  Charge order in underdoped and optimally doped high-$T_\mathrm{c}$
superconductors Bi$_{2}$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$ (Bi2201) is
investigated by Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS). We
have directly observed charge density modulation in the optimally doped Bi2201
at momentum transfer $Q_{\|} \simeq 0.23$ rlu, with smaller intensity and
correlation length with respect to the underdoped sample. This demonstrates
that short-range charge order in Bi2201 persists up to optimal doping, as in
other hole-doped cuprates. We explored the nodal (diagonal) direction and found
no charge order peak, confirming that charge order modulates only along the
Cu-O bond directions. We measured the out-of-plane dependence of charge order,
finding a flat response and no maxima at half integer \emph{L} values. This
suggests there is no out-of-plane phase correlation in single layer Bi2201, at
variance from YBa$_2$Cu$_3$O$_{6+x}$ and La$_{2-x}$(Ba,Sr)$_x$CuO$_4$.
Combining our results with data from the literature we assess that charge order
in Bi2201 exists in a large doping range across the phase diagram, i.e. $0.07
\lesssim p \lesssim 0.16$, demonstrating thereby that it is intimately
entangled with the antiferromagnetic background, the pseudogap and
superconductivity.
