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The focal length of a lens is defined by the thin lens equation, and it can be interpreted as a measure of the inverse strength of the lens. If you make a lens's optical surfaces more strongly curved, or if you increase its index of refraction, it gets stronger, and the focal length goes down. When you change do and di so as to maintain focus, the focal length f normally stays constant; this is what justifies interpreting it as a fixed property of the lens. (As pointed out in a comment, some lenses do contain moving parts that allow them to automatically change their focal length, but this is a side issue.)

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So what's the correct term for “stuff at this distance will be in focus” then? (I.e., the thing you change with the focus ring.) If I want stuff 3 meters away to appear sharp, what parameter have I set to 3 meters?

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A possible source of confusion is that in many cases when you're doing photography, do is much greater than di. Under these conditions, di is approximately the same as f. Therefore some people may be under the impression that the focal length is defined as the distance from lens to sensor. But in reality, when you change the focus on your camera, di changes while f stays the same.

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The term for the distance to the subject in focus is the focus distance and is measured from the image plane (sensor/film plane). The distance from the lens to the subject is called the working distance which can be significantly less within the context of macro photography. The zone which appears in focus either side (front and back) of the subject is the depth of field. This varies with the aperture - depth of field increases as the aperture gets smaller (f-number gets larger). All else being equal, depth of field is greater at f/4 than at f/2.

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Focal length is the distance between the lens and the sensor when the subject is in focus, not the distance to the subject.

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So if you focus on an object 3 meters away with a focal length of 18mm and aperture of f/11, everything from 1m to infinity will be in focus. However, if you focus on the same subject with the same aperture with a focal length of 135mm, the near focus limit is 2.9m and the far focus limit is 3.1m - the depth of field is only 20cm deep, in other words.

Generically, in optics, this is called the object distance. In photography it can also be referred to as the focal distance.

I was under the impression that the “focal length” of a lens is the distance at which stuff appears in-focus. (E.g., perhaps I set the camera so that objects 3 meters away appear sharp, and anything nearer or further than that is blurry.) But everything I've read seems to suggest that focal length is actually a slightly odd way of describing the field of view of the lens, and actually nothing to do with focus at all. (?)

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everything I've read seems to suggest that focal length is actually a slightly odd way of describing the field of view of the lens

The specific answer to the core of your Title Question, "the term for the distance", is: Infinity. Infinity is the (imagined) subject distance in front of the optical center of the lens that corresponds to an in-focus image on the sensor when it is spaced behind the lens at the nominal focal length. The engraved "focal length" which appears somewhere on the lens housing is a hypothetical specification of the nominal distance from the lens to the sensor when (imagined) subjects at infinity appear "in-focus" on the sensor. For a simple axi-symmetric double convex lens, the measuring reference point is the optical center (also called the geometric center) of the lens. To focus images of real subjects closer than infinity, the lens must be moved away from the sensor, towards the subject. In this situation, the focus (not focal) distance is always longer than that number engraved on the barrel (the focal distance). Thus, the nominal focal length is a convenient label to characterize the focus properties of the lens assembly. For compound lens assemblies there is no easily found reference point. The reference point is the center of a hypothetical single element with the same focal length. In this case the technique to determine the reference point is very complicated. It is up to the reader to investigate further. In answer your second question regarding: “stuff at this distance will be in focus”, the term is "subject to lens distance". Given f = focal length, u = subject to lens distance, and v = sensor to lens distance, the following formula represents the relationship: 1/f = ( 1/u ) + ( 1/v ). At infinity, 1/u approaches zero.

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