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I've been browsing some Digital Holography papers these days, and have come across this fundamental question.

When you reconstruct the complex amplitude for the object image, you use e.g. Fresnel Transform to simulate diffraction.

The thing is, one of the parameters in this process is the distance, d, between the holographic plate (or CCD) and the object.

However, the whole point of Digital holography, I believe, is to find out the depth profile of the object, that is to say, the value of d.

We wouldn't have to do DH in the first place if we had known the precise (down to nanometric realm) value of the distance!

Could anyone clarify this for me?

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You can reconstruct the wavefront at any distance $z$. If you choose $z$ the same as the location of the object $d$, it will appear in focus. If you choose $z$ larger or smaller than $d$, it will look like out of focus (blurred). This has the same effect as changing the focus on a usual optical camera.

In fact this focus thing is equivalent to a lens: the the Fresnel transform includes a term identical to the phase modulation of a lens of focal length $z$. The image obtained is the same that a lens would provide.

Digital holography can actually do more than that. In some conditions, you can obtain the phase of the wavefront at any depth, thus infer the deformation of that wavefront at a precision better than the light wavelength. This is useful for studying deformations of microscopic objects.

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So you can reconstruct at any depth (as long as it's not too far out of focus), then propagate forwards and backwards and find out what the focal distance actually is. SO the algorithm can infer $d$ - I believe this is what you're saying - it's just not spelt out. BTW I don't believe you have to delete anything - your answer is sound - just spell out the inference of $d$ bit as the OP was asking this. – WetSavannaAnimal aka Rod Vance Aug 23 '13 at 6:58
I used a wrong definition of $d$. Now corrected. You can reconstruct the wavefront at any distance $z$. If $z=d$ it is in focus. You may indeed be able to write an algorithm that finds $d$ by obtaining the best focus. – fffred Aug 23 '13 at 7:10
Okay, so I see that the choice of right d is a matter of focus. – Seong Joon Oh Aug 23 '13 at 7:36
Let me explain how I have understood your comments. Suppose we are trying to microscope a bump with height 200nm. The top of the bump is 200000000.nm away from the hologram and the background is 200000200nm away from the hologram. If you set d=200000000nm, the focal point is upon the top of the bump, so the reconstructed image does not show a focused image of the bottom area. – Seong Joon Oh Aug 23 '13 at 7:45
On the other hand, if you set d=200000200nm, you will get a reversed result. So either way, you can never get an exact calculation of the wavefront. But, these considerations are all trivial because 200nm is such a small amount besides there are so many extrinsic factors that blurs the image focus by a greater amount. i.e. the miscalculation due to wrong focus is within the experimental error, as long as d is fairly exact (e.g. within 1mm). Am I right? – Seong Joon Oh Aug 23 '13 at 7:51

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