After can be omitted from a modifying clause. However, the earlier time is then expressed in the form of the verb—a past participle. Past Nonfinite.
1- After being burglarized, Anne is very cautious.
Having been burglarized, Anne is very cautious.
2- After closing the windows, Anne locked the front door.
Having closed the windows, Anne locked the front door.
3- After setting the alarm, Anne locked the front door.
Having set the alarm, Anne locked the front door..
Participle clauses are a form of adverbial clause which enables us to say information in a more economical way. We can use participle clauses when the participle and the verb in the main clause have the same subject. For example:
Waiting for John, I made some tea.
Waiting for John, the kettle boiled. [This would suggest that the kettle was waiting for John!]
Participle clauses can be formed with the present participle (-ing form of the verb) or past participle(third form of the verb). Participle clauses with past participles have a passive meaning:
Shouting loudly, Peter walked home. [Peter was shouting]
Shouted at loudly, Peter walked home. [Someone was shouting at Peter]
If we wish to emphasise that one action was before another then we can use a perfect participle(having + past participle):
Having won the match, Susan jumped for joy.
Having been told the bad news, Susan sat down and cried.