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Conservation laws of the complex short pulse equation and coupled complex short pulse equations
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作者 Lihua Zhang Bo Shen +2 位作者 Shuxin Han Gangwei Wang Lingshu Wang 《Communications in Theoretical Physics》 SCIE CAS CSCD 2022年第7期49-56,共8页
In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models ar... In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases.Based on the linearizing operator and adjoint linearizing operator for the two real systems,adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair(SA) method. Relationships between the nonlinear selfadjointness method and the SA method are investigated. 展开更多
关键词 complex short pulse equation coupled complex short pulse equations SYMMETRIES adjoint symmetries conservation laws SA method
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Skin Effect of Reversely Switched Dynistor in Short Pulse Discharge Application 被引量:1
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作者 Lin Liang Yue-Hui Yu 《Journal of Electronic Science and Technology of China》 2009年第2期146-149,共4页
The skin effect in the reversely switched dynistor (RSD) devices is investigated in this paper. Based on the plasma bipolar drift model of the RSD, the current density distributions on the chip are simulated with co... The skin effect in the reversely switched dynistor (RSD) devices is investigated in this paper. Based on the plasma bipolar drift model of the RSD, the current density distributions on the chip are simulated with considering the skin effect. The results indicate that the current density on the border can be several hundred to a thousand A/cm2 higher than that in the center of the chip. The skin effect becomes more prominent as the voltage increases and the inductance decreases in the main circuit. The phenomenon that most of a certain group of chips break over on the border has proved the existence of the skin effect. 展开更多
关键词 Index Terms-pulsed power switch reverselyswitched dynistor (RSD) short pulse skin effect.
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Comparative study of travelling-wave and numerical solutions for the coupled short pulse (CSP) equation
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作者 Vikas Kumar R. K. Gupta Ram Jiwari 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第5期60-66,共7页
The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry ge... The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find reductions in terms of system of ordinary differential equations, and exact solutions of the CSP equation are derived, which are compared with numerical solutions using the classical fourth-order Runge-Kutta scheme. 展开更多
关键词 coupled short pulse (CSP) equation Lie symmetric analysis Runge-Kutta scheme exact solutions
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Reciprocal transformations of the space-time shifted nonlocal short pulse equations
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作者 Jing Wang Hua Wu Da-Jun Zhang 《Chinese Physics B》 SCIE EI CAS CSCD 2022年第12期14-19,共6页
Reciprocal transformations of the space-time shifted nonlocal short pulse equations are elaborated.Covariance of dependent and independent variables involved in the reciprocal transformations is investigated.Exact sol... Reciprocal transformations of the space-time shifted nonlocal short pulse equations are elaborated.Covariance of dependent and independent variables involved in the reciprocal transformations is investigated.Exact solutions of the space-time shifted nonlocal short pulse equations are given in terms of double Wronskians.Realness of independent variables involved in the reciprocal transformations is verified.Dynamics of some obtained solutions are illustrated. 展开更多
关键词 reciprocal transformation space-time shifted nonlocal short pulse equation covariance of variable solution
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Exact Solutions to Short Pulse Equation
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作者 FU Zun-Tao ZHENG Ming-Hua LIU Shi-Kuo 《Communications in Theoretical Physics》 SCIE CAS CSCD 2009年第3期395-396,共2页
In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.
关键词 Jacobian elliptic function the short pulse equation TRANSFORMATION
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Diffractive Interaction for Short Pulses in 2×2 System
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作者 袁明生 《Journal of Shanghai Jiaotong university(Science)》 EI 2008年第4期499-508,共10页
This paper studies the asymptotic behavior of the diffractive interaction of pulses with two linear phases of length ε ≤ 1 in 2 × 2 semilinear strictly hyperbolic system with constant coefficiients. By a formal... This paper studies the asymptotic behavior of the diffractive interaction of pulses with two linear phases of length ε ≤ 1 in 2 × 2 semilinear strictly hyperbolic system with constant coefficiients. By a formal analysis, it derives problems of profiles in the expansion of the pulse like solution with respect to the length s, and obtains that the leading profile satisfies a nonlinear Schrodinger type system. The problems of profiles are solved, and the formal expansion is justified. It is observed that there is interaction between two phases starting at the third order profiles. 展开更多
关键词 short pulses diffractive interaction 2 × 2 system
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Global smooth solutions of 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
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作者 Bingbing Ding Zhouping Xin Huicheng Yin 《Science China Mathematics》 2026年第3期667-736,共70页
For the short pulse initial data with a first-order outgoing constraint condition and optimal orders of smallness,we establish the global existence of smooth solutions to 2D quasilinear wave equations with higher-orde... For the short pulse initial data with a first-order outgoing constraint condition and optimal orders of smallness,we establish the global existence of smooth solutions to 2D quasilinear wave equations with higher-order null conditions.Such kinds of wave equations include 2D relativistic membrane equations,2D membrane equations,and some 2D quasilinear equations which come from the nonlinear Maxwell equations in electromagnetic theory or from the corresponding Lagrangian functionals as perturbations of the Lagrangian densities of linear wave operators.The main ingredients of the analysis here include looking for a new good unknown,finding some key identities based on the higher-order null conditions and the resulting null frames,as well as overcoming the difficulties due to the slow decay of solutions to the 2D wave equation,so that the solutions can be estimated precisely. 展开更多
关键词 quasilinear wave equations short pulse initial data higher-order null condition inverse foliation density good unknown null frame
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Demonstration of full-scale spatiotemporal diagnostics of solid-density plasmas driven by an ultra-short relativistic laser pulse using an X-ray free-electron laser
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作者 Lingen Huang Michal Smíd +42 位作者 Long Yang Oliver Humphries Johannes Hagemann Thea Engler Xiayun Pan Yangzhe Cui Thomas Kluge Ritz Aguilar Carsten Baehtz Erik Brambrink Engin Eren Katerina Falk Alejandro Laso Garcia Sebastian Gode Christian Gutt Mohamed Hassan Philipp Heuser Hauke Hoppner Michaela Kozlova Wei Lu Josefine Metzkes-Ng Masruri Masruri Mikhail Mishchenko Motoaki Nakatsutsumi Masato Ota Ozgül Oztürk Alexander Pelka Irene Prencipe Thomas R.Preston Lisa Randolph Martin Rehwald Hans-Peter Schlenvoigt Ulrich Schramm Jan-Patrick Schwinkendorf Sebastian Starke Radka Stefaníková Erik Thiessenhusen Monika Toncian Toma Toncian Jan Vorberger Ulf Zastrau Karl Zeil Thomas E.Cowan 《Matter and Radiation at Extremes》 2026年第1期6-19,共14页
Understanding the complex plasma dynamics in ultra-intense relativistic laser-solid interactions is of fundamental importance for applications of laser-plasma-based particle accelerators,the creation of high-energy-de... Understanding the complex plasma dynamics in ultra-intense relativistic laser-solid interactions is of fundamental importance for applications of laser-plasma-based particle accelerators,the creation of high-energy-density matter,understanding planetary science,and laser-driven fusion energy.However,experimental efforts in this regime have been limited by the lack of accessibility of over-critical densities and the poor spatiotemporal resolution of conventional diagnostics.Over the last decade,the advent of femtosecond brilliant hard X-ray free-electron lasers(XFELs)has opened new horizons to overcome these limitations.Here,for the first time,we present full-scale spatiotemporal measurements of solid-density plasma dynamics,including preplasma generation with tens of nanometer scale length driven by the leading edge of a relativistic laser pulse,ultrafast heating and ionization at the main pulse arrival,the laser-driven blast wave,and transient surface return current-induced compression dynamics up to hundreds of picoseconds after interaction.These observations are enabled by utilizing a novel combination of advanced X-ray diagnostics including small-angle X-ray scattering,resonant X-ray emission spectroscopy,and propagation-based X-ray phase-contrast imaging simultaneously at the European XFEL-HED beamline station. 展开更多
关键词 preplasma generation spatiotemporal diagnostics understanding complex plasma dynamics x ray free electron laser planetary scienceand conventional diagnosticsover solid density plasmas ultra short relativistic laser pulse
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Generation of tail-free short pulses using high-pressure CO_(2) laser 被引量:4
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作者 Yue Lu Ziren Zhu +4 位作者 Jinzhou Bai Xinjun Su Rongqing Tan Jinghan Ye Yijun Zheng 《Chinese Optics Letters》 SCIE EI CAS CSCD 2022年第5期14-18,共5页
The spectral linewidth of a transversely excited pulsed CO_(2) laser is broadened at high working pressures.This phenomenon causes a decrease in the upper-level lifetime such that the pulse width is significantly comp... The spectral linewidth of a transversely excited pulsed CO_(2) laser is broadened at high working pressures.This phenomenon causes a decrease in the upper-level lifetime such that the pulse width is significantly compressed.Although the tail part of CO_(2) laser pulses owns a non-negligible proportion of total energy,it has minor effects during the interaction process between photons and materials due to its low amplitude.Thus,it is of great significance to yield the tail part and generate a narrow pulse in most applications.In this study,a continuously tunable pulsed CO_(2) laser with a low nitrogen proportion in the mixture is developed to generate tail-free short pulses;a minimum pulse width of 30.60 ns with a maximum pulse energy of 481 mJ is synchronously achieved at a pressure of 7 atm,and the estimated peak power is above 15 MW.A numerical simulation is also conducted for comparison with the experimental results.The contribution of the spectral gain toward the compression of the pulse width is discussed in the last section of this paper. 展开更多
关键词 short pulse continuous tuning pulse compression
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Pulsed LD side-pumped MgO:LN electro-optic cavity-dumped 1123 nm Nd: YAG laser with short pulse width and high peak power 被引量:1
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作者 Yang Bai Bing Bai +5 位作者 Diao Li Yanxiao Sun Jianlin Li Lei Hou Mingxuan Hu Jintao Bai 《High Power Laser Science and Engineering》 SCIE CAS CSCD 2018年第1期20-26,共7页
We report a cavity-dumped 1123 nm laser with narrow pulse width and high peak power by an Mg O: LN crystal electrooptic(EO) modulator. Based on the structural optimization design of a folded biconcave cavity using the... We report a cavity-dumped 1123 nm laser with narrow pulse width and high peak power by an Mg O: LN crystal electrooptic(EO) modulator. Based on the structural optimization design of a folded biconcave cavity using the 808 nm pulsed laser diode(LD) side-pumped ceramic Nd: YAG rod, output pulses with maximum pulse energy and peak power up to39.6 m J and 9.73 MW were obtained, corresponding to 100 Hz repetition rate and 4.07 ns pulse width. The instabilities of pulse width and pulse energy were ±1.55% and ±2.06%, respectively. At the highest repetition rate of 1 kHz, the pulse energy, pulse width, and peak power were 11.3 mJ, 5.05 ns, and 2.24 MW, respectively. The instabilities of pulse width and pulse energy were ±2.65% and ±3.47%, respectively. 展开更多
关键词 CAVITY-DUMPED high peak power MGO LN EO 1123nm laser short pulse width
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A new method on diagnostics of muons produced by a short pulse laser 被引量:1
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作者 Feng Zhang Boyuan Li +3 位作者 Lianqiang Shan Bo Zhang Wei Hong Yuqiu Gu 《High Power Laser Science and Engineering》 SCIE CAS CSCD 2017年第3期5-9,共5页
Muons produced by a short pulse laser can serve as a new type of muon source having potential advantages of high intensity, small source emittance, short pulse duration and low cost. To validate it in experiments, a s... Muons produced by a short pulse laser can serve as a new type of muon source having potential advantages of high intensity, small source emittance, short pulse duration and low cost. To validate it in experiments, a suitable muon diagnostics system is needed since high muon flux generated by a short pulse laser shot is always accompanied by high radiation background, which is quite different from cases in general muon researches. A detection system is proposed to distinguish muon signals from radiation background by measuring the muon lifetime. It is based on the scintillator detector with water and lead shields, in which water is used to adjust energies of muons stopped in the scintillator and lead to against radiation background. A Geant 4 simulation on the performance of the detection system shows that efficiency up to 52% could be arrived for low-energy muons around 200 MeV and this efficiency decreases to 14% for high-energy muons above 1000 MeV. The simulation also shows that the muon lifetime can be derived properly by measuring attenuation of the scintilla light of electrons from muon decays inside the scintillator detector. 展开更多
关键词 DIAGNOSTICS Geant4 simulation muon source short pulse laser
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Grain refining in weld metal using short-pulsed laser ablation during CW laser welding of 2024-T3 aluminum alloy 被引量:2
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作者 Masaki Kasuga Tomokazu Sano Akio Hirose 《International Journal of Extreme Manufacturing》 2019年第4期34-41,共8页
The 2024 aluminum alloy is used extensively in the aircraft and aerospace industries because of its excellent mechanical properties.However,the weldability of 2024 aluminum alloy is generally low because it contains a... The 2024 aluminum alloy is used extensively in the aircraft and aerospace industries because of its excellent mechanical properties.However,the weldability of 2024 aluminum alloy is generally low because it contains a high number of solutes,such as copper(Cu),magnesium(Mg),and manganese(Mn),causing solidification cracking.If high speed welding of 2024 aluminum alloy without the use of filler is achieved,the applicability of 2024 aluminum alloys will expand.Grain refining is one of the methods used to prevent solidification cracking in weld metal,although it has never been achieved for high-speed laser welding of 2024 aluminum alloy without filler.Here,we propose a short-pulsed,laser-induced,grain-refining method during continuous wave laser welding without filler.Bead-on-plate welding was performed on a 2024-T3 aluminum alloy at a welding speed of 1 m min−1 with a single mode fiber laser at a wavelength of 1070 nm and power of 1 kW.Areas in and around the molten pool were irradiated with nanosecond laser pulses at a wavelength of 1064 nm,pulse width of 10 ns,and pulse energy of 430 mJ.The grain-refinement effect was confirmed when laser pulses were irradiated on the molten pool.The grain-refinement region was formed in a semicircular shape along the solid–liquid interface.Results of the vertical section indicate that the grain-refinement region reached a depth of 1 mm along the solid–liquid interface.The Vickers hardness test results demonstrated that the hardness increased as a result of grain refinement and that the progress of solidification cracking was suppressed in the grain refinement region. 展开更多
关键词 2024 aluminum alloy hot cracking laser welding grain refinement dendrite fragmentation short pulsed laser laser ablation
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Short pulse propagation in dense dispersion compensated systems
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作者 肖燕 郭淑琴 +2 位作者 薛文瑞 李仲豪 周国生 《Chinese Optics Letters》 SCIE EI CAS CSCD 2003年第4期205-207,共3页
The short Hermite-Gaussian optical pulse transmission over 1440 km in a dense dispersion compensated system is investigated based on numerical simulation. In the simulation, compensation is made not only for the group... The short Hermite-Gaussian optical pulse transmission over 1440 km in a dense dispersion compensated system is investigated based on numerical simulation. In the simulation, compensation is made not only for the group-velocity dispersion but also for the third order dispersion. It is demonstrated that the pulse with reasonably lower power can propagate steadily in net zero, positive and negative dispersion system. 展开更多
关键词 in on be as that short pulse propagation in dense dispersion compensated systems of link TOD FIGURE length
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A Mutual Pulse Injection-Seeding Scheme for Optical Short Pulse Generation
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作者 D. N. Wang 《光学学报》 EI CAS CSCD 北大核心 2003年第S1期649-650,共2页
A mutual pulse injection-seeding scheme is developed to produce wavelength tunable optical short pulse generation. The sidemode suppression ratio obtained is more than 31 dB over the wavelength-tuning rang of 18 nm.
关键词 for in In of NM A Mutual pulse Injection-Seeding Scheme for Optical short pulse Generation THAN
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The implementation and data analysis of an interferometer for intense short pulse laser experiments
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作者 Jaebum Park Hector A.Baldis Hui Chen 《High Power Laser Science and Engineering》 SCIE CAS CSCD 2016年第3期60-66,共7页
We present an interferometry setup and the detailed fringe analysis method for intense short pulse(SP) laser experiments.The interferometry scheme was refined through multiple campaigns to investigate the effects of p... We present an interferometry setup and the detailed fringe analysis method for intense short pulse(SP) laser experiments.The interferometry scheme was refined through multiple campaigns to investigate the effects of pre-plasmas on energetic electrons at the Jupiter Laser Facility at Lawrence Livermore National Laboratory. The interferometer used a frequency doubled(λ=0.527 μm) 0.5 ps long optical probe beam to measure the pre-plasma density, an invaluable parameter to better understand how varying pre-plasma conditions affect the characteristics of the energetic electrons. The hardware of the diagnostic, data analysis and example data are presented. The diagnostic setup and the analysis procedure can be employed for any other SP laser experiments and interferograms, respectively. 展开更多
关键词 Mach-Zehnder interferometer optical interferometry picosecond probe plasma density short pulse laser
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High-order harmonics generation by few-cycle and multi-cycle femtosecond laser pulses
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作者 运晨霞 滕浩 +6 位作者 张伟 詹敏杰 韩海年 钟欣 魏志义 王兵兵 侯洵 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第12期301-304,共4页
This paper investigates experimentally high-order harmonic generation (HHG) of neon gas with 5-fs and 25-fs driving laser pulses. It has been demonstrated that the cutoff energy of the harmonic extreme ultraviolet p... This paper investigates experimentally high-order harmonic generation (HHG) of neon gas with 5-fs and 25-fs driving laser pulses. It has been demonstrated that the cutoff energy of the harmonic extreme ultraviolet photons is extended to 131 eV and the HHC spectrum near the cutoff region becomes continuum as the driving laser pulse duration is 5 fs; whereas much lower cutoff photon energy and discrete harmonic spectrum near the cutoff region are presented as the laser pulse duration is 25 fs. The results can be explained by the fact that neutral atoms can be exposed to more intense laser field before they are depleted by ionization because of the extremely short rising time of the few-cycle pulse. The 5-fs driving laser pulse paves the way of generation of coherent x-ray in the water window and single attosecond pulse. 展开更多
关键词 high harmonics generation short pulse super-continuum spectrum
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The Critical Power of Short Pulse Initial Data on the Global Existence or Blowup of Smooth Solutions to 3-D Semilinear Klein-Gordon Equations
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作者 Jindou Shen Huicheng Yin 《Acta Mathematica Sinica,English Series》 2026年第3期568-582,共15页
It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations u+u=F(u,∂u)with cubic nonlinearities.However,for the short pulse initial data(u,∂_(t)u)(0,x)=(δ^(ν+1)u... It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations u+u=F(u,∂u)with cubic nonlinearities.However,for the short pulse initial data(u,∂_(t)u)(0,x)=(δ^(ν+1)u_(0)(x/δ),δ^(ν)u_(1)(x/δ))withν∈R and(u_(0),u_(1))∈C_(0)^(∞)(R),which are a class of large initial data,we establish that whenν≤-1/2,the solution u can blow up in finite time for some suitable choices of(u_(0),u_(1))and cubic nonlinearity F(u,∂u);when ν>-1/2,the smooth solution u exists globally.Therefore,ν=-1/2 is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations. 展开更多
关键词 Klein-Gordon equation short pulse initial data global solution blowup critical power
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Inhomogeneous Dielectric Target Properties for Increased Non-thermal Pressure in Laser Boron Fusion by CPA-Pulses
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作者 Heinrich Hora Warren McKenzie 《Journal of Energy and Power Engineering》 2020年第8期273-278,共6页
Based on the documentation of the invited paper and the subsequent discussion at a virtual conference,discoveries are indicated,which are summarized in the following paper for further evaluation on the topic of non-th... Based on the documentation of the invited paper and the subsequent discussion at a virtual conference,discoveries are indicated,which are summarized in the following paper for further evaluation on the topic of non-thermal forces using terms of extremely powerful components of Maxwell’s stress tensor at the interaction of CPA(chirped pulse amplification)laser pulses in the fusion plasmas of hydrogen with the isotope 11 of boron.This is caused by a recoil mechanism given by the Fresnel formulas of the suppression of the reflectivity of inhomogeneous plasma given by optical constants of the plasma properties. 展开更多
关键词 Clean nuclear fusion of hydrogen H with 11B Fresnel formulas for inhomogeneous plasmas optical constants in plasmas excluding unnecessary temperatures of hundred million degrees for thermal fusion pressures by non-thermal pressured from extreme laser pulses CPA of ultra-extremely short pulses
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Fast repetition rate fs pulsed lasers for advanced PLIM microscopy
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作者 Sviatlana Kalinina Alexander Jelzow +1 位作者 Tobias Plotzing Angelika Rück 《Journal of Innovative Optical Health Sciences》 SCIE EI CAS 2019年第5期86-94,共9页
Simultaneous metabolic and oxygen imaging is promising to follow up therapy response,dis-ease development and to determine prognostic factors.FLIM of metabolic coenzymes is now widely accepted to be the most reliable ... Simultaneous metabolic and oxygen imaging is promising to follow up therapy response,dis-ease development and to determine prognostic factors.FLIM of metabolic coenzymes is now widely accepted to be the most reliable method to determine cellular bioenergetics.Also,oxygen consumption has to be taken into account to understand treatment responses.The phosphorescence lifetimne of oxygen sensors is able to indicate local oxygen changes.For phosphorescence lifetime imaging(PLIM)dyes based on ruthenium(I)coordination com-plexes are useful,in detaill TLD1433 which possesses a variety of different triplet states,enables complex photochemistry and redox reactions.PLIM is usally reached by two photon exci-tation of the drug with a femtosecond(fs)pulsed Ti:Sapphire laser working at 80 MHz repe-tition rate and(time-correlated single photon counting)(TCSPC)detection electronics.The interesting question was whether it is possible to follow up PLIM 1using faster repetition rates.Faster repetition rates could be advantageous for the induction of specific photochemical reactions because of similar light doses used normally in standard CW light treatments.For this,a default 2p-FLIM-PLIM system was expanded by adding a second fs pulsed laser("helixx")which provides 50 fs pulses at a repetition rate of 250 MHz,more than 2.3 w average power and tunable from 720 nm to 920 nm.The laser beam was coupled into the AOM instead of the default 80 MHz laser.We demonstrated siuccessful applications of the 250 MHz laser for PLIM which correlates well with measurements done by excitation with the conventional 80MHx laser source. 展开更多
关键词 Multiphoton microscopy axygen consumption phosphorescence lifetime microscopy fast repetition rate excitation short laser pulses Ru(I)complexes
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Experimental Investigation on Complex Structures Machining by Electrochemical Micromachining Technology 被引量:10
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作者 刘勇 朱荻 +2 位作者 曾永彬 黄绍服 余宏兵 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2010年第5期578-584,共7页
Electrochemical micromachining (EMM) technology for fabricating micro structures is presented in this article. By applying ultra short pulses, dissolution of a workpiece can be restricted to the region very close to... Electrochemical micromachining (EMM) technology for fabricating micro structures is presented in this article. By applying ultra short pulses, dissolution of a workpiece can be restricted to the region very close to the electrode. First, an EMM system for meeting the requirements of the EMM process is established. Second, sets of experiments is carried out to investigate the influence of some of the predominant electrochemical process parameters such as electrical parameters, feed rate, electrode geometry features and electrolyte composition on machining quality, especially the influences of pulse on time on shape precision and working end shape of electrode on machined surface quality. Finally, after the preliminary experiments, a complex microstructure with good shape precision and surface quality is successfully obtained. 展开更多
关键词 ELECTROCHEMICAL MICROMACHINING ultra short pulse working end shape surface quality complex structure
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