Where Is Phase Velocity in Minkowski Space?

In the special theory of relativity, massive particles can travel at neither the speed of light c nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The idea could be a mathematical device or physical simplification. By contrast, the preceding notion of wave-group duality has two velocities: a group velocity vg and a phase velocity vp. In light vp = vg = c;but it follows from special relativity that, in massive particles, vp > c. The phase velocity is the product of the two best measured variables, and so their product constitutes internal motion that travels, verifiably, faster than light. How does vp then appear in Minkowski space? For light, the spatio-temporal Lorentz invariant metric is s2=c2t2−x2−y2−z2, the same in whatever frame it is viewed. The space is divided into 3 parts: firstly a cone, symmetric about the vertical axis ct > 0 that represents the world line of a stationary particle while the conical surface at s = 0 represents the locus for light rays that travel at the speed of light c. Since no real thing travels faster than the speed of light c, the surface is also a horizon for what can be seen by an observer starting from the origin at time t = 0. Secondly, an inverted cone represents, equivalently, time past. Thirdly, outside the cones, inaccessible space. The phase velocity vp, group velocity vg and speed of light are all equal in free space, vp = vg = c, constant. By contrast, for particles, where causality is due to particle interactions having rest mass mo > 0, we have to employ the Klein-Gordon equation with s2=c2t2−x2−y2−z2+mo2c2. Now special relativity requires a complication: vp.vg = c2 where vg c and therefore vp > c. In the volume outside the cones, causality due to light interactions cannot extend beyond the cones. However, since vp > c and even vp >> c when wavelength λ is long, extreme phase velocities are then limited in their causal effects by the particle uncertainty σ, i.e. to vgt ± σ/ω, where ω is the particle angular frequency. This is the first time the phase range has been described for a massive particle.

Study on the Superluminal Group Velocity in a Coaxial Photonic Crystal

In this paper, the superluminal group velocity in a coaxial photonic crystal is studied. The simulation of the effective refraction index in coaxial photonic crystal is performed. The group velocity is calculated based on the transmission line equations and compared with experimental results.

G-type dispersion equation under suppressed rigid boundary:analytic approach

This paper studies dispersion of a G-type earthquake wave under the influence of a suppressed rigid boundary. Inside the Earth, the density and rigidity of the crustal layer and the mantle of the Earth vary exponentially and periodically along the depth. The displacements of the wave are found in the individual medium followed by a dispersion equation using a suitable analytic approach and a boundary condition. The prominent effect of inhomogeneity contained in the media, the rigid boundary plane, and the initial stress on the phase and group velocities is shown graphically.

Studies on phase and group velocities from acoustic logging

It is still argued whether we measure phase or group velocities using acoustic logging tools. In this paper, three kinds of models are used to investigate this problem by theoretical analyses and numerical simulations. First, we use the plane-wave superposition model containing two plane waves with different velocities and able to change the values of phase velocity and group velocity. The numerical results show that whether phase velocity is higher or lower than group velocity, using the slowness-time coherence （STC） method we can only get phase velocities. Second, according to the results of the dispersion analysis and branch-cut integration, in a rigid boundary borehole model the results of dispersion curves and the waveforms of the first-order mode show that the velocities obtained by the STC method are phase velocities while group velocities obtained by arrival time picking. Finally, dipole logging in a slow formation model is investigated using dispersion analysis and real-axis integration. The results of dispersion curves and full wave trains show similar conclusions as the borehole model with rigid boundary conditions.

Design and analysis of a wideband photonic crystal waveguide with low group-velocity and low dispersion

The slow light propagation in a line waveguide in the two-dimensional triangular photonic crystal has been numerically studied, based on which a wideband photonic crystal waveguide with low group-velocity and low dispersion is proposed. The numerical simulation analysis shows that it is possible to maximize the group index and minimize the group-velocity dispersion in wide bandwidth by increasing the radius of the basic air hole and changing the position of the first two rows of air holes in photonic crystal waveguides. Such a photonic crystal waveguide exhibits low group velocity and low group-velocity dispersion over a broad wavelength range. A larger group index-bandwidth product is achieved in this type of waveguide structure. The numerically computed results present the normalized bandwidth as 0.32%, 0.48% and 0.642% corresponding to the group index of 85, 58 and 45, respectively.

Two Concepts in Optics of Anisotropic Dispersive Media and Polariton Case in Coordinate-Invariant Way

Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.

Group velocity dispersion effect on the collapse of femtosecond laser pulses in air: a revised form for Marburger's law

The influence of group velocity dispersion（GVD） on the self-focusing of femtosecond laser pulses is investigated by numerically solving the extended nonlinear Schr？dinger equation. By introducing the GVD length LGVDinto the semi-empirical, self-focusing formula proposed by Marburger, a revised one is proposed, which can not only well explain the influence of GVD on the collapse distance, but also is in good agreement with the numerical results, making the self-focusing formula applicable for more cases.

槽波层析成像方法在煤田勘探中的应用--以河南义马矿区为例

在地下煤田的开发中,工作面内的小构造、异常体、煤层厚度变化等是需要解决的关键问题,而槽波探测则为这些问题的解决提供了重要的物探方法.本文对河南义马矿区11061工作面进行槽波透射法测量,在巷道显示的煤层厚度变化为1.5～8m,从理论频散曲线分析速度与厚度关系,确定125Hz频率槽波主要用于观测厚度约为2～5m的煤层厚度变化;有效提取了684个频散曲线,并分别拾取了125Hz时槽波群速度与走时,采用走时层析成像方法获得工作面内煤层速度、厚度以及高应力区分布特征,回采验证了结果的正确性.

中国东部海域地壳-上地幔瑞利波速度结构研究

为了进一步了解中国东部沿海及相邻海域的地壳－上地幔结构特征，对该区域的构造演化历史、地震活动及深部构造等方面研究提供一些基础资料，利用３１个数字地震台记录的高质量瑞利波资料，采用一种新的混合路径频散的网格反演方法（Ｏｃｃａｍ方法），对中国东部海域瑞利波群速度横向不均匀分布进行了初步研究．根据反演得到的１０—１５０ｓ共３６个中心周期的群速度分布特征，以及几个典型地点的剪切波速度结构的深度变化，对研究区域内各构造单元的划分以及它们在速度结构和上地幔低速层埋深等方面的特征进行了讨论。

透射边界高频耦合失稳机理及稳定实现--SH波动

就大型近场波动的高效数值模拟而言,稳定实现高阶人工边界是一个尚未圆满解决的问题.本文针对使用多次透射公式的SH波动集中质量有限元模拟,依据GKS定理的群速度解释,进一步阐明了人工边界与内域离散格式耦合所导致高频失稳的机理,即两者支持群速度指向内域的外行高频平面谐波,波动能量自发地从人工边界进入內域,从而导致失稳,而这类谐波是由集中质量有限元离散引入的.本文提出了消除此种耦合失稳的一种方法:通过修改有限元刚度阵来改变内域离散格式,并保证修改格式的精度不低于原有格式的精度.理论分析和数值实验表明此法能稳定实现透射边界.本文研究结果具有推广应用前景.

余弦-高斯激光脉冲在单模光纤中的色散效应

首次提出了余弦-高斯激光脉冲,并证明了该脉冲可由两个相位相反的线性啁啾的高斯激光脉冲叠加得到。由NLS方程出发,对余弦-高斯激光脉冲在色散介质中的色散效应进行了详细的研究。研究结果表明,初始啁啾并不影响入射场的光强分布;当脉冲参数a=1.13时,余弦-高斯激光脉冲的初始脉宽有最小值,且比高斯激光脉冲的初始脉宽要窄;余弦-高斯激光脉冲的展宽因子与传输距离、色散长度LD、激光脉冲参数a、初始啁啾参量C和β2的符号有关,余弦-高斯激光脉冲都比高斯激光脉冲的展宽要快,且参数a越大展宽就越明显。

基于光纤中四波混频效应的全光串-并转换研究

为了实现高速信号的降速处理,设计并通过实验演示了一种高速全光串-并转换系统。在方案中,利用时钟脉冲自身的频谱宽度和光纤中四波混频的高速响应特性,可以从一个光分频时钟脉冲出发,利用光纤中群速度色散(GVD)致脉冲展宽效应,把一个重复频率为10 GHz的时钟窄脉冲在时域上展宽,并通过光纤中的四波混频过程,将一路40 Gb/s的归零(RZ)码信号转换成为4路10 Gb/s信号,完成串-并转换功能。该方案响应速率高,对波长和码率透明,并具有很大的转换路数可拓展性。

Analysis of high-frequency local coupling instability induced by multi- transmitting formula - P-SV wave simulation in a 2D waveguide

Local coupling instability will occur when the numerical scheme of absorbing boundary condition and that of the field wave equation allow energies to spontaneously enter into the computational domain. That is, the two schemes support common wave solutions with group velocity pointed into the computation domain. The key to eliminate local coupling instability is to avoid such wave solutions. For lumped-mass finite element simulation of P-SV wave motion in a 2D waveguide, an approach for stable implementation of high order multi-transmitting formula is provided. With a uniform rectangular mesh, it is proven and validated that high-freqaency local coupling instability can be eliminated by setting the ratio of the element size equal to or greater than x/2 times the ratio of the P wave velocity to the S wave velocity. These results can be valuable for dealing instability problems induced by other absorbing boundary conditions.

Simulations of Effect of High-Index Materials on Highly Birefringent Photonic Crystal Fibres

Novel highly birefringent photonic bandgap fibres （PBGFs） are obtained by filling of a high index material in the air holes of total internal reflection birefringent photonic crystal fibres. The effect of the filling high index material on the transmission characteristics has been theoretically investigated. The photonic bandgap has been achieved by using plane-wave method. Moreover, the phase and group modal birefringence have been studied by a full-vector finite-element method. Numerical results show that very high group and phase modal birefringence with magnitude of order of 10^-2 and 10^-3 has been respectively acquired, which is much higher than those of the non-filled fibres. Furthermore, strong coupling between surface modes and the fundamental modes has been found in the bandgap of the birefringent PBGFs, whose effect on the birefringence and confinement loss has also been discussed.

Influence of Initial Chirp on Propagation of Super-Gaussian Pulse inside Fiber

Under the condition of combined effects of group--velocitydispersion and self- phase modulation, the step Fourier method isused to simulate the propagation of initial chirped super-Gaussianpulses inside fiber. The initial chirp influences the shapes of superGaussian pulses in propagation process, and positive and negativechirps have different effects. For the existing of initial chirp, thesplits of pulses and the spreading speed move ahead and increase.When the amplitude of super-Gaussian pulses increases by 1.4 times,in the range of │C│<1.5, pulses can keep good shapes along theirpropagation distance.

Theoretical Calculation of Ultrashort-Pulse Optical Parametric Generation

A general theory of optical parametric generation that accounts for pump depletion, loss, phase mismatch, group-velocity mismatch among the pump, signal and idler pulses, and intrapulse group-velocity dispersion is proposed for coherent ultrashort pulses with arbitrary shapes and carrier chirps. The coupled differential equations are numerically solved using a symmetric split step beam-propagation method. The general solutions of these equations are obtained and the optical parametric generation process is theoretically investigated. Results show that the major factors, which remarkably affect the optical parametric conversion efficiency and durations of the pulses in phase-matched structure, are the group velocity mismatch and the intrapulse group velocity dispersion.

Determination of the Crust and Upper-Mantle Structure in Anatolia by Surface Wave Data

In this study, the crust and upper mantle structure of Anatolia have been investigated by measuring the group velocity dispersion data of discriminated seismic surface waves. In the scope of the study, it has selected the profiles between six stations located in western Anatolia of Bogazici University Kandilli Observatory Earthquake Research Institute, national network of Turkey, and records of an earthquake (having about 10&#176;epicentral distance) occurred in the eastern of Anatolia have been used. Firstly, surface wave discrimination filter based on the polarization properties has been applied tothree-component recordsand emphasized to surface waves. Then the group velocities have been calculated by multiple filter technique. A five-layered crustal model having total thickness of 38 - 40 km and Pn-wave velocity of 8.00 km/sec in the upper-mantle has been determined through inversion of surface wave group velocity dispersion data in the period range of 10 sec to 60 sec.

玻色-爱因斯坦凝聚体的光学色散关系

最近的研究表明,玻色-爱因斯坦凝聚体(Bose-Einstein Condensate,BEC)可作为量子电介质材料对光场产生反作用,实现光场-物质波的协同操控.然而BEC的色散性质还没有被研究.为此,解析得到了BEC对大失谐光的一阶色散和二阶色散的计算公式.数值计算表明,BEC的折射率以及二阶色散系数与红、蓝失谐的性质有关:在红失谐时,折射率大于1,且二阶色散是正常色散;在蓝失谐时,折射率小于1,二阶色散为反常色散.二阶色散系数会随着失谐量的改变而剧烈变化,当失谐量在GHz数量级时,表现为强色散介质.一阶色散和红、蓝失谐的性质关系不大,随着失谐量的增加,一阶色散减小,相应的群速度增加.因此,对于超短脉冲光,BEC是一种新型的色散介质.

Induced Diffraction in Phase-Mismatched Second-Harmonic Generation

We show analytically that in phase-mismatched second-harmonic generation, an effective diffraction is induced at the second-harmonic （SH） frequency. Numerical simulation results agree with the analytical predictions. Compared to the case of linear propagation, the effect of the overall diffraction at the SH frequency becomes doubled due to the induced diffraction, which causes an interesting result that the SH beam width will be larger than that of the fundamental field.

利用啁啾飞秒激光脉冲调控金薄膜中传输表面等离激元的群延迟色散

准确评估与灵活调控表面等离激元(surface plasmon polaritons,SPP)的传输特性对基于等离激元效应设计的高速-小型化器件具有重要意义.本文从理论上推导了SPP在不同厚度金膜表面传播的群速度色散.当金膜厚度小于40 nm时,随着膜厚度的增大,SPP的群速度色散显著减小;金膜厚度为40-60 nm时,SPP的群速度色散下降趋势变缓,并在厚度大于60 nm后保持恒定.利用时域有限差分方法,数值模拟了不同传播距离下SPP的电场时间演化.通过比较不同传输距离下SPP近场强度与入射光色散量的关系确定SPP的群速度色散,并得到接近理论推导的结果,进而提出利用负啁啾脉冲激发SPP来补偿传输中产生的群延迟色散,实现不同传输距离SPP场振幅、脉宽的调控.此外,利用定制的SPP激发金属纳米天线,通过调控入射脉冲色散量与SPP在金膜中传播产生群延迟色散的传播特性相结合的手段,实现在纳米信号接收器中热点位置的fs时间灵活时空相干控制.这对于利用等离激元效应设计和控制微型片上集成系统具有重要意义.