Measuring Focal Length, Aperture and F-Stop

I've been asked how I measure the focal length and f-stop of my homemade lenses. The answer is: That depends. It depends on whether I am measuring a single element lens, in which case there is a simple way to directly measure focal length and f-stop, or I am measuring a multi-element lens, in which case I use a comparison method. Let's take the simple case first.

Single element lenses with positive focal lengths, (that is, they focus light, rather than diverge it) can be directly measured to determine their focal lengths and apertures. You'll need a piece of paper, a ruler, and a distant light source such as a lamp across the room (do NOT use the sun for this). Project the light from the lamp through the lens and onto the paper. Adjust the distance between the lens and the paper until the image is at its sharpest. Now just measure the distance from the lens to the image on the paper. This is the focal length of the lens, or close to it. Ideally, the light source should be an infinite distance away, but I find that a light across the room gives me a pretty good approximation. (If you insist on precision, then also measure the distance from the object to the lens and use the formula given above.)

Measuring Focal Length, Aperture and F-Stop

Measuring the aperture of a single element lens is even easier. Just measure the diameter of the lens, excluding any opaque border or mount. The f-stop (or the term I prefer: focal ratio) is simply the focal length divided by the aperture.

Example: Suppose that when you projected a distant light source through a lens and onto a piece of paper, the distance between the lens and the focused image was about 6 inches (150 mm). Suppose also that this lens is about 2 inches (50 mm) in diameter. The focal ratio of the lens is 6 divided by 2 (or 150 divided by 50) which equals f/ 3.0.

OK, that was too easy. Multi-element lenses are not that easy. First of all, you cannot determine the focal length of a multi-element lens simply by measuring the distance from the lens to a focused image. Doing that, doesn't tell you the focal length, however it does tell you something interesting. It tells you the back focal length, which is also important because that's where the film/sensor needs to be to capture focused images. But I'm getting off topic.

To understand the measurement of focal length, do the following experiment: Take that awful zoom kit lens that came with your camera (or any zoom lens), and starting with the lens set to its shortest focal length, measure the distance from the lens mount to a focused image just as we did above for a single element lens. If the lens was made for a SLR or DSLR, then you will get a measurement in the range of 45 mm. This is the back focal length. Now adjust the zoom to the maximum focal length and take the measurement again. What's this? It doesn't change. But did you notice that something else did change? The size of the image changed. So the key to measuring focal length of a complex lens is to measure the image size. It's possible to do this with an image projected on a piece of paper. But there's a better way: using photographs to compare the image size with a lens of a known focal length.

Measuring Focal Length, Aperture and F-Stop

You've probably figured it out by now, but let me spell it out, step-by-step. Put your camera on a tripod and point it at an object with distinct edges. I like to use a bookcase. Now take two photos: one using your mystery lens, and one using a reference lens with a known focal length. If your reference lens is a zoom, try to adjust it so that the object appears to be the same size as it is with the mystery lens. If you are able to get a match, just look at where the zoom is set. That is the focal length of your mystery lens.

However, if your mystery lens seems to be outside the range of your zoom, or if you are comparing it to a fixed (prime) lens, then there are a few more steps. Choose the biggest object that is clearly defined in both images. I like to use the distance between two shelves in my bookcase. Get your ruler out again because you will need to measure the size of this object as it appears in each of your two photos. I usually do a crude measurement using the LCD on the back of my camera. A more accurate measurement can be done by uploading them to your computer. Either way, just make sure you are comparing full size images shown at the same size. The focal length of your mystery lens is calculated by multiplying the focal length of your reference lens by the ratio of the two measurements.

Example: suppose your reference lens has a focal length of 50 mm. And suppose that when you measure the distance between two shelves in the image taken with the mystery lens you get 35 mm, while in the image taken with the reference lens, the distance between two shelves measures 25 mm. This means that your mystery lens has a focal length of 50 x (35 / 25) = 70 mm.

By the way, there's absolutely no reason why the above method can't be used with film. It just won't be as fast.

Measuring Focal Length, Aperture and F-Stop

Now what about f-stop? Once again, I prefer to measure the effective focal ratio (F-stop) of a homemade or mystery lens by comparing it to a reference lens. Use a reference lens with a focal length that is as close as possible to your mystery lens. Mount your camera on a tripod and point it at a scene in which the light level is not changing. I generally do this inside, under artificial light. Starting with the mystery lens and the camera set to manual, adjust the shutter speed (and ISO if necessary) to get a correctly exposed photo. Now, leaving the shutter speed and ISO unchanged, try to get a matching photo using your reference lens by adjusting only the aperture. The histogram is very helpful. Keep in mind that if you are testing a homemade lens, it probably has much lower contrast than your reference lens, which will cause your histogram to be "scrunched together." You're looking for an approximate match of the centers of the curves. Just as when we were testing focal length, there are two possible outcomes. If you find an approximate match, you're done. Just read off the f-stop from the reference lens.

However, if your mystery lens is outside the aperture range of your reference lens, (most likely faster), then you're not quite done. I frequently have this case when I am using that nasty kit lens as my reference lens. The trick is to adjust the shutter speed or ISO until you have the closest match and then use this as an offset to the aperture reading on the reference lens.

Let's take an example: Suppose that the image from the mystery lens is brighter than the reference lens even when the reference lens is at its widest aperture (lowest f-stop). Suppose that the shutter speed and ISO (for both the mystery lens shot and the reference lens shot) is 1/60 second at ISO 400. Let's assume that your reference lens aperture is f/ 5.6. You notice that when you change the shutter speed of the reference shot to 1/15 second, you get an approximate match in image brightness between the mystery lens image and the reference image. Since each time you change the shutter speed by a factor of two is equivalent to a 1 stop change in aperture, changing from 1/60 to 1/15 is a two stop difference. So your mystery lens is two stops faster than f/ 5.6, which is f/ 2.8. (A one f-stop change is a factor of 1.4. Standard f-stops are 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32.)

Measuring Focal Length, Aperture and F-Stop

By the way, adjusting ISO rather than shutter speed will work just as well. Doubling the ISO is equivalent to a one stop aperture change. Therefore in the example above, increasing the ISO in the reference shot from 400 to 1600 should have produced the same results as decreasing the shutter speed from 1/60 to 1/15 second. If you choose to do this measurement using a flash (which I do not recommend), then adjusting ISO is your only option because adjusting shutter speed will not work.

This method gives you an approximate effective focal ratio which takes into account all the optical properties of the lens including light loss due to glass and coating quality.

One final note: If you've built a working SLR/DSLR lens with a focal length that measures less than 15 mm or a focal ratio that measures under f/ 0.75 , please let me know as soon as possible. ;-)

OK, back to the lab.