The concept of an extended fractional Fourier transform (FRT) is suggested. Previous FRT’s and complex FRT’s are only its subclasses. Then, through this concept and its method, we explain the physical meaning of any optical Fresnel diffraction through a lens: It is just an extended FRT; a lens-cascaded system can equivalently be simplified to a simple analyzer of the FRT; the two-independent-parameter FRT of an object illuminated with a plane wave can be readily implemented by a lens of arbitrary focal length; when cascading, the function of each lens unit and the relationship between the adjacent ones are clear and simple; and more parameters and fewer restrictions on cascading make the optical design easy.
© 1997 Optical Society of America
Original Manuscript: October 2, 1996
Revised Manuscript: March 27, 1997
Manuscript Accepted: June 11, 1997
Published: December 1, 1997
Jianwen Hua, Liren Liu, and Guoqiang Li, "Extended fractional Fourier transforms," J. Opt. Soc. Am. A 14, 3316-3322 (1997)