Analytical structure of an apertured vector Gaussian beam in the far field

Based on the angular spectrum representation of the Maxwell’s equations and the complex Gaussian expansion of the aperture function, the structure of an apertured vector Gaussian beam in the far field is presented in the integral form. By means of the method of stationary phase, the analytical vectorial structures are obtained. According to the analytical expressions, the characteristics of vectorial structure of an apertured Gaussian beam are investigated in the far field. The influence of a linearly polarized angle on the vectorial structure is also studied in the far field. This research provides a novel approach to further comprehend the vectorial property of an apertured Gaussian beam.     

Vectorial structure of the far-field of an apertured four-petal Gaussian beam

Based on the theorem of vectorial structure and the method of stationary phase, an analytical vectorial structure of the far-field of an apertured four-petal Gaussian beam has been derived without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured four-petal Gaussian beam are also presented in the far-field reference plane, respectively. The energy flux distributions of the TE term, the TM term, and the apertured four-petal Gaussian beam are graphically demonstrated in the far-field plane. The dependences of the energy flux distributions of the TE term, the TM term, and the apertured four-petal Gaussian beam on the f-parameter, the truncation parameter, and the beam order are also examined.     

Far-field structural property of a Gaussian beam diffracted by a phase aperture

Based on the angular spectrum representation of an arbitrary electromagnetic beam and the method of stationary phase, an analytically vectorial structure of a Gaussian beam diffracted by a phase aperture has been derived in the far-field. Moreover, the derivation is performed without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured Gaussian beam are also presented in the far-field, respectively. The influence of the phase delay on the energy flux distributions of the TE term, the TM term, and the apertured Gaussian beam is discussed in the far-field.     

Analytical vectorial structure of an apertured hollow Gaussian beam in the far field

Based on the vector angular-spectrum and the complex Gaussian expansion of the aperture function, the structure of an apertured hollow Gaussian beam in the far field is given in a integral form. In virtue of the method of stationary phase approximation, the analytical vectorial structures are derived. Starting from the analytical expressions, the propagation properties of apertured hollow Gaussian beams with different order in the far field are illustrated graphically. In addition, the influence of the truncation parameter on far field distribution is studied detailedly. This research can shed light on the further understanding of the vectorial property of an apertured hollow Gaussian beam.     

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:      

Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence

The propagation properties of apertured laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) through atmospheric turbulence are studied in detail both analytically and numerically. The analytical expressions for the average intensity, power in the bucket (PIB) and Strehl ratio (S _R ) of apertured laser beams with AMs and PFs propagating through atmospheric turbulence are derived. It is found that the worse the phase fluctuation and the higher the amplitude modulation are, the less laser beams are affected by turbulence. Furthermore, apertured Gaussian beams are more sensitive to turbulence than apertured laser beams with AMs and PFs. The average intensity of apertured laser beams with AMs and PFs may be even larger than that of apertured Gaussian beams due to turbulence. In particular, the influence of turbulence on the average maximum intensity of apertured laser beams with PFs and AMs may become serious if an unsuitable truncated parameter is chosen, which should be avoided in practice.     

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.     

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 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.     

Transformation of a hollow Gaussian beam into a ring-shaped beam using a phase aperture

Propagation of a hollow Gaussian beam diffracted by a circular phase aperture is studied without making the paraxial approximation. The analytical expression of the intensity of the apertured hollow Gaussian beam is presented in the far field. The influences of the truncation parameter and the order of hollow Gaussian beam on the intensity distributions are discussed. It is shown that a circular ?-phase aperture can be used to transform a hollow Gaussian beam into a ring-shaped beam in the far field with the appropriate parameters.     

Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a slightly misaligned optical system in a cylindrical coordinate system, an analytical formula for a Laguerre–Gaussian beam passing through such an optical system is derived. Furthermore, an approximate analytical formula is derived for a Laguerre–Gaussian beam passing through an apertured slightly misaligned optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Some analytical formulas are also given for a flattened Gaussian beam by expanding its field as a superposition of a finite series of Laguerre–Gaussian beams. PACS 42.25.Bs; 41.85.Ew; 41.85.Ct     

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.     

Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma

In this paper, diffraction pattern of a vortex carrying beam with a Gaussian background has been studied by using Fresnel-Kirchhoff diffraction integral, in the presence of third-order coma. Results of intensity distribution and encircled energy at the Gaussian plane have been presented for two values of the topological charge. Positional shift and splitting of the dark core have been investigated in detail. It is noticed that the diffraction pattern of a beam with double topological charge is affected more by comatic aberration in comparison to the beam with single topological charge. We have also verified our results by using the optical transfer function approach. Propagation of an apertured Gaussian background vortex beam through a π-phase shifter has also been studied for two values of the topological charge.     

Approximate analytical expressions of apertured broadband beams in the far field

The approximate analytical expressions of the apertured broadband beams in the far field with Gaussian and Laguerre-Gaussian spatial modes are presented.For the radially polarized Laguerre-Gaussian beam,the result reveals that the electromagnetic field in the far field is transverse magnetic.The influences of bandwidth(Γ) and truncation parameter(C 0) on the transverse intensity distribution of the Gaussian beam and on the energy flux distribution of radially polarized Laguerre-Gaussian beam are analysed.     

Propagation of a general-type beam through apertured aligned and misaligned ABCD optical systems

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, analytical formulae for the electric field of a general-type beam propagating through apertured aligned and misaligned ABCD optical systems are derived using the generalized Collins formulae, which provide a convenient way of studying the propagation of a variety of laser beams, such as Gaussian, cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, flat-topped, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, higher-order annular Gaussian, Hermite-sinh-Gaussian and Hermite-cos-Gaussian beams, through such optical systems. As numerical examples, the propagation properties of a cos-Gaussian beam through an apertured aligned or misaligned thin lens are studied.     

Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system

Propagation of a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry through an apertured optical system is investigated. By expanding the hard aperture function as a finite sum of complex Gaussian functions, some approximate analytical propagation formulas are derived for a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry propagating through an apertured paraxial general astigmatic (GA) optical system or an apertured paraxial misaligned stigmatic (ST) optical system. The derived formulas are very fast to compute. The results obtained by using the approximate analytical expressions are in a good agreement with those obtained by direct numerical integration. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of a circular or non-circular flat-topped beam through an apertured general optical system.     

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.     

Fractional Fourier transform of apertured paraboloid refracting system

The limitation of paraxial condition of paraboloid refracting system in performing fractional Fourier transform acts like an aperture, which makes the system different from ideal systems. With aperture expanded as the sum of finite complex Gaussian terms, a more practical approximate analytical solution of fractional Fourier transform of Gaussian beam in an apertured paraboloid refracting system is obtained and also numerical investigation is presented. Complicated and practical fractional Fourier transform systems can be constructed by cascading several apertured paraboloid refracting systems which are the simplest and the most basic units for performing more precise transform.