I agree that the averaging "feels wrong" at first glance, maybe especially so when coming from some other technical background. I had the same initial reaction. Regarding waiting for someone with real knowledge, Abbe's ghost, etc... science has been progressing largely based on predecessors' writings, which can even be preserved beyond their deaths. The selection of papers I mentioned is nothing special, just three I saved on my computer a few years ago when I wondered the same thing as Harry and researched it briefly. I can't really judge how correct or authoritative they are but if you don't like them then many other sources can be found with searches like "microscope resolution condenser aperture" and following references forward and backward.
1950, H. H. HOPKINS AND P. M. BARHAM, The Influence of the Condenser on Microscopic Resolution
1952, JOHN R. BAKER, Remarks on the Effect of the Aperture of the Condenser on Resolution by the Microscope
1991, D. J. GOLDSTEIN, Resolution in light microscopy studied by computer simulation
It is not what I would call a rule-of-thumb, rather an exact solution in a specific, simple physical case which can be considered an approximation in physical cases more complex than the one it was derived from. Not just internet forum hearsay or purely empirical (how I generally interpret "rule-of-thumb") or whatever. Here are some excerpts from Goldstein's paper:zondar wrote: ↑Sat Mar 30, 2024 5:00 pmThe common equation for resolution of a condenser+objective combination, as seen in the Thor Labs link, is highly idealized, containing only 3 variables (wavelength and the NA's), and hence seems more like a simple rule-of-thumb than one that can deal with real-life complexities (such as the subject that the light is passing through, changes in the RI of the medium(s) through which the light passes, etc.).
What I still wonder is how the formula (essentially averaging the two NA's) can ever be considered correct?
The rule (sometimes attributed to Abbe) that resolving power is proportional to the mean of NA and NA_c is correct for oblique coherent illumination in the case of a grating object, provided NA_obl does not exceed NA. In the case of two isolated objects the rule is only approximately correct, but applies even if NA_obl is greater than NA.
Following Martin (1966, p. 230) a comparison of resolution under different conditions is given using K: K = x⋅NA/λ where x is the interval of a grating, or separation of two objects, just resolved with an objective of given numerical aperture (NA), and λ is the wavelength of the light in vacuum. Assuming that resolving power is directly proportional to NA, the ultimate limit of resolution under given conditions is K wavelengths of light measured in the medium between object and objective.
The relationship between microscopic resolving power, NA and the obliquity of coherent illumination was first adequately explained on the basis of diffraction theory by Abbe (Abbe, 1873; Lummer & Reiche, 1910). K is 1.0 if the object is a diffraction grating illuminated with axial coherent light and 0.5 (the resolving power is doubled) if oblique coherent illumination just enters the objective aperture. Resolution with other types of specimen under various conditions has been investigated by many recent workers, some of whom are cited below, but is less widely understood.
Note that Abbe's and Goldstein's analyses of image resolution involve both:The present work indicates that the ‘rule’ is in fact precisely correct for a grating object provided the obliquity NA_obl of the coherent illumination does not exceed NA, and is approximately correct for two coherently illuminated line objects even if NA is smaller than NA_obl.
- An object being imaged which redirects some light outside the original, undisturbed double cone of illumination. (By diffraction in this case.)
- Taking into account the effect on image formation of light redirected beyond the original illumination cone angle but still falling within the objective entrance cone angle.
Did you read this somewhere? If so please give the source because I think it must be wrong or misunderstood. Or is it your own deduction? In that case I think you would want to start by explaining how you define image resolution in the absence of any subject being imaged. The idea that the inclusion of a subject to be imaged is somehow confounding the analysis of image resolution seems pretty nonsensical to me.
Baker in 1952 anticipated your complaint:
In addition, Hopkins and Barham themselves provided a graph summarizing all the math:The treatment by Hopkins and Barham is quite general. Many microscopists would probably like to know the conclusions to be drawn from their work in terms of practical microscopy, and it is the main purpose of the present paper to provide this information. When an object is too small to be resolved by a particular objective, another of higher aperture is usually substituted until the oil-immersion lens of N.A. 1.3 or 1.4 is reached. These two numerical apertures are therefore of particular importance in matters connected with resolution, and they alone will be considered here.
The curve of Figure 2 shows the variation of K with s. If the influence of the aperture of the condenser on resolution is ignored, K=0.61 for all values of s. This is the broken straight line in Figure 2. The broken curved line shows K as a function of s on the basis of the rule attributed to Abbe. According to this the effective aperture is the mean of the apertures of the condenser and the objective. It leads to an absurd result if the numerical aperture of the objective tends to zero. On the other hand it gives a rough approximation to the present result if 0.5 < s < 1.0. When one remembers the other factors (such as scattered light, contrast in the object) which influence the resolution of the microscope, it is not surprising that the Abbe rule has been acceptable in practice.