• I agree with what BetterSense says, except for one quibble, which is that the term "Numerical Aperture" is not correctly used by BetterSense in this context. Numerical Aperture is essentially equivalent to f-number in the sense that both numbers are ways of expressing the angular aperture of a lens.

The f-number is twice the cotangent of the half angle that defines the angular aperture of the lens. (This is strictly true only if the principal surfaces are planes.) The Numerical Aperture is the sine of the half angle.

In the small angle approximation sin(angle)=~1/cot(angle), hence 2*f-number=~1/numerical aperture.

At lower f-numbers this approximation breaks down.

Someone can check my math, to make sure I have the factors of 2 placed correctly, but in any case I believe the functional relationships are correct.

I think what BetterSense meant to say was that the depth of field depends on the lens diameter, not the numerical aperture. (I'm overlooking some subtleties here, such as telephoto or retro-focus designs, in which the physical lens diameter does not equal the effective optical diameter.)