Angle-Independent Nongyrotropic Metasurfaces
aa r X i v : . [ phy s i c s . c l a ss - ph ] M a r Angle-Independent Nongyrotropic Metasurfaces
G. Lavigne and C. Caloz Polytechnique Montr´eal, Department of Electrical Engineering, Blvd. Edouard-Montpetit, H3T 1J4,Montr´eal, [email protected]
Abstract – We derive a general condition for angle-independent bianisotropic nongyrotropicmetasurfaces and present two applications corresponding to particular cases: an angle-independentabsorber/amplifier and an angle-independent spatial gyrator.
I. I
NTRODUCTION
Metasurfaces are generally designed for a given specification, namely the incident, reflected and transmittedfields with specific angles, frequencies and polarizations [1, 2, 3]. Specifically, general bianisotropic metasurfaceshave been shown to have intricate and diverse angular scattering properties [4]. It would be particularly desirablefor many applications to devise metasurfaces that would exhibit the same response for different excitations.Here, we theoretically derive bianisotropic nongyrotropic metasurfaces whose response is angle-independent.Based on this result, we study two potential applications of angle-independent bianisotropic metasurafaces: anabsorber/amplifier and a spatial gyrator. II. T
HEORETICAL D ERIVATION
We consider the problem of an uniform bianisotropic metasurface surrounded by air placed in the xy plane z = 0 . Such a metasurface can be efficiently modelled as a zero-thickness discontinuity in space using generalized sheettransition conditions (GSTCs) and surface susceptibility tensors [5]. The tangential susceptiblity-based GSTCsmodel of such a metasurface is ˆ z × ∆ H = jωǫχ ee E av + jkχ em H av , (1a) ∆ E × ˆ z = jkχ me E av + jωµχ mm H av . (1b)where the symbol ∆ and the subscript ’av’ represent the differences and averages of the tangential electric ormagnetic fields at both sides of the metasurface, and χ ee , χ mm , χ em , χ me are the bianisotropic susceptibility tensorscharacterizing the metasurface.We seek here a general condition for metasurface angle-independent transformation, with the only restriction ofnongyrotropy to avoid excessive complexity at this point. For the sake of conciseness, we limit the investigation tothe s polarization, and a similar treatment naturally applies to the p polarization. Under such conditions, Eqs. (1)reduce to ∆ H x = jωǫχ yy ee E y, av + jkχ yx em H x, av , (2a) ∆ E y = jkχ xy me E y, av + jωµχ xx mm H x, av , (2b)where the field differences and averages are related to the forward-incidence transmission and reflection coeffi-cients ( T and R ) as ∆ H x = ( T − (1 + R )) H x, i , H x, av = 12 ( T + R + 1) H x, i , (3a) ∆ E y = ( T − (1 − R )) E y, i , E y, av = 12 ( T + 1 − R ) E y, i . (3b)he ratio of the tangential electric and magnetic fields in (3) as a function of the incidence angle θ can be writtenas E y, i H x, i = η cos θ , (4)where η is the wave impedance in free space. Inserting (3) into (2), respectively dividing the resulting equationsby H x, i and by E y, i and substituting (4) yields the following expression for the forward transmission and reflectioncoefficients in terms of the susceptibilities: T = (2 + jkχ yx em )( − j + kχ xy me ) − jǫµω χ yy ee χ xx mm j η cos θ ǫχ yy ee ω + 2 µχ xx mm ω cos θη − j (4 + k χ yx em χ xy me − ǫµω χ yy ee χ xx mm ) , (5a) R = 2( k ( χ xy me − χ yx em ) + η cos θ ǫχ yy ee ω − µχ xx mm ω cos θη ) j η cos θ ǫχ yy ee ω + 2 µχ xx mm ω cos θη − j (4 + k χ yx em χ xy me − ǫµω χ yy ee χ xx mm ) . (5b)Following the same procedure for the backward case yields T = ( − jkχ yx em )(2 j + kχ xy me ) − jǫµω χ yy ee χ xx mm j η cos θ ǫχ yy ee ω + 2 µχ xx mm ω cos θη − j (4 + k χ yx em χ xy me − ǫµω χ yy ee χ xx mm ) , (6a) R = 2( k ( χ yx em − χ xy me ) + η cos θ ǫχ yy ee ω − µχ xx mm ω cos θη ) j η cos θ ǫχ yy ee ω + 2 µχ xx mm ω cos θη − j (4 + k χ yx em χ xy me − ǫµω χ yy ee χ xx mm ) . (6b)Inspecting (5) and (6) reveals that a metasurface transformation generally has an angle dependance that isassociated with the cos θ and θ coefficients in the above expressions of the reflection and transmission coeffi-cients. However, these coefficients only affect the χ yy ee and χ xx mm susceptibilities. Therefore, imposing the restriction χ yy ee = χ xx mm = 0 removes the angular dependance. Practically, even if the χ yy ee and χ xx mm susceptiblities are not ex-actly zero but still negligeable compared to the χ yx em and χ xy me susceptiblities, the response of the metasurface willbe essentially independent of the angle, except towards grazing angles θ → ∞ . Inserting this condition into (5)and (6) yields then the angle-independent scattering coefficients T = (2 j − kχ yx em )( − j + kχ xy me )4 + k χ yx em χ xy me , R = 2 jk ( χ xy me − χ yx em )4 + k χ yx em χ xy me , (7a) T = ( − j − kχ yx em )(2 j + kχ xy me )4 + k χ yx em χ xy me , R = 2 jk ( χ yx em − χ xy me )4 + k χ yx em χ xy me . (7b)III. A NGLE -I NDEPENDENT A BSORBER /A MPLIFIER
An extremely useful application of such an angle-independent metasurface would be that of an angle-independentabsorber .Inspecting (7) shows that the reflection at both sides of the metasurface is suppressed by imposing χ yx em = χ xy me since this leads to R = R = 0 . Note that such a condition implies nonreciprocity since as it violates thereciprocity condition χ em = − χ T me , which reveals that nonreciprocity is a fundamental condition for metasurfaceangle-independent absorption. The corresponding foward and backward transmission coefficients reduce then to T = − k ( χ yx em ) + 4 jkχ yx em + 44 + k ( χ yx em ) , (8a) T = − k ( χ yx em ) − jkχ yx em + 44 + k ( χ yx em ) . (8b)Equations (8) are analysis equations giving the transmission coefficients through a metasurface of susceptibility χ yx em . For design, we need an inverse, synthesis equation. Such an equations is obtained by solving (8a) for aspecific T , T , spec , which yields A bianisotropic metasurface was demonstrated as thin-absorber was theoretical and experimentally demonstrated in [6] and [7], respec-tively. However, this metasurface absorber is not angle-independent. xy me = χ yx em = 2 ik − T , spec T , spec , (9)whose insertion in(8b) yields T = 1 T , spec . (10)Equation (10) reveals that specifying T via (9) results in having T being the opposite of this T . This physicallymeans that an angle-independent absorption corresponding to T , spec in the forward excitation implies a gain of thesame level in the opposite direction. Therefore, the overall metasurface needs to be active. Such a metasurface maybe realized by integrating transistors in the metasurface [6]. The topic of transistor-based metasurface absorber waspreviously discussed in [7]. IV. A NGLE -I NDEPENDENT S PATIAL G YRATOR
Another interesting application would be an angle-independent spatial gyrator. Given the fundamental natureof the gyrator as nonreciprocal component [8], one may easily anticipate that an angle-independent spatial couldenable many unique devices. Such a gyrator may be realized by specifying zero reflection and T = 1 e iπ/ , sothat T = 1 e − iπ/ according to (10), which indeed corresponds to a π phase difference between the forward andbackward transmissions. The corresponding susceptibilities are found from (9) as χ xy me = χ yx em = 2 k . (11)V. C ONCLUSION
We have derived a general condition for an angle-independent nongyrotropic metasurface, and theoreticallyderived two fundamental applications: an angle-independent absorber/amplifier and an angle-independent spatialgyrator. R
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