How did Darth Vader discover Luke is his Son?

Darth Vader and Luke Skywalker Jedi Fight

At what point did Vader have the realization that the young Jedi was his son? It is widely known that Vader didn't know this in A New Hope. So, how did he come to know it in The Empire Strikes Back? According to the lore of the films, Queen Amidala was actually made to look pregnant at her funeral on Naboo. The idea is that the children died as well, thus keeping them safe from Darth Vader. However, in the Empire Strikes Back, the Emperor lets Darth Vader know that he does indeed have a son. According to the expanded universe around these films, in this particular case a comic book, Luke, Han, and Leia, after the battle of Yavin, find themselves inside an Imperial weapons factory. It is here that Luke Skywalker first battles Vader. In the battle, Vader sees his old lightsaber and wants to know more about this person who attacked him. He sends Boba Fett to figure this out, and that is when he goes to Tatooine and comes back with the name Skywalker. Again, this is not technically part of the film world, though it is considered canon. If a Darth Vader spin-off movie was made, it certain could become more clear, and well known to those who don't follow the comics. In fact, there are aspects of this shared universe that could finally come to light and possibly make all the films in the Star Wars saga more coherent.

Darth Vader's Sith Lord Apprenticeship Needs to Be Explored.

Darth Vader's Sith Lord Apprenticeship

We know Darth Vader is a badass. We know that he can hold up his hand and choke someone from distances much further away than physics says they can. His skills with a lightsaber never waver. Even when he seems like he's hurt or troubled in a battle, it is all part of a plan. As if Lord Vader wants his opponents to feel a false sense of bravado. How did he get this way? Did Vader have any problems with his combat training? As he started off following the ways of the Force, did he ever have any issues with what he was learning? Was his "sacrifice" of his past life as Anakin Skywalker enough to show his dedication? How did he get along with the other students who were also undergoing the same training? What kinds of relationships did Darth Vader form? Did he even have to undergo this training considering how the Emperor essentially fast tracked him to his current position? Again, this is a part of Darth Vader's lore that could serve to truly humanize him if it were put on the big screen in a standalone movie. It also might bolster the credibility of Revenge of the Sith if, in some way, a spin-off movie about Darth Vader played as a companion piece to the third film in the trilogy.