How do lenses produce 2-dimensional images, if a lens bends all incoming rays of light to intersect at the focal point? Shouldn't this produce a single dot of light on a screen placed at the focal length?

This is basically the standard diagram that always shows up in textbooks:

std geometric optics pic

I know this doesn't happen in real life--I used to use telescopes pretty frequently for work. The most in-focus image would be the one with the smallest diameters for the stars, which, of course, we think of point sources at infinity. But--the light from all stars (despite being at infinity) is not all focused at a single point. Instead, each star's light is focused at its own $(x,y)$ point on a 2D image.

I can't reconcile the theory as I understand it with any of my real-life experience with optics. What am I missing?

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    $\begingroup$ Don't you actually get a single dot of light at the focal point? Pictures are taken at a distance before/after it, not at it exactly, as far as I understand. $\endgroup$ – Alex Leonov Aug 13 '15 at 3:47
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    $\begingroup$ In my "physics 3" course, they had us measure the focal length of a lens by finding the most "in-focus" image we could produce of the filament of a light bulb. When moving the lens and screen back and forth along an optical bench to find the best image, I never remember actually seeing a single focused dot. (This was years ago, though, so my memory might not be 100%) $\endgroup$ – Kristin Aug 13 '15 at 3:56
  • $\begingroup$ Right. Disregard my previous comment. Take a look at this picture upload.wikimedia.org/wikipedia/commons/thumb/c/c2/… and this page en.wikipedia.org/wiki/Cardinal_point_(optics)#Focal_planes $\endgroup$ – Alex Leonov Aug 13 '15 at 4:10
  • $\begingroup$ That actually happens in real life, for a ideal lenses ;) $\endgroup$ – Derek 朕會功夫 Aug 13 '15 at 15:59
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    $\begingroup$ A lense does not focus all incoming light to a point. It only focusses (as you diagram correctly shows) all light that comes in the same direction on a point. Thus, different points in the focal plane collect light from different directions. $\endgroup$ – Walter Aug 14 '15 at 17:21

...if a lens bends all incoming rays of light to intersect at the focal point? Shouldn't this produce a single dot of light...?

(In your diagram, the source image is at infinity. I will continue the analysis along that idea.)

It is true that all rays parallel to the axis focus to that single dot. Not all rays, however, are parallel to the axis:

enter image description here

Rays coming from different angles focus to different points. That is how an image is formed.

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    $\begingroup$ Okay, so while we think of as stars as being "at infinity," since they're actually a finite distance away, their light actually comes in at slightly different angles, and as such is focused at different points on the detector? $\endgroup$ – Kristin Aug 13 '15 at 4:22
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    $\begingroup$ @Kristin Not as such, for all but the biggest telescopes each star focuses to a point. But the light from the next star over is coming from a slightly different direction and that focuses onto a different point. For extended bodies, such as a person the light from the nose comes from a different direction than the light from the eyes and the light from the shoulder and so on. Each of those focus to a different point on the film/CCD in a way that mimics the physical arrangement of the parts imaged. $\endgroup$ – dmckee --- ex-moderator kitten Aug 13 '15 at 4:30
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    $\begingroup$ @Kristin Stars emit light radially in all directions. The light from each star illuminates the lens on all locations. Now, look at the diagram in my answer: there is a red star and a green star. Light from each illuminates all locations on the lens. Because of the difference in angles, though, the light from the green star focuses at a point above the focal point for the red star. $\endgroup$ – John1024 Aug 13 '15 at 5:34
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    $\begingroup$ I think this diagram also explains why images are often inverted when focused through a simple lens. $\endgroup$ – AaronLS Aug 14 '15 at 20:49
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    $\begingroup$ @Kristin no, they don't enter the lens at different locations. Light from both stars hits every point on the surface of the lens. But light from star A is focused on image point A, and light from star B is focused on image point B, because they enter the lens from different directions. $\endgroup$ – hobbs Aug 15 '15 at 5:06

A convex lens does not focus all the rays to a single point. It focuses the all axis parallel rays on the focal point. It also focuses all the rays emanating from a given point to a corresponding point on the other side. That point is the image of the original point.

The standard diagram shows that a lens sends all axis parallel incoming rays to the focal point. Parallel rays can be thought of as emanating from a point at inifinity. So you can think of focal point as the image of a point at infinity.

More general ray diagrams look like this: enter image description here

The full rules for drawing ray diagrams as given in the Hyperphysics article on the subject:

  1. A ray from the top of the object proceeding parallel to the centerline perpendicular to the lens. Beyond the lens, it will pass through the principal focal point. For a negative lens, it will proceed from the lens as if it emanated from the focal point on the near side of the lens.
  2. A ray through the center of the lens, which will be undeflected. (Actually, it will be jogged downward on the near side of the lens and back up on the exit side of the lens, but the resulting slight offset is neglected for thin lenses.)
  3. A ray through the principal focal point on the near side of the lens. It will proceed parallel to the centerline upon exit from the lens. The third ray is not really needed, since the first two locate the image.
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Your diagram shows parallel light beams originating from infinity. Light entering the eye in the real world isn't all parallel.

In the real world, all the light that bounces off a single point will hit the sensor (eye-cone, digital sensor, etc) at a single point, assuming that point is in focus. However, light from different points will strike the sensor at different places, creating the image that you see.

In-focus points

If the point is not in focus, it will be spread out on the sensor in a circle. Photographers call this the circle of confusion.

Out-of-focus point

Playing around with this tool should make it pretty obvious. I took these images from my answer to the Photography.SE question How does aperture work without “cropping” the image hitting the sensor?

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Truth be told: geometric optics is wrong. It is sufficient for small angles and not too complicated setups, but is still an approximation.

The first thing: Normal lenses do have not a focal point, but something which resembles more a line, the so-called caustic. The correct shape for getting a perfect point on the line crossing the middle of the lens (I call that axis now) is not a spherical shape, but a complicated curve, the cartesian ovoid. So you won't get a point image from normal optical surfaces in the first place; there are aplanatic lenses, but they are extremely expensive.

Second: Even if you have an aplanatic lens, still light is not a ray, but a wavefront. A perfect planar wavefront which passes an aplanatic lens forms still not a point, but a so-called airy disc. You see that very well that if you increase magnification; at some point the image gets more and more fuzzy, the airy discs of the point images are starting to overlap and ruin the image. The only possibility to avoid that is to increase the size of the lens or mirror which makes the airy disc smaller (That is the reason astronomers need big, big lenses and mirrors).

But even that is not sufficient enough if you want really high-quality, in that case you need the exact light equations which are described by the Maxwell equations.

Now we need an image, I am using the open source image of the German Wikipedia under "Linsengleichung":
Geometric optic image

As you see, if the source of the light rays is near the lens, the rays are not running parallel and their image does not develop in the focal point f, but in increased distance. You are using two rays from the same point: One parallel to the axis which crosses the focal point and one which runs from the point exactly through the lens. Where they cross is the image of the source point and this is not the focal point !

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    $\begingroup$ Truth be retold: wave optics is also wrong. It is sufficient for moderate intensities and not too high frequencies, but is still an approximation. $\endgroup$ – Ruslan Aug 13 '15 at 14:23

There is a fallacy in your question which might be the source of your confusion. It is not all light that gets focused to a point, it is only rays parallel to the lens axis!
For the case of two (point source) stars, if the rays of one star are parallel to the lens axis, the rays of the other star will not be (not co-linear), therefore its image will appear at a different point than the first star.

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