Two and Three Dimensional Incommensurate Modulation in Optimally-Doped   Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
  X-ray scattering measurements on optimally-doped single crystal samples of the high temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ reveal the presence of three distinct incommensurate charge modulations, each involving a roughly fivefold increase in the unit cell dimension along the {\bf b}-direction. The strongest scattering comes from the well known (H, K$\pm$ 0.21, L) modulation and its harmonics. However, we also observe broad diffraction which peak up at the L values complementary to those which characterize the known modulated structure. These diffraction features correspond to correlation lengths of roughly a unit cell dimension, $\xi_c$$\sim$20 $\AA$ in the {\bf c} direction, and of $\xi_b$$\sim$ 185 $\AA$ parallel to the incommensurate wavevector. We interpret these features as arising from three dimensional incommensurate domains and the interfaces between them, respectively. In addition we investigate the recently discovered incommensuate modulations which peak up at (1/2, K$\pm$ 0.21, L) and related wavevectors. Here we explicitly study the L-dependence of this scattering and see that these charge modulations are two dimensional in nature with weak correlations on the scale of a bilayer thickness, and that they correspond to short range, isotropic correlation lengths within the basal plane. We relate these new incommensurate modulations to the electronic nanostructure observed in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ using STM topography. 
J.P. Castellan, B.D. Gaulin, H.A. Dabkowska, A. Nabialek, G. Gu, Z.   Islam, X. Liu, and S.K. Sinha