The factor of nonparaxial Hermite–Gaussian beams and related problems

On the basis of the second-order moment of the power density and in the use of the series expansion, the expressions for the beam width, far-field divergence angle and M² factor of nonparaxial Hermite–Gaussian (H–G) beams are derived and expressed in a sum of the series of the Gamma function. The theoretical results are illustrated with numerical examples. The M² factor of nonparaxial H–G beams depends not only on the beam order m, but also on the waist-width to wavelength ratio w₀/λ. The far-field divergence angles of nonparaxial H–G beams with even and odd orders approach their upper limits θ_max=63.435 and 73.898, respectively, which results in M²<1 as w₀/λ→0. For the special case of m=0 our results reduce to those of nonparaxial Gaussian beams. Some problems related to the characterization of the nonparaxial beam quality are also discussed.     

高斯涡旋光束的光束传输因子和峭度参数

基于强度二阶矩定义, 导出了高斯涡旋光束光束传输因子即M² 因子的解析表达式, 高斯涡旋光束的M² 因子唯一取决于拓扑电荷数n. 数值计算表明, 高斯涡旋光束的M² 因子随着拓扑电荷数n的增大而增大. 基于强度高阶矩, 还导出了高斯涡旋光束经傍轴ABCD光学系统传输时峭度参数的解析表达式, 高斯涡旋光束的峭度参数取决于拓扑电荷数n、参数δ、矩阵元A和矩阵元D. 在自由空间传输时, 高斯涡旋光束的峭度参数仅取决于拓扑电荷数n和参数δ. 自由空间传输时, 高斯涡旋光束峭度参数的变化规律为: 峭度参数随参数δ的增大先减小而后趋向于一最小值, 随拓扑电荷数n的增大而减小. 这一研究有助于高斯涡旋光束的实际应用.     

Measurement of beam quality factor (M) by slit-scanning method

Beam quality factor (M²) and far-field scattering angle of LD end-pump Nd : YVO₄ laser were measured by slit-scanning method. The experimental results showed that the laser operated on a multi-mode state. The corresponding analytical treatments for slit-scanning method and M² factor measurement also were presented in this paper.     

厄米-高斯光束的M2因子矩阵

提出了厄米-高斯光场的M²因子矩阵.引入束半宽平方的交叉项、M²因子的交叉项,理论推导出了在同一坐标系下光场旋转一定角度后的M²因子矩阵,数值模拟了与M²因子矩阵有关的各参数随光场旋转角度变化的规律,给出了光场的M²因子矢量点随光场旋转角度变化的轨迹曲线.计算结果与理论推导结果相符,证实了利用M²因子矩阵可以将旋转前后的二维厄米-高斯光场用旋转矩阵统一起来.该方法可推广到对一般的二维高阶高斯光束的光束质量的理论分析上,具有普适性,对光束质量的实际测量有重要的理论指导意义. 关键词： M²因子矩阵')" href="#">M²因子矩阵 厄米-高斯光束 非对称激光束 矩阵光学     

Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams

The analytical expression for the beam propagation factor (M²-factor) of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the M²-factor of GSM array beams depends on the beam number, the relative beam separation distance, the beam coherence parameter, the type of beam superposition, and the strength of turbulence. The turbulence results in an increase of the M²-factor. However, for the superposition of the intensity the M²-factor is less sensitive to turbulence than that for the superposition of the cross-spectral density function. The M²-factor of GSM array beams is larger than that of the corresponding Gaussian array beams. However, the M²-factor of GSM array beams is less affected by turbulence than that of the corresponding Gaussian array beams. For the superposition of the cross-spectral density function a minimum of the M²-factor of GSM array beams may appear in turbulence, which is even smaller than that of the corresponding single GSM beams.     

Beam quality of radial Gaussian Schell-model array beams

Taking the Rayleigh range z_R and the M²-factor as the characteristic parameters of beam quality, the beam quality of radial Gaussian Schell-model (GSM) array beams is studied. The analytical expressions for the z_R and the M²-factor of radial GSM array beams are derived. It is shown that for the superposition of the cross-spectral density function z_R is longer and the M²-factor is lower than that for the superposition of the intensity. For the two types of superposition, z_R increases and the M²-factor decreases with increase in beam coherence parameter, and both z_R and the M²-factor increase with increase in inverse radial fill-factor. For the superposition of the cross-spectral density function, z_R increases and the M²-factor decreases with increase in beam number, while for the superposition of the intensity both the z_R and M²-factor are independent of the beam number.     

矢量非傍轴厄米-拉盖尔-高斯光束的光束质量

基于矢量Rayleigh-Sommerfeld衍射积分，推导出矢量非傍轴圆偏振厄米-拉盖尔-高斯(HLG)光束的远场解析表达式.矢量非傍轴圆偏振高斯光束可作为一般公式的特例给出.将桶中功率(PIB)概念推广到非傍轴范畴，用以描述矢量非傍轴光束的远场光束质量，其中光强用时间平均坡印廷矢量的z分量取代.数值计算和分析表明，矢量非傍轴HLG光束的PIB不仅与束腰宽度与波长之比w₀/λ有关，而且还与α参数，模指数n和m以及所取桶的尺寸有关. 关键词： 矢量非傍轴厄米-拉盖尔-高斯光束 矢量Rayleigh-Sommerfeld衍射积分 桶中功率     

The beam propagation factor and far-field distribution of Bessel-modulated Gaussian beams

The explicit expressions for the beam propagation factor (M ² factor) and far-field distribution of Bessel-Modulated Gaussian beams with quadratic radial dependence are derived. The results are analyzed with numerical example.     

Effects of the deformation of reflection volume Bragg gratings on the M-factor of super-Gaussian laser beams

When high-power laser beams pass through a volume Bragg grating (VBG), the surface of VBG occurs distortion because of thermal effects, and it results in the decline of the grating’s performance. Considering distortion of the surface induces differences of the index and period, the scalar wave equations for super-Gaussian (SG) laser beams propagating in the reflection VBG have been solved by using finite-different and sparse matrix methods. The changes of the total power reflection coefficient, the beam width, the far-field beam angle and the M²-factor for the laser beams through the reflection VBG with deformation have been analyzed quantitatively. It can be shown that the deformation of the VBG affects evidently on the intensity and phase distributions, the total power reflection coefficient, the beam width, the far-field divergence angle and the M²-factor. The beam width, the far-field divergence angle, and the M²-factor of the output beams increase significantly with the distortions of the grating. The influence of the distortions of the VBG on the power reflection coefficient, the far-field divergence angle and the M²-factor of the output beams are smaller for higher beam orders.     

An alternative approach to determine the spot-size of a multi-mode laser beam and its application to diode laser beams

An alternative approach is suggested to determine the spot-size of a multi-mode laser beam. It has been shown by simulations that the suggested approach can give the beam quality factor and characteristic radius with less than 5% error. Unlike the power content method, the proposed method is applicable to the beams even with diameter one tenth of the CCD size. The new approach has been applied to a multi-mode diode laser output and it is shown that the ABCD matrix analysis can be used for beam propagation, with the measured parameters of the laser.     

Does the beam width affect the generalized M factor of hard-edged diffracted beams?

A detailed study of the generalized M² factor of hard-edged diffracted beams based on the truncated second-order moments method, asymptotic analysis and self-convergent beam width approach is performed. The dependence of the generalized M² factor on the parameters characterizing the spatial profile, and beam truncation, etc. is analyzed.     

Propagation characteristics of the Yb-doped fiber lasers after spectral beam combining by the VBGs

The propagation model for the spectral beam combining system of five Yb-doped fiber laser beams by the volume Bragg gratings (VBGs) has been built up. Considering the surface distortion by thermal effect, the intensity of an individual beam passing through the spectral beam combining system by the VBGs with deformation has been analyzed numerically and iteratively by using finite-different and sparse matrix methods, and the intensity of combined beam has been studied by the principle of incoherent superposition. Consequently, according to the intensity second-order moments method, the M²-factor of the combined beam has been calculated. The effects of deformation of the VBGs on the intensity, the combining efficiency, the M²-factor and the power in the bucket of the combined beam have been analyzed quantitatively. It can be shown that the effects of the deformation of the VBGs on the intensity distribution of combined beam become more and more obvious with the increasing of the deformation of the VBGs. The results show that the surface distortion of the VBGs due to high-power lasers dramatically degrades both the beam quality of the combined beam and the combining efficiency of the spectral beam combining system.     

Far-field intensity distribution, M factor of beams generated by Gaussian mirror resonator

We investigated M² factor and far-field distribution of beams generated by Gaussian mirror resonator. And we found usable analytical expressions of the M² factor and the far-field distribution intensity with respect to variation of diffraction parameters. Particular attention was paid to the parameters such as mirror spot size and reflectance of the Gaussian mirror.     

Effect of absorbed pump power on the quality of output beam from monolithic microchip lasers

The dependence of the beam propagation factor (M ² parameter) with the absorbed pump power in the case of monolithic microchip laser under face-cooled configuration is extensively studied. Our investigations show that the M ₂ parameter is related to the absorbed pump power through two parameters (α and β) whose values depend on the laser material properties and laser configuration. We have shown that one parameter arises due to the oscillation of higher order modes in the microchip cavity and the other parameter accounts for the spherical aberration associated with the thermal lens induced by the pump beam. Such dependency of M ² parameter with the absorbed pump power is experimentally verified for a face-cooled monolithic microchip laser based on Nd³⁺ -doped GdVO₄ crystal and the values of α and β parameters were estimated from the experimentally measured data points.     

单口径相干合成系统激光光束的M2因子研究

单口径相干合成系统激光光束的光束质量是一个亟待解决的重要问题.基于二阶矩定义, 文中给出了单口径TEM₀₀, TEM₀₁及TEM₁₀两两相干光束M²因子的解析表达式, 并比较分析了束腰宽度、传输距离、振幅之比,以及源场位置矢量对相干光束M²因子的影响, 得到了诸如源场位置参量d₁<100λ时,各相干光束M²因子恒定,反之, 其随位置参量d₁的增大而增大等一些结论.最后,文章对两TEM₀₀模相干光束M²因子的 部分理论进行了实验验证.     

柱坐标系中截断光束的广义M2因子

推导出截断圆对称光束二阶矩的传输方程，它们与无截断光束的二阶矩满足相似的传输定律.因此，基于广义截断二阶矩方法，将直角坐标系中截断二维光束的二阶矩特征参数推广到柱坐标系中的截断圆对称光束，类似方法定义的广义M²因子（M²_G因子）是一个传输不变量.对理论的自洽性作了物理解释.推导出柱坐标系中截断超高斯光束的二阶矩参数和M²_G因子.对一些有意义的特殊情况作了讨论，并以数值计算例作了说明. 关键词： 光束描述 截断光束 圆对称性 广义截断二阶矩(GTSM) 2因子（M²_G因子）')" href="#">广义M²因子（M²_G因子）     

截断部分相干厄米-高斯光束的广义M2G因子

基于广义截断二阶矩法,推导出了截断部分相干厄米-高斯(H-G)光束的广义M²_G因子的解析表达式. 截断完全相干H-G光束、截断高斯谢尔模型(GSM)光束以及截断高斯光束可以做为本文结果的特例给出. 研究表明:截断部分相干H-G光束的广义M²_G因子与截断参数δ,模阶数m以及相干参数α有关. 当δ非常小时,M^{2关键词： 截断部分相干厄米高斯光束 M2_G因子')" href="#">广义M2_G因子 广义截断二阶矩法}     

The M factor of partially coherent ultrashort pulsed beams

Starting from the space-time Wigner distribution function and taking the Gaussian Schell-model pulsed beam as a typical example, the M² factor of partially coherent ultrashort pulsed beams is studied. It is shown that the M² factor increases with increasing bandwidth and decreases with increasing spatial correlation. Furthermore, for chirped pulse, the M² factor increases as the chirp parameter increases.     

非傍轴矢量高斯光束的传输

运用非傍轴光束传输的矢量矩理论，对非傍轴矢量高斯光束的传输特性进行了系统的研究.结果表明，基于二阶矩定义的横向光束宽度在光束传播过程中满足简单的双曲线变化规律，并且给出了光束传输因子的解析表达式.就高度非傍轴情形，进一步给出了简洁的计算公式，在高斯光源线度趋向零的极限情形下，横向的最大发散角为90°.同时，还推广到了傍轴情形，得到了与原有傍轴公式稍有区别的结果，而且光束传输因子始终保持略大于1最后，对非傍轴矢量高斯光束和非傍轴标量高斯光束的传输进行了比较，结果显示对于线度在两个波长范围之内的高斯光源发散 关键词： 矢量高斯光束 光束传输 非傍轴 二阶矩