  Zero-index metamaterials (ZIMs) feature a uniform electromagnetic mode over a
large area in arbitrary shapes, enabling many applications including
high-transmission supercouplers with arbitrary shapes, direction-independent
phase matching for nonlinear optics, and collective emission of many quantum
emitters. However, most ZIMs reported till date are passive, with no method for
the dynamic modulation of their electromagnetic properties. Here, we design and
fabricate a magnetically tunable ZIM consisting of yttrium iron garnet (YIG)
pillars sandwiched between two copper clad laminates in the microwave regime.
By harnessing the Cotton-Mouton effect of YIG, the metamaterial was
successfully toggled between gapless and bandgap states, leading to a "phase
transition" between a zero-index phase and a single negative phase of the
metamaterial. Using an S-shaped ZIM supercoupler, we experimentally
demonstrated a tunable supercoupling state with a low intrinsic loss of 0.95 dB
and a high extinction ratio of up to 30.63 dB at 9 GHz. Our work enables
dynamic modulation of the electromagnetic characteristics of ZIMs, enabling
various applications in tunable linear, nonlinear, quantum and nonreciprocal
electromagnetic devices.
