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.     

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.     

Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence

The propagation properties of apertured laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) through atmospheric turbulence are studied in detail both analytically and numerically. The analytical expressions for the average intensity, power in the bucket (PIB) and Strehl ratio (S _R ) of apertured laser beams with AMs and PFs propagating through atmospheric turbulence are derived. It is found that the worse the phase fluctuation and the higher the amplitude modulation are, the less laser beams are affected by turbulence. Furthermore, apertured Gaussian beams are more sensitive to turbulence than apertured laser beams with AMs and PFs. The average intensity of apertured laser beams with AMs and PFs may be even larger than that of apertured Gaussian beams due to turbulence. In particular, the influence of turbulence on the average maximum intensity of apertured laser beams with PFs and AMs may become serious if an unsuitable truncated parameter is chosen, which should be avoided in practice.     

截断的有振幅调制和位相畸变光束的等效曲率半径

高功率激光通常具有振幅调制和相位畸变. 采用统计光学方法推导了截断的有振幅调制和位相畸变光束在大气湍流中传输 的等效曲率半径R的解析公式.研究表明:随着位相畸变参量、振幅调制参量和光 束截断参量的增大,光束在自由空间中的 R增大,但R受湍流的影响也会增大; 并且高斯光束在自由空间中的R最大, 但其受湍流影响也最大. 因此,在大气湍流中传输到足够远时, 截断的有振幅调制和位相畸变光束的R就要比高斯光束的大.特别地, 相对等效曲率半径R_r随传输距离为非单调变化, 存在一个最小值, 即在该位置处R受湍流的影响最大. 此外, 达到 R_r最小值所需传输距离随光束位相畸变和振幅调制的加剧而增大.     

厄米-高斯光束在非Kolmogorov大气湍流中的传输性质

基于广义惠更斯-菲涅尔原理和非Kolmogorov(非K)谱,推导出了厄米-高斯光束在非K大气湍流中传输的束宽、角扩展以及M²因子的解析表达式.数值计算表明,在传输距离比较远(如z≥3 km)时,厄米-高斯光束的束宽、角扩展和M²因子随广义指数参量α的增大而增加直到α=3.11时达到最大值后再随α的增大而减小;随湍流的内尺度l₀的减小而增大;随外尺度l₀的增加而增大(3.6<α<4).但是当广义指数参量α在3<α<3.6区间取值时,束宽和M²因子几乎不随外尺度的增加而变化.     

基于M2因子测量的大气湍流参数的确定方法

基于广义惠更斯-菲涅尔原理以及大气湍流理论,推导出部分相干光束在大气湍流中传输的光束传输M ^{2因子的解析表达式.定量分析了表征大气湍流参数的折射率结构常数 C2_n}和涡旋内尺度 l _{0对 M ^{2因子的影响,并由此提出了一种通过实验测量大气湍流中光束的 M 2因子,进而确定出大气湍流参数的新方法.研究结果表明,由于大气湍流对相干性好的光束影响更为明显,在测量中可采用具有高相干性的基模高斯光束作为测量光源,而测量装 关键词： 2因子')" href="#">光束传输 M 2因子 大气湍流参数 湍流折射率结构常数 湍流涡旋内尺度}}     

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.     

Changes of propagation characteristic parameters for partially coherent Hermite-Gaussian beams in a turbulent atmosphere

Based on the extended Huygens-Fresnel principle and the definition of second-order moments of the Wigner distribution function (WDF), the analytical formulas for the root-mean-square (rms) spatial and angular widths and the M²-factor of partially coherent Hermite-Gaussian (PCHG) beams propagating through atmospheric turbulence are derived. The major factors affecting on the M²-factor are also discussed in detail. The results show that the rms spatial and angular widths and the M²-factor increase with the propagation distance and the strength of turbulence. However, the relative rms spatial and angular widths and the relative M²-factor of PCHG beams with a higher order, lower coherence and longer wavelength are less affected by the turbulence.     

双曲余弦高斯列阵光束在湍流大气中的光束传输因子

本文推导出了双曲余弦高斯(ChG)列阵光束在湍流大气中的光束传输因子( M ^{2因子)的解析公式,并采用相对 M 2因子研究了湍流对 M 2因子的影响.研究表明,在湍流大气中 M 2因子不再是一个传输不变量,湍流使得 M 2因子增大.非相干合成情况下, M 2因子随着传输距离、光束参数、相对子光束间距和子光束数目的增大而增大.相干合成情况下, M 2因子随光束参数和相对子光束间距的增大呈现振荡上升.相干合成情况下的 M 2因子比 关键词： M2因子)')" href="#">光束传输因子(M2因子) 光束质量 双曲余弦高斯列阵光束 大气湍流}     

Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence

The analytical expressions for the effective Rayleigh range z_R of Gaussian array beams in turbulence for both coherent and incoherent combinations are derived. It is shown that z_R of Gaussian array beams propagating through atmospheric turbulence depends on the strength of turbulence, the array beam parameters and the type of beam combination. For the coherent combination z_R decreases due to turbulence. However, for the incoherent combination there exists a maximum of z_R as the strength of turbulence varies. The z_R of coherently combined Gaussian array beams is larger than that of incoherently combined Gaussian array beams, but for the coherent combination case z_R is more sensitive to turbulence than that for the incoherent combination case. The larger the beam number is, the longer z_R is, and the more z_R is affected by turbulence. For the coherent combination z_R is not monotonic versus the relative beam separation distance, and the effect of turbulence on z_R is appreciable within a certain range of the relative beam separation distance.     

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}     

M~2-factor of high-power laser beams through a multi-apertured ABCD optical system

Based on the generalized truncated second-order moments, an approximate analytical formula of the beam propagation factor M~2 of high-power laser beams passing through the optical system with multiple hard-edged apertures is deduced. Numerical examples of the beams passing through an aperture-spatial filter are enclosed, and the influences of amplitude modulations(AMs) and phase fluctuations(PFs) on the beam propagation quality of high-power laser beams passing through the multi-apertured ABCD optical system are considered and discussed. It is shown that PFs are able to degrade the beam propagation quality of laser beams more than AMs when the high-power laser beams passing through the aperture-spatial filter, furthermore, one or two aperture-lens optical systems configured appropriate aperture parameters are both able to upgrade the beam propagation quality of high-power laser beams. The M~2 factor of Gaussian beam passing through the multi-aperture optical system is a special case in this paper.     

激光光束传输因子M2的一些问题

本文讨论与M²因子有关的一些问题,指出在近轴近似条件下由光束的二阶强度矩定义的M²因子满足M²≥1,其中只有对基模高斯光束等式才成立.由光束的功率通量值定义的M²因子(M_pc²)有可能小于1,M_pc²的大小取决于所定义的光斑半径内包含的光功率的百分数.通过计算光场的二阶矩,我们获得了轴向相干叠加的高斯光束的M²因子的解析解.     

The Beam Propagation Factor of Nonparaxial Truncated Flattened Gaussian Beams

By using the second-order moment of the power density, the beam width, far-field divergence angle and M² factor of nonparaxial truncated flattened Gaussian (FG) beams are derived analytically. It is shown that the M² factor of nonparaxial truncated FG beams depends not only on the truncation parameter δ and beam order N, but also on the initial waist-width to wavelength ratio w₀/λ. The far-field divergence angle approaches an asymptotic value of θ_max=63.435° when the truncation parameter δ → 0. For the special cases of N = 0 and δ → ∞ our results reduce to those of nonparaxial truncated Gaussian beams and nonparaxial untruncated FG beams, respectively.     

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.     

非Kolmogorov大气湍流对高斯列阵光束扩展的影响

本文推导出了高斯列阵光束在非Kolmogorov大气湍流中传输的瑞利区间zR、湍流距离zT和远场发散角θ的解析表达式,研究了非Kolmogorov湍流的广义指数α和列阵光束的合成方式对高斯列阵光束扩展的影响.研究表明:不论相干还是非相干合成高斯列阵光束,zR,zT和θ均随着α的增加而呈非单调变化.当α=3.108时,zR和zT取极小值,而θ取极大值,即当α=3.108时高斯列阵光束扩展最厉害,光束扩展受湍流影响也最厉害.非相干合成高斯列阵光束扩展比相干合成的要大,但受非Kolmogorov湍流影响却要小.特别值得指出的是:当自由空间光束衍射较小时,有zTzR,即在瑞利区间范围内大气湍流就对光束扩展有影响;而当自由空间光束衍射较大时,有zTzR,即在瑞利区间范围内大气湍流对光束扩展几乎没有影响.     

Beam propagation factor of partially coherent beams in gain or absorbing media

The definition of second-order intensity moments in the spatial domain and spatial frequency domain has been generalized for the case that the linear gain or absorbing media are included, where the wave number is generally complex. The formula for beam propagation M²-factor of partially coherent beams in linear gain or absorbing media has been given. The partially coherent flat-topped Schell-model beam is taken as an example. The closed-form expression for the beam propagation M²-factor of partially coherent flat-topped beam in gain or absorbing media has been derived. The changes of the M²-factor in media have been discussed with numerical examples. It can be shown that the M²-factor of flat-topped Schell-model beams in gain or absorbing media depends on the coherent parameter β, the coherent length σ₀, the beam order M, the propagation distance B, the imaginary part of the wave number K_i, as well as the real part of the wave number K_r.     

Investigation on z-scan experiment by use of partially coherent beams

Z-Scan experiment using partially coherent beams is investigated theoretically. We derive the expression for the cross-spectral density of the Gaussian Schell-model (GSM) beam passing through the nonlinear thin sample. Then the influences of the aperture radius and the spatial degree of coherence on the z-scan curves are analyzed. We find that transmittance difference ΔT_p-v between the peak and valley in z-scan curves decreases gradually with the increment of the aperture radius, which is similar to the case of the fully coherent beams. It is also shown that ΔT_p-v is getting bigger with the increment of the spatial degree of coherence α₀, and when α₀ is larger than a certain value, ΔT_p-v becomes almost unchanged. The results show that the z-scan experiment with partially coherent beams may provide a feasible method for measuring the spatial degree of coherence.     

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）     

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.