What is the reference point that the focal length of a lens is calculated from?

I understand the idea of a focal length for a single lens, i e the distance from the lens to the point at which parallel rays of light converge. However, in the case of a photographic lens with multiple lens elements, where exactly is the focal length of the lens as a whole measured from?

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From top: Positive/convex lens, negative/concave lens, concave mirror, convex mirror.

Parallel rays entering the lens will focus at some point (F), and the focal length (f) is given by the distance between the center of the lens (the optical center) and the focus point.

So the reference point is the optical center of a single element.

OK, but what about multi-element lenses?

For multi-element lenses there is no reference point you can find easily. As David says, the reference point is the center of a hypothetical single element with the same focal length.

This reference point can be anywhere - in front of the first element, inside the lens, or behind the last element.

How can the optical center be shifted outside the lens?

Telephoto group: Most commonly by using what's called a telephoto group:

In this diagram there are two element groups. The first group (to the left) acts like a "normal" (convex or positive) lens, causing the rays (blue lines) to converge. The second group (to the right) is the telephoto group, acting as a negative lens that spreads the rays.

The net effect is that the focus point will "see" the equivalent of a single positive element much further away (indicated by red dotted lines). It's the optical center of this hypothetical "equivalent single element" (H') that is the reference point for measuring the focal length (f').

Inverted telephoto: You can swap the groups to put the telephoto group in front. Then you get a (wide angle) lens where the distance between the last element and the focus point can be greater than the focal length. This construction is called a retrofocus lens.

Mirrors: You can also use mirrors. Mirror lenses "reuse" their physical length by bouncing the rays back and forth. Again the focus point will "see" the equivalent of a single element much further away.

Mirror lens, here combined with telephoto group

Why would you want to do that?

For long tele lenses, it's because a standard design would give a lens that is physically too long to be convenient:

500mm tele without telephoto group. A 500mm would have to be at least 50 cm (20") long.

For wide angle lenses, it's to give more space between the lens and the image sensor. As an example, there are 10mm lenses for DSLRs, but 10mm between the sensor and the lens wouldn't leave enough room for the mirror. So ultrawide lenses are generally designed as retrofocus lenses.

7.5mm fisheye without retrofocus. Note the tube sticking out from the lens mount to get the elements close enough to the film. Mounting the lens required mirror lock-up, and you couldn't use the viewfinder or the built-in metering while the lens was mounted. (Image from B&H)

Then how can I check the focal length of my lens?

In short:

• take a picture of two distant points
• measure the angle between the points
• measure the distance between the points on the image sensor (count the pixels between the points in the photo, and derive the sensor distance from resolution and sensor size)
• focal length = distance-on-sensor-in-mm / angle-in-degrees * (180 / pi)

Sources:

Images: Fisheye lens from B&H, other images courtesy Wikipedia.

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Thank you for a really thorough answer! – SoftMemes Apr 10 '12 at 22:20

To expand on the Nodal point:

Simple definition: it is at the center of the equivalent single element meniscus lens which has the same focal length and aperture as the compound lens under test.

The most practical aspect of the Nodal Point of a lens is that it is the point around which the lens can be swung without producing any movement in the image. For a simple one-element lens, the node is in the center of the lens; for a folded optics system such as a Schmidt-Cassegrain telescope, the node is way out in front of the lens.

It is a bit tricky to set up a physical test to find the nodal point of a lens. I'll try to explain it without visual aids:

Consider a tripod mount. The vertical axis of rotation of the tripod head is going to be the place where we are going to put the lens's node. To do this, the lens has to be mounted on a plate that allows its distance from the tripod axis to be varied. In addition, the image must be projected onto a screen not attached to the lens (so the camera body cannot be used!). We look for the combination of screen position and slider setting where the tripod head can be rotated and the only effect on the image is vignetting as the optical axis of the lens moves into / out of being in line with the screen.

A real practical implication comes into play when you are taking panoramic shots. For there to be the least distortion between images - and hence better stitching or tracking - the camera has to be turned around its nodal point. So putting your pocket camera - or even your fancy pro DSLR - on the tripod directly, using its body's mount point, is going to mean you are rotating around somewhere between the nodal point and the image plane, except possibly with some wide angle lenses.

Generally speaking, for an apparent 1:1 fov lens (50mm on a full size imager, or on a 35 mm film body), the node will be somewhere near the middle of the lens. For a zoom lens at full zoom, it is likely to be out in front, and for a wide angle it will be somewhere behind the lens center. Whichever you are using, the node will be at the stated focal length in front of the camera's imager.

(this was the subject of one of my more intense Physics labs back in the day, so even tho I got an A for it there is a 45 year memory gap twixt now and then, but I think I have the general details right still).

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"When a photographic lens is set to "infinity", its rear nodal point is separated from the sensor or film, at the focal plane, by the lens's focal length."

Which leads to the question... What is a nodal point? Also answered by Wikipedia as:

"The front and rear nodal points have the property that a ray aimed at one of them will be refracted by the lens such that it appears to have come from the other, and with the same angle with respect to the optical axis."

With a complex structure that is a modern camera lens, the rear nodal point isn't always going to be in the same place, so the question of exactness will depend on the lens.

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Thank you for the effort, but unfortunately, I still don't quite get it. Could, in theory, even say outside the physical lens itself? – SoftMemes Mar 26 '12 at 7:39
@Freed - You would have to consult a physicist, I'm not an expert in optics. – John Cavan Mar 26 '12 at 10:40
John, I appreciate that, but surely someone here has to be. :) – SoftMemes Mar 26 '12 at 15:04
@Freed - You assume that will lead to clarity... :) I would note that physical lens lengths are usually smaller than their focal length (take a look at some of the pancake lenses out there) so that has some implication to all of this. The physics of this is beyond me, I'd rather take pictures anyways... :) – John Cavan Mar 26 '12 at 15:38
The focal length is not necessarily related to the physical size of the lens. A "telephoto" lens has a longer focal length than the physical lens size, that is what telephoto means. There are also "retro focal" lenses, that are physically longer than their focal length. – Pat Farrell Mar 31 '12 at 15:47