Need Value for NA Formula

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linuxusr
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Need Value for NA Formula

#1 Post by linuxusr » Wed May 05, 2021 12:01 am

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.
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patta
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Re: Need Value for NA Formula

#2 Post by patta » Wed May 05, 2021 6:46 am

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.

Some links on homemade apertometers
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
Apertometer diagram photos.jpg (101.56 KiB) Viewed 2909 times

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Re: Need Value for NA Formula

#3 Post by 75RR » Wed May 05, 2021 10:17 am

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.
.

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Greg Howald
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Re: Need Value for NA Formula

#4 Post by Greg Howald » Fri May 07, 2021 12:31 am

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

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Re: Need Value for NA Formula

#5 Post by linuxusr » Fri May 07, 2021 4:03 am

@patta

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

@75RR

ditto
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linuxusr
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Re: Need Value for NA Formula

#6 Post by linuxusr » Mon May 10, 2021 4:26 am

@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.

Please advise!

In your diagram, the green line is the eye and the Bertrand lens is held between the eye and the eye tube, focusing the same image I looked at closer to my eye?
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linuxusr
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Re: Need Value for NA Formula

#7 Post by linuxusr » Mon May 10, 2021 11:11 pm

OP:

@patta was helping me to find theta by mechanical means. I'm still waiting to hear back as I had some questions. Meanwhile, I thought I would try to find theta mathematically.

The formula for NA is:
(n)[sin(theta)]

n = 1 = refractive index for air, so we can forget that as I am trying to find theta for my 10x objective, and the reason I'm doing that is to get a better understanding of N.A.

The NA for my 10x is 0.25, so if I do:

0.25 = sin(x) and solve for x in degrees, I get 14.47 (I dropped off 5 place values).

So the angle for my 10x from optical axis to the outside of the cone is 14.47 degrees.
To check that, I do:

x = sin(14.47) and I get 0.249 which is my NA! If I hadn't dropped off those place values, I probably would have gotten 0.25. So it's pretty easy to find theta mathematically.
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Re: Need Value for NA Formula

#8 Post by patta » Tue May 11, 2021 11:56 am

Hello
yes, your last Theta-NA calculation is correct; NA=Sin(Theta); so you get Theta from the NA by inverting this formula. That is why the Theta is not written in the objective specifications, it can be calculated from the NA. I thought was clear.

Measuring the Theta, will be useful to check if the manufacturer is cheating, of see what's happening when you do "wrong" things, like using a 100x oil immersion objective with water. Or if you use strange objectives and setups.

For the look-through-the-objective method, you shouldn't measure the diameter of the lens, but the width (angle) of the image you see trough it.
Like, for your 10x 0.25, it should be possible to see the whole screen of the computer (say 14" diagonal) from about that distance (14"). The screen size is R* Sin(theta), the screen distance is R*Cos(theta). R is the distance objective-to-screen-corner, and disappears when you calculate the NA or theta. Theta = Inverse Tangent (Screen size / Distance). To understand better, make a drawing yourself, it always help in trigonometry. But in practice this method is difficult and tricky, as the image inside the objective is very small. To pick up and enlarge this image, you may use a camera with macro lens, or maybe the 4x objective as magnifying glass. Or one 10x eyepiece, reverted.

The "apertometer" method, if the distance was correct, should have worked.
The "20" likely means NA 0.20. If is possible to see a bit more beyond the "20" circle, then the aperture is a bit more than 0.20, likely 0.25. All fit!

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Re: Need Value for NA Formula

#9 Post by farnsy » Fri May 14, 2021 7:52 pm

Greg Howald wrote:
Fri May 07, 2021 12:31 am
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
NA is not sin times theta. Sin() is a function that takes as an argument an angle and gives back the ratio of the length of opposite side of a right triangle to the length of the hypotenuse of that triangle. The inverse function of sin() is arcsin(). If you give it the ratio, it will give you the angle. In this case, theta=arcsin(NA). That is what patta said and is the answer to the OP's question. On a scientific calculator, arcsin() is sometimes written as sin() with a -1 superscript. Make sure your calculator is set to degrees, not radians, if that is what you expect the answer to be expressed in.

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