Quadratic Nonlinear Interactions Research Articles

Abnormally large radiational S2 tides in the Gulf of Tonkin and implications for other diurnal resonant seas

Radiational tides derived from solar radiation are important components of ocean tides. Understanding dynamical features of radiational tides can be beneficial for further promoting the accuracy of water level prediction. Generally, radiational S2 amplitudes are only 10%–30% of gravitational S2 amplitudes, which has been verified by numerous studies. However, a previous study indicated that radidational S2 amplitudes can reach 60% of gravitational S2 amplitudes in the Gulf of Tonkin and related physical processes are unknown. In this paper, we revisit this abnormal phenomenon and find that such large radiational S2 amplitudes are not real, but artifacts induced by the inadaptation of the method based on the credo of smoothness. Specifically, observed semi-diurnal tidal admittances in the Gulf of Tonkin are not smooth due to strong nonlinear interactions between large main diurnal tides. We quantitatively estimate and remove the contributions of quadratic and quartic nonlinear interactions to main semi-diurnal tides and find that the real ratio of radiational S2 amplitude to astronomical S2 amplitude is only 0.17 at Beihai, which is basically consistent with previous model results. Our study highlights the importance of eliminating the interference of nonlinear interactions to semi-diurnal tidal admittances in all diurnal resonant sea areas like the Gulf of Tonkin and the Gulf of Thailand when extracting radiational S2 tides from sea level observations.

Soliton molecules in Fermi–Pasta–Ulam–Tsingou lattice: Gardner equation approach

We revisit the Fermi–Pasta–Ulam–Tsingou lattice (FPUT) with quadratic and cubic nonlinear interactions in the continuous limit by deducing the Gardner equation. Through the Hirota bilinear method, the fundamental as well as multi-soliton, particularly two-, three- and four-soliton, solutions are obtained for the Gardner equation. Based on these multi-soliton solutions, we show the excitation of an interesting class of table-top soliton molecules in the FPUT lattice through the velocity resonance mechanism. Depending on the condition on the free parameters, which determine the bond length between the soliton atoms, we classify them as dissociated and synthetic type molecules. The main feature of the table-top soliton molecules is that they do not exhibit oscillations in the coalescence region. This property ensures that they are distinct from the soliton molecules, having retrieval force, of the nonlinear Schrödinger family of systems. Further, to study the stability of the soliton molecule we allow it to interact with a single (or multi) soliton(s). The asymptotic analysis shows that their structures remain constant, though the bond length varies, throughout the collision process. Then, from the relative separation distance calculation we also observe that the bond length of each of the soliton molecules is not maintained during soliton molecule–molecule interaction while their structures are preserved. In addition, we consider the FPUT lattice with quadratic nonlinear interaction and FPUT lattice with cubic nonlinearity as sub-cases and point out the nature of the soliton molecules for these cases also systematically. We achieve this based on the interconnections between the solutions of the Gardner, modified Korteweg–de Vries and Korteweg–de Vries equations. Finally, in order to validate the existence of all the soliton structures, especially reported for the Gardner equation in the continuous limit, we simulate the corresponding FPUT chain and verified these various structures numerically and analytically as well. We believe that the present study can be extended to other integrable and non-integrable systems with applications in fluid dynamics, Bose–Einstein condensates, nonlinear optics, and plasma physics.

Nonlinear propagation of quasiplanar shear wave beams in soft elastic media with transverse isotropy.

Model equations are developed for shear wave propagation in a soft elastic material that include effects of nonlinearity, diffraction, and transverse isotropy. A theory for plane wave propagation by Cormack [J. Acoust. Soc. Am. 150, 2566 (2021)] is extended to include leading order effects of wavefront curvature by assuming that the motion is quasiplanar, which is consistent with other paraxial model equations in nonlinear acoustics. The material is modeled using a general expansion of the strain energy density to fourth order in strain that comprises thirteen terms defining the elastic moduli. Equations of motion for the transverse displacement components are obtained using Hamilton's principle. The coupled equations of motion describe diffraction, anisotropy of the wave speeds, quadratic and cubic plane wave nonlinearity, and quadratic nonlinearity associated with wavefront curvature. Two illustrative special cases are investigated. Spatially varying shear vertical wave motion in the fiber direction excites a quadratic nonlinear interaction unique to transversely isotropic soft solids that results in axial second harmonic motion with longitudinal polarization. Shear horizontal wave motion in the fiber plane reveals effects of anisotropy on third harmonic generation, such as beam steering and dependence of harmonic generation efficiency on the propagation and fiber directions.

Experimental observation of resonance manifold shrinking under zonal flow shear.

In two-dimensional turbulent systems the redistribution of energy can be described by quadratic nonlinear three-wave interactions, which are limited by resonance conditions. The set of coupling modes can be understood as resonant manifold. It has been predicted by theory that, in the presence of a shear flow, the resonant manifold in wave-number space shrinks in time favoring large-scale structures. The phenomenon of manifold shrinking in the presence of shear flows is studied the first time experimentally in drift wave turbulence at the stellarator TJ-K by bicoherence analysis. By estimating effective mode numbers characterizing the width of the manifold, it is demonstrated that increasing shear leads to a shrinking of the resonance manifold.

Resonance and beating phenomenon in a nonlinear rigid cylindrical acoustic waveguide: The axisymmetric mode

A rigid-walled semi-infinite circular cylindrical waveguide enclosing a weakly nonlinear fluid is considered. A quadratic nonlinear interaction is assumed as a model. A flat rigid piston generates an axisymmetric harmonic pressure that is prescribed as the input at the finite end of the waveguide. The objective is to study the modal interactions of waves. A regular perturbation method is used to separate the linear (primary) and the quasilinear (second harmonic) equations. First, linear solutions are established. Their quasilinear interactions then lead to modal interactions. Typically, the linear wavenumbers at the quasilinear order generate homogenous wavenumbers. Intersections of both these wavenumbers in the wavenumber-frequency plane are considered as resonances since the resulting pressure grows in amplitude in the axial propagating direction. The conditions under which the solutions become resonant or non-resonant are presented. In this last case, a beating phenomenon occurs with distance. It is found that the resonances are rare, except for the plane wave. Generally, the forced linear wavenumbers and the quasilinear generated wavenumbers acquire numerical values close to each other and create the beating phenomenon.

Higher-Order Interactions in Quantum Optomechanics: Analysis of Quadratic Terms

This article presents a full operator analytical method for studying the quadratic nonlinear interactions in quantum optomechanics. The method is based on the application of higher-order operators, using a six-dimensional basis of second order operators which constitute an exactly closed commutators. We consider both types of standard position-field and the recently predicted non-standard momentum-field quadratic interactions, which is significant when the ratio of mechanical frequency to optical frequency is not negligible. This unexplored regime of large mechanical frequency can be investigated in few platforms including the superconducting electromechanics and simulating quantum cavity electrodynamic circuits. It has been shown that the existence of non-standard quadratic interaction could be observable under appropriate conditions.

Higher-Order Spectral Analysis to Identify Quadratic Nonlinearities in Fluid-Structure Interaction

Hydrodynamic forces on a structure are the manifestation of fluid-structure interaction. Since this interaction is nonlinear, these forces consist of various frequencies: fundamental, harmonics, excitation, sum, and difference of these frequencies. To analyze this phenomenon, we perform numerical simulations of the flow past stationary and oscillating cylinders at low Reynolds numbers. We compute the pressure, integrate it over the surface, and obtain the lift and drag coefficients for the two configurations: stationary and transversely oscillating cylinders. Higher-order spectral analysis is performed to investigate the nonlinear interaction between the forces. We confirmed and investigated the quadratic coupling between the lift and drag coefficients and their phase relationship. We identify additional frequencies and their corresponding energy present in the flow field that appear as the manifestation of quadratic nonlinear interaction.

Effects of dual wavenumber dispersion solutions on a nonlinear monochromatic wave-current field

An analytical solution for quadratic nonlinear wave-current interactions is derived using a perturbation method. In contrast to existing formulations, it accounts for dual wavenumber solutions of the linear problem for a single frequency and their nonlinear interactions. It is demonstrated that even a small amount of energy present on the second solution will introduce significant quantitative and qualitative differences in the surface elevation. These differences include a shorter wavelength component moving with a lower wave celerity, nonlinear spatial focusing, and a mean water level that varies in space. The second solution is also shown to produce a mechanism which may easily result in localized violations of wave breaking criteria. These weakly nonlinear effects are expected to be relevant to a variety of engineering applications and physical processes.

Gravitational-wave tail effects to quartic non-linear order

Gravitational-wave tails are due to the backscattering of linear waves onto the space-time curvature generated by the total mass of the matter source. The dominant tails correspond to quadratic non-linear interactions and arise at the one-and-a-half post-Newtonian (1.5 PN) order in the gravitational waveform. The ‘tails-of-tails’, which are cubic non-linear effects appearing at the 3 PN order in the waveform, are also known. We derive here higher non-linear tail effects, namely those associated with quartic non-linear interactions or ‘tails-of-tails-of-tails’, which are shown to arise at the 4.5 PN order. As an application, we obtain at that order the complete coefficient in the total gravitational-wave energy flux of compact binary systems moving on circular orbits. Our result perfectly agrees with black-hole perturbation calculations in the limit of extreme mass ratio of the two compact objects.

Squeezing a Surface Plasmon through Quadratic Nonlinear Interactions

Quantum plasmonics presents a new insight into quantum photonic science and technology. The unique properties of surface plasmon polaritons (SPPs) provide immense potential for quantum control of light in ultracompact systems. The quantum behavior of SPPs should be described by corresponding SPP states, which have attracted a lot of investigative interest recently. In this work, we focus on the interaction of single quanta of SPPs. The generation of a squeezed SPP state through parametric down conversion (PDC) is investigated. Due to intrinsic loss of an SPP, this system deviates from those based on traditional bulk optics. Unlike the previous negative image, it is surprising to find that the role of loss is partly positive. As an illustration, the squeezing process of an SPP in a hybrid plasmonic waveguide system is calculated and shown. The degree of squeezing could reach above 7 dB with a propagation length of only about 12 μm in a single path. Moreover, the spectrum of squeezing has a band of 481.7 TH...

Quasi-parametric amplification of chirped pulses based on a Sm^3+-doped yttrium calcium oxyborate crystal

One inherent characteristic of quadratic nonlinear interaction is that it allows both forward and backward energy transfer among the three interacting waves. This backconversion effect, universal in all the parametric processes, is detrimental when a unidirectional energy transfer is desired and limits the conversion efficiency. We report a family of quadratic nonlinear interactions, quasi-parametric amplification (QPA), in which the idler wave is depleted by the introduction of a material loss and only the signal is amplified. In contrast to optical parametric amplification (OPA), the QPA scheme can inhibit the backconversion effect and thus enable ideal chirped-pulse amplification with high conversion efficiency and broad gain bandwidth. We have numerically proved the feasibility of this new scheme, and experimentally realized it by using a Sm3+-doped yttrium calcium oxyborate crystal that is highly absorptive at the idler wavelength and transparent at the pump and signal wavelengths. Amplification of broadband chirped pulses, corresponding to a pump depletion of 70% and a signal efficiency of 41%, has been achieved in a typical Gaussian pump case, exceeding the results of the previously reported state-of-the-art OPA. The proposed QPA scheme will be a promising approach for efficiently amplifying chirped pulses to unprecedented powers.

A simplified model for quadratic parametric processes considering pump depletion in a layered quasi-phase-matching structure

Quadratic parametric processes in a one-dimensional nonlinear structure were investigated by using a nonlinear transfer matrix and iterative algorithm. Our method can provide an accurate description in the regime of pump depletion. Detailed analysis shows iteration is important for achieving a precise result. In order to simplify the complex iterative derivation, we develop a simplified model for quasi-phase-matched (QPM) second harmonic generation. We extend the simplification strategy to treat other quadratic nonlinear interactions in a QPM structure. The validity of simplification is demonstrated by numerically comparing conversion efficiencies. Our method provides a simple and efficient way to investigate second-order parametric processes in QPM structures.

Managing orbital angular momentum in second-harmonic generation

We propose a convenient approach for managing orbital angular momentum (OAM) of second-harmonic photons generated by a quadratic nonlinear interaction under noncolinear type-I phase matching condition. Since the OAMs of two fundamental photons can be flexibly controlled by using computer-generated holograms, we were able to realize arbitrary combination of both OAMs and experimentally confirm the OAM conservation selection rule ${\ensuremath{\ell}}_{2\ensuremath{\omega}}={\ensuremath{\ell}}_{\ensuremath{\omega}}^{\ensuremath{'}}+{\ensuremath{\ell}}_{\ensuremath{\omega}}^{\ensuremath{'}\ensuremath{'}}$ in general cases. In particular, our approach could be used to generate the second-harmonic photon states with fractional and odd-order OAMs.

A cascaded second-order approach to computing third-order scattering of noncollinear acoustic beams

A theoretical development of the third-order nonlinear scattering of sound from two noncollinear ultrasonic beams produced by baffled piston sources is presented. The third-order intermodulation (IM3) frequency components are derived by exploiting cascaded second-order nonlinear effects where the quadratic nonlinear interaction of second-order frequency components with first-order (primary) frequency components is considered. It is shown that cascaded second-order interactions generate intermodulation frequency components that are equivalent to those generated by cubic nonlinear effects. Comparison of measured and modeled amplitude sweeps demonstrate the three-to-one gain in decibels of the amplitudes of the third-order intermodulation frequency components to that of the primary frequency components. Measurements are also presented for the farfield interaction of distantly spaced sources, which results in a highly focused ultrasonic parametric array. Also considered are the nearfield interaction of closely spaced sources, which results in scattering, with good agreement to the theory.

Theory and experiments on multistep parametric processes in nonlinear optics

We present a comprehensive model for the description of different types of parametric interactions, associated with simultaneous phase-matching of several optical processes: the so-called multistep parametric interactions. Our approach is based on a recently derived single-wave broadband equation that is able to describe general quadratic nonlinear optical interactions and can be solved with modest computational effort. We compare theoretical results with experiments on simultaneous second- and third-harmonic generation performed in periodically poled lithium tantalate crystals.

Dependences of the group velocity for femtosecond pulses in MgO-doped PPLN crystal

Theoretical investigation on the group velocity control of ultrafast pulses through quadratic cascading nonlinear interaction is presented. The dependences of the fractional time delay as well as the quality factor of the delayed femtosecond pulse on the peak intensity, group velocity mismatch, wave-vector mismatch and the pulse duration are examined. The results may help to understand to what extent some optical operation parameters could have played a role in controlling the ultrashort pulses. We also predict the maximum achievable pulse delay or advancement efficiency without large distortions. A compact solid medium integrating multiple functions including slowing light, wavelength conversion or broadcasting on a single chip, may bring significant practicality and high integration applications at optical communication band.

Management of the orbital angular momentum of vortex beams in a quadratic nonlinear interaction

Light intensity control of the orbital angular momentum of the fundamental beam in a quadratic nonlinear process is theoretically and numerically presented. In particular we analyzed a seeded second harmonic generation process in presence of orbital angular momentum of the interacting beams due both to on axis and off axis optical vortices. Examples are proposed and discussed.

Proposal for a Tunable and Switchable Multiwavelength Laser With Variable Spacing Based on Broadband Cascaded Quadratic Nonlinearity

A scheme to realize a tunable and switchable multi-wavelength laser with variable wavelength spacing is proposed based on broadband cascaded quadratic nonlinear interactions in an aperiodically poled MgO-doped lithium niobate (MgO:APPLN) waveguide. Four output wavelengths can be generated simultaneously from the laser and can be switched by tuning the operation temperature of MgO:APPLN. Additionally, the four output wavelengths of the laser can be selected by launching two seeded lights with different wavelengths and wavelength spacing. This device, capable of wavelength tunable function, will facilitate applications in the fiber sensing and wavelength routing for dense wavelength-division-multiplexed (WDM) networks.

Theoretical analysis of power swapping in quadratic nonlinear medium

Nonlinear optical effects in quadratic nonlinear mediums for the traditional frequency conversion generally involve two or three incident optical waves and transfer the power of the input signal wavelength to another new generated wavelength. The authors find that bidirectional power swapping can be carried out by proper waveguide length and power arrangement when two signals and two continuous-wave pumps simultaneously take part in the quadratic nonlinear interactions. Theoretical analyses referring to four incident optical waves are presented, taking into account the cascaded sum- and difference-frequency generation in a periodically poled lithium niobate waveguide. We apply such theory to ultrafast data information swapping and derive analytical solutions clearly indicating the swapping condition. An example of bidirectional ultrafast data information swapping is demonstrated.

All-Optical Ultrawideband Monocycle Generation Using Quadratic Nonlinear Interaction Seeded by Dark Pulses

We propose and demonstrate a new approach to all-optical generation of ultrawideband (UWB) monocycle pulses using the quadratic nonlinear interaction seeded by dark pulses in a periodically poled lithium niobate (PPLN) waveguide. When feeding two dark optical pulses with proper relative time delay to participate in the sum-frequency generation (SFG) in a PPLN, the parametric depletion effect accomplishes the tailoring of the dark pulse to be a UWB monocycle pulse at a single wavelength. Pairs of polarity-inverted UWB monocycle pulses are generated in the experiment with the central wavelength, 10-dB bandwidth, and fractional bandwidth conforming to the UWB definition by the U.S. Federal Communications Commission (FCC, part 15).