Holography Transmission Equations Part II
Back to Holography Transmission Equations Part One.
Lateral Magnification The ratio of the image distance to the object distance in lensed imagery gives the magnification ratio; how big is the image compared to its original size. There are a variety of equations useful for figuring this out, but the simplest to visualize and most like the holographic equation is:
When image distance = object distance, magnification ratio = 1:1, the Xerox machine case. In normal picture taking situations, a magnification < 1 or minification takes place; a whole human figure is reduced down to the size of a piece of film or digital chip, with object distance being much greater than image distance. (The denominator in the equation is bigger than the numerator.) A projector has a short object distance compared to its image distance, and the magnification is >1. (Values of numerator and denominator reversed.) Spare me and don’t ask about what happens when an object is at infinity today.
This equation refers to the object’s dimensions in the two planes parallel to the holoplate; the longitudinal, or z dimension magnification is an evil equation and is coming next. It relates the original object wavefront curvature, Robj to it’s current position, Rimg. But it can also apply to what happens when the replay wavelength is different than the exposing one. And our little friend m pops up and although not pertinent to the art of H1 masters it is useful for cases when there are multiple image orders, like in a Gabor hologram and its close relative, the zone plate.
So if we put our properly exposed and developed laser transmission hologram back in the plateholder where it was made once again like in the examples above, we can calculate:
Image is the same size as the object, since we haven’t moved anything.
In the previous section, the image distance was calculated when the reference beam was moved in closer. (Scenario refresher: reference beam in at 45 degrees from infinity or collimated, object plane 100 mm from plate.) Plugging in these object distances in response to moving in the reference beam closer, we get magnifications of these new object distances over 100 mm, since the ratio of wavelengths is unity and so is m.
- R ill = 10 m,
- R out = 99 mm;
- mag = .99
- R ill = 1 m,
- R out = 90.9 mm;
- mag = .909
- R ill = 100 mm,
- R out = 50 mm;
- mag = .5
If the object were a 10 by 10 grid of squares 10 mm (1 cm) on a side, the when the reference was at 10 m, the grids would shrink to 9.9 mm on a side; at 1 m, about 9.1 mm squares; and with that very close reference, they drop down to 5 mm mini-squares.
Couldn’t there be magnification just like in the lens examples above? There was none possible in the previous hologram because the reference beam during recording was collimated, parallel and coming from infinity, and there is no longer R ill to plug in and let the equation “grow” in replay.
Sharp-eyed readers might realize that there is no R ill in the Magnification Equation. But it’s necessary to find R img for the latter, and so the example will include solving for image distance based on reference replay position, then to be plugged into the magnification equation.
The parameters for this hologram will be Robj, the point on the object of interest, positioned 10 cm behind the holographic plate; and a Rref of one meter, a distance easily attainable in a sandbox or other small tabletop isolation system, for the reader to verify themselves. It could also be the basis for a Science Fair Project, something on the order of “Experimental Verification of Predicted Object Magnification in Transmission Holograms.” Or just looking at any transmission hologram by moving it back and forth in an expanding illuminating beam will also bring a sense of relevance to these mathematical meanderings.
1/R out = (lambda ill/lambda exp)(1/R obj - 1/R ref) + 1/R ill
= (633 nm/633 nm) X (1/100 mm - 1/1000 mm) + 1/1000 mm
= 1 X (.01 - .001) + .001
R out = 100 mm
As usual this is the replay configuration that duplicates the recording, so the object distance in the hologram is the same as in real life, and it’s safe to say that magnification is = 1.
Halving the reference distance: 1/R out = (lambda ill/lambda exp)(1/R obj - 1/R ref) + 1/R ill
= (633 nm/633 nm) X (1/100 mm - 1/1000mm) + 1/500 mm
= 1 X (.01 - .001) + .002
R out = 90 mm compared to R obj of 100 mm yields a magnification ratio of 90/100 = .9 or 90%. This didn’t yield an object magnification of 50%! This equation works exponentially, not linearly.
To see where the reference beam needs to come from for a magnification of 50%, (50 mm object distance, or R out) we need to solve for R ill: 1/R out = (lambda ill/lambda exp)(1/R obj - 1/R ref) + 1/R ill
= (633 nm/633 nm) X (1/100 mm - 1/1000 mm) + ?
= 1 X .01 - .001 + ?
1/R ill = .011,
R ill = 90.9 mm
Going in the other direction, doubling the R ill distance, 1/R out = (lambda ill/lambda exp)(1/R obj - 1/R ref) + 1/R ill
= (633 nm/633 nm) X (1/100 mm - 1/1000 mm) + 1/2000 mm
= 1 X (.01 - .001) + .0005
R out = 105.2631 mm, rounded to 105, for a 5% increase in image size.
The largest possible virtual image magnification would be when the R ill is brought all the way to infinity, so 1/R ill drops out, and the object distance is 1/.009, or 111.111…
An exercise for the reader would be to think of how to tweak the holographic set up (reference, object distance, and replay distance) for the maximum possible magnification, kind of like a holographic magnifying glass!
Another tricky transformation that this equation implies is the change in object size when the wavelength is changed. This was Gabor’s impetus for developing holography; to shoot the hologram with short wavelengths, like electron waves or X-rays, and then replay it in the optical domain for an extreme magnification, namely on the order of the ratio of the wavelengths. Replaying a hologram shot with 6 nm X-Rays with a 589 nm sodium vapor light (this is pre-laser era!) would yield approximately a 100X magnification!
Reading the early papers of Gabor and Rogers they did attempt to experimentally verify this by exposing holograms with the blue line of a mercury vapor lamp and replaying it with its green line! Theoretically satisfying, the big practical stumbling block was being able to supply a point source of the really short lambdas!