In May 1940 the Aeronautique Militaire was grouped together with the Observer Service and Anti-Aircraft Artillery to form the Territorial Air Defense Corps (Defense Aeronautique du Territoire or DAT).
Believe to have formed part of SHAPE after WW2.
SHAPE = Supreme Headquarters Allied Powers Europe (Allied Command Operations), created 1951.
The Supreme Headquarters Allied Powers Europe (SHAPE) is the military headquarters of the North Atlantic Treaty Organization's (NATO) Allied Command Operations (ACO) that commands all NATO operations worldwide.
I do not have much information on this patch.