非偏振贝塞尔高斯光束的球散射

无衍射光束球散射性质的研究目前一般采用贝塞尔光束,但是贝塞尔光束在物理上是不可实现的。贝塞尔高斯光束作为近似无衍射光束,是亥姆霍兹方程在傍轴条件下的解,并且可以用激光振荡器直接产生,但其光束宽是有限的。应用傅里叶变换,平面波谱展开和球面矢量波函数展开法,推导了非偏振贝塞尔高斯光束的球散射远场的无量纲散射函数。通过数值模拟,对非偏振的贝塞尔高斯光束与贝塞尔光束,高斯光束的球散射远场进行了比较,比较发现:当球散射体偏离光轴时,非偏振贝塞尔高斯光束跟贝塞尔光束散射远场的差异主要是散射强度的差异,但是散射极点所在的方向基本保持不变;贝塞尔高斯光束和贝塞尔光束的散射在光束圆锥角方向上比较显著,但高斯光束的前向散射比较显著。     

带宽对脉冲贝塞尔光束传输的影响

从惠更斯-菲涅耳衍射积分公式出发得到具有高斯型频谱分布的脉冲贝塞尔光束频率域的传输表示式,并使用傅里叶变换最后推导了脉冲贝塞尔光束时间域的传输公式。根据得到的公式通过具体数值计算对脉冲贝塞尔光束在自由空间中的传输进行了研究。计算结果表明脉冲光束的带宽对脉冲贝塞尔光束传输中的脉冲时间延迟和脉冲宽度产生影响。具体表现为脉冲光束带宽的增宽使脉冲时间延迟增大,并使脉冲时间波形的后沿拖长,从而导致脉冲宽度增宽。脉冲时间延迟和脉冲宽度还随传输距离发生变化。脉冲光束传输距离越大,脉冲时间延迟越大,脉冲宽度变得越宽。     

用多项式近似逆滤波函数实现场曲修正

根据光电成像系统传递函数,确定逆滤波函数对获得图像进行反降质恢复.将连续逆滤波函数按泰勒级数展开,用多项式近似表示.对逆滤波函数的多项式作反傅里叶变换,得到图像恢复在空间域中的近似运算表达,即图像信号及其各阶导数的线性组合,而不是复杂的反卷积操作.该方法特别适合空间移变系统的图像恢复.详细分析了这种方法的原理,推导了运算公式,并给出了对场曲图像处理的结果.     

三类边界条件下的贝塞尔-傅里叶矩的图像分析

提出了一种基于三类边界条件下贝塞尔函数的构造矩—贝塞尔-傅里叶矩,其定义在极坐标系下,可视为一种整体正交复数矩;该矩的正交多项式有较多零点,且多数呈均匀分布.通过对26个大写字母二值图像的重构及1260幅灰度蝴蝶图像的分类实验来验证所提出矩的有效性,同时提取三类边界条件下贝塞尔-傅里叶矩的不同阶矩作为图像分析的特征值(图像描述质)来表征图像.理论分析和实验结果表明:与正交傅里叶-梅林矩和泽尼克矩相比,三类边界条件下贝塞尔-傅里叶矩更适用于图像的分析和旋转不变性的目标识别,且图像重构准确度及不变性识别准确性均更优.     

圆孔夫琅禾费衍射的光强度研究

本文从惠更斯-菲涅尔原理出发,得到圆孔夫琅禾费衍射的光振幅分布的积分表达式,应用贝塞尔函数展开式对积分式进行了计算,得出了衍射光的强度分布,从而推出爱里斑的半角公式.该文还用Mathematica软件对结果进行了图示和数值计算.     

小瓣数贝塞尔声束的二次谐波

采用高斯展开法,研究了具有三个瓣的小瓣数贝塞尔声束的基波和二次谐波的传播性质.根据准线性近似下Khokhlov-Zabolotskaya-Kuznetsov方程的积分形式解,分析了媒质的声吸收(声衰减)对波束形状的影响.结果表明:吸收参量对二次谐波径向分布有很大的影响.当瓣数较少时,实际的有限孔径贝塞尔声束二次谐波仍然具有理想无限大孔径贝塞尔声束的主要特征,在近场无衍射区,径向几乎无衍射. 关键词： 二次谐波 贝塞尔声束 声吸收 有限孔径声束     

离散傅里叶方法分析环缝透镜产生无衍射光束

使用离散傅里叶方法分析环缝透镜系统产生无衍射(贝塞尔)光束。将基尔霍夫衍射积分公式与菲涅耳衍射积分公式转换成傅里叶变换的形式,在环缝透镜系统中用傅里叶方法描述观察面的光强分布,对衍射面进行数据抽样,导出观察面光强分布的离散化傅里叶公式;通过设置相关参数并使用MATLAB软件进行数值模拟,得到不同位置下的贝塞尔光强分布。设计环缝透镜系统进行实验验证,数值模拟和实验结果匹配度较高,表明离散傅里叶方法应用于环缝透镜系统是可行的。     

广义参量Wigner分布函数:一种新的信号双域表示方法

本文基于分数傅里叶变换的概念,提出了由三个广义参量p1、p2和p3所表征的Wigner分布函数-广义参量Wigner分布函数,然后对其性质进行了讨论,并指出广义参量Wigner分布函数也属于一般的Cohen双线性类,而部分相干光场的互谱密度、传统的Wigner分布函数以及分数Wigner分布函数都可以作为广义参量Wigner分布函数在其广义参量取特殊值时的特例而得到,表明广义参量Wigner分布函数具有比传统的Wigner分布函数更强的信号表征能力.     

用Laplace变换求Hartmann势的精确解及波函数的递推关系

本文求解了在球坐标下Hartmann势的Schrdinger方程,得到了能量方程和归一化的波函数.用Laplace变换使径向的二阶微分方程退化为一阶微分方程,直接积分后用级数展开,应用Laplace逆变换得出本征函数.讨论了径向本征函数的像函数的递推关系,从而得出径向波函数的递推关系.     

从振动波形研究贝塞尔函数和勒让德函数的渐近关系

通过圆平面和球面上对称振动的分析,比较了零阶贝塞尔函数和勒让德函数曲线,发现两者形状有明显的相似性.在进行振幅变换后,两者曲线几乎重合.从数学和物理角度解释了圆平面和球面上振动波形相似的原因,导出了零阶贝塞尔函数和勒让德函数的渐近关系,并得到了贝塞尔函数零点的一个估计公式.     

水平层状各向异性介质中电磁场并矢Green函数的一种高效算法

利用高阶窗函数结合连分式展开等技术研究并建立一种水平层状各向异性介质中电磁场并矢Green函数的快速有效算法. 首先借助于高阶窗函数将构成并矢Green函数的Sommerfeld积分转化成广义快速下降路径上积分,并给出高阶窗函数Hankel变换的一种新的更高阶幂级数展开式以及严格的Lommel函数表达式,以满足在全空间上高精度计算并矢Green函数的要求. 在此基础上,用Bessel函数的零点将积分路径划分成一系列小区间并通过改进的自适应Gauss求积公式确定各个小区间上的积分值,然后引入连分式展开法对各个区间上的积分值求和,从而使整个积分的收敛效率得到大大提高. 最后通过数值结果验证本方法的有效性. 关键词： 高阶窗函数 连分式展开 并矢Green函数 层状各向异性介质     

On the probability density of the envelope of a sum of random vectors

A most efficient way to calculate the probability density for the projection of the sum of n sinusoidal waves with differing amplitudes and phases that are uniform in (0, 2π) is to expand it in a Fourier series. The corresponding series for the envelope of such a series is known to be a Fourier-Bessel series. It is difficult, however, to calculate moments of the envelope from this series. It is shown that by relating the probability density of the envelope to that of the projection one can find moments of the envelope in an easily computable form. It is also possible to calculate an approximate form for the density function for the envelope amplitude near its maximum value. A generalized expansion is developed, valid when the probability density function depends on both angle and radial distance.     

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     

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.     

Generating Generalized Bessel Equations by Virtue of Bose Operator Algebra and Entangled State Representations

With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.     

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 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.     

用改进的传输线算法计算水平层状横向同性地层中海洋可控源电磁响应

利用二维Fourier变换与电磁场分解技术将层状横向同性地层中Maxwell方程转化成两个独立的关于横磁(TM)波和横电(TE)波的传输线方程; 借助传输线理论与叠加原理, 仅利用电流源传输线Green函数得到TM波和TE波的解, 改进传输线算法, 建立横向同性地层中频率-波数域电流源电场和磁场并矢Green函数的新算法与新的解析表达式, 提高海洋可控源电磁响应数值模拟效率. 在此基础上, 利用传输线Green函数的基本解以及边界条件, 推导出广义反射系数与振幅递推公式, 得到各个地层中传输线Green函数的解析解; 然后利用Fourier逆变换与Bessel公式将海洋可控源电磁响应表示为Sommerfeld形式的积分, 借助三次样条插值与Lommel积分公式快速计算其数值解. 通过数值模拟结果考察工作频率以及地层各向异性电阻率变化等对海洋电磁响应的影响. 关键词： 传输线法 横向同性地层 海洋可控源电磁 Sommerfeld积分     

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.     

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.