within the spectral representation theory. Both components are assumed to be weakly nonlinear with a displacement ( D )—electric field ( E ) relation of the form D =ϵ i E +χ i| E | 2 E , where ϵ i is the linear dielectric constant and χ i is the third-order nonlinear response of the ith component ( i=1,2). The effective field-dependent dielectric response of the composites is defined as ϵ ̃ e=ϵ e+χ e| E 0| 2+η e| E 0| 4 , where ϵ e , χ e and η e are the effective linear dielectric constant, the effective third-order and fifth-order nonlinear susceptibilities, respectively. By taking into account the correction to the nonlinear local fields, general expressions are derived for χ e and η e to the second order of χ i . We perform numerical calculations on Au/SiO 2 composites with three typical microstructures, described by the Maxwell–Garnett approximation, Bruggeman effective medium approximation and differential effective medium one. Our formulae are then generalized to systems of arbitrary nonlinearity. In the dilute limit, the formulae reduce to those in previous works, as expected. For large volume fractions, our theoretical results are in good agreement with simulation data in nonlinear random resistor networks.

Biexcitonic nonlinear response in GaAs thin film studied by degenerate four-wave mixing

Nonlinear optical response of a high-quality GaAs thin film of 200 nm thickness was studied by time-resolved degenerate four-wave mixing (DFWM) using pico-second excitation pulses at low temperature (5 K). The DFWM signals show a strong nonlinearity due to biexciton at a slightly lower energy of exciton resonance. The polarization dependence of the DFWM also reveals that higher-order nonlinear response is dominated by biexciton processes.

Experimental investigation of response stability and transition behaviour of a nonlinear ocean structural system

This study experimentally investigates the nonlinear response stability and transition behaviour of a submerged, moored ocean structural system which consists of a spherical buoy and attached multi-point mooring lines. The system is excited by a periodic wave field in a closed channel. System nonlinearities include complex geometric restoring (stiffness) force and coupled fluid–system interaction exciting forces. Experimental set-up, operating procedures and analysis of the measured results are presented. Characteristic motions observed include harmonic, subharmonic and ultraharmonic responses, which demonstrate a signature of the intricate pattern of the nonlinear global behaviour. Good agreements between the measured and most predicted responses are demonstrated in both time and frequency domains. These results confirm the validity of the analytical model presented and calibrate the accuracy of the existing numerical predictions. Primary and secondary resonances in the response are identified via frequency response curves. Response bifurcation cascades are observed in the experimental results and the possible existence of higher-order nonlinear responses is inferred.

Higher order nonlinear response in random resistor networks: numerical studies for arbitrary nonlinearity

Higher order nonlinear response in random resistor networks: numerical studies for arbitrary nonlinearity