I love my Canon EF 85mm f/1.2L II USM lens. Or was it a 1.4? It turns out that even though it says f/1.2 on the barrel, the story is more complex than that.
F-stops are the bread and butter of photography; they quantify and control some of the most essential parameters in the making of an image, namely how much light gets to the sensor and your depth of field. An f-stop is simply the ratio of the focal length of the lens to the diameter of the entrance pupil. For example, with my 85mm f/1.2 lens, which has a 72mm entrance pupil:
This ratio simply represents how wide the opening of a lens is as compared to how long it is and thus, how much relative light it lets in. Though slightly confusing to beginning photographers, a lower f-stop means more light can be let in, because the relative diameter of the lens is greater. This affects both low-light abilities and depth of field. Lenses with a wide maximum aperture (or low minimum f-number) are highly sought-after for their enhanced low-light abilities and subject-isolating depth of field; they also tend to be of a higher general quality at normal apertures.
What the F(-stop)?
So, why did I call my 85mm f/1.2 lens a 1.4? Because it's not a 1.2. It's a 1.4. Actually, it's both. Here's the issue: nothing, including high quality, optics-level glass, transmits light perfectly. Every material absorbs and reflects some of the light hitting it, meaning that the amount of light that reaches the sensor is always less than the amount of light that enters the lens. Lens manufacturers work hard to ensure that the glass they use has a high "transmittance," which quantifies how transparent a material is by representing what factor of light that strikes the material passes through it, rather than being reflected or absorbed. A hypothetically perfectly transparent material would have a transmittance of 1 (100%). A modern camera lens has a transmittance 0.6-0.9 (60-90%), while top-level lenses may exceed that. Have you ever looked into the interior of your lens? Most lenses have 10-15 pieces of glass in them, but you certainly don't notice that many, right? That's because manufacturers seek to make that glass as transparent as possible to maintain a high transmittance.
Talking about transmittance gives rise to a natural analog to the f-stop, the T-stop. The T-stop is defined as the f-stop divided by the square root of the transmittance:
If you'd like to geek out with me for a hot second, notice that since the transmittance is always positive and strictly less than one, its square root is also always positive and strictly less than one and thus, the reciprocal of the square root is strictly greater than one. If you'd prefer I shut up about math and just tell you the result: the T-stop will always be greater than the f-stop. In other words, a lens will always perform more slowly than it's labeled to perform.
The Great Conspiracy
So, have lens manufacturers been pulling the wool over our eyes for years? No, of course not. Someone much smarter than me worked this out years ago. The fact is that the f-stop is a geometric quantity that is unaffected by how good or bad a lens' glass is. It will always be the ratio of the focal length to the entrance pupil diameter, whether you're using the latest L-glass or a repurposed toilet paper tube. The T-stop, on the other hand, is a measure of real-world performance. You might wonder then why manufacturers don't specify the T-stop of a lens instead of the f-stop. Some manufacturers do, actually. In particular, cinema lenses are always labeled with T-stops instead of f-stops. Whereas we can tolerate minor variations in still photography, the intricate lighting, multiple camera setup, and very nature of movies being a series of quickly projected images demands both accuracy and precision.
There's also another argument for labeling still lenses with f-stops: depth of field is determined by the f-stop, not the T-stop. Depth of field is a geometric property and is thus, unaffected by the amount of light coming through the lens. Another reason is that the effect is relatively small in modern lenses, typically on the order of a third or half-stop. Thus, it's easier to consider the difference negligible, rather than to follow the potentially confusing route of labeling lenses with the same geometric aperture by different stops. Remember my 85mm f/1.2? It has a T-stop of 1.4; so, when it comes to depth of field and isolating a subject, it behaves like an f/1.2 lens, but when it comes to light-gathering abilities, it behaves like an f/1.4 lens (a third of a stop slower).
How Fast Can We Go?
Did you think we were done with the math? Sorry (not sorry).
In this equation, Nmin is the minimum possible aperture, n is the index of refraction (a measure of how much light is bent by a material) of the medium between the lens and the film or sensor (typically air), and theta is the half-angle of the maximum cone of light that can reach the film or sensor (in other words, theta measures how wide a field of view the sensor can theoretically see). Note that because the index of refraction for air is one and sine is bounded above by one, Nmin is bounded below by one-half.
If you'd prefer the "OMG does this guy ever stop talking about math?" version, a lens that operates in air can never be faster than f/.5. However, the maximum cone shrinks as the image plane moves farther from the rear element of a lens; thus, the maximum aperture is normally around f/1.2 for modern DSLRs. This is also likely why maximum aperture lenses typically have their rear element in the seemingly precariously exposed position at edge of the lens, rather than being recessed. Because mirrorless cameras can move the image plane closer to the rear element (no swinging mirror to get in the way), the maximum cone of light is larger for a mirrorless camera and thus, the maximum aperture is wider, thus why we see those crazy f/0.95 lenses.
All lenses perform just a bit more slowly than they're labeled to. This is because even the high-quality glass used by manufacturers absorbs and reflects some light, preventing all of it from getting to the sensor. As such, the T-stop, an alternative to the f-stop, can be thought of as the "true" f-stop in terms of light-gathering ability. Depth of field is unaffected, however. If you plan to do video work, you should always be thinking in terms of T-stops.