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.     

Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture

Based on the generalized Collins diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expressions of Bessel-Gaussian beams and QBG beams passing through a paraxial ABCD optical system with an annual aperture are derived. As special cases, the corresponding closed-forms for the unaperture or circular aperture or circular black screen cases are also given. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Numerical examples are given to illustrate the propagation properties of Bessel-Gaussian and QBG beams.     

Roughness and gradient parameters for characterising shape uniformity of laser beams

We define six parameters, the roughness parameter, the local gradient parameter, the directionless gradient parameter, the integral gradient parameter, the directionless integral gradient parameter and the absolute gradient parameter for characterising the uniformity of a laser beam in its transversal section. For shape-invariant beams the roughness parameter is propagation invariant. The absolute gradient parameter is propagation invariant for symmetric shape invariant beams. As an example, five of the defined parameters are calculated for a dummy irradiance. The values of the roughness parameter and the absolute parameter are calculated and tabulated for symmetric Hermite–Gaussian, Laguerre–Gaussian, spherical Gaussian, flattened Gaussian (at the waist plane) and super Gaussian beams (at the waist plane). The results are discussed.     

Propagation of vectorial Gaussian beams behind a circular aperture

Based on the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian functions, an approximate analytical expression for the propagation equation of vectorial Gaussian beams diffracted at a circular aperture is derived and some special cases are discussed. By using the approximate analytical formula and diffraction integral formula, some numerical simulation comparisons are done, and some special cases are discussed. We find that a circular aperture can produce the focusing effect but the beam becomes the shape of ellipse in the Fresnel region. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum.     

非傍轴矢量高斯光束的圆屏衍射

利用矢量瑞利衍射积分公式，推导出非傍轴矢量高斯光束圆屏衍射的解析表示式.非傍轴矢量高斯光束圆屏衍射的轴上场分布、远场表示式、自由空间中的传输公式，以及傍轴近似下高斯光束圆屏衍射的菲涅耳和夫琅禾费衍射公式可以作为一般公式的特例统一处理.数 值计算和比较实例说明了非傍轴矢量高斯光束的光强分布和远场特性.分析表明，在圆屏衍 射中，f参数和截断参数决定光束的非傍轴行为. 关键词： 传输光学 非傍轴矢量高斯光束 圆屏衍射 矢量瑞利衍射积分公式     

Propagation of vectorial nonparaxial Gaussian beams through an annular aperture

Starting from the vectorial Rayleigh diffraction integrals, the nonparaxial propagation of vectorial Gaussian beams through an annular aperture is studied. The analytical propagation expressions are derived, which permit us to treat the on-axis field and far field of vectorial nonparaxial Gaussian beams diffracted at the annular aperture, the nonparaxial diffraction at a circular aperture and a circular disc as our special cases in a unified way. The validity of our treatment is confirmed by direct numerical integration of the Rayleigh formulae. It is shown that the f-parameter and annular obscuration affect the beam nonparaxiality in the case of diffraction at the annular aperture.     

Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.     

The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture

The rigorous electromagnetic theory of the diffraction of vector beams by an aperture is proposed and numerically evaluated by the diffraction of vector Gaussian beams by a circular aperture. The results are compared with those, by using the vector Rayleigh–Sommerfeld diffraction integrals and angular-spectrum expression theory, showing a good consistency. The numerical calculation shows that the result calculated by the rigorous theory is much more precise than those calculated by the integral method in diffraction near field, and a highly consistency reaches by the three methods in diffraction far field. A further extension of the theory is discussed.     

Propagation properties of Gaussian beams through the anamorphic fractional Fourier transform system with an eccentric circular aperture

As known, it is important for the propagation of Gaussian beams in optics. In this paper, based on the expanding a hard-edge circular aperture function as a finite sum of complex Gaussian functions and the scalar Rayleigh-Sommerfeld diffraction formula, an approximation analytical solution for Gaussian beams propagating through the anamorphic fractional Fourier transform system with an eccentric circular aperture is performed. Then, the detailed numerical calculation for the two-dimensional Gaussian beams in the above-mentioned optical system is presented. The simulation also shows that different location and size of aperture result in the change of diffracted field, including location, intensity and width. All these characteristics help us understand Gaussian beams propagation better.     

Propagation characteristics of the rectangular flattened Gaussian beams through circular apertured and misaligned optical systems

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expression for the output field distribution of a rectangular flattened Gaussian beam passing through a circular apertured and misaligned paraxial ABCD system is derived. The result brings more convenient 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 a rectangular flattened Gaussian beam through a circular apertured and misaligned optical system.     

Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams

Based on the relations between Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) modes and by introduced the complex Gaussian expansion method for two dimensional rectangular aperture, the approximate analytical propagation expressions of the rotational symmetrical LG beams along with their even and odd modes through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. As special cases of the results, the corresponding closed-forms of the circular aperture diffracted LG beams and non-truncated LG beams are also given. Numerical examples are given to prove the validity of this approximate analytical method and illustrate the propagation properties of the rectangular hard-edged aperture diffracted LG beams.     

平顶高斯光束经含失调圆孔光阑的失调光学系统的传输特性

为分析平顶高斯光束通过光学系统传输时圆孔光阑失调和光学元件失调对平顶高斯光束传输特性的影响,利用失调圆孔光阑的近似展开式和适用于失调光学系统的广义衍射公式,得出了平顶高斯光束经含失调圆孔光阑的失调光学系统传输的近似解析式,给出了输出光束场分布与光束参量、光阑孔径尺寸、光阑和光学元件失调量等的定量关系.针对特定光学系统定量分析了各失调量对输出光束场分布的影响,结果表明各元件失调都对输出光束强度分布产生较大影响.但在各失调量较小的情况下,透镜失调对输出光束传输特性的影响比光阑失调对输出光束传输特性的影响更明显.     

Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture

Based on the vectorial Rayleith-Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.     

Kurtosis parameter of truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems

On the basis of the propagation equation of truncated standard and elegant Laguerre–Gaussian beams, the closed-form propagation expressions for the kurtosis parameter K of standard and elegant Laguerre–Gaussian beams passing through apertured optical systems are derived, it is shown that the kutosis parameter K of standard and elegant Laguerre–Gaussian beams depend on order p and index m of Laguerre polynomial, the beam-truncated parameter δ and the generalized Fresnel number F. Detailed numerical examples are given to illustrate the analytical results.     

拉盖尔-高斯截断级数展开法的应用

 用拉盖尔-高斯截断级数模拟了任意半径的圆孔硬边光阑窗口函数,并应用于研究高斯光束和零阶贝塞尔-高斯光束通过硬边光阑的衍射。对高斯光束和零阶贝塞尔-高斯光束的轴向和横向光强分布的计算结果表明：当积分面不是很靠近光阑时，用参数最佳化选取的拉盖尔-高斯截断级数法与用直接数值积分所得结果符合很好。     

Propagation characteristics of linearly polarized Bessel-Gaussian beams through an annular apertured paraxial ABCD optical system

By means of Collins diffraction integral formula in the paraxial approximation and based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical expression for linearly polarized Bessel-Gaussian beams passing through a paraxial ABCD optical system with an annular aperture has been derived. The results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly. By using the analytical expression and the diffraction integral formula some numerical simulations are done to illustrate for the propagation characteristics of a linearly polarized Bessel-Gaussian beam through an optical system with an annular aperture.     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.     

Analytical formula for a circular flattened Gaussian beam propagating through a misaligned paraxial ABCD optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial ABCD optical system in the cylindrical coordinate system, analytical formula for a circular flattened Gaussian beam propagating through such optical system is derived. Furthermore, an approximate analytical formula is derived for a circular flattened Gaussian beam propagating through an apertured misaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Numerical examples are given.     

Interference of intersecting singular beams

A vorticity of the light field created by interference of two intersecting Laguerre–Gaussian singular beams is analysed. It is demonstrated that the number and location of the vortices present in the field depend on the propagation length as well as on the topological charges of the individual beams, their intersection angle and amplitude ratio.     

Propagation properties of anomalous hollow beams in a turbulent atmosphere

Propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere is investigated in detail. Based on the extended Huygens–Fresnel integral, analytical formulae for the average irradiance of circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere are derived. The irradiance and spreading properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere and in free space are studied numerically. It is found that circular and elliptical anomalous hollow beams at short propagation distance in turbulent atmosphere have similar propagation properties to those of free space, while at long propagation distance, circular and elliptical anomalous hollow beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The conversion from an anomalous hollow beam to a circular Gaussian beam becomes quicker and the beam spot spreads more rapidly for a larger structure constant, a shorter wavelength and a smaller waist size of the initial beam.