Fourier shifting theorem in diffraction

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The discussion revolves around 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 theorem suggests that the diffraction pattern remains in its original position, with only a change in spatial phase. This seems counterintuitive, as one might expect the diffraction pattern to shift with the aperture. The original poster initially sought clarification on this concept but later indicated they resolved the confusion themselves. The discussion highlights the complexities of interpreting the effects of translation on diffraction patterns.
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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