3 Scattering and the Spatial Frequency Representation COLIN JR SHEPPARD Division of Bioengineering, 9 Engineering Drive 1, National University of Singapore, Singapore, 117576, and Department of Diagnostic Radiology, National University of Singapore, 5 Lower Kent Ridge Road, Singapore 119074. 3.1. Introduction This chapter describes surface scattering in terms of direction cosines, or, equiv- alently, in terms of three-dimensional (3D) spatial frequencies. 1 This approach results in simplified expressions and provides a physical interpretation. It is also directly applicable to investigation of 3D imaging (including holography, tomog- raphy, microscopy, interferometry, surface profiling and shape measurement). It is notable that different areas of optics tend to have their own adherents with their own literature, and few connections between the areas are exploited. Imag- ing is usually based on diffraction theory, which is distinguished from scatter- ing theory mainly on the basis that diffraction takes place from objects large compared with the wavelength and scattering from structures of the order of the wavelength in size. This distinction breaks down for microscopic imaging where the resolution limit can be sub-wavelength. Both diffraction and scatter- ing also have a geometrical optics limit for large structures, smooth surfaces, and so on. 3.2. Plane Waves The amplitude of a scalar plane wave at the point r = x i + y j + z k can be written as (the time dependence exp (−iωt ) is suppressed) U (r) = exp[in 0 k ( px + qy + sz )] = exp(in 0 k p · r), (1) where k = 2π/λ, n 0 is the refractive index of the medium, and the vector p is p = pi + q j + s k, (2) with p, q , s direction cosines so that s =±  1 − ( p 2 + q 2 ), (3) 60