Can I have full screen in the desktop app actually go as-fullscreen-as-possible?

I'm running the desktop app on OS X, and the phone I'm streaming from is a Galaxy S6. Full-screening the desktop app, either through the 4 arrows button or the green window button (standard with OS X) makes the window go full screen, but doesn't actually expand its contents.

Since I'm doing a public presentation with the app, this would be useful.

I uploaded another image to make it clear that the rectangle in the centre is the phone screen itself.
(Yes, I'm using this to demo stuff with GearVR Apart from the full-screen issue, it's the best solution I've found to far. )