$n=2$ deflection is $\hat{f}_2^{({\rm Ca})}=\kappa^2/4$ in the limit of $\kappa\rightarrow 0$. Considering the higher-order effect with respect to $\kappa$, [@mag2003] derived $\hat{f}_n^{({\rm Ca})}=\frac{1}{4}\kappa^2$ $\{1+\frac{3}{8}\kappa$ $(1+\frac{3}{8}\kappa+\frac{73}{64}$ $\kappa^2)\}$. For $\kappa < \sim 0.7$, such a higher-order $\kappa$ correction is responsible for the enhancement of the $n=2$ deflection from the leading-order one as seen in the better agreement with the measured and simulated deflections. However, the correction is not sufficient for the narrower gap $\kappa> \sim 0.7$, and thus the theoretical underestimation of the $n=2$ deflection becomes more serious with $\kappa$. Further, the theory does not cover the considerable increase in the higher-order $n\geq 3$ deflections with $\kappa$ as demonstrated by both the measurement and the simulation.

The modal deflection $\hat{f}_n^{({\rm Ca})}$ is linked to the migration velocity and force as decomposed into $$V_{B3}^{({\rm Ca})}=\sum_{n=2}^\infty V_{B3,n}^{({\rm Ca})},\ \ 
F_M^{({\rm Ca})} = \sum_{n=2}^\infty F_{M ,n}^{({\rm Ca})},$$ which are $$\frac{V_{B3,n}^{({\rm Ca})}}{V_{B1}}=\frac{F_{M ,n}^{({\rm Ca})}}{F_{DC}},
\label{eq:vb3nca_fm}$$ $$F_{M ,n}^{({\rm Ca})}=
\oint_{S_B}\!\!\!\!\!{\rm d}^2{\bm x}\ 
{\cal L}\left(
\hat{f}_n^{({\rm Ca})}P_n^1(\cos\theta)\cos\phi
\right).$$

Figure \[fig:modal\](a) shows the contribution of the modal deflection to the migration velocity for the modes $n=2$ and $n=3$. For small $\kappa$, the simulation result is consistent with the analytical solution ([@mag2003]), which considers only the $n=2$ deformation to cause 