I have seen two different definitions for the f-number of a universal lens. Are they consistent?

I assume that the subjects are far away (no macro, the magnification is low), and we have a lens that is corrected for coma and spherical aberration.

I mean f-number in the sense that it indicates the brightness of a lens that has no internal losses.

The 1 / (2 * Numerical_Aperture) definition results in f / 0.5 as the limit; The F / D definition is not.

I searched: https://books.google.it/books?id=UY6QzgzgieYC&lpg=PP1&dq=nakamura%20The%20F-number%20of%20a%20lens%20(%20F%20)%20is%20expressed%20as% 20half% 20the% 20opening% 20angle & hl = it & pg = PA25 # v = onepage & q = The% 20F number% 20of% 20a% 20lens & f = false

Supplement – Clarification attempt:

My background:

I have a mathematics degree with an applied / physical focus, so I understand Trig Identities & Approximations, though my work does not take up so much of my math training.

I have not studied optics since graduation.

As a photographer, I understand the everyday use of F-numbers when photographing non-macro subjects and that T-numbers are sometimes more relevant. I am aware of the changes in the effective f-number in macro cases, but I do not actually make macros.

Confusion and question:

The question relates to photographic lenses that are at least somewhat corrected for coma and spherical aberration, are focused almost infinitely, with negligible magnification and negligible internal losses in a medium having a refractive index near 1 at points on the sensor near the axis of the lens.

The most common formula for the f-number is: N = f / D

Numerical aperture ("NA") numerical aperture formulas in combination with numerical aperture formulas sometimes appear to yield results for f-number ("N") that differ from N = f / D when f / D is small (say f-number <2).

How should these contradictory results be reconciled?

The NA approach makes it clear that there is a lower limit to the f-number at 0.5 because the cone angle of the light that hits the center of the sensor can not exceed 180 degrees. This lower limit is not immediately apparent from the N = f / D formula.

My confusion is for small f-numbers above this f-number of 0.5.

As I said, I do not know much optics. I wonder if the inconsistencies are related to the assumed shape of the "second major plane" of the lens.

If the half-cone angle is θ & # 39 ;, I seem to get different values for θ & # 39 ;, depending on the assumed shape of the second principal plane:

- If the second principal plane is assumed to be flat, I get tan θ = = D / 2f
- If the second principal plane is assumed to be spherical with the radius f, I obtain sin θ = = D / 2f

Perhaps, as indicated in the comments, none of the forms is a *very* An accurate representation of a real lens and an accurate response can only be predicted by ray tracing.

In any case, sin θ = = D / 2f is a better approximation than tan θ #. = D / 2f for a universal lens?

(For slow lenses, θ ~ = sin θ = = tan θ ~ = D / 2f, where & im is in radians.)

I do not really understand this, but I read that a (near) spherical second major plane is desirable to correct for spherical aberrations.

If NAi = n sin θ und and f-number = 1 / (2 * NAi):

- If sin θ = = D / 2f, we get f-number = (1 / n) (f / D), even for fast lenses
- If tan θ = = D / 2f, we obtain f-number = (1 / n) (f / D) sqrt (1+ (D / 2f) ^ 2)