Resolving neighboring distant points: Fresnel zone plates

favalora · Post by **favalora** » Mon Sep 08, 2008 6:49 pm

Hello -

There is a question in a different thread regarding how the parameters of one's laser impacts your hologram's ability to reconstruct "deep" images. I am curious what other factors influence this. Certainly this has been answered before in elementary textbooks (though I searched three, including not-so-elementary ones, and can't find the answer.)

For instance, what does a theoretically perfect Gabor-type (i.e. directly on-axis) hologram look like of a scene with a single point? It's a Fresnel zone plate, I suppose a sinusoidal one. What does the hologram of two points look like assuming the points are on the same normal vector to the hologram? I'm not sure, but one interesting thing to consider is simply treating them as point sources.

Here is the result of playing around with Mathematica. Did I make some awful error here? I'm assuming the interference fringes add linearly (so I just divided by the number of sources). I suppose what this is showing is that there is useful information - low-frequency beat modulation of an underlying high spatial frequency - distant from the center of the fringe pattern.

This suggests that your film quality is important - you need a setup that can capture the intended contrast between beams, and also handle the frequency needed to reconstruct well along depth.

Oh, and, this confirms the well-known notion that points close to the hologram will be recorded best. (That is, the information is relatively "broad", with predominantly low-spatial frequency information.)

Ok - so - where did I completely mess this up? (Colin I hope I can delete this thread in the future so Google doesn't capture my gaffes forever...)

-g

Ed Wesly · Post by **Ed Wesly** » Mon Sep 08, 2008 8:06 pm

Gregg Favalora wrote:

Here is the result of playing around with Mathematica.

There is a plug-in for Mathematica called Optica, written by a fellow holographer, Don Barnhart. I am not sure of exactly how to get the program, which would undoubtedly prove useful for this problem and most other optical engineering concerns..

The answer might be found in the Benton book Holographic Imaging. I got it for < $100 on Amazon. It is almost all equations, being mainly a mathematical development of holography. I haven't encountered any set up tips or secret processing formuale yet, but I just waded through the section on zone plates, which is pertinent to the above discussion. This is a book to be read slowly and deliberately. If you are math-phobic, you can leave this one off your Xmas list!

favalora · Post by **favalora** » Mon Sep 08, 2008 10:33 pm

Hi Ed -

I agree; I like Benton & Bove's book a lot. Although I don't have the hardcover, I had an uncharacteristic moment of foresight & downloaded all of the chapters while it was still a Media Lab course.

g

Ed Wesly · Post by **Ed Wesly** » Tue Sep 09, 2008 12:55 am

Yeah, why buy the cow when you can get the milk for free! The book does flow a lot smoother than the downloads. I don't know if the class notes are still on-line, but if anyone wants to PM me care of this forum I will send them the pdf's.

Colin Kaminski · Post by **Colin Kaminski** » Tue Sep 09, 2008 5:11 pm

Gregg,

Thank you for starting this thread. I am very interested in the answer to this question as well. I believe there is a fall off of resolution do to depth but I have not seen any equations to describe it.

favalora · Post by **favalora** » Thu Sep 11, 2008 10:20 am

Colin,

I see that many of the forum members have commented that it is difficult to buy holographic film. Does it turn out that there are very few types of film to buy in the first place, implying that my point has limited practical relevance?

g

JohnFP · Post by **JohnFP** » Thu Sep 11, 2008 10:49 am

DCG has unlimited resolution, well not really unlimited but surely much better then silver halides.

It would be good to take some low res film, high res film and some DCG and test. Maybe even throw in some High resolution aireal film for super low res if doing in line transmission.

Example: Slavich VRP-M (low res), Slavich PFG-03C (high res) and DCG
Expose all with the same wavelenth (green) and set up using a ruler or yardstick as the object (thanks again Jeffrey).

Colin Kaminski · Post by **Colin Kaminski** » Thu Sep 11, 2008 5:09 pm

favalora wrote:Colin,

I see that many of the forum members have commented that it is difficult to buy holographic film. Does it turn out that there are very few types of film to buy in the first place, implying that my point has limited practical relevance?

g

There are few films available but we are making more and more at home. I think this is one of the best topics we can talk about. I have read all of the books on holography and the resolution away from the film plane is not discussed to my knowledge. I have studied the effects of aberrations away from the film plane. So for a theoretical discussion we need to talk about a transmission hologram replayed with the same laser and with no shrinkage. Shrinkage and replay bandwidth will cause aberrations that will cloud this discussion.

Ed Wesly · Post by **Ed Wesly** » Thu Sep 11, 2008 10:40 pm

I've finally gotten a chance to chime in again. I think that if what you want to know is what is the smallest thing that can be resolved at a certain distance, the same calculations apply to finding the resolution of a hologram as do finding the diffraction limited resolution of a lens! It's all about the angle between the extreme rays from the edge of the optic, or the f/number, which is like the formula for finding fringe spacing. d = lambda/sin theta, theta being the angle formed by the top of the hologram to the object point to the bottom of the hologram. This angle could vary depending if you used the big side, the short side, or the diagonal of a rectangular hologram. But as the object point gets further from the holo, the angle will become more acute, causing larger fringes or speckle, meaning less resolution.

This is just off the top of my head, as I am proctoring a test right now, without access to my library, but I think it should suffice.

And no, I am not going to say where is Dinesh when we need him!

Joe Farina · Post by **Joe Farina** » Fri Sep 12, 2008 9:40 am

Colin Kaminski wrote:I have read all of the books on holography and the resolution away from the film plane is not discussed to my knowledge.

I have definitely noticed this effect also, and it certainly occurs in DCG, and if an explanation exists in the literature, I missed it. (Just to clarify the actual observed effect, I think (?) it differs from the blurring caused by an extended replay source, rather it's like a graininess or noisiness when objects get futher away from the emulsion plane.)