Study of the Angular Spectrum of a Bianisotropic Refractive Metasurface at a Dielectric Interface
Guillaume Lavigne Karim Achouri Christophe Caloz
physics.optics





Study of the Angular Spectrum of a Bianisotropic Refractive Metasurface at a Dielectric Interface
    Guillaume Lavigne and Christophe Caloz
Department of Electrical Engineering
 Polytechnique Montréal

 Montréal, Québec, Canada

Karim Achouri
Nanophotonics and Metrology Laboratory 
 École Polytechnique Fédérale de Lausanne

 Lausanne, Vaud, Switzerland



    January 19, 2018
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We present an initial study of the angular spectrum of a bianisotropic refractive metasurface at an interface between two dielectric media. In this study, we report on the existence of three distinct angular regions: a) a rotated transmission cone, b) a modified total internal reflection region, and c) a new total retro-reflection region.





§ INTRODUCTION


Metasurfaces have recently received considerable interest due to their unprecedented ability to control electromagnetic fields and resulting virtually unlimited applications <cit.>.

One of the most fundamental operations enabled by metasurfaces is generalized refraction <cit.>, which has recently been demonstrated to require bianisotropy <cit.> in order to avoid undesired diffraction orders.

Here, we present a preliminary study of the angular spectrum of such a bianisotropic refractive metasurface placed at the interface between two dielectric media, and reveal the existence of three regions.




§ METASURFACE SYNTHESIS


We use a susceptiblity-GSTCs (Generalized Sheet Transition Conditions) metasurface synthesis method <cit.> to design the bianisotropic refractive metasurface. Assuming a purely transverse metasurface, the GSTCs read



    ẑ×Δ𝐇 = j ωϵ_0 χ_ee·𝐄_av +  j k_0 χ_em·𝐇_av ,


    Δ𝐄×ẑ   = j ωμ_0 χ_mm·𝐇_av +  j k_0 χ_me·𝐄_av ,



where the Δ symbol and the 'av' subscript represent the differences and averages of the tangential fields on both sides of the metasurface, and χ_ee, χ_em, χ_me, χ_mm are the bianisotropic surface susceptibility tensors describing the metasurface. This method consists in the following three steps: 1) specifying the total fields on both sides of the metasurface, 2) calculating the difference and average tangential fields, and 3) inserting these fields into Eqs. (<ref>) to obtain the susceptibility functions.

We consider a metasurface placed at the interface between two dielectric media, denoted a and b, with parameters ϵ_a, η_a, k_a and ϵ_b, η_b and k_b, respectively, as shown in Fig. <ref>. In this system, a plane wave incident from medium a at an angle θ_a is refracted, without any diffraction orders, into medium b at an arbitrary angle θ_b.


This problem was previously solved for a general bianisotropic refractive metasurface for TM polarization in <cit.>. The corresponding susceptibility functions, derived in that work, are




    χ^xx_ee =
    -4 k_a k_b  T  sin (α x)/ϵ_0 ω(T  βcos (α x)+η_a k_b k_az+η_b k_a k_bz T ^2),


    χ^xy_em =
    2 j (T  γcos (α x)-η_b k_b k_az+η_a k_a k_bz T ^2)/k_0 (T  βcos (α x)+η_a k_b k_az+η_b k_a k_bz T ^2),


    χ^yx_me =
    2 j (T  γcos (α x)-η_b k_b k_az+η_a k_a k_bz T ^2)/k_0 (T  βcos (α x)+η_a k_b k_az+η_b k_a k_bz T ^2),


    χ^yy_mm =
    -4 η_a η_b k_az k_bz  T  sin (α x)/μ_0 ω(T  βcos (α x)+η_a k_b k_az+η_b k_a k_bz T ^2),


where α=k_ax-k_bx, β = η_a k_b k_az+η_b k_a k_bz and γ = η_b k_a k_bz-η_a k_b k_az. The transmission coefficient T =√((η_bcosθ_a)/(η_acosθ_b)) is obtained by enforcing power conservation across the metasurface such that all incident power is transmitted.



These susceptibilities correspond to a periodic bianisotropic metasurface, which is passive, lossless and reciprocal. The metasurface can be tailored to accommodate arbitrary media surrounding it since its susceptibilities are a function of η_(a,b) and k_(a,b). It adds the momentum K_x = k_bx-k_ax to the wave vector of the incident wave k_ax, which is akin to the phase gradient in the generalized laws refraction and reflection <cit.>.




§ ANGULAR SPECTRUM STUDY


We shall see that the metasurface-interface admits three distinct asymmetric (due to the metasurface momentum) angular regions, as indicated in Fig. <ref>: a) a transmission region, corresponding to a rotated escape cone, b) a total internal reflection region, very different from that of the conventional interface, and c) a total retro-reflection region, non existing in the conventional interface.

We consider here a bianisotropic metasurface interface synthesized to refract a plane wave incident at normal incidence (θ_a=0^∘) in a dielectric medium a with ϵ_ra=4 into a plane wave propagating at an angle θ_b = 25^∘ in air (medium b). We shall now present GSTC-FDFD simulations <cit.> for the three aforementioned angular regions.

A simulation for the transmission region is shown in Fig. <ref>(a), specifically for the excitation corresponding to the synthesis specification. As in the case of a phase-gradient metasurface <cit.>, this transmission region is delimited by the two critical angles




    θ_c1 = arcsin(K_x+n_b/n_a),
     θ_c2 = arcsin(K_x-n_b/n_a),


which derives from the momentum added by the metasurface to the wavevector, K_x = k_bx-k_ax, and correspond to θ_c1=45.3^∘ and θ_c2=-16.8^∘ in our example.

Figure <ref>(b) shows a simulation for the total internal reflection region. The scattering in this region is similar to that of a conventional interface, except for its asymmetry. This region is composed of two sectors, delimited by delimited by θ_c1 and θ=+π/2 and by θ_c2 and an angle θ_rr, that is specific to the metasurface interface.

The angle θ_rr corresponds to the limit where the incident wave would be reflected at θ=+π/2 and beyond which the wave also becomes evanescent in the incident medium. Its value is given by




    θ_rr= arcsin(2K_x/n_a-1),


which numerically corresponds here to θ_rr=-35.3^∘. We observe by full-wave simulation, without explanation so far, that this phenomenon results total retro-reflection beyond θ_rr, as shown in Fig. <ref>(c). This is an interesting and potentially useful new property of this bianisotropic metasurface interface.





§ CONCLUSION


We presented an initial study of the angular spectrum of a bianisotropic refractive metasurface interface at an interface between two dielectric media, and reported on the existence of three scattering regions. This study provides an insight in the extra effects achievable at an interface hosting a metasurface.



IEEEtran


