F ratio and image brightness

The f -ratio of a telescope or camera lens is the ratio of focal length to aperture:

Aperture diameter

One of the most basic principles of photography is that the brightness of the image, on the film or sensor, depends on the f-ratio. That's why, in daytime photography, we describe every exposure with an ISO setting, shutter speed, and f-ratio.

To understand why this is so, remember that the aperture tells you how much light is gathered, and the focal length tells you how much it is spread out on the sensor. If you gather a lot of light and don't spread it out much, you have a low f-ratio and a bright image.

The mathematically adept reader will note that the amount of light gathered, and the extent to which it is spread out, are both areas whereas the f -ratio is calculated from two distances (focal length and aperture). Thus the f -ratio is actually the square root of the brightness ratio. Specifically:

Old f-ratio 2

Relative change in brightness = -

So at f /2 you get twice as much light as at f /2.8 because (2.8/2)2 = 2. This allows you to take the same picture with half the exposure time, or in general:

Exposure time at new f -ratio =

Old f-ratio

New f-ratio 2

For DSLRs, this formula is exact. For film, the exposure times would have to be corrected for reciprocity failure.

Low f -ratios give brighter images and shorter exposures. That's why we call a lens or telescope "fast" if it has a low f -ratio.

Comparing apples to oranges

A common source of confusion is that lenses change their f -ratio by changing their diameter, but telescopes change their f -ratio by changing their focal length.

For instance, if you compare 200-mm f /4 and f /2.8 telephoto lenses, you're looking at two lenses that produce the same size image but gather different amounts of light. But if you compare 8-inch (20-cm) f /10 and f /6.3 telescopes, you're comparing telescopes that gather the same amount of light and spread it out to different extents.

Lenses have adjustable diaphragms to change the aperture, but telescopes aren't adjustable. The only way to change a telescope from one f -ratio to another is to add a compressor (focal reducer) or projection lens of some sort. When you do, you change the image size.

That's why, when you add a focal reducer to a telescope to brighten the image, you also make the image smaller. Opening up the aperture on a camera lens does no such thing because you're not changing the focal length.

There are no focal reducers for camera lenses, but there are Barlow lenses (negative projection lenses). They're called teleconverters, and - just like a Barlow lens in a telescope - they increase the focal length and the f-ratio, making the image larger and dimmer.

What about stars and visual observing?

The received wisdom among astronomers is that f-ratio doesn't affect star images, only images of extended objects (planets, nebulae, and the like). The reason is that star images are supposed to be points regardless of the focal length. This is only partly true; the size of star images depends on optical quality, focusing accuracy, and atmospheric conditions. The only safe statement is that the limiting star magnitude of an astronomical photograph is hard to predict.

The f -ratio of a telescope does not directly determine the brightness of a visual image. The eyepiece also plays a role. A 20-cm f /10 telescope and a 20-cm f /6.3 telescope, both operating at x 100, give equally bright views, but with different eyepieces. With the same eyepiece, the f /6.3 telescope would give a brighter view at lower magnification.

Why aren't all lenses f /i?

If low f-ratios are better, why don't all telescopes and lenses have as low an f -ratio as possible? Obviously, physical bulk is one limiting factor. A 600-mm f /1 camera lens, if you could get it, would be about two feet in diameter, too heavy to carry; a 600-mm f /8 lens is transportable.

A more important limitation is lens aberrations. There is no way to make a lens or mirror system that forms perfectly sharp images over a wide field. This fact may come as a shock to the aspiring photographer, but it is true.

Consider for example a Newtonian telescope. As Sir Isaac Newton proved, a perfect paraboloidal mirror forms a perfect image at the very center of the field. Away from the center of the field, though, the light rays are no longer hitting the paraboloid straight-on. In effect, they are hitting a shape which, to them, is a distorted or tilted paraboloid, and the image suffers coma, which is one type of off-axis blur. A perfectly made Cassegrain or refractor has the same problem.

The lower the f-ratio, the more severe this problem becomes. An f/10 paraboloid is nearly flat and still looks nearly paraboloidal when approached from a degree or two off axis. An f /4 paraboloid is deeply curved and suffers appreciable coma in that situation. For that reason, "fast" telescopes, although designed for wide-field viewing, often aren't very sharp at the edges of the field.

Complex mirror and lens systems can reduce aberrations but never eliminate them completely. Any optical design is a compromise between tolerable errors. For more about aberrations, see Astrophotography for the Amateur (1999), pp. 71-73.

Incidentally, f /1 is not a physical limit. Canon once made a 50-mm f /0.95 lens. Radio astronomers use dish antennas that are typically f /0.3.

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