## Confirmation of interpretation of NA

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ray_parkhurst
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### Confirmation of interpretation of NA

My understanding is that then NA of a lens in air is NA = sin(Theta) where Theta is the half-angle of the widest rays coming through the lens at infinity focus. This is used to calculate the infinity aperture by f# = 1/2NA, and the effective aperture by EA = m/2NA. I will assume this is correct.

Now, this implies that the NA of a lens is independent of the magnification, correct? There is no concept of "effective NA".

When focused closer than infinity, the widest rays coming through the lens to the sensor are indeed at a smaller angle. If NA were recalculated with this smaller angle, it would be a smaller number, sort of an "effective NA". There are folks over on depreview.com who are insisting that this recalculation is correct, and that NA is indeed equivalent to effective aperture through the 1/2NA formula, with the NA recalculated based on the non-infinity focus. I can't find a reference that says this is not true, so I just want to make sure based on the collective knowledge here.

Or maybe my assumptions are incorrect, and NA does change with magnification. That would be unfortunate for my sanity, but I'm willing to accept such a setback.

Or perhaps there are different definitions of NA? Please let me know if I have been barking up the wrong tree, or if I am correct that NA, like f#, are constants for a particular lens system.

rjlittlefield
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I regret the impact on your sanity, but yes, NA can change with magnification.

Based on your quick summary, it sounds to me like the folks at dpreview have it exactly correct.

But before spending any more time on explanation, I'd like to read firsthand exactly what it is that they're saying.

--Rik

ray_parkhurst
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rjlittlefield wrote:I regret the impact on your sanity, but yes, NA can change with magnification.

Based on your quick summary, it sounds to me like the folks at dpreview have it exactly correct.

But before spending any more time on explanation, I'd like to read firsthand exactly what it is that they're saying.

--Rik

I was adamant in the thread that NA is a constant. I hate having egg on my face...

rjlittlefield
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I was adamant in the thread that NA is a constant. I hate having egg on my face...
Thanks for the reference. Reading that thread got so painful that I started skimming. Not only did you put egg on your face in the first place, but then you persisted in smearing it around despite an assortment of warning signs.

I am tempted to wonder, if I try explaining it to you, will you listen any better to me than you did to the two dpreview Veteran Members who tried to explain it to you?

--Rik

ray_parkhurst
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rjlittlefield wrote:

I was adamant in the thread that NA is a constant. I hate having egg on my face...
Thanks for the reference. Reading that thread got so painful that I started skimming. Not only did you put egg on your face in the first place, but then you persisted in smearing it around despite an assortment of warning signs.

I am tempted to wonder, if I try explaining it to you, will you listen any better to me than you did to the two dpreview Veteran Members who tried to explain it to you?

--Rik

rjlittlefield
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OK. I'll try to make this brief. I'll also try to write in very different style from what bclaff and Pixel Pooper wrote over at dpreview. But my intention is that all of my content is consistent with theirs. If that does not appear to be the case, then please ask (but don't argue).

NA is a description of light paths at a particular place, for a particular configuration of the optics. It is numerically equal to the sine of the half-angle of the ray cone -- at that place, in that configuration. If you change the configuration, for example by changing focus, then typically the NA will change also. If you look at a different place, for example at sensor plane versus object plane, then again you will get a different value.

At the same place, for the same configuration, EA = 1/(2*NA), and you can take that as one of several equivalent definitions of "effective aperture".

It is a convenient fact that for any well corrected optical system, if you measure NA and magnification at each of two different image planes, then m1*NA1 = m2*NA2. In particular, if the focused image on sensor is m times larger than the object, then NA at the object will be m times larger than NA at the sensor. This is the reason for our commonly used formula that EA_sensor = mag / (2*NA_object), which relates the sensor-side effective F-number to the object-side NA.

It should be clear from all of the above that NA and EA are highly subject to change.

So why do we ever label a lens with a specific value of NA or F-number? The answer is that those are approximations for convenience. Microscope objectives are designed to be used in one specific configuration; the NA that is engraved on their side is the object-side NA in that configuration. Other lenses are labeled as if they were going to be used at infinity; their nominal F-number is the sensor-side EA at infinity focus. This convention is so traditional that it's applied even in the case that the lens cannot be used at infinity focus and because of pupil ratio it will never act like you'd expect an F-whatever lens to act. Caveat user, or something like that.

I have no idea where you got the idea that NA was fixed and was attached to infinity focus.

So I have to ask, does the above explanation help make any different sense of whatever you read that made you think that NA is fixed?

--Rik

ray_parkhurst
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### Mea Culpa, and Farewell

Yes, that does help, and it makes sense. I don't know how I got it so wrong, but it is not the first time I have done such a thing for such a relatively simple concept. Sorry to have created another "cringe moment" for you. I seem to remember quite a few of these, and I appreciate your excellent explanations of my mistakes over the years.

Thanks also to Lou, Robert, Chris S, and others who have been supportive in my various quests and ponderings.

I've decided to step back from my postings going forward. I joined this forum nearly a decade ago hoping to improve my knowledge of photomacrography in support of my numismatic photography hobby, and am happy with what I have been able to achieve.