The following is information related to the question but doesn't fully answer it. There's a bit of a confusion on where the "blue dome" feature originated. Here's the answer from a production viewpoint, as near as I can figure it.
The blue dome feature on the NX-01 governor appears to be a visual "nod" to a similar feature on the Motion Picture Enterprise and related vehicles. Andrew Probert, the final designer for TMP Enterprise, called it the "deflector crystal" labelling it so here:
Basically he thought that the impulse drive on the refit was powered by the matter antimatter reactor which, from it's name "intermix shaft" and swirly lights along its hole length, you can get the idea worked very differently from a TMP warp core.
Here's a Trekyards interview with Probert to hear his original thinking.
Recall that he did all this BEFORE Sternbach and Okuda (with some knowledge of real world fusion rocket concepts like DAEDALUS) assigned the Galaxy-class' system of fusion rockets that later fandom assumes is the "one true way" to do impulse drives.
What the crystal did was redirect matter antimatter intermix exhaust out the back of the impulse drives for thrust. The idea that the matter/antimatter intermix needed to be working right for impulse flight is hinted at on screen by Scotty's line in TMP when they're about to leave dock:
"Intermix set, Bridge, impulse power at your discretion."
So that's another wrinkle in the tech canon for you to mull over. As a real world space propulsion engineer having multiple ways of doing the "same thing" adds verisimilitude for me...but your mileage may vary. I find a lot of fans I know dislike "discontinuities" like this.
For those interested the linear intermix shaft always reminded me of a certain resemblance to a linear laser confinement fusion device picture in Chen et al:
I found this idea here: