Stopping down an imaging lens improves image quality – most of the time. When stopping down too much, the image quality gets worse again. This short article gives a simple intuitive explanation of why that is and when it makes sense to stop down and when it does not.
Stopping Down
The following imaging system creates an imperfect image of its object:
The reason for the imperfect image creation is simple: The marginal rays (red) do not intersect with the chief ray (blue) at the image plane, failing to create a sharp image. Instead, the marginal rays intersect far behind the image plane, thus creating a blurry image at the image plane.
Removing those problematic marginal rays removes the image blur. This is done by blocking them with an aperture. This process, called “stopping down the lens,” looks like this:
Now, the new marginal rays (green) perfectly intersect with the chief ray at the image plane. The image is as sharp as it can get. Stopping down the lens improved image quality. The only apparent downside is perhaps somewhat longer exposure times, as less light is now forming the image.
If exposure time is not much of a concern, the implication seems to be that it never really makes sense to have large apertures, as small apertures create better images. Small apertures remove problematic rays. However, when stopping down the lens too much, the image becomes blurry again, like this:
At first, this finding is surprising. If all the rays intersect at the image plane and form the image at the correct plane, why would the image blur? The reason is that for the image to not blur, it must contain a certain amount of detail. This detail can get lost in one of two ways: either aberration causes the detail to get lost (as was the case above) or the detail never gets transported from the object to the image in the first place (as is now the case). Larger apertures transport more detail than smaller apertures.
This concept is perhaps easier understood by plotting the system’s MTF curves in all three scenarios. They would look like this:
The amount of detail contained in an image can be approximated by the MTF50 value, the frequency at which the MTF descends to 50%. Larger MTF50 values mean the image contains more detail, while smaller MTF50 values mean the image contains less detail.
In the no-aperture scenario (red), aberrations reduce the MTF and thus drastically drop the MTF50 value. In the aperture scenario (green), only aberration-free rays form the image, creating an overall much higher MTF and thus a higher MTF50 value. In the small-aperture scenario (blue), only aberration-free rays form the image. However, the system’s overall MTF resolving power is so drastically reduced by the small aperture that the aberration-free imaging alone doesn’t help: the MTF is low and so is the MTF50 value. There is practically no difference in resolving power between the aberrated and the heavily stopped-down system.
A Stopped Down Telescope
Some systems are more aberrated than others, and stopping those systems down is more effective in improving image quality. Consider, for example, the following 50 mm telescope, realized in three different variants:
One variant uses a 50 mm plano-convex element (PCX), a second variant a 50 mm double convex element (DCX), and a third variant an achromatic doublet (DBL) to create an image of an infinity-focused object at a 3° field illuminated with white light.
Just as before, a large aperture introduces a lot of aberration and creates a bad focus, while a small aperture introduces little aberration and thus creates a good focus. Like this:
Each variant’s resolution (or MTF50 value) differs with the aperture size. This resolution also differs between different field points. For aperture sizes between 0 mm and 10 mm, these are the resolutions in units of cyc/°:
From the graph, it becomes immediately obvious that the DBL variant of the telescope is far superior to the PCX and DCX variants. Not only does it have far higher resolution, it also supports much larger aperture sizes, both on-axis and in the field. The optimal aperture sizes and their corresponding resolutions are:
| Variant | Field | Aperture Diameter (mm) | Resolution / MTF50 (cyc/°) |
| PCX | 0° (on-axis) | 2.8 | 26.1 |
| 3° | 2.9 | 7.4 | |
| DCX | 0° (on-axis) | 2.9 | 26.8 |
| 3° | 5.7 | 9.3 | |
| DBL | 0° (on-axis) | 6.8 | 63.0 |
| 3° | 2.9 | 23.6 |
For a telescope to have good-enough resolution for the human eye, it is a good idea to have it resolve at least 23 cyc/°. Only the doublet variant of the telescope can provide that over the entire field at a 2.9 mm aperture.
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