Optics

Hi Guys

I hope this is in the right section, it's kind of old homework, albeit my daughters. I was hoping someone might help me with an insomnia problem...

My daughter had a science project recently with the theme of 'Secret Messages'. Our original idea was a variation on a 'peep-show' shot glass we saw in a restaurant in France -- obviously the variation being to replace the naked couple with a more suitable hidden message.

In case you're not familiar --and don't take your kids to the kind of seedy joints we do, apparently-- it looks like a regular shot glass when empty, but when full of a clear liquid (in this case creme de menthe) an image is revealed in the bottom of the glass (in this case reverse cowgirl).

Due to our surprise at being given such a thing in a family restaurant, I didn't really examine the workings (of the glass, or the image).

We tried to recreate the effect, guessing the key was a convex-plano lens, but could not get the image to completely disappear. Around the house we only had a lens from an old torch and I'm not sure it was suitable...

...long one short.

She ended up doing something else (and got a pretty good grade), but it's been bugging me since; hence the sleepless nights.

Can anyone explain how one would, even theoretically, go about making it work. I assume the parameters must be more demanding than our half baked attempt: lens tolerance, distance from lens to image, or even distortion of the image (?), etc.

Many thanks in advance.

I knew that the French were more relaxed about easily accessible porn than the English or Americans. This is a new one to me, though...

The following sounds plausible to me. I've done no real experiments or maths, so take with a pinch of salt. Anyone with references or actual experimental data feel free to shoot me down.

My guess is that to re-create this you actually want a plano-concave surface, and that no optically useful lens will be steep enough to pull off the trick.

The shot glasses we've got at home are almost cylindrical, but the space for the liquid is almost bullet-shaped. For simplicity's sake, assume that the hollow is actually conical and the base is flat (hope the ASCII art comes out ok):

Start with the glass empty. A light ray coming up from the base will strike the surface at quite a steep angle. As you may know, light bends away from the normal when it moves from a medium with a high refractive index (like glass) to one with a lower refractive index (like air). If the angle is steep enough, total internal reflection occurs, and the light ray doesn't pass the surface but gets reflected and, in this case goes out of the side of the shot glass. You, looking down into the glass, don't see the image. If you trace a ray backwards down from your eye, it gets refracted by the glass and goes diagonally downwards out of the side of the glass, so you would see a distorted image of a part of the table surface next to your glass.

Now pour in a liquid. Since this has a much higher refractive index than air, this changes the critical angle for total internal reflection. A ray coming up from the base now continues up to your eye, and you can see a distorted version of a photo stuck to the bottom of the glass. You could choose to print the photo slightly distorted so that the optically induced distortion undoes the printed distortion, if you think your viewers are going to care.

In practice, a shot glass will not have a precise conical shape - it's more likely to be bullet-shaped. This means that you probably could have seen an ultra-small version of the photo in the very centre of the bottom of the empty glass if you'd known what you were looking for. You could probably also have seen a distorted view of part of the image if you'd looked into the bottom of the side.

In terms of doing the experiment with your kids, the following might work - no guarantees though! Try buying a shot glass with a bullet-shaped internal volume and the thickest base you can. If the base isn't flat, turn it upside down, fill the space with water, cover with cling-film and turn the glass right side up again. Put it on top of a photo and see if it vanishes. If it does, fill the glass and see if it appears.

If you actually try this, please post and let me know how (if) it works out.

Thanks for getting back.

To confuse matters further, it was a Chinese Restaurant.

I see what you're getting at with a concave lens. I guess the principle is the refractive effect of the water cancelling out the refraction of the lens -- I suppose either convex or concave might work, at least in principle. There's loads on the the net about using refraction to make 'things' disappear; I'm surprised I could find nothing on this principle.

I think urine sample bottles are the shape you describe. I'll see if I can get hold of one and give it ago. I expect, being plastic, that the refractive qualities won't have much effect though.

I did find a picture --that I can't link to. Google peepshow shot glasses... it's okay, they are quite tame -- of some that give a pretty good idea of construction. It looks like the image would be on the base, with a gap between it and the lens. The shot glasses we were given where china and very nicely made.

We also had some success with a glass egg cup with a clear marble in the bottom... at least enough for the kids to get the general idea.

Cheers

This post has been edited by BillIam on Mar 23 2011, 08:09 PM

Similar item:

http://www.worldwidefred.com/peepshow.htm

http://ajp.aapt.org/resource/1/ajpias/v76/i9/p856_s1

Related technology:

Beer steins with Risque Lithophanes

Glass stained with porn! That must be The Unholy Grail.

rpenner: "Related technology" - I love the terminology.

BillIAm: The shot glasses in the link provided by rpenner do indeed use a convex lens. As the joke about theorists goes, that's even easier to explain...

The thin lens formula is a good starting point, even if the lens is not remotely thin and it doesn't strictly apply. For a lens with focal length f and an object a distance u to the left of the lens, an image is created a distance v to the right of the lens. The relationship is

1/f = 1/u + 1/v

If you position the object (the naughty picture) slightly closer to the lens than the focal length (i.e. u<f, but not very much less) then 1/f-1/u is small and negative, so v is large and negative. Since the magnification is given by v/u, we can determine that the image is large (because the magnification is large), upside-down (because the magnification is negative) and appears to be a long way (because v is large) below (because v is negative) the bottom of the shot glass.

Effectively, you are looking at a fifty foot (say) high copy of the photo a hundred feet (say) away through a hole the size of the bottom of the shot glass - so you don't see much. You can't move your eye closer to the hole because the sides of the shot glass get in the way and you can't move your eye around to scan around the image for the same reason.

Even worse, if you care to trace out the rays from a slightly off-axis point on the picture (using the full n1.sin(i) = n2.sin(r ) formula) you'll find that even less gets through the lens than would get through a hole of the same size because the phenomenon of total internal reflection kicks in and reflects some of the rays off to the side.

Even worse still, the lens is spherical and lenses that are this far from flat need to be parabolic. This means that the image will suffer from horrible spherical aberration (see the explanation at Wikipedia, and an example of mild - compared to this situation - spherical aberration in figure 6 on this page), so even if you could find a position where you ought to be able to see a part of the image off axis, you'll probably only see smear (smeared smut, I suppose...).

When you fill the glass with liquid the refractive index step at the boundary drops and the optical power of the lens falls off. In terms of the thin lens equation above, u remains constant (because we don't move the photo) but f gets much larger. As f gets larger, v becomes smaller for constant u. The image remains upside down and slightly magnified, but that's fine. Spherical aberration drops off too, but I bet the image quality isn't much cop unless they were very careful with the refractive index of the glass. It's porn on the bottom of a shot glass, so image quality is probably not high on the list of priorities.

I hope that makes sense - happy to expand on it if it doesn't.

In terms of doing this one at home, I'd go with a margarine tub with a hole cut in the bottom of it. Cover the hole with a hemispherical transparent piece of plastic (I'm thinking a blister pack for a spherical toy a couple of inches in diameter) with the hemisphere upwards. Glue it down and make sure the seal is water tight. Put the thing you want to hide (it'll need to be waterproof) on the bottom of a washing-up tub and fill with water. Push the margarine tub into the water surface, making sure that the blister pack is full of water and the tub itself is not. Look downwards through the blister. If you can see the thing you want to hide and it's right-side-up, push the tub deeper into the water until it vanishes. If it's the wrong way up, pull the tub upwards until it vanishes - you might need to add more water in this case. Pour water into the margarine tub and the object should appear as soon as the blister pack is covered.

Again, let me know if that doesn't make sense and I'll try to explain better. Also, I've just made this experiment up and would love to know if you try it and it works. My son's a bit young for this kind of thing, but I'm counting the days...

Kino, thanks. We'd looked at most of those links before, but your post explains the details nicely. We haven't tried the marg tub in a bowl of water thing yet, but we have had some success on our original scheme.

We found a small, clear, plastic bottle with a longish neck and then cut the bottom off. Then we filled the neck with clear resin, with the 'message' embedded in the middle. Then we made a convex plano lens, again with clear resin, and attached it at the point where the neck opens out into the main body of the bottle (convex toward the base (or where the base used to be)).

As it happened we got the distance between message and lens right first time, but the cylinder of resin in the neck could, in theory, be adjusted by sliding it up or down in the neck.

Works great. We're now hoping our son gets the same project in a few years.

Cheers

Thanks for letting us know - glad you got it working. Your method is better than mine - less messy. I guess any lens will work as long as you place the object just inside the focal point.

If you're doing this again, you could save yourself some potential messing around by measuring the focal length of the lens first. Rays from infinity are focused at the focal distance. Indoors on a bright day you should be able to form an image of the windows on a piece of paper (don't do this in direct sunlight, unless you want a fire for some reason). If the windows are a lot further away from the lens than the screen is then you can pretend the windows are at infinity, and then the lens-screen distance is approximately the focal length. Close enough for government work, anyway, and it'll at least let you know if your bottle is roughly the right size.