overexposing DCG

Dichromated Gelatin.
Joe Farina
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overexposing DCG

Post by Joe Farina »

I didn't realize that overexposure could have such a drastic effect on DCG until I saw this:
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Looks like a good reason to use varied (test-strip, etc.) exposures frequently. The above graph is from "Lasers and Holography" by Mehta and Rampal (1993), which is a good book, by the way. The graph was reprinted from a paper by Chang and Leonard ("Dicromated gelatin for the fabrication of holographic optical elements," Applied Optics 1979). The wavelength for exposure was 514nm, and the material was standard DCG.

Rallison made a comment which confirms this behavior, in "DCG and other non-silver holographic materials," Lake Forest notes 1994: "at 514 or 532nm, 50 to 100mJ may be necessary....Gross overexposure will cause a decrease in efficiency all the way to zero."
Dinesh

overexposing DCG

Post by Dinesh »

Yes, this is well known. This is why dcg does not have an exposure figure such as silver, the actinic energy does not have a sharp peak. In silver, the mechanism is the dissociation of the silver salt, not polymerisation. However, these curves are very dependent on the Bragg structure. The curves shown in the diagram are probably for planar gratings. Just recently, I've been shooting confocal paraboloid gratings and the figures are different.

I'd be a little suspicious of curve B. It implies that the modulations flattens out, then reforms at the same spatial frequency with the same delta n. This implies some sort of hysteresis. I'm not sure how you'd take a dcg hologram inside a liquid gate, wouldn't you be developing it as you were shooting it?
BobH
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overexposing DCG

Post by BobH »

Looks like these curves are from a transmission grating. And I'm also curious how they got the material "inside a water gate".
Joe Farina
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overexposing DCG

Post by Joe Farina »

They were all transmission holograms, though the authors claimed that the results could "basically be applied to reflection holograms with slight adjustments." I don't understand the water gate either, nor what they mean by "angular modulation" in the 1979 paper:
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Joe Farina
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overexposing DCG

Post by Joe Farina »

To confirm what Dinesh and Bob were saying, those curves in the above paper were for simple transmission gratings with a beam ratio of 1:1, the spatial frequency was supposed to be 837 lines/mm, the emulsion thickness 15 microns, and the exposing wavelength 514.5nm.

I found a 1988 dissertation which has a somewhat similar curve (attached below) for exposure vs. diffraction efficiency, though it's much "compressed." It does show a sharp rise and fall in the beginning, and then another rise and fall. The complete paper can be found here:

http://arizona.openrepository.com/arizo ... sip1_m.pdf

There is a second curve on that same page (second attachment) which really surprised me, it shows the diffraction efficeincy versus the index modulation (delta n). I always thought there was a simple linear relationship between modulation and efficiency (?)
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Dinesh

overexposing DCG

Post by Dinesh »

The rise and fall and rise of the de did have me puzzled for a while. But, I think it can be explained by non-linear and interference effects. Ideally, the physical, actinic reaction must follow the E field of the interference pattern (actually, the E^2 field). Since, in the ideal two-collimated-beam geometry, this is a sinusoidal pattern, the physical parameters, the density in the case of dcg, must also follow a sinusoidal function. However, in dcg, the cross linking function is continuous, so as you pour energy into the medium, the cross linking continues. Thus, as you expose for longer and longer, the surrounding material around the strict sinusoidal profile starts to increase in density. The result is that the sinusoidal function of the interference pattern is translated into more and more of a square wave profile with slanted sides - a sort of truncated triangle. Also, the variation in density surrounding the sinusoidal profile is continuous. When a reconstruction beam impinges on this sort of pattern, the reconstruction is a summation of reconstrruction from all the sinusoidal components of the physical profile. You get a Fourier breakdown on reconstruction, with the coefficiants of the Fourier being dependent on the density profile around the sinusoid. These Fourier components will then interfere with each other. So, as you expose, the density profile starts to "expand", giving rise to Fourier components. Depending on the rise in density modulation, these Fourier components react with each other and, given a particular set of coefficiants, this might result in destructive interference. As you increase the exposure, the density variations surrounding the sinusoid increase and so the coefficiants of the Fourier start to increase and this may result in constructive interference of the component waves. I suspect that if you continued to increase the exposure, you'd continue to get these increases and decsreases, as the various Fourier components mutually interfered, with an envelope of a slowly falling exponential.

This does not happen to the same extent in silver halide because the ionisation energy of the silver halide has a sharper peak - it's not continuous. Therefore these non-linear effects, along with the concommitant interference effects, are not so noticeable.
Joe Farina wrote: I always thought there was a simple linear relationship between modulation and efficiency (?)
No. For a lossless, dielectric transmission grating (the ideal dcg grating), the efficiency is given by:

eta = {sin^2(nu^2 + eta^2)}/{1 + (xi/eta)^2}
nu = (pi*delta n*d)/{lambda*(c(r)*c(s)^0.5}

Equations 42 and 43 in the Kogelnik paper. You can see that the modulation is carried in the nu factor (c(r) and c(s) are slant factors to take the slant of the fringes into account). Thus, the efficiency of a transmission grating is a sin^2 function of the modulation (to 1st order).

For a lossless reflection grating, the efficiency is:

eta = [ {1 + (1 - (xi/nu)^2)}/sinh^2[(nu^2 - eta^2)]^0.5]^(-1)

Again, the modulation is carried by the nu factor, but this time it's not a simple function. Note that the slant of the Bragg planes is also carried by the nu factor. For unslanted Bragg planes (which is not the case for display holograms), the efficiency is:

eta = tanh^2{(pi*delta n*d)/(lambda*cos(theta))}

This is the efficiency function that Rallison and others often quote, but this does not hold for display holograms, since display holograms (by definition) have slanted planes.

A couple of points:
These are for lossless, pure phase gratings. No real hologram is ever completely lossless, there's always some absorption.
The entire Kogelnik analysis is based on the fact that the modulation is very low. He bases his RCWT on the Helmholtz equation and ignores the second derivative since the changes in the index are very, very low. However, in modern materials, the index variation is not that slow, so the Kogelnik analysis does have to be modified. Strict RCWT, based on the Helmholtz eqaution should not completely ignore the second derivates. It may be that there may not be analytical solutions and so you have to use a numerical approach. Also, RCWTs associates a scalar with the wave intensity, in other words the polarisation of the waves is not taken into account (apart from the Malus effect from Malus' Law, which Kogelnik does touch upon). There were a number of papers in the 80's - notably Moharam and Gaylord - who did tackle this issue and came up with numerical models.

All of these rigorous Coupled Wave Theories are based on a Cartesian system. I'm curious whether it's possible to reformulate a RCWT in generalised coordinates. A long time ago, sometime around 1989 or 1990, I did reformulate Kogelnik into elliptical coordinates, but I only showed it to Steve Hart (now of Holorad). However, that analysis showed a very much simplified expression for the various efficiencies. This sort of made sense, since most holographic geometries are based on elipses and not on a Cartesian grid.
Joe Farina
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overexposing DCG

Post by Joe Farina »

Thanks Dinesh, that was very interesting and quite helpful. If a material like DCG generates a fringe structure during exposure, then that fringe structure must have some kind of effect on the later portions of that same exposure. I've always glossed over this because I don't know how to deal with it. But I'm gaining a few insights now.

If a display DCG hologram has a similar behavior (DE vs. exposure time) as what was shown in the two graphs for the transmission gratings, then there would be a strong incentive to expose only long enough to get to the top of the first "sharp" peak and no further. On a practical level for display DCG, this might mean testing various exposure levels on a single plate, and keeping everything else constant.
Joe Farina
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overexposing DCG

Post by Joe Farina »

These are a couple more graphs from a 1987 Applied Optics paper by Georgekutty and Liu, "Simplified dichromated gelatin hologram recording process."

The first one shows a similar double-dip effect compared to Chang and Leonard. The second one shows the effect of different spatial frequences, but I assume they're all transmission gratings.

In the paper, they had a good comment regarding DCG holography in general:

"Moreover, one can often get frustrated if any essential step is messed up....and no good HOE is produced at the end."
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Dinesh

overexposing DCG

Post by Dinesh »

Joe Farina wrote:Moreover, one can often get frustrated if any essential step is messed up....and no good HOE is produced at the end.
True enough! The trick is in knowing the 'essential step'. I often describe dcg holography, especially with HOEs, as juggling a large number of balls while walking across a rickety bridge. Every parameter is dependent on every other parameter, while you have these unknown environmental effects to which dcg is far more sensitive than silver. Over the years, I've developed an intuition for dcg, but sometimes I'm thrown for a loop also!
Joe Farina
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overexposing DCG

Post by Joe Farina »

I've been reading the Newell thesis, and it has some very interesting information regarding the overexposing of DCG. In particular, it relates to reflection holograms rather than transmission. As Dinesh has pointed out, it seems like almost all of the papers on DCG are describing effects in transmission holograms (optical elements) and have these in mind, even though it's not stated in the papers. I think (?) Dinesh also said (not in this thread, but in another, if memory serves) that reflection DCG does not fall off in efficiency drastically as the earlier (presumably transmission holos) graphs would suggest. (If I'm wrong here, or if someone else said it, my apologies.)

The Newell graphs suggest a very different picture for reflection holograms in DCG, with rising modulation with exposure to a maximum point, then a gentle slope downward. The downward slope is a bit more pronounced with 20-micron films as opposed to 5-microns. These graphs are shown as figures 3.14 and 3.15 in http://ora.ox.ac.uk/objects/uuid%3Adf0f ... TTACHMENT1

I did some strip-exposure tests on ~25-micron DCG, and indeed, there was a decrease in image brightness after the exposure reached a certain point. This decrease might correspond to graph 3.14 (20-micron) but perhaps a bit more pronounced.

So, apparently, if someone is working in reflection mode, they don't need to worry that "oh my gosh, if I overexpose an extra 30 seconds, the image might disappear." ;)

A couple of other things that Newell mentioned were that bandwidth tends to increase with overexposure (don't remember if he said why) and a blue-shift (shrinkage) tends to be introduced. Also, I think it's fairly well known that noise tends to increase as the exposure gets longer.
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