After a really lot of googling, I have found some info. I don't know the part number of the optointerrupter I'm using, but 3 randomly picked datasheets were for ones working at 940 nm. Thanks to the info on this page, I know that this wavelength falls onto the onset of eye's opacity. That shouldn't be a problem, but shifting to 850 nm is desirable. It seems that most eye-exploring istruments work at 850 nm.
Second is that the red-eye peak seems to be small enough to be completely obscured by the changes in reflections off lenses and white portions of the eye caused by movement of the detector. So, I probably have to get rid of them somehow... Maybe, doing some calculations will help to figure out by how much I have to suppress unwanted reflections.