Showing papers on "Parametric oscillator published in 2010"
TL;DR: In this article, a fully integrated, CMOS-compatible, multiple-wavelength source with high differential slope efficiency at only a few tens of milliwatts of continuous-wave power is presented.
Abstract: Integrated multiple-wavelength laser sources, critical for important applications such as high-precision broadband sensing and spectroscopy1, molecular fingerprinting2, optical clocks3 and attosecond physics4, have recently been demonstrated in silica and single-crystal microtoroid resonators using parametric gain2,5,6. However, for applications in telecommunications7 and optical interconnects8, analogous devices compatible with a fully integrated platform9 do not yet exist. Here, we report a fully integrated, CMOS-compatible, multiple-wavelength source. We achieve optical ‘hyper-parametric’ oscillation in a high-index silica-glass microring resonator10 with a differential slope efficiency above threshold of 7.4% for a single oscillating mode, a continuous-wave threshold power as low as 54 mW, and a controllable range of frequency spacing from 200 GHz to more than 6 THz. The low loss, design flexibility and CMOS compatibility of this device will enable the creation of multiple-wavelength sources for telecommunications, computing, sensing, metrology and other areas. Through optical ‘hyper-parametric’ oscillation in a high-index silica glass microring resonator, scientists demonstrate a fully integrated CMOS-compatible low-loss multiple-wavelength source that has high differential slope efficiency at only a few tens of milliwatts of continuous-wave power. The achievement has significant implications for telecommunications and on-chip optical interconnects in computers.
695 citations
TL;DR: A ZnGeP(2)-based optical parametric oscillator (OPO) with 22 W of output power in the 3-5 µm range and a beam quality factor M(2) ≈1.4 is reported on.
Abstract: We report on a ZnGeP(2)-based optical parametric oscillator (OPO) with 22 W of output power in the 3-5 µm range and a beam quality factor M(2) ≈1.4. The OPO uses a novel V-shaped 3-mirror ring resonator that allows two passes of the beams through the same nonlinear crystal. The pump is a 39 W hybrid Tm:fiber laser/Ho:YAG laser.
113 citations
TL;DR: In this article, two methods to change a quantum harmonic oscillator frequency without transitions in finite time are described and compared, one based on a transitionless tracking algorithm and the other based on engineering an invariant of motion.
Abstract: Two methods to change a quantum harmonic oscillator frequency without transitions in a finite time are described and compared. The first method, a transitionless-tracking algorithm, makes use of a generalized harmonic oscillator and a non-local potential. The second method, based on engineering an invariant of motion, only modifies the harmonic frequency in time, keeping the potential local at all times.
105 citations
TL;DR: Experiments in which the nonaxisymmetric sectorial oscillations of water drops have been excited using acoustic levitation and an active modulation method and these oscillations are excited by parametric resonance find the oblate initial shape of the water drops essential to this kind of excitations.
Abstract: We report experiments in which the nonaxisymmetric sectorial oscillations of water drops have been excited using acoustic levitation and an active modulation method. The observed stable sectorial oscillations are up to the seventh mode. These oscillations are excited by parametric resonance. The oblate initial shape of the water drops is essential to this kind of excitations. The oscillation frequency increases with mode number but decreases with equatorial radius for each mode number. The data can be well described by a modified Rayleigh equation, without the use of additional parameters.
104 citations
TL;DR: In this article, a simple prototype interaction between the inflaton and any additional fields can lead to isolated bursts of particle production during inflation (for example from parametric resonance or a phase transition).
Abstract: Interactions between the inflaton and any additional fields can lead to isolated bursts of particle production during inflation (for example from parametric resonance or a phase transition). Inflationary particle production leaves localized features in the spectrum and bispectrum of the observable cosmological fluctuations, via the Infra-Red (IR) cascading mechanism. We focus on a simple prototype interaction ${g}^{2}(\ensuremath{\phi}\ensuremath{-}{\ensuremath{\phi}}_{0}{)}^{2}{\ensuremath{\chi}}^{2}$ between the inflaton, $\ensuremath{\phi}$, and isoinflaton, $\ensuremath{\chi}$; extending previous work on this model in two directions. First, we quantify the magnitude of the produced non-Gaussianity by extracting the moments of the probability distribution function from lattice field theory simulations. We argue that the bispectrum feature from particle production might be observable for reasonable values of the coupling, ${g}^{2}$. Second, we develop a detailed analytical theory of particle production and IR cascading during inflation, which is in excellent agreement with numerical simulations. Our formalism improves significantly on previous approaches by consistently incorporating both the expansion of the universe and also metric perturbations. We use this new formalism to estimate the shape of the bispectrum from particle production, showing this to be distinguishable from other mechanisms that predict large non-Gaussianity.
102 citations
TL;DR: In this article, the authors investigated the dynamic stability of an axially accelerating viscoelastic beam undergoing parametric resonance by considering the effects of shear deformation and rotary inertia.
76 citations
TL;DR: In this article, the effects of system nonlinearities on degenerate parametric amplifiers are investigated, and it is shown that parametric amplification can be realized in nonlinear, dynamic-range limited systems, such as resonant micro- or nanosystems, but at the expense of performance degradation.
Abstract: This work investigates the effects of system nonlinearities on degenerate parametric amplifiers. A simple, Duffing-type nonlinearity is appended to a representative equation of motion for a mechanical or electromechanical parametric amplifier, and classical perturbation methods are used to characterize the resulting effects on the amplifier’s frequency response and performance. Ultimately, the work demonstrates that parametric amplification can be realized in nonlinear, dynamic-range limited systems, such as resonant micro- or nanosystems, but at the expense of performance degradation. Additionally, it is shown that nonlinear amplifiers can be operated above their linear instability threshold but that doing so results in bistable amplified responses.
73 citations
TL;DR: In this article, the authors established an explicit analogy between the dynamical Casimir effect and the photon emission of a thin non-linear crystal pumped inside a cavity and proposed a system based on a type-I optical parametric oscillator (OPO) to simulate a cavity oscillating in vacuum at optical frequencies.
Abstract: We establish an explicit analogy between the dynamical Casimir effect and the photon emission of a thin non-linear crystal pumped inside a cavity. This allows us to propose a system based on a type-I optical parametric oscillator (OPO) to simulate a cavity oscillating in vacuum at optical frequencies. The resulting photon flux is expected to be more easily detectable than with a mechanical excitation of the mirrors. We conclude by comparing different theoretical predictions and suggest that our experimental proposal could help discriminate between them.
59 citations
TL;DR: The method is generally applicable to measurements based on resonant detection, increases the information content of the measurement without requiring a large detection bandwidth, and optimally uses the enhanced sensitivity near resonance to extract information and minimize error due to detector noise.
Abstract: We describe a method of analysis which allows for reconstructing the nonlinear disturbance of a high Q harmonic oscillator. When the oscillator is driven with two or more frequencies, the nonlinearity causes intermodulation of the drives, resulting in a complicated spectral response. Analysis of this spectrum allows one to approximate the nonlinearity. The method, which is generally applicable to measurements based on resonant detection, increases the information content of the measurement without requiring a large detection bandwidth, and optimally uses the enhanced sensitivity near resonance to extract information and minimize error due to detector noise.
54 citations
TL;DR: In this article, the analytical solution of the problem of two two-level atoms with degenerate two-photon transitions interacting with a single-mode radiation field in the presence of a parametric amplifier term is presented.
Abstract: The analytical solution of the problem of two two-level atoms with degenerate two-photon transitions interacting with a single-mode radiation field in the presence of a parametric amplifier term is presented. The purity of the atomic state has been used to measure the degree of entanglement between the atom and the field. The temporal evolution of variance and entropy squeezing as well as atomic inversion for the single-atom case are studied. It has been shown that maximum squeezing for the variance and entropy squeezing occurs when the ratio between the amplifier coupling λ3 and the field frequency ω equals 0.26. Increasing the value of the ratio λ3/ω further leads to the vanishing of squeezing from the system. It is also noted that the existence of the coupling parameter results in the system never reaching the pure state except at the points of revival times. The Q function has been also considered to give more information in the phase space about the system. These aspects are sensitive to changes in the amplifier parameter.
46 citations
TL;DR: In this article, an approach for parametric excitation of large-amplitude stable oscillations of a microstructure operated by a parallel-plate electrode is presented. But the experimental results indicate that the suggested approach can be efficiently used for excitation in various types of microdevices where stable resonant operation combined with robustness and large vibrational amplitudes are desirable.
Abstract: In this work we report on an approach allowing efficient parametric excitation of large-amplitude stable oscillations of a microstructure operated by a parallel-plate electrode, and present results of a theoretical and experimental investigation of the device. The frame-type structure, fabricated from a silicon on insulator (SOI) substrate using deep reactive ion etching (DRIE), consists a pair of cantilever-type suspensions connected at their ends by a link. The time-varying electrostatic force applied to the link by a parallel-plate electrode is transformed into a periodic tension of the beams, resulting in the modulation of their flexural stiffness and consequently the mechanical parametric excitation of the structure. The lateral compliance of the beams allows for large-amplitude in-plane oscillations in the direction parallel to the electrode while high axial stiffness prevents undesirable instabilities. The lumped model of the device, considered as an assembly of geometrically nonlinear massless flexures and a rigid massive link and built using the Rayleigh?Ritz method, predicted the feasibility of the excitation approach. The fabricated devices were operated in ambient air conditions by a combination of a steady (dc) and time-dependent (ac) components of voltage and the large-amplitude responses, up to 75 ?m, in the vicinity of the principal parametric and primary resonances were registered by means of video acquisition and image processing. The shapes of the experimental resonant curves were consistent with those predicted by the model. The location and size of the instability regions on the frequency?voltage plane (parametric tongues) were quantitatively in good agrement with the model results. Theoretical and experimental results indicate that the suggested approach can be efficiently used for excitation of various types of microdevices where stable resonant operation combined with robustness and large vibrational amplitudes are desirable.
TL;DR: In this article, the problem of two two-level atoms in interaction with a non-degenerate parametric amplifier is considered and the analytical solution of the wave function is obtained and used to derive the density matrix operator.
Abstract: The problem of two two-level atoms in interaction with a nondegenerate parametric amplifier is considered. The analytical solution of the wavefunction is obtained and used to derive the density matrix operator. The temporal evolution of the atomic inversion, the degree of entanglement, as well as the variance and entropy squeezing are discussed. In our analysis, we assumed that the atomic systems are in the excited states and the fields in the squeezed-pair coherent state. It has been shown that the coupling parameter, λ3 (the coupling between the two fields), gets more effective for the case in which the q-parameter is not equal to zero. Also for a strong coupling parameter λ3 the superstructure phenomenon can be observed. In the meantime, as we increase the value of the coupling parameter, the entanglement between the atoms and the fields gets stronger. Also it has been shown that in the presence of the parametric amplifier term, the system never reaches the pure state except during revival periods.
TL;DR: In this article, it is shown that if the force excites the system inside the principal parametric resonance, then for a small excitation amplitude a resonance curve includes an internal loop.
Abstract: Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.
TL;DR: A widely tunable synchronously pumped coherent source based on the process of narrowband parametric amplification in a dispersion-shifted fiber that yields oscillations at 1970 to 2140 nm-the longest reported wavelength for a fiber parametric oscillator.
Abstract: We describe a widely tunable synchronously pumped coherent source based on the process of narrowband parametric amplification in a dispersion-shifted fiber. Using an experimental fiber with a zero-dispersion wavelength of 1590nm and pump wavelengths of 1530 to 1570nm yields oscillations at 1970 to 2140nm—the longest reported wavelength for a fiber parametric oscillator. The long-wavelength oscillations are accompanied by simultaneous short-wavelength oscillations at 1200 to 1290nm. The parametric gain is coupled to stimulated Raman scattering. For parametric oscillations close to the Raman gain peak, the two gain processes must be discriminated from each other. We devised two configurations that achieve this discrimination: one is based on the exploitation of the difference in group delay between the wavelengths where Raman and parametric gain peak, and the other uses intracavity polarization tuning.
TL;DR: A continuous-wave singly resonant optical parametric oscillator pumped by a widely tunable titanium-doped sapphire ring laser with Spectroscopic capability is demonstrated by measuring parts of the photoacoustic absorption spectrum of NH(3) near 3196 cm(-1).
Abstract: We report a continuous-wave singly resonant optical parametric oscillator pumped by a widely tunable titanium-doped sapphire ring laser. It produces up to 0.8 W of mid-infrared power. The wavelength can be tuned in a few seconds from 2.5 to 3.5 µm or from 3.4 to 4.4 µm and scanned up to 40 GHz without mode-hops by only changing the pump beam wavelength. Spectroscopic capability is demonstrated by measuring parts of the photoacoustic absorption spectrum of NH3 near 3196 cm−1.
TL;DR: A continuous intracavity frequency-doubled singly resonant optical parametric oscillator is stabilized to the side of the transmission peak of a medium finesse Fabry-Perot cavity to lead to a stability of a few kilohertz with respect to the locking etalon.
Abstract: A continuous intracavity frequency-doubled singly resonant optical parametric oscillator (OPO) is stabilized to the side of the transmission peak of a medium finesse Fabry–Perot cavity. The narrow bandwidth of the frequency noise of this OPO allows this simple scheme to lead to a stability of a few kilohertz with respect to the locking etalon. The system, operating in the visible domain, remains locked for more than 1h.
TL;DR: This work generates squeezed state of light at 860 nm with a monolithic optical parametric oscillator, which makes it possible to drive many monolithic cavities simultaneously by a single laser.
Abstract: We generate squeezed state of light at 860 nm with a monolithic optical parametric oscillator. The optical parametric oscillator consists of a periodically poled KTiOPO4 crystal, both ends of which are spherically polished and mirror-coated. We achieve both phase matching and cavity resonance by controlling only the temperature of the crystal. We observe up to −8.0±0.2 dB of squeezing with the bandwidth of 142 MHz. Our technique makes it possible to drive many monolithic cavities simultaneously by a single laser. Hence our monolithic optical parametric oscillator is quite suitable to continuous-variable quantum information experiments where we need a large number of highly squeezed light beams.
TL;DR: In this article, the use of optical fiber for optical parametric chirped pulse amplification to amplify sub-picosecond pulses was demonstrated for the first time, and the peak power of the signal was amplified from 93 mW to 10 W.
Abstract: We experimentally demonstrate, for the first time to the best of our knowledge, the use of optical fiber for optical parametric chirped pulse amplification to amplify subpicosecond pulses. We use this system to amplify a subpicosecond signal at 1595 nm generated by a fiber-optical parametric oscillator. The 750-fs signal from the oscillator output is stretched to 40 ps, amplified by an all-fiber optical parametric amplifier and then compressed to 808 fs. The peak power of the signal is amplified from 93 mW to 10 W.
TL;DR: An electrical distributed parametric oscillator is presented to realize a passive CMOS frequency divider with low phase noise and there is a good agreement among analysis, simulation, and 10-MHz measurement results.
Abstract: We present an electrical distributed parametric oscillator to realize a passive CMOS frequency divider with low phase noise. Instead of using active devices, which are the main sources of noise and power consumption, an oscillation at half of the input frequency is sustained by the parametric process based on nonlinear interaction with the input signal. To show the feasibility of the proposed approach, we have implemented a 20-GHz frequency divider in a 0.13-μm CMOS process. Without any dc power consumption, 600-mV differential output amplitude is achieved for an input amplitude of 600 mV. The input frequency ranges from 18.5 to 23.5 GHz with varactor tuning. The output phase noise is almost 6 dB lower than that of the input signal for all offset frequencies up to 1 MHz. There is a good agreement among analysis, simulation, and 10-MHz measurement results. To the best of our knowledge, this is the first passive frequency divider in a CMOS process.
23 May 2010
TL;DR: In this article, a 600MHz thin film bulk-acoustic wave resonator (FBAR)-based differential oscillator fabricated in a 0.13µm CMOS process is presented.
Abstract: A 600MHz thin film bulk-acoustic wave resonator (FBAR)-based differential oscillator fabricated in a 0.13µm CMOS process is presented. The oscillator employs a crosscoupled pair with an FBAR resonator tank providing high Q source degeneration to realize frequency oscillation at the series resonance. The measured phase noise is −126 and −150dBc/Hz at 10kHz and 1MHz frequency offsets respectively; the integrated RMS jitter from 10kHz to 20MHz is 50fs. The oscillator achieves a frequency drift of 50ppm over the temperature range from 25 to 110 °C, providing the potential for quartz replacement in some applications. The figure-of-merit (FOM) of the oscillator is 214dB.
TL;DR: In this article, a dynamical model for the Ising-bloch transition is proposed and investigated, and the uniform solutions supported by the uniform states are characterized and classified into two types: motionless and propagative states, rising through a non-variational Ising−Bloch transition.
TL;DR: A picosecond optical parametric oscillator (OPO) that is synchronously pumped by a fiber-amplified gain-switched laser diode and can be switched to higher repetition rates up to ~1GHz is demonstrated.
Abstract: We demonstrate a picosecond optical parametric oscillator (OPO) that is synchronously pumped by a fiber-amplified gain-switched laser diode. At 24W of pump power, up to 7.3W at 1.54µm and 3.1W at 3.4µm is obtained in separate output beams. The periodically poled MgO-doped LiNbO3 OPO operates with ~17ps pulses at a fundamental repetition rate of 114.8MHz but can be switched to higher repetition rates up to ~1GHz. Tunabilty between 1.4µm and 1.7µm (signal) and 2.9µm and 4.4µm (idler) is demonstrated by translating the nonlinear crystal to access different poling-period gratings and typical M2 values of 1.1 by 1.2 (signal) and 1.6 by 3.2 (idler) are measured at high power for the singly resonant oscillator.
TL;DR: In this article, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated, assuming a polynomial constitutive model to describe the restitution force of the oscillator.
Abstract: In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.
TL;DR: In this article, the authors measured frequency and dissipation-quadrature noise in superconducting lithographed microwave resonators with sensitivity near the vacuum noise level using a Josephson parametric amplifier.
Abstract: We measure frequency- and dissipation-quadrature noise in superconducting lithographed microwave resonators with sensitivity near the vacuum noise level using a Josephson parametric amplifier. At an excitation power of 100~nW, these resonators show significant frequency noise caused by two-level systems. No excess dissipation-quadrature noise (above the vacuum noise) is observed to our measurement sensitivity. These measurements demonstrate that the excess dissipation-quadrature noise is negligible compared to vacuum fluctuations, at typical readout powers used in micro-resonator applications. Our results have important implications for resonant readout of various devices such as detectors, qubits and nano-mechanical oscillators.
TL;DR: In this paper, the excitation of thin planar liquid sheets formed by impinging two collinear water jets to acoustic waves was studied at varying frequencies and sound pressure levels (SPLs).
Abstract: The excitation of thin planar liquid sheets formed by impinging two collinear water jets to acoustic waves was studied at varying frequencies and sound pressure levels (SPLs). Experiments were conducted over a range of liquid velocities that encompassed the stable and flapping regimes of the sheet. For a given frequency, there was a threshold value of SPL below which the sheet was unaffected. The threshold SPL increased with frequency. Further, the sheet was observed to respond to a set of specific frequencies lying in the range of 100–300 Hz, the frequency set varying with the Weber number of the liquid sheet. The magnitude of the response for a fixed pressure level, characterized by the reduction in the extent of the sheet, was larger at lower frequencies. The droplet sizes formed by the disintegration of the sheet reduced with an increase in the measured response and the drop-shedding frequency was near the imposed frequency. Model equations for inviscid flow and accounting for the varying pressure field across the moving liquid sheet of constant thickness was solved to determine the linear stability of the system. Numerical solution shows that the most unstable wavelengths in the presence of the forcing to be smaller than in the absence, which is in line with observations. Both the dilatational and sinuous modes are coupled at the lowest order and become significant for the range of acoustic forcing studied. The model calculation suggests that the parametric resonance involving the dilatational mode may be responsible for the observed instability although the model was unable to predict the observed variation of threshold SPL with frequency.
TL;DR: In this paper, a three-beam structural system with attached mass is considered, and its multidegree-of-freedom discretized model for the structure undergoing planar motions is carefully studied.
Abstract: Nonlinear dynamics of elastic structures with two-mode interactions have been extensively studied in the literature. In this work, nonlinear forced response of elastic structures with essential inertial nonlinearities undergoing three-mode interactions is studied. More specifically, a three-beam structural system with attached mass is considered, and its multidegree-of-freedom discretized model for the structure undergoing planar motions is carefully studied. Linear modal characteristics of the structure with uniform beams depend on the length ratios of the three beams, the mass of the particle relative to that of the structure, and the location of the mass particle along the beams. The discretized model is studied for both external and parametric resonances for parameter combinations resulting in three-mode interactions. For the external excitation case, focus is on the system with 1:2:3 internal resonances with the external excitation frequency near the middle natural frequency. For the case of the structure with 1:2:5 internal resonances, the problem involving simultaneous principal parametric resonance of the middle mode and a combination resonance between the lowest and the highest modal frequencies is investigated. This case requires a higher-order approximation in the method of multiple time scales. For both cases, equilibrium and bifurcating solutions of the slow-flow equations are studied in detail. Many pitchfork, saddle-node, and Hopf bifurcations appear in the amplitude response of the three-beam structure, thus resulting in complex multimode responses in different parameter regions.
TL;DR: A high-energy optical parametric oscillator synchronously pumped by a 7.19 MHz, Yb:fiber-amplified, picosecond, gain-switched laser diode that maintains a compact design through the use of an intracavity optical fiber is reported.
Abstract: We report a high-energy optical parametric oscillator (OPO) synchronously pumped by a 7.19 MHz, Yb:fiber amplified, picosecond, gain-switched laser diode. The 42m long ring cavity maintains a compact design through the use of an intracavity optical fiber. The periodically poled MgO-doped LiNbO3 OPO provides output pulse energies as high as 0.49 µJ at 1.5 µm (signal) and 0.19 µJ at 3.6 µm (idler). Tunability from 1.5 to 1.7 µm and from 2.9 to 3.6 µm is demonstrated, and typical M2 values of 1.5 x 1.3 and 2.8 x 1.9 are measured for the signal and idler, respectively, at high power.
TL;DR: In this article, the authors demonstrate parametric oscillation of discrete polariton states in a single squared GaAs/GaAlAs micropillar cavity with a continuous wave pump laser.
Abstract: We demonstrate parametric oscillation of discrete polariton states in a single squared GaAs/GaAlAs micropillar cavity. Resonantly exciting a selected polariton mode with a continuous wave pump laser, parametric oscillation is evidenced on the two neighbored modes (signal and idler). Abrupt switch-off of the device is observed under high excitation. We present comprehensive results concerning the power-dependence of the energy and linewidth of the emission resonances as well as their far-field patterns. Quantum Monte Carlo calculations give a quantitative understanding of the physics of this micro-optical parametric oscillator involving fully confined discrete polariton modes.
TL;DR: The carrier-envelope-offset (CEO) frequency of pulses from a femtosecond optical parametric oscillator was stabilized for 30 min without the need for f-2f self-referencing in either the OPO or its pump laser.
Abstract: The carrier-envelope-offset (CEO) frequency of pulses from a femtosecond optical parametric oscillator (OPO) was stabilized for 30 min without the need for f–2f self-referencing in either the OPO or its pump laser. Interference between the high-frequency modes of the pump supercontinuum and the modes of the non-phase-matched pump+signal sum-frequency-mixing pulses provided the beat signal used for locking. The −3 dB bandwidth of the locked CEO-frequency was measured as 1.1 kHz, and the cumulative phase error recorded over 1 s was 0.38 rad, representing 1-order-of-magnitude improvement in comparison to previous results.
TL;DR: In this paper, the nonlinear vibration model of a moving strip between two mill with time-dependent tension was established, and approximate solutions were obtained using the method of multiple scales, and the stable regions and unstable regions for parametric resonance were determined with different cases.
Abstract: The strip with a time-dependent tension moves, namely a harmonically varying tension about a constant initial tension. The nonlinear vibration model of moving strip between two mills with time-dependent tension was established. Approximate solutions were obtained using the method of multiple scales. Depending on the variation of the tension, three distinct cases arise: frequency away from zero or two times the natural frequency, frequency close to zero, frequency close to two times the natural frequency. For frequency close to zero and away from zero and two times the natural frequency, the system is always stable. For frequency close to two times the natural frequency, the stability is analyzed respectively when the trivial solution exists and the nontrivial solution exists. Numerical simulation was made on some 1660 mm tandem rolling mill, and the stable regions and unstable regions for parametric resonance are determined with different cases. The rolling speed and the thickness of strip have strong influences on the stability of principle parametric resonances. But the distance between two mills has little influence on the stability of principle parametric resonances.