Michael, thank you for the additional information and insight about Frédéric's setup. Clearly it is a huge advantage to speak all three languages: English, French, and Optics.
As I see this, the key concept is that changing the physical length of an optical segment has only a small effect on focus and magnification, if the affected segment is near infinity focus.
Let me see if I can illustrate, using fewer optical elements.
What I've set up here is a nearly traditional infinity setup using a Mitutoyo 5X objective (40 mm) with a 125 mm Raynox DCR-250 placed close behind it.
The setup would be exactly traditional, except that I've defocused the Raynox so that it gives magnification 0.2 when focused by itself on a nearby object.
Then, moving backward the camera and Raynox as a unit, I reach this situation.
If the Raynox were focused at infinity, this move would have no effect on focus. But because the Raynox is focused someplace else, the move does have some effect.
Using the focus block to measure how much the focus has shifted, I find that a 165 mm movement of the rear assembly (camera + Raynox) has shifted the focus point by only 0.75 mm.
This corresponds to a finite difference of delta w / delta e = 0.75/165 = 0.0045 .
I have not checked the derivation of your formulas, but I do notice that my measured value is pretty close to the 0.0041 that I calculate from your formulas given the same inputs. A slight deviation from m2 = 0.2 could easily account for that, and I did not take care to get the magnification exact.
From this it is seen that setting the second stage to a small magnification yields very small stacking steps on the object side.
Yes. In fact at high magnification the steps may become so small as to be troublesome. Before running the numbers, the first setup that I tried was using a 20X objective (10 mm) and a 200 mm lens instead of the 125 mm. What I observed experimentally was that in my setup focus shift then was dominated by flex in the mechanics, as the weight of the camera shifted backward. I was surprised by that, but when I run your calculation, the result I get is that dw/de = 0.0001, which means that my 165 mm of rear movement would be expected to accomplish only 16.5 microns of focus shift. Given that an appropriate step size for a 20X NA 0.42 objective is around 2 or 3 microns, that would be a rather short stack!
If i look ath the illustrations i wonder whether it is necessary to combine the componon with a tube lens
Even with the infinite objective, I question whether the tube lens is appropriate in these setups. You may have noticed that mine doesn't have one. That was not just to make it simpler. In order for the setup to work, the output from the front optics has to be diverging at a rate that matches the focus of the rear lens. Used by itself, the objective has to be dragged away from its design point to accomplish that. If I added the tube lens, as in Frédéric's setup, the objective would have to be dragged even farther away from its design point. It's a case of "too many parts".
While I am impressed by the ingenuity of this focusing method, I think it's important for other readers to realize that the main advantage of this setup -- perhaps its only advantage -- is that it allows making small focus steps with low precision mechanics.
Compared to the traditional approach of using a microscope focus block or a precision screw drive to simply change subject-to-lens distance, this method has some non-trivial drawbacks. I expect that Frédéric has thoroughly mastered the technique with his own equipment, but as illustrated by Soldevilla's experience (and my own, for that matter!), other people attempting to adopt and adapt this technique may find themselves running into unexpected difficulties.
As an editorial opinion, I think this point is particularly important because the very high quality of Frédéric's mineral images may lead other people to spend time trying to make his focusing methods work for them, when their time would be far better spent on studying his methods for illumination and staging.