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Numerical Aperture
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rjlittlefield



Joined: 06 May 2005
Posts: 727
Location: Richland, WA, USA

PostPosted: Tue Apr 11, 2006 9:03 am    Post subject: Reply with quote

S. Alden wrote:
I like pictures (illustrations)... Laughing

I do too, and that's no laughing matter! Very Happy

Now if only I could figure out how to quickly construct illustrations and embed them in these discussions. Hhmm...

Anyway, the link that you posted has great illustrations and a lot of helpful detail.

It also highlights a couple of other issues about explanations.

First issue is illustrated by the fact is that while their formula for NA looks correct, there's a glaring error in the accompanying explanation. They write that
Quote:
...multiplied by n, the refractive index of the medium between the cover slip and the objective. When a lens is designed to be used dry n = 0 but when a lens is intended to be joined to the prep with immersion oil (oel) the refractive index is 1.515.

That would imply that all dry lenses have NA=0, which clearly 'tain't so. In fact, dry lenses have n=1 (or real close to it), the refractive index of air being real close to 1, not 0.

I have seen this happen pretty often.

I think that what happens is the asides and informal explanations get written too quickly and not proofread enough. Whatever the cause, I always consider them suspect. (Of course, I consider everything suspect, especially stuff I write myself, and I'm not laughing about that either!)

Second issue is that sometimes the explanations seem plausible but don't accurately or completely reflect what's going on.

I'm pretty sure that's the case when n>1, as in oil immersion.

The usual explanation is that the oil increases the cone angle. Certainly that's part of what happens (by avoiding total internal reflection at glass-to-air transitions --- yet another advanced topic), but it still doesn't get at the fundamental issue, which (I think!) is the reduction of wavelength at the subject being observed.

A more complete explanation would talk about why it's important to have high refractive index in the mountant as well.

Offhand I don't recall seeing an explanation like that, although it's sort of alluded to in Sue's link -- "Ideally the mounting medium will also have a refractive index of 1.515 as will the coverslip." Maybe I'll see a better one now that I'm on the lookout for it. It's an issue of practical significance, though. Perhaps an article on mounting subjects intended for very high magnification would be a good place to look for it.

Third issue is that at some level of theory a bunch of formulas appear that all purport to be describing the same thing, but they're all slightly different and it's unclear why the differences exist. Usually (I think) the different formulas represent different approximations and assumptions, but hardly anybody spells out what those approximations and assumptions are. And if they did, I suppose our eyes would glaze over.

For example, on the page that you link, it's claimed that resolution = 1.22 lambda / (NAobj + NAcdn). At http://www.microscopyu.com/articles/formulas/formulasresolution.html, the same formula is listed, but also the formulas lambda/(2*NA) and 0.61*lambda/NA. Personally, I think a better formula is probably 0.61*lambda/min(NAobj+NAcdn), since it's the minimum cone angle that matters. But I don't recall having seen anybody quote that one although they probably have.

Another long post, sorry.

Quick summary:
- pictures are good
- explanations are good, but don't take them as gospel
- formulas are good too, but don't get wedded to any particular one.

Hope this is helpful,
--Rik
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rene



Joined: 17 Mar 2005
Posts: 125

PostPosted: Tue Apr 11, 2006 9:57 am    Post subject: Reply with quote

Rik, think about NA as angle (aperture), then go on about gratings which show maxima(first order, second order and so on) and then tell students that in order to resolve that grating microscopically, you will need to catch at least the first order max, the more the better, thus, a wide angle. That's intuitive. You can use this analogy to show oblique lighting and after that sunk in, the cone aperture from the condenser finishes the story.

BTW, the diatom mountant has nothing to do with resolution, the surface lowest in RI is the bottleneck. The contrast issue for diatoms would confuse it.

Rene.
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rjlittlefield



Joined: 06 May 2005
Posts: 727
Location: Richland, WA, USA

PostPosted: Tue Apr 11, 2006 11:52 am    Post subject: Reply with quote

Rene,
You wrote:
...the cone aperture from the condenser finishes the story.

I'm not sure about that. Maybe it depends on your point of view.

Here's a thought experiment. Suppose I have an ideal condenser that outputs a full 180-degree cone of light, and a lens that likewise accepts a full 180-degree cone. Also suppose that my subject is a grating with zero thickness.

What I think happens is that if I surround the grating/condenser/lens with air, then the resolution I get corresponds to NA 1.0, but if I surround with water it jumps to 1.33, standard oil 1.515, and so on.

The only way I know to understand this is in terms of shortening the wavelength in higher R.I. medium. Trying to think just in terms of cone angle doesn't make sense because that doesn't change in this thought experiment.

rene wrote:
...the diatom mountant has nothing to do with resolution, the surface lowest in RI is the bottleneck. The contrast issue for diatoms would confuse it.

I think we agree here, but maybe not.

The problem that the diatom valve has R.I. too close to standard medium is a contrast issue, not related to resolution.

But I think that if you embed the diatom in high R.I. mountant and manage to maintain the same cone angle as for low R.I., then you would get higher resolution as discussed above.

Is this not correct? I'm certainly not equipped to run the experiment physically!

--Rik
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rene



Joined: 17 Mar 2005
Posts: 125

PostPosted: Tue Apr 11, 2006 3:14 pm    Post subject: Reply with quote

Sorry Rik, reacted on an earlier message from you where you were asking yourself how to tell students about NA.

Quote:
But I think that if you embed the diatom in high R.I. mountant and manage to maintain the same cone angle as for low R.I., then you would get higher resolution as discussed above.

Still, your whole system is limited to the lowest RI, the glass slide, coverslip, objective lens glass. A 'higher' cone than can be sustained by the glass coverslip is relected back to the sample. Even when you immerse (without coverslip) with high refractive medium directly against the lens, it will only sustain a cone limited by the RI of the frontlens. Zeiss made some 1.6NA lenses around 1900 or so, famous for that!
Nevertheless, there just isn't much room to play.

Quote:
Personally, I think a better formula is probably 0.61*lambda/min(NAobj+NAcdn), since it's the minimum cone angle that matters.


Nope. Again I have to refer to my explanation. The cone (when maximal) doubles the resolution compared to lighting by a parallel bundle. In your example the illumination light would then be sin0=0 and therefore resolution 0?!

Hope that clarifies things, keep asking questions.

Rene.
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rjlittlefield



Joined: 06 May 2005
Posts: 727
Location: Richland, WA, USA

PostPosted: Tue Apr 11, 2006 4:27 pm    Post subject: Reply with quote

Rene,
You wrote:
Still, your whole system is limited to the lowest RI, the glass slide, coverslip, objective lens glass. A 'higher' cone than can be sustained by the glass coverslip is relected back to the sample. Even when you immerse (without coverslip) with high refractive medium directly against the lens, it will only sustain a cone limited by the RI of the frontlens. Zeiss made some 1.6NA lenses around 1900 or so, famous for that!
Nevertheless, there just isn't much room to play.

I agree. The issue I'm wrestling with is how to explain/understand the increased resolution. As long as everything is dry including the subject, it works OK to think in terms of cone angle and wavelength of the illumination. But when immersed in R.I. > 1, it still seems to me that the cone angle explanation doesn't work by itself and you have to appeal to shorter wavelength also.

Actually, things get really interesting when I try to think in detail about the usual case where the subject is mounted in high R.I. stuff but the condenser and lens are dry. I think that what's actually going on is that a wide cone coming from the condenser is converted into a narrower cone, at correspondingly shorter wavelength, when it enters the slide, then turns back into a wide cone when it leaves the coverslip. Oiling the condenser and lens allows the cone to be wide even at the subject, where it really matters. In all cases, at the subject, the wavelength is lambda/r, where r is the R.I. of the mountant. So the challenge, in some sense, is to explain how come that shorter wavelength at the subject does not always result in higher resolution out the back of the lens!

Notice that I'm not concerned with how to compute the correct answer. The formulas work fine for that. What I'm concerned with is trying to understand why that answer is correct, in terms of the underlying physics. It's a good question why I feel compelled to do that. Maybe I am just burdened with a "gotta-know" pathology.

Quote:
In your example the illumination light would then be sin0=0 and therefore resolution 0?!

I don't think so. The formula is sin(a) where a is the half-angle of the cone. In my thought experiment, a would be 90 degrees, sin = 1. Did I miss something here?

Rene wrote:
Rik wrote:
Personally, I think a better formula is probably 0.61*lambda/min(NAobj+NAcdn), since it's the minimum cone angle that matters.

Nope. Again I have to refer to my explanation. The cone (when maximal) doubles the resolution compared to lighting by a parallel bundle.

Hhmm, OK, I'll buy that. I didn't think it through completely. So I guess the simpler formulas quoted by Microscopy U must carry an unstated assumption that the illumination cone is maximal. And also I guess the general formula would be 1.22*lambda/(NAobj+min(NAcdn,NAobj)). Or is it even more subtle than that?

--Rik
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rene



Joined: 17 Mar 2005
Posts: 125

PostPosted: Wed Apr 12, 2006 7:46 am    Post subject: Reply with quote

rjlittlefield wrote:
Rene,
I agree. The issue I'm wrestling with is how to explain/understand the increased resolution. As long as everything is dry including the subject, it works OK to think in terms of cone angle and wavelength of the illumination. But when immersed in R.I. > 1, it still seems to me that the cone angle explanation doesn't work by itself and you have to appeal to shorter wavelength also.
--Rik


Rik, look at what happens through a coverglass, the original cone (from air to glass and back to air) doesn't change when it is back in air. So you would have to compare cones in similar RI, or calculate via RI what the new cone would be in angle. It's not only cone width, but cone width + RI that's determining resolution, and that's exactly what you see coming back in the formule

NA (read: resolution)=n sin(a)

Hope that's clear?

If you bring in max allowable cone from slide to high refractive media, indeed things could have been better from resolution standpoint where a bigger cone could be used in high RI. In this case however the slide glass (or condenser glass, or whatever) limits the cone to a theorethical max NA of 1.5, practically around 1.45, and mostly 1.3-1.4 for oil immersion objectives. The diatom mountant could indeed sustain a 'higher' cone, but whatever extra above NA 1.5 will be cut off when entering the coverglass. Or the immersion oil. Or the frontlens.

Quote:
Quote:
In your example the illumination light would then be sin0=0 and therefore resolution 0?!

I don't think so. The formula is sin(a) where a is the half-angle of the cone. In my thought experiment, a would be 90 degrees, sin = 1. Did I miss something here?


I think you mentioned parallel light ('normal lighting' eg directly from a lamp at infinity). Then a=0, sina=0, resolution is only dependant on objective NA.
With a half cone from the condenser (NAcond=0.5 NAobj) the resolution is 1.5 times that number.

Hope that helped. Now I'm on holidays, so if you get stuck, get something from your GP Wink

Rene.
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rjlittlefield



Joined: 06 May 2005
Posts: 727
Location: Richland, WA, USA

PostPosted: Wed Apr 12, 2006 9:18 am    Post subject: Reply with quote

Rene,

Thanks for the discussion!

I understand & agree with everything that I hear you saying now. I'm still not sure what are the simplest ways to think about what's going on at various levels of detail, but that's a problem that can just sit quietly in my head for a while.

You wrote:
I think you mentioned parallel light ('normal lighting' eg directly from a lamp at infinity).

Ah! I think we have here an example of another issue with explanations: did the reader hear what the writer intended to say?

Reviewing the posts, I can't find the words "normal" or "parallel" until you used them. What I wrote was "Suppose I have an ideal condenser that outputs a full 180-degree cone of light". I was intending to describe a source that uniformly illuminates a half-sphere; apparently to you my words evoked a parallel beam of light. An accompanying picture would have helped (I hope!). But no matter, I think we're clear now.

Frez, I hope you appreciate what an interesting discussion your "beginner's question" touched off.

Truly, it was a good thing that Ken moved it! Very Happy

--Rik
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nzmacro
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Joined: 20 Apr 2004
Posts: 1604
Location: New Zealand

PostPosted: Mon Apr 17, 2006 3:40 am    Post subject: Reply with quote

Yes, but can you guy's make a decent cup of coffee, thats the question Cool Wink , oops, wrong forum Shocked
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