Propagation characteristics of laser beams with amplitude modulations and phase fluctuations in apertured fractional Fourier transforming system

The apertured fractional Fourier transforming system is introduced and applied to treat the propagation of Gaussian beams with amplitude modulations and phase fluctuations. Based on the treatment that a rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the mutual intensity distribution of laser beams with amplitude modulations and phase fluctuations passing through the apertured fractional Fourier transforming system are obtained. Some numerical simulations are illustrated for their propagation properties.     

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 properties of cosh-squared-Gaussian beam through fractional Fourier transform systems

Based on Collins integral formula and the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of cosh-squared-Gaussian beam passing through ideal and apertured FRFT systems have been studied, and some comparisons between using the methods of analytical formula and diffraction integral formula have been done. Further, the studies indicate that the normalized intensity distributions on FRFT plane depend on the fractional order, truncation parameter and initial beam parameter Ω. Variations of normalized intensity distributions with FRFT order are periodic: when the impact of aperture cannot be ignored, the variation period is 4; and when the impact of aperture can be ignored, the variation period is 2.     

Nonparaxial propagation of a vectorial apertured off-axis Lorentz beam

Based on the vectorial Rayleigh--Sommerfeld integral formula and the complex Gaussian expansion of the hard-edge aperture function, an analytical propagation expression for a nonparaxial vectorial off-axis Lorentz beam passing through a rectangular aperture is derived. The unapertured case, the far field expression and the scalar paraxial result are also presented as special cases of the general formulae, respectively. Some numerical examples are also given to show the propagation characteristics of a nonparaxial vectorial off-axis Lorentz beam through a rectangular aperture. It is indicated that the f parameter, the off-axis displacement and the truncation parameter all play an important role in determining nonparaxial propagation behaviour.     

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.     

The propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian beams passing through fractional Fourier transformation systems

Based on the Collins diffraction integral formula and irradiance moments, the propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian (HChG) beams passing through fractional Fourier transformation systems with spherically aberrated lenses are analyzed in detail. The closed-form expressions for the propagation equation and kurtosis parameter of HChG beams passing ideal fractional Fourier transformation systems are also obtained. By using an efficient algorithm some numerical calculations are also performed in this paper. The results show that the spherically aberrated lenses greatly influence on the intensity distribution and kurtosis parameter of HChG beams passage through fractional Fourier transformation systems. The intensity distribution and kurtosis parameter of HChG beams passage through two types of Lohmann's systems with spherically aberrated lenses, respectively, are no longer equal even in the case of the same fractional order and the same spherical aberration coefficients. The fundamental periods of the two types of Lohmann's systems with spherically aberrated lenses are different, 4 for the type I and 2 for the type II. Therefore, the two optical setups for implementing the fractional Fourier transformation are no longer equivalent in the presence of spherically aberrated lenses.     

Propagation of Lorentz and Lorentz-Gauss beams through an apertured fractional Fourier transform optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derive approximate analytical formulae for Lorentz and Lorentz-Gauss beams propagating through an apertured fractional Fourier transform (FRT) optical system. As an application example, properties of a Lorentz-Gauss beam in the FRT plane after propagating through a squarely or annularly apertured FRT optical system are studied numerically. The results obtained using the approximate analytical formula are in good agreement with those obtained using numerical integral calculation. The FRT optical system provides a convenient way for laser beam shaping.     

Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of an anomalous hollow beam (AHB) passing through an apertured fractional Fourier transform (FRT) system are derived. By using the approximate analytical formulae and diffraction integral formulae, the propagation properties of an AHB in circular and rectangular apertured FRT system are studied numerically. The results show that this method provides a convenient tool for studying the propagation properties of an AHB passing through apertured FRT system, and the apertured FRT system can be applied to laser beam shaping conveniently.     

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.     

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.     

Laser beam brightness of apertured optical resonators

We model the fundamental mode of a circularly symmetric plano-concave resonator where one mirror is apertured by a super-Gaussian aperture, while the other is apertured by a π/2 phase aperture. The aim of the paper is to find the best values of the apertured cavity parameters allowing the highest output beam brightness. For that we take into account loss level, fundamental mode volume and beam quality factor M². We demonstrate that for a given loss level, the phase aperture is able to increase the laser beam brightness, with a rate of about 60% when its diameter is properly chosen.     

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.     

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.     

Coincidence fractional Fourier transform with a stochastic electromagnetic Gaussian Schell-model beam

Based on classical theory of optical coherence and polarization, coincidence fractional Fourier transform (FRT) with a stochastic electromagnetic Gaussian Schell-model (EGSM) beam is investigated. Dependences of the quality and visibility of the coincidence FRT pattern of an object on the degree of polarization and correlation coefficients of the EGSM beam are studied numerically. It is shown that the quality and visibility of the coincidence FRT pattern of an object are determined by the polarization and coherence of the EGSM beam together.     

基于全光时域分数阶傅里叶变换的光脉冲最小损伤传输新方法

提出了一种基于全光分数阶傅里叶变换(optical fractional fourier transformation,OFRFT)的关于色散和自相位调制对光脉冲传输影响的分析方法,并基于该方法给出了一种新的光脉冲在光纤中传输最小损伤的方法,通过在发射端将输入的高斯光脉冲做OFRFT然后再送入到光纤中去传输,可以提高光脉冲抵抗色散和非线性的能力,仿真发现当色散占主要作用的时候,负阶数的分数阶傅里叶变换可以很大程度上减小光脉冲的展宽;当自相位调制作用占主要的时候,正阶数的分数阶傅里叶变换可以消除自相位调制作 关键词： 光脉冲传输 分数阶傅里叶变换 色散 自相位调制     

A novel Fourier based 3D full-vectorial beam propagation method

In this paper a Fourier based full-vectorial beam propagation method (BPM) with perfectly matched layer (PML) absorbing boundary conditions is presented. The method starts with the slowly varying envelope coupled wave equations, including coordinate-stretching type PMLs, which are discretised using the Galerkin method with Fourier basis functions in the transversal direction. The proposed method is then assessed in a wide variety of situations, including 2D propagation of TE and TM waves and 3D full-vectorial situations. The obtained results show that good accuracy can be obtained even for complex strongly-guiding waveguides with a small number of basis functions.     

分数傅里叶变换面上余弦-高斯光束的变换特性

 利用Wigner分布函数的方法，研究了余弦-高斯光束的分数傅里叶变换特性。导出了余弦-高斯光束在分数傅里叶变换面上光强分布和束宽的解析计算公式，并对此进行了数值模拟计算。研究表明：分数傅里叶变换阶数对余弦-高斯光束的光强分布有明显影响，余弦-高斯光束的轴上光强随分数傅里叶变换阶数呈周期性变化，束宽随分数傅里叶变换阶数也呈周期性变化，周期为2；对给定调制参数的余弦-高斯光束，通过适当选取分数傅里叶变化阶数可以获得平顶的光强分布。     

Dual beam encoded extended fractional Fourier transform security hologram with in-built repositioning

This paper describes a simple method for making dual beam encoded extended fractional Fourier transform (EFRT) security holograms. The hologram possesses different stages of encoding so that security features are concealed and remain invisible to the counterfeiter. These concealed and encoded anticounterfeit security features in the security hologram can only be read through a key hologram. Key hologram also facilitates in-built repositioning of security hologram. The method of fabrication, the principle of reconstruction and the experimental results are presented.