The physics behind Google Glass' "prism"

Question

In writing a bachelor's thesis about applicable use cases for Google Glass in retail I also strive to explain the physics behind Glass' optics. So far I've come to the following conclusions:

1. Glass must use a concave mirror (the reflection layer in this image) to magnify the image projected by it's LCoS Display and project it at infinite distance
2. In addition to the concave mirror, it must use some sort of either one-way-mirror or beam splitter (shown here as the diagonal piece) to reflect the concave mirror image to the user

So here's what I think happens (I'm not proficient in physics at all, so please don't mind the inaccurate rays ^^)

Assuming this is how everything works, here's what I don't get yet:

1. How does the concave mirror project the magnified virtual image into a distance of according to the Google FAQs around 2.5 meters or 8 feet in front of one's eye? Does it have something to do with the distance from the focal point of the mirror to the actual light source?
2. Assuming 3 is actually a one-way mirror (or semi-transparent), speaking in terms of ray orientation, a perceived distance of 2.5m should equal nearly parallel rays, right? In that case, the mirror should already be reflecting parallel rays, right? If that's the case, how does the concave mirror, which enlarges the image, produce parallel rays to place the image at infinity?
3. Looking through Glass on your head gives you the enlarged image at infinity. Looking through the other side gives a horizontally mirrored small version of the original image. Why is this? I thought if 3 a semi-transparent mirror, it should reflect the enlarged image of 2, thus one should also be able to see the enlarged image through the other side. Instead, this is what happens: I could explain the fact that you see the small original version of the image due to the fact that the mirror is semi-transparent and thus reflects 50% of the original image back out the other side. But if it is semi-transparent, why can't I see the enlarged image through the backside as well?

I'm sorry if these seem like basic questions to you, I'm trying really hard to understand all the inner workings of the device and have not had any real physics education since 8th grade now..

If you need any more pictures or a more detailed explanation of specific parts, I can deliver all of that :)

Already a profound thanks for helping me out!

Answer 1

So taking your questions one at a time 1) The concave mirror will form a virtual image if the object is placed closer to the mirror than the focal point of the mirror. The formula for the position of the final image is $s'=\frac{sf}{s-f}$ where s is the object-mirror distance and f is the focal length of the mirror. You can see that if $s<f$ this will be negative which implies a virtual image. The virtual image means the rays do not actually meet but appear to come from a point. In the diagram I have drawn real rays as solid and virtual rays dotted. So in the google glass case the LCoS screen must be closer to the mirror than the focal point (indeed from your photo of the prism you can see the curved edge is very slight implying quite a long focal length - on my diagram I rather exaggerated the curvature). The presence of the 45 degree mirror turns the real rays toward the eye while maintaining their relative angles and so the virtual rays now appear to come from a point straight ahead of the viewer.

2) You're right that at 2.5m the rays from the virtual image will be nearly parallel but it's still not infinity. Parallel rays would only result if the LCoS screen was actually at the focal point. At 2.5 m the normal eye can comfortably focus on the image when needed while allowing it to blur out when focusing at more distant or closer objects in the real world.

3) To the mirror: It is hard to tell but this is probably just formed by having two slightly different refractive index materials. Ideally lower on the right than on the left. This means the reflection from concave mirror to the eye will be by total internal reflection and so will be quite bright. Where as the spurious first reflection (shown by narrow lines in the diagram) will be a dim partial reflection. It is the image from this reflection you see when looking from the wrong side. You don't see the other image because the rays from it are directed away to the eye. It is possible there is a further partial reflection from the eye-side flat surface but you are not likely to see this because its rays will be mostly reflected back toward the concave surface by TIR. It's very likely this surface is in any case anti-reflection coated to stop just such spurious images. If they had used a partially silvered mirror instead of a prism then you might see some light from the main image on the wrong side due to dust scattering on the mirror surface but it would be a very blurred image. Using a bonded prism pair, probably assembled in a clean-room, this shouldn't ever happen here.

It's a pretty clever little arrangement! There is a neat little java app here where your can play with the imaging properties of concave mirrors. It's a little old so I had to add it to my exception list as modern Java complained it was a security risk.