@3Roomlab,
I can only comment on my method of selecting a close-up lens.... no maths needed
As I previously stated in another thread, I first came across the use of a biconvex close-up lens when repairing video presenters (document cameras) a couple of years ago. The cameras were standard Sony modules that could not focus close enough for the document imaging applications. The manufacturer solved this by fitting a removable No.2 Close-up Lens. The Focal Length (FL) of that lens was the perfect distance for the camera to focus correctly on a document 40cm away. This was effectively providing the camera with compensation for its long sightedness.... a bit like fitting reading glasses to it ! I tried that as well.....a set of cheap reading glasses had the same effect as the No.2 CU lens.
I used that example to track down suitable lenses for my Fire fighting Thermal cameras. These are fixed focus and designed for distance working of more than 2m. I found ZnSe and Gallium Arsenide lenses would work at the required thermal camera wavelength, with ZnSe more common, and with higher transmittance figures. Fortunately CO2 laser cutting machines use such ZnSe lenses with nice short focal lengths....perfect for close up work on PCB's. At $30 delivered they were affordable so I bought some to play with. On arrival I placed a 20mm diameter 100mm FL ZnSE lens in front of my PM695 cameras lens and noted the excellent performance. I then tried it in front of the ARGUS 2 fire fighters camera and its performed equally as well. I found that the closer to the objective lens that you could place it, the better. You have to also consider the dimensions and FOV of the objective. If the ZnSe lens is moved too far away from the objective, the FOV will exceed the lens diameter with negative effects on the image produced.
When I bought my E4 I just placed my 100mm FL ZnSe lens in front of the E4 lens and all worked well. I shared the news with this forum and I remain impressed with the excellent lens holders that have been designed by clever people and now the aluminium design that we see here.... Superb ! I have been using 19mm and 20mm diameter lenses to ensure I didn't have FOV issues but as has been shown in this thread, a 12mm lens is possible and far more compact that my lens holders.
There may be some maths that may be applied to the use of these lenses but I just did it by test. The No.2 lens on the document camera was the same diameter as the objective and around 5mm from it, so Sony seemed to think closer is better
Aurora