Lenses theory

[Tarzan]

Hi everybody

I'm currently trying to design my own HMD, in my own style. No offense guys, all your HMD are awesome, but I prefer trying to find my own components rather to order the same ones as everybody (Palmer's magic lenses, Donegan...and so on).
In my opinion, the most difficult part of the design is the lenses. I've found many models on the market that I want to test, but I can't figure if the lenses I found could work, or if I am completely wrong and I need to refocus my sourcing (no need to throw mony out the window ). That's why I have some theorical questions concerning the lenses.
I 've been surfing a lot on google and MTBS, but I wasn't able to find the answers.

1. We found different specifications on the lenses everywhere : some speak about focal lengh, others about FOV and the last ones about magnification. Thoses specs are linked right? Speakinig of a single lens, could you please confirm that SHORT focal length = LARGE fov = LOT of magnification = MORE convex/concave shape (and the opposite LONG focal length = NARROW fov = FEW magnification = LESS convex/concave shape)?

2. Is there a way to have the formula between Focal length, FOV, and magnification? Does this fromula depends on the shape of the lens, or is it a general formula? if there is no formula; is there a way to approximate thoses values?

3. Is it right to think that the diameter of the lens have an effect on the FOV? I mean, if we imagine a gigantic lens, and we place the eye in the center, the FOV will be narrow right?

4. Some are using aspheric lenses and others don't. From what I've read, aspheric lenses correct the distortions that classic convec lenses have (not a single focus point). Distortion is not always bad for doing HMD, as long as it is well managed. So what about other type of lenses? (double convex, double concave, convex-concave, concave-flat...)

I have my opinion on those questions, but it's only intuitive answers. I would please have your opinion, scientific ones as well as expertises ones.
I hope your answers and opinions will also help other who would like to join the community of HMD builders

Thanks a lot

[Fredz]

Is there a way to have the formula between Focal length, FOV, and magnification?

M = f / (f -d)

M : magnification
f : focal length
d : distance between lens and object (screen in this case)

See : http://en.wikipedia.org/wiki/Lens_(optics)#Thin_lens_equation

For the FOV it'll also depend on the distance between the eye and the lens, but I don't know an easy way to calculate it, see my second answer.

Does this fromula depends on the shape of the lens, or is it a general formula? if there is no formula; is there a way to approximate thoses values?

You could try with the thin lens approximation (see previous link), I tried but I didn't get correct results, especially for wide FOVs.

To calculate it correctly you need to use the Snell-Descartes law for refraction, but you need to know the refractive index of the lens and its surface profile. Generally the lenses used are aspheric, so you can use the formula shown here : http://en.wikipedia.org/wiki/Aspheric_l ... ce_profile

But manufacturers don't give these informations, so you'd have to estimate them yourself. If you don't estimate them correctly I guess your calculations would be useless. With the right parameters you should also be able to calculate the FOV for a given eye-lens distance.

I've been trying to do this, for now I've got a basic HTML/Javascript page that calculates the FOV for a given display size and lens focal length using the thin lens approximation, but as I said the results don't seem to be correct.

Capture-aspheric.html - Chromium.png

I'm now trying to do the same for the 5x aspheric lens, I took a photo of the lens from the side and I'll try to estimate its parameters using the photo and this app : http://www.arachnoid.com/OpticalRayTracer/

I'll also try to estimate the refractive index with a laser pointer I recently bought, but that may be complicated, if possible at all.

Is it right to think that the diameter of the lens have an effect on the FOV?

I think it'll limit the FOV.

I mean, if we imagine a gigantic lens, and we place the eye in the center, the FOV will be narrow right?

I don't think so, but anyway it's not possible to have gigantic magnifying lenses, there is a physical limit to their size (which is quite small, can't remember where I've read that).

what about other type of lenses? (double convex, double concave, convex-concave, concave-flat...)

For single lens arrangements I think only convex lenses would be useful. For LEEP arrangements with two lenses you may have a look here : http://www.vrtifacts.com/hmds/leep-on-the-cheap/

[Fredz]

Maybe this thread should be moved to the VR section, more chances to get valuable feedback there.

[cybereality]

OK, no prob. Moved the thread to the VR R&D section.

[Fredz]

Thanks Cyber.

[Tarzan]

Hello everybody

Thank you Fredz for your answers. Everything you said quite confirmed what I thought.
Anyway, I'm still disappointed to see that there is no efficient way (i mean for amateurs like us) to calculate precisely the results produced by simple unique lenses.

M = f / (f -d)

M : magnification
f : focal length
d : distance between lens and object (screen in this case)

Well, so LOT of magnification = SHORT focal length. If think we can so easily add = LARGE FOV to the equation.

You could try with the thin lens approximation (see previous link), I tried but I didn't get correct results, especially for wide FOVs.

The problem, in my opinion, is because we are not working with thin lenses. The lenses we are looking for for HMDs are large ones (to produce a big FOV)

To calculate it correctly you need to use the Snell-Descartes law for refraction, but you need to know the refractive index of the lens and its surface profile. Generally the lenses used are aspheric, so you can use the formula shown here : http://en.wikipedia.org/wiki/Aspheric_l ... ce_profile

But manufacturers don't give these informations, so you'd have to estimate them yourself. If you don't estimate them correctly I guess your calculations would be useless. With the right parameters you should also be able to calculate the FOV for a given eye-lens distance.

Yeah, manufacturers don't give those informations, especially for lenses that are not made for "precise" optic applications... too bad

I've been trying to do this, for now I've got a basic HTML/Javascript page that calculates the FOV for a given display size and lens focal length using the thin lens approximation, but as I said the results don't seem to be correct.

Nice try, and good job

I'm now trying to do the same for the 5x aspheric lens, I took a photo of the lens from the side and I'll try to estimate its parameters using the photo and this app : http://www.arachnoid.com/OpticalRayTracer/

I will certainly give a try at this app when I will have received my lenses

Tarzan wrote:
Is it right to think that the diameter of the lens have an effect on the FOV?
I think it'll limit the FOV.

Tarzan wrote:
I mean, if we imagine a gigantic lens, and we place the eye in the center, the FOV will be narrow right?
I don't think so, but anyway it's not possible to have gigantic magnifying lenses, there is a physical limit to their size (which is quite small, can't remember where I've read that).

thanks for the infomation about gigantic lenses. I didn't knew it.

But I still can't figure out the effect of the diameter of the lens on the FOV. Let's not speak about gigantic lenses, but about small lenses. We have 2 lenses :
- 50 mm diameter, 30 mm focal length
- 5 mm diameter (around the size of the pupil), 30 mm focal length
If we look straigth into the lenses, without turning the eye globe (wich my cause bad results in the case of the small lens), will the visible result be the same ?

I will give you my opinion, so we will be able to discuss. In my mind, the result is yes and no. It depends on the type of lense :

If we consider aspheric lenses, the result for both lenses will be the same, because the main factor important here is the curvature of the lens. Because the 2 lenses have the same focal length, I assume they have the same curvature.The the light trajectories will be the same, and so the effect.

If we consider 2 common lenses, the result will no be the same. According from what I read online, the problem with non-aspheric lenses, is that they don't have a unique focal point. If you draw the light trajectories, the more far from the center of the lens the ray passes, the more far from the theorical focal point it will arrive. I the case of the 5mm diameter lens, an important part of the rays will pass far from the center of the lens. So the focal point will not be clear and the image produced blurry. This is not the case with the 5cm diameter lens because most of the rays pass near the center. The image will get a better quality.

I am right to think that, or do I miss something ? In that case, we can consider non-aspheric lenses good for HMD devices as long as they are enoug big (how much big is what I wonder...)

For the building of my HMD, I have found many lenses. Those lenses may or may not have a good quality, I don't know yet. Some are in acrylic, some in glass. I have very few information on it :
- diam. 20.8mm, f.l. 18 (convex)
- diam. 22mm, f.l ? (convex aspheric)
- diam. 25mm, f.l 16.5 (convex)
- diam. 25mm, f.l 28 (convex)
- diam. 25mm, f.l 28 (convex aspheric)
- diam. 27.4mm, f.l 25 (convex)
- diam. 30mm, f.l 30 (convex)
- diam. 32mm, f.l 11 +-2 (convex)
- diam. 25mm, f.l 28 (convex)
- diam. 33mm, f.l 32 (convex)
- diam. 33mm, f.l 38 (convex)
- diam. 33.5mm, f.l 22+-2 (convex)
- diam. 36mm, f.l ? (convex)
- diam. 37mm, f.l 28 (convex)
- diam. 38mm, f.l 25.4 (convex aspheric)
- diam. 41.8mm, f.l 63 (convex)
- diam. 49.2mm, f.l 29 (convex)
- diam. 50mm, f.l 28 (fresnel)
- diam. 50mm, f.l 30 (convex)
- diam. 69mm, f.l 45 (convex)
- diam. 74.2mm, f.l 50 (convex)

In your opinion, should I get rid of some or do they seem to fit for a 7 inch display ?

Thank you for everything

[Fredz]

The problem, in my opinion, is because we are not working with thin lenses. The lenses we are looking for for HMDs are large ones (to produce a big FOV)

I thought aspheric lenses were closer to the thin lens approximation since they don't exhibit spherical abberation like standard lenses do. Maybe it's only an error in my calculations though, the only way to be sure is to try with aspheric lenses equations.

I am right to think that, or do I miss something ? In that case, we can consider non-aspheric lenses good for HMD devices as long as they are enoug big (how much big is what I wonder...)

With big standard lenses the main problem will be the weight, even more if they're made of glass (which seems common for such lenses). I think aspheric lenses are the best candidates for now since they are cheap, are often made from plastic (so they are lightweight) and they don't suffer from spherical abberation. Fresnel lenses could also work equally well it seems according to a recent post in the DIY HMD thread.

For the building of my HMD, I have found many lenses. Those lenses may or may not have a good quality, I don't know yet. Some are in acrylic, some in glass. I have very few information on it :
[...]
In your opinion, should I get rid of some or do they seem to fit for a 7 inch display ?

Until I manage to implement a good lens simulation for HMDs I really can't answer that question. I don't know much about optics in HMD, but one aspect that is often mentionned is the exit pupil. I'm not sure lenses with a small diameter could provide a large exit pupil, that's probably why DIY HMDs are all using large lenses (50mm diameter).

[geekmaster]

... Fresnel lenses could also work equally well it seems according to a recent post in the DIY HMD thread. ...

Not equally well. Good quality lenses are always better than fresnel lenses.

As mentioned in my posts in the other thread, fresnel lenses with a fine pitch (many grooves per inch) work much better than I expected (when stacked in layers as described), and would be great for a simple "quick and dirty" DIY Rift prototype. For sure, they provide a quick and inexpensive (and locally available) resource to try while waiting for your "good" lenses to arrive...

However, the way I stacked these fresnel lens elements actually gives me a "supernatural" FoV (making my nose invisible with a larger than normal FoV), so in some ways they may be better than ordinary aspheric lenses. Of course, quality custom lenses should give the same effect while providing superior image quality.

FYI, the referenced post (and follow-ups) starts here:
http://www.mtbs3d.com/phpBB/viewtopic.p ... 335#p95633

[cadcoke5]

Just a comment that, while flat display technology has, and is, and will be undergoing major improvements. But, another commentator in another thread said that lenses are pretty much stagnate, and so have remained a major bottleneck to HMD.

I imagine the lens system is the thing to work on first, since all the display electronics and actual display can be simulated by pasting a picture on a piece of cardboard. Then if the lens does not work out, you have not wasted the effort on the other stuff.

Joe Dunfee

[kwx]

Is there a way to have the formula between Focal length, FOV, and magnification?

M = f / (f -d)

M : magnification
f : focal length
d : distance between lens and object (screen in this case)

I think this is the wrong formula for the HMD use case - this magnification calculation is for projecting an image onto a screen, where you want light emanating from a point to be collected at a single point at the other side of the lens.

Just to add to the confusion, the "magnification factor" usually shown for magnifying glasses or loupes is something different also. A 10x magnification means that you can hold the object at 25mm instead of the standard 250mm, so it looks 10x bigger. The focal length is equal to 250mm / (magnification factor), so a 10x lens has a 25mm focal length.

For a HMD, the lenses are supposed to make the screen appear further away than it actually is, so that the eye can focus on it comfortably even though the screen is only a couple inches away. To keep things simple, let's say we want to have the screen look right when the eye is set to focus at infinity. This means that light rays coming from a single point of the display all become parallel on the other side of the lens(*), and you get that when the distance from the lens to the screen equals the focal length.

Here's a diagram:


Ignoring lens distortion, the field of view then depends on the following factors:

• Distance from lens to screen - see above, we'll assume that this equals the lens focal length "f".
• Distance from the eye to the lens, "e".
• Size of the lens - the diagram uses "r" for the lens radius, this is half the diameter.
• Size of the screen image ("s" = distance from viewport center to edge) - this doesn't actually affect the field of view directly, but it's a limiting factor. If your field of view only lets you see a small part of the screen, you're wasting pixels. If you can see out beyond the edges of the screen, the excess field of view doesn't contribute to immersion since you can't display anything there. (That's what happens in the image on the right of the diagram.)

Based on the diagram, the resulting formulas are:

Code: Select all

FOV = 2 * atan(r / e)

Code: Select all

s = f * r/e

So let's say I'm using a 5x lens (f=50mm) with 40mm diameter (r=20mm), and have my eye 15mm from the lens (e=15mm). Then the potential FOV = 2 * atan(20mm / 15mm) = 106 degrees.

Then the next question is if this FOV is usable. The viewport radius is s = 50mm * 20mm / 15mm = 66mm, or diameter 132mm. If I draw two images side by side with the centers separated by my pupil distance of 68mm, the needed screen width is 68mm + 2 * 66mm = 200mm. My Nexus 7 only has a 150mm screen width, so I'd actually be seeing empty space around the edges of the display. Unless I'd want to strap a 10" tablet to my face, I'd need a stronger lens or accept a smaller FOV with my eye further away from the lens.

Turning it around, let's say I want to reach s=41mm so that the viewports fill my N7 screen. Then the eye distance e = f * r / s = 50mm * 20mm / 41mm = 24mm, and FOV = 2 * atan(20mm / 24mm) = 80 degrees.

These numbers seems to be a pretty good match for my actual experiments. What do you think? If you have built a headset, I'd be curious to know how your numbers work out.

(*) Footnote: Your eye is then responsible for focusing the parallel rays onto a single point on your retina, same as it does when looking at a distant object. Since I'd consider hardware modifications to eyes to be beyond the scope of typical DIY projects, we don't really need to include this in the overall calculations.

[android78]

@kwx - Nice write up. You are still missing one point. Your eyes don't rotate around the lens, but a point behind the lens. So, while the calculations may be correct for looking straight ahead through the lens, they will be off somewhat when looking to the side, top or bottom. I think this is one of the problems of having a high magnification lens so close to the eye (without it being a contact lens), because even if you have it perfect focus in the middle, once you move your eye, the focus will shift slightly, since your eye is moving closer or further away from the lens.
This issue becomes worse, the closer to your eye that you have the lens too.

[kwx]

You are still missing one point. Your eyes don't rotate around the lens, but a point behind the lens. So, while the calculations may be correct for looking straight ahead through the lens, they will be off somewhat when looking to the side, top or bottom.

You're right, though the calculation should still work for estimating peripheral vision FOV while looking straight ahead. A rough approximation would be to reduce the effective lens radius to compensate - rotating a 12mm radius eyeball by 45 degrees moves the pupil laterally by 12mm * sin(45) = 8mm.

In any case, really close distances won't work since the distance "e" is measured between the optical lens centers, and your cornea would bump into the lens surface well before that reaches zero. But I think the formula is still useful as an upper bound, i.e. I now know that I won't realistically get more than ~80deg FOV given the lenses and screen size I was using.

The (probably obvious) conclusion is that a large FOV needs lenses with a large usable diameter fairly close to the eyes, and sufficiently short focal length to have the screen viewports fill your view.

[cadcoke5]

The (probably obvious) conclusion is that a large FOV needs lenses with a large usable diameter fairly close to the eyes, and sufficiently short focal length to have the screen viewports fill your view.

I think this is why it is good to go to a larger display that is further away from the eye. Of course, then the possibility of using a single display split down the middle is shot out of the water, unless you use some sort of system, like using prisms, to bend the eye's view outward. Then the simplicity of the system is also shot out of the water.

I think the best compromise is for the display to use something like a LCD shutter system or polarized display to allow the left and right views to occupy the same display. The idea of a larger display that is further away is more practical since the advent of the pocket projector or the soon to-be-available OLED displays. Both systems can be used to make a lightweight curved display.

Why can't such a system be used with a simple pair of strong reading glasses? True, it does not fully fill your view with an in-focus display. But in the real-world, people who wear glasses experience that anyway, and seem to be OK with it. There are reading glasses that cover more of the eye than the typical end-of-nose style. I use some for viewing my computer screen.

To test the idea, buy a strong pair of reading glasses from a $1 store. Apply red and cyan gel over each eye, and print out a rec/cyan 3D image the size of your intended display. Then, you can see how it will work.

Joe Dunfee

[kwx]

I think the best compromise is for the display to use something like a LCD shutter system or polarized display to allow the left and right views to occupy the same display. The idea of a larger display that is further away is more practical since the advent of the pocket projector or the soon to-be-available OLED displays. Both systems can be used to make a lightweight curved display.

That's an interesting approach, but it has the downside of cutting the framerate in half. Based on what I'd been reading, it seems that even 60Hz isn't quite enough to be fully immersive, and 85Hz+ would be preferable. Finding affordable displays that can handle twice that would be challenging.

Why can't such a system be used with a simple pair of strong reading glasses? True, it does not fully fill your view with an in-focus display. But in the real-world, people who wear glasses experience that anyway, and seem to be OK with it. There are reading glasses that cover more of the eye than the typical end-of-nose style. I use some for viewing my computer screen.

Hm - typical off-the-shelf reading glasses seem to go up to +3.5 diopter or so, and that's a focal length of 286mm. (f = 1m / diopter.) For comparison, I've just ordered a pair of 7x magnifiers for my DIY kit, that's a 36mm focal length or +28 diopters. Not sure if a display at >20cm distance would be practical, somehow I'm now visualizing something like this:



I guess it could work with projectors on a very lightweight screen, and a pair of projectors could use polarized light to avoid cutting the framerate in half. If you do exeriment in that direction, I'd be curious how it turns out.

[cadcoke5]

typical off-the-shelf reading glasses seem to go up to +3.5 diopter or so, and that's a focal length of 286mm. (f = 1m / diopter.) For comparison, I've just ordered a pair of 7x magnifiers for my DIY kit, that's a 36mm focal length or +28 diopters. Not sure if a display at >20cm distance would be practical

I think that is only if you assume the eye is focused at infinity. If you can compromise on that issue, then a +3.5 dopter can certainly allow a person with normal vision to focus closer. But, I don't understand the math enough to calculate what apparent distance it is.

Joe Dunfee

[kwx]

I think that is only if you assume the eye is focused at infinity. If you can compromise on that issue, then a +3.5 dopter can certainly allow a person with normal vision to focus closer. But, I don't understand the math enough to calculate what apparent distance it is.

True, but you need to be careful for accomodation/convergence mismatch. If the screen is displaying distant objects but the eye needs to focus at a close distance, this can appear unnatural and in extreme cases lead to inability to fuse the images. This probably varies with the individual, but personally I seem to tolerate too-distant focus much better than too-close focus.

If I understand the math right, the diopter values are just reciprocal meters, and you can add and subtract them for basic calculations. For example, a 3 diopter lens would move the effective focus distance from 0dpt (1/0 m, infinity) to 3dpt (1/3 m) or from 2dpt (0.5m) to 5dpt (0.2m).

[mcdohl]

hello all, I found a shop with double convex lenses and double concave lenses. The diameter 50 mm and focal lenght from 100mm to 500mm. would be appropriate the 50mm diameter with 300mm focal lenght would give good result for a 7 "? double concave or double convex

[kwx]

hello all, I found a shop with double convex lenses and double concave lenses. The diameter 50 mm and focal lenght from 100mm to 500mm. would be appropriate the 50mm diameter with 300mm focal lenght would give good result for a 7 "? double concave or double convex

A double concave lens spreads light instead of focusing it, these are used for myopia correction but aren't suitable for a head-mounted display. (Except as part of multi-lens optics, but that's significantly more complex.) Try double convex, and I'd recommend something around 50mm focal length for a 7" tablet.

[mcdohl]

I finally bought 2" aspherical lenses. If something works, buy it

[drifterXF]

Aah, I just saw that there is an older thread already covering the lense problematic.

Can anyone provide feedback regarding the screen-coverage within the FOV when using the common 2" pocket lenses at focal distance (50mm) with a 5.6" and a 7" screen?

I'd expect the unused, black border-area around the actual image/screen to be quite big when using these lenses with a 5.6" display.