Nonlinear Mixing Research Articles

Intrinsic photomixing detector based on amorphous silicon for envelope mixing of optical signals

In this work, a promising device for direct optical envelope mixing, the Intrinsic Photomixing Detector (IPD) based on hydrogenated amorphous silicon, is reported. The IPD directly generates a photocurrent proportional to the nonlinear mixing of two optical modulation envelope functions. Experiments illustrate efficient mixing in the visible range at low light levels down to ϕ1 = 4.36 mW/cm2 (444 nm) and ϕ2 = 1.03 mW/cm2 (636 nm). Modulation frequencies exceeding the MHz range are demonstrated. Electro-optical simulations identify defect-induced electrical field screening within the absorber to cause the nonlinear mixing process, opening-up the opportunity to tailor devices toward application-specific requirements. The IPD functionality paves the way toward very simple but high-performance photodetectors for 3D imaging and ranging for direct optical convolutional sensors or for efficient optical logic gates. Using amorphous silicon provides a photodetector material base, which can easily be integrated on top of silicon electronics, enabling fill factors of up to 100%.

Theory of localization-hindered thermalization in nonlinear multimode photonics

Our society’s appetite for ultra-high bandwidth communication networks and high-power optical sources, together with recent breakthroughs in mode multiplexing/demultiplexing schemes, forced the photonics community to reconsider the deployment of nonlinear multimode systems. These developments pose fundamental challenges stemming from the complexity of nonlinear mode-mode mixing by which they exchange energy in the process towards an equilibrium Rayleigh-Jeans (RJ) distribution. Here we develop a universal one-parameter scaling theory for the relaxation rates of out-of-equilibrium excitations towards their RJ thermal state. The theory predicts an exponential suppression of the rates with increasing disorder due to the formation of stable localization clusters resisting the nonlinear mode-mode interactions that tend to separate them. For low optical temperatures, the rates experience a crossover from linear to nonlinear temperature dependence which reflects a disorder-induced reorganization of the low frequency eigenmodes. Our theory will guide the design of nonlinear multimode photonic networks with tailored relaxation-scales.

A Multi-Box Behavioural Nonlinear Mixer Model

ABSTRACT This paper presents a new behavioural mixer model in a multi-box structure. In the proposed model, as well as amplitude and phase nonlinearity, the memory effect is also modelled. The multi-box structure is constructed in line with the Hammerstein model to characterise both fundamental signals, intermodulation distortion (IMD) and harmonic distortion (HD) for mixers in particular and all frequency translating devices in general. A testbed is constructed to measure both amplitude and phase characteristics. A good match between the measured and the calculated results is obtained using curve fitting and optimisation algorithms. In order to model the memory effect, a finite impulse response (FIR) filter is used. The measured IM3 imbalance is successfully modelled with this filter. For fundamental components, amplitude and phase errors are less than 0.1 dB and 1 degree, respectively. For IM3 components, amplitude and phase errors are less than 4 dB and 10 degrees, respectively. For the overall accuracy of the model, measurements are conducted with two-tone and modulated signals, and normalised mean square error (NMSE) values are calculated as −31.5 dB and −29.3 dB, respectively. The results are obtained for a wide range of input power span, including the compression region where nonlinearity is strongly pronounced. The proposed model is capable of accurately representing nonlinearity and memory effect for mixers and frequency translating devices in general and thus may be used for system-level modelling and linearisation applications.

Impact of longitudinal phase-matching variations on three-wave nonlinear interactions

A general study of three-wave nonlinear mixing in the presence of longitudinal variations in phase-matching conditions is presented. The efficiency of second-harmonic generation and optical parametric amplification is quantified using a normalized set of equations and a polynomial description of the wave-vector mismatch as a function of the longitudinal coordinate. These modeling results are used to estimate the impact of spatial variations in wave-vector mismatch experimentally obtained for five partially deuterated potassium dihydrogen phosphate crystals. The longitudinal inhomogeneities in the properties of crystals of similar quality are not expected to have a significant impact on their use for second-harmonic generation and optical parametric amplification, but the efficiency of nonlinear processes in crystals with larger variations could decrease.

Modelling Spectral Unmixing of Geological Mixtures: An Experimental Study Using Rock Samples

Spectral unmixing of geological mixtures, such as rocks, is a challenging inversion problem because of nonlinear interactions of light with the intimately mixed minerals at a microscopic scale. The fine-scale mixing of minerals in rocks limits the sensor’s ability to identify pure mineral endmembers and spectrally resolve these constituents within a given spatial resolution. In this study, we attempt to model the spectral unmixing of two rocks, namely, serpentinite and granite, by acquiring their hyperspectral images in a controlled environment, having uniform illumination, using a laboratory-based imaging spectroradiometer. The endmember spectra of each rock were identified by comparing a limited set of pure hyperspectral image pixels with the constituent minerals of the rocks based on their diagnostic spectral features. A series of spectral unmixing paradigms for explaining geological mixtures, including those ranging from simple physics-based light interaction models (linear, bilinear, and polynomial models) to classification-based models (support vector machines (SVMs) and half Siamese network (HSN)), were tested to estimate the fractional abundances of the endmembers at each pixel position of the image. The analysis of the results of the spectral unmixing algorithms using the ground truth abundance maps and actual mineralogical composition of the rock samples (estimated using X-ray diffraction (XRD) analysis) indicate a better performance of the pure pixel-guided HSN model in comparison to the linear, bilinear, polynomial, and SVM-based unmixing approaches. The HSN-based approach yielded reduced errors of abundance estimation, image reconstruction, and mineralogical composition for serpentinite and granite. With its ability to train using limited pure pixels, the half-Siamese network model has a scope for spectrally unmixing rock samples of varying mineralogical composition and grain sizes. Hence, HSN-based approaches effectively address the modelling of nonlinear mixing in geological mixtures.

Experimental demonstration of an optics-based 4-PSK half-adder using nonlinear wave mixing.

We experimentally demonstrate an optics-based half-adder of two 4-phase-shift-keying (4-PSK) data channels using nonlinear wave mixing. The optics-based half-adder has two 4-ary phase-encoded inputs (i.e., SA and SB) and two phase-encoded outputs (i.e., Sum and Carry). The input quaternary base numbers {0,1,2,3} are represented by 4-PSK signals A and B with four phase levels. Along with the original signals A and B, the phase-conjugate signal copies A* and B*and phase-doubled signal copies A2 and B2 are also generated to form two signal groups SA(A, A*, A2) and SB(B, B*, B2). All of the above signals in the same signal group are (a) prepared in the electrical domain with a frequency spacing of Δf and (b) generated optically in the same IQ modulator. When combined with a pump laser, group SA mixes with group SB in a periodically poled lithium niobate nonlinear (PPLN) device. At the output of the PPLN device, both the Sum (A2B2) and the Carry (AB + A*B*) are simultaneously generated with four phase levels and two phase levels, respectively. In our experiment, the symbol rates can be varied between 5 Gbaud and 10 Gbaud. The experimental results show that (i) the measured conversion efficiency of two 5-Gbaud outputs is approximately -24 dB for Sum and approximately -20 dB for Carry, and (ii) the measured optical signal-to-noise ratio (OSNR) penalty of the 10-Gbaud Sum and Carry channels is <10 dB and <5 dB, compared with that of the 5-Gbaud channels at the BER of 3.8 × 10-3.

Few-femtosecond phase-sensitive detection of infrared electric fields with a third-order nonlinearity

Measuring an electric field waveform beyond radio frequencies is often accomplished via a second-order nonlinear interaction with a laser pulse shorter than half of the field’s oscillation period. However, synthesizing such a gate pulse is extremely challenging when sampling mid- (MIR) and near- (NIR) infrared transients. Here, we demonstrate an alternative approach: a third-order nonlinear interaction with a relatively long multi-cycle pulse directly retrieves an electric-field transient whose central frequency is 156 THz. A theoretical model, exploring the different nonlinear frequency mixing processes, accurately reproduces our results. Furthermore, we demonstrate a measurement of the real part of a sample’s dielectric function, information that is challenging to retrieve in time-resolved spectroscopy and is therefore often overlooked. Our method paves the way towards experimentally simple MIR-to-NIR time-resolved spectroscopy that simultaneously extracts the spectral amplitude and phase information, an important extension of optical pump-probe spectroscopy of, e.g., molecular vibrations and fundamental excitations in condensed-matter physics.

Augmented GBM Nonlinear Model to Address Spectral Variability for Hyperspectral Unmixing

Spectral unmixing (SU) is a significant preprocessing task for handling hyperspectral images (HSI), but its process is affected by nonlinearity and spectral variability (SV). Currently, SV is considered within the framework of linear mixing models (LMM), which ignores the nonlinear effects in the scene. To address that issue, we consider the effects of SV on SU while investigating the nonlinear effects of hyperspectral images. Furthermore, an augmented generalized bilinear model is proposed to address spectral variability (abbreviated AGBM-SV). First, AGBM-SV adopts a generalized bilinear model (GBM) as the basic framework to address the nonlinear effects caused by second-order scattering. Secondly, scaling factors and spectral variability dictionaries are introduced to model the variability issues caused by the illumination conditions, material intrinsic variability, and other environmental factors. Then, a data-driven learning strategy is employed to set sparse and orthogonal bases for the abundance and spectral variability dictionaries according to the distribution characteristics of real materials. Finally, the alternating direction method of multipliers (ADMM) optimization method is used to split and solve the objective function, enabling the AGBM-SV algorithm to estimate the abundance and learn the spectral variability dictionary more effectively. The experimental results demonstrate the comparative superiority of the AGBM-SV method in both qualitative and quantitative perspectives, which can effectively solve the problem of spectral variability in nonlinear mixing scenes and to improve unmixing accuracy.

Normalized ground states for fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

In this paper, we study the existence of normalized ground states for nonlinear fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities in R3. To overcome the special difficulties created by the nonlocal term and fractional Sobolev critical term, we develop a perturbed Pohožaev method based on the Brézis–Lieb lemma and monotonicity trick. Using the Pohožaev manifold decomposition and fibering map, we prove the existence of a positive normalized ground state. Moreover, the asymptotic behavior of the obtained normalized solutions is also explored. These conclusions extend some known ones in previous papers.

Analysis of large time asymptotics of the fourth‐order parabolic system involving variable coefficients and mixed nonlinearities

In a general domain, we discuss an initial value problem for the symmetry generalized ‐coupled system of fourth‐order parabolic equations with space–time‐dependent diffusion coefficients and mixed nonlinearities. For the power‐type nonlinearities containing the sub‐critical Sobolev exponents, we achieve the local existence, a unique continuation, global existence, or finite‐time blow‐up of solutions to the system in time‐weighted Sobolev space. For the logarithmic nonlinearities containing the critical Sobolev exponents, we investigate the local existence in time‐weighted Sobolev space.

Assessment of assorted soliton solutions and impacts analysis of fractional derivatives on wave profiles

The nonlinear geometric optics, neutral scalar masons, unidirectional propagation of small amplitude, long surface gravity waves, long waves with mixed nonlinearity and dissipative influence, and some other real-world circumstances are explained by the time-fractional Nizhnik-Novikov-Veselov and the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equations. Using the fractional wave transformation, the nonlinear models are converted to nonlinear equations of a single wave variable. Diverse comprehensive, typical, and some standard wave solutions in the form of rational, trigonometric, hyperbolic functions and their assimilations to the stated models are investigated in this article using the (F′/F,1/F)-expansion approach. When the parameters are assigned to definite values, the general waves yield a variety of shapes, including periodic, kink, bell-shaped, anti-kink, periodic-type solitons, etc. The impact of the fractional parameter α on wave patterns has also been studied. It is seen that the soliton shape changes with the change of the fractional order derivative α. Through three and two dimensional plots; the physical properties of different soliton solutions are examined. The results demonstrate the approach is suitable for investigating a range of nonlinear fractional systems in the sense of beta derivative.

Terahertz generation by beat-wave mechanism of two Gaussian laser beams with corrugated noble-metal nanospheres in external magnetic field

An analytical model is presented that deals with THz generation by nonlinear mixing of two Gaussian laser beams with different frequencies ω1 and ω2 and wave vectors k1→ and k2→simultaneously propagating through an array of spatially corrugated noble-metal nanoparticles placed in the presence of a static magnetic field where ponderomotive nonlinearities are operative. The two co-propagating laser beams exert a nonlinear ponderomotive force on the plasma electrons, causing them to acquire nonlinear oscillatory velocity. This velocity leads to the generation of nonlinear macroscopic current density at the beat frequency ω (ω1-ω2) which further gives rise to strong terahertz radiation. The beat frequency lies in the THz region and density ripple provides the necessary phase-matching conditions. The externally applied magnetic field enhances the nonlinear coupling between the electric field of the lasers and causes resonant interaction of the laser beam with electrons of the NPs. THz radiations being non-invasive and biologically innocuous, play an important role in medical imaging, biomedical and pharmaceutical fields, and THz spectroscopy.

Visible to near-infrared reflectance and Raman spectra of evaporites from sulfate-chloride Mars analogue brines

While much attention has been given to the identification and characterization of single-phase salts on Mars, relatively little has been applied to mixed evaporative assemblages. Given the likely existence of these assemblages on Mars (e.g., basin deposits) and the lack of data on their spectral signatures, here we present an experimental study of multicomponent S and Cl-bearing Mars-relevant brines. We modeled, synthesized and evaporated brines in the laboratory under both martian and terrestrial (P, T, pCO2) environmental conditions, and characterized the resulting precipitates using Visible–Near Infrared (VNIR), Raman spectroscopy, XRD and SEM-EDS. We compared these results to mineral assemblages calculated using the FREZCHEM thermodynamic model. For mixed brines primarily containing Na+, K+, Mg2+, Ca2+, SO42− and Cl−, epsomite (MgSO4•7H2O) and bischofite (MgCl2•6H2O) overwhelm VNIR and Raman spectra, while anhydrous crystalline salts and Na- and K-sulfates are unidentified. Mg-sulfates are identifiable in the VNIR and Raman even at low or no modeled mass abundance in an evaporative assemblage and are often the only clearly identifiable salt. These results imply that regions of Mg-sulfate identification on Mars may have only minor amounts of Mg-sulfate present, and significant amounts of halides or other sulfates may be undetectable. This may be due to a combination of late-stage Mg-sulfate precipitation and non-linear spectral mixing. Raman is more sensitive than VNIR to the identification of Ca-sulfate salts in these mixed assemblages. We predict that high abundances of mixed chloride and sulfate salts species will be identified as the Curiosity Rover continues to explore the Sulfate Unit of Gale Crater, and note that the Perseverance rover offers the first opportunity to identify such mixed assemblages in Jezero Crater with this combination of techniques.

Barotropic Instability during Eyewall Replacement

Just before making landfall in Puerto Rico, Hurricane Maria (2017) underwent a concentric eyewall cycle in which the outer convective ring appeared robust while the inner ring first distorted into an ellipse and then disintegrated. The present work offers further support for the simple interpretation of this event in terms of the non-divergent barotropic model, which serves as the basis for a linear stability analysis and for non-linear numerical simulations. For the linear stability analysis the model’s axisymmetric basic state vorticity distribution is piece-wise uniform in five regions: the eye, the inner eyewall, the moat, the outer eyewall, and the far field. The stability of such structures is investigated by solving a simple eigenvalue/eigenvector problem and, in the case of instability, the non-linear evolution into a more stable structure is simulated using the non-linear barotropic model. Three types of instability and vorticity rearrangement are identified: (1) instability across the outer ring of enhanced vorticity; (2) instability across the low vorticity moat; and (3) instability across the inner ring of enhanced vorticity. The first and third types of instability occur when the rings of enhanced vorticity are sufficiently narrow, with non-linear mixing resulting in broader and weaker vorticity rings. The second type of instability, most relevant to Hurricane Maria, occurs when the radial extent of the moat is sufficiently narrow that unstable interactions occur between the outer edge of the primary eyewall and the inner edge of the secondary eyewall. The non-linear dynamics of this type of instability distort the inner eyewall into an ellipse that splits and later recombines, resulting in a vorticity tripole. This type of instability may occur near the end of a concentric eyewall cycle.

Metasurface spatial filters for multiple harmonic signals

Abstract Nonlinear frequency mixings have shown an alternative way to create new electromagnetic sources in frequency ranges that are difficult to access with conventional techniques. To simultaneously use the fundamental frequency pump beam and multiple harmonic signals generated in the same channel, a device capable of separating each frequency component is required. Here, we propose and experimentally demonstrate metasurface-based spatial filters for the pump frequency and multiple harmonic frequencies. The metasurface was designed using eight different split ring resonator-based phase elements with 45° phase spacing, which allows wavefront shaping. The metasurface designed to have a one-dimensional gradient phase array produces cross-polarized reflection waves with different beam steering angles at the third- and fifth-harmonic frequencies (15 and 25 GHz) and operates as a metallic mirror at the fundamental frequency of 5 GHz. Our work suggests a new method to enable simultaneous use of broadband multi-frequency sources based on nonlinear frequency mixing.

A note on inhomogeneous fractional Schrödinger equations

We study some energy well-posedness issues of the Schrödinger equation with an inhomogeneous mixed nonlinearity and radial data iu˙−(−Δ)su±|x|ρ|u|p−1u±|u|q−1u=0,0<s<1,ρ≠0,p,q>1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ i\\dot{u}-(-\\Delta )^{s} u \\pm \\vert x \\vert ^{\\rho} \\vert u \\vert ^{p-1}u\\pm \\vert u \\vert ^{q-1}u=0, \\quad 0< s< 1, \\rho \ eq 0, p,q>1. $$\\end{document} Our aim is to treat the competition between the homogeneous term |u|^{q-1}u and the inhomogeneous one |x|^{rho}|u|^{p-1}u. We simultaneously treat two different regimes, rho >0 and rho <-2s. We deal with three technical challenges at the same time: the absence of a scaling invariance, the presence of the singular decaying term |cdot |^{rho}, and the nonlocality of the fractional differential operator (-Delta )^{s}. We give some sufficient conditions on the datum and the parameters N, s, ρ, p, q to have the global versus nonglobal existence of energy solutions. We use the associated ground states and some sharp Gagliardo–Nirenberg inequalities. Moreover, we investigate the L^{2} concentration of the mass-critical blowing-up solutions. Finally, in the attractive regime, we prove the scattering of energy global solutions. Since there is a loss of regularity in Strichartz estimates for the fractional Schrödinger problem with nonradial data, in this work, we assume that u_{|t=0} is spherically symmetric. The blowup results use ideas of the pioneering work by Boulenger el al. (J. Funct. Anal. 271:2569–2603, 2016).

Accounting for non-ideal mixing effects in the hydrogen-helium equation of state

Context. The equation of state for hydrogen and helium is fundamental for studying stars and giant planets. It has been shown that because of interactions at atomic and molecular levels, the behaviour of a mixture of hydrogen and helium cannot be accurately represented by considering these elements separately. Aims. This paper aims at providing a simple method to account for interactions between hydrogen and helium in interior and evolution models of giant planets. Methods. Using on the one hand ab initio simulations that involve a system of interacting hydrogen and helium particles and pure equations of state for hydrogen and helium on the other, we derived the contributions in density and entropy of the interactions between hydrogen and helium particles. Results. We show that relative variations of up to 15% in density and entropy arise when non-ideal mixing is accounted for. These non-ideal mixing effects must be considered in interior models of giant planets based on accurate gravity field measurements, particularly in the context of variations in the helium-to-hydrogen ratio. They also affect the mass-radius relation of exoplanets. We provide a table that contains the volume and entropy of mixing as a function of pressure and temperature. This table is to be combined with pure hydrogen and pure helium equations of state to obtain an equation of state that self-consistently includes mixing effects for any hydrogen and helium mixing ratio and may be used to model the interior structure and evolution of giant planets to brown dwarfs. Conclusions. Non-linear mixing must be included in accurate calculations of the equations of state of hydrogen and helium. Uncertainties on the equation of state still exist, however. Ab initio calculations of the behaviour of the hydrogen-helium mixture in the megabar regime for various compositions should be performed in order to gain accuracy.

Debonding detection in FRP-strengthened concrete structures utilising nonlinear Rayleigh wave mixing

This paper reports an investigation on the use of Rayleigh wave mixing method for the debonding detection in concrete structures strengthened with externally bonded fibre-reinforced polymer (FRP) composite plates. A three-dimensional (3D) finite element model is developed to simulate the propagation of Rayleigh waves in FRP-strengthened concrete structures. In this model, the frequency domain of the transmitted waves shows the generation of second harmonics and combinational harmonics on account of debonding. The numerically simulated results are then experimentally validated with carbon fibre-reinforced polymer (CFRP)-plated concrete blocks containing two different sizes of debonding. Based on the experimentally verified numerical model, parametric studies are then conducted, where a nonlinear debonding crack growth parameter is proposed. The nonlinear Rayleigh wave mixing method is proven to be practical, reliable, and sensitive for detecting debonding at bonded concrete-FRP interfaces in FRP-strengthened reinforced concrete (RC) structures.

Parametric amplification based on intermodal four-wave mixing between different supermodes in coupled-core fibers.

Parametric amplifiers relying on the nonlinear four-wave mixing process are known for their signature symmetric gain spectrum, where signal and idler sidebands are generated on both sides of a powerful pump wave frequency. In this article we show analytically and numerically that parametric amplification in two identically coupled nonlinear waveguides can be designed in such a way that signals and idlers are naturally separated into two different supermodes, hence providing idler-free amplification for the supermode carrying signals. This phenomenon is based on the coupled-core fibers analogue of intermodal four wave-mixing occurring in a multimode fiber. The control parameter is the pump power asymmetry between the two waveguides, which leverages the frequency dependency of the coupling strength. Our findings pave the way for a novel class of parametric amplifiers and wavelength converters, based on coupled waveguides and dual-core fibers.

Applications of nonlinear four-wave mixing in optical communication

In this paper I will briefly review some applications of nonlinear four-wave-mixing in optical fibers and silicon-nitride waveguides. This includes ultrafast all-optical waveform sampling and ‘noiseless’ phase-sensitive optical amplification which have been demonstrated to enable signal recovery at very low optical powers.