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I need to find the precise collection cone angle of an arbitrary objective at an arbitrary (but known and finite) object-side focal length. This is typically given as the "numerical aperture", defined as sin(entrance pupil radius / focal length). In other words, I need to find a way to get the precise diameter of the entrance pupil.

I am aware of the relationship magnification = (object NA / image NA), and thanks to a calibration image and the data sheet of my CCD, I actually do know the magnification to three decimal places. I am not sure if this is helpful at all.

I am having great trouble finding useful (and correct) literature on this topic, as this is obviously irrelevant to "normal" photography. (Background: I am trying to generalize formulae in interferometry, which rely heavily on collection angles but assume simple thin lenses.)

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This is not "obviously irrelevant to normal photography" at all; we just don't normally worry about the sort of precision that you'll need to deal with. There are two numbers that we ordinarily take at face value, knowing that they're slight fibs: the focal length of the lens (which is usualy rounded to a "friendly" value except on data sheets), and the rather imprecise f-stop value that represents the entrance pupil diameter in a way that is most useful for everyday photography (which is more concerned with effective quantitative light transmission and effective depth of field than with precise geometry).

The focal length we can solve, but the only circumstances in which the approximation stamped on the lens isn't close enough for jazz is when we need to worry about precise reproduction ratios and the calculations we need to make to achieve them. From a photographer's point of view, this has traditionally been limited to the province of large-format macro and micro photography, and, frankly, the simple process of measuring (directly) on the ground glass is less error-prone in the field than calculating and setting up according to the calculations. The equipment is large and tends to use bellows (which may have calibrated vernier indicators on the rails, but there are mechanical offsets to consider between the rail and lens board and between the rail and the camera back/film plane, and both of those are subject to mechanical stresses that change with the orientation of the camera). You don't have that limitation; you can use a rather precise helical (screw) focus, leaving only thermal expansion to compensate for if conditions change, and can include a reliable light-based measuring system if stable dimensions are going to be a problem.

The other factor, and the core of your question, is the nut of the problem. The aperture iris will only be round when the lens is used wide open (or you're using Waterhouse stops or something similar), and it's usually buried inside the lens. (Yes, modern irises are "round", but not round round They're good enough for bokeh balls, but not good enough for Euclid.) To top that, it's not the physical iris opening that we need to worry about, it's the size of the entrance pupil, and that's just a virtual image of the physical iris, which may be larger or smaller than the entrance pupil.

For ordinary photography, we are only concerned with compensating for bellows draw (the change in effective f-stop and T-stop caused by focusing closer). We can make that compensation using only the magnification (since it depends on ratios rather than absolute values) or using only focus extension and the experimentally-derived (or spec sheet) focal length (since the entrance pupil remains constant, the same ratio applies). The actual depth of field is usually determined by inspection; the main concern is for correct exposure, and if we can get that to within 1/3 of a stop or better, we stop worrying about decimal places right there. Again, the magnification is usually a matter of direct measurement in technical (film) photography, since it is the most critical element. (And even then there is usually a measurement reference placed in the object field, such as a forensic corner scale, so that enlargement and reduction of the resulting photograph is easily determined and compensated for.)

But back to the main question at hand: determining the entrance pupil. I'm afraid that since it is an optical phenomenon (the iris of the lens as it is seen through the lens) the only real solution is to measure what you can see as precisely as you can — or rely on a data sheet, if you can get one. And don't expect the value to be better than approximate unless the lens is wide open or uses Waterhouse (or similar) stops rather than a mechanically-variable iris.

  • We can assume the lens is wide open, yes. – sebastian_k Jul 18 '14 at 16:51

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