Propagation of a four-petal Gaussian vortex beam through a paraxial ABCD optical system

Based on the Collins diffraction integral formula, an analytical expression of a general four-petal Gaussian vortex beam passing through a paraxial ABCD optical system is derived by means of the mathematical technique. As a numerical example, the normalized intensity distribution of a four-petal Gaussian vortex beam propagating in free space is graphically demonstrated. The influences of beam order and topological charge on the normalized intensity distribution are discussed in detail. This research is useful to the optical trapping, optical communications, and beam splitting techniques, etc.     

Spectral shift of partially coherent flattened multi-Gaussian beams with a hole passing through ABCD optical systems

Based on flattened multi-Gaussian beam model and partially coherent theory, the expression for cross-spectral density of partially coherent flattened multi-Gaussian beam with a hole was given. The analytical expression for on-axis spectrum of partially coherent flattened multi-Gaussian beam with a hole passing through a paraxial ABCD optical system was derived. The spectral shifts of the partially coherent flattened multi-Gaussian beam with a hole propagating in free space and passing through the lens have been analyzed. The effects of the beam profile, spatial coherence parameter and the system parameters on the relative spectral shift have been discussed. Our results show that the inner radius of flattened multi-Gaussian beam with a hole determines its relative spectral shift in near field and its outer radius determines that in far field. For the flattened multi-Gaussian beam with a hole passing through a lens, the on-axis relative spectral shift changes sharply near the focal plane with the increase in Fresnel number and the spatial coherence parameter.     

Matrix formulation of axis-symmetric laser beams through a paraxial ABCD system containing hard-edged apertures: A comparative study

A matrix formulation is presented, which enables us to study the propagation of axis-symmetric beams through a paraxial optical ABCD system containing hard-edged aperture. Numerical calculation results of super-Gaussian beams passing through a multi-aperture-lens system are given to illustrate the advantage of the method. A comparison of the matrix formulation, complex Gaussian expansion and direct numerical integration of the Collins formula is made, where the propagation of apertured Laguerre-Gaussian beams is chosen as an illustrative example. It is shown that the matrix formulation provides satisfactory results in both Fraunhofer and Fresnel regions, and reduces the computational time greatly in comparison with the direct integration. However, this method is suited only to axis-symmetric optical beams and systems. By using the complex Gaussian expansion discrepancies exist in the near zone closer to the aperture, but usually its computational efficiency is higher than the matrix formulation.     

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.     

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.     

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.     

Four-petal Gaussian beams and their propagation

A new form of laser beams called four-petal Gaussian beams is introduced. Based on the Collins integral, two kinds of analytical propagation expressions for this new kind of beams through a paraxial ABCD optical system are derived. The propagation properties of the four-petal Gaussian beams are studied and illustrated with numerical examples. At the source plane the beam has four-petals; the space among the petals is determined by the beam order. In the far field the beam evolves into a number of mirror symmetric petals and the petals of higher order beams can be equally spaced.     

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.     

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     

A detailed study of Mathieu-Gauss beams propagation through an apertured ABCD optical system

Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu-Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu-Gauss beams in (y-z) and (x-z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu-Gauss beam is similar to that of Bessel-Gauss beam. Furthermore, the focalization of the Mathieu-Gauss beams through a thin lens is illustrated and analyzed with numerical results.     

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.     

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 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 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.     

偏心厄米--高斯光束通过失调一阶光学系统的传输特性

使用广义惠更斯—菲涅尔衍射积分,增广矩阵和Wigner分布函数方法,研究了偏心厄米--高斯光束通过失调非对称一阶光学系统的传输特性,证明了偏心厄米--高斯光束通过一阶光学系统时保持其封闭性,光束的二阶矩矩阵按通常的规律变换,一阶矩矩阵遵循几何光线的变换规律,光束传输因子保持不变。     

Propagation of elliptical Gaussian beam passing through apertured paraxial optical systems

The propagation of elliptical Gaussian beam passing through paraxial optical systems with aperture is investigated analytically by using tensor method. The approximate formula for propagation of elliptical Gaussian beam through hard apertured optical systems is derived based on the fact that the circ function can be expanded into a finite sum of complex Gaussian functions. The derived formula provides a convenient tool for treating the propagation and transformation of elliptical Gaussian beam through apertured optical systems. As an application example, the propagation properties of elliptical Gaussian beam through apertured fractional Fourier systems are discussed.     

Propagation of off-axial cosine-Gaussian beams passing through an apertured and misaligned optical system

A closed-form expression for the one-dimensional off-axial cosine-Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system is derived. As special cases, the corresponding closed-forms for the off-axial or non off-axial cosine-Gaussian beams passing through apertured or unapertured and misaligned or aligned optical systems are also given. The obtained results could be straightforward to the two-dimensional case or sine-Gaussian beams.     

Multiplication rule of similarity transformations in the coherent state representation and its mapping onto quantum optical generalized ABCD law

We find that classical similarity transformations in the coherent state representation projects onto the similarity transformation operators (STO), these operators constitute a loyal representation of symplectic group. Remarkably, the multiplication rule of the STOs naturally leads to the quantum optical generalized ABCD law, which is the quantum mechanical correspondence of the classical optical ABCD law. Throughout the whole derivation, the technique of integration within an ordered product (IWOP) of operators is employed.     

HfO2/SiO2高反射膜的缺陷及其激光损伤

用原子力、Normaski和扫描电子显微镜等分析仪器,对高损伤阈值薄膜常采用的HfO2光损伤所形成的孔洞,与镀制过程中形成的孔洞形貌相似,激光再损伤能力也相似。低能量密度的激光把节瘤缺陷变为孔洞缺陷是激光预处理提高薄膜损伤阈值的原因之一。