A Star Wars Question for Troy

I've been watching the movies to prep for watching Force Awakens for the first time, and I've got a question: How did Vader know he was Luke's father?  I know the last name is one obvious way, but there could be other Skywalkers in the galaxy, right?  As far as Anakin knew when he became Vader, the babies died with Padmé.  Is there anything in the lore that explains this further?