Prof. Sat is not popular with/among his students.
Which usage is correct here and why?
"Popular among" and "popular with" are both correct. In this case, I would suggest "popular with".
"Popular with" is usually used for people, e.g. "Popular with the girls", "Popular with the working class..."
"Popular among" is usually used with ideas among people, eg. "The new policy is not going to popular among the members of the party...."
The phrase "popular with" just generally seems to be more popular with people regardless whether it's the popularity of people or things.
Compare the ngram for "it is popular among" vs "it is popular with":
To that of "he is popular among" vs "he is popular with":