Averaging condenser and objective numerical aperture when predicting resolution

[hans]

The Gold Standard for specimens that demonstrate resolving capabilities is the Pluerosigma diatom. It is very easy to see the impact of reducing the NA of the objective by closing the condenser.

You will be better off using diatoms as resolution targets.

I don't have any diatom slides but I think that mosquito larva part was adequate for what I was trying to demonstrate. The point was that with only basic equipment (condenser with iris, 4x objective as a reference to set 0.1 condenser NA, 40x dry objective, prepared slide assortment, cheap stage micrometer, photo through eyepiece) anyone can check for themselves and see that the idea to use the lesser of condenser and objective NA when predicting resolution doesn't work well at all if the condenser NA is much smaller than the objective NA.

[Macro_Cosmos]

There seemed to be some confusion on this in the papers and other stuff I looked at, but generally as far as I understood the averaging rule (especially if used as an approximation for real-world specimens) is only supposed to apply if NA_cond <= NA_obj.

Yeah, that is where it falls short as a general rule and it gets confusing as it gets passed on. The following are my personal opinions. I am not against anyone wanting to use this as a general and easy rule. There are plenty of these in photography, such as so called lens compression and equivalent focal length and apertures.

My issue with such a formula:
Suddenly we need to add a bunch of conditions. The technical uses here might know, but a general user will likely not care to find out.

Conditions: RI of the oil, the condenser must be oiled to the slide, the coverslip thickness will have an adverse affect, using 550nm as the assumed wavelength and so on.

What happens to water immersion objectives? The objective mates to the coverslip with water, but the condenser must be oiled to the slide...? Is that going to yield 1.2 NA of "resolution"? The RI is different. Practically, the same applies to air as well.

Generally, we are talking about 1.4 NA, what about 1.5? All of a sudden, temperature matters and so does coverslip thickness.
How about 1.6 and 1.7? Now the type of oil matters and the material of the coverslip matters, on top of the stuff above.

I prefer practical results. My two cents.

[hans]

I agree about all the practical difficulties of getting a really accurate prediction. However I would also point out that theoretical predictions, even if not extremely accurate, help troubleshoot practice. All those conditions and variables you point out are potential mistakes to be made. Theory can be useful to give advice like "if you don't get within x% of whatever theoretical limit you are probably doing something wrong." It it not always easy to just test and see unless someone else who knows how to not make any mistakes has already tested something very similar.

[Phill Brown]

Vintage diatom slides are generally not expensive.
SEM images will help to determine if you are seeing aberrations or resolved structures.

[apochronaut]

Resolution is affected by multiple elements.
An N.A. .90 cone of white light is modified differently when passing through a .90 N.A. objective than when passing through a 1.40 N.A. objective. It also splits at the front lens and does not behave like a monochrome ray does in a simple illustration as is often portarayed in books or illustrations. Although it splits in both front lenses, a .90 objective will perform some subsequent corrections at it's extreme periphery, wheras a 1.40 will correct a .90 cone of light mostly in a more capable zone of each lens. At disperaion the red light assumes a wider angle than the blue light and in a higher N.A. objective the lens systems are designed to correct at an even wider angle than the refracted .90 light. In an N.A. .90 objective and this is obviously dependant on the individual objective's design to some degree, the light entering from the greatest angle will be diverged and converged in zones of the lenses most vulnerable to accompanying aberrations and distortions.
In a 1.40 objective there will be less or no vignetting of marginal wider angle refracted rays and more complete convergence at the extreme periphery of the ray bundle. A 1.40 objective will likely pass more light and be prone to less internal scatter. Essentially, it will utilize the incoming rays more efficiently and impose lower levels of concomitant variables on the system.
This is why a microscope system with combined condenser, objective N.A.s is often referred to as having an N.A. of about such and such, or more or less , or effectively and will perform at a higher level than would be predicted by the condenser N.A.
Interestingly, the math associated with resolution in a microscope objective was formulated to explain the perception of resolution as viewed through instrumentation, not the other way around. In each system commonly referenced, Abbe's, Rayleigh's or Sparrow's : the definition of perceived resolution is different, thus the math used to describe each, different. Rayleigh's criterion evolved from astronomical observations of adjacent stars which are light emitting sources. The same criterion is not used for microscopes. A light emitting source is included : the condenser. At the time Abbe published his paper on microscope resolution , and he published no equations, Zeiss were not using condensers. In fact, it has been reported that customers were complaining that they had to modify Zeiss microscopes by adding condensers from other makers. It is not surprising then, that the condenser was not considered a factor in the resolution of the objective. Sparrow defined resolution more liberally than either Rayleigh or Abbe by defining a not resolved condition , rather than an is resolved condition.

[hans]

Vintage diatom slides are generally not expensive.
SEM images will help to determine if you are seeing aberrations or resolved structures.

Yeah, I was going to buy some a while back but then got busy with other things and stopped watching my eBay searches.

In any case, I think all the suggestions to use diatoms are missing the point. To observing the effect of reducing condenser smaller than objective aperture we don't necessarily need to be starting at the extreme ultimate limits of optical resolution using legendarily difficult to resolve diatoms, achromatic aplanatic condensers, apochromatic objectives, immersion oil, blue or UV light, etc. I think it will be more useful to discuss experiments that are widely accessible, like starting at 0.66 dry then reducing the condenser down to 0.1 and observing the effect as in the two photos I posted earlier.

Quick question regarding diatoms to test resolution, are there species where the spacings are consistent enough across individuals that people are using them for resolution test without measuring? Or are people always measuring the spacing on their individual specimen before doing resolution tests?

[hans]

At the time Abbe published his paper on microscope resolution , and he published no equations, Zeiss were not using condensers. In fact, it has been reported that customers were complaining that they had to modify Zeiss microscopes by adding condensers from other makers. It is not surprising then, that the condenser was not considered a factor in the resolution of the objective.

There is an interesting paper that tries to decipher some of the history:
1981, H. Köhler (different Köhler), On Abbe's Theory of Image Formation in the Microscope

Of the objects whose microscopic images Abbe calculated (see p. 1692), only one example is treated below according to equations (11) and (12).

<section heading> Two narrow slits at a finite distance

There are then two subsections "(a) Coherent light" and "(b) Incoherent light" where Köhler derives these equations (16) and (20) respectively:

kohler-abbe-fourier-eq16.png (21.13 KiB) Viewed 2221 times

kohler-abbe-fourier-eq20.png (11.31 KiB) Viewed 2221 times

Abbe treated equations (16) and (20) for the real case with the then available methods, and arrived, of course, at the same results. Equations (16) and (20) can therefore be interpreted according to the original images in the book by Lummer and Reiche: figure 4 shows the results for coherent illumination. It is obvious that in incoherent light the double-slit structure is resolved with a slit separation of Δ = λ / (2 sin σ). In coherent light the double-slit structure with the same separation is still not resolved. If in oblique coherent light the cosine in the third term assumes the value 0 or -1, the double-slit structure is also resolved at a separation of Δ = λ / (2 sin σ).

Then later concludes:

In Abbe's time the essential difference between image formation in coherent and incoherent light was a completely new piece of knowledge, which represents another important result of Abbe's efforts.

So it sounds like Abbe did understand quite a lot about the basic physics that makes analyzing the case NA_cond < NA_obj difficult. No idea how that lines up historically with Zeiss condenser production.

[apochronaut]

My post really wasn't about Abbe and I don't doubt whether he understood basic physics or not. I don't think Zeiss would have kept him around if he was lax in that department. He undoubtedly guided Zeiss towards incorporating condensers into their better microscopes and of course had the most ubiquitous condenser made, named after him. However , it doesn't appear that condensers played a role in his thinking up to 1883 and he never published a resolution formula. Lummer and Reiche published a formula derived from his writings and lectures, after his death.

My post actually sought to throw some light on why higher N.A. objectives aren't necessarily limited by a condenser acting like a gate valve. Abbe was just part of the fabric of the theoretical thinking in that regard.

[Free2Fish]

Quick question regarding diatoms to test resolution, are there species where the spacings are consistent enough across individuals that people are using them for resolution test without measuring? Or are people always measuring the spacing on their individual specimen before doing resolution tests?

I think they're very consistent on Kemp's 8 form test plate. The Amphipleura pellucida have 37 striae/10 um and the interval between striae is .27 um.

Harry

[hans]

I didn't intend that post to be argumentative or contradictory.

My post actually sought to throw some light on why higher N.A. objectives aren't necessarily limited by a condenser acting like a gate valve. Abbe was just part of the fabric of the theoretical thinking in that regard.

I can't say I follow you reasoning but I agree with the conclusion. I posted those excerpts related to coherent vs. incoherent illumination because I think they one of Abbe's "parts of the fabric" particularly relevant to explaining mathematically the complicated manner in which condenser aperture affects resolution rather than acting as a simple "gate valve".

[StargazerX5]

Hi hans,

Thanks for the discussion. I also have collected papers over the years including the ones you referenced earlier.

The Abbe rule averaging NAobj & NAcond has never seemed to be a useful one. At best it is a first approximation only. Given the range of specimens, stained and unstained, diffraction gratings, diatoms, onion mitosis and others the scope of applicability is so large the rule is hardly more than a historical curiosity. The graphs you posted above certainly demonstrate that.

In book 2 (ref 1) of Zieler writes on pg 55:

"There is still another reason for the improvement of the image quality under conditions of multidirectional illumination with reduced NA. The best approach to an explanation of this is by referring again to the optical conditions of truly "critical" illumination. A reduction of the NA of the illumination causes an increase in the diameters of the diffraction discs which are the images of single points of the light source. Throughout a relatively large circular area of a diffraction disc coherence prevails. When the diameter of that area is greater than the limit of the resolving power of the objective, light waves from adjacent object points, separated by a distance only slightly larger than the limit of its resolving power, can still interfere with each other, resulting in greater contrast in the image. This increased contrast, however, is limited to object detail very close to the limit of the resolving power, whereas the general character of the image is still that of optical fidelity of reproduction. The diffracted light waves proceeding from these object points can still pass through the entire NA range of the objective, including the peripheral zones through which the light illuminating the object cannot pass. Therefore, the resolving power of the objective is still higher than that which would prevail if the NA of the objective had been reduced to match the reduced NA of the illumination."

Note that when Zieler refers to "critical" he is referring to critically focused - not Nelsonion illumination. I included this excerpt as I noticed you had made some coherence arguments. Zieler books are not as mathematical as L. C. Martins magnum opus (and his papers) you referred to above.

I don't know if you have run across Abbe's discussion on diatoms or not but he did examine them and reported measurements of Pleurosigma angulatum, Surirella gemma and Frustulia rhomboides. But he also makes clear that a full description of objective performance must include other specimens.

Although I usually employ ach-apl condensers in my work, its quite amazing that one can achieve high resolution results with the Abbe condenser at least visually. A good reference paper to a short history of condensers can be found in reference 2 below. If you do not have access then pm me and I will forward it to you. I also note that with the ubiquitous Abbe one can also elucidate details formerly hidden with small amounts of condenser defocus! Simulating that would be difficult to say the least.

(1) "The Optical Performance of the Light Microscope 1972 Vol2
H. Wolfgang Zieler

(2) "Thoughts on the substage condenser"
Quekette Journal of Microscopy, 1999, 38, pg 305-309
Gilbert Hartley

Glenn

[hans]

I had not seen either of those references, thanks Glenn. Zieler's book seems quite obscure but it is on archive.org. Not freely available but with an account it can be "borrowed" in 1 hour increments for online reading.

Regarding critical vs. Nelsonian illumination, I have not read much about the history but had in mind that the terms were basically interchangeable. I do see what you mean that Zieler appears to be making a distinction between the two but it is not clear to me from reading quickly the rest of chapter 4 exactly what distinction he is making. Do you follow it better?

It is good you mentioned LC Martin, although I think some mix up because I did not cite him earlier, just Hopkins and Barham. I had apparently found his book before (already had it marked on archive.org where it can also be borrowed like Zieler) but forgotten about it and had not thought to check if there was anything on effects of illumination NA in there. Martin has a short summary of the Hopkins/Barham result but also references an experimental result and reproduces their graph with the experimental result included:

Experimental trials described by Arnulf, Dupuy, and Flamant (5) support the theoretical conclusions reached by Hopkins and Barham on the grounds of coherence theory, though the visual criterion of resolution for point images was generally found to involve K-factors somewhat lower than those of the theoretical curve as seen in the figure.

I could not find the paper but the exact citation Martin gives is:
Arnulf, Dupuy, Flamant (1953) Revue d'Optique 32, 529.

Finally, regarding the defocused Abbe condenser, I don't normally use an Abbe condenser either but remember reading somewhere how the large spherical aberration prevents filling the entire NA range of high-magnification objectives at any single focus setting. Perhaps the specific defocus setting you are referring to results in only the higher-NA periphery of the objective aperture being filled similar to circular oblique lighting which seems to be popular for diatoms?

[apochronaut]

Finally, regarding the defocused Abbe condenser, I don't normally use an Abbe condenser either but remember reading somewhere how the large spherical aberration prevents filling the entire NA range of high-magnification objectives at any single focus setting. Perhaps the specific defocus setting you are referring to results in only the higher-NA periphery of the objective aperture being filled similar to circular oblique lighting which seems to be popular for diatoms?

That is why when you focus the condenser for Köhler illumination you focus using the maximum possible aperture rather than closed down as many on line sources suggest.
COL is popular for observing diatoms and has been used as a method for achieving resolution of diatom structures where it otherwise might be difficult but it isn't necessarily a method used for TESTING resolution because it relies on contrast as a highlighting tool and if the test involves comparison of objectives, it is difficult to standardize. I know it has been used, though. Any tests I have done were done using a subject with fairly regular intervals of known parameters and lighting as accurately repeatable as possible. Harry mentions the Amphipleura Pellucida on Kemp's test plate which is good for high N.A. testing but even with it, details at quite low N.A.s show up under phase, DF and DIC , which are not visible under BF, so variables of contrast are playing a role in some proportion to the inherent N.A. of the system. I suppose that if one chose to test or compare objectives or condensers under only one of those contrast regimes, that would improve validity but in my mind I think for the purest testing of the resolution capability of a condenser/objective combination, it should be done in BF at full condenser aperture on the same microscope stand using the same illumination adjustment.

[StargazerX5]

Hans,

In the first post of this thread you wrote:
"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, ..."

You did not give an explicit reference to Martins text but it was clear where you got it from in your quotation.

I'm busy today in the backyard, but if I find the time I may direct you to references of the unfortunate confusion between critical focus and critical illumination. I can say that in the presence of any spherical aberration (SA) the best focus is not the gaussian point but combined with a little defocus a better quality image results (before or after the gaussian point depending on the sign of the SA).

Glenn

[hans]

Oh yeah, Goldstein was referencing Martin's book there, I didn't make that connection earlier. (I haven't tried to exhaustively follow references backward or forward.)

I believe I read somewhere a rule of thumb that viewers typically judge subjective best image quality to be around either 1/3 or 2/3 (don't remember which) of the way from paraxial focus to circle of least confusion. I would be interested to read more about critical focus vs. critical illumination if you get a change to dig up those references.

Apo, I mostly agree, the analyses discussed so far are for amplitude objects, not phase objects that require some additional optical contrast method. (Except maybe arguments like Abbe's based on diffraction orders from gratings can apply to phase gratings as well?) However I think it would be confusing to try to lump too much together into a single "inherent NA" number. In the case of phase and DIC the physics of image formation are fundamentally different enough that they are given completely separate treatments as far as I've seen. For example that book on phase contrast by Alva Bennett and others from American Optical has a fairly long, mathematical analysis of image formation and resolution specific to phase contrast and separate from the bright field case. (I have not tried to understand it at all.) And I agree also, "purest" test of analyses like Hopkins and Barham would be in bright field using a pure amplitude object that has minimal phase effects, like maybe a very thin stained tissue section if a real world sample was desired as opposed to some sort of synthetic resolution test target. It's too bad the Arnulf, Dupuy, Flamant paper doesn't seem to be available, would be interesting to see what they actually tested.

[MichaelG.]

This is way too heavy for me, but you clever chaps might find it of interest:
https://opg.optica.org/oe/fulltext.cfm ... 9-27-26249

PDF is freely downloadable

MichaelG.

[hans]

Looks interesting Michael but above my head as well.

I came across some relevant comments from A.E. Conrady in 1923. First on the thought to use the lesser of the NAs by analogy with Airy's analysis of the telescope and how that doesn't work:

When applied to the microscope, Airy's theory of the spurious disc explains some of the observed phenomena fairly satisfactorily. As the smallness of the image depends on the effective aperture, i.e. that part of the lenses which is really filled with light, there should be a maximum of resolving power when the object-glass is completely filled with light and a reduced resolving power if only a part of the full aperture is utilised. This is qualitatively in accordance with experience, but not quantitatively, for it is found that a microscope objective retains one-half of its maximum resolving power if only a very small axial illuminating pencil is employed; by the theory the resolving power should be reduced to a very small fraction of its maximum value. Light scattered by the structure of the object and thus slightly utilising the space not filled with direct light was naturally adduced in explanation; but it is impossible thus to account for the fact that -- with all kinds of objects -- the resolving power is just half that obtainable from the lens when completely and uniformly filled with light.

Then later on the difficulty of doing analysis more realistic than Abbe's:

But if the light-transmitting aperture attains a size of the order of a wave-length or if there are a number of apertures within small distances of each other, then there will be interference effects between the light from different points of the aperture or apertures, resulting in the producing of some type of diffraction spectra. In the vast majority of cases the result is hopelessly complicated and, moreover, inaccessible to theoretical discussion, inasmuch as the latter would require the minute structure of the object to be known with absolute certainty.

From article "Microscope, Optics of the" which he wrote for:
1923, Glazebrook, A Dictionary of Applied Physics Vol. IV Light--Sound--Radiology

[Macro_Cosmos]

Diatoms are adequate, but not quantitative. It fails as a decent reference and vintage slides were prepared with vintage techniques, making them somewhat unsuitable. As a general toll, sure, it works well enough. I love diatoms, I have invested lots of time and money in them but the idea of test diatoms is really a past thing. Modern methods involve soulless charts and nanorulers. They are boring but reliable and reproduceable. The art is gone.

The reason for this is that results are derived from averages among typical diatom species. There are differences in striae distances even among diatoms within the same sample. You need a SEM of that exact diatom to validate the results. Such slides do exist, they are expensive.

Vintage slides typically have the frustules manipulated on the slide itself rather than the coverslip. This will induce spherical aberration and impede results. Modern slides use high RI mountants, >1.74 typically. The results you get are a decent reference of resolving power. Most samples are mounted in neutral balsam or simply water, where the RI is ~1.51 and 1.33 respectively.

[StargazerX5]

Conrady was one of the greats in optical design (lots of geometry and algebra) but it would not be until the French physicist Duffieux put fourier optics on a sound foundation that image formation with its complexities could be analyzed. Even then its surprising how long it took for his discoveries to be widely known.

On diatoms etc. Abbe wrote (ref 1) on pg 802

"Objects which show a regular striation, and in general all regular periodic
structures, are particularly insensible to the residuary defects of the Objectives, because they produce only a limited number of isolated diffraction-beams, and thus leave the greatest part of the objective's aperture entirely unemployed. In observing an object with only one set of parallel lines which are near the limit of the resolving power of the objective, only two small portions of the aperture are simultaneously utilized, one by the direct beam, the other at the opposite edge of the opening by the diffracted beam, as may be ascertained by a glance at the objective’s clear aperture. All defects and aberrations of the system which inhere in the inactive portions, do not exist for the image in that case, whilst they will at once become effective when those objects of a very complicated and irregular structure, which produce a continuous and widely spread out diffraction-pencil, are observed. This consideration will show that the ordinary test-objects of the Microscope-particularly lined objects, and in a somewhat less degree all kinds of diatom markings-are the most unsuitable preparations for a proper judgment of the performance of the instrument in regard to the general conditions of scientific work inasmuch as the latter are always much less favorable than those of diatom observations."

When Abbe writes of "residuary defects" he is referring to aberrations.
The practicing microscopist should understand when examining diatoms they are not 'real' but useful for comparing objectives and illumination techniques.

Zernike (ref 2) had an interesting result on the role of condensers and
'resolution' where he writes on pg 794-5:

"The only practical use of an illuminating lens therefore lies in its condensing property, that is, a much smaller source will be sufficient with the lens. Especially in the case of microscope condensers, it has often be presumed that a high degree Of correction must be of advantage for the resolving power. Our result proves on the contrary that this has no influence. In ordinary microscopic observations, where the necessary brightness is easily attained, the condenser may even as well be replaced by an illuminated white surface, subtending the necessary angle as seen from the object. The microscopic observation of transparent objects is thus influenced by the illuminating aperture as well as by the objective aperture. The diffraction image corresponding to the first aperture determines the range of the coherence in the object, while that of the second aperture determines the overlapping of the object-points in the image. For equal illuminating and observing apertures the result will be that two points just separable by the objective are incoherently illuminated. The transparent object will then appear practically in the same way as a self-luminous one."

Glenn

(1)
Abbe, Ernst (1883)
"The Relation of Aperture and Power in the Microscope (continued)"
Journal of the Royal Microscopical Society. 2. 3 (1). London, UK: Williams & Norgate: 790–812

(2)
Zernike, Frits (1938)
"The concept of degree of coherence and its application to optical problems"
Physica Volume 5, Issue 8, Pages 785-795

[Chas]

I looked at the pinholes in a rather decayed sheet of generic mylar mirror-film, with a metallic objective last night and the sharpest/finest images came from using the (achro) condenser iris shut right down... I thought this was down to a 'run of the mill' 40x 0.65 Vickers objective.
But this morning I came across a paper by Siedentopf who was looking at mercury droplets ( the inverse of the pinholes I was looking at, I think) and likely not carried out using poor objectives :

Sied second.jpg (127.67 KiB) Viewed 1283 times

The table:

Siedentopf March 20 1929 Table crop.jpg (40.7 KiB) Viewed 1244 times

From Journal of the Royal Microscopical Society 1929:

Reference .jpg (13.03 KiB) Viewed 1266 times

... in this volume, also, Berek comments on the higher resolution that can be obtained with annular brightfield illumination.