Forgetting that coordinate scaling or coordinate rotations break shift-invariance. Chapter 3: Foundations of Scalar Diffraction Theory
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The core of Goodman's work is the idea that optical systems can be treated as linear invariant systems. This allows us to apply the same mathematical tools used in electrical engineering—like the Fourier transform—to the propagation of light.
A shift in the input plane coordinates results in a corresponding shift in the output plane coordinates. This allows the system to be characterized by an Impulse Response, known in optics as the Point Spread Function (PSF). 2. Scalar Diffraction Theory introduction to fourier optics goodman solutions work
Introduction to Fourier Optics by Joseph W. Goodman: Solutions and Work
( 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] )
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tl(x,y)=exp[−jk2f(x2+y2)]t sub l open paren x comma y close paren equals exp open bracket negative j k over 2 f end-fraction open paren x squared plus y squared close paren close bracket
“Use the Fourier transform of rect = sinc. Then intensity is sinc²... done.”
The backbone of Fourier optics is the two-dimensional Fourier transform. It maps a complex field distribution from the spatial domain to the spatial frequency domain The core of Goodman's work is the idea
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The or equation you are trying to solve? Whether you are analyzing a coherent or incoherent system?