Well, as far as I can gather, Cortana used the last of her energy inside the Didact's ship to provide a sort of "safe" area for the Chief after he detonated the nuke and defeated the Didact. It would appear that this safe area was probably the light-bridge/Cortana herself (you saw how Cortana could manipulate it to allow herself and her mood-spikes to come out of it) and Cortana had just made a kind of room out of what was left of its/her energy to save the Master Chief in one final effort. As the room collapsed around them, it signalled Cortana's death - but probably not her last appearance, at least, Halo fans all hope not. The "data field" as you put it disappeared and the Master Chief was picked up by a squad of ODSTs in a Pelican. If you like it, you can get halo 4 helmet amazon  also.

That was her using the last of her energy to do the one thing she had always wanted to do, but was physically incapable of. Another attempt of her trying to do this was visible in "Origins Part II", an episode in Halo: Legends, in about 2556, when Master Chief was still in cryo.

Hope this sort of helped! As you can gather, not many people (besides 343) probably really know what happened, and the programmers have yet to explain themselves.

Source(s):

Personal experience from playing the game myself, collecting master chief helmet and being an avid Halo fan!