基于回转椭球波函数的光学系统分析

Optical System Analysis Based on Prolate Spheroidal Wave Functions

回转椭球波函数在有限空间域和无限空间域内都是一组完备正交函数集,适合分析孔径尺寸有限的实际光学系统。线性正则变换是一种重要的时频分析工具,同时菲涅耳变换也是一种特殊的线性正则变换,因此线性正则变换可以模拟一个光学系统。研究了有限空间域和有限频域条件下的回转椭球波函数的补偿线性正则变换,以补偿线性正则变换模拟一个二维光学系统,并以回转椭球波函数作为信号函数,分析了信号通过该系统的能量损失情况,根据椭球波函数本征值性质,其本征值反映了椭球波函数的能量保存比。数值计算结果表明信号函数通过该系统的能量比与通过函数本征值的能量比的估计值一致,表明了该方法的有效性。

The prolate spheroidal wave function is a set of complete orthogonal functions set in both the finite and infinite space domain, which is suitable to analyze the practical optical systems with finite aperture size . The linear canonical transform is a kind of important time-frequency analysis tool, and the Fresneltransform is a special case of linear canonical transform. So, the linear canonical transform can model optical system. The offset linear canonical transform of prolate spheroidal wave functions in finite space domain and frequency domain is instituted, and the offset linear canonical transform models a two dimension optical system. Using the spheroidal wave functions as signal functions, the signal＇s energy loss after passing through the system is analyzed. Based on prolate spheroidal wave functions＇ eigenvalue property, the eigenvalue reflects the energy-preservation ratio. The numerical calculation result suggests that the signal function＇s energy ratio after passing through the system is consistent with the energy ratio＇s estimation value gained by the function＇s eigenvalue, which indicates the validity of this method.