Fourier shifting theorem in diffraction

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SUMMARY

The discussion centers on the Fourier shifting theorem in the context of diffraction patterns produced by a one-dimensional aperture illuminated by a plane wave. When the aperture is translated along the x-direction, the diffraction pattern remains at its original position, with only a change in the spatial phase. This counterintuitive result highlights the distinction between the physical position of the aperture and the mathematical representation of its diffraction pattern. The original poster resolved their confusion regarding this phenomenon during the discussion.

PREREQUISITES
  • Understanding of Fourier transforms in optics
  • Knowledge of diffraction patterns and their formation
  • Familiarity with plane wave illumination
  • Basic concepts of spatial phase in wave optics
NEXT STEPS
  • Study the implications of the Fourier shifting theorem in optical systems
  • Explore the mathematical derivation of diffraction patterns from apertures
  • Investigate the effects of aperture translation on diffraction in higher dimensions
  • Learn about phase shifts in wavefronts and their impact on imaging systems
USEFUL FOR

Optical physicists, engineers working with imaging systems, and students studying wave optics will benefit from this discussion on the Fourier shifting theorem and its implications in diffraction.

blue_leaf77
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Suppose one has a simple aperture in one dimension across x direction (1D aperture). Illuminated by plane wave, this aperture will produce certain diffraction pattern which, at sufficiently large distance, is just the aperture's Fourier transform, and we place a detector to measure it. Now this aperture is translated along x direction by let's say ## a ##, so that the center is now located at ## a ##. According to Fourier shifting theorem, the diffraction pattern of this translated aperture will still be located at its previous position (before translation), with only minor difference that the spatial phase is tilted at some angle. But intuitively this is not true, if the aperture is translated so is the diffraction pattern. So how does it turn out counterintuitive like this? I hope somebody can solve this.
 
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Ah never mind, I solved it already.
 

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