Introduction to Equilibrated Space-Time-Varying Electromagnetic Systems Sajjad Taravati and Ahmed A. Kishk Department of Electrical and Computer Engineering Concordia University Montreal, Quebec, Canada Sajjad.taravati@concordia.ca Abstract—This study introduces the equilibrated spatiotemporally-varying media as a new class of highly efficient nonreciprocal electromagnetic systems with zero space-time local reflections, zero photonic band gap and enhanced linear nonreciprocity. We show that equilibrium in the electric and magnetic properties of space-time-modulated media brings about enhanced nonreciprocity accompanied with larger sonic regime interval, in comparison with the conventional space-time permittivity-modulated media. This provides extra design freedom to achieve strong nonreciprocity by a weak pumping strength. Moreover, it is demonstrated that the width of photonic band gaps in periodic space-time permittivity- and permeabilitymodulated media is proportional to the absolute difference between the electric and magnetic pumping strengths. space-time modulated medium are indicated by magenta circles. However, as we switch on the magnetic permeabilitymodulation with the same pumping strength ( δε = δμ = 0.5), as a result of the equilibrium in the electric and magnetic properties of the medium, photonic band gaps disappear. In addition, Fig. 1 demonstrates the enhanced nonreciprocity, Δβ/Δβ+, of equilibrated space-time modulated medium. Keywords—Space-time; nonreciprocity; photonic band gap; refractive-index I. INTRODUCTION Conventional periodic space-time permittivity-modulated medium was first introduced as a travelling wave parametric amplification and recently is utilized for nonreciprocal wave transmission [1]-[5]. This topic has acquired a surge of scientific interest thanks to the peculiar way of Lorentz reciprocity breaking in such medium. Conventional spacetime permittivity-modulated medium is endowed with nonreciprocal periodic photonic transitions, whereas they suffer from local space-time impedance mismatch, moderate nonreciprocity, and periodic photonic band gaps. This study first describes the fundamental limitations of the conventional space-time permittivity-modulated medium, and then, introduces and characterizes the equilibrated space-time permittivity- and permeability-modulated medium. The equilibrium is introduced between electric and magnetic properties of the medium by identical periodic spatiotemporal modulation of the electric permittivity and magnetic permeability of the medium. II. SPACE-TIME PERMITTIVITY- AND PERMEABILITYMODULATED SYSTEM To best investigate the effect of equilibrium in electric and magnetic properties of spatiotemporally modulated medium, we may analyze their dispersion diagrams [4]. Figure 1 compares the analytical solution for the dispersion diagrams of the conventional space-time permittivity-modulated medium, with δε = 0.5 and δμ =0, with the equilibrated space-time permittivity- and permeability-modulated medium, i.e. δε = δμ =0.5. In the figure, the photonic band gaps of the conventional Fig. 1. Comparison of dispersion diagrams of conventional spacetime-modulated medium, i.e. δε = 0.5 and δμ = 0 [4], with equilibrated space-time-modulated medium, i.e. δε = δμ = 0.5. References [1] Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics, vol. 3, pp. 91 – 94, Jan. 2009. [2] S. Taravati and C. Caloz, “Space-time modulated nonreciprocal mixing, amplifying and scanning leaky-wave antenna system,” in IEEE AP-S Int. Antennas Propagat. (APS), (Vancouver, Canada), 2015. [3] S. Taravati and C. Caloz, “Mixer-duplexer-antenna leakywave system based on periodic space-time modulation,” IEEE Trans. Antennas Propagat., vol. 65, pp. 442– 452, Feb. 2017. [4] S. Taravati, N. Chamanara, and C. Caloz, “Nonreciprocal electromagnetic scattering from a periodically space-time modulated slab and application to a quasisonic isolator,” Phys. Rev. B, vol. 96, p. 165144, Oct. 2017. [5] S. Taravati, “Self-biased broadband magnet-free linear isolator based on one-way space-time coherency,” Phys. Rev. B, vol. 96, p. 235150, Dec. 2017.
