I Understanding magnification of concave mirrors for near-eye situation?

[nikosb]

TL;DR Summary

Does the magnification of a virtual image depend on the distance between the eye and a concave mirror?

Hi,

I am working on a project for a near-eye display where I need to magnify and project far out a real image using a mirror. I have some off the shelves mirror that I am experimenting with but I can't seem to get any perceived magnification at all when I use a concave mirror. To illustrate the problem I created the following setup shown below. I have two mirrors side by side, one flat and one spherical mirror with radius of curvature of 130cm and focal length of 65cm.

I use a phone to display a real image of the letter "A". The phone is placed at a distance of approximately 70cm away from the mirrors. Using another phone I took a photo of the reflected image on both mirrors.

As you can see from the images above the virtual image reflected off the concave is clearly a lot bigger, 2-3 times.

Next, I took another photo but this time I moved the camera very close to the mirrors, about 2cm, to simulate a near-eye situation as shown in the illustration below.

Please note that the I didn't move the object, the distance between the real image "A" and the mirrors is the same as before, approximately 70cm. In the photo comparison below you can see the perceived virtual image doesn't appear to be magnified at all by the concave mirror.

I have the following questions:

1. Why does the concave mirror fail to magnify the image when the eye is very close to the mirror? I thought the magnification depends only on the distance between the object and the mirror and the focal length. How does the distance between the eye of the observer and the mirror affect the magnification?

2. How can I actually magnify the image when the mirror is close to the eye? Do I have to use a different mirror, like an aspheric?

3. Since the distance between the object the and the mirror is greater than the focal length why does a virtual image still forms? I thought only real images form in that situation.

 

[sophiecentaur]

Read this link. It shows you how all distances and curvature affect what you see. If you look at a concave mirror from beyond its focal length the image will be real and inverted. If you go in closer , through the focal plane, the image passes through an exploding version and ends up erect, right way up and bigger. If you have a concave mirror to hand then you can do the experiment yourself very easily.

Many homes have a shaving / makeup mirror with a concave face. Perfect for this sort of thing.

 

[nikosb]

Read this link. It shows you how all distances and curvature affect what you see. If you look at a concave mirror from beyond its focal length the image will be real and inverted. If you go in closer , through the focal plane, the image passes through an exploding version and ends up erect, right way up and bigger. If you have a concave mirror to hand then you can do the experiment yourself very easily.

Many homes have a shaving / makeup mirror with a concave face. Perfect for this sort of thing.

I've read that link. This theory applies to on-axis image formation. Even if a real image forms an observer has to be on-axis and behind the real image to see it. If one is off-axis and in front of the real image (between the real image and the mirror) then one doesn't see the real image. I haven't found any information for concave mirrors and off-axis image formation.

 

[nikosb]

I've read that link. This theory applies to on-axis image formation. Even if a real image forms an observer has to be on-axis and behind the real image to see it. If one is off-axis and in front of the real image (between the real image and the mirror) then one doesn't see the real image. I haven't found any information for concave mirrors and off-axis image formation or near-eye situations.

 

[Ibix]

Worth noting that as you get very close to a spherical mirror the behaviour for a given object must tend towards indistinguishable from a flat mirror, for the same reason that the Earth looks flat when standing on it.

Is the camera autofocusing? Because that might be making a difference to what's imaged over and above the simple mirror optics.

 

[sophiecentaur]

This theory applies to on-axis image formation.

The on axis condition also applies off axis (with finite errors, of course) If it didn't, then no optical devices would work. You could tilt the lens a bit and the focus would not overlap the path from the object but you will still see a magnified image, albeit a bit distorted.
Better still, do what reflecting telescope designers do. They have a small, secondary mirror near the entry point of the tube at 45 degrees and the image is viewed at the side of the optical tube. You omit the eyepiece lens and place your eye closer than the focal point. The image is hard to see as you need good control to chase it without the help of an eyepiece lens.

If you want to bring your eye close enough to the main mirror for magnification you need the eyepiece tube to be placed nearer to the mirror.

 

[Drakkith]

1. Why does the concave mirror fail to magnify the image when the eye is very close to the mirror? I thought the magnification depends only on the distance between the object and the mirror and the focal length. How does the distance between the eye of the observer and the mirror affect the magnification?

I believe the crux of your issue is a misunderstanding of your system here. The diagrams on the page that Sophie linked to, and the ones you've likely seen, are for simple systems with single a optical component (a single mirror) focusing light down onto a surface. You do not have just a single mirror. You have a mirror and then the optical system that is your camera. The simple rules shown in the diagrams get thrown out the window once you introduce additional optical components. In general, the image might end up smaller, larger, inverted, distorted, or something else depending on what exact setup you have after the mirror.

One way to model this would be to turn the mirror into a positive lens, add another positive lens after the first to represent your camera's optical assembly, and then add the camera sensor shortly after that as a surface that the image is focused onto. There might be a simpler way to go about things than this, but it's been years since I took my optical engineering classes and I can't remember.

You're also putting the camera very close to the mirror, where it intercepts the light well before it can come to focus to form a real image in the air and well before it has even converged much. I think this is the reason you're getting very little magnification. My guess is that you need a stronger mirror if you want to place the camera at such a short distance from the mirror.

 

[Drakkith]

I've read that link. This theory applies to on-axis image formation. Even if a real image forms an observer has to be on-axis and behind the real image to see it. If one is off-axis and in front of the real image (between the real image and the mirror) then one doesn't see the real image. I haven't found any information for concave mirrors and off-axis image formation.

As Sophie said, there is no difference between being on and off-axis except for a variable degree of degradation due to aberrations and distortion.

 

[sophiecentaur]

You have a mirror and then the optical system that is your camera.

With a camera it may be that manual focussing can select what hits the sensor coherently. Have you done a diagram to scale? You could post it for us all to have a better idea.

 

[Drakkith]

With a camera it may be that manual focussing can select what hits the sensor coherently.

What does 'coherently' mean here? Are you using that word as it is used in optics, or a more general way?

 

[sophiecentaur]

What does 'coherently' mean here? Are you using that word as it is used in optics, or a more general way?

The way a lens produces an image is (ideally) to make all optical paths from a point on the Object arrive in phase (same time) at a point in the image plane. That's an 'optical' definition of "coherently", IMO.

 

[sophiecentaur]

The simple rules shown in the diagrams get thrown out the window once you introduce additional optical components.

The eye is an 'additional component' and a simple camera lens behaves the same way. When there is a magnified virtual image, the camera 'sees it' the same as we do but not if autofocusing insists on a different plane for the image sensor. If the field of view from the camera contains objects at very different distances from the virtual magnified image then the camera won't know what to do. OTOH, a camera works fine with many mirror images; it's a common trope in cinematography.