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Mathematical formulas for N/A and f aperture value.

 
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harisA



Joined: 03 Jul 2011
Posts: 387
Location: Greece

PostPosted: Thu May 26, 2016 10:32 am    Post subject: Mathematical formulas for N/A and f aperture value. Reply with quote

1)Which is the formula that gives the f number of an objective of known N/A and focal distance.How this formula changes for a combination of an objective and a tube lens of known focal distance and aperture.

2)What is the best way to reduce the n/a of an objective) (a)close the aperture of the condenser,b) put an iris at the back of the objectives.
What is the difference of these methods?

Thank you in advance for your help.
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rjlittlefield
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Joined: 01 Aug 2006
Posts: 17390
Location: Richland, Washington State, USA

PostPosted: Thu May 26, 2016 5:16 pm    Post subject: Reply with quote

At the camera, the effective f-number of any objective that is used in its correct focus arrangement is:

f_eff = magnification / (2*NA)

For example an NA 0.25 objective at 10X has an effective f-number of f_eff = 10/(2*0.25) = f/20.

If you put a nominal 10X NA 0.25 objective with a short tube lens so as to give 5X, then its effective f-number becomes f_eff = 5/(2*0.25) = f/10.

------

Putting an iris at the back of the objective will block light on all paths, exactly the same as if you used a smaller NA objective.

Closing the aperture of the condenser will block the edges of the objective from seeing direct light, but light diffracted or diffused by the subject can still go through the entire aperture.

For a highly transparent subject they are about the same. For a dense subject that causes the light to spread out a lot, closing the condenser aperture may have little effect except to reduce the light intensity.

--Rik
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harisA



Joined: 03 Jul 2011
Posts: 387
Location: Greece

PostPosted: Fri May 27, 2016 3:16 am    Post subject: Reply with quote

Rik thank you so much for your fast answer.

Kind Regards

Haris
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Hokan



Joined: 06 Apr 2007
Posts: 57
Location: Sacramento, CA, USA

PostPosted: Sat May 28, 2016 11:53 am    Post subject: Converting effective f number to NA? Reply with quote

So would the same formula, (solving for NA rather then effective f number), be the generally accepted formula to use?

Hokan
_________________
Nikon SMZ-U Stereomicroscope, SOM, (Son Of Multiphot), a DIY macro/micro rig.w/120, 65, 35, 19mm Macro-Nikkors, 28mm f/1.8 Ultra-Micro-Nikkor, 55mm f/1.2 CRT Nikkor, and far too many other lenses and objectives to list.
Zeiss Axiomat, incident BF, DF, DIC, Pol, Phase.
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rjlittlefield
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Joined: 01 Aug 2006
Posts: 17390
Location: Richland, Washington State, USA

PostPosted: Sat May 28, 2016 12:13 pm    Post subject: Reply with quote

Hokan, I don't understand your question. What problem are you trying to solve? What is the application? If you had a formula, how would you use it?

--Rik
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Hokan



Joined: 06 Apr 2007
Posts: 57
Location: Sacramento, CA, USA

PostPosted: Sat May 28, 2016 2:15 pm    Post subject: Formula for f number Reply with quote

Hi Rik,

Back in my earlier days of photomicrography when I was suffering from NAS, (in remission now), I acquired a Nikon Ultra Micro Nikkor 30mm, effective aperture, according to Nikon, of f/1.2, and magnification 1/25X. Now that I have my DIY macro rig up and running I plan on reversing the lens and seeing how well the lens works.

The lens has a overall working distance of 810mm, object area of 50mm, image area of 2mm when used as designed. So my inspiration is to reverse the lens and do some test step photographs, (with Zerene of course). So I was just curious how the lens's specifications compared to a good 25X Nikon microscope objective.

In 1964 Nikon was claiming a 1,250 lines/mm resolution through out the entire 2 mm image area. Using a 546 nm, green, wavelength.

Hope I answered your question.

Regards,

Hokan
_________________
Nikon SMZ-U Stereomicroscope, SOM, (Son Of Multiphot), a DIY macro/micro rig.w/120, 65, 35, 19mm Macro-Nikkors, 28mm f/1.8 Ultra-Micro-Nikkor, 55mm f/1.2 CRT Nikkor, and far too many other lenses and objectives to list.
Zeiss Axiomat, incident BF, DF, DIC, Pol, Phase.
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rjlittlefield
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Joined: 01 Aug 2006
Posts: 17390
Location: Richland, Washington State, USA

PostPosted: Sat May 28, 2016 3:06 pm    Post subject: Reply with quote

Ah, got it! Thank you for the excellent and detailed description.

For your application, a better formula is the one that relates effective f-number and NA on the same side of the lens:

Quote:
NA = 1/(2*f_eff)

By this formula, a lens that is effective f/1.2 on the camera side is also NA 0.4167 on the camera side.

When you reverse it, and run at the complementary magnification, the lens will be NA 0.4167 on the subject side, very reasonable for a 25X objective.

Effective f-number on the camera side in that setup will then be 25/(2*0.4167) = f/30.

Effective f-numbers on each side of the lens are always related simply by the magnification, so it is no coincidence that 30 = 25*1.2 .
(f_eff_camera = magnification * f_eff_subject)

The claim of 1,250 lines/mm can be cross-checked against the MTF formulas quoted at http://www.photomacrography.net/forum/viewtopic.php?p=124831#124831.

Using those formulas with lambda = 0.000564 mm, I calculate that cutoff frequency nu0 = 1526 lp/mm, and MTF(1250 lp/mm) = 0.09 .

So, the resolution they're quoting is consistent with 9% MTF with perfect optics. That would definitely be visible with a high contrast target, so that looks like a reasonable spec.

--Rik
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Hokan



Joined: 06 Apr 2007
Posts: 57
Location: Sacramento, CA, USA

PostPosted: Sat May 28, 2016 9:58 pm    Post subject: Formula for f number Reply with quote

Rik,
I very much appreciate your quick response and understandable explanations.

I have written your formulas into my studio journal for future use.

Your ability to explain how your formulas can be used while at the same time showing how they are related to the physics of optics and light is most, how shall I say; illuminating! ???? Laughing

With kind regards,
Hokan
_________________
Nikon SMZ-U Stereomicroscope, SOM, (Son Of Multiphot), a DIY macro/micro rig.w/120, 65, 35, 19mm Macro-Nikkors, 28mm f/1.8 Ultra-Micro-Nikkor, 55mm f/1.2 CRT Nikkor, and far too many other lenses and objectives to list.
Zeiss Axiomat, incident BF, DF, DIC, Pol, Phase.
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rjlittlefield
Site Admin


Joined: 01 Aug 2006
Posts: 17390
Location: Richland, Washington State, USA

PostPosted: Sat May 28, 2016 10:28 pm    Post subject: Reply with quote

And I greatly appreciate your kind words and engaging pun!

--Rik
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