Addendum - Attempt to clarify:
My background:
I have maths degrees, with an applied/physics bias, so I understand trig identities & approximations, though my work doesn't use so much of my maths education.
I haven't formally studied optics since high school thin spherical lens stuff.
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'm aware of changes in effective f-number in macro cases, but I don't really do macro.
Confusion, and question:
The question concerns photographic lenses at least somewhat corrected for coma and spherical aberration, focussed near infinity, with negligible magnification, negligible internal losses, in a medium of refractive index close to 1, at points on the sensor close to the axis of the lens.
The most common formula given for f-number is: N = f/D
Formulae for f-number involving Numerical Aperture ("NA"), combined with formulae for Numerical aperture sometimes appear to give results for f-number ("N") which differ from N = f/D when f/D is small (say f-number < 2).
How should these conflicting results be reconciled ?
The NA approach makes it clear that there's a lower bound on f-number, at 0.5, because the cone angle of the light hitting the centre of the sensor cannot exceed 180 degrees. That lower bound is not immediately clear from the N = f/D formula.
My confusion is for small-ish f-numbers above this f/0.5 limit.
As I said, I don't know much optics. I wonder if the inconsistencies are related to the assumed shape of the "Second principal plane" of the lens.
If the half-cone-angle is 𝜃', I seem to get different values for 𝜃', depending on the assumed shape of the Second Principal Plane:
- If the second principal plane is assumed flat, I get tan 𝜃' = D/2f
- If the second principal plane is assumed spherical, with radius f, I get sin 𝜃' = D/2f
Perhaps, as hinted at in the comments, neither shape is a very accurate representation of a real lens, and an accurate answer can only be predicted by ray-tracing.
Is any case, is sin 𝜃' = D/2f likely to be a better approximation than tan 𝜃' = D/2f for a general-purpose photographic lens ?
[For slow lenses, 𝜃' ~= sin 𝜃' ~= tan 𝜃' ~= D/2f, where 𝜃' is in radians]
I don't really understand this, but I read that a (near) spherical second principal plane is desirable to correct spherical aberrations.
If NAi = n sin 𝜃', 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 get f-number = (1/n)(f/D)sqrt(1+(D/2f)^2)
