## Need Value for NA Formula

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linuxusr
Posts: 52
Joined: Sun Jun 24, 2018 9:00 pm

### Need Value for NA Formula

Hello All,

Now that I understand the practical reason for the N.A. being imprinted on the objectives, i.e. evaluating if magnification is useful dependent on the eyepiece magnification--NA(1000) = useful magnification.

Now I want to take my understanding of NA one step further. I figure the best way to understand the concept is to do the math. I post here because this sub-forum is the closest to optics.

The formula is:

NA = sin(theta)

I already know the NA of my objectives, so I want to complete the formula in reverse--sin(theta) = ?
and see if the value I arrive at is the same as that printed on my objective.

I think that theta will be a positive integer representing the number of degrees (of a given objective) between the optical axis and the outward side of the light cone between the condenser and the focal plane of the specimen.

The problem is that I cannot find the value of theta. I have looked at the specs of many objectives and I never see specs for theta. Also, I cannot find any chart.

So how do I find theta for my objectives?

As a side note, according to one lecture I heard, the angle of the cone is theta, so the angle to the optical axis is theta/2.

Once I complete the formulas for all my objectives, I'll have a clearer idea of the relationship between angles and resolution and depth of field, if I want to add that. I'll probably make a table.
Nikon AlphaPhot 2 < Zeiss Primostar 3, Full Koehler

patta
Posts: 52
Joined: Sun May 10, 2020 6:01 am
Location: Stavanger Norway

### Re: Need Value for NA Formula

The Theta is directly related to the NA, so it is not needed to write it in the spec. Unless the NA stated is wrong...
Yes, the Theta is the max angle at which the light enter the objective. May use the inverse formula for the sine, Theta= ARCSEN(NA)
I call theta the angle of the half-cone, so whole cone angle is 2*theta.
Others call the full cone angle theta, so the NA will be NA=SIN(theta/2), Theta = 2*ARCSIN(NA). Just choose one of the conventions!

You can observe/measure directly the theta of your objective with a tool "apertometer".

You can make an apertometer yourself, or you can still easily estimate/measure the theta with just a ruler or a goniometer (works for dry objectives only):
Unscrew the objective
Look at the objective from the back, at some distance like 10cm
You will notice that you see through the objective, like from a door peephole, upside-down
This is the objective aperture
If you can estimate how wide is the view, this is 2*theta, the max aperture.

The Funsci link has also extensive geometric /trigonometric explanations.
http://www.microscopy-uk.org.uk/mag/art ... ometer.pdf
https://www.quekett.org/wp-content/uplo ... Lenses.pdf
http://microscope-antiques.com/hxapertometer.html
http://www.funsci.it/files/n-47-Misura- ... 5tu6a1.pdf
https://www.binomania.it/phpBB3/viewtop ... pertometer

My current apertometer (recommended, minimal effort) (not for oil immersion objectives)
Open one of the above links with the phone; zoom into one of the illustrations; put the phone under the microscope; rack the focuser so the objective is at the right distance from the phone (about 14mm in my case; you should measure the image (b) with a ruler and calculate the height (a) so Theta & NA are correct); center the phone. Look through the microscope and read the value at the extreme.
(needs "Amici/Bertrand lens", or is also doable by removing the eyepiece and looking directly)
Apertometer diagram photos.jpg (101.56 KiB) Viewed 195 times

75RR
Posts: 7753
Joined: Sun Oct 12, 2014 2:34 am
Location: Estepona, Spain

### Re: Need Value for NA Formula

linuxusr wrote:
Wed May 05, 2021 12:01 am
Now that I understand the practical reason for the N.A. being imprinted on the objectives, i.e. evaluating if magnification is useful dependent on the eyepiece magnification--NA(1000) = useful magnification.
Calculating useful magnification is certainly something that should be done when putting together a microscope system, however, the NA on the objective is really all about the resolution that it is capable of.

Here is a video with a very good explanation of resolution - do note at the end of the video the reference to Nyquist and sensor pixel size in capturing max resolution in photomicrography.
.

Zeiss Standard WL (somewhat fashion challenged) & Wild M8
Olympus E-P2 (Micro Four Thirds Camera)

Greg Howald
Posts: 553
Joined: Tue Oct 20, 2020 6:44 am

### Re: Need Value for NA Formula

I admit that I'm no mathematician. I can barely spell the word. But. If NA equals sin multiplied by theta, then theta equals NA divided by sin and sin equals NA divided by theta. Don't know if that helps you or not but I wish you every success.
Greg

linuxusr
Posts: 52
Joined: Sun Jun 24, 2018 9:00 pm

### Re: Need Value for NA Formula

@patta

Thanks much for your robust post. Need some time to dig in. I will post reply down the line.

@75RR

ditto
Nikon AlphaPhot 2 < Zeiss Primostar 3, Full Koehler

linuxusr
Posts: 52
Joined: Sun Jun 24, 2018 9:00 pm

### Re: Need Value for NA Formula

@patta

I have begun to apply your steps. Using 10x as example, you said that the diameter of the small hole from the back would be 2(theta) or maximum aperture. Using a ruler, I estimated the diameter to be
5/36. 5/36 over 2 = 0.694. This should be the angle for theta. Using the formula for NA, sin(0.694) = 0.12. But my NA for that objective is 0.25. I'm getting 1/2 of the value of what should be my NA.

For the NA, I also used my 10x and the same aperture diagram that you used. Looking through the eye tube I centered on a circle with a 10 inside. At 9 o'oclock and at 3 o'oclock were the number 20, respectively. This is what a friend told me as my vision was too limited. I don't know what these values mean.