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For some extended context, this is how Heads Up Displays (HUDs) and Reflector Sights work:

Reflector sight diagrams

However, as far as a I know, a lens will take collimated light and focus it to a very small point called the focal point. And since the reverse is also true, only the focal point can be collimated and "focused to infinity" to create the virtual image in a HUD.

Convex lens diagram

The reflector sight diagrams clearly show a source reticle that is larger than a focal point, and HUDs in general usually use an LCD display to create the initial image.

Although modern HUDs use several optical elements to correct for aberrations and other distortions, most early HUDs and all reflector sights used a single large convex lens to collimate the image.

Is this merely an issue of lens diameter? Would using a larger lens allow for the focal point to be larger in diameter too which would allow for a small display to be completely collimated?

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  • $\begingroup$ An image is a collection of many "small points of light" in different locations. I don't quite understand the question. Are you asking how geometrical optics describes how a lens focuses an off axis object point to an off-axis image point? For that you have to consider ray bundles that are neither parallel to the optical axis nor parallel to themselves, so the last image of a lens focusing a parallel ray bundle is insufficient. $\endgroup$ Commented May 25, 2023 at 2:18
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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
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    Commented May 25, 2023 at 2:28
  • $\begingroup$ @FlatterMann I'm asking how can a lens collimate an off axis image point since that's how a HUD works. The problem is a lens shouldn't be able to do such a thing because an off axis image point would relate to an off axis object point, in which case the ray path would be in a offset direction as seen in this example: physics.stackexchange.com/questions/200078/… $\endgroup$ Commented May 25, 2023 at 2:48
  • $\begingroup$ Why can a lens not map one plane on another? That's just how lenses work. One lens can also map a plane with object points onto a bundle of rays with different angles relative to the optical axis. That's an image "at infinity". A second lens can then focus that ray bundle onto a second plane. That second lens is, in this case, in your eye and the image plane is your retina. I still don't understand your question. $\endgroup$ Commented May 25, 2023 at 6:20
  • $\begingroup$ @FlatterMann In that case let me rephrase the question a bit. Can a lens have a focal point that has a known radius/size? I'm trying to figure out how a lens can project more than just the focal point to infinity, since that is how a reflector sight works. It takes the base image, and then using a convex lens displays it an infinity. My question is how does it achieve this if the focal point is smaller than the image to be displayed? $\endgroup$ Commented May 25, 2023 at 6:47

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A lens cannot collimate an image. It can take the light from each point of the image into a collimated beam in a different direction.

Your eye can take a collimated beam and focus it to a point on your retina. It can take beams in different directions and focus each to a different point, thus reconstructing the image.


Edit - A ray diagram will help.

This is from Holographic Combiners Improve Head-Up Displays. It shows a real HUD. This is a little more complex than ideal for our purposes.

enter image description here

At the far right is a screen that shows an image to be displayed to the pilot as he looks through the cockpit windshield. For our purposes, it is an object.

All the lenses on the right produce a magnified image at the field stop. For our purposes, this image is the object we care about. Points in the plane of the field stop emit light. That light is just what we would see if a real (magnified) object was at the field stop.

The lens on the left is the one you have been talking about. The blue rays in the center come from the center point of the image. An on axis collimated beam generated from that point passes through the eye box.

The green and red rays are from off axis image points. They too generate collimated beams that pass through the eye box and different angles.

Here is another not quite ideal ray diagram of a compound microscope from Bill Casselman's page at the University of British Columbia. Ignore the pink rays. The yellow rays look like our heads up display.

The point is that your eye will focus off axis collimated light to an off axis image point.

enter image description here

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  • $\begingroup$ If that is the case for all lenses, how does a HUD or Reflector Sight collimate an image as opposed to a point? $\endgroup$ Commented May 25, 2023 at 4:01
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    $\begingroup$ OP said, "collimate," but they probably meant to say, "focus." $\endgroup$ Commented May 25, 2023 at 11:53
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    $\begingroup$ Thank you the edit was exactly what I was looking for. I was under the false assumption that the eyebox was cylindrical with an infinite length, so I didn't know how the rays coming from an off axis would fit. The diagram and explanation shows that they converge at an eybox area which explains how complete images can appear collimated. $\endgroup$ Commented May 25, 2023 at 21:04
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We wouldn't normally say that an optic collimates an image. The HUD collimates the light from each point in the source separately, producing a separate collimated bean from each point. Unlike a collimator, these beams go in different directions. It thus transforms the location of a point on the source into the direction from which the light enters the eye.

The eye then performs the opposite transformation, projecting the beams coming from different directions onto different points on the retina.

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