The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner distribution, and the ambiguity function are described, leading to a model for effects of Fresnel number in terms of a rotation in phase space. The uncertainty principle is discussed, including the significance of the beam propagation factor M<sup>2</sup> and the width of optical fiber beam modes. Calculation of the moments in terms of the modulus and the phase of the illuminating wave is presented, and the use of the Kaiser–Teager energy operator is also described.
© 2000 Optical Society of America
(060.2270) Fiber optics and optical communications : Fiber characterization
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.2590) Fourier optics and signal processing : ABCD transforms
(140.3300) Lasers and laser optics : Laser beam shaping
(260.1960) Physical optics : Diffraction theory
Colin J. R. Sheppard and Kieran G. Larkin, "Focal shift, optical transfer function, and phase-space representations," J. Opt. Soc. Am. A 17, 772-779 (2000)