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I'm really confused about it. If such a lens is necessarily focused at infinity and does not have a plan of focus, then, does it mean it can produce a clear picture on a screen, no matter how close it is to the lens ? Eg., I want to build a telephoto with two positive meniscus (which creates an afocal system), each one having 4000mm of focal length. Then, I would have two questions:

  1. How should I calculate what is the resulting focal length ? Would it be halved as it is expected in any focal system with two identical positive lenses ? Like this:

F = (f1 * f2) / (f1 + f2 -d)

Where f1 and f2 are the two lenses' respective focal lengths and D is the space separating them.

  1. At what distance should I put my screen from the lens in order to produce a clear picture ? Would there be a certain back focal length ?

Sorry if I can be unclear. I'm an amateur that got really interested in the optics of lenses, and english isn't my native language.

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An afocal lens does not produce an image on a screen. Diverging rays from an object or screen point keep diverging on the other side of the lens, just not in a straight continuation. The point of an afocal lens is to change the geometry of the virtual image you are seeing. To get a screen image, you'll still need a convex imaging lens.

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Per the wikipedia article for an afocal system d=f1+f2, where d is the distance between the lenses, not the distance to the subject. Your equation then gives F as infinity, saying that you cannot project an image on the screen. In your example you would space the two lenses 8 meters apart and parallel rays coming in would be parallel going out. The lens combination would do nothing except invert the image.

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The front focal point of the first lens is imaged at the rear focal point of the second lens... i.e. a subject at infinity would be focused at 4000mm with the two lenses spaced 8000mm apart.

The first image is the typical afocal magnification system w/o consideration of front/rear focal distances.

If instead we consider a subject distant enough to be considered in focus and a point light source (i.e. at infinity) we get the second drawing. Which is then redrawn in the third drawing as an afocal imaging system (w/o focus mechanism).

enter image description here

https://spie.org/publications/fg01_p18_afocal_systems

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