ta-form + 後 「あと」 ＝ after ～

A ta-form precedes あと. The meanings of あとで、　あとに and あと　are similar.

a) 　ご飯を食べた後で、勉強した。 [I studied after I had a meal.]
ｂ）　ご飯を食べた後に、勉強した。 [I studied just after I had a meal.]
ｃ）　ご飯を食べた後、勉強した。  [I studied after I had a meal.]

(a), (b) and (c) have almost the same meanings; however, there are slight differences. The time to study in (a) can be the unspecified time (the time span is wide) after a meal, but the time to study in  (b) must be the specified one (the time span is narrow) after it.

Therefore, ご飯を食べた後で、しばらく気分が悪かった。 [I felt sick for a while after I had a meal.] can be possible, but you cannot say ご飯を食べた後に、しばらく気分が悪かった。 [I felt sick for a while just after I had a meal.]. (c) does not have any usage limitations like (a) or (b), but (a) and (b) are more often used than (c) in daily communication.