A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary real-order fractional Fourier transforms and Fresnel diffraction. Its computational complexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the sampling resolutions in both x and u space and zoom in on any portion of the data of interest. Computational results are compared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10−12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate near-field patterns directly behind the aperture, 0≤z<d2/λ. It compensates another algorithm for Fresnel diffraction that is limited to z>d2/λN [J. Opt. Soc. Am. A 15, 2111 (1998)]. Experimental results from waveguide-output microcoupler diffraction are in good agreement with the calculations.
© 2000 Optical Society of America
(050.1590) Diffraction and gratings : Chirping
(050.1940) Diffraction and gratings : Diffraction
(070.2590) Fourier optics and signal processing : ABCD transforms
(350.6980) Other areas of optics : Transforms
Xuegong Deng, Bipin Bihari, Jianhua Gan, Feng Zhao, and Ray T. Chen, "Fast algorithm for chirp transforms with zooming-in ability and its applications," J. Opt. Soc. Am. A 17, 762-771 (2000)