Fig. 2: Cloaking with conformal maps.
a Scheme of a conformal map in action space near the branch point. The original \({{{\mathcal{F}}}}_{-y}^{{{\mathcal{A}}}}\)-fence running over the cut is split into two branches near the branch point after the map is applied and travels around the yellow-cloaked region. Due to the branch point and the finite radius of curvature of the border at the branch point, there are two red regions adjacent to the branch point where the local rotation of the pattern \(\left\vert \arg [dc(z)/du(z)]\right\vert \, > \, \pi /4\) is too large such that fence points in control space corresponding to this region are no longer encircled by the control loop. If one wants to use the blue \({{{\mathcal{F}}}}_{+y}^{{{\mathcal{A}}}}\)-fence lines for transport, this sets a lower bound κ > κc for the curvature of the cloak border at the branch point. b Conformally mapped and analytically continued (yellow) cloak pattern according to map c()(z) together with some of the conformally mapped fence lines \({{{\mathcal{F}}}}_{\pm \! x,\pm \! y}^{{{\mathcal{A}}}}\) and gates of the original pattern. The cloak shape works for arbitrary large sizes. c Control space \({{\mathcal{C}}}\) of the cloaked pattern in b) with the rotated fence points \({{{\mathcal{F}}}}_{\pm \! x,\pm \! y}^{{{\mathcal{C}}}}\) outside the cloak. Fence points of the four different fence families are well separated from each other, allowing the control loop to wind around one family without excluding any family member nor including any member of a different fence family. Supplementary video 2 shows the opening of the boat-shaped cloak and the associated broadening of the fence points in control space as we increase the cloak size R.
