Proving that a lens performs a perfect two-dimensional Fourier transform.
Chapter 5 introduces the thin lens as a phase transformation element, while Chapter 6 analyzes the frequency response of generalized imaging systems. The Thin Lens Transformation Goodman models a thin lens as a quadratic phase factor: Proving that a lens performs a perfect two-dimensional
In a standard 4f system architecture, the physical layout is structured as follows: tl(x,y)=exp[−ik2f(x2+y2)]t sub l open paren x comma y
Convert physical apertures into mathematical functions (Rect, Circ, Gaus). Problem 2-14 introduces the Wigner distribution, a powerful
tl(x,y)=exp[−ik2f(x2+y2)]t sub l open paren x comma y close paren equals exp open bracket negative i k over 2 f end-fraction open paren x squared plus y squared close paren close bracket
: This chapter lays the mathematical foundation. Problem 2-4 introduces the concept that a sequence of two Fourier transforms can produce an "image" with magnification, a crucial idea for understanding imaging systems. Problem 2-8 explores the conditions under which a simple cosinusoidal object yields a cosinusoidal image, providing deep insight into the nature of image formation. Problem 2-14 introduces the Wigner distribution, a powerful concept for analyzing signals in both space and frequency. The problems here are designed to build an intuitive as well as a mathematical understanding. Problems 2-1, 2-2, and 2-3, for example, rigorously prove fundamental properties of Dirac delta functions and Fourier transforms.
Solution: Using the lens equation and the definition of magnification, we get: