## magnified hologram

Holography related topics.
tonyr
Posts: 10
Joined: Sat Jun 08, 2019 5:46 pm

### Re: magnified hologram

Din wrote:
Mon Jul 01, 2019 8:06 am
MT= dXi/dxo= 1/(1 +/-zo/mu*zc–zo/zr)(Meyer; paraxial)
MT= (cos(alpha)o/cos(alpha)i){1/(1 +/-zo/muzc–zo/zr)} (Champagne; non-paraxial)

The longitudinal magnification is:
ML= dZi/dzo= -(1/mu){1/(1-zo[(1/muzc) + (1/zr)}2 = -(1/mu)M
If I'm fixing typos correctly we have:
MT = dxi/dxo = 1/(1 ± (1/mu)*zo/zc - zo/zr)
ML = dzi/dzo = -(1/mu)*1/(1 - (1/mu)*zo/zc - zo/zr)^2 = -(1/mu)*MT^2

If we have collimated playback (zc=infinity) at same wavelength (mu=1) then
MT = 1/(1 - zo/zr)
ML = -1/(1 - zo/zr)^2

So it will be magnified (not shrunk) if the reference beam diverges from a location on same side as object but farther away. If that location is twice as far from the plate as the object for instance, we will get 2x transverse magnification and 4x longitudinal.

lobaz
Posts: 243
Joined: Mon Jan 12, 2015 6:08 am
Location: Pilsen, Czech Republic

### Re: magnified hologram

tonyr wrote:
Mon Jul 01, 2019 6:56 pm
Thanks lobaz that is extremely helpful, not just the result but how you derived it.
You are very welcome. However, note that I wanted to keep the formulas as simple as possible. As Dinesh correctly pointed, magnification should be calculated by taking d(xO)/d(xI) rather than my simplistic xO/xI. Anyway, if you understand the principle, you can derive it yourself or check the articles Dinesh mentioned:

Magnification and Third-Order Aberrations in Holography
Reinhard W. Meier
Journal of the Optical Society of America Vol. 55, Issue 8, pp. 987-992 (1965) doi: 10.1364/JOSA.55.000987

Nonparaxial Imaging, Magnification, and Aberration Properties in Holography
Edwin B. Champagne
Journal of the Optical Society of America Vol. 57, Issue 1, pp. 51-55 (1967) doi: 10.1364/JOSA.57.000051

(PM me if you don't have access)

Also note that the formulas (also posted by Dinesh) use different symbols than I used.
tonyr wrote:
Mon Jul 01, 2019 6:56 pm
So the bottom line is you can only get perfect magnification by stretching the holographic pattern, and even then you need to use either a higher diffractive order or different reconstruction wavelength.
Something like that. Note that in display holography, perfect magnification is not necessary; the formulas just give you insight what is actually happening.
tonyr wrote:
Mon Jul 01, 2019 6:56 pm
Separate wavelengths aren't that practical, at most ~2x magnification if you record with blue and playback with red.
Exactly. On the other hand, changing wavelength was the Gabor's original idea behing the holography.
tonyr wrote:
Mon Jul 01, 2019 6:56 pm
Not sure higher orders are practical either considering weakness like you said. 10x magnification sounds out of the question.
Agree.
tonyr wrote:
Mon Jul 01, 2019 6:56 pm
Not to mention, how would you even implement the stretch? Use a lens to magnify and reimage the holographic pattern onto a new plate? Has that been done?
I am not sure, but I think it something was done using "exotic" waves, like microwaves or sound waves. Maybe Dinesh knows more. Anyway, magnifying/reimaging fringes for display application sounds hard.

lobaz
Posts: 243
Joined: Mon Jan 12, 2015 6:08 am
Location: Pilsen, Czech Republic

### Re: magnified hologram

tonyr wrote:
Mon Jul 01, 2019 7:16 pm
So it will be magnified (not shrunk) if the reference beam diverges from a location on same side as object but farther away. If that location is twice as far from the plate as the object for instance, we will get 2x transverse magnification and 4x longitudinal.
Maybe
I always have to draw a few rays to get the idea what is going on. Once I know the result qualitatively, I can believe the results from the formulas.

tonyr
Posts: 10
Joined: Sat Jun 08, 2019 5:46 pm

### Re: magnified hologram

Thanks to both of you for the references.
lobaz wrote:
Tue Jul 02, 2019 2:59 pm
changing wavelength was the Gabor's original idea behing the holography.
Looked up Gabor's original paper. Sure enough, he mentions these concepts there. Maybe that's where I should have started huh?

Here is the end of the paper:
If the principle is applied to electron microscopy, the
dimensions in the optical synthetizer ought to be
scaled up in the ratio of light waves to electron waves,
that is, about 100,000 times. One must provide an
illuminating system which is an exact optical imita-
tion of the electronic condenser lens, including its
spherical aberration. To avoid scaling-up the diagram,
one has to introduce a further lens, with a focal
length equal to the distance of the object from the
photographic plate in the electronic device, in such
a position that the plate appears at infinity when
viewed from the optical space of the point focus.
Work on the new instrument, which may be called
the 'electron interference microscope', will now be
taken in hand.
Sounds like the idea is to use a lens in situ to magnify the pattern for use at optical
wavelength.

By the way, world's first hologram from the paper, Fig 2d, only ~70 years ago:
gaborhologram.png (470.42 KiB) Viewed 3841 times

BobH
Posts: 373
Joined: Tue Jan 06, 2015 10:26 pm
Location: Mesa, AZ

### Re: magnified hologram

One can make a hologram of an object seen through a magnifying glass. There is also the popular images of microscopes and telescopes where one can look through the holographic eyepiece and see a magnified image of some object. One can also make a hologram of a lens and it will work to magnify objects. Or, one can magnify holographic images with holographic optical elements.