Goodman’s solutions work because they move from "ray tracing" to "Fourier transforming." When you design a spectrometer or a telescope, ask: What is the Optical Transfer Function (OTF) of this system?
Goodman assumes continuous functions. The moment you digitize a Fourier transform (FFT), you must respect the Nyquist limit. Fix: Ensure your aperture width ( \Delta x ) and wavelength ( \lambda ) satisfy ( \Delta x < \lambda z / (N \Delta x) ) in Fresnel simulations. introduction to fourier optics goodman solutions work
To illustrate what good solutions work looks like, consider a typical problem from Chapter 4 (Fresnel Diffraction): Goodman’s solutions work because they move from "ray
Problem 4.3 (paraphrased): A plane wave of wavelength λ illuminates an aperture with field transmittance t(x,y) = rect(x/a) rect(y/b). Using the Fresnel diffraction integral, derive the intensity pattern at a distance z. Problem 4
Even "correct" solutions can be misleading if you don't understand the context.