Talk:Numerical aperture

Merge proposal
The two definitions should be merged into a single page, replacing this disambiguation page. The definitions are all fundamentally the same--the NA formula for a fiber in terms of the indices of refraction theoretically should be the same as the NA as defined by the acceptance angle of the fiber (which would be the "correct" optical definition). Unfortunately, it seems that the federal government has confused this in their standards, making the approximate theoretical formula the definition of NA for optical fibers. This ambiguity could all be addressed in a single page.--Srleffler 06:48, 16 November 2005 (UTC)

symbolic syntax
Is it so common to write out # as a variable name? I took a class and we used $$f_\#$$, which seems more like usual mathematical conventions. Potatoswatter 22:58, 13 March 2007 (UTC)


 * is by far the most common notation, and is officially the standard notation in photography (e.g. ASA standard PH2.12-1961 American Standard General-Purpose Photographic Exposure Meters). Note, though, that this notation is not a variable name in the usual sense, which helps to explain its odd form. One writes that the f-number of a camera lens is 2.5, not that some variable equals 2.5. Additionally, though it looks like a fraction it is best not thought of that way since when one writes 2.5 one means that the f-number is 2.5, not that one should consider the focal length divided by 2.5 (which would be the diameter of the entrance pupil).


 * If one is doing a lot of mathematics with f-numbers in a science or engineering class, it is convenient to assign a more conventional variable. $$N$$ is commonly used, but $$f_\#$$ seems like a good notation as well.


 * The notation is discussed at f-number, and there is a brief description of the history of this odd notation at f-number.--Srleffler 03:51, 14 March 2007 (UTC)


 * The notation # is horrible. I would change it to $$~D~$$. dima 07:35, 15 June 2007 (UTC)

-

english?

this article is a very hard read.. I feel like Im reading a math book or an issue of scientific american... most of the people on here are not math prodigy's.. anyway to rewrite this in laymans terms..lol `Tracer9999 02:39, 23 August 2007 (UTC)

Light gathering capacity
"Lenses with larger numerical apertures also collect more light and will generally provide a brighter image." - and, moreover, a better signal-to-noise ratio. I'd like to know a bit more about this - how exactly does light-gathering vary with NA? This is of increasing interest in microscopy, where we've recently seen improvements in NA at the top end from 1.40, the state of the art 5-10 years ago, to 1.45, now common although far from standard, and recently 1.49, at sell-a-graduate-student-into-slavery prices. The improvement in resolution is small, about 6%, but people reckon the better lenses make a significant difference, so i assume it's to do with light-gathering capacity. Or it's a big con.

The 1.49 lenses are marketed specifically for Total internal reflection fluorescence microscopes; i don't know if that's because the TIRF physics is especially sensitive to NA, or because TIRF microscopes are just such bastards generally that you need a great lens to get useful data.

-- Tom Anderson —Preceding unsigned comment added by 128.40.81.185 (talk) 16:04, 26 October 2007 (UTC)


 * TIRF lenses must have a high NA to achieve the critical angle required for total internal reflection of the excitation light. An added bonus is you collect more of the fluorescence emisson from a fluorophore because of the high NA. Objective-based TIRF requires high NA lenses but prism-based TIRF can be used with whatever lens you've got, although it will affect your light collecting abilities. I have been investigating the geometry behind light collection with a microscope (using glass coverslips) and, interestingly, I do not think that two lenses with the same NA means that they must collect the same amount of light from a fluorophore. The resolution of course will be the same, but my opinion, based on my inspection of the geometry, is that the light collecting could be different depending on the medium in which the fluorophore is located relative to the medium in which the lens is working. Also, there are weird effects when the fluorophore is very close to the surface of an interface (see Fluorescence Microscopy of Living Cells Axelrod Chapter 15: Emission Of Fluorescence At An Interface) which will change the results drastically.

132.239.25.220 (talk) 02:52, 22 August 2011 (UTC)

Working f-number
Anyone who reads this article AND the f-number article will get confused by a couple of things:
 * 1) This article uses the optical convention, $$m=-o/i$$, while the f-number article uses the photograph convention, $$m=o/i$$ ignoring the inversion of the image.  So the two articles give definitions of working f-number with opposite sign conventions.
 * 2) Neither article explains adequately what happens in macro photography.  In that case, the image plane is significantly behind the focal point.  Thus the light-gathering capacity is not adequately measured by the f-number, which assumes they coincide.  In this case, the angle subtended by the entrance pupil, seen from the image plane, is less than the angle subtended from the focal point.  They differ by a factor $$1+m$$ (in the photographic sign convention).

I can clarify point 1 fairly easily, but I'm not sure what the best way to explain macro photography is. As I understand it, working f-number is needed in macro photography because the "light gathering" happens further behind the focal point... but maybe it ALSO matters that the light from the object is not collimated? Can someone help with this? ǝɹʎℲxoɯ ( contrib ) 00:08, 25 January 2008 (UTC)


 * I returned the formula at f-number to the conventional form. I had the correct sign there originally but another editor changed it. I think he forgot that m is typically negative. I made some other adjustments there. Of particular note: someone had asserted there that the NA was being defined. In fact the reverse is true: the working f-number is defined as 1/(2NA).


 * I'm not sure I understand point 2. Is the light-gathering capability of the lens in macro photography correctly described by the working f-number? I would have thought so. Do you assert otherwise? I thought the working f-number took into account the displacement of the image plane from the focal plane. This displacement is not particular to cameras, but is generally true of finite-conjugate optical systems (where working f/# is used). --Srleffler (talk) 01:00, 25 January 2008 (UTC)


 * I also don't understand point 2. The 1+m (or 1-m) factor should take care of it.  As to the sign convention, I agree we should keep our text in agreement with out cited sources.  I don't recall if it was I who changed it, but I would generally prefer to use the positive m convention in the f-number article, because that's what I see in photography sources, and conversely the negative m in the the numerical aperture article, because that's what's used in optics.  Dicklyon (talk) 01:17, 25 January 2008 (UTC)


 * For the sign convention, it's fine with me now that it's consistent! I should change angle of view too, since I used the opposite convention there.


 * For point 2, I agree with you guys that the formula is correct. I'm just confused about the explanation given in the article.  As I see it, the light-gathering ability is reduced due to displacement of the image plane, which I think you agree with me on.  But this article gives a different explanation:
 * The f-number describes the light-gathering ability of the lens in the case where the marginal ray before (or after) the lens is collimated... In optical design, the finite distance between the object and the lens must often be considered. In these cases, the working f-number is used instead.
 * The article seems to claim that it's the decreased distance to the object plane which explains the working f-number, whereas we think that it's the increased distance to the image plane. Which is right?  ǝɹʎℲxoɯ ( contrib ) 01:21, 25 January 2008 (UTC)


 * I'd say the latter makes more sense, even though they're probably equivalent. And I wouldn't call it the "light-gathering ability"; it's that the gathered light gets spread out more when the image plane is further from the exit pupil.  In fact, the NA should be a ratio involving exit pupil and size, which are not very closely related to focal length; at the focal plane, the ratio comes out the same; at other image planes, it's less obvious to me what happens. Anyone have a generally correct formula for this?  Is it really just dependent on m?  Perhaps so. Dicklyon (talk) 04:07, 25 January 2008 (UTC)


 * Right, it's not the light-gathering ability per se, good point. How would the two notions of working f-number be quivalent?  When you say "the NA should be a ratio involving exit pupil and size", what size are you referring to?  As I see it, the NA should be the ratio of exit pupil to image distance... but I may be missing something here.  The current text bothers me the more I stare at it.  I don't really see what collimation has to do with it. ǝɹʎℲxoɯ ( contrib ) 04:27, 25 January 2008 (UTC)


 * Sorry, fingers outraced brain. I meant to type "ratio between exit pupil size and distance".  I think the collimation really just means the image plane has to be moved; too indirect for my taste, too. Dicklyon (talk) 04:34, 25 January 2008 (UTC)


 * I think I probably wrote that (I haven't checked), and that what I had in mind was not that the change in object position causes the difference in the working f-number, but rather that the object not being at infinity is the sign that you need to consider working f-number rather than the regular f-number. A photographer doesn't know where the image plane is relative to the rear focal plane, but she does know where the object is. I agree that it is the change in image plane position that explains why we need the working f-number.--Srleffler (talk) 05:54, 25 January 2008 (UTC)

I'm not too sure about the version of the formula with the absolute value of m. That works only if working f-number is not used in cases where a lens has positive magnification (in the usual sign convention). Is the formula inapplicable there? I don't recall. At the least, this form of the equation appears to be original research, unsupported by any citation.--Srleffler (talk) 05:09, 25 January 2008 (UTC)


 * Hmm... I don't think numerical aperture would really make sense in the case where you have a virtual image. I don't really see it as OR, but just an attempt to use consistent terminology and conventions across the optics and photography articles.  There may not be any other encyclopedias or other works that are so broad in scope, so it's a challenge to use consistent notation across overlapping but non-identical fields of study.  ǝɹʎℲxoɯ ( contrib ) 05:17, 25 January 2008 (UTC)


 * Yeah, I figured that was too clever to get by you. There are a few sources that might back this up.  How would you handle a mirror imaging system if not this way? Dicklyon (talk) 05:20, 25 January 2008 (UTC)


 * What is the problem with mirrors, exactly? Single mirrors also have negative magnification whenever they form a real image.--Srleffler (talk) 06:06, 25 January 2008 (UTC)


 * Oh, sorry, I was confusing the mirror situation with something else. Never mind, I retract my supposed need for an absolute value there.  Still, it wouldn't hurt; might be more clear; or we should at least mention the alternate convention if we don't do that.  Dicklyon (talk) 06:10, 25 January 2008 (UTC)

Definition of working f-number
Until today, the article said that the working f-number was defined in terms of the numerical aperture:


 * $$N_\mathrm{w} = {1 \over 2 \mathrm{NA}} \approx (1-m)\, N .$$

Today's edits changed this to
 * $$\frac{1}{2 \mathrm{NA}} \approx N_\mathrm{w} = (1-m)\, N .$$

One of these two statements is incorrect. I believe the reference cited in the text supports the previous version. I'll check that when I have a chance, and if so will restore the previous formula and supporting text, unless someone can offer a solid reference to support the other definition. It's possible we have a discrepancy in the sources here, since (as often happens in optics) people take the approximations as exact so often they forget what relations are approximate and which are exact. I am pretty sure that optical design software uses the first definition of working f-number.--Srleffler (talk) 05:29, 13 January 2009 (UTC)


 * I'll look for sources, too. I always understood the "working f-number" to be the ratio of distance to diameter, and the formula seemed to have the approximation in the wrong place.  This came up because I was trying to add the object-side NA, which has a good approximate relationship to f-number and magnification as shown (unless I got that wrong, too, which is quite possible as I was playing kind of loose with the equations as you can see from my sloppy edit history).  I'll try to work it out, but appreciate any help you can give. Dicklyon (talk) 05:43, 13 January 2009 (UTC)


 * OK, here's the problem: most sources don't admit that there's an approximation (they effectively stick to paraxial approximations), so don't help us decide which side the approximation is on.  This book (another SPIE Field Guide) does it the way I would, defining working f-number as f-number times (1-m), the ratio of lens diameter to distance from focal plane; but it doesn't talk about numerical aperture.  Several others, like the Grievenkamp one we cite, define working f-number as 1/2NA; but only this guy throws in the approximation in this funny place. This one can be used as a source for the equation I added; but they make it an equality, which is inconsistent when tan theta is not equal to sin theta.  So this is clear evidence of ignoring the approximation in stating the definition; object-side and image-side NA can't be exactly related by the magnification no matter which definition you accept, right?  But the ratios of the diameters to distances are so related. Dicklyon (talk) 06:26, 13 January 2009 (UTC)


 * I think it would be a shame to settle on a definition that forces us to put the approximation in the strange place the Grievenkamp puts it and implies that the working f-number for an infinitely distant object is only approximately equal to the f-number. So can we accept the other field guide's definition, that comes with a specific statement that the former degenerates to the latter? Dicklyon (talk) 06:34, 13 January 2009 (UTC)


 * Ray (also here) and Shannon define it my way but call it "effective f-number". This guy has a slightly malformed def for effective f-number that's more like Grievenkamp's. Anyway, the evidence suggests that Grievenkamp knew there had to be an approximation in there somewhere, but put it in a place that made a lot of things wrong.  Dicklyon (talk) 06:52, 13 January 2009 (UTC)


 * I looked up the definition Zemax uses for working focal length, in the program's manual. As I recalled, they use the same definition as Greivenkamp. This is important not only due to the approximation, but also because the NA and the working f-number are defined in terms of real rays—to get them one traces a marginal ray through the actual optical system without making the paraxial or thin lens approximations. Greivenkamp also specifies that the NA is based on real, not paraxial, rays. On the other hand, the conventional f-number and $$(1-m) f/\#$$ are paraxial properties. Note also the text above the formula in Greivenkamp: he is explicit that he is using the relation between NA and f-number to define the working f-number. It's not merely a casual or arbitrary choice of symbol in the equation.


 * Clearly, we have here a divergence in the definition of "working f-number" used by different authors. I suspect the difference varies by field, with optical designers using the NA-based definition, which lends itself to ray tracing, and others using the paraxial definition. Neither is wrong, but they are incompatible. I wonder if the usage correlates at all with the usage of "working f-number" vs. "effective f-number". --Srleffler (talk) 13:24, 14 January 2009 (UTC)


 * I'm with Dicklyon on this one. Once we have that f-number = focal_length / aperture_diameter, and we also have that NA = sin(theta), then N = 1/(2NA) is only an approximation, good to the same extent that sin(theta) and tan(theta) are approximately equal.  If we also want "working f-number" = f-number at infinity focus, and I surely do, then the approximation has to be where today's edit puts it.  RikLittlefield (talk) 08:39, 13 January 2009 (UTC)


 * By the way, the edits that Dick made here were prompted by an issue that I raised to him in offline email. In microscopy, most NA's are in object space, while in photography most f-number's are in image space.  But this distinction is seldom called out.  A colleague of mine who was trying to understand NA versus f-number overlooked the difference and got confused by an apparent conflict in the equations.  I asked Dick to take a look at how to keep other people from making the same mistake. RikLittlefield (talk) 18:31, 14 January 2009 (UTC)


 * And in so doing I found it hard to make a sensible object-side NA correspond to the image-side NA with the approximation where it was. I fixed to what I thought was right, without checking the cited source, for which I apologize.  But having now checked sources, it's clear there a bit of confusion around exactly how the NA and "working f-number" are defined. Dicklyon (talk) 18:49, 14 January 2009 (UTC)


 * H. Lou Gibson (1969) Photomacrography p. 61 uses effective aperture like fe/50 equal to (E+1) times the relative aperture (here he's using E as enlarger magnification and talking about the enlarger's effective f-number from the print side to get a diffraction blur). And in Closeup Photography (1970) p.31 "the effective f-number, fe/-, is the one at which the lens is really working.  The two are related by the following formula:  fe-number = (m+1) fr-number."  (where the sub r is for "relative f-number", the usual f-number. Dicklyon (talk) 20:44, 14 January 2009 (UTC)


 * The third (2000) edition of Modern Optical Engineering by Warren J. Smith defines image-side and object-side NA and says they're related by the absolute value of the magnification. But then it goes on to say "The term 'working f-number' is sometimes used" and defines it as the infinity f-number times (1-m).  Can't both be exact, but here the working f-number is defined more sensibly at least. Dicklyon (talk) 20:51, 14 January 2009 (UTC)


 * Rudolf Kingslake (1951) Lenses in Photography defines f-number and effective f-number like I do "effective f-number of a lens is equal to its true f-number multiplied by (1+)" (p.99), but on p.98 defines f-number both as focal length over aperture diameter and as 1/2sin(theta'); he has a footnote about how the second principal plane is actually part of a sphere, so the relevant angle theta' comes from a triangle with hypotenuse f, in which case these can actually both be correct! (also, he sets f in italic face, except in the figure where he was using sans serif and doesn't both to even oblique it).  Dicklyon (talk) 21:59, 14 January 2009 (UTC)


 * I should have read his footnote first, starting on p.97: "It is a common error to suppose that the ratio Y/f is actually equal to tan(theta'), and not sin(theta') as stated in the text. The tangent would, of course, be correct if the principle planes were really plane.  However, the complete theory of the Abbe Sine Condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius f centered about the focal point, ..."  So, we have a bit of an explanation that might help us understand the near-identical meanings of the different definitions.  However, it's still not quite right when the lens is focus close up if this is the story; a good macro lens takes care of that, of course.  And for a thin lens, arguing over the difference may be irrelevant since a thin lens can never be usable at low f-number.  Oh, well, at least Kingslake understood what was going on. Dicklyon (talk) 22:08, 14 January 2009 (UTC)


 * Dick, thanks for pointing out that footnote. Kingslake (1992) Optics in Photography contains what appears to be exactly the same footnote, except that it is on page 107.  That point about the second principal "plane" actually being a sphere had escaped my attention until now. RikLittlefield (talk) 00:07, 15 January 2009 (UTC)


 * Allen R. Greenleaf (1950) Photographic Optics p. 24 makes another very good point. Using F for the f-number, and F' for "equivalent f number", he points out that "Illuminance varies inversely as the square of the distance between the exit pupil of the lens and the position of the plate or film.  Because the position of the exit pupil usually is unknown to the user of a lens, the rear conjugate focal distance is used instead; the resultant theoretical error so introduced is insignificant with most types of photographic lenses. ..."  Anyway, the point here is that the definition that I was wanting to use as exact is based on a model that's too simple to ever be more than approximate.  So maybe the NA-based way of defining effective or working f-number is really the way to go. Dicklyon (talk) 00:39, 15 January 2009 (UTC)

I added some stuff about what we found to the article. See if it's satisfactory. I still need to add the book publishers when I look at them again tomorrow. Some of the other sources might still be useful to add, too. Dicklyon (talk) 07:54, 16 January 2009 (UTC)


 * The additions are very helpful. I tweaked the wording to clarify how Kingslake's footnote applies to the formulas.  I also changed "angular acceptance" to "angular aperture" because in photography the f-number describes the angles of rays coming out of the lens, not going in.  And I deleted the phrase "in air" because it's not needed and is also too restrictive.  —Preceding unsigned comment added by RikLittlefield (talk • contribs) 19:14, 18 January 2009 (UTC)

Signed versus unsigned magnification
I would like to re-open the discussion of how to treat magnification.

Since the other contributors to this page probably don't know me, a bit of introduction may help. I work a lot at the boundary between microscopy and macro photography. It's a constant challenge helping people to understand the relationships between resolution, diffraction, numerical aperture, the lens's nominal f-number, and the working or effective f-number.

This effort is hampered by differences in convention between various fields.

The handling of "magnification" is one particularly vexing difference.

In photography and microscopy, the convention is that magnifications are unsigned numbers and inversion is addressed as a separate property. See for example Kingslake, "Optics in Photography", page 31, and Nikon's microscopyu.

Negative versus unsigned magnification is a problem for people trying to bridge the fields. The formulas for unsigned magnification look very different from those for negative magnification. In addition, proper handling of negative numbers is a classic stumbling block for people who don't work math every day. It's even a stumbling block for me, and I do work math every day, in addition to teaching it to other people. I see from the history comments that negative magnification has been a problem for other editors of this page as well.

My guess is that most people who read these Wikipedia pages will be coming from domains like microscopy and photography where magnifications are unsigned.

If that's true, then the pages would be more helpful if they also used unsigned magnification, while calling out that other fields (optical engineering) treat inversion via negative magnification and have different formulas as a result. On quick scan, making the change to unsigned magnification seems to affect only about half a dozen pages at this time, so it's not a huge editing task. I'll be happy to help, if we can reach consensus.

I would also be happy to try writing a summary of how this stuff fits together for macro photography, but that would have to use the unsigned convention to be consistent with reference sources. RikLittlefield (talk) 21:44, 14 January 2009 (UTC)


 * I come from a physics background and work in optics, so I prefer signed magnification. It is conventional in optics, and using unsigned magnification would make Wikipedia's treatment different from that used in elementary optics texts. I would think that readers who are interested in mathematical treatments of optics would be as likely to be taking a high school or freshman university physics course, as to be microscopists or photographers. I am also concerned that the change in approach might make the treatment less flexible. Unsigned magnification makes sense in systems where the image is either always inverted or never inverted. I suspect it becomes much less convenient in a general treatment of optics, where systems can be either way. I feel that the Wikipedia articles on general optics concepts should stay general-purpose, and oppose changes that lean them too far towards supporting only a photography audience.


 * Maybe we can address these issues some other way? The optics articles could discuss sign convention issues more than they do. Another option would be to have a separate treatment of optics that is focused on the needs of photographers, perhaps at Science of photography, or in a new article.


 * NB magnification is not always negative in the signed-magnification convention. I have therefore renamed this section for clarity.--Srleffler (talk) 03:30, 15 January 2009 (UTC)


 * It's hard to satisfy everyone with overlapping articles when different fields adopt different conventions. It might be OK, though, to include alternative formulae with absolute value of m in the formula, like $$1+|m|$$. Dicklyon (talk) 05:51, 15 January 2009 (UTC)


 * Those are all good points.


 * We don't have direct data about who visits Wikipedia articles, but a proxy measure might be provided by Internet search. At this moment, Google search on just "numerical aperture" returns Wikipedia articles as the first two hits.  Of the next 20 hits, 19 relate specifically to fiber optics, microscopy, or lithography.  The lone exception is an application-neutral Melles Griot Optics Guide.  That source notes that magnification is the ratio of two NA's, which I presume would produce a positive number.  Yahoo search provides similar results, with the exception that it finds a different page by Melles Griot.  That page does not define magnification, but does provide equations that require positive magnification to produce the correct result.  Google search on "numerical aperture" magnification (both terms together) produces results that are even more lopsided.  Aside from Wikipedia pages, all of the first 100 hits relate to imaging domains where magnification is conventionally unsigned.


 * I have no doubt that signed magnification is a powerful and unifying tool in the hands of an optics system designer. However, the search engine results noted above suggest that it is not the most commonly encountered convention.


 * I agree that Wikipedia articles on general optics concepts should stay general-purpose. But I believe this goal can be accomplished while still keeping them well aligned with other frequently referenced sources.


 * A more definite discussion of sign convention issues would be very helpful but I believe it is not sufficient. The formatting of Wikipedia articles emphasizes formulas over text, and readers look to formulas for concise summaries.  For the reader's benefit, I believe formulas in Wikipedia should both use the same sign conventions and be formatted to look the same as formulas in other sources.  If forced to choose between $$1-m$$, $$1+|m|$$, and $$m+1$$, this argument would select $$m+1$$.  My feeling about $$1+|m|$$ is that it's an interesting compromise, but I don't recall seeing that form elsewhere.  This raises two issues: unfamiliarity to readers and possibly original research.


 * I have reviewed the Wikipedia pages that seem most relevant to this discussion (Magnification, Lens (optics), Curved mirror, Numerical aperture, F-number). My impression is that only a few places are problems, particularly the two formulas in Numerical aperture and one formula in F-number that use $$m$$ on the right hand side.


 * Perhaps we could display both forms of these few equations, each paired with a few words about the convention that it assumes. This would clarify both the role and importance of the convention and also provide readers with good hooks into whichever literature they are familiar with.  The added cost could be only a few column-inches of space, while the added value seems much more than that.


 * Another option is to use dual presentation only on the Numerical Aperture page. It has a strongly math flavor and also sits more clearly on the divide between design and use of optics.  The F-number page seems strongly oriented to photography; those readers might be well served by using just the photographic convention, referenced to a corresponding source. --RikLittlefield (talk) 19:13, 15 January 2009 (UTC)


 * I agree with Rik on this; signed magnification is for optics specialists, not for the typical reader, and it shouldn't be hard to mention both conventions where relevant. Other articles that use magnification include Depth of field, where it is positive, as in all the relevant sources. Dicklyon (talk) 04:52, 16 January 2009 (UTC)

Subject NA versus image NA
The article doesn't mention this distinction. It should. —Ben FrantzDale (talk) 00:02, 29 September 2010 (UTC)


 * Actually, the distinction between object and image NA is mentioned, in the section "Working" or "effective" f-number. --Srleffler (talk) 02:27, 29 September 2010 (UTC)

Just a small change
I changed the symbol "D" to "w0" in the definition of NA in the laser physics heading, since D is used previously in the article and I believe a simple change of symbol avoids confusion/ambiguity, especially to those who do not read the page in it's entirety. I may have caused someone offence by doing so since they swiftly returned the equation to it's previous form (inside 40mins of my edit, well played that man). Is my suggestion really that bad?Dtlloyd (talk) 22:16, 20 April 2011 (UTC)
 * I reverted your edit because D is equal to 2×w0, so your edit made the equation incorrect (if one uses the conventional meaning of w0). I didn't notice that D was already in use in the article for something else. That would clearly make it necessary to change one of the variables.--Srleffler (talk) 02:34, 21 April 2011 (UTC)
 * My mistake. Thank you for clearing this up.Dtlloyd (talk) 08:29, 21 April 2011 (UTC)

A Note on Objectivity
When stating errors the objective approach gives the error in an estimate relative to that of the true value. --Jbergquist (talk) 00:16, 22 March 2012 (UTC)
 * It turns out in this case that the "true" value is not so true, and for many lenses the "approximate" value presented is more accurate than the value presented as "true". I've amended the text to reflect this better.--Srleffler (talk) 04:10, 22 March 2012 (UTC)

NA vs f-number
N=f/D is unreliable as a definition of f-number and most often only an approximation. For instance N is not equal to f/D in the typical thin lens diagram shown in the article, as explained by Kingslake in the quoted paragraph. N is on the other hand always equal to 1/[2nsin(theta)] for a practical photographic lens, per Kingslake and others. That's the definition that should be used at the beginning of the section. It follows that N = 1/2NA exactly, which should shorten the section considerably, increasing clarity and removing uncertainty. Can page admins make the relative changes? I will make similar comments to the f-number pageJack Hogan (talk) 16:09, 3 March 2016 (UTC)
 * There is indeed an unreliable approximation here, but it's not $N = f/D$, it's the thin lens. See the discussion on f-number page, and continue the discussion there rather than here if necessary. — Edgar.bonet (talk) 10:02, 4 March 2016 (UTC)

Improvement request
Could someone rewrite the 'General Optics' preface? It is very unclear. I couldn't understand it. Specifically, "maximal half-angle of the cone of light that can enter or exit the lens". What does it mean? Say, we have an object that touches the lens at the center. (Let us assume the lens is thin.) Then this half-cone is 90°. Clearly this is not what is intended. Lockywolf (talk) 03:29, 7 September 2017 (UTC)
 * You missed the line in the second paragraph, "The NA is generally measured with respect to a particular object or image point and will vary as that point is moved.". So, yes if your object is touching the lens the half-cone is 90°. That's not a particularly interesting case, since the lens isn't going to form an image there.--Srleffler (talk) 05:00, 7 September 2017 (UTC)

Dead link to reference
The link to the reference: Cargille, John J. (1985). "Immersion oil and the microscope" (2nd ed.) is dead. There is however a working link on and the paper itself (PDF) can be downloaded from. I don't know how to fix it (which I of course had done if I only had known how - but, as it now is, I can't even mark the link as dead!) as there only is a template "reflist", and where the references are hidden in cyberspace is completely unknown to me (a really stupid system not to include the references in the article - in my "humble" opinion...). So, now it is somebodyelse's problem, not mine! Episcophagus (talk) 06:27, 15 October 2019 (UTC)
 * Thanks for pointing that out. I fixed the link. The references are embedded in the article itself, but they are embedded between special tags at the location in the text where the references is referred to, rather than at the end of the article where the reference actually appears. If you click the "edit" link beside the section heading for the part of the article where the reference first appeared, you see embedded in the text: ""
 * This is the bit that actually makes the reference work. There are other referencing systems in use on Wikipedia. Some of them actually do embed the references in the "reflist" at the end, but the system used in this article is the most common one.--Srleffler (talk) 01:34, 17 October 2019 (UTC)
 * Thanks. I see, works like . I wasn't aware of that. Episcophagus (talk) 09:50, 17 October 2019 (UTC)

Proposed merge of Optical unit into Numerical aperture
Under-maintained stub without enough content to keep as a standalone article. Topic is closely related to the proposed target. –LaundryPizza03 ( d c̄ ) 23:16, 7 April 2021 (UTC)
 * I don't think this article is a good merge target for that topic. Perhaps it belongs in a microscopy article somewhere?--Srleffler (talk) 17:30, 10 April 2021 (UTC)
 * Diffraction-limited system might be a better target. Optical unit is rather in need of improvement too (even if merged). Klbrain (talk) 11:27, 13 September 2021 (UTC)
 * I cleaned up Optical unit to the best of my ability, but there's not a lot of sourcing. Added a discussion on the talk page of that article to discuss what should be done with it. PianoDan (talk) 21:08, 9 December 2021 (UTC)
 * Without further discussion, I am closing the merge request, and leaving Optical unit where it is. PianoDan (talk) 17:12, 15 December 2021 (UTC)