I got bored so I decided to give these weapons some lore to help them fit a bit more.

The Institute Defender: The Institute Defender was created when the institute found a crashed vertibird that came from the west, it was NCR. With the dead inhabitants came their weapons. Intact was an advanced looking plasma pistol, which they brought back to the Institute to reverse engineer, they also cleaned up the wreckage looking for more tech. Nothing else survived. They gave the plasma defender they found to Kellogg, but he dumped it somewhere in Fort Hagen, as he was too attached to the 44 magnum his mother had given him.

The Beam Rifle: The Beam Rifle is the Institute's own version of the Combat Rifle. Meant to pack a punch for more dangerous targets, like the Brotherhood. The tech is based off of the AER, but still has a slower rate of fire than the AER. Bored scientists in Advanced Systems decided to convert the gun into a nitrogen synthesizer, so rather than creating a fun gimmick, they made a deadly weapon.

The Tesla Cannon: Created in anticipation for the brotherhood. The disappearance of Dr. Zimmer, the worry was up that the Brotherhood caught onto the Institute, and knowing the Brotherhood's obsession with hoarding technology, they had to act. This created the Tesla Cannon, a prototype heavy weapon that could shoot bolts of lightning from the power of a fusion core. One Tesla Cannon was lost during a field test in the commonwealth. Some say it was passed around until it made it's way to The Capital Wasteland, right into the hands of a Brotherhood Outcast Sentinel.

The Sonic Emitter: The Sonic Emitter is a big mystery. While the tech is pretty much stolen from The Big MT, no one knows how the Institute made it to Big MT. All people know is that it was stolen and reverse engineered as a last minute effort against Maxon's Brotherhood's arrival to the Commonwealth. Making a strong Anti-Power Armor weapon. One malfunctioning synth was given a sonic emitter and sent on a mission into the waste, the synth had gone rogue and killed the other synths in the squad, ran away, and found some raiders that modified the sonic emitter to it's max, causing the projectiles to explode, and if you're lucky enough to survive the explosion your ears end up bleeding. The gun was painted in blood from all it killed, including some poor dude that tried to stop the sonic emitter by putting his hand on the speaker, in summary, that guy doesn't have a hand anymore. The rogue synth was soon captured, scrapped, the raider gang slaughtered, and the modified sonic emitter used as a warning to check all synths before missions.