Influence of the chirp on the intensity distributions of an apertured pulse

Starting from the Huygens–Fresnel diffraction integral and the Fourier transform, the propagation expression of a chirped pulse passing through a hard-edged aperture is derived. Using the obtained expression, the intensity distributions of the pulse with different chirp in the near and far fields are analyzed in detail. Due to the modulation of the aperture, many intensity peaks emerge in the intensity distributions of the chirped pulse in the near field. However, the amplitudes of the intensity peaks decrease on increasing the chirp, which results in the smoothing effect in the intensity distributions. The beam smoothing brought by increasing the chirp is explained physically. Also, it is found that the radius of the intensity distribution of the chirped pulse decreases when the chirp increases in the far field.     

大气湍流对多色高斯-谢尔模型光束扩展的影响

基于广义惠更斯-菲涅耳原理,采用Rytov相位结构函数二次近似和硬边窗口函数的复高斯展开法,推导了受光阑限制的多色高斯-谢尔模型(GSM)光束在大气湍流中的二阶矩束宽公式。研究表明:二阶矩束宽随着大气湍流折射率结构常数、源光谱带宽和光束传输距离的增加而增大,随着光束截断参数和光束相干参数的增加而减小;并且,当源光谱带宽越大、光束截断参数和光束相干参数越小,则湍流对受光阑限制的多色GSM光束扩展的影响就越小。关键词:      

大气湍流对多色高斯-谢尔模型光束扩展的影响

基于广义惠更斯-菲涅耳原理,采用Rytov相位结构函数二次近似和硬边窗口函数的复高斯展开法,推导了受光阑限制的多色高斯-谢尔模型(GSM)光束在大气湍流中的二阶矩束宽公式。研究表明：二阶矩束宽随着大气湍流折射率结构常数、源光谱带宽和光束传输距离的增加而增大,随着光束截断参数和光束相干参数的增加而减小;并且,当源光谱带宽越大、光束截断参数和光束相干参数越小,则湍流对受光阑限制的多色GSM光束扩展的影响就越小。关键词：      

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.     

Narrowing effect of focused polychromatic laser with small Fresnel number

Starting from Huygens-Fresnel diffraction integral, the field representation of focused polychromatic laser beams is derived. Intensity distributions of the focused polychromatic laser are investigated with different Fresnel number and bandwidth at focal plane. It is found that the width of transverse intensity distribution of the focused polychromatic laser decreases with increasing the bandwidth at the focal plane and zero local time, and the transverse intensity presents more obvious narrowing effects with smaller Fresnel number. A physical explanation for the narrowing effects is given based on a simplified expression.     

Focal shift in Hermite–Gaussian beams based on the encircled-power criterion

The focal shift in Hermite–Gaussian beams, whose axial irradiance vanishes, is studied based on the encircled-power criterion, and the results are illustrated numerically. The related problems are discussed.     

Influence of the comatic aberration of an apertured lens on the focused polychromatic Gaussian beams

The focusing of polychromatic Gaussian beams by an apertured lens with comatic aberration is investigated in this paper. We study the influence of the comatic aberration and truncation parameter on the intensity distribution at the focal plane, and singularities and spectral behavior near the focal plane in detail. The simulation results show that the maximum intensity at the focal plane will shift and split with the gradual increment of comatic aberration. We also find that the decrease of truncation parameter will counteract the influence of comatic aberration, which leads to the intensity variation. Furthermore, the singularity transformation and spectral anomalies are discussed. It is found that with the increment of comatic aberration, some singularities are split into several new singularities, and by combining some of these new singularities with other singularities near the focal plane, they vanish or form new singularities with the continuing increase of comatic aberration. The decreasing value of truncation parameter will cause similar phenomena.     

Studies of propagation characteristics of elliptical Gaussian beams through circular apertured optical systems

An approximate analytical propagation formula of the elliptical Gaussian beams through a paraxial ABCD optical system with a circular aperture is obtained on the basis of the expansion of a hard-edged aperture into a finite sum of complex Gaussian functions. And the parameter characteristics of the truncated beams including the beam propagation factor and the kurtosis parameter are also studied in detail by using the second-order-moments method.     

Effect of quartic-phase aberrations on the focal switch of Hermite–Gaussian beams

The focal switch of Hermite–Gaussian beams diffracted at an aperture and subsequently focused by a spherically aberrated lens is studied. Our main attention is focused on the effect of quartic-phase aberrations on the behavior of the focal switch. It is shown that quartic-phase aberrations affect the relative focal shift Δz_f, turning position s_1,t, and relative transition height Δz_sw. Apart from a critical maximum truncation parameter α_c,max, there is a critical minimum truncation parameter α_c,min. Within the region α_c,min<α<α_c,max the focal switch can take place, but quartic-phase aberrations give rise to a decrease of α_c,max-α_c,min in comparison with the aberration-free case.     

Propagation of the Hermite–Gaussian beams through misaligned optical system with a circular aperture

By means of expanding a hard-edged aperture into a finite sum of complex Gaussian functions, the approximate analytical formula of one kind of higher-order Gaussian beams called the Hermite–Gaussian beams (HGBs) passing through circular apertured and misaligned optical system is obtained in this paper. The result provides more convenience for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of the HGBs through the circular apertured optical systems.     

Far-field properties and beam quality of vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

The far-field properties and beam quality of vectorial nonparaxial Hermite–Laguerre–Gaussian (HLG) beams are studied in detail, where, instead of the second-order-moments-based M² factor, the extended power in the bucket (PIB) and βparameter are used to characterize the beam quality in the far field and the intensity in the formulae is replaced by the z component of the time-averaged Poynting vector S_z. It is found that the S_z PIB and βparameter of vectorial nonparaxial HLG beams depend on the mode indices n, m, αparameter and waist-width-to-wavelength ratio w₀/λ and the PIB and βparameter are additionally dependent on the bucket's size taken.     

Generalized beam propagation factor of hard-edged circular apertured diffracted Bessel–Gaussian beams

Based on the truncated second-order moments method on the cylindrical coordinate systems and the incomplete gamma function, an analytical expression of the generalized beam propagation factor (M_G² factor) of hard-edged circular apertured diffracted Bessel–Gaussian beams is derived and illustrated numerically. It is shown that the M_G² factor of hard-edged diffracted BGBs mainly depends on the truncation parameter δ and the beam parameters m and η. The results can be reduced to that for the non-truncated Bessel–Gaussian beams case and that for the truncated fundamental Gaussian beams case under certain conditions, respectively. The power fraction is also discussed analytically and numerically.     

Lens transformations of high power solid-state laser beams

Beam parameters of 300 W-class CW Nd:YAG lasers with double pump cavities have been investigated. The calculations have demonstrated that in the strongly pumped operating laser the amplitude of stresses in the rod can be decreased by a factor of approximately 1.5. A multibeam model of high power beam structure has been proposed. It has been shown theoretically and experimentally that 1.4–1.8 times reduction of the beam parameter product of a complex non-Gaussian beam may be achieved by placing a lens with a strength and location determined by laser parameters. An integrated parameters product has been proposed to describe the complex beams. In the range of possible pumping levels (3–10 kW) the output full divergence angle changed from 2 up to 23–28 mrad but in image space of the lens the variations of this value were no more than ±12%.     

Application of the Wigner distribution function to complex-argument Hermite– and Laguerre–Gaussian beams beyond the paraxial approximation

The Wigner distribution function (WDF) is applied to study the propagation of complex-argument Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams beyond the paraxial approximation. The analytical expressions for their intensity distributions in free-space propagation are derived, which are expressed in terms of Hermite polynomials for nonparaxial complex-argument HG beams and in terms of the sum of finite Hermite polynomials for nonparaxial complex-argument LG beams. A detailed comparison of the WDF approach, series expansion method and paraxial expressions is made, which shows that in the paraxial regime the WDF approach and series expansion method deliver consistent results with that of paraxial expressions. Beyond the paraxial approximation, the WDF approach offers convergent results, whereas the series expansion method has a limited applicable range, within which it gives consistent results with that of WDF approach but beyond which it gives unrealistic and divergent results.     

Vectorial structure of Hermite–Laguerre–Gaussian beam in the far field

Starting from Maxwell's equations, a Hermite–Laguerre–Gaussian (HLG) beam is decomposed into the TE and TM terms by using the vector angular spectrum representation. By means of the method of stationary phase, the analytical TE and TM terms are presented in the far field. The energy flux distributions of the TE and TM terms are also investigated and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated. This research reveals the internal vectorial structure of HLG beam and may provide a new approach to the manipulation of laser beams.     

Propagation of elegant Laguerre–Gaussian beams through an annular apertured paraxial ABCD optical system

Based on the generalized Huygens–Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of elegant Laguerre–Gaussian beams passing through a paraxial ABCD optical system with an annular aperture is derived. Meanwhile, the corresponding closed-forms for the unapertured, circular apertured or circular black screen cases are also given. The obtained results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Some numerical examples are given to illustrate the propagation properties of elegant Laguerre–Gaussian beams.     

多色部分相干偏心光束在non-Kolmogorov湍流中的传输

推导了多色部分相干偏心光束在non-Kolmogorov 湍流中传输的总光强、轴上光谱、相干度的解析表达式, 研究了光束偏心参数β 、湍流广义指数α和源光谱带宽Ω对激光传输特性的影响. 研究表明: β越大, 则光束重心偏离传输轴越远, 相干度的不对称性越明显, 但是, β对轴上光谱几乎没有影响; 湍流广义指数α对总光强、 轴上光谱和相干长度的影响是非单调的, 当α=3.1时, 湍流对光束传输特性的影响最大. 值得指出的是: 在某些传输距离处, 不同α对应的轴上光谱位移量相同; 在某些传输距离处, 轴上光谱位移量为零, 且该传输距离与Ω无关, 但湍流使得该传输距离缩短. 所得结论对多色部分相干偏心光束在 湍流大气中传输的相关应用具有重要意义. 关键词： non-Kolmogorov湍流 多色部分相干偏心光束 光谱强度 相干度     

Changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian beams in turbulence

The unified theory of coherence and polarization is applied to an investigation of the changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian (EHG) beams propagating through atmospheric turbulence. Based on the extended Huygens–Fresnel principle, the 2×2 electric cross-spectral density matrix of partially coherent EHG beams in turbulence is expressed analytically by using a quadratic approximation of Rytov's phase structure function. The analytical expressions for the degree of polarization, the degree of coherence and the spectrum of partially coherent EHG beams in turbulence are derived, and the corresponding analytical results of electromagnetic Gaussian Schell-model (EGSM) beams can be obtained when the order number of the Hermite polynomial is zero. Some interesting results are obtained, which are illustrated by using numerical examples.     

管板式大功率横流CO2激光横模测量及数值模拟

为了提高激光稳定性和激光加工质量,实验测量了大功率横流CO2激光横模,并理论分析了横模形成机理。针对大功率横流CO2激光器管板式电极结构,由麦克斯韦方程给出电场的横向分布,并通过数值计算得到了激光强度的横向分布。实验结果与数值模拟得到的激光能量分布基本一致。结果表明,在给定放电电压条件下,电极结构和气体流动速率决定了激光横模峰值大小,激光峰值的横向位置取决于气体流动的速率。     

Self-similar collapse and blow up of laser beams in nonlinear Kerr media—Revisited

Some features of the blow up and collapse phenomena characteristic of the focusing nonlinear Schrödinger equation are discussed. Particular attention is given to recently found solutions describing self-similar collapse in the form of ring-shaped structures. The characteristic parameters of such collapsing rings are found and particular single-humped initial conditions giving rise to ring formation are also discussed. The possibility of self-similarly collapsing beams with parabolic intensity profiles is investigated and used to infer the blow up distance of initially Gaussian beams. The result is shown to be in good agreement with previous empirical results for the blow up distance.