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I was reading about Holograms, and I read something to the effect of the following statement -

"In a hologram, any part of the hologram with sufficient size contains the whole of the stored information"

What exactly is the mathematics behind this? Does it have to do with the frequency domain representation of the object? Also, if someone could point me to a good source for reading about the math behind this stuff, i'd appreciate it very much. Thank you.

ramseysdream111
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  • Very good question. I also read this, and I nowadays believe this is wrong. Since holograms basically map amplitude to phase, I would say that the lateral bandwidth goes down, decreasing the resolution of the hologram if one only takes a part of the hologram. But i dont really know – lalala Nov 20 '19 at 18:07
  • @lalala thanks. thats interesting. Btw, do you know some text i can refer to that explains this without necessarily going into the implementation level details? – ramseysdream111 Nov 20 '19 at 18:11
  • Unfortunately no. I would also be intested in good sources (even implementation level) – lalala Nov 20 '19 at 18:21
  • It is a Fourier transform. Each area of it has the info on the entire object, but the amount of the info (resolution) depends on the size of the area. – safesphere Nov 20 '19 at 18:57

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This doesn't really answer your question but it might be interesting. Here is an image from Gabor's 1949 "Microscopy by Reconstructed Wavefronts" (which I think was basically the first paper on holography):

Clearly there is some pretty strong spatial corellation between the image and the hologram. So while it might be true that each bit of the hologram has some information about the whole image, it seems likely that different pieces of the hologram would display different parts of the image with different resolution. Maybe modern methods are different?

Mainly posting this because the question reminded me of this cool picture. Someone who actually knows about holograms please correct me.

Joe Schindler
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It is not necessarily correct to say that every piece of a hologram contains information about every part of the image. A great example is a hologram of a scene in which a large object is in front of a smaller object, thereby obscuring the smaller object as seen from some angles. Peeking through some parts of the hologram, you will see only the large object.

It is correct, though, if the scene is two-dimensional and every part of the object can be seen from every part of the hologram plane.

S. McGrew
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  • Okay, but how does that work mathematically? – ramseysdream111 Nov 21 '19 at 06:24
  • A Fourier representation of the interference pattern that forms a hologram really amounts to an angular spectrum of the scene. For a 2D scene of limited size far away, the angular spectrum of the scene is limited to a finite region of angle space. If another 2D scene is placed beyond the first, the first 2D scene blocks a portion of the angular spectrum of the 2nd scene that would otherwise be available to form the hologram, but the specific portion of the spectrum that's blocked depends on location along the recording plane. – S. McGrew Nov 21 '19 at 14:08
  • Okay, thank you! – ramseysdream111 Nov 21 '19 at 20:08