Introduction To Fourier Optics Goodman Solutions Work -

PSF = np.abs(np.fft.fftshift(np.fft.fft2(pupil)))**2

Step 1 – Fresnel integral: ( U(x,y,z) = \frace^ikzi\lambda z e^i\frack2z(x^2+y^2) \iint t(\xi,\eta) e^i\frack2z(\xi^2+\eta^2) e^-i\frac2\pi\lambda z(x\xi+y\eta) d\xi d\eta )

Step 2 – Approximation for large z (Fraunhofer): The quadratic phase factor inside the integral ( e^i\frack2z(\xi^2+\eta^2) \approx 1 ) when ( z \gg \frack(a^2+b^2)2 ).

Step 3 – Separable integrals: ( U = \frace^ikzi\lambda z e^i\frack2z(x^2+y^2) \left[ \int_-a/2^a/2 e^-i2\pi x\xi/\lambda z d\xi \right] \left[ \int_-b/2^b/2 e^-i2\pi y\eta/\lambda z d\eta \right] ) introduction to fourier optics goodman solutions work

Step 4 – Evaluate: Each integral yields ( a \cdot \textsinc(a x/\lambda z) ) and ( b \cdot \textsinc(b y/\lambda z) ).

Step 5 – Intensity: ( I(x,y,z) = \left( \fracab\lambda z \right)^2 \textsinc^2\left( \fraca x\lambda z \right) \textsinc^2\left( \fracb y\lambda z \right) )

Why this is good: It shows approximations, separability, and units. A novice learns when the Fresnel → Fraunhofer transition occurs. PSF = np

Before understanding the solutions, one must respect the problem. Goodman’s text is unique because it refuses to separate the math from the physics.

Most students pick up the book expecting a simple repetition of Fresnel and Fraunhofer diffraction. Instead, Chapter 1 introduces the linear systems approach. Suddenly, a pinhole camera is a convolution kernel; a lens is a quadratic phase factor. By Chapter 5, you are using the ambiguity function to analyze partially coherent light.

The core difficulty: Goodman writes for the "radar engineer" as much as the "optics engineer." He visualizes light as a complex amplitude passing through a series of linear filters. The Fourier transform is no longer just a math tool; it is the physical mechanism of diffraction. Chapter 4 (Fresnel and Fraunhofer Diffraction) is typically

So, when we ask "how do the solutions work?" we are really asking: "How do we map physical optics onto linear systems theory?"

Chapter 4 (Fresnel and Fraunhofer Diffraction) is typically where students get stuck. The transition from the Rayleigh-Sommerfeld diffraction integral to the statement “The diffraction pattern is the Fourier transform of the aperture” is mathematically elegant but physically abstract. Goodman’s problems force you to prove this—not just state it.

Before discussing solutions work, one must understand the pedagogical hurdles the textbook presents.

When reviewing a solution, ask yourself: