Deep petrosal nerve
The deep petrosal nerve transmits post-ganglionic sympathetic fibers from the internal carotid plexus to the Vidian nerve on its way to the pterygopalatine ganglion.
Gross anatomy
The nerve begins at the internal carotid plexus and runs alongside the lateral aspect of the internal carotid artery through the carotid canal. It pierces the fascia which covers the foramen lacerum before joining the greater petrosal nerve to form the Vidian nerve.
The Vidian nerve then passes through the pterygoid canal on its way to synapse with the pterygopalatine ganglion. The fibers which came from the deep petrosal nerve will go on from here to provide sympathetic innervation to the vasculature of the nasal and oral mucosa.