[05:00]

For a smaller design and lower unique part count, I've decided to use a convex and concave lens system to collimate the light from the display. For some reason, I thought the 55mm square fresnel lens I found on AliExpress had a focal length of 35mm, but it's twice that at 70mm. I also found a 21mm concave lens with a focal length of 17.835mm. The result:

Yeah this was quite a bit of a "snap back to reality" moment. The reduced beam is still much too large to use, and the overall system length of 54mm also needs to be reduced.

I found a 100x100mm fresnel lens with a focal length of only 35mm, so it's possible to obtain a smaller focal length. I'm going to assume I can get a 60x60mm fresnel lens with a 30mm focal length.

Next iteration:

I've changed the cross sectional axis so that the longest length is in view. Reading this webpage on combining lenses, I used the separated lenses equation. I've seen somewhere else that concave lenses have a negative focal length, so -17.835 is what I've used as the second lens.

If the first concave lens is 23.8mm from the fresnel, the focal point for the system is almost 46mm. Then, by placing the second identical concave 17.835mm away from this new focal point, I should get a parallel beam.

As you might be able to notice, a small change in the distance between fresnel1 and concave1 causes a larger change in the distance between concave1 and concave2. It doesn't seem that the resultant image size is affected as much though.

The target was 7mm, but a sub 9mm image should suffice too, requiring a beamsplitter of >=18mm.

It also seems that a assembly size under 30mm is possible. However, a problem arises when it comes time to expand the beam again, since the size would be twice as big and the 21mm lenses are barely enough already.

[some searching later...]

Oh wow:

This is one nice looking lens, and the company even offers a customisation service. Why AliExpress search didn't show this to me sooner, I don't know. 

There's also concave lenses here, and the listing says that the one below has a edge thickness of 5.2mm. The 21mm lens I found earlier only lists the centre thickness, so it's very likely that the above light path wouldn't work when the edge thickness is taken into account.

[About 1 hour later]

These lenses exist. Along with the F30mm fresnel, this is the next iteration of the optical path.

The dual 18mm diameter lenses reduce the 52mm square image down to 3.1mm. Merging 3.1mm beams is also going to be... interesting. After all the mergers, the beam size is doubled to 6.2mm. It then goes through a 3rd 18mm lens and then a larger lens before hitting another 30mm focal length fresnel lens. For the D43mm lens, an image of infinite distance is almost identical in size to the input. For the D40.5mm lens, an image of 5.97m distance is 13% larger.

I think I'm going to go with the D43 lens. Abberations are still a question mark, but hopefully they're not too bad considering the input image is approximately equal to the output image in terms of size and distance. 

It looks like it's about £5 per lens, so that's 5 * (4 * 3 + 2 * 3) = 5 * 6 * 3 = £90 in optics. Not great, but looking how much some singular sets of glasses cost, not terrible. If I use the 120Hz display driver board on AliExpress, the combined cost would lie somewhere around £400, depending on if there's a sale or not. This makes me think that the full BOM could be £599.

[5:20] There's a cheaper solution than using the D43 lens:

Image size: 52.75mm

[13:45] Turns out a fresnel lens focal point of 30mm was a good starting value, as 28 or 32 gives me an image way too large or small. Looking at IPD charts on the internet, I think targetting a 55mm IPD instead of 60 would be more inclusive. I'll see if I can get something like a 55x60mm or 55x65mm F30 fresnel lens so that I can hold the lens on the 2 shorter faces.