In the process of scripting a short instructional video about Time-Lapse Photography and dealing with Aperture Flicker, I recently started digging deeper into the physics of lens optics to try and better understand the relationship between Aperture and Depth of Field. I was never very good at physics in high school but in my research I came across this description written by the late Gary W. Sims of the Stonehaven Laboratory. It is by far the best and most accessible description of this relationship I have ever found. It also explains how variable aperture lenses work as well as the price differential between standard zoom lenses and fixed aperture lenses like those in the Canon L Series. It's still complicated and requires a measure of persistence but I wanted to share it here if for no other reason than to make it accessible somewhere beyond the random Photo.net message board where I found it.
As you've probably heard, a very small hole in a surface like a thin wall will display a focused image (of a landscape say) on another flat surface placed behind the wall. No lens is required. That's called a "pinhole camera" in English. Basically, the light from any given part of the scene has only one path to follow through the hole to the image plane behind the "wall."
Actually, many paths exist, but they are so tightly grouped together that they fall within so small an area on the image that it appears to be in complete focus without help from a lens. The smaller the hole, the better this appearance of focus. (Up to a point. Different topic.) But a small hole admits little light. So the scene must be in bright sunlight, and the area behind must be a fairly dark room for our eyes to pick up the faint image projected. Most film is less sensitive than our eyes, so the problem is worse when we want to capture the image for posterity. So we must make the hole bigger to capture enough light.
When we make the hole bigger, light from any given portion of the scene has more paths to follow. The optics of a lens bring each of those paths to the same point on the image plane -- or that's what we try to accomplish. The bigger the hole, the harder the problem of designing a lens that will bring each path to the same point for the three critical frequencies of light. (Light of different frequencies is refracted by a different angle through any given optical material.)
A factor you may not have noticed yet is that lenses with longer focal lengths tend to have smaller maximum aperture numbers. That arises from another effect. A lens of short focal length has a wide angle of acceptance of light. That's roughly a right angle (90 degrees or Pi/2 radians) for a "wide-angle" lens. A "standard" lens takes in a little less than 45 degrees, and a very "long" lens is down around ten degrees acceptance angle. The wide lens is accepting light from most of the scene. The "standard" lens accepts well under half the light that actually reaches the lens, and a long lens accepts almost none of the light.
Of course, that's our intent. We use a lens of longer focal length to restrict our image to a smaller part of the scene. But the effect of accepting light from so little of the scene is that it takes a bigger hole to capture enough light from that part of the scene to expose the film. Picture it this way: The film is the same size, but the part of the scene that must provide light to expose that film area is much smaller when we use a longer lens. That means the hole must be bigger if the focal length is longer if we want to expose the film in the same length of time.
If we measured aperture in millimeters, photographers would go crazy trying to compensate for the focal length of each lens to figure exposures. So we use a ratio. We say that a lens has an aperture of f/4 to mean that the hole is 1/4 the focal length. So a 50mm lens (standard in 35mm format) at f/4 will have a hole 50/4 or 12.5 mm across. At the same aperture number, a 100mm lens will have a hole of 100/4 or 25mm diameter. That gives the longer lens a hole with four times the area, so it can capture the same energy from an area of the scene that is four times as small.
Incidentally, when you price zoom lenses, you will notice that expensive zooms have a single maximum aperture number, like the Canon 28-70mm, f/2.8L that costs about U$1350. (The L stands for lust I understand.) Less expensive zooms, like the Canon 24-85mm (at U$350) have a range of aperture numbers specified. For that example, the range is f/3.5 to f/4.5.
That might give the impression that the less expensive lens closes down its aperture as it zooms, but in fact the opposite is happening. The hole actually expands from about 7 mm at the short focal length to 19 mm at the longest focal length, but the aperture ratio is decreasing because the optics are limited in the size hole they can properly focus. To hold an f/3.5 aperture at the long lens the hole would have to be 30% wider, or 24 mm across. The professional class lens starts at a 10 mm hole -- already half as big as the less expensive lens ever reaches -- and it expands to a 25 mm hole at the long end. Canon breaks off the short zoom at 70 mm, but they have a 70-200 mm that picks the range at the same aperture ratio of f/2.8. When that other lens reaches 85 mm focal length at f/2.8, it's hole is over 30 mm across. This is more than half again as wide as the 19 mm hole that the optical design of the less expensive lens can tolerate. At the long end, a Canon 400mm, f/2.8 lens has a hole that is 143mm across (call it 5.6 inches if you prefer). That's big enough to put your arm through.
Managing the optical paths through such a wide hole is not trivial, but even more important, when a wide hole and a wide angle of acceptance combine, the optical calculations keep supercomputers very busy. Thus the Canon 50mm,f/1.0L is a stunning achievement in "consumer" optics with a 50mm hole and an angle of acceptance of 40 degrees. ("Consumer" is relative. This time, I just mean it doesn't take a government to own one at "only" U$2,500 retail.)
The absolute size of the hole, and the angle of acceptance, are two of the dominant factors in designing a lens, and meeting that challenge requires sophisticated manufacturing after inordinately expensive design processes. That's why you'll find the smallest aperture numbers associated with the most expensive lenses.
On the other end of the scale, the design challenge of a big hole with a narrow angle of acceptance is less for something like a prime lens at say 200mm -- but a big hole means a wide lens. That means each element in the lens is physically larger, requires more high-quality glass (et al), and more surface area to polish, coat, and so forth. You can plot a neat curve that will predict the price of a lens and its weight very nicely from the maximum physical aperture it reaches.
The smallest hole is mechanically easy to achieve since we just swing the diaphragm blades closer together. Over quite a range, this just makes the lens' job easier, since the sheave of paths the light can follow is much narrower. Optical limits do arise at the smallest holes, but an aperture of say f/16 is easy to manage in the design and manufacture of a lens. Canon lenses typically provide f/22, and f/45 is available on the longest lenses (where that is not really a very small hole in absolute dimensions).
Gary W. Sims Stonehaven Laboratory