orbital contributions in the two-dimensional reciprocal space at ${\bf Q}_z=0$ (Fig. 2d). The intraorbital, but interband, scattering process between $\Gamma(0,0)\leftrightarrow M(1,0)$ shown in Fig. 2d favors the transversely lengthened vertices [@jhzhang10]. This momentum anisotropy is compatible with the experimentally observed elliptically shaped low-energy spin excitations in superconducting BaFe$_{2-x}T_x$As$_2$ [@jtpark; @clester; @hfli; @hqluo12] and spin waves in BaFe$_2$As$_2$ (Fig. 2g) [@lharriger]. Upon electron-doping to enlarge the electron pockets near $M(1,0)/(0,1)$ and shrink the hole pockets near $\Gamma(0,0)$, the mismatch between the electron and hole Fermi pockets becomes larger (Figs. 2e), resulting in a more transversely elongated ellipse in the low-energy magnetic response (Fig. 2h). Indeed, this is qualitatively consistent with the doping evolution of the low-energy spin excitations [@jtpark; @hqluo12; @msliu12].

For hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$, one should expect enlarged hole Fermi pockets near $\Gamma(0,0)$ and reduced electron pockets near $M(1,0)/(0,1)$, as shown in Fig. 2c. Based on first principles calculations, spin excitations for optimally hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ at $x=0.4$ should have longitudinally elongated ellipses [@jtpark], and gradually evolve into incommensurate magnetic scattering (elastic and/or inelastic) with increasing $x$ due to poor nesting between the hole and electron Fermi pockets [@castellan]. INS experiments on single crystal Ba$_{0.67}$K$_{0.33}$Fe$_2$As$_2$ [@chenglinzhang] indeed confirm that the low-energy spin 