I often am but go back to bed between 0400 and 0500. My REM sleep is about sun up. I hate that.
I wish I was more of a Lark and less of an Owl. It just isn’t going to work out that way though. I have a friend who is 90 and he is still an Owl. His perfect day is to get up about 0730 or 0830, start working at 0900, work until about 1400, lunch, go to about 1800 or 1900, dinner, then work again until about 2100 to midnight or 0100.
It basically means after midnight but before daybreak. In English we have no word for it. We refer to it as in the "wee wee hours" or "small hours" of the morning.
My dad used to tell us kids that that was when you wanted to be at home and off the streets because all of the drunks, criminals and weirdoes were out and about between midnight and dawn. Good advice, Dad.