What is the definition of focal length for a camera lens?

Question

I came across many articles where the focal length of a camera lens is defined as the distance between the optical centre of the lens and the sensor. One such article is here:

https://photographylife.com/what-is-focal-length-in-photography

https://av.jpn.support.panasonic.com/support/global/cs/dsc/knowhow/knowhow11.html

Quoting from the first page, focal length measures 'the distance, in millimetres, between the optical centre of the lens and the camera’s sensor (or film plane).'

But, according to the lens law (1/v + 1/u = 1/f), when an object is at infinite distance (very far away), the distance from the lens where the image of that object forms is the focal length.

So, for instance, if an object is quite close to the lens and we change the distance between the lens and the sensor such that the object is focused on the sensor, the said distance does not remain same as f in the lens equation (since u is not large to the extent that 1/u tends to infinity). In that case, the distance between the lens and the film has to be uf/(u-f), which is different from f.

I am new to these concepts, so am I missing something?

Would much appreciate some insights in this regard.

Answer 1

You are not missing anything, you are exactly correct!

The confusion comes about from terminology and an omission of rigor that would be distracting for most purposes. When the articles you mentioned, and many like them, say "focal length measures 'the distance, in millimetres, between the optical centre of the lens and the camera’s sensor (or film plane),'" they leave out the implied subject at infinity portion.

Realistically the difference between subjects at infinity and pretty damn close is insignificant when you consider the sensor plane / subject distance ratios as applied to the lens law equation you quoted.

The terminology becomes fractious when you are talking about the imaging distance of a subject at close range. When the image is in-focus is that the focal length? It's the length required for focus but the definition of focal length is for subject at infinity.

It's really just word games, the lens law math is reality.

Fortunately for photographers, assuming a constant effective focal length equal to the defined infinite distance definition approximates well within necessary parameters through a very large range of practical use.

This difference becomes more significant with very close or Macro Photography. To quote from Effective Aperture and Macro

Put simply, when working at greater magnifications, roughly 1:2 or more, the displayed aperture on your lens or in your camera will be slightly different from what the true f-stop is, and this number will continue to change as the magnification of your shot increases. This is due to the focal length of the lens beginning to change as focus extension changes; since the lens is physically further away from the sensor or film, there is a change in exposure and f-stop.

Answer 2

Focal length is a fixed property (i.e., constant) of a lens element, lens group, prime lens, or for a given zoom setting of a zoom lens. It has nothing to do, directly, with where the lens is located with respect to the subject, or image/sensor plane of the camera.

The focus distances (subject, and image) are the distances from the lens to the subject, and from the lens to the image plane, respectively.

The thin lens equation you mention, relates the subject focus distance u, image focus distance v, and the fixed focal length ƒ. The only time the lens is positioned at an image focus distance v equal to ƒ is when the subject is at infinity. When the subject in focus is closer than infinity, the lens is further from the lens than the distance ƒ.

When the subject you are photographing is reproduced 1:1 on the image sensor (if your lens is able to focus that close; i.e., at that magnification level), then the subject focus distance is 2ƒ, and also the image focus distance is 2ƒ (this comes from setting u = v in the thin lens equation and solving for v).

Answer 3

Focal length is measured from the point where parallel rays from the source begin to converge to a focused point on the image plane. This point is the optical center (H' in the diagrams).

For the rays entering the lens to be essentially parallel the source must be some distance away... that distance is described as being infinity, but it obviously is not. And that distance is greater for lenses of larger diameter; so that all rays leaving the source at a significant angle diverge farther and miss the lens entrance (in reality the rays are not entirely parallel).

For a simple lens this is easy... it is the center of the lens to the image plane.

For a wide angle retrofocus lens you extend the diameter of the front opening (defines the limits of parallel rays) backwards. The focal length is less than the physical length of the lens.

And for a telephoto lens you extend the diameter of the objective element opening forward until it intercepts the image cone. The focal length is greater than the lens itself.

Answer 4

The focal length of a lens is a measurement made when the lens is imaging a far distant object such as a star. Such an object is said to be at an infinite distance. We focus the image of this target object on a screen. If the lens is a simple symmetrical single piece of glass, we measure from the center of the lens to the focused image. Likely this measurement will be made indoors on an optical bench. The target will be an artificial star. This is a tiny lamp that emits a parallel beam of light.

Camera lenses are rarely simple, they are a complex array of lenses conceived to mitigate image marring aberrations. This makes the measuring point difficult to locate. A special optical bench known as a nodal slide is needed. The complex lens has two cardinal measuring points. The focal length is measured from the rear nodal, the object distance is measured from the front nodal.

If the object being imaged is closer than infinity, the projected image cast by the lens becomes elongated. This happens because the lens has limited ability to refract (bend) light waves. As the camera approaches the object, its image becomes blurred, now we must adjust focus to retain a sharp image. This is accomplished by increasing the lens to film / image sensor distance. This elongated projection distance changes names from focal length to back focus distance.

The locations of the front and rear nodal are difficult to find without proper tools. The lens maker chooses their location based on need. They can be inverted, and they can fall in air forward or rearward of the glass in the lens barrel. These accommodations are made to shorten the barrel or make room for reflex mirror or allow a short focus lens to focus on the film / sensor.