I'll have a look and see what I can find, I have written about 1000 articles on underwater photography, so it is not always easy to find everything - and I have to say that a lot of these types of tests were just shared on the forums. Also I did most of this testing in the first years of shooting full frame digital underwater, which was 17 years ago now, which makes things tricky.
Before the Nauticam 140mm dome existed, I commissioned, the late, Ken Sullivan to make me such a dome for full frame shooting. It was an approximate 6” dome. I still have and use it. There is a passing reference to it in this article from 10 years ago:
https://wetpixel.com/articles/review-and-field-notes-subal-nd4/P3
I definitely did a lot of tests with it when I first started using it - to determine what aperture I was happy using it at. I remember concluding f/14 was pretty comparable to f/13 for image quality with a 230mm dome (on full frame). Which is the aperture I have used with it, ever since. And with the Nauticam 140mm, which pretty much has the same profile.
Oh, and I just found the review of the Nauticam 140mm dome from 2014:
https://wetpixel.com/articles/review-nauticam-140mm-dome-port-by-alex-mustard
which was part of the Nikon D750 review:
https://wetpixel.com/articles/field-review-nikon-d750-and-nauticam-na-d750/P1
I would just add that we had small domes, long before we had big domes. The bigger ones were created for a reason.
Alex
Thanks for the link to the article
The Sigma 15mm has a specific design. The lens is smaller than other fisheye from Canon or Nikon 8-15mm but the working distance is the same. Here is a link to the lens design
Note how the PD is set 16.34mm from the front of the lens while the distance from the image plane is I-P=78.89mm
Considering a working distance of 150mm and taking out I-P results in 71.11mm
This means the dome inner radius must be at least 71.11 otherwise the lens won't be able to focus right on the dome when housed.
Nauticam 140mm dome whose radius is 69mm just falls short of this specifications.
In addition a smaller dome like Zen DP100 (100mm on the outside but 94mm on the inside) will not work well with this lens which will still not be able to focus as close as required. The smaller dome will only result in less useable working range for the lens. Definitely not a good combination for this lens. I would also add not a good choice of minidome for the Nikon 16mm Canon 16mm fisheye all lenses that do not focus close enough to work well with small domes and really requires larger ports.
However the Canon 8-15mm or the Nikon 8-15mm are much larger lenses with the same working distance from the image plane of the Sigma 15mm also more expensive if I can add.
Looking at the canon and doing the same calculations of the sigma results in a minumum radius of just 39.55mm to focus right on the glass. I would therefore conclude that users of those 8-15mm lenses (nikon is pretty much the same) can actually use a very small dome and the camera when house will focus right on it. I am not suggesting this is the right thing to do however if you wanted to get very close you can do it with those lenses.
In terms of depth of field you need to take into accoun that the focus distance is taken from the image plane not from the centre of the dome. So if you consider the same example of the Canon 8-15mm you need to add 108mm to the virtual image plus the space inside the dome
Assuming Nauticam 140mm dome this is 108+69+(69*3)=384mm infinity
If you take a dome with a radius of 12cm (similar to the nauticam 230 wide angle port) this is 108+120+(120*3)=588mm infinity
None of those ports will have enough depth of field at f/22 to reach the infinity point when the target is touching the dome
Now some other theoretical calculations
Imagine a target at half meter with a 6" dome this will be around 12.5 cm from the dome surface however this means 30cm from the focal plane. At f/8 this already covers 0.43m which is larger than the infinity point
The same target at half meter with a 10" dome is actually 18cm from the surface however this means 40cm from the focal plane with the infinity point at 58cm. I can cover that range at f/6.3
As the target moves further away the requirements on depth of field drop.
So the larger dome results in a 2/3 stops increase in depth of field for a 1.66x size increase. Looking at weight for reference to the nauticam 140mm fisheye dome and the 230mm wide angle port the weight is double
Obviously if you are interested in the sharpness front to bottom larger is better and if your lens does not focus that close larger is mandatory
Generally all the calculations I am running under the assumption the dome is properly placed at the entrance pupil show that on fisheye lenses improvement in depth of field in stops is pretty much log2(radius1/radius2).
So a dome with 11cm radius will have 2/3 stops over one of 7cm radius. And the difference between 4.33" and 5 1/2" is 1/3 BUT you need to make sure your lens can focus. The Canon/Nikon 8-15mm will work with any size you like ultimately it becomes a compromise between size/cost/quality
I am still puzzled by the 230mm wide angle port being called at times a fisheye dome when it does not cover 180 degrees but that is another story
