Hankel transform of focus wave modes

The Hankel transform generally used to analyse the transverse behaviour of Gaussian beams also supplies new classes of scalar focus wave modes. Received: 4 January 1999 / Revised version: 4 March 1999 / Published online: 2 June 1999     

A quasi-discrete Hankel transform for nonlinear beam propagation

We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion (DQDHT) in this paper. By making use of the property that the zero-order Bessel function derivative J' ₀(0)=0, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parseval theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.     

On the general expressions of finite Hankel transform

The general expressions of finite Hankel transform are naturally deduced with the help of the property of Bessel functions.The equations in this paper can degenerate into three kinds of boundaries since all the coefficients in the boundary conditions are taken into consideration.The results can be adopted in solving physics problems involving the finite Hankel transform.     

Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform

We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively.     

Eigenmodes of fractional Hankel transform derived by the entangled-state method

Using the entangled-state method in quantum mechanics, we find that the eigenmodes of the fractional Hankel transform are two-variable Hermite-Gaussian functions that can be rewritten in a clearer form as Laguerre polynominals.     

离散汉克尔变换及其在高效激光整形中的应用

发展了离散零阶汉克尔变换并应用到光场均匀化的相位元件设计,结果显示出比二维傅氏变换更为理想和精确.     

Propagation characteristics of elegant Laguerre-Gaussian beams in the fractional Hankel transform plane

The fractional Hankel transform (FRHT) system is applied to study the transformation properties of elegant Laguerre-Gaussian beams (ELGBs). An analytical formula is derived for the FRHT of ELGBs based on the definition of the FRHB. By using the derived formula and the definition of the kurtosis parameter, the analytical propagation expression for the kurtosis parameter of ELGBs in the FRHT plane is derived. Some detailed numerical examples are given to illustrate the analytical results. The results show that the intensity distribution and the kurtosis parameter of elegant Laguerre-Gaussian beams in the fractional Hankel transform plane not only depend on order p and index m of Laguerre polynomial, but also evolve periodically with fractional order p of fractional system.     

Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation

Starting from a complex fractional Fourier transformation [Opt. Lett. 28, 680 (2003)], it is shown that the integral kernel of a fractional Hankel transformation is equivalent to the matrix element of an appropriate operator in the charge-amplitude state representations; i.e., the fractional Hankel transformation is endowed with a definite physical meaning (definite quantum-mechanical representation transform).     

基于Dini展开的高阶Hankel变换及其在光束传输中的应用

基于Dini级数展开, 导出了p(>0) 阶准离散Hankel变换(the pth-order quasi-discrete Hankel transform based on Dini series expansion, pDQDHT)算法, 并给出了该算法在光束传输中的应用. 通过不同输入函数分别对pDQDHT算法进行测试及光束通过透镜传输的应用实例, 结果表明: pDQDHT算法不仅精度高于现有的Hankel变换算法, 可以进行多次正、逆变换, 能广泛应用于光束分步传输问题, 而且执行速度也与一般的快速Hankel变换算法相当. 关键词： Dini级数展开 高阶Hankel变换 光束传输     

离散Hankel变换

首次提出了离散Hankel变换,从理论上给出了离散Parseval定理和变换矩阵。引入了抽样权重因子。并讨论了抽样原则及S因子对计算精度的影响。     

The forward and backward projection of acoustic fields from axisymmetric ultrasonic radiators using impulse response and Hankel transform techniques

A generalized impulse response formulation to evaluate the harmonic pressure field of ultrasonic planar vibrators having axisymmetric nonuniform surface velocity distributions is presented. The harmonic pressure is expressed as a Fourier transform of a generalized impulse response which is a function of the spatially nonuniform velocity of the vibrator. A backward projection method is then developed to reconstruct the normal surface velocity of axisymmetric vibrators from harmonic field pressures using an angular spectrum or Hankel transform formulation. The numerical accuracy of the backward projection technique is evaluated using the impulse response formulation to evaluate the pressure fields for several velocity distributions on disk vibrators. Experiments were performed to reconstruct the velocity distributions over the surface of a uniformly driven piezoelectric ceramic disk and ceramic ring using farfield measurements of the complex pressure. The experimental results were in good agreement with theoretical results based on the electrode patterns of the transducers.     

On Quantum Mechanical Hankel Transform and Its Applications

Based on our preceding works of how to relate the mathematical Hankel transform to quantum mechanical representation transform and how to express the Bessel equation by an operator identity in some appropriate representations we propose the concept of quantum mechanical Hankel transform with regard to quantum state vectors. Then we discuss its new applications.     

On Quantum Mechanical Hankel Transform and Its Applications

Based on our preceding works of how to relate the mathematical Hankel transform to quantum mechanical representation transform and how to express the Bessel equation by an operator identity in some appropriate representations we propose the concept of quantum mechanical Hankel transform with regard to quantum state vectors. Then we discuss its new applications.     

基于快速傅里叶变换和汉克耳变换的逆阿贝尔变换

激光等离子体实验中的轴对称的Ｘ光的体发射强度及光学全息干涉的测量,需要采用逆阿贝尔（Ａｂｅｌ）变换。用快速傅里叶变换（ＦＦＴ）和汉克耳（Ｈａｎｋｅｌ）变换的算法数值求解逆阿贝尔变换,具有精度高、可在频域内滤波的特点。     

The representation of bound state wavefunctions by spherical Hankel functions

The representation of a nucleon bound state by a single Hankel function is generalised such that the bound state is expanded as a linear combination of Hankel functions of the samel-value. The singularity at the origin due to the use of Hankel functions is removed. The arguments and coefficients of the expansion are determined by a variational method.     

利用汉克尔变换设计高斯光束整形衍射元件的应用研究

衍射光学元件由于可以实现对高斯光束的整形而被重视,其通常的设计方法为G-S算法,由于使用傅里叶变换运算量大、费时长,将快速汉克尔变换应用到这些算法中可以极大地提高运算速度,节省运算时间,为设计复杂的光束整形元件提供了高效、可行的方法。本文利用该种方法设计针对中心波长为775 nm、光束束腰口径为6 cm的激光器,成功设计了一个具有二阶相位的折衍混合光学元件。仅单独这一片元件,既可在距离其35 m处得到一半径为200μm的圆形平顶光斑,均方根误差D0.021。当抽样值取2~15时,在普通PC机上运行时间仅为20.05 s,大大节省了优化设计时间(整个优化设计过程往往需要几十次甚是上百次这种运算)。同时利用离子刻蚀技术加工了该折衍混合元件,并进行了实际测试,结果与设计值基本相符,整形效果较好。这种单片的整形元件不仅整形效果好,还有利于与激光器的集成,简化系统的调节。     

A Generalized Hankel Transformation Derived by Parametrized Entangled State Representations

Using the parametrized entangled state representations we have found a generalized Hankel transformation with the integral kernel being a combination of Bessel functions. This generalized Hankel transformation corresponds to the appropriate quantum mechanical representation transformation.     

A Generalized Hankel Transformation Derived by Parametrized Entangled State Representations

Using the parametrized entangled state representations we have found a generalized Hankel transformation with the integral kernel being a combination of Bessel functions. This generalized Hankel transformation corresponds to the appropriate quantum mechanical representation transformation.