  Using the public data from the Herschel wide field surveys, we study the
far-infrared properties of optical-selected quasars from the Sloan Digital Sky
Survey. Within the common area of $\sim 172~deg^2$, we have identified the
far-infrared counterparts for 354 quasars, among which 134 are highly secure
detections in the Herschel $250~\mu m$ band (signal-to-noise ratios $\geq5$).
This sample is the largest far-infrared quasar sample of its kind, and spans a
wide redshift range of $0.14{\leq}z\leq 4.7$. Their far-infrared spectral
energy distributions, which are due to the cold dust components within the host
galaxies, are consistent with being heated by active star formation. In most
cases ($\gtrsim80$\%), their total infrared luminosities as inferred from only
their far-infrared emissions ($L_{IR}^{(cd)}$) already exceed
$10^{12}~L_{\odot}$, and thus these objects qualify as ultra-luminous infrared
galaxies. There is no correlation between $L_{IR}^{(cd)}$ and the absolute
magnitudes, the black hole masses or the X-ray luminosities of the quasars,
which further support that their far-infrared emissions are not due to their
active galactic nuclei. A large fraction of these objects
($\gtrsim50\text{--}60\%$) have star formation rates $\gtrsim
300~M_{\odot}yr^{-1}$. Such extreme starbursts among optical quasars, however,
is only a few per cent. This fraction varies with redshift, and peaks at around
$z\approx2$. Among the entire sample, 136 objects have secure estimates of
their cold-dust temperatures ($T$), and we find that there is a dramatic
increasing trend of $T$ with increasing $L_{IR}^{(cd)}$. We interpret this
trend as the envelope of the general distribution of infrared galaxies on the
($T$, $L_{IR}^{(cd)}$) plane.
