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Images Formed by Ideal Lenses and Mirrors |
We'll consider the real and/or virtual images formed from a scene by ideal positive and negative lenses, and by ideal positive and negative mirrors in turn. We'll also consider the images formed by lenses when a real image is "projected through" them, so that the "scene" is "behind" the lens.
In each case, we'll be using the properties we described for ideal lenses and mirrors. In particular, we will be assuming that an image is formed in each case. The derivations done here are just to pin down the location and size of the image, which we can do by tracing two rays to one point on the image in each case; from that, plus the assumption that an image will be formed, we can obtain all the information we need. In the figures, the "scene" consists of a simple vertical arrow, and it's located to the right of the illustration. For lenses, the viewer (or "eyepoint") is somewhere off to the left, unless we mention otherwise. For mirrors, the viewer is located to the right.
Throughout the page, we're going to use the following labels on the diagrams:
1. Positive Lens, Scene in Front of Lens, Beyond Focus (inverted real image)
(Also see properties of an ideal positive lens.) The "scene" (a simple arrow) is to the right of the lens, with an image formed on the left side of the lens (figure 1). (This is the behavior of the front ("objective") lens in binoculars, simple refractor telescopes, microscopes, and simple camera lenses.) l1 is the distance from the lens to the scene, and l2 is the distance to the image formed by the lens. The magnification is defined as the ratio of the image size to the scene size:(1a)
There are a number of useful formulas for the magnification, which we'll now extract from the figure.
The green ray passes through the center of the lens and is not bent. Comparing the triangles marked with green arcs, we see
(1b)
The blue ray passes through the right focus, and hence will be bent to exit parallel to the axis. Comparing the triangles marked with blue arcs, we see
(1c)
Finally, the red ray in figure 1 enters from the right going parallel to the axis, so it will be bent to pass through the left focus. Comparing the similar triangles marked with red arcs, we also see
(1d)
Finally, the angles subtended by the scene and the image, as viewed from the center of the lens, are identical, as we can see by looking at the green ray. This is a common feature of all the images formed by lenses and mirrors.
When the scene is at infinity (l1 = infinity), the image is formed at the left focus of the lens (l2 = f). The angles subtended by the scene and the image are equal when viewed from the center of the lens, as we can see from the two angles marked with green arcs in figure 1. Hence the size of the image in this case is:
(1e)
For small angles, which is all we're typically dealing with in a telescope, this is almost exactly
(1f)
Figure 1: Real image formed by a positive lens |
2. Positive Lens, Scene in Front of Lens, Within Focus (erect virtual image)
The scene (a simple arrow, again) is to the right of the lens, but closer to the lens than the right focus (figure 2). A virtual image is formed to the right of the scene; it must be viewed by looking through the lens from the left. (This is the configuration of a simple magnifying glass or loupe. It's also how mild "reading glasses" work. The eyepiece on a simple astronomical telescope is also configured this way. Because the primary lens on a telescope projects an inverted image, and the eyepiece doesn't invert it "back", when we look through an astronomical telescope the image we see is inverted.) The actual light rays are shown as solid colored lines. The "apparent paths" of the rays from the head of the arrow in the scene are traced back to where they meet at the head of the arrow in the image. The paths which are traced back are shown as dotted lines.The magnification is once again defined to be
(2a)
Looking at the green ray which passes through the center of the lens in figure 2, both the scene and the image lie in the triangle marked with a green arc which is formed by the green ray and the green dotted line "traced back" to the image. Comparing the similar triangles they form, we see
(2b)
From the triangle marked with a red arc, whose top is formed by the red ray from the head of the scene arrow and the dotted line traced back to the right focus, we can see
(2c)
Finally, inspecting the triangle marked with a blue arc, we can see that
(2d)
When the scene is on the focus (l1 = f), the magnification is infinite, but the image is pushed out to infinity; it still subtends the same angle as the scene, when they're both viewed from the center of the lens.
Figure 2: Virtual image formed by a positive lens |
3. Positive Lens, Scene Behind Lens (erect real image)
The "real scene" is actually someplace off to the right of the diagram in figure 3. An image of it, which we call the "projected scene", is generated by an optical system of some sort, and projected from right to left, through the lens. (This configuration is used in the front lens of a simple teleconverter or Barlow lens. It's also how the lenses of your eyes are operating if you look at a distant object through glasses for farsightedness; in that case the lenses of the glasses are actually projecting images through the lenses of your eyes.)If we removed the lens, the scene would appear as shown at the left of figure 3. With the lens there, however, it's shifted and scaled; to find out how we trace the rays which would have gone to the head of the projected scene arrow. The actual light rays are shown as solid lines. The paths they would have followed are shown as dotted lines leading to the head of the scene arrow.
Again, we define the magnification,
(3a)
By inspecting the triangle marked with a green arc, we can see that
(3b)
From the blue triangle, we can see that
(3c)
And finally, from the triangle with the red arc, we can see that
(3d)
When the projected scene is infinitely distant (off to the left), all rays leading to a particular point on it are parallel, and the situation is identical to the case where the real scene is infinitely distant, but off to the right (and inverted).
Figure 3: Positive lens forming real image from scene projected through the lens |
4. Negative Lens, Scene in Front of Lens (erect virtual image)
(Also see properties of an ideal negative lens.) In figure 4, we've shown the scene on the right, and the (virtual) image is formed between the right focus and the lens. (This is the arrangement of the lenses in glasses for nearsightedness.)This is actually described nicely by case (1) with a negative focal length for the lens. None the less, in this section we'll find the formulas for use when all values are given as lengths (and are treated as positive or zero).
The actual rays are shown as solid lines. We've traced the the paths "back to the image" with dotted lines, and the lines leading to the foci are also shown with dotted lines.
Again we define
(4a)
and by inspecting the green ray we again see that
(4b)
From the triangle formed by the blue ray, we see that
(4c)
And from the triangle formed by the red ray, we see
(4d)
These were all identical to the equivalent formulas for case (3), a positive lens forming a real image from an image projected through the lens. The geometry of the two cases is identical, though of course the paths followed by the light rays are somewhat different.
When the scene is infinitely distant, the image is formed on the focus of the lens (l2 = f). Again, the angles subtended by the image and the scene are equal, as viewed from the center of the lens. Consequently, if the angle subtended by the image is θ, we again have the image size as:
(4e)
Figure 4: Negative lens forming virtual image |
5. Negative Lens, Scene Behind Lens, Beyond Focus (inverted virtual image)
As with case (3), in figure 5 we're projecting the scene through the lens. If the lens were not there, the scene would appear at the left, as shown by the "(Projected) Scene" in the figure. However, the lens blocks the light coming in from the right, and forms an inverted virtual image in place of the real image. (This is the configuration used in the eyepieces of simple opera glasses, where the projected image is formed by a positive lens in front.)This arrangement is also covered by case (3) with a negative focal length and a projected scene. In this section, however, we'll derive the formulas for use when all values are lengths, given as positive or zero.
Solid lines indicate light rays, while dotted ones show the paths the rays would have taken had the lens not been present. Dotted lines are also used to "backtrack" rays to the location of the virtual image.
As always, we have
(5a)
and checking the green ray shows that
(5b)
From the triangles marked with blue arcs, we can see that
(5c)
And from the triangles marked with red arcs, we see
(5d)
These formulas are identical to the ones from case (1), a positive lens forming a real image. The geometry is identical, too, save that the diagrams are flipped relative to each other.
When the projected scene is on the focus of the lens, the virtual image is at infinity, and subtends the same angle as the scene, as viewed from the center of the lens.
Figure 5: Negative lens, scene behind lens, forming inverted virtual image |
6. Negative Lens, Scene Behind Lens, Within Focus (erect real image)
This is the same as case (5), save that we've moved the scene closer to the lens, so that it's between the lens and the rear focus (figure 6). (This configuration is used on the rear lens of a simple teleconverter or Barlow lens, where the scene is projected through the lens by a positive lens placed in front.)This case is actually identical to case (2) with a negative focal length, and a direct image (not projected).
The magnification is:
(6a)
From the triangle marked with a green arc, we also have:
(6b)
From the red ray, which forms the triangle marked with the red arc, we have
(6c)
From the blue ray, which forms the triangle marked with a blue arc, we see
(6d)
Figure 6: Negative lens, scene behind lens, forming erect real image |
A mirror's behavior is identical with that of a lens, save that everything is reflected. Figures 1 through 6 describe the behavior if we just fold them in the middle, along the line marked "lens". Because of that, and because making these diagrams takes quite a bit of time, we're not going to provide illustrations for the mirror cases; we'll refer to the diagrams we used for the lenses and just repeat the resulting formulas.
There are six more cases for mirrors -- the same six cases we already treated for lenses, but reflected. We'll define m, h, l, and f the same way we defined them for lenses.
7. Positive (Concave) Mirror, Scene in front, beyond focus (inverted real image)
See figure 1 (but fold it over along the "lens" line), and see the discussion for case (1). This is the arrangement used in the primary mirror of a telescope, as well as the primary mirror of a mirror telephoto lens for a camera. We repeat the results of case (1) here with no further comment:(7a)
(7b)
(7c)
(7d)
The formula for image size is very important for telescopes. When the scene lies at infinity and subtends angle θ, the image size will be:
(7e)
8. Positive Mirror, Scene in front, within focus (erect virtual image)
See figure 2 (but fold it along the line marked "lens"), and see the discussion for case (2). Note that the virtual image appears behind the mirror -- you must look into the mirror to see it.A positive mirror with the scene closer to the mirror than the focal length is a "magnifying mirror"; it's how makeup mirrors are configured. Here are the results from case (2), repeated:
(8a)
(8b)
(8c)
(8d)
9. Positive Mirror, Scene behind mirror (erect real image)
By "scene behind the mirror", we mean a real image is projected by an optical system of some sort, and a mirror is interposed in the path. If the mirror weren't there, the scene would appear behind the mirror's actual position. I can't think of a common example which uses a positive mirror in this arrangement.See figure 3, and imagine it folded along the "lens" line, and see the discussion of case (3). Results, copied from case 3:
(9a)
(9b)
(9c)
(9d)
10. Negative (concave) Mirror, Scene in front (erect virtual image)
See figure 4, but fold it along the "lens" line, and see the discussion for case (4). But note that the virtual image appears behind the mirror -- you must look into the mirror to see it.Examples of this configuration abound. Christmas tree balls, those shiny spheres people put in their gardens, the bumpers of old cars -- just about any mirrored surface that's wrapped around something will provide an example of this.
Results from case (4), copied:
(10a)
(10b)
(10c)
(10d)
The size of the image of a distant object, reflected in such a mirror, will be:
(10e)
11. Negative Mirror, Scene behind mirror, beyond focus (inverted virtual image)
See figure 5, folded along the "lens" line, and see the discussion of case (5). Again keep in mind that, when we say "scene behind the mirror", we mean the scene is projected to a spot that would lie behind the mirror, if the mirror weren't there. The distance from the mirror's location to the plane of the projected scene would be longer than the focal length of the mirror.Note that the virtual image appears behind the mirror -- you must look into the mirror to see it. I can't think of a common example of this arrangement (but see case (12) for a related arrangement which is widely used).
Results, copied from case (5):
(5a)
(5b)
(5c)
(5d)
12. Negative Mirror, Scene behind mirror, within focus (erect real image)
See figure 6, folded along the line marked "lens", and see the discussion of case (6). Again, the scene is projected along a path which would form an image behind the location of the mirror, if the mirror were not there. In this case, the projected scene would lie no farther from the mirror's position than f.This configuration is used in the secondary mirror of some telescopes. By using a short focal length primary and a negative secondary, it's possible to build a compact telescope with the same optical properties as a far larger Newtonian reflector. (But such scopes typically don't use parabolic mirrors. They generally combine a "corrector plate" -- a thin front lens -- with either spherical or hyperbolic surfaces on the mirrors. That's way beyond the 'scope' of this page, however.)
Results, copied from case (6):
(6a)
(6b)
(6c)
(6d)
Page created on 09/18/2007