傅里叶自函数的维格纳分布及应用

傅里叶变换是现代光学发展的重要理论工具。自1991年Caola首次定义傅里叶自函数以来1,它在光学领域的应用研究日趋活跃。本文首先对傅里叶自函数定义进行扩展,再讨论其维格纳分布函数及其矩,研究它们在光学中的应用。最后推导出傅里叶自函数应用于光学变换器成象时的变换矩阵。

THE WIGNER DISTRIBUTION FUNCTION OF SELF FOURIER FUNCTION AND ITS APPLICATION

A self-Fourier function (SFF),according to Caola,is a function that is its own Fourier transform,i.e.f^F(u=fu).In this paper,the definition of self-Fourier function is generalized into f^F (u)=a₁f(a₂u),such that it can include a larger number of functions.For example,apart from exp (-πx²),exp (-dx²) with an arbitrary α can also be included under the generalized definition of SFF. The Wigner distribution function (WDF) of a one dimensional signal g (x) can be defined as two equivalent forms.WDF and the total energy of SFF is denoted.Fig.1 shows the WDF display of a SFF for the maximal-contrast mode.On the basis of the WDF property that area in the Wigner display is equivalent to SW (the space bandwidth product).One can recalculate SW_SFF=(Δx-Δx_c)Δx_c/2. Particular form of the WDF for the generalized definition of SFF is indicated as W(x,u)=|a₁|²∫f(a₂u+a₂u’/2)f (a₂u-a₂u’/2) exp (-2πixu’) du’.The moments of the WDF for the generalized definition of SFF is discussed.The different order moments of the WDF have very important applications,e.g. second-order moment of spatial coordiate x² is a measure of the spatial extent of the beam and second-order moment of the angular coordiate u² is a measure of its angular divergence.The beam quality factor Q=x²u²=(xu)² etc. The general form of the optical system matrix for which an input SFF transforms to an output SFF is found.Finally,as example of application,the bright soliton of the electromagnetic wave is treated for the generalized SFF.