Tuning of a nonlinear metamaterial band gap by an external magnetic field

We consider tuning of permeability of a nonlinear metamaterial with resonant conductive elements by an external varying magnetic field. We propose two tuning regimes: slow tuning, when the permeability follows adiabatically the field variations, and high-frequency tuning, when changes of the pump microwave amplitude and/or frequency affect the permeability. We demonstrate how the metamaterial band gap can be tuned and describe the resulting metamaterial switching between transmitting, reflecting, and absorbing states. The details appear to depend drastically on the type of nonlinear components inserted into the resonant conductive elements. We perform practical estimates for typical diodes available for the microwave frequency range. We predict that the transmittance of a metamaterial slab can be modulated by several orders of magnitude already using a slab with thickness equal to one microwave wavelength in vacuum.