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.     

The beam quality of nonparaxial Hermite-sine-Gaussian beam

Based on the theory of the second intensity moment of nonparaxial scalar beam and the method of angular spectrum, the expressions for the far-field divergence angle, waist width and M² factor of nonparaxial Hermite-sine-Gaussian (HSiG) beams are derived. Calculation and analysis show the dependence of the far-field divergence angle of nonparaxial HSiG beams on the parameter w₀/λ, as well as on order m, And it with even and odd orders approach 73.898° and 63.435° as w₀/λ → 0. With increasing order m and the parameter w₀/λ, the waist width increase monotonously, which is same as paraxial case. But nonparaxial M² factor is different from paraxial case, it cannot only less than 1, but also approach 0 as w₀/λ → 0.     

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.     

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.     

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     

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.     

Beam quality of truncated laser beams with amplitude modulations and phase fluctuations in turbulence

The closed-form expressions for the Rayleigh range z_R and the M²-factor of truncated laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) in turbulence are derived, and the beam quality is studied by taking the z_R and the M²-factor as the characteristic parameters of beam quality. The M²-factor of truncated laser beams with AMs and PFs is always larger than that of truncated Gaussian beams both in free space and in turbulence. However, in turbulence the beam quality of truncated laser beams with AMs and PFs may be better than that of truncated Gaussian beams if the z_R is taken as the characteristic parameter of beam quality. For laser beams with AMs and PFs in turbulence, the beam quality expressed in terms of z_R is consistent with that in terms of the M²-factor versus the phase fluctuation parameter α, but not versus the intensity modulation parameter σ_A. The beam quality of truncated laser beams with AMs and PFs is less sensitive to turbulence than that of truncated Gaussian beams. The beam quality of laser beams with smaller α and larger σ_A is less affected by turbulence than those with larger α and smaller σ_A.     

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.     

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.     

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.     

InnoSlab混合腔输出光束质量的理论研究

对InnoSlab Laser用混合腔的输出光束质量问题从热透镜效应和腔镜的不对准两个方面的影响进行理论分析,并从远场光强分布、远场发散角、光腰半径、光束质量因子M²值和桶中功率曲线等几个角度进行了讨论,特别是离轴非稳腔方向上腔镜倾斜对光束质量的影响,利用M²值和PIB曲线两个参数共同评价非稳腔的光束质量,对目前并无统一标准的非稳腔光束质量评价问题提供了有意义的参考. 关键词： InnoSlab 混合腔 光束质量 桶中功率     

Propagation properties of partially coherent modified Bessel–Gauss beams

Starting from the propagation law of partially coherent light, the analytical propagation equations of partially coherent modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived and illustrated with typical application examples. Furthermore, by using the intensity moments method and integral transformation technique, the important characteristic parameters, including the beam width, far-field divergence angle, M² factor and kurtosis parameter of partially coherent MBGBs, are expressed in a closed and simple form. As a result, some basic properties of MBGBs and the dependence of the M² factor and kurtosis parameter on the spectral degree of coherence and beam order are illustrated both analytically and numerically.     

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.     

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}}     

Irradiance-moments characterization of off-axis Gaussian beam combinations by means of the Wigner distribution function

The irradiance-moments characterization of off-axis Gaussian beam combinations is presented in use of the Wigner distribution function (WDF). Analytical expressions for the propagation through an optical paraxial ABCD system, beam propagation factor (M² factor), beam width, far field divergence angle and kurtosis parameter of the resultant beam are derived and illustrated with specific numerical examples. It is reexamined that a flat-topped irradiance profile of the resultant beam can be achieved at a certain plane by appropriately choosing beamlet parameters. As compared with previous work, our results show the advantage of using the WDF and irradiance-moments methods.     

M factor invariance through the geometric etendue for Gaussian Schell model sources

Taking advantage of the relation of the M² factor for Gaussian Schell model sources in terms of the global coherence parameter, derived by Santarsiero et al., we have shown in this paper the invariance of the M² factor through its connection with geometrical Etendue of the pencil, along each independent coordinate.     

M factor and kurtosis parameter of super-Gaussian beams passing through an axicon

The effects of axicons on the M² factor and kurtosis parameter of super Gaussian beams(SGBs) are studied in detail. The closed-form expression for the M² factor of SGBs after passing through an axicon is derived, and the reason why the convergent and divergent axicons give rise to the same increase of the M² factor is explained physically from the similarity of the resulting irradiance distributions. The analytical propagation equation of the K parameter of SGBs passing through an axicon followed by a paraxial optical ABCD system is given, some interesting special cases are discussed. In particular, it is found that even for the Gaussian special case theK parameter is no longer a constant due to the effect of the axicon. Numerical examples are also given to illustrate our analytical results.     

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.     

Propagation invariants of twisted Gaussian Schell-model beams

The second-order moments method is used to study the M² factor and intrinsic astigmatism of twisted Gaussian Schell-model (GSM) beams. It is shown that the M² factor of twisted GSM beams defined by the determinate of the 4×4 variance matrix is a propagation invariant and is independent of the beam twist, whereas the twist affects the intrinsic astigmatism of twisted GSM beams.