We report the first results of a systematic investigation to characterize blazar variability power spectral densities (PSDs) at optical frequencies using densely sampled (5--15 minutes integration time), high photometric accuracy ($lesssim$0.2--0.5%) R-band intranight light curves, covering timescales ranging from several hours to $sim$15,minutes. Our sample consists of 14 optically bright blazars, including nine BL Lacertae objects (BL Lacs) and five flat-spectrum radio quasars (FSRQs) which have shown statistically significant variability during 29 monitoring sessions. We model the intranight PSDs as simple power--laws and derive the best-fit slope along with uncertainty using the `power spectral response method. Our main results are the following: (1) on 19 out of 29 monitoring sessions, the intranight PSDs show an acceptable fit to simple power-laws at the rejection confidence $leq$ 90%; (2) for these 19 instances, the PSD slopes show a large range from 1.4 to 4.0, consistent with statistical characters of red (slope$sim$2) and black (slope$geq$3) noise stochastic processes; (3) the average PSD slopes for the BL Lacs and FSRQs are indistinguishable from one another; (4) the normalization of intranight PSDs for individual blazar sources which were monitored on more than one occasion turns out to be consistent with one another with a few exceptions. The average PSD slope, 2.9$pm$0.3 (1$sigma$ uncertainty) is steeper than the red-noise type character of variability found on longer timescales (many decades to days), indicative of a cutoff in the variability spectrum on timescales around a few days at the synchrotron frequencies of the emission spectrum.
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