shift has no local maximum.

   $W$($\mu$m)   $W/\lambda$   $x_{m,{\rm cr}}$   $x_{m,f}$   $\delta_{r,f}$   $x_{m,b}$   $\delta_{r,b}$   $x_{m,{\rm md}}$   $W_{r,{\rm md}}/W$
  ------------- ------------- ------------------ ----------- ---------------- ----------- ---------------- ------------------ --------------------
    $\infty$      $\infty$         0.08648         0.08654        4214.2        0.08642      $-7956.8$             –                   –
       320         505.53          0.08647         0.08717        494.72        0.08581      $-527.20$          0.08647             1.72054
       160         252.76          0.08646         0.08788        258.86        0.08517      $-255.52$          0.08647             1.72957
       80          126.38          0.08640         0.08939        135.19        0.08396      $-122.74$          0.08647             1.73259
       40           63.19          0.08617         0.09286        72.20         0.08173       $-57.48$          0.08648             1.73621
       20           31.6           0.08530         0.10287        40.69         0.07790       $-25.45$          0.08652             1.74811
       10           15.8           0.08246            –             –           0.07196       $-10.10$          0.08683             1.78945

  : \[tab:tab1\] Summary of the numerical results for varying beam widths $W$. $x_{m,{\rm cr}}$ is the normalized critical thickness of the metal layer. $x_{m,f}$ and $x_{m,b}$ are the normalized thicknesses for the maximal forward and backward GH shifts, while $\delta_{r,f}$ and $\delta_{r,b}$ are the corresponding maximal forward and backward GH shifts normalized by $\lambda$. $x_{m,{\rm md}}$ is the normalized thickness corresponding to the maximum distortion, namely, the maximum $W_r$. $W_{r,{\rm md}}/W$ is the normalized value of the maximum distortion.

Our result can be understood from the theory of the GH shift in multilayered structures, where the sign of the GH shift is determined by the competition between intrinsic damping and radiative 