Attack of the Blinth

I finally saw Revenge of the Sith today. Not much to say about it; it wasn't a totally terrible conclusion to the franchise (although Darth Vader's final line is almost unforgivably bad). Lots of eye candy, and some neat glimpses into various worlds and civilizations in the Star Wars universe. And it wrapped up most of the loose ends of the story, with some nice connections to Episode IV. (Although they didn't show Yoda landing on Dagobah, unless I missed it.)

One thing I hadn't noticed but read about on Wikipedia is that Obi-Wan picks up Anakin's lightsaber after fighting him, which he eventually gives to Luke in Episode IV. Is this how Darth Vader figures out that Luke is his son, because he recognizes his lightsaber in the duel in Episode V? I couldn't find any other references to when exactly Vader figures out Luke's identity.