Aperture review

The Error

I have written about this topic in an earlier post, but I found that still this concept is not fully clear in my mind. So, I did some research, and I present an improved version for this topic.

In Wikipedia article, the definition is very clear of what is an “f-stop” also known as f-number:
“The f-number is dimensionless and is usually expressed using a lower-case hooked f with the format f/N, where N is the f-number.”
https://en.wikipedia.org/wiki/F-number

What is my often made Error ?
I mistakenly believed that the aperture is same as f-number, and indeed there are interchangeable terms, but in mathematical terms, the two, are very distinct: aperture diameter value is different than the f-stop value. There is a clear distinction between the NOTATION and the actual value !

N=f/d and is noted as “f/N“.

Example: an “f/4” f-stop means that the value of N is 4 and it does not mean that the f-stop is 1/4 !

https://www.geogebra.org/geometry/gvh78mx4?embed
https://www.geogebra.org/geometry/gvh78mx4

f-stop smaller than 1 !

f-stop can also be smaller than 1, in which case the aperture diameter is greater than the focus distance of the lens. But there is a problem with these lenses. Because the angle at which the light hits the sensor is getting bigger (angle made by horizontal axis and rays from lens edges to sensor),there are some limits in building a lens that can bend the rays so profound. There are also practical difficulties in taking photos, because the DOF will be extremely shallow. (for a small sized lens !)

Example of lenses with N<1 are the Carl Zeiss Super-Q-Gigantar 40mm f/0.33 or the Carl Zeiss Planar 50mm f/0.7:
Carl Zeiss Super-Q-Gigantar 40mm f/0.33: The Fastest Lens Ever Made? | PetaPixel
https://en.wikipedia.org/wiki/Carl_Zeiss_Planar_50mm_f/0.7

There are ways to bypass some of the construction difficulties of these type of lenses, but is not going to look exactly pretty. Check out this video: https://youtu.be/9cT0jXI7l4E?si=wXJHmer6poRrRy22

Shape of the lens

Another note I want to highlight, for which again I got a bit confused, is that the focus of the lens is not the same as the radius of the lens.

Thin lens optics and formulas:

Here, the author is showing that the focal is not same as the center of the lens: (start at minute 13)

https://en.wikipedia.org/wiki/Lens

https://www.microscopyu.com/tutorials/perfectlens

>> aperture relation to DOF: smaller aperture f/N = greater DOF, and inverse
>> aperture and sensor size -> apparent DOF > there is no apparent DOF !

Also, there is a limit at 0.5 that is explained by a Quora user as following:
Alan Robinson says: “For a well-corrected lens with both image space and object space in air (or vacuum), the F-number cannot be less than 0.5. This is a hard theoretical limit, derived from geometric ray optics, and linked to the second law of thermodynamics.” https://www.quora.com/How-low-can-a-camera-lens-aperture-go-Is-it-possible-to-get-to-an-f-stop-of-a-limit-at-0-infinitely-closer-or-is-there-some-hardware-limitation#:~:text=For%20a%20well%2Dcorrected%20lens,the%20second%20law%20of%20thermodynamics.