Collimatinglens

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

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.

Laser collimation

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.

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.

Laser collimatinglens

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.

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.

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.

Collimatinglensvs focusinglens

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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?

How to collimatelight

Collimated lightsource

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.

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.

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.

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.

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.

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.