## F/# as an angle

i have recently come across an issue with designing an optical system to interface with an Oriel MS257 monochrometer and found people referring to the divergence of the instrument by the F/# listed in the MS257's spec sheet - an F/3.9 at the input slit. This didn't make sense to me as F/# is defined as the ratio of a lens's focal length to its aperture.

After some digging, it turns out that optics people use F/# and numerical aperture (NA) as ways of expressing a cone angle which, for the rest of this post, i will refer to as θ. If we define the half angle using tangent, θ/2 is defined by the ratio of its opposite length to its adjacent length which easily correlate to half a lens' aperture and its focal length respectively. Using basic trigonometry we get:

where d is the length of the aperture and fl is the focal length. We can then massage this equation such that it is in terms of F/#:

NOTE:

So drawing from the MS257 example where we have F/3.9 at the input, that translates to an input angle of 14.61°, and any source that we input into the monochrometer must be focusing at the input slit at an angle of 14.61°.

If we want to relate F/# to NA, we must start with the definition of numerical aperture which is defined by the equation:

where n is in index of refraction and in this situation it is ≈1, and φ is the half cone angle which we have been referring to as θ/2. We defined θ above and when we combine these two equations we get:

To again use our MS257 example of F/3.9, the input slit of the monochrometer has an NA of 0.127.

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