Subharmonic Generation Research Articles

Two-mode subharmonic generating system with coherent and thermal light

The statistical and squeezing characteristics of the light generated by a two-mode subharmonic generating system in a cavity pumped by two-mode coherent light and coupled to a two-mode thermal reservoir through a single-port mirror have been studied in this study. We construct the c-number Langevin equations linked to the normal ordering using the master equation for the system under discussion, and we establish the correlation characteristics of the noise forces. The mean and variance of the photon number of the cavity light are then determined using the solutions of the resulting differential equations. We have discovered that the cavity light mode’s photon statistics are super-Poissonian because the variance of the photon number is more significant than the mean of the photon number. Additionally, using the same solutions, we were able to extract the antinormally ordered characteristic function, which is used to calculate the Q function. In addition, the photon number distribution for the cavity mode is computed using the Q function. The variance of the plus and minus quadrature of the two-mode cavity light generated by the two-mode subharmonic generating system is also obtained. It is discovered that the cavity mode’s squeezing characteristics are unaffected by the two-mode driving coherent light, however, the thermal light has the effect of decreasing the squeezing of the cavity light and the squeezing takes place in the plus quadrature. Finally, we discovered that when the cavity is coupled to a vacuum reservoir, there is a 50%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} maximum squeezing of the two-mode cavity at a steady state and at the threshold. The nondegenerate subharmonic generator generates a two-mode cavity light that is 50%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} entangled.

Coupled modes analysis of stimulated Brillouin scattering in the regimes of nonlinear ion acoustic waves

The nonlinear saturation of stimulated Brillouin scattering (SBS) in long scale length plasmas is studied in detail through coupled mode equations. Our model incorporates harmonic and subharmonic generation of ion acoustic waves (IAWs), as well as nonlinear Landau damping and the nonlinear frequency shift of IAWs induced by particle trapping. Numerical simulations are carried out across various IAW wavenumbers (k a λ De ) and electron-ion temperature ratios (Z i T e /T i ) within different SBS instability regimes. The results demonstrate that our model can distinguish the importance of each effect contributing to the nonlinear behavior in SBS under different plasma conditions. Furthermore, we examine the scaling of SBS reflectivity with laser intensity under conditions relevant to inertial confinement fusion.

Ambient Pressure Sensitivity of the Subharmonic Response of Coated Microbubbles: Effects of Acoustic Excitation Parameters.

The sensitivity of the acoustic response of microbubbles, specifically a strong correlation between their subharmonic response and the ambient pressure, has motivated development of a non-invasive subharmonic-aided pressure estimation (SHAPE) method. However, this correlation has previously been found to vary depending on the microbubble type, the acoustic excitation and the hydrostatic pressure range. In this study, the ambient pressure sensitivity of microbubble response was investigated. The fundamental, subharmonic, second harmonic and ultraharmonic responses from an in-house lipid-coated microbubble were measured for excitations with peak negative pressures (PNPs) of 50-700 kPa and frequencies of 2, 3 and 4 MHz in the ambient overpressure range 0-25 kPa (0-187 mmHg) in an in vitro setup. The subharmonic response typically has three stages-occurrence, growth and saturation-with increasing excitation PNP. We find distinct decreasing and increasing variations of the subharmonic signal with overpressure that are closely related to the threshold of subharmonic generation in a lipid-shelled microbubble. Above the excitation threshold, that is, in the growth-saturation phase, subharmonic signals decreased linearly with slopes as high as -0.56 dB/kPa with ambient pressure increase; below the threshold excitation (at atmospheric pressure), increasing overpressure triggers subharmonic generation, indicating a lowering of subharmonic threshold, and therefore leads to an increase in subharmonic with overpressure, the maximum enhancement being ∼11 dB for 15 kPa overpressure at 2 MHz and 100 kPa PNP. This study indicates the possible development of novel and improved SHAPE methodologies.

Time evolution of ambient pressure sensitivity of microbubbles subharmonic response as a function of hydrostatic pressure and the filling gas

The ultrasound contrast microbubbles have shown a great ability to produce nonlinear oscillations in response to acoustic excitation which results in the generation of harmonics and subharmonics of the excitation. The subharmonic response of ½ order changes significantly with a change in hydrostatic pressure in the physiological range. This property is being investigated for subharmonic aided pressure estimation (SHAPE), a noninvasive alternative to current methods for vascular pressure measurement in critical organs. In this study, the ambient pressure sensitivity of the subharmonic response of microbubbles with different gas cores and shells was investigated over 25–700 kPa acoustic pressure and the hydrostatic pressure range of 0–25 kPa. The subharmonic showed different trends of increase and decrease with hydrostatic pressure. The sensitivity has shown gradual change over time which was observed to be related to the hydrostatic pressure magnitude and the gas core. These findings could help to calibrate the SHAPE.

Generation of subharmonics in acoustic resonators containing bubbly liquids: A numerical study of the excitation threshold and hysteretic behavior

In this paper we study the generation and behavior of subharmonics in a bubbly liquid confined in an acoustic resonator, through numerical simulations carried out at finite-amplitude acoustic pressure. Several configurations in terms of resonator length and driving frequency are considered here. Our results show that these frequency components, created from a higher-frequency signal at the source (ultrasound), are due to the nonlinearity of the medium at high acoustic-pressure amplitude and to the configuration of the resonator (geometry and boundaries). We also show that they have an amplitude-threshold dependence, which is in concordance with the literature. The response of these subharmonics to different sequences of pressure amplitudes also reveals the hysteretic nature of the bubbly liquid.

Ultra-broadband mid-infrared frequency combs produced by optical subharmonic generation

Optical frequency combs have revolutionised accurate frequency and time measurements and have enabled broadband and simultaneously high resolution spectroscopic measurements that were not previously possible. This paper is an overview of the main results of the previously performed work, describing a new approach to extending frequency combs to the mid-infrared ‘molecular signature’ range using a subharmonic generator based on an optical parametric oscillator operating in degenerate mode. Such an instrument acts as an efficient frequency divider that rigorously down-converts and augments the spectrum of a pump laser frequency comb while maintaining its coherence. Our recent result is the demonstration of a subharmonic system with a two-octave spectrum, 3 – 12 μm, which covers vibrational resonances for most molecular species. Potentially, through frequency division in the coherent subharmonic optical parametric amplifier regime, this method can be used to obtain intense long-wavelength pulses for high-field physics applications.

Sensitivity of the subharmonic responses from contrast microbubbles to ambient pressure

Microbubbles are well-known ultrasound contrast agents which can produce nonlinear oscillation leading to harmonic and subharmonic responses. There has been an interest in utilizing this phenomenon for noninvasive pressure estimation and its use in diagnosis of diseases such as portal hypertension. The subharmonic response of a microbubble display three different regimes with increasing excitation: occurrence, growth, and saturation. In this study, subharmonic responses of lipid-shelled microbubbles were investigated with varying ambient pressures at different excitation parameters. We observed that subharmonic sensitivity to ambient pressure varied differently in different excitation ranges. We also noted that applying overpressure could induce subharmonic generation where there was no subharmonic without overpressure. At occurrence stage, the sensitivity of subharmonic to ambient pressure was significantly higher. This property can be utilized for improved pressure estimation with high sensitivity even at low acoustic excitation.

Improved Sensitivity of Ultrasound-Based Subharmonic Aided Pressure Estimation Using Monodisperse Microbubbles.

Subharmonic aided pressure estimation (SHAPE) has been shown effective for noninvasively measuring hydrostatic fluid pressures in a variety of clinical applications. The objective of this study was to explore potential improvements in SHAPE sensitivity using monodisperse microbubbles. Populations of monodisperse microbubbles were created using a commercially available microfluidics device (Solstice Pharmaceuticals). Size distributions were assessed using a Coulter Counter and stability of the distribution following fabrication was evaluated over 24 hours. Attenuation of the microbubble populations from 1 to 10 MHz was then quantified using single element transducers to identify each formulation's resonance frequency. Frequency spectra over increasing driving amplitudes were investigated to determine the nonlinear phases of subharmonic signal generation. SHAPE sensitivity was evaluated in a hydrostatic pressure-controlled water bath using a Logiq E10 scanner (GE Healthcare). Monodisperse lipid microbubble suspensions ranging from 2.4 to 5.3 μm in diameter were successfully created and they showed no discernable change in size distribution over 24 hours following activation. Calculated resonance frequencies ranged from 2.1 to 6.3 MHz and showed excellent correlation with microbubble diameter (R2 > 0.99). When investigating microbubble frequency response, subharmonic signal occurrence was shown to begin at 150 kPa peak negative pressure, grow up to 225 kPa, and saturate at approximately 250 kPa. Using the Logiq E10, monodisperse bubbles demonstrated a SHAPE sensitivity of -0.17 dB/mmHg, which was nearly twice the sensitivity of the commercial polydisperse microbubble currently being used in clinical trials. Monodisperse microbubbles have the potential to greatly improve the sensitivity of SHAPE for the noninvasive measurement of hydrostatic pressures.

Advances in nonlinear ultrasonic detection of microcracks

<p indent="0mm">A large number of key components are widely operated under extreme conditions of high temperature and high pressure in power, nuclear energy, aerospace and other important industrial fields. Microdefects are inevitably generated in these key components during the process of manufacturing. Due to the periodic loading and thermal stress, microcracks are generally initiated from these microdefects at the location with stress concentration. It has been found that the initiation and propagation of fatigue microcracks usually occur prior to 70%–90% of fatigue life. It is difficult to observe the evidence before the formation of macrocracks, which may result in catastrophic accidents. Thus, the risk of sudden failure of key components increases greatly. Therefore, the accurate detection of microcracks is crucial for guaranteeing the safe operation of key components. Based on the diffraction, reflection and transmission, traditional ultrasonic testing method is merely validated to effectively detect macroscopic defects within half a wavelength. The nonlinear ultrasonic testing method is found to be sensitive to microcracks, as high order harmonics, sub-harmonics, DC components and frequency mixing waves can be generated due to the strong nonlinear interaction microcracks and ultrasonic waves. With the outstanding effectiveness to detect microcracks, the nonlinear ultrasonic testing method has attracted much attention in recent years. In this work, the research on detection of microcracks using nonlinear ultrasonic testing method is systematically reviewed in terms of theoretical models, numerical simulations and experimental measurements. Firstly, according to the breathing effect and sliding mechanism of microcrack surfaces, we present several theoretical models of nonlinear interaction between ultrasonic waves and microcracks, namely contact acoustic nonlinearity (CAN), bi-linear stiffness model, nonlinear spring model, Hertz contact theory, and rough contact surface model. The advantages and disadvantages of these theoretical models are discussed comparatively. Two mechanisms are mainly considered for the nonlinear interaction of microcracks and ultrasonic waves, namely the macroscopic opening and closing of crack surfaces and the microscopic contact of rough microcracks surfaces. Secondly, we introduce numerical simulation studies on the nonlinear ultrasonic generation induced by microcracks. The applicability is analyzed for several simulation methods, such as finite element method, spectral finite element method, finite difference method, boundary element method and local interaction simulation approach, and the generation and propagation of nonlinear ultrasonic waves are analyzed as well. The effect of microcrack geometries on the nonlinear ultrasonic waves is then summarized. The microcrack geometries are considered mainly clouding the microcrack length, width, depth, density and direction. Furthermore, we present the experimental measurements on the generation of nonlinear ultrasonic waves induced by microcracks and on the localization and imaging of microcracks based on the nonlinear ultrasonic waves. The generation of high order harmonics, sub-harmonics, DC components and frequency mixing waves was observed in the specimens with microcracks. The nonlinear ultrasonic detection is presented to evaluate fatigue microcracks with various microcrack length, width, depth, density and direction. Based on this phenomenon, nonlinear ultrasonic mixing method, transducer array imaging method and nonlinear ultrasonic phased array method are introduced to localize the microcracks. Finally, further studies on evaluation of microcracks using nonlinear ultrasonic waves are proposed, including the comprehensive physical model for actual microcracks in engineering components, robust detection method, and quantitative evaluation of microcracks and service life.

Noise generation in a bilayer ferromagnet-piezoelectric heterostructure at the converse magnetoelectric effect

The paper describes first observation of parametric generation of noise in a composite multiferroic resonator. Resonator of a disk shape contains two mechanically coupled layers, one of which is the amorphous ferromagnet (FM) FeBSiC and the other is the piezoelectric (PE) lead zirconate titanate. DC magnetic field of 0–100 Oe is applied parallel to the plane of the resonator. Resonator is excited in the frequency range f = 2–10 kHz by a harmonic electric field with amplitude of up to 330 V cm−1 applied to the PE layer. Changes in the magnetization of the resonator caused by the converse magnetoelectric effect were recorded using an electromagnetic coil. With an increase in the excitation field to the threshold value, the parametric generation of harmonics and subharmonics with a discrete spectrum is observed, which then turns into a stochastic mode and a continuous spectrum of frequencies is generated. Noise density is hysterically dependent on the excitation field and non-monotonically depends on the dc magnetic field. Theory of parametric generation of the noise in a multiferroic resonator is developed, taking into account excitation of acoustic resonances and magnetoacoustic nonlinearity of the FM layer of the resonator. Theory, qualitatively describes the main characteristics of noise generation.

Early-time behavior of quantum subharmonic generation

A few years ago Avetissian {\it et al.} \cite{Avetissian2014,Avetissian2015} discovered that the exponential growth rate of the stimulated annihilation photons from a singlet positronium Bose-Einstein condensate should be proportional to the square root of the positronium number density, not to the number density itself. In order to elucidate this surprising result obtained via a field-theoretical analysis, we point out that the basic physics involved is the same as that of resonant subharmonic transitions between two quantum oscillators. Using this model, we show that nonlinearities of the type discovered by Avetissian {\it et al.} are not unique to positronium and in fact will be encountered in a wide range of systems that can be modeled as nonlinearly coupled quantum oscillators.

Quantum analysis of sub harmonic generation with two-mode coherent light

In this work the statistical and squeezing properties of light-driven by sub-harmonic generation with two-mode coherent light are studied. With interaction Hamiltonian of both two-mode coherent and sub harmonic generation, we have driven master equation of system under consideration. From the master equation, the solution of the C-number Langevin equation is derived. It helps us to solve quadrature variance, quadrature squeezing, mean, and variance of photon number for light produced by sub-harmonic generation with the two-mode coherent light state. And the result shows that; the squeezing occurs in plus quadrature with the maximum squeezing of 87%. The photon statistics of the system under consideration is subpoissonian in which both mean &amp; variance are increasing as kappa increase.

Broadband high-resolution molecular spectroscopy with interleaved mid-infrared frequency combs

Traditionally, there has been a trade-off in spectroscopic measurements between high resolution, broadband coverage, and acquisition time. Originally envisioned for precision spectroscopy of the hydrogen atom in the ultraviolet, optical frequency combs are now commonly used for probing molecular ro-vibrational transitions throughout broad spectral bands in the mid-infrared providing superior resolution, speed, and the capability of referencing to the primary frequency standards. Here we demonstrate the acquisition of 2.5 million spectral data points over the continuous wavelength range of 3.17–5.13 µm (frequency span 1200 cm−1, sampling point spacing 13–21 MHz), via interleaving comb-tooth-resolved spectra acquired with a highly-coherent broadband dual-frequency-comb system based on optical subharmonic generation. With the original comb-line spacing of 115 MHz, overlaying eight spectra with gradually shifted comb lines we fully resolve the amplitude and phase spectra of molecules with narrow Doppler lines, such as carbon disulfide (CS2) and its three isotopologues.

Parametric Generation of Subharmonics in a Composite Multiferroic Resonator

Parametric generation of subharmonics in a composite multiferroic resonator is observed and investigated. The resonator has the form of a disk and contains two mechanically coupled layers, one of which is amorphous ferromagnet $\mathrm{Fe}\text{\ensuremath{-}}\mathrm{B}\text{\ensuremath{-}}\mathrm{Si}\text{\ensuremath{-}}\mathrm{C}$ and the other piezoelectric lead zirconate titanate. The resonator is placed inside two planar electromagnetic coils with orthogonal axes. A static magnetic field of 0--100 Oe is applied parallel to the plane of the resonator. The resonator is excited in the frequency range f = 9--10 kHz by either a harmonic magnetic field with an amplitude of up to 5 Oe generated by one of the coils, or a harmonic electric field with an amplitude of up to 500 V/cm applied to the piezoelectric layer. When the pump field is above a certain threshold, generation of a subharmonic of half-frequency (f/2) is observed for three different excitation methods. The first two employed either the direct magnetoelectric effect or the converse magnetoelectric effect, while in the third a transformer system is utilized. The subharmonic is generated in a limited range of pump frequencies and its amplitude is a nonlinear function of both the pump-field amplitude and the strength of static magnetic field. A theory of parametric generation of the subharmonic in a multiferroic resonator is developed, taking into account the magnetoacoustic nonlinearity of the ferromagnetic layer of the structure and excitation of acoustic resonances near the pump and subharmonic frequencies. The theory qualitatively describes the main characteristics of the subharmonic generation.

Nonlinear wave-wave interactions in the brain

Neural field theory of the corticothalamic system is used to analyze nonlinear wave-wave interactions in steady state visual evoked potential responses. The nonlinear power spectrum is analytically calculated by convolving the linear power spectrum with itself and other factors. Periodic sine and square wave stimuli are used to generate steady state visual evoked potential responses and to study stimulus-driven nonlinear corticothalamic dynamic interactions. Moreover, we use dual sine drives to analyze the driven dynamics. Numerical analysis shows that the nonlinear power spectrum embodies key nonlinear features, including harmonic and subharmonic generation, entrainment of the alpha rhythm to periodic stimuli at the drive frequency, sum and difference frequencies due to wave-wave coalescence and decay. Further, the scaling properties of the key phenomena observed in nonlinear interactions are studied, verifying some of the theoretical predictions for these being generated by three-wave processes.

Optimisation of the transmit beam parameters for generation of subharmonic signals in native and altered populations of a commercial microbubble contrast agent SonoVue®

The aim of this work was to establish the optimum acoustic characterisation approach and insonation transmit beam parameters for subharmonic signal generation with ‘native’ and ‘altered’ populations of a commonly-used microbubble contrast agent. Dynamic contrast-enhanced (DCE) ultrasound is a non-invasive method of imaging the microvasculature, typically implemented using harmonic imaging. Subharmonic imaging, in which echoes at half the fundamental frequency are detected, detects signals which are generated by the ultrasound contrast agents (UCAs) but not by tissue. However, optimal transmission parameters and furthermore, the optimum acoustic characterisation method have not been established. The subharmonic response of ‘native’ and ‘altered’ UCA, altered through decantation, was investigated at transmit centre frequencies 1.8–5 MHz and pulse lengths 1–8 cycles. The ‘altered’ UCA had reduced polydispersity (1–4 µm: 82% bubble volume), compared to ‘native’ (4–10 µm: 57% bubble volume). A custom-built narrow-band acoustic characterisation system was found to be more appropriate for acoustic characterisation compared to the commonly used broadband pulse-echo approach. Both UCA generated the highest subharmonic signal at pulse length of 3-cycles. The maximum ‘native’ subharmonic signal was generated at a transmit centre frequency of 1.9 MHz, corresponding to a subharmonic at 0.95 MHz. This optimal frequency increased in the ‘altered’ population to 2.3–2.5 MHz, bringing the subharmonic above 1 MHz and hence into a range amenable to clinical abdominal imaging transducers. The use of subharmonic signal detection coupled with a modified UCA size distribution has potential to significantly improve the quantification sensitivity and accuracy of DCE ultrasound imaging.

1Pb2-4 Study on the characteristics of aluminum-alloy fatigue cracks and the behavior of subharmonic generation

1Pb2-4 Study on the characteristics of aluminum-alloy fatigue cracks and the behavior of subharmonic generation

Mode-locking dynamics of corticothalamic system responses to periodic external stimuli

A physiologically based continuum model of the corticothalamic system is applied to explore mode-locking dynamics of ongoing cortical activity subject to periodic drive. The ongoing cortical activity is poised near a Hopf bifurcation. Time-dependent center manifold reduction shows that the nonlinear dynamics of driven corticothalamic system with delays can be described by an inhomogeneous low-dimensional equation near the critical point of the Hopf bifurcation. This driven normal form captures the main features of experimentally observed nonlinear interactions between brain activity and periodic stimuli. The effects of underlying physiological parameters on the type of Hopf bifurcation and onset dynamics are then studied, which allows the determination of phase-locked structures for various regions in the parameter space of the drive. It is shown that the system can adjust its oscillation regime to the frequency content of stimuli. The key features of entrainment of the alpha rhythm to periodic stimuli, including harmonic and subharmonic generation, are also explored. The theoretical predictions of these nonlinear responses are confirmed by numerical simulations. Further, photosensitive seizures are predicted for physiologically based model parameters arising from mode locking to the frequency of periodic stimulus, and their harmonics due to higher order synchronization states. These results indicate that the approach used here provides a powerful framework for the study of nonlinear interactions between brain activity and flicker by incorporating periodic driving into the corticothalamic system.

Resonant excitation and all-optical switching of femtosecond soliton molecules

The emergence of confined structures and pattern formation are exceptional manifestations of concurring nonlinear interactions found in a variety of physical, chemical and biological systems[1]. Optical solitons are a hallmark of extreme spatial or temporal confinement enabled by a variety of nonlinearities. Such particle-like structures can assemble in complex stable arrangements, forming "soliton molecules"[2,3]. Recent works revealed oscillatory internal motions of these bound states, akin to molecular vibrations[4-8]. These observations beg the question as to how far the "molecular" analogy reaches, whether further concepts from molecular spectroscopy apply in this scenario, and if such intra-molecular dynamics can be externally driven or manipulated. Here, we probe and control such ultrashort bound-states in an optical oscillator, utilizing real-time spectroscopy and time-dependent external perturbations. We introduce two-dimensional spectroscopy of the linear and nonlinear bound-state response and resolve anharmonicities in the soliton interaction leading to overtone and sub-harmonic generation. Employing a non-perturbative interaction, we demonstrate all-optical switching between distinct states with different binding separation, opening up novel schemes of ultrafast spectroscopy, optical logic operations and all-optical memory.

Schrödinger cat states and steady states in subharmonic generation with Kerr nonlinearities

We discuss general properties of the equilibrium state of parametric down-conversion in superconducting quantum circuits with detunings and Kerr anharmonicities, in the strongly nonlinear regime. By comparing moments of the steady state and those of a Schr\"odinger cat, we show that true Schr\"odinger cats cannot survive in the steady state if there is any single-photon loss. A delta-function 'cat-like' steady-state distribution can be formed, but this only exists in the limit of an extremely large nonlinearity. The steady state is a mixed state, which is more complex than a mixture or linear combination of delta-functions, and whose purity is reduced by driving. We expect this general behaviour to occur in other driven, dissipative quantum subharmonic non-equilibrium open systems.