I don't think it's that important that your alarm clock for an REM period. REM, with the exception of stage 1, is the lighest sleep you have (see below). If allowed to wake without any external stimuli (i.e. alarm clock) you are most likely to wake directly from REM. I don't know about you, but I usually feel the most rested when I am allowed to wake without the alarm clock. So waking from REM might not be so good.



FILD requires you to be really tired because you need to fall back asleep quickly. It's tricky because if your alarm clock is loud enough to wake you from stage 3 or 4 (these according to my psych. book are the most difficult to wake from) it might be loud enough to startle you further awake. BUT, If you aren't startled by it you are likely to be at your sleepyest when woken from stage 4 sleep. You are also likely to be disoriented and maybe forget to FILD.

My advice would be:
If your alarm scares you awake, keep it low and let it wake you
If you alarm doesn't startle you, turn it up and make it wake you, you are more likely to be tired enough to FILD quickly this way.

experimentation is probably the best way, but still—this might be worth a try.