No idea, I said "experiment", so it was not for photography at all. I was not trying to mislead anyone, it was obviously applicable to my experiment only.
I would not be allowed to show anything. It is very basic, one can say it was identical but due to the nature of the objective, a hotspot was induced. The hotspot was not present without the beamsplitter. When a relay lens is focused to infinity, light rays are parallel, no hotspot should occur, right? That is why it was quite puzzling, but then, the objective lens was not some kind of ordinary thing designed as an infinity objective. You can think of it more like some kind of lens coupling system, or mounting a finite objective to a relay lens. Altering the distance between the objective and the relay lens would alter its focus. The objective must be placed at a specific distance away from the relay lens, otherwise it will not generate an image. The relay lens images what the objective lens projects. It is called "Binocular Indirect Ophthalmoscopy (BIO)" if you are interested in seeing photos of the retina. In our case, the binoculars was replaced with a detector.
My best guess was that the hotspot was a combination of ceiling lights, and both the beamsplitter and the objective lens itself, which was placed in front of a lens focused to infinity. Using a cube beamsplitter, the hotspots were at least minimalised. With the lights turn off, lots of the glare was gone. I chose to house the optical train in light bafflers because turning off the lights violated safety codes
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Polarisers were an issue because light was scarce, however, the scarcity of light was due to the boost-buck converter being faulty. It ended up being too bright even with polarisers. Installing an iris also eliminated much of the glare, however, one central hotspot remained. My solution was to simply take two images, one with the objective lens tilted, which changed the location of the hotspot. Now, blend the two together. The supervisors did not seem to like this solution.
We did not use a beamsplitter cube because of its size. We needed a beamsplitter that was around 60mmx60mm, a beamsplitter cube of that size would be forbiddingly expensive.
Well, none of this really matters in photomacrography. I do have an arsenal of beamsplitters, but I would likely not have any time to produce a sound comparison/experiment until longer holidays. What is there to compare anyway, image quality maybe? It is hard since my cube based one is a DIY setup and plate based one is pre-made into the microscope I use. My DIY job is likely going to underdeliver. I have yet to acquire the parts necessary for Köhler illumination.
I have used both plate and cube beamsplitters with infinity systems and I have not seen hotspot/glare/ghosting issues within the objective's accepted projection circle on a full-frame camera. Outside of it, sure, degradation is huge, more exaggerated, but still none of these issues unless at low magnifications (<4x), such as 2x, which makes what you have shown here very useful.