【作者】 王长涛；

【导师】 吕百达；

【作者基本信息】 四川大学 ， 光学， 2002， 硕士

【摘要】 分数傅立叶变换自引入光学领域以后，它在光学，图象处理，信号滤波等方面的应用研究备受人们重视。将分数傅立叶变换的阶数由实数域扩展到复数域是非常自然的。本文的主要目的在于研究复数阶傅立叶变换光学实现及其数值计算方法。主要工作包括： 1．根据复数阶傅立叶变换的定义和厄米—高斯光束在复ABCD光学系统中传输规律，推导出了复数阶傅立叶变换在复ABCD光学系统中实现须满足的一般条件，并给出了可以实现一般复数阶傅立叶变换的光学装置。 2．大多数的复ABCD光学系统无法实现复数阶傅立叶变换。如果在复系统的输入和输出面上附加合适的透镜或(和)高斯光阑，则可以将其转换为复数阶傅立叶变换光学系统。 3．利用菲聂耳衍射积分公式，证明了在增益损耗介质中相隔一定距离的两个复球面上的光场分布可以用复数阶傅立叶变换联系起来。根据这一结论，光束在某些满足一定条件的充满增益或损耗介质的高斯反射镜腔的传输，可以视作连续的复数阶傅立叶变换过程。 4．由于复数阶傅立叶变换一般无法直接表示为傅立叶变换的形式，故无法直接利用快速傅立叶变换(FFT)算法作数值计算。利用复数阶傅立叶变换的阶数可加性，将其分解为实数阶傅立叶变换和阶数实部为1的复数阶傅立叶变换两步操作，则可以通过两次快速傅立叶变换得到数值计算结果。更多还原

【Abstract】 Since Fractional Fourier transforms were firstly introduced in optics in 1993, their application has been exploited in many research domains, such as beam propagation, image processing and signal filtering etc. The extension of real-valued order Fourier transforms to complex regime is natural. This dissertation is devoted to investigate complex-order fractional Fourier transforms and their optical implementation. The main work of the dissertation is summarized as follows.1. Based on the propagation formula of Hermite-Gaussian beams in complex ABCD optical systems, the condition for the implementation of complex-order Fourier transforms is derived. A simple optical setup is proposed to realize general complex-order Fourier transforms.2. It is hard to implement complex-order Fourier transforms for general complex ABCD optical systems, because of the severe conditions. However, we can make a complex-order Fourier transform system out of any complex ABCD optical system, if two lenses or (and) Gaussian apertures with appropriate parameters are appended to the input and output planes.3. By using of the Fresnel diffraction integral, it is demonstrated that the optical fields on two complex-spherical surfaces in a uniform gain medium separated by a distance can be related by complex-order Fourier transforms. The beam propagation between two Gaussian reflectivity mirrors filled with gain medium can be regarded as a complex-order Fourier transform process.4. Complex-order Fourier transforms can not be generally written in terms of the conventional Fourier transform, and so the numerical calculation can not be performed through the Fast Fourier transform (FFT) algorithm. One solution of the problem is to divide the transform into two passages, a real-valued order fractional Fourier transform and a complex-order Fourier transform with the real part of the order being one. Each of the passages can be calculated by single FFT. So numerical calculation results of complex-order Fourier transforms can be obtained after two steps of FFT.更多还原