How do the f-number formulas f/D and 1/(2NA) relate, and why does NA suggest a limit near f/0.5?
Asked 9/19/2016
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For a lens focused near infinity, with small magnification and ignoring transmission losses, I usually see the photographic f-number defined as N = f/D, where D is the entrance pupil diameter. But optics texts also relate f-number to numerical aperture, often as approximately N ≈ 1/(2NA) (or more exactly in air, using the marginal ray angle). These seem to diverge for very fast lenses, and the NA form appears to imply a lower limit around f/0.5.
How should these formulas be reconciled for ordinary photographic lenses near the optical axis? In particular, when are f/D and 1/(2NA) effectively equivalent, and why does the NA-based expression make the lower limit on f-number more obvious?
Originally by Photography Stack Exchange contributor. Source · Licensed CC BY-SA 4.0
Photography Stack Exchange contributor
9y ago
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Your formula f-number = 1/(2*NA*), where NA is the numerical aperture is not an accurate reflection of the formula presented in the book you reference: f-number = 1/(2sinΘ'), unless one assumes the index of refraction is equal to exactly 1. The refractive index of a vacuum is 1. The refractive index of air at standard temperature and pressure is 1.000277. Although a thin lens will satisfy the n=1.000277 requirement, no compound lens is perfectly corrected for aberrations such as coma and spherical aberration. Thus f-number ≈ 1/(2*NA*) is the actual formula.
f-number = f/D where f is the focal length and D is the diameter of the entrance pupil is equivalent to f-number ≈ 1/(2*NA*) within the limitations of the maximum angle at which light entering the lens is allowed to pass through the lens. I think where some get off track is that they assume if the front of the objective is enlarged that the entrance pupil will necessarily also enlarge to the increased size of the objective.
Originally by user15871. Source · Licensed CC BY-SA 4.0
user15871
9y ago
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They are consistent once the assumptions are made clear.
For ordinary photography at small magnification and near the optical axis, the working photographic definition is N = f/D with D the entrance pupil diameter. This correctly predicts image irradiance for an ideal lossless lens.
Using numerical aperture, the more exact relation is tied to the marginal ray angle θ on the image side: N ≈ 1/(2NA), with NA = n sin θ and in air n is very close to 1, so this becomes essentially N ≈ 1/(2 sin θ).
For modest angles, sin θ ≈ tan θ, and geometrically tan θ ≈ D/(2f), so the NA form reduces to N = f/D. That is why both formulas agree for typical lenses.
They begin to differ for extremely fast lenses because the small-angle approximation breaks down. The NA form then shows the physical limit more clearly: since sin θ cannot exceed 1, the ideal lower bound is about N = 0.5 in air. The f/D expression does not display that limit by itself because it is the paraxial/geometric form.
So: use f/D for normal photographic practice, and view the NA expression as the more fundamental high-angle form that explains the f/0.5 limit.
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