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Assume that you go to sleep at 0:00 am and wake up at 3:00 am.
Why do you feel more tired (compared to your tiredness at 23:59) when you wake up at 3:00 am? Why does sleep makes us more tired in this case? Is there a critical sleeping period for a person, below which makes him/her more tired instead. Also it is confusing that day time sleep doesn't make the same effect.
Sleep proceeds in cycles of REM and NREM, usually four or five of them per night, the order normally being N1 → N2 → N3 → N2 → REM. There is a greater amount of deep sleep (stage N3) earlier in the night, while the proportion of REM sleep increases in the two cycles just before natural awakening. (Wikipedia)
Sleep follows a certain cycle which repeats throughout the night. Being forced to wake up when in the deep sleep stage (N3) usually leads to you being more tired for the rest of the day (although usually it only last for an hour at most). This is known as sleep inertia.
Day time short nap doesn't work the same because you woke up before you reach the deep sleep stage. This kind of short nap is called power napping.
Sleep cycle is not considered controversial among people who studied sleep, and is backed by EEG scans which shows that the different stages have different brain wave pattern.
Some relevant researches showing sleep inertia:
http://www.ncbi.nlm.nih.gov/pubmed/21167917
Sleep inertia (SI) denotes a period of hypovigilance, confusion and impaired cognitive and behavioral performance that immediately follows awakening. Based on the observation that the reactivation of some cortical areas is faster than other upon awakening, here we examined regional differences between presleep and postsleep waking period.
http://www.ncbi.nlm.nih.gov/pubmed/22734577
Twenty-four healthy young men (Protocol 1: n = 12, mean age = 25.1 yrs; Protocol 2: n = 12, mean age = 23.2 yrs) were provided with nap opportunities of 20-, 40-, and 60-min (and a control condition of no nap) ending at 02:00 h after ∼20 h of wakefulness (Protocol 1 [P1]: simulated night work) or ending at 12:00 h after ∼30 h of wakefulness (Protocol 2 [P2]: simulated extended operations). A 6-min test battery, including the Karolinska Sleepiness Scale (KSS) and the 4-min 2-Back Working Memory Task (WMT), was repeated every 15 min the first hour after waking. [...] WMT [Working Memory Task] performance was impaired (slower reaction time, fewer correct responses, and increased omissions) on the first test post-nap, primarily after a 40- or 60-min nap
http://www.ncbi.nlm.nih.gov/pubmed/21075238
The benefits of brief (5-15 min) naps are almost immediate after the nap and last a limited period (1-3h). Longer naps (> 30 min) can produce impairment from sleep inertia for a short period after waking but then produce improved cognitive performance for a longer period (up to many hours).
http://www.ncbi.nlm.nih.gov/pubmed/20636214
However, one potential disadvantage of napping is that awakening from naps is disturbed by sleep inertia, which has also been found to impair performance and/or mood, transiently. The authors examined the effects of the timing and length of a night-shift nap on sleep inertia in a laboratory setting.
