OK, so then try this explanation...

There is nothing very special about a microscope objective. It has a short focal length, and it's optimized for best image quality over a small field, but other than that, "it's just a lens".

So if we know the objective's F-number when considered as an ordinary lens, then we can use the tables for ordinary lenses.

This is where things get a little tricky. The aperture of an objective is specified in terms of its "numerical aperture" (NA), and we have to convert that to some equivalent F-number.

You have already found the calculator at

http://extreme-macro.co.uk/microscope-o ... calculator , which purports to convert between NA and effective F-number. But what you don't realize, and what I do not see written on the page, is that the calculator works only when the objective is used

*as intended*. So yes, a 4X NA 0.10 objective is effective f/20 --

*when the objective is used at 4X*. But when you change the tube length to get less magnification, that calculator will no longer give you the correct answer when used in the obvious way.

Instead, what we can do is to take the calculator's answer

*in the correct case*, f/20, think of the objective as "just a lens", and calculate what the nominal F-number of that lens must be.

I assume you know that for ordinary lenses, focused only by extension, the usual formula is that effective F-number = nominal F-number * (magnification+1). (There are some other assumptions in behind that formula, but for this discussion I will ignore those.) So then, we know that 20 = nominal F-number * (4+1), and then a little algebra tells us that nominal F-number = 4. In other words, the microscope objective is just an ordinary f/4 lens, extended to give 4X magnification, and thus effective f/20.

Now that we know what the objective looks like when considered as an ordinary lens, we can calculate using the formulas for ordinary lenses.

One such calculator is

http://extreme-macro.co.uk/focus-stacking/#calculator , but I do not like that one for the reasons explained earlier, to wit, it gives numbers that will produce focus banding in the final image.

A better calculation is the formula that produces Table 2-A at

https://zerenesystems.com/cms/stacker/d ... romicrodof . So to take your example, if we now plug 2X and f/4 into that calculation, the answer that comes out is 0.0792 mm.

All of this is the exact same approach that I wrote earlier, just with more words to explain the rationale.

You wrote:

According to the "NA to Effective f-stop calculator" in extreme-macro.co.uk the Effective f-stop = 10

Yes, but that's a misuse of the calculator. At 2X, your 4X-NA0.1-finite-on-short-tube will actually have effective f-stop = 4*(2+1) = f/12.

Nominal f-number = Effective f-stop / (1 + Magnification) so: Nominal f-stop = 10/(4+1) = 2

That equation works when correctly applied. But you have misapplied it in two different ways. First, you have used the wrong effective f-stop, which you got from misusing the "NA to Effective f-stop calculator". Second, even using the correct effective f-stop, you have to plug in the corresponding magnification. If you had plugged in f/10 (not quite right) and 2X, then you would have gotten 10/(2+1) = 3.33, which is not quite right by the same amount. If you had some way to find the right value, effective f/12, and plugged in 2x instead of 4x, you would have gotten 12/(2+1) = 4. But there are no calculators that would have given the correct value f/12, so we have to go about the whole process as I explained above.

I found on the net the equation:

One of my great frustrations in life is the large number of times when somebody reaches into the literature, grabs a formula that may or may not apply, plugs in some numbers that may or may not be appropriate, does some arithmetic that may or not be correct, ends up getting a number that does not correspond to reality, and then very reasonably asks "what did I do wrong". Often the first thought that crosses my mind is "Where do I start?"

This is not intended to be a personal criticism. I can totally understand how you got to where you did get, and I consider this conversation to be just an apparently necessary part of the learning process. I wish I knew how to write some explanation that would quickly and concisely give a newcomer enough understanding to get everything right by themselves. But I'm not there yet, and to be honest I don't feel a lot closer than I did 5 years ago. Thank you for your patience.

--Rik