Abstract

Continuum extensions of common dual pairs of operators are presented and consolidated, based on the fractional Fourier transform. In particular, the fractional chirp multiplication, fractional chirp convolution, and fractional scaling operators are defined and expressed in terms of their common nonfractional special cases, revealing precisely how they are interpolations of their conventional counterparts. Optical realizations of these operators are possible with use of common physical components. These three operators can be interpreted as fractional lenses, fractional free space, and fractional imaging systems, respectively. Any optical system consisting of an arbitrary concatenation of sections of free space and thin lenses can be interpreted as a fractional imaging system with spherical reference surfaces. As a special case, a system departing from the classical single-lens imaging condition can be interpreted as a fractional imaging system.

© 2003 Optical Society of America

Simplified fractional Fourier transforms

Soo-Chang Pei and Jian-Jiun Ding
J. Opt. Soc. Am. A 17(12) 2355-2367 (2000)

Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

Haldun M. Ozaktas, Billur Barshan, David Mendlovic, and Levent Onural
J. Opt. Soc. Am. A 11(2) 547-559 (1994)

Fractional Fourier optics

Haldun M. Ozaktas and David Mendlovic
J. Opt. Soc. Am. A 12(4) 743-751 (1995)

Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms

Soo-Chang Pei and Jian-Jiun Ding
J. Opt. Soc. Am. A 20(3) 522-532 (2003)

Perspective projections in the space-frequency plane and fractional Fourier transforms

İ. Şamil Yetik, Haldun M. Ozaktas, Billur Barshan, and Levent Onural
J. Opt. Soc. Am. A 17(12) 2382-2390 (2000)