Nonreciprocal Phase Gradient Metasurface: Principle and Transistor Implementation
Guillaume Lavigne Christophe Caloz



















 









§ 
[hang]
  1ex


 §.§ 
[hang]
  1ex













fancy
 
 







=0.4cm






    _#1 #1øωεμ∇·̇× Re Imł#1#̊1(<ref>)
    outdir=./
    
    
    
    #1
       #1 
    
    
    Nonreciprocal Phase Gradient Metasurface: Principle and Transistor Implementation
        G. Lavigne^1 and C. Caloz^1
        
    =================================================================================
    
    fancy
    
    
    
    ^1Polytechnique Montréal, Department of Electrical Engineering, Blvd. Edouard-Montpetit, H3T 1J4, Montréal, Canada  
    
    guillaume.lavigne@polymtl.ca
     
    
    Abstract  – We introduce the concept of nonreciprocal nongyrotropic phase gradient metasurfaces. Such metasurfaces are based on bianisotropic phase shifting unit cells, with the required nonreciprocal and nongyrotropic characteristics. Moreover, we present a transistor-based implementation of a nonreciprocal phase shifting subwavelength unit cell. Finally, we demonstrate the concept with a simulation of a 6-port spatial circulator application. 
    
    
    
    § INTRODUCTION
    
    
    Metasurfaces have been shown to enable a myriad of wave transformations <cit.>. The vast majority of the works reported so far in this area have focused on reciprocal structures. Integrating of nonreciprocity <cit.> in metasurfaces inherently bears potential for even more sophisticated and useful metasurface wave transformations. A few nonreciprocal metasurfaces have already been recently reported, based either on time-varying <cit.> or on transistor-loaded structures <cit.>.
    
    We introduce here the concept of nonreciprocal nongyrotropic phase gradient metasurfaces based on nonreciprocal phase shifting unit cells. Moreover, we propose a transistor-based implementation of such a structure, and demonstrate it in a 6-port spatial circulator.
    
    
    
    
    § OPERATION PRINCIPLE
    
    
    The proposed metasurface is based on nonreciprocal phase shifting particles arranged in such a way as to exhibit a phase gradient for wave deflection in space, as illustrated in fig:Concepts (a). In the case of this figure, the phase gradient has been set only in one direction of propagation, so that the wave trajectory is deflected in that direction, while being unaltered in the opposite direction.
    
    
    
    
    
    § NONRECIPROCAL PHASE SHIFTING METASURFACE
    
    
    For simplicity, we assume here nongyrotropy, and we will report on the gyrotropic case elsewhere. Except for this restriction, we allow the metasurface to be bianisotropic, which leads to the tangential susceptiblity-based generalized sheet transition conditions (GSTCs) <cit.>
        ẑ×Δ𝐇 = j ωϵχ_ee𝐄_av +  j k χ_em𝐇_av ,
    
    
        Δ𝐄×ẑ   = j k χ_me𝐄_av + j ωμχ_mm𝐇_av,
    
    where the symbol Δ and the subscript 'av' represent the differences and averages of the tangential electric or magnetic fields at both sides of the metasurface, and χ_ee, χ_mm, χ_em, χ_me are the bianisotropic susceptibility tensors characterizing the metasurface.
    
     Figure <ref>(b) plots the FDFD-simulated <cit.> fields propagating across a uniform nonreciprocal phase shifting metasurface with phase difference Δϕ = 180^∘, corresponding to a spatial gyrator.  The corresponding susceptiblities for an x-polarized incident wave are obtained from (<ref>) as
    
    
        χ_ee^xx = tan(ϕ_12)+tan(ϕ_22)/ηωϵ,   χ_mm^yy = η(tan(ϕ_12)+tan(ϕ_22))μω, 
           χ_em^xy = tan(ϕ_12)-tan(ϕ_22)k,   χ_me^yx = tan(ϕ_12)-tan(ϕ_22)k,
    
    where ϕ_1  and ϕ_2 correspond to the phase shifts in the opposite directions of propagation.
    
    Figure <ref>(c) plots the full-wave simulated scattering parameters for a transistor-based implementation of a uniform nonreciprocal phase shifting metasurface. The unit cell of this metasurface is composed of three metallic layers supported by two Rogers3003 dielectric substrates and has a period of ∼λ_0 /4.5 at the operating frequency. A transistor (modeled by an ideal isolator) connects the first and third metallic layers with the second layer, which is acting as the ground. The unit cell provides a 90^∘ phase difference between the forward and backward transmissions with around -3 dB transmission. Optimization work is still required for higher performance.
    
    
    
    
    § NUMERICAL DEMONSTRATION WITH A SPATIAL CIRCULATOR
    
    
    Figure <ref> presents FDFD simulation results for the nonreciprocal phase gradient metasurface of fig:Concepts(a) for a phase gradient period of √(2)λ_0 corresponding to the 45^∘ deflection of a normally incident wave. In this simulation setup, 6 ports are placed in the directions corresponding to the 6 propagating diffraction orders of the supercell period. This metasurface operates as a 6-port spatial circulator with circulation following the sequence 1 → 4 → 3 → 5 → 2 → 6 → 1. This only represents an example; other devices can be realized by appropriately designing the forward and backward phase gradients, and even manipulating the transmission and reflection magnitudes.
    
    
    
    Another promising application of the proposed nonreciprocal phase gradient metasurfaces is the realization of antennas with nonreciprocal radiation patterns <cit.> . For instance, an antenna placed at the port 5 in fig:circulator would transmit straightforwardly towards port 2, but receive from port 3 at 45^∘.
    
    While the transistor-based implementation demonstrated in fig:Concepts(b) is currently restricted to microwave regime, it may soon extend to the optical regime following recent advances in the development of optical transistors <cit.>. In the meanwhile, the proposed nonreciprocal phase gradient concept can be readily applied at optical frequencies using alternative nonreciprocal technologies such as for instance space-time modulation <cit.>.
    
    
    
    
    § CONCLUSION
    
    
    We have presented the concept of nonreciprocal phase gradient metasurfaces based on nonreciprocal phase shifting unit cells, proposed a related transistor-based implementation, and demonstrated the concept in a 6-port spatial circulator device. We have also pointed out that this technology has a potential for a diversity of other applications.
    
    ieeetr
