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Why aperture affects depth of field?

Yonatan Feldman , Dec 22, 1997; 09:32 p.m.

Looking through the rwo physics books that I own, I was unable to find any information on how depth of field works. Why does depth of field decrease with larger apertures and increase with smaller ones?

Responses

Erik Addison , Dec 23, 1997; 09:35 a.m.

At large apertures, image forming light converges obliquely on the film plane. At small apertures it approximates more closely to parallel rays. If a focussed object moves nearer to or further from the camera, the focus point is displaced from the film plane to a point in front of or behind it. The magnitude of defocusing depends on the breadth of the cone of light projected from the rear element & truncated by the film plane. A broad cone (wide aperture) spreads the image more for a given displacement than a narrow cone.

Small apertures do not make an out of focus image perfectly sharp, only sharper than it would have been at a wider aperture but I believe there is an objective criterion for computing depth of field involving circles of confusion.

P. Aing , Dec 23, 1997; 10:36 p.m.

Here is one of the best "with the hands" explanation of the function an aperture. I found it in EOS list and it was written by Gary W. Sims :

"As you've probably heard, a very small hole in a surface like a thin wall will display a focussed 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.

> Therefore shouldn't the lowest aperture (i.e the largest f number) > be limiting for standard lenses compared to good quality primes? > I'm not sure exactly what you mean, but 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).

Regards,

Gary W. Sims Stonehaven Laboratory"

(BTW, I don't really know if I legally have the right to insert such a big quotation. I think it is more convenient than to give a pointer to the EOS mailing list database)

P. Aing , Dec 24, 1997; 08:23 p.m.

Just a precision : Gary W.Sims gave his retroactive permission for my last posting.

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