Question About Cylindrical Holograms.

Light and its behaviour and properties
obscura
Posts: 1
Joined: Mon Oct 19, 2020 9:05 pm

Question About Cylindrical Holograms.

Post by obscura » Wed Oct 21, 2020 1:31 am

Hi, I'm new to holography. I came across this video (https://youtu.be/yVzk7bbQOA8?t=843) which shows a cylindrical film encompassing the object which the interference pattern is recorded.
If one was to cut the resulting hologram film strip, turning it from a cylinder into a flat plane, would viewing the hologram at a viewing angle perpendicular to the hologram result in seeing a "fish-eye" distorted image of the object, and does translating ones viewing position, while maintaining a perpendicular viewing angle to the hologram have the effect of rotating the object? If there is "fish-eye" distortion, would it be possible to enclose the object in a spherical or cylindrical lens to offset this distortion?

BobH
Posts: 402
Joined: Tue Jan 06, 2015 10:26 pm
Location: Mesa, AZ

Re: Question About Cylindrical Holograms.

Post by BobH » Wed Oct 21, 2020 9:18 pm

You have it essentially right, but in practice a computer generated image would be used (instead of a sculpture), and that image would be digitally predistorted to give you what you want. There would also be a huge problem with lighting the resulting film, unless the reference beam used to make the hologram was also massively pre-distorted. I'm really oversimplifying the issues here.

Din
Posts: 259
Joined: Thu Mar 12, 2015 4:47 pm

Re: Question About Cylindrical Holograms.

Post by Din » Thu Oct 22, 2020 11:49 am

obscura wrote:
Wed Oct 21, 2020 1:31 am
Hi, I'm new to holography. I came across this video (https://youtu.be/yVzk7bbQOA8?t=843) which shows a cylindrical film encompassing the object which the interference pattern is recorded.
If one was to cut the resulting hologram film strip, turning it from a cylinder into a flat plane, would viewing the hologram at a viewing angle perpendicular to the hologram result in seeing a "fish-eye" distorted image of the object, and does translating ones viewing position, while maintaining a perpendicular viewing angle to the hologram have the effect of rotating the object? If there is "fish-eye" distortion, would it be possible to enclose the object in a spherical or cylindrical lens to offset this distortion?
No, you won't get just pincushion distortion, if by 'pincushion' you mean barrel distortion. Barrel distortion is just one of the 5 Seidel aberrations, and depending on the size of the flattened hologram, you will get all 5 at some positions, if you use the same source to reconstruct (illuminate) the hologram. You cannot get rid of all the Seidel aberrations at once, so you cannot pre-distort the image by software along the entire hologram.

Consider that in order to correctly reconstruct (illuminate) the hologram, you have to use a source as close as possible to the original reference beam used to record it. If you record with a laser (as usual) and reconstruct with white light (also as usual), you're already deviating from the recording conditions. So, it's true to say that all display holograms have aberrations. However, when viewing a display hologram, it's not just your eyes that are evaluating the hologram, it's also your brain. So, if you see something that looks like a cat, and it's slightly distorted, you're not going to notice. One of my mentors, Kaveh Bazargan, used to say, "The eye forgives a lot". In a discussion with a professor of optics from Cambridge University, we both agreed that the eye is not a camera. This is not true in a technical hologram, where all aberrations are noticed. Usually, the Seidel aberrations are given in terms of lens aberrations, but, there is an analogy of the Seidel aberrations to holograhy, so long as the reconstruction wavelength is the same as the recording wavelength. If the wavelength is different, the holographic Seidel aberrations diverge from the lens aberrations, because ray optics considers λ = 0. So, when viewed in white light, the holographic aberrations vary quite a bit from the lens aberration equations

Getting back to your cylindrical hologram, flattened out, consider the original cylinder reconstructed by a point source (white light) along the central axis, below the cylinder; in a cylindrical coordinate system, we may consider the base of the cylindrical hologram to lie in the x-y plane, extending up to some value along the z axis to z₀ (z₀ = the height of the hologram). In this case, the illumination source maybe be placed at some (cylindrical) coordinate -zc. The illumination is at the same angle to the centre of the hologram at all angles, ie, along the cylindrical coordinate φ, the reconstruction angle, θ, is constant at the centre and varies along the z axis; that is, it's symmetric wrt φ and varies along z.

Now, you flatten the hologram. If you maintain the reconstruction source in the same relative position to the centre of the hologram, ie keep the same θ, then the centre strip will be illuminated in the same way as the original cylinder. If we switch to Cartesian coordinates, let the hologram be in the x,y plane, with y being "up" the hologram, and x "along" the hologram. If we centre the Cartesian system at the centre of the hologram, then the hologram coordinates along the y axis is now +/- (z₀/2), and along the x axis +/-πR, where R is the radius of the original cylinder. The illumination source is now at x = 0; y = {zc + (z₀/2)} and z = R/2 (yes, I've turned the hologram upside down to get rid of the annoying negative quantities :) .

But, now, do you see that the illumination source relative to any coordinate (x, y)will be twisted to some angle so that the azimuthal angle φ' and the radial angle θ' is now at

φ' = arcsin{x/r]
θ' = arcsin[y/r']
r² = x² + {zc + (z₀/2)}²+ R/2
r'² = x² + {zc + (z₀/2)}²+ R/2

In effect, you are moving the reconstruction, and twisting it with reference to the hologram, and any ray from the reconstruction source to any strip on the hologram will be twisted, by some angle. As you move the reconstruction source away from it's recording position, the third order Seidel aberrations get worse. The image will twist, elongate and perhaps stretch. There will also be pincushion and barrel ('fish eye') distortions. Eventually, for a real holographic medium, the image will disappear at some angle. Ideally, the hologram is reconstructed by Ramanj Nath diffraction, where diffraction occurs at all angles. But, a real hologram has Bragg components. We measured the angle at which the hologram disappears at about 20 degrees for a client interested in this number. But, it varies according to the emulsion thickness.

If you're interested, here is a paper on holographic aberrations. Unfortunately, it's behind a pay wall, so even though I'm the author, I don't think I'm the owner of the paper ;) . So, I don't think I can post the whole paper:

https://iopscience.iop.org/article/10.1 ... 5/1/012031

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