I work for an airline that allows controlled rest in the flight deck. Per our SOP it is for 40 minutes so as not to induce sleep inertia. Honestly though, between the time getting ready to rest and the time afterwards to get back into the picture, it takes around an hour total time.
It's a fantastic tool to avoid serious fatigue in our flying. We often fly night turn-arounds with 12 hours duty starting at 8pm or long haul flights of 7-8 hours starting around the same time. On those flights the controlled rest is very necessary. Never once had a passenger voice a concern over it. We take pillows, blankets, eye shades, and ear plugs into the cockpit for this. The FAs at our airline are trained to call us every half hour if we don't call first. That prevents the two pilot asleep problem. Not to mention the aircraft we fly has an advisory and caution system if no pilot inputs are made for more than 20 minutes.
In over ten years of domestic flying in the USA I never felt the need to nap. Luckily the majority of my flying, well 99.9% of it, was between 6am and midnight. For long haul international without relief crews it's an absolute must. For any red-eye flying it should be allowed.
The FAA and the U.S. airlines really need to get onboard with the science of fatigue managment for both this issue and flight and duty time limitations.
Typhoonpilot