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.     

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.     

Propagation of a hollow Gaussian beam through a paraxial misaligned optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a paraxial misaligned optical system in the cylindrical coordinate system, we obtain an analytical formula for a hollow Gaussian beam passing through a paraxial misaligned optical system. Furthermore, we also obtain the approximate analytical formula for a hollow Gaussian beam passing through a paraxial circularly apertured misaligned optical system by expanding the hard aperture function into a finite sum of complex Gaussian functions. As a numerical example, the propagation properties a hollow Gaussian beam through a misaligned thin lens are studied numerically.     

洛伦兹光束经光阑失调傍轴光学系统的传输

基于广义衍射积分公式和光阑函数的复高斯展开,导出了一洛伦兹光束经一个带圆形光阑失调傍轴光学系统的近似解析传输公式.作为一般公式的特例,还给出了洛伦兹光束经一无光阑失调傍轴光学系统的解析传输式.作为数值计算的例子,运用所得到的公式分析了洛伦兹光束经带光阑失调薄透镜的传输特性.结果表明：不同强度的衍射即圆形光阑半径的大小明显影响衍射光束的归一化强度分布及其传输变化规律. 关键词： 洛伦兹光束 失调傍轴光学系统 光束传输     

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.     

Propagation properties of the beam generated by Gaussian mirror resonator passing through a paraxial ABCD optical system

By expanding a hard aperture function into a finite sum of complex Gaussian functions, approximate propagation formula is derived in the situation that the beam generated by Gaussian mirror resonator passes through a paraxial ABCD optical system with an annular aperture. The corresponding forms for a circular aperture and a circular black screen are also given. Some numerical simulations are shown to illustrate propagation properties and focusing properties of the beam passing through a paraxial ABCD optical system with the three different kinds of aperture.     

Propagation of generalized Mathieu–Gauss beams through paraxial misaligned optical systems

Based on the generalized diffraction integral, we derive an analytical formula for generalized Mathieu–Gauss beams (gMGBs) passing through an apertured misaligned optical system. Furthermore, we use the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions to establish an approximate propagation equation of gMGBs through paraxial circularly apertured optical system. As an example, the propagation of ordinary and modified zeroth order MGBs through a misaligned thin lens is studied numerically.     

Approximate analytical expressions of Laguerre-Gaussian beams passing through a paraxial optical system with an annular aperture

On the basis of the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of Laguerre-Gaussian beams passing through an annular apertured paraxial ABCD optical system is derived. Meanwhile, the corresponding closed-forms for the unapertured or circular apertured or circular black screen cases have also been given. Numerical examples are given to illustrate the propagation characteristics of Laguerre-Gaussian beams.     

慧差对线偏振贝塞耳-高斯光束聚焦场的影响

利用Richards-Wolf矢量衍射积分公式,获得线偏振贝塞耳-高斯光束经具有初级慧差的高数值孔径系统聚焦后的三维光场复振幅函数,根据复振幅函数模拟了不同慧差系数下线偏振贝塞耳-高斯光束各分量及总光场在焦平面上的光强分布模式。研究结果表明,初级慧差使线偏振贝塞耳-高斯光束聚焦光场在焦平面的分布由双轴对称变成单轴对称,光强模式发生改变;各分量及总光场的光强模式中的光斑位置随初级慧差系数的增大偏离光轴;大的初级慧差使焦平面出现干涉条纹。     

Propagation properties of partially polarized Gaussian Schell-model beams through an axis-unsymmetric paraxial ABCD system

By using the beam coherence-polarization (BCP) matrix approach, analytical propagation equations of partially polarized Gaussian Schell-model (PGSM) beams through an axis-unsymmetric paraxial optical ABCD system are derived, which enable us to study the propagation-induced polarization changes and irradiance distributions at any propagation distance of PGSM beams through axis-unsymmetric systems within the framework of the paraxial approximation. Detailed numerical results for a PGSM beam passing through a bifocal lens are presented to illustrate the propagation properties of PGSM beams. A comparison with the previous work is also made.     

Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture

Based on the generalized Huygens-Fresnel diffraction integral, a recurrence propagation equation of Hermite-Gaussian (H-G) beams through a paraxial optical ABCD system with hard-edge aperture is derived, which permits us to obtain the analytical propagation expression for H-G beams of any order. The advantages of our analytical results are analyzed, and the application is illustrated with numerical examples.     

Propagation characteristics of the two-dimensional off-axial Hermite-cosh-Gaussian beams through rectangular apertured and misaligned optical systems

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for the two-dimensional off-axial Hermite-cosh-Gaussian beams passing through a rectangular apertured and misaligned paraxially ABCD optical system has been derived. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral directly. Some numerical simulations are also illustrated for the propagation characteristics of a two-dimensional off-axial Hermite-cosh-Gaussian beam through a rectangular apertured ABCD optical system.     

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.     

The elliptical elegant Laguerre–Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new kind of laser beam called the elliptical elegant Laguerre–Gaussian beam (EELGB) is defined by using tensor method. By using the generalized diffraction integral formulas for light beam passing through paraxial optical system, the analytical propagation formulas for EELGB passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, the propagation properties of EELGBs in free space propagation and through a misaligned thin lens are studied.     

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derived an approximate analytical formula for a partially coherent twisted anisotropic Gaussian Schell-model (AGSM) beam propagating through an apertured paraxial general astigmatic (GA) optical system by use of a tensor method. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. Our formulas avoid time-consuming numerical integration and provide a convenient and effective way for studying the propagation and transformation of a partially coherent twisted AGSM beam through an apertured paraxial GA optical system.     

Property study of coherent annular beam generated by the annular phase-locking laser

An annular phase-locking laser with annular gain media is presented. In this paper, the light field distribution of coherent annular beam (CAB) can be given by an analytical expression based on the Collins formula. Typical numerical examples are given to illustrate the application of our analytical result and the propagation property of a CAB through a paraxial ABCD optical system.     

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.     

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.     

Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, some approximate analytical formulae for the cross-spectral density of a partially coherent flattened Gaussian beam (FGB) propagating through apertured aligned and misaligned ABCD optical systems are derived based on the generalized Collins formula. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. As a numerical example, the focusing properties (including average irradiance distribution and focal shift) of a partially coherent FGB focused by an apertured thin lens are studied, and it is found that the focusing properties of a partially coherent FGB are closely related to its initial coherence and the aperture width. Our results provide an effective and fast way for studying the paraxial propagation of a partially coherent FGB through apertured ABCD optical systems.     

Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture

Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture.