We already know one way of saying “after” in Japanese: by using the te form.

コーヒーをんで、仕事しごとはじめました。
I drank coffee and then started work.

If we want to emphasis that A happened after B then we can add から to the te form of the first clause.

コーヒーをんでから仕事しごとはじめました。
After I drank coffee I started work.

We can also use the following structure.

【Prior Action】あと【で or に】【Subsequent Action】

The verb in the prior action must be in the standard simple past tense. This gives rise to the same “issue” we saw when using “before”; namely, that the tense of the first clause is not dependent on the time of the action. This means need to form sentences with literal translations like, “After I drank tea I will phone my friend” to get the correct grammar in Japanese.

ちゃんだあとで、友達ともだち電話でんわします。
After I drink tea I will phone my friend.
試験しけんわったあとに、受験者じゅけんしゃ同時どうじ部屋へやを出ました。
After the examination finished the candidates left the room at the same time.

To be clear, the following is incorrect.

ちゃあとで、友達ともだち電話でんわします。()

We use the possessive particle to connection 後 to nouns.

かいあと、すぐにかえってきてました。
I came back home and slept straight after the drinks.

あとで vs 〜あと

This is actually quite a frustrating one to explain because in the many cases both are natural. Used on its own to mean “later” 後で is typically used.

あと電話でんわします。
あと電話でんわします。()
I’ll call you later.

Generally speaking, で is used to signify the simple chronology of events. に is used to emphasis that the succeeding clause is a natural result consequence of the preceding.

べたあとで、友達ともだち電話でんわする。
食べた後に、友達に電話でんわする。()
After I’ve eaten I will call my friend.
あめったあとで、れた。()
あめったあとに、れた。
After rain fell the weather cleared.

あとで vs 〜てから

Compared with あとで, ~てから places importance on the following three things:

1: Order, i.e. that B must happen after A.
日程にってい確認かくにんしてから、ホテルを予約よやくする。
日程にってい確認かくにんしたあとで、ホテルを予約よやくする。()
Book the hotel after you’ve confirmed the itinerary.

Clearly the fact that you will book the hotel after checking the itinerary is not a coincidence; order is important and so ~てから is the more natural sentence.

2: Intention of the prior action.
けんかしてから、あやまる。()
けんかしたあとで、あやまる。
Apologise after an argument.

The first sentence suggests that you intended to have the argument and then apologise; whereas the second is more natural because it simply states that the apology happened after the argument without insinuating any prior intention (with is presumably the case).

3: Continuity of the final action.
結婚けっこんしてから、ずっと東京とうきょうんでいる。
結婚けっこんしたあとで、ずっと東京とうきょうんでいる。()
Since we got married we’ve always been living in Tokyo.
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