Hello. I’m wondering if someone could shed some light on how to calculate the objective back focal plane diameter.

I’m using an Olympus IX81, 60x NA 1.2 water lens. I assume the tube lens focal length is 180mm. If mag = tube lens focal length / objective focal length, I derive the objective f = 180mm/60 = 3mm.

For small angle, N.A = n*half objective pupil diameter/objective f. So, objective pupil diameter, D = 2 * N.A. * objective f / n. D = 5.4mm. Can I assume the objective back focal plane diameter equals D, 5.4mm?

I am not sure if you want to know how to obtain the BFP diameter or, specifically, how to calculate it.
Re calculations, bear in mind that the diameter of the visible front aperture of your lens may not all be used for imaging - there may be redundant glass there due to the practicalities of optical lens manufacture and mounting and there may be apertures (‘diaphragms’) built into the objective to limit light passage through only the ‘reliable’ optical diameter of a lens according to optical design specs.
For these reasons If I wanted to obtain the BFP size I would directly measure it rather than try to calculate it from measurements of visible glass surfaces because direct measurement of the BFP circumvents this (and other) unknowns that are due to the practicalities of design and manufacture.
It would be interesting to hear what others think about that.

Thanks for your thoughts. I’d also appreciate it if you could share how to measure the BFP diameter. I was going to compare the calculated BFP diameter with the measured one. The way I would measure it is to use a point source to shine from the sample side and put a vertical plate in the lens back, align all three on the lens axial axis. Then, measure the spot diameter. Would this work?

For calculating the D, I didn’t physically measure the diameter of the lens front glass part. Instead, I used N.A = n*half D/f objective. Mag = f tube lens/ f objective. This can circumvent the point you mentioned, considering other optical elements in the objective that restricting light passing into the objective. I’m not sure if I can equal this calculated D (effective objective diameter) to the BFP diameter. Any thoughts?

A point source won’t focus at the BFP. A collimated plane wavefront source will.
Without special equipment I would do it like this:
If you have a camera that can zoom in to the back of the lens with variable focus, simply point the lens to a distant high contrast object (like white clouds in a blue sky - but no sun!) and focus the camera on the image of that distant object. Your camera will then be focussed on the BFP. Keep that focus fixed then take your objective lens with the camera still attached, put a cover slip on it with its immersion medium between the lens and coverslip and - keeping everything vertical so the coverslip doesn’t slide off - point this assembly close to an extended diffuse source (like a light box but just a sheet of paper on a desk might suffice if your camera is sensitive enough) close up till it fills the field of view (i.e. the light disc in the BFP doesn’t get any bigger as you move closer). That white disc you see at the back of the objective with this setup is the full available BFP. For such a high NA lens the object you focus on in the first step above doesn’t need to be really distant like the clouds - some far away buildings or trees may suffice - because the focal depth is very deep in the BFP of high NA lenses (in contrast to the depth of focus for a specimen in the image plane - which is very narrow).
To calibrate this (for measurement of BFP diameter in mm) there are a number of ways but one way is to remove the camera from the system and place a ruler or graticule at about the same distance from it as the the back of the objective was from the camera in your in situ setup and adjust that distance slightly till the graticule/ruler is in focus. Obviously keep the camera lens focus fixed at the same position it was in when you were measuring the BFP light disc - don’t focus using the camera lens.
There are other ways but you will need a bit more specialist equipment. Take care not to scratch the front lens of your objective if using the method above.
Also, the above assumes you want the actual mm diameter of the BFP disc - not the angular diameter or NA. That would require a different setup (traditionally an Abbe apertometer but there are other ways).

I got the idea of using camera to focus onto the BFP deep inside the objective. But I’m confused about the following question. Since the objective is an infinity objective, the light beam that exits the objective should be infinity parallel beam. The full BFP light disc should be preserved along the optical axis. Can I just put a plate behind the objective and measure the light disc diameter? It should be the same as the full BFP diameter regardless of how far away (in theory) it is from the BFP.

… assuming the point source that the objective is imaging is infinitesimally small, and focused precisely in the front focal plane, and sufficiently bright to see it. Which are hard to achieve in practice

Beams and wave fronts are two different things. You can have a diverging or converging beam composed entirely of parallel wavefronts. The wavefronts exiting an infinity objective will be parallel (plane) for every point across the luminous field of view in the specimen plane when the objective is focussed on that plane. But, as @talley alluded to, the beam formed by those parallel wavefronts will only be parallel if you only consider one single dimensionless point - and that is assuming theoretical perfection and a self-lumious dimensionless point (which in practice is quite difficult). But I don’t see how achieving that will help you measure the BFP because if you focus on a single parallel wavefront - you see a point!

The equation for BFP diameter is D = 2 * NA * EFL. The diameter of the BFP is 7.2mm for an Olympus 60x/1.2.

Whether the actual internal aperture is exactly this is a separate question. But my wager is that it is pretty close; if it were drastically different then the advertised NA would be different.

The refractive index comes into the calculation of the collection cone on the sample side, but the BFP size doesn’t depend on it.

Conveniently Olympus gives as a specification the axial position of the BFP, in this case 19.1mm inside the mounting flange (just under 26mm from the sample-side focal plane). Many Olympus objectives have this BFP distance so that Kohler illumination is preserved when changing between objectives.

If you place a point source in the sample-side focal plane with angular spread equal to the collection angle and use the correct immersion medium – this immersion medium is part of the optical design of the objective – then on the camera side of the objective you should observe a collimated beam of light with diameter matching the BFP diameter. If you run this in reverse and input a collimated beam, the objective should focus the beam to a diffraction-limited spot.

Lateral displacement of the point source from the center position will change the angle of the output collimated beam with the focal length being the proportionality constant. A well-corrected lens transforms positions at one focal plane to angles at the other, and vice versa. This is the basis of Fourier optics but is very convenient way of tracing rays out by hand and understanding conceptually what the lens is doing to the light.

@JonD . Thanks for sharing. Is the Fourier optics example the same principle for TIRF illumination?

Can you also share the source of BFP diameter, 7.2mm, of Olympus 60x 1.2? I used a similar equation but my calculation involves immersion media, water in this case. That’s why I got 7.2/1.33 = 5.4mm. Why is BFP diameter immersion media independent? If one uses wrong immersion media, wouldn’t it cause the effective BFP diameter to change?

Thanks @P_Tadrous and @talley . I think you are right I’m confused about the parallel beam and wavefront. Is the output of an infinity objective, parallel beam or parallel wavefront? I was thinking it’s emitting parallel beam so that one can insert optical elements in the infinity space for advanced applications.

The BFP diameter equation does not include refractive index: 2 * NA * EFL. For your example 2 * 1.2 * 180mm/60 = 7.2mm.

Remember that the immersion medium is part of the design of the objective lens and if you don’t use it (or deviate significantly from it) then the optical performance probably will be degraded significantly. Especially for high-NA objectives.

If you did use a different immersion medium (say you tried to use a water objective in air) then I believe that the EFL would change as well.

Another way of thinking about this is that there is an internal aperture stop that doesn’t change size with external circumstances. I admit that there is a subtle difference between the aperture stop and the back focal plane but they are closely related.

I’ll be doing a video on this subject in the near future with illustrated examples so you might want to subscribe to my YT channel! Direct answer to your question: It is the wavefronts that are plane (i.e. straight and parallel) - the beam is divergent unless you are only imaging a single dimensionless point. Because the wavefronts are plane, you can intercept them at any point up the tube - the trouble is, unless the angle is straight up (specimen is a point source dead centre of the lens) you will intercept less and less of all off-axis wavefronts for a fixed diameter optical tube the further away you go from the BFP of the objective and image quality will degrade. So ‘infinity optics’ is convenient for not requiring intermediate optics for each attachment, but it has its limitations too. Infinity objectives are not absolutely better in all respects to finite tube length objectives. Another ‘bad’ point for infinity objectives (from my personal point of view) is that it gave license to all the commercial microscope manufacturers to deviate and diverge from a unified standard - but that is more to do with commercial business decisions and marketing than physics. I would really like to see an open source standard for infinity-corrected microscopes emerge in the near future (if it hasn’t already - let me know please).

BTW I liked your video on spherical aberrations - you explained this issue very clearly.
I am currently putting out some C Programming tutorials because I want my audience to be able to understand the code in my future videos on optics simulations, Abbe’s diffraction theory and image processing - because up till now I’ve been explaining microscopy in terms of ye olde fashionde ray optics - but those are nearly all out now.