Scattering matrices being used incorrectly

We found that the vast majority of the literature using scattering matrices in semi-analytical methods (i.e. method of lines, rigorous coupled-wave analysis, eigen-mode expansion, mode matching, etc.) are defining the four terms S11, S12, S21, and S22 incorrectly. It has been convention for many decades that S11 is reflection and S21 is transmission, yet most of these papers use S11 as transmission and S21 as a reflection quantity. Why perfectly correct mathematically, it is very confusing and not consistent with convention.

We recently published a paper on this topic and also contributed some new concepts for implementing scattering matrices more easily and more efficiently. The paper is open access and freely available:

https://www.jpier.org/PIERB/pier.php?paper=11083107

Let me know your thoughts.