Averaging condenser and objective numerical aperture when predicting resolution

[hans]

Continued from Do I need another objective? #53

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

The 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?

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:

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.

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.

Note that Abbe's and Goldstein's analyses of image resolution involve both:

1. An object being imaged which redirects some light outside the original, undisturbed double cone of illumination. (By diffraction in this case.)
2. 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.

That is, I still think that the net resolution should always be limited by the lesser of the two NA's, at least assuming there's no subject to confound things.

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.

I did take a look at the Hopkins and Barham paper noted above, and there's no way I'd have the patience to work through all that math!

Baker in 1952 anticipated your complaint:

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.

barker-table-1.png (100.92 KiB) Viewed 2708 times

In addition, Hopkins and Barham themselves provided a graph summarizing all the math:

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.

hopkins-barham-figure-2.png (33.42 KiB) Viewed 2708 times

[PeteM]

Thanks for taking the time to dig up those sources, Hans.

We both likely suspect the answer gets even more interesting if the subject isn't a grating, but something like a stained tissue.

[apochronaut]

Using math as the final arbiter to describe complex systems is only as good as the number of variables that can be included in the calculations. Those must be envisioned in their totality in order to make sense of the math but in a complex system such as is being assessed here , how can that be done? I would argue that the participants in this discussion, myself included, are far from understanding the number of variables that are required in order to accurately assess this question adequately in terms of mathematics. You can get a rough estimation but any accurate understanding will be elusive. The onus here is being put on the condenser, without due regard to elements of light transmission , condenser function and objective acceptance, that are outside the calculation of N.A., yet affect N.A. Such factors that affect light transmission as focus, reflection ,diffraction and scatter in the real system of the condenser/slide complex as well as not easily quantifiable aberrations and distortions in the condenser lens systems all play a role in exerting an influence on the final N.A. and resolution of the objective.
Since perception too is heavily involved in any analysis of a microscope system and in fact one's perception is the ultimate end assessment tool of the system, it makes as much sense to rely on empirical and measured information as it does math, if not more, since perceptually one is forced to include unknown or unmeasured variables in the analysis. They are still there, even if the math doesn't include them. All one has to do is arrange a straightforward, precisely managed DF imaging set up using as fine an iris equipped objective as is available, then compare that to BF at full aperture with the same sample and it becomes immediately obvious that where the condenser N.A. exceeds that of the objective, or the exact opposite of what is being rummaged through in these two threads, some version of homogenization is taking place. .90 in Df yields considerably more information than it does in BF.

What ever math evolves to be used, is fixed and undeniable as a language. Perception not so.This is the main argument against using perception as an assessment tool, yet we use perception or interpretation all the time in almost every scientific experiment. No doubt, different people will perceive the same situation differently but to aid in gaining some uniformity in perception we can employ some simple measuring tools. For the assessment of resolution , micrometers and subjects with known precise graduations help immeasurably.
So, I go back to my earlier statement in the prefacing thread that the theory behind whether a condenser works like a valve or not can be tested empirically by placing another tool, a variable valve in the objective : an iris diaphragm. It is very easy utilizing this simple tool, to see that a condensing objective system works more like a variable control valve in the system, not as a shutoff valve.

[Free2Fish]

Despite my failure to do so after a quick look it seems to me a clever group could design an experiment to solve or at least better understand our dilemma.
I think the math for condenser NA and separately, objective NA is well estabished and understood.
Where all the ambiguity comes in is when the two are considered a system in which Condenser NA is lower than objective NA.
As I understand it, as long as condenser NA is > objective NA we all agree there is enough collimated light to get the designed NA out of the objective.
Is there a way to quantify the NA, or any other measurement that might approximate or emulate it once the light has passed through the objective? Or am I entirely out to lunch?

Harry

[apochronaut]

1) The math is only well established for the variables of the system that are included in the math.If one thinks that they are fully cognizant of all the variables and that they have a formula to explain them all, then they are either out to lunch or they aren't spending enough time looking in a microscope and too much time doing math.
2) In DF , where the condenser NA must exceed the objective N.A. by some if not a considerable margin just for the mechanism to work at all, the resolution exceeds expectations. Some DF systems can resolve extremely fine detail, finer than is possible in BF , at much higher N.A.s.

[Free2Fish]

I’m not sure what your point is since I agree with both. Nor do I want to get into arguments about the theory or mathematics behind microscopy optics, you clearly know more about both than I do.
I’m merely exploring whether there is a practical way to determine the effect of using a low NA dry condenser with a high NA oil objective.
Perhaps examining areolae or striae on one of the diatoms with varying objectives and condensers?

Harry

[apochronaut]

I am all for practical tests and practical solutions.
So, missing from the discussion are some of those variables that don't get included in the math, such as how well corrected the condenser is, the diameter of the lenses, image circle described etc. In order to lessen the effects of those variables , one has to choose the condensers they are comparing carefully. While you can use any condenser in any microscope both the front and rear focal lengths of condensers vary as well, based on the condenser front lens to field iris distance. In order to do a really valid comparison between a .90 and oiled condenser and form a solid empirical decision on performance potential , the condensers should both be factory original to the stand. However, given the scarcity of some high N.A. condensers , if you were to find one that you can make work, it is probably better than a kick in the pants until such time as the one you want shows up, so comparing a 1.40 achromat aplanat from anywhere to your Nikon .90 achromat aplanat will be just as valid, if that is the 1.40 achromat you are using.
For years I did not have a factory 1.40 achromat aplanat and used a 1.40 achromat aplanat from a 1930's research stand alternately with a .90 achromat aplanat. I then found some N.O.S. Reichert barrels which contained the 1.40 top( back) lens and built a couple of condensers in a couple of those shells using surplus lenses from Silo. One has an apochromatic front lens and seems comfy with water or glycerin immersion, the other seems to want dry, so neither can actually utilize that 1.40 back lens but they both seem somewhat better than the .90 achromat aplanat, while a little less capable than the factory oil 1.40 that I eventually obtained but I learned a lot making them, especially the water/glycerin one.
Testing wise, diatoms are probably the most useful from a practical standpoint because they are cheap. High precision high resolution test targets are excrutiatingly expensive.
You may be able to find an 8 form test plate , formerly made by Klaus Kemp, that someone is selling or one from the Diatom Lab. They both have a range of diatoms with features that challenge objectives in ascending resolution levels.
Over time certain slides you own, that contain various markers that you will become familiar with , can be used quickly as measuring sticks simply due to familiarity and the assessment almost intuitive.

[Free2Fish]

I have a couple of Zeiss apl universal condensers, a .63 NA version and a 1.4 NA version. Unfortunately my high NA objectives are phase objectives, further muddying the waters. I tried a quick test yesterday and the results weren't noteworthy.
I also have Kemp's 8 form test plate should I be able to do something worthwhile.
My problem now is that the weather is getting nicer up here and I'll soon be engaged in outdoor activities, relegating microscopy to a secondary role till next fall/winter.

Harry

[apochronaut]

Phase objectives tend to sacrifice resolution for contrast enhancement, so they aren't the best choice for assessing condenser performance.

[Chas]

Is the table in here of any help? :
https://archive.org/details/microscope0 ... 8/mode/2up
The long and short of it seems to be that for every 0.1 of NA you resolve ~10,000 lines per inch using 'white light' (Fraunhofer Line E is actualy green)

It would be nice to know how the illuminated fraction of the back of the objective relates to NA ..is it a linear thing?

[hans]

I'm thinking that a lot of the confusion around this is coming from too closely equating the concepts of resolution and NA and using the terms somewhat interchangeably. NA is an arbitrarily-defined quantity which useful because when you express other quantities in terms of NA the formulas/equations are often simpler in form. The simplification of form is particularly dramatic in cases involving resolution. (I say "simplification of form" because I don't want to imply that anything is fundamentally simpler, just easier to write down.) We could carry out all this discussion of resolution without ever mentioning NA, we would just be saying "sine" and "refractive index" a lot more.

Harry's original question, for which I created this new thread, was explicitly theoretical:

I’ve wondered about the formula that suggests that when condenser NA is lower than Objective NA the result is an average of the two. Intuition suggests that NA should revert to the lower of the two.
Does anyone know if there is a mathematical basis for the averaged formula?
I suspect that averaging formula is an educated guess and that system NA will be difficult to predict. Light is very unruly and reflection, refraction and diffusion are all in play between the condenser and objective. Isn’t that what makes DF possible?

In my opinion differences between dry and oiled condensers that deliver the same NA to the specimen plane are not relevant to the basic, theoretical question. Obviously there is a practical importance, but it would probably be better to have that discussion in a separate thread to avoid confusion.

Also, I had missed Harry's point about darkfield. It is a good point, along with Pete's line of questioning, which zondar should consider.

<several posts>

I don't think most of these points you are making contradict anything above, nor do I see them adding much useful explanation in the direction of Harry's original question. It does sound like you personal experience is consistent with the case (two pinholes) analyzed by Hopkins and Barham which shows resolution remaining relatively good (better than the "Abbe rule" averaging predicts) especially in the extreme limit as condenser aperture is restricted toward zero.

It would be useful to hear whether the following numbers sound reasonable based on practical experience of yourself or anyone else who wants to comment:

• 1.4 NA objective with 0.9 NA illumination gives 13% worse resolution.
• 1.4 NA objective with 0.1 NA illumination (about as small as most condenser diaphragms go, I think, to match typical 4x 0.1 NA objectives) gives 35% worse resolution.

(These numbers are further summarized by me from Baker's table.) They seem reasonable to me based on casual observation -- closing down the condenser iris excessively gives a lot of "contrasty" ringing/halo artifacts and dimmer image but doesn't actually impact resolution very much.

And just to be clear, apo, differences between theory and practice have not been disputed by anyone here and are also acknowledged in the papers, for example:

Baker:

It must be remarked that ordinary microscopical images are affected by a factor separate from those concerned in the theoretical basis of the table. If any object is of markedly different refractive index from the surrounding mounting-medium, it will tend (as is well known) to appear with a Becke line round it, and this will necessarily interfere with the perfection of the image. The effect will be increased if a narrow cone of light is used for illumination, whether that cone is central (that is, central light of low numerical aperture) or not (oblique light of high numerical aperture). Whenever one opens the iris diaphragm, this undesirable Becke effect will be reduced. There are theoretical reasons, however, for supposing that it is usually undesirable to use an illuminating cone of quite so high a numerical aperture as the objective, even if the extreme marginal zone of the latter be assumed to be perfectly corrected for spherical aberration (see Oettle, 1950). This accords with the experience of most practical microscopists. Oettle's explanation is that when direct light enters the marginal zone of an objective, only half of the diffracted rays are concerned in image-formation, the remainder being too oblique to enter the lens-system.

Goldstein:

The apparent inter-object distance may be greater than its value according to geometrical optics (cf. Fig. 2a, d), and a pair of points a certain distance apart may be ‘resolved’ whilst other pairs which are further apart may not be. Interpretation of the images may therefore be difficult or impossible without prior or additional knowledge, and theoretical ‘resolution’ may be of little practical use. An improvement in resolution could in theory also be obtained in bright-field microscopy with axial coherent illumination, if one object element retards more than another.

[apochronaut]

Could you point me to Harry's point about DF, I missed that too.

[wabutter]

I still have not figured out how to post images into these threads, b/c if have some that will graphically show what I discuss below. It seems there is some room to simplify the discussion. Although the subject question of Averaging the condenser and objective NA to predict resolution is being queried, It seem the question to ask is what impact does changing the condenser aperture have on the resolution of the image.

It is generally recognized by all of the major manufactures that maximum resolution is calculated ( 0.61 *lambda/NA). Lambda being the wavelength of light. The shorter the wavelength the greater the resolution. It is also recognized that the NA of the condenser must be equal to or great than that of the objecitve. As NA is defined as the mathamatical calculation of the cone of light collected by the condenser and receive by the objective. As mentioned by Apochronaut in an early post, any discussion of resolution can not move forward with considering variables, of which the most important is the sample. When we look at microscope systems, we must also recognize that we are discussiong a diffraction limited system. What is most important is what ultimately happens in real world.

When the condenser diaphragm is reduced so it is smaller than the back aperture of the objective, the resolution is decreased and the contrast is increased. Hence there is conflict in a microscope system between resolution and contrast. You can have maximum resolution by matching the back aperture of the ojbective to the condenser aperture, but there might not be enough contrast to resolve things. That is why specimens are stained. it helps create contrast. Keep in mind the condenser aperture is conjugate to ojective back aperture, so you can see it and adjust it by removeing an eyepeice, looking down the eyepeice tube.

Becasue the NA is exspressed as NA = n*sin u where n is the refractive inces of the medium tha u is in and u = the 1/2 the angular angle of the cone of light presented ot the condenser.

Lets also keep in mind, that all of the criteria for Koehler illumination must be met in order to acheive real or even theoretical maximum resolution in a microscope system. So in practical terms what does this mean. If you were to have a 40x objective with the following properties., you could experience the following resulting resolution. @ 546nm of light, This is used in the calculations because it is in the middle of the visible spectrum. The values below represent if the condenser aperture matches the objective NA

40x 0.65 resolution 0.510 micrometers, Theoretical resolution 0.418 micrometers. by reducing the condenser aperture to 7/8th the back aperture the result is resolution of 0.556 micrometers
40X 0.75 resolutino 0.442 micrometers Theoretical resolution 0.362 micrometers by reducing the condenser aperture to 7/8th the back aperture the result is resolution of 0.480 micrometers
40x 0.85 resolution 0.393 micrometers Theorectical resolutino 0.322 micrometers. by reducing the condenser aperture to 7/8th the back aperture the result is resolution of 0.442 micrometers
40x 1.0 oil resolution 0.331 micrometer Therectical reaolution 0.272 micromters. by reducing the condenser aperture to 7/8th the back aperture the result is resolution of 0.357 micrometers

it is generally an accepted practice to set the condenser to 7/8 of the back aperture to acheive the best contrast an resolution for most stained samples. A simpe way to set the aperture diaphragm is to open it wide open, as you observe your sample slowly clopse the aperture until you see contrast begin. Keep in mind you will need to do this for each objective a the NA is different.

Specimen preperation, mounting material thickness, stain density, and focus of the condenser all impact the resolution of the sample that can be legitimately be obsreved in the microscope .

BTW, when using a incident light microscope system, the NA of the objective automatically matches the NA of the condenser. They are infact the same thing.

[MichaelG.]

I still have not figured out how to post images into these threads …

.
It would be very helpful if you could ^^^

The process is relatively simple, but only permits the use of small file sizes

At the bottom of the composition-pane, there is a tab labelled Attachments via which you can insert images “inline” at the current location of the cursor.

Do please give it a try

MichaelG.

[Free2Fish]

It would be useful to hear whether the following numbers sound reasonable based on practical experience of yourself or anyone else who wants to comment:

• 1.4 NA objective with 0.9 NA illumination gives 13% worse resolution.
• 1.4 NA objective with 0.1 NA illumination (about as small as most condenser diaphragms go, I think, to match typical 4x 0.1 NA objectives) gives 35% worse resolution.

Hi Hans, thanks for the effort involved in constructing this thread. Your numbers loosely jive with apo’s assertion that there is a drop of 1 nm of resolution per drop of .01 NA. If my arithmetic is correct that works out to about 21% vs your 13%. Certainly in the ball park.

I suspect different people will take different things from this thread. I’m comfortable with not understanding the relationship between low NA condensers and high NA objectives and believe it is much more complicated than a simple formula can convey. Perhaps like trying to predict the weather.

And we haven’t even included quantum effects, knowing the photon is a quantum particle.

Harry

[hans]

Yeah, the situation with partially coherent illumination is just inherently pretty difficult to analyze, it seems. I would guess two reasons for the averaging rule persisting despite wide acknowledgement that the physical basis for it (oblique coherent illumination of a grating object) gives only a "somewhat usable but not good" approximation to real-world illumination and specimens:

1. The difficulty of predicting resolution when NA_cond < NA_obj means there just aren't any other simple, physics-based formulas available.
2. Using condenser NA much smaller than objective NA is avoided in practice. Wayne mentioned a 7/8 rule of thumb and I have also seen 2/3 suggested but never smaller. Using dry 0.9 condenser with 1.4 objective is also about 2/3. In that range the averaging rule is not too bad and even though it predicts worse resolution than is actually possible you would need to be doing everything else perfectly to realize the extra resolution and notice the discrepancy. Basically what Hopkins and Barham point out in the excerpts I quoted earlier: "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." (s it the the ratio NA_cons/NA_obj.)

Maybe excessive beating of a dead horse by this point, or mainly for my own benefit, but anyways I had some more thoughts regarding:

I’ve wondered about the formula that suggests that when condenser NA is lower than Objective NA the result is an average of the two. Intuition suggests that NA should revert to the lower of the two.

In other words, should we be using min(NA_obj, NA_cond) rather than (NA_obj + NA_cond) / 2 to predict resolution? I had the same feeling and intuition at first and I think this is where it comes from: If we are imagining double cones of light propagating undisturbed from condenser to objective then min(NA_obj, NA_cond) is the largest NA that can pass both apertures. Visualized in that context of an undisturbed double cone, the quantity (NA_obj + NA_cond) / 2 doesn't make much sense because physically it would partially blocked by the smaller aperture. But predicting resolution is an entirely different and more complex problem than just predicting the size of double cone that can pass the system undisturbed and (NA_obj + NA_cond) / 2 is only ever used (as far as I know) in the context of predicting resolution. It is never implied that a physical cone of light with NA = (NA_obj + NA_cond) / 2 actually exists anywhere in the system. Maybe that's all too specific to my own internal stages of confusion to be helpful, but I suspect it's why min(NA_obj, NA_cond) seems to have so much intuitive appeal.

So there is some intuitive appeal but 0.61 * λ / min(NA_obj, NA_cond) predicts even worse resolution than the usual 1.22 * λ / (NA_obj + NA_cond) (which is already underestimating) especially when condenser NA is much smaller then objective NA. The disagreement is large enough that it seems like it should be easy to observe on just about any microscope without using anything exotic, for example:

40X 0.66 with condenser also set to 0.66 showing part of a mosquito larva with ridges spaced at roughly 1 to 1.5 um or about 2-3 times the theoretical limit, photos are 75 um across with 75 nm pixel pitch on subject, poor image quality with lots of spherical aberration haze because it's a low-quality prepared slide with thick, grainy mountant.

obj0.66-cond0.66.jpg (75.68 KiB) Viewed 2398 times

Same except condenser reduced to 0.1, difficult (for me at least) to subjective judge loss of resolution due to the very different appearance of the image, but pretty clearly not 6.6 times worse as would be predicted by using min(NA_obj, NA_cond) instead of the average. Using the lower of the NAs (condenser 0.1) to predict resolution would mean the spacing of the ridges should now be 2-3x smaller than the theoretical resolution limit and not resolved at all.

ojb0.66-cond0.10.jpg (150.37 KiB) Viewed 2398 times

[zondar]

Figure's 58 and 59 in the volume cited above illustrates why a glass-air interface strictly limits the NA to a theoretical maximum of 1, and how oil (with approximately the same index of refraction as glass) allows up to about NA=1.5:

https://archive.org/details/microscope0 ... 0/mode/2up

So why is a large aperture (high NA) even desirable in the first place? Why is that better than a stopped-down aperture (low NA), say just with more light to compensate? The relationship between NA and resolution is related to diffraction, e.g. the ratio of the periphery (which changes linearly with radius) to area of a diaphragm (which changes with the square of the radius) gets larger (worse) as the radius is stopped down. The text also tries to explain here in fig. 49:

https://archive.org/details/microscope0 ... 8/mode/2up

Well, except that the explanation gets kind of nutty in places ("rays of light travel in direct straight lines only when there is a sufficient body of them traveling together", etc.), but ok, it's a quite old volume.

Anyway, at a glass-air interface such as the above example, all light attempting to exit the lens outside the 82 degree included angle is lost to the objective's perspective, reflected back into the condenser. Provided that there is nothing else of note between the CD and OBJ, then the net NA must strictly be the lesser of NAcd and NAobj.

But then we do have to consider the subject being imaged, too. If it's something like the diffraction grating mentioned in an earlier post, which would be a very high contrast target indeed, then diffraction in the subject can and likely will spread the light out enough to fill a high-NA objective's acceptance angle.

I think the answer is simple: The "average of the two" formula is not exact, and in reality is nothing more than a rule of thumb for an average setup with an average target. You would get the lesser of the two if you image nothing at all, and up to the objective's NA if imaging an especially contrasty and active target like a diffraction grating of the appropriate spacing, or a luminous target as mentioned elsewhere, and somewhere in the middle (the average) with an "average" target. Is there still more to it than this?

[MichaelG.]

… Is there still more to it than this?

I think the “more to it” is probably Modulation Transfer Function
.
I am feeling too old and tired at the moment to get involved in this discussion but, if my memory serves, this is a big step in the right direction when trying to understand the difference in performance of various optics.

Back in the early days of [camera] ‘Brand Loyalty’ it became understood that Leica lenses had excellent resolution, but somehow looked a little soft … whereas the Zeiss competitors had perhaps a little less resolution but ‘looked sharper’

Only when MTF measurements were done on comparable lenses did the explanation start to emerge.

If anyone wants a Rabbit-Hole to explore … that might be it.

MichaelG.
.

Ref. https://www.edmundoptics.com/knowledge- ... tf-curves/

[Free2Fish]

Old and tired....me too! I spent a couple of days last week reading up on MTF and decided I'd rather be focusing my microscope on a water droplet.
If anyone wants more info than he/she needs on MTF a book that covers the subject in terms of microscopy is "Better Microscopy, Vol III" by D.J. Jackson.

Harry

[wabutter]

These were the images I wanted to share in my earlier post. I think they can help visualize the objective NA/Condenser NA relationship. Unfortunately it does not show what happens to working distance as the NA increases. But you can see the change in tne Angular Aperture Angle increasing as the NA increases. This is generated from and App on my phone called resolution. It is highly interactive and provides a slider for the NA on the objective as well as the Condenser. Wavelength can also be selected along with different immersion medea. Water, oil, Glycerin, Silicon.
Sorry I don't recall who made the comment above, but I think rather than averaging the NA, NA objective +NA Condenser/2 the better outcome is realized by taking the lowest NA and using that as the base line for computation. A 1.40 objective with a 0.95 condenser would never realize a resolution of a 1.17 average, the best you could hope for would be a .95 as that would be the maximum cone of light sent the objective.

Image 1 1.0oil.png (150.69 KiB) Viewed 2378 times

Image-1 0.75.png (156.54 KiB) Viewed 2378 times

Image-1 0.85.png (157.64 KiB) Viewed 2378 times

Image-1.png (154.99 KiB) Viewed 2378 times

[hans]

Sorry I don't recall who made the comment above, but I think rather than averaging the NA, NA objective +NA Condenser/2 the better outcome is realized by taking the lowest NA and using that as the base line for computation. A 1.40 objective with a 0.95 condenser would never realize a resolution of a 1.17 average, the best you could hope for would be a .95 as that would be the maximum cone of light sent the objective.

Ok now it really seems like everyone (myself included) has this intuition at first glance. In the previous post I gave some rambling speculation about where, for me personally at least, I think the intuitive idea that we should use the lower NA to predict resolution comes from. (And to point out again, the lowest NA is indeed what governs the NA of light passing undisturbed from condenser through objective with no specimen present.) But this whole thread has been about discussing how that intuition to use min(NA-obj, NA_cond) to predict resolution seems to disagrees with more careful thinking about the physical situation including analyses in published papers as well as with peoples empirical experience and observations.

[zondar]

… Is there still more to it than this?

I think the “more to it” is probably Modulation Transfer Function

MichaelG.

MTF is a way to characterize lens performance as seen in the image plane vs. distance off-axis from the center. I.e., the corners of a photo are generally less sharp and more distorted than the center. Stopping down usually improves this non-uniformity, and MTF plots (e.g. for camera lenses) usually include two or three curves plotting performance at different f-stops because of this. The problem is that stopping down ultimately reduces resolution in the center, too, where you likely care about it the most in a microscope.

Side note: Camera lenses are often softer when wide-open, except for some very expensive lenses that are highly optimized for wide-open performance, and so most benefit from stopping down to a mild degree. You will see this in MTF curves that improve even in the center when stopped down from f/1.4 to say f/4. But microscope objectives, with very few if any exceptions, are always designed for the highest possible performance at only one aperture - wide open at full NA.

Anyway, I don't see it as obvious that "MTF" has much to do with the more fundamental question being discussed here, i.e. how a limited condenser's NA also limits the objective's performance. Sure, the corners might have less distortion when effectively stopped down, but the resolution in the center is definitely going to be worse! But I'm sure rabbits can be chased in this direction too.

I'm sticking with "it ain't no more than a rule-of-thumb for average setups with average subjects."

[Chas]

(As Hans' two photographs show) as Conrad Beck says:

..a small almost transparent body may be rendered invisible
by illuminating it with a full solid cone of light from the
condenser, and it may be necessary to use a small angle of light
to render it visible, an imperfect image being better than none at
all.

Written before phase-contrast et al. obviously

[hans]

..,and it may be necessary to use a small angle of light
to render it visible, an imperfect image being better than none at
all.

And most importantly, doing so (even with very small condenser NA) does not reduce resolution nearly as much as 0.61 * λ / min(NA_obj, NA_cond) predicts. It appears to me also still better than the usual 1.22 * λ / (NA_obj + NA_cond) which predicts roughly a factor of 2 worse resolution. To me the result looks closer to what the more sophisticated analysis done by Hopkins and Barham predicts.

[zondar]

Can anyone provide a clear mathematical argument showing that the resolution of a simple condenser plus objective scenario (with no subject to confound matters) is not strictly limited to Min(NAcd, NAobj)?

Until then, the confounding factor causing all the debate is the object being imaged. That can be anything from virtually transparent, to a high-contrast pattern such as a diffraction grating, to fluorescing particles, etc.

I see cause for the result to range from Min(NAcd, NAobj), say when the subject barely refracts light at all, all the way up to simply NAobj, say when observing fluorescing beads.

[PeteM]

The long-standing relationship of numerical aperture relationship to resolution is basically a geometric one - assuming every photon of light at some wavelength/energy is fully constrained within some cone of light. By that simple math, and using ray tracing, I think we all agree that (whether it is the condenser or the objective) the smallest cone of illumination rules. Pretty much what Hans said at the outset.

Of course, having a specimen involved is the whole point of microscopy.

It may also be that real illuminators, field lenses, condensers, blank slides, and objectives find small ways to let some random photons go their own way. Likely the smallest of effects, even if it's possible to illuminate a bit beyond the NA-prescribed cones of light.

Then there's "super resolution microscopy," which goes beyond the diffraction limit . . . That one requires a photon-emitting specimen, which is kind of the "to what extent" question noted in the other thread.

[hans]

I'm having trouble following the points you are making and many of the things you are bring up in newer posts are things I already replied to earlier. I copied/pasted quite a bit of stuff below to try to get it in one place and keep track of what you are talking about.

Can anyone provide a clear mathematical argument showing that the resolution of a simple condenser plus objective scenario (with no subject to confound matters) is not strictly limited to Min(NAcd, NAobj)?

Earlier:

That is, I still think that the net resolution should always be limited by the lesser of the two NA's, at least assuming there's no subject to confound things.

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.

It sounds like this is your own deduction, so then how do you propose to define resolution with no subject present? Let me try another way of rephrasing to highlight why I find the idea nonsensical: Say you have some kind of perfect, wave-accurate microscope simulator so you don't have to wait for someone else to provide a clear mathematical argument for you. You run it with NA_obj = 1.25 and NA_cond = 1.25 with nothing in the specimen plane and it produce blank image. Repeat with NA_cond = 0.9, again blank image. NA_cond = 0.1, still blank image, and so on... What else do you even do with the simulator without including a specimen? What do you conclude the resolution is?

Until then, the confounding factor causing all the debate is the object being imaged. That can be anything from virtually transparent, to a high-contrast pattern such as a diffraction grating, to fluorescing particles, etc.

I see cause for the result to range from Min(NAcd, NAobj), say when the subject barely refracts light at all, all the way up to simply NAobj, say when observing fluorescing beads.

It's good we have some agreement that the subject is important, but I don't see where you're going with this. Earlier I posted photos at objective NA 0.66 showing a reduction of condenser NA from 0.66 to 0.1 with relatively small loss of resolution, nowhere near the factor of 6.6 worse resolution predicted by 0.61 * λ / min(NA_obj, NA_cond). I don't have a ton of practical microscopy experience but I have looked at most of the usual hobbyist stuff and don't recall ever observing the dramatic loss of resolution predicted by 0.61 * λ / min(NA_obj, NA_cond). (And it is pretty common to observe the restricted condenser NA case in the process of switching objectives before adjusting the condenser iris.) I would say the appearance of that mosquito larva part with severely restricted condenser NA is pretty typical across a wide variety of common (for hobbyists at least) real world specimens.

Do you have in mind any particular type of specimen, real or idealized, and being more specific than "barely refracts light at all", that you are thinking should show 0.61 * λ / min(NA_obj, NA_cond) loss of resolution under restricted illumination NA? From my limited, personal, non-expert, non-professional use of microscopes over the last few years I would guess the behavior of typical specimens is closer to the partially-coherently-illuminated pinholes in the analysis done by Hopkins and Barham than anything else that has been discussed.

<regarding MTF>

Agree that MTF is sort a distraction here. Talking about results in terms of MTF vs. PSF vs. Rayleigh/Sparrow criteria vs. whatever else doesn't change the underlying physics which is the hard part.

I'm sticking with "it ain't no more than a rule-of-thumb for average setups with average subjects."

There was never any disagreement about averaging being an approximation when applied to "average setups with average subjects" either here or in the papers. Goldstein pointed out very clearly (and I even included that excerpt above) that the averaging rule is exact only for a grating object under coherent oblique illumination and is an approximation when applied to real-world specimens. I said I would not call it a "rule-of-thumb" but that is an imprecise term anyways. What I mainly objected to was the characterization as some kind of internet forum nonsense when in fact it comes from some analysis done from first principles by one of the most famous/influential/respected microscopy theoretician of all time:

I've heard "an average of the condenser and the objective's NA" here more than once, but that's nonsense. The net, in fact, can be no greater than the lesser of the two.

The idea to use the average NA to predict resolution doesn't come from internet forums. The oldest reference I saw was Ernst Abbe himself in 1873. (I didn't try to find it or read it because it's in German, only read what some newer papers summarized about it.)

Regarding air gaps:

<stuff about air gaps limiting illumination NA to a little less than 1, link to book by Conrad Beck>

Earlier:

Once the cone of light exits the condenser and hits air, you are stuck with a theoretical maximum net NA of 1...

Air does limit the illumination NA to 1 or less, but what is "net NA"?

So if there's a single glass-air interface in the condenser+objective path, whether at the condenser's output or the objective's input, then the net NA of the condenser+objective is also limited to a theoretical maximum of 1, and the common formula estimating the resolution of the combination wouldn't work anymore.

Confused what you mean by "anymore", when are you saying it would work? Only if there is no air gap present? If so then I still have the same question as in previous reply, how is having illumination NA limited by an air gap different than simply setting the condenser diaphragm NA slightly less than one?

(And yes, the situation isn't much different than stopping down a diaphragm, which reduces resolution too.)

I though this was already settled. For the purposes of this discussion why can we not simply assume the microscope is being used in a normal manner with oil used as necessary anywhere NA should exceed ~0.9 so that illumination NA is being limited by condenser aperture alone. That would avoid all the confusing distraction about air gaps.

[wabutter]

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.

[Macro_Cosmos]

I personally refuse to use this formula simply because a 1.4 NA Apl-Achr condenser fully open coupled with a 4x at 0.1 NA will not yield 0.75 NA. You will be better off using diatoms as resolution targets.

[hans]

I personally refuse to use this formula simply because a 1.4 NA Apl-Achr condenser fully open coupled with a 4x at 0.1 NA will not yield 0.75 NA.

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. Goldstein says:

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.