It is recognized that standing wave effects appearing in large-area,very-high-frequency capacitively coupled plasma(CCP)reactors cause center-high plasma non-uniformity.Using a high-frequency magnetic probe,we present...It is recognized that standing wave effects appearing in large-area,very-high-frequency capacitively coupled plasma(CCP)reactors cause center-high plasma non-uniformity.Using a high-frequency magnetic probe,we present a direct experimental diagnostic of the nonlinear standing waves and bulk ohmic electron power absorption dynamics in low pressure CCP discharges for different driving frequencies of 13.56,30,and 60 MHz.The design,principle,calibration,and validation of the probe are described in detail.Spatial structures of the harmonics of the magnetic field,determined by the magnetic probe,were used to calculate the distributions of the harmonic current and the corresponding ohmic electron power deposition,providing insights into the behavior of nonlinear harmonics.At a low driving frequency,i.e.13.56 MHz,no remarkable nonlinear standing waves were identified and the bulk ohmic electron power absorption was observed to be negligible.The harmonic magnetic field/current was found to increase dramatically with the driving frequency,due to decreased sheath reactance and more remarkable nonlinear standing waves at a higher driving frequency,leading to the enhancements of the ohmic heating and the plasma density in the bulk,specifically at the electrode center.At a high driving frequency,i.e.60 MHz,the high-order harmonic current density and the corresponding ohmic electron power absorption exhibited a similar node structure,with the main peak on axis,and one or more minor peaks between the electrode center and the edge,contributing to the center-high profile of the plasma density.展开更多
Based on the basic equations of hydrodynamics,the nonlinear acoustic wave equation is obtained.By taking into account the boundary condition and properties of nonlinear standing wave,the equation is solved through per...Based on the basic equations of hydrodynamics,the nonlinear acoustic wave equation is obtained.By taking into account the boundary condition and properties of nonlinear standing wave,the equation is solved through perturbation method,and the stable expressions of fundamental wave and second harmonic are presented.The sound pressures in an ultrasonic cleaner are measured by hydrophones,and the relationship between the received voltages of hydrophones and the output voltages of the ultrasonic generator is researched.The study shows the existence of the nonlinear effect of liquid and analyzes the frequency spectrum of the received signals by hydrophones,by which the fundamental wave,second and high order harmonics are found coexisting in the bounded space filled with liquids.The theory and experimental results testify the existence of the nonlinear standing wave in liquid.Owing to the restricted applicability of perturbation method,the theoretical results of the fundamental wave and second harmonic are good only for the weak nonlinear phenomenon.展开更多
Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times sati...Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor＇s series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.展开更多
基金financially supported by National Natural Science Foundation of China(NSFC)(Nos.12005035 and 11935005)China Postdoctoral Science Foundation(Nos.2020M670741 and 2021T140085)+2 种基金Fundamental Research Funds for the Central Universities(No.DUT20LAB201)National Science Foundation(No.PHY-1500518)Department of Energy Office of Fusion Energy Science(No.DE-SC0001939)for financial support。
文摘It is recognized that standing wave effects appearing in large-area,very-high-frequency capacitively coupled plasma(CCP)reactors cause center-high plasma non-uniformity.Using a high-frequency magnetic probe,we present a direct experimental diagnostic of the nonlinear standing waves and bulk ohmic electron power absorption dynamics in low pressure CCP discharges for different driving frequencies of 13.56,30,and 60 MHz.The design,principle,calibration,and validation of the probe are described in detail.Spatial structures of the harmonics of the magnetic field,determined by the magnetic probe,were used to calculate the distributions of the harmonic current and the corresponding ohmic electron power deposition,providing insights into the behavior of nonlinear harmonics.At a low driving frequency,i.e.13.56 MHz,no remarkable nonlinear standing waves were identified and the bulk ohmic electron power absorption was observed to be negligible.The harmonic magnetic field/current was found to increase dramatically with the driving frequency,due to decreased sheath reactance and more remarkable nonlinear standing waves at a higher driving frequency,leading to the enhancements of the ohmic heating and the plasma density in the bulk,specifically at the electrode center.At a high driving frequency,i.e.60 MHz,the high-order harmonic current density and the corresponding ohmic electron power absorption exhibited a similar node structure,with the main peak on axis,and one or more minor peaks between the electrode center and the edge,contributing to the center-high profile of the plasma density.
基金supported by the National Natural Science Foundation of China(Grant Nos.10674090 and 10974127)
文摘Based on the basic equations of hydrodynamics,the nonlinear acoustic wave equation is obtained.By taking into account the boundary condition and properties of nonlinear standing wave,the equation is solved through perturbation method,and the stable expressions of fundamental wave and second harmonic are presented.The sound pressures in an ultrasonic cleaner are measured by hydrophones,and the relationship between the received voltages of hydrophones and the output voltages of the ultrasonic generator is researched.The study shows the existence of the nonlinear effect of liquid and analyzes the frequency spectrum of the received signals by hydrophones,by which the fundamental wave,second and high order harmonics are found coexisting in the bounded space filled with liquids.The theory and experimental results testify the existence of the nonlinear standing wave in liquid.Owing to the restricted applicability of perturbation method,the theoretical results of the fundamental wave and second harmonic are good only for the weak nonlinear phenomenon.
文摘Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor＇s series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.
