数值求解三维含时Schrodinger方程及其应用

发展了一套高精度、高效率的伪谱方法,以非微扰的方式求解真实原子三维含时Schrodinger方程.该方法选用二阶劈裂算符作为时间演化算子,分别选择能谱表象和坐标表象作为含时波函数演化的两个表象.在坐标表象下波函数的径向部分使用库仑波函数离散变量表象来离散;角向波函数展开在两维的Gauss-Legendre-Fourier格点上.以H原子的光激发和光电离过程为例,进行了数值计算并和解析解进行了比对.结果表明二者符合很好.该方法很好地处理了库仑奇点问题.还计算了强激光辐照H原子的多光子电离过程,并和其他的数值方案进行了比较.结果表明,在计算收敛的前提下本方法计算效率更高.

Abstract：

We present an accurate and effective pseudospectral method for solving the three-dimensional time-dependent Schrodinger equation involving the Coulomb potential. In this method, the Hamiltonian is evaluated by exploiting the two representations of the wave function. One is a grid representation, in which the angular dependence of the wave function is expanded in a two-dimensional Gauss-Legendre-Fourier grid in the coordinate space of polar and azimuthal angles. The radial coordinate is discretized using a discrete variable representation constructed from the Coulomb wave function (CWDVR) . The other is a spectral representation, in which the wave function is expanded in a set of square integrable functions chosen as the eigenfunctions of a zero-order Hamiltonian. The time of propagation of the wave function is calculated using the well-known second-order split-operator method implemented through the transform between the grid and spectral representations. Calculations on the photo-absorption strength of hydrogen atom are presented to demonstrate the accuracy of present method in low energy limit by the time-dependent wave-packet propagation method. As another example, the present method is applied to multiphoton ionization of H atom. For a wide range of field parameters, ionization rates calculated using the present method are in excellent agreement with those from other accurate numerical calculations. The new algorithm will be found more efficient than the close coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.     

数值求解三维含时Schr?dinger方程及其应用

发展了一套高精度、高效率的伪谱方法,以非微扰的方式求解真实原子三维含时Schrdinger方程.该方法选用二阶劈裂算符作为时间演化算子,分别选择能谱表象和坐标表象作为含时波函数演化的两个表象.在坐标表象下波函数的径向部分使用库仑波函数离散变量表象来离散;角向波函数展开在两维的Gauss-Legendre-Fourier格点上.以H原子的光激发和光电离过程为例,进行了数值计算并和解析解进行了比对.结果表明二者符合很好.该方法很好地处理了库仑奇点问题.还计算了强激光辐照H原子的多光子电离过程,并和其他的数值方案进行了比较.结果表明,在计算收敛的前提下本方法计算效率更高. 关键词： 三维含时Schrdinger方程 库仑奇点 强场 含时波包传播     

Laser pulse transient method for measuring the normal spectral emissivity of samples with arbitrary surface quality

A laser pulse transient method for measuring normal spectral emissivity is described. In this method, a laser pulse (λ=1064 nm) irradiates the top surface of a flat specimen. A two-dimensional temperature response of the bottom surface is measured with a calibrated thermographic camera. By solving an axisymmetric boundary value heat conduction problem, the normal spectral emissivity at 1064 nm is determined by using an iterative nonlinear least-squares estimation procedure. The method can be applied to arbitrary sample surface quality. The method is tested on a nickel specimen and used to determine the normal spectral emissivity of AISI 304 stainless steel. The expanded combined uncertainty of the method has been estimated to be 18%.     

基于时变功率谱的激光器调谐瞬时线宽测量方法研究

半导体激光器的线宽通常采用激光外差测量技术,通过差拍信号的功率谱密度函数来确定,受傅里叶变换方法的限制,得到的均是在一定时间段内的静态平均线宽。为了获得半导体激光器在电流调谐过程中的瞬时线宽特性,提出了利用时变功率谱获知调谐瞬时线宽的相干和非相干测量方法,并分别进行了理论分析和实验验证。首先对半导体激光器输出光信号及差拍信号进行了时间-频率域下的数学描述,确定了时变功率谱与调谐瞬时线宽的关系;其次,针对差拍信号的趋向性特征,提出了趋势局部均值分解方法,并研究了利用分解出的乘积函数建立差拍信号及激光器输出光信号的时变功率谱的方法;最后利用非相干和相干测量法分别获得了分布反馈式半导体激光器在50~51及50~100 mA锯齿波电流调谐过程中的瞬时线宽。     

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieves asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.     

Determination of the amplitude and the phase of the elements of electric cross-spectral density matrix by spectral measurements

The amplitudes and the phases of the elements of electric cross-spectral density matrix are determined experimentally for a pair of points in the cross-section of an expanded laser beam. A modified version of the Young’s interferometer is used as an experimental tool, which separates the beams emerging from the double-slit widely and provides ease in insertion of polarizers and half wave rotators in individual beams. To determine these complex elements of the cross-spectral density matrix, the experimentally obtained values of the spectral densities at an off-axis point are put in the mathematical expressions derived by us using the spectral interference law. The four complex generalized Stokes parameters are also determined using the linear combinations of the matrix elements. This unique but simple experimental approach for determining both the two-point parameters might provide a means to investigate the polarization and the coherence properties of the random electromagnetic beams on propagation.     

On the possibility of high-energy neutron diffraction in a crystal in the field of laser radiation

For the first time the high-energy neutrons scattering in a crystal is considered under the influence of an external laser wave field. The process is inelastic with respect to the laser wave and at the same time it is elastic with respect to a crystal. The possibility of high-energy neutron diffraction is illustrated—wavelength of neutrons is less than the lattice period. The method is based on the multiphoton interaction of the neutron anomalous magnetic moment with the field of laser radiation by using the Farri representation. The neutron-phonon interaction is considered to be a perturbation described by the Fermi pseudopotential.     

Fresnel integral in the complex form: Its new representation,interpretation, and application in problems of wave beam diffraction

A new representation for the complex Fresnel integral, allowing for its transformation into the trigonometric form, and its new interpretation based on the 2D Cornu spiral and a 3D spiral with a smooth envelope surface have been obtained. Within this representation the integral is separated into a constant part and a rapidly oscillating additive. The former is responsible for the geometrical-optical field transport, and the latter implements diffraction. This separation is exact. Based on this approach, we have proposed and implemented a new method for solving the problems of wave beam propagation in the near-field zone on a large-scale grid at large optical distortions of the wavefront, with a transport of the initial-wave amplitude and phase. The amplitude is approximated by a step or linear function, and the phase is approximated taking into account the local linear tilt and focusing of the wavefront. The physical meaning of the solutions obtained is established. The approach proposed allows one to perform regular calculations of wave field propagation in the near-field zone on a large-scale grid (up to ∼10 points within aperture) at large initial wavefront aberrations (up to ∼100λ or more).     

Kinetics of the Raman instability in laser plasma

The phase-space evolution in a non-relativistic and homogeneous laser plasma in the presence of the stimulated Raman scattering is studied. Transform method is used for a solution of the set of partial differential equations which consists of the Vlasov equation and of the full set of Maxwell equations in a 1D model. Numerical instability of the Fourier-Hermite representation is described and discussed. To overcome numerical instabilities during the simulation, a simplified Fokker-Planck collision term is employed. In the collisionless case the solution is pushed to the practicable limit and the initial phase of particle trapping and acceleration in the potential wells of the electrostatic wave accompanying the Raman backscattered wave was recorded. Also the growth of the electrostatic partner of the Raman forward scattered wave was observed.     

The thermoelastic excitation of air-solid interface waves using the pulsed laser

In the 1920s, the solid-solid interface wave, Stoneley wave, was first studied by Stoneley. From the 1930s to 1940s, the fluid-solid interface waves, usually called Scholte wave or Scholte-Stoneley wave, were studied by Cagniard and Scholte respec-tively[1]. The Scholte wave corresponds to the real root of the fluid-solid interface secu-lar equation, which is usually called the Scholte equation, and the velocity of Scholte wave is only slightly lower than the longitudinal velocity of the f…     

激光场中类氢原子的电离本征值

光场下类氢原子的Schrdinger方程可用缀饰势方法求解.波动方程展开为Floquet分波后,可以得到弱光场或强光场下近似的径向波函数和复的电离本征值,然后计算了共振能量和半宽度.     

关于层子模型中的电磁形式因子与结构波函数的积分表示

虽然从B-S方程出发,用谐振子位势来讨论介子的质谱,可以得到较好的结果,但是当用这样得到的近似波函数来计算电磁形式因子时,却给出形式因子对类空的q²为复数。本文指出,出现上述不合理结果的原因,在于所得的近似波函数不具有正确的对P₀的解析性。为了保证这种解析性同时又使波函数保持协变形式,比较适当的方法是,利用戴逊证明的定理将波函数写成积分表示。另外,根据波函数在x=0点为有限的物理要求,我们还给出了积分表示中的谱函数所应满足的一些求和条件。     

Broadband (1000 cm−1) multiplex CARS spectroscopy: Application to polarization sensitive and time-resolved measurements

A multiplex CARS spectrometer based on a cw diode-pumpedQ-switched Nd: YLF laser, a broadband dye-laser and a multichannel spectrum detection system has been constructed. Excellent mode characteristics of the laser beams and high pulse repetition rate (2 kHz) have resulted in good signal-to-noise ratio achieved with a few seconds accumulation time. A 1000 cm^–1 wide spectral range is covered in a single CARS spectrum with an expanded bandwidth of the dye laser. A thin-jet sampling method is used in order to avoid the phase-matching limitation. The efficient spectral intensity normalization by the reference (CCl₄) nonresonant spectrum and subsequent computer fitting have been implemented. The performance of the system is demonstrated by two different experiments. First, the polarization sensitive measurements (PS-CARS) of cyclohexane show its potential for accurate Raman depolarization ratio determination and for detection of weak (overlapped) Raman bands. Second, the transient resonance CARS measurement of the lowest excited triplet state of all-trans retinal indicate its feasibility to time-resolved CARS spectroscopy of fluorescent excited states.     

Dynamics of atoms in strong laser fields I: A quasi analytical model in momentum space based on a Sturmian expansion of the interacting nonlocal Coulomb potential

In this study we present a model that we have formulated in the momentum space to describe atoms interacting with intense laser fields. As a further step, it follows our recent theoretical approach in which the kernel of the reciprocal-space time-dependent Schrödinger equation (TDSE) is replaced by a finite sum of separable potentials, each of them supporting one bound state of atomic hydrogen (Tetchou Nganso et al. 2013). The key point of the model is that the nonlocal interacting Coulomb potential is expanded in a Coulomb Sturmian basis set derived itself from a Sturmian representation of Bessel functions of the first kind in the position space. As a result, this decomposition allows a simple spectral treatment of the TDSE in the momentum space. In order to illustrate the credibility of the model, we have considered the test case of atomic hydrogen driven by a linearly polarized laser pulse, and have evaluated analytically matrix elements of the atomic Hamiltonian and dipole coupling interaction. For various regimes of the laser parameters used in computations our results are in very good agreement with data obtained from other time-dependent calculations.     

Galerkin时空耦合谱元法求解声波动方程

提出了时空耦合谱元方法,并将其用于带第一类边界条件的非齐次一维、二维、三维波动方程的求解。分别采用四边形、六面体和超六面体作为计算单元,在每个单元内采用Chebyshev多项式的极值点作为Lagrange插值节点,并且探讨了区域剖分方式对计算精度的影响。时空耦合谱元法能够得到精度很高的数值结果,并且其色散随时间推移是稳定的;当总网格节点数相同时,不同的网格剖分方式所得数值误差不同,当空间方向Chebyshev多项式的阶数较高和时间方向Chebyshev多项式的阶数较低时,得到的数值精度较高;在总节点数相同的情况下,与时间全域方式相比,逐时间子区域方式计算所需要的时间更经济,两种方式可以得到相同的精度。结果表明:时空耦合谱元方法使时空方向精度相匹配,可以提高整体精度;空间方向的Chebyshev多项式对数值精度起主要影响作用;时间子区域方式的采用可以扩大问题的计算区域。      

Propagation of the ultrashort laser pulses through the quantum nonlinear medium with resonant properties

The propagation of ultrashort few-cycle laser pulses through a quantum nonlinear medium with resonant properties is studied using the method of slowly varying amplitudes with allowance for dispersion effects. A self-consistent system of nonlinear wave equation for the evolution of laser field and the nonstationary Schrödinger equation that determines the material polarization response is numerically solved. Specific features of the propagation of laser pulse caused by a relatively large spectral width and the nonadiabatic character of the laser pulse are established. The rise of the slowly dumping polarization at the eigenfrequency of the medium and the consequent stretching of the laser pulse are observed. The role of the resonant absorption of the laser energy in the medium is analyzed. The nonlinear spectral broadening is demonstrated for the laser pulse propagating through the medium.     

Demonstration of a spatial-spectral holographic LIDAR range-Doppler processor

We present a new approach to laser interferometric Doppler and ranging (LIDAR) processing using spatial-spectral holography (SSH). In this approach, broadband optical signals from a random noise or frequency-modulated laser are transmitted and reflected off remote targets. The return signals interfere spatially and spectrally with a local copy of the original transmit signal in an SSH medium, resulting in spectral gratings that have a spectral period inversely proportional to the LIDAR target's range and a position proportional to the target's Doppler (or velocity). These gratings are subsequently read out by a slowly chirped source onto a parallel detector array, and the velocity and range of the targets are inferred. We present the theoretical framework that describes the function of the LIDAR processor, as well as proof-of-concept experimental results.     

High-power picosecond LiB3O5 optical parametric oscillators tunable in the blue spectral range

The paper reports on an experimental investigation and numerical analysis of noncritically and critically phasematched LiB₃O₅ (LBO) optical parametric oscillators (OPOs) synchronously pumped by the third harmonic of a cw diode-pumped mode-locked Nd:YVO₄ oscillator–amplifier system. The laser system generates 9.0 W of 355-nm mode-locked radiation with a pulse duration of 7.5 ps and a repetition rate of 84 MHz. The LBO OPO, synchronously pumped by the 355-nm pulses, generates a signal wave tunable in the blue spectral range 457–479 nm. With a power of up to 5.0 W at 462 nm and 1.7 W at 1535 nm the conversion efficiency is 74%. The OPO is characterized experimentally by measuring the output power (and its dependence on the pump power, the transmission of the output coupler and the resonator length) and the pulse properties (such as pulse duration and spectral width). Also the beam quality of the resonant and nonresonant waves is investigated. The measured results are compared with the predictions of a numerical analysis for Gaussian laser and OPO beams. In addition to the blue-signal output visible-red 629-nm radiation is generated by sum-frequency mixing of the 1.535-μm infrared idler wave with the residual 1.064-μm laser radiation. A power of 1.25 W of 1.535-μm idler radiation and 5.7 W of 1.064-μm laser light generated a red 629-nm output power of 2.25 W. Received: 2 February 2000 / Revised version: 28 July 2000 / Published online: 22 November 2000     

O(4) Expansion of the Ladder Bethe-Salpeter Equation

The Bethe-Salpeter amplitude is expanded on a hyperspherical basis thereby reducing the original four-dimensional integral equation into an infinite set of coupled one-dimensional ones. It is shown that this representation offers a highly accurate method to determine numerically the bound state solutions. For generic cases only a few hyperspherical waves are needed to achieve convergence, both for the ground state as well as for radially or orbitally excited states. The wave function is reconstructed for several cases and in particular it is shown that it becomes independent of the relative time in the nonrelativistic regime. Received July 18, 1996; accepted in final form October 2, 1996     

Poincaré representation of polarization-shaped femtosecond laser pulses

The usage of Poincaré phase space for the representation of polarization-shaped femtosecond laser pulses is discussed. In these types of light fields the polarization state (i.e. ellipticity and orientation) changes as a function of time within a single laser pulse. Such deliberate variation can be achieved by frequency-domain femtosecond pulse shaping in which two polarization components are manipulated individually. Here it is shown how these light pulses can be represented as temporal trajectories through the ellipticity-orientation (Poincaré) phase space, whereas conventional light (either continuous-wave or pulsed) is determined by only one specific Poincaré location. General properties of parametric Poincaré trajectories are discussed, and their relation to experimentally accessible pulse-manipulation parameters (i.e. amplitudes and phases) determined. Specifically, it is shown how the maximum rate by which a given polarization state can be turned into a different one (at significant intensity levels) is limited by the spectral laser bandwidth. Apart from their direct usage in polarization-shaped pulse representation, Poincaré trajectories also form the basis for intuitive quasi-three-dimensional renderings of the electric field profile. There, the temporal evolution of polarization, intensity, and chirp is directly apparent in a single illustration. Received: 10 December 2002 / Published online: 24 April 2003 RID="*" ID="*"Corresponding author. Fax: +49-931/888-4906, E-mail: brixner@physik.uni-wuerzburg.de