I am hoping to get help with some optical calculations, for DIY lathe work to modify a darkfield (microscopy) funnel stop. I will gladly compensate the time required for such calculations. Thank you!
Basically, the darkfield funnel stop works similar to an exit pupil after all lenses in an objective and it stops down the numeric aperture of an objective to produce bright subject image on black background. I would like to know what funnel opening size we should use, for darkfield NA 1.0, 0.92 and 0.83.
The objective I want to work on is a LOMO water immersion apo 65x NA 1.1, which has no factory iris for darkfield (DF). Its NA 1.1 works, for some tiny diatoms in DF, with a NA 1.2/1.4 oil darkfield condenser, but that NA is too high for many thick subjects (I would get halos).
I currently have a DF funnel stop, made by AO Spencer for a different apo objective (90x NA 1.4), that fits my LOMO objective and stops it down to around NA 0.7 (I have done careful diatom resolution test to come up with that NA value, its DF resolution is definitely less than NA 0.75 and slightly better than NA 0.65). But I want to increase the stopped-down NA back up to NA 0.83, 0.92 and 1.0, by opening up the current aperture.
The DF funnel stop that I have is rare (I may not be able to obtain another one). My machinist friend, who has no microscope to check DF, is located in another state. So I would like to get accurate numbers as to how many millimeters we should make the funnel opening be (that way, we do not have to go back and forth).
Pictures of the metal funnel is attached herein, along with a sketch. Its current opening size is 3.05 mm (for stopping down NA 1.1 to NA 0.7).
Here is some data that may help optical calculations:
The 65x NA 1.1 objective has 160 mm tube length (focal length is 160/65 = around 2.46 mm). It is 33 mm long (from thread to tip). The DF funnel tip sits 15 mm below the thread, while last lens sits 20-21 mm below thread (ideally they should be very close, like the other iris objective that I have). The objective has a rear diaphragm of around 7.8 mm diameter, sitting at 4.7 mm above thread (23.7 mm above the last lens). There is nothing else between the rear diaphragm and last lens.
Please refer to the attached drawing for dimensions of the funnel stop. The narrow end of the DF funnel stop has inner aperture opening of 3.05 mm in diameter. The wider end has opening of 10.3 mm in diameter.
Our idea of modification is the open up the funnel stop from NA 0.7 (3.05 mm) to NA 1.0 (what ?mm). Then use a muzzle cap to stop it further down to NA 0.92 (?mm) and 0.83 ?mm). The wide end of the muzzle cap will snap onto the original Funnel's narrow end. The narrow end of the muzzle cap will be about 2 mm away from the rear lens.
If cost is too much, we will only make one muzzle cap and pick NA 0.87 as stopped-down NA. Again, muzzle cap end will be 2 mm from rear lens and we would like to know what diameter should we use there.
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And here is some extra details and explanations that may NOT be needed for optical calculations:
I have a water immersion achromat objective with iris 30x NA 0.65-0.9. Its iris also works as an exit pupil. For this one, the NA vs aperture opening diameter relation does not seem linearly proportional. Iris setting at NA 0.65 uses an aperture of about 18 mm diameter, while NA 0.9 uses aperture of around 86 mm. Focal length of this 30x objective is around 5.33mm and it is again 33 mm long from thread to tip. Its iris sits at 13.2 mm below thread, while last lens sits at 13.6 mm below thread.
The darkfield condenser that I use is a Bausch and Lomb paraboloid oil condenser, with light cone NA of 1.2-1.4.
I do not know the math for calculating exit pupil diameter vs numeric aperture / f stop number. Not sure if the relation is linearly proportional. I looked at Wikipedia's NA vs f number and related formulas, but they are based on entrance pupil (not exit pupil): https://en.m.wikipedia.org/wiki/Numerical_aperture
I am not aware of any other affordable DF solution for water immersion objectives. There are oil immersion iris objectives, but I do not want those. Alternative option is to use a LOMO 85x NA 1.0 achromat water immersion objective that costs around $80, but I prefer to use the apo objective that I already have.
My machinist friend will use a precision manual lathe with collet holder, which allows concentricity within about 0.01mm. He will use some small reamers and drills for the work.
Thank you again!
