Angle-Independent Nongyrotropic Metasurfaces

TL;DR

This paper derives conditions for angle-independent bianisotropic nongyrotropic metasurfaces and demonstrates their applications as an absorber/amplifier and a spatial gyrator, enabling angle-insensitive electromagnetic control.

Contribution

It provides a general condition for designing angle-independent nongyrotropic metasurfaces and presents two novel applications demonstrating their practical utility.

Findings

Derived a general condition for angle-independent metasurfaces.

Designed an angle-independent absorber/amplifier.

Developed an angle-independent spatial gyrator.

Abstract

We derive a general condition for angle-independent bianisotropic nongyrotropic metasurfaces and present two applications corresponding to particular cases: an angle-independent absorber/amplifier and an angle-independent spatial gyrator.

Equations

$$\hat{z}\times \Delta \mathbf{H}=j\omega ϵ{\overline{\overline{\chi }}}_{\text{ee}}{\mathbf{E}}_{\text{av}}+jk{\overline{\overline{\chi }}}_{\text{em}}{\mathbf{H}}_{\text{av}},$$

$$\Delta \mathbf{E}\times \hat{z}=jk{\overline{\overline{\chi }}}_{\text{me}}{\mathbf{E}}_{\text{av}}+j\omega \mu {\overline{\overline{\chi }}}_{\text{mm}}{\mathbf{H}}_{\text{av}}.$$

$$\Delta H_x=j\omega ϵ{\chi }_{\text{ee}}^{yy}{E}_{y,\text{av}}+jk{\chi }_{\text{em}}^{yx}{H}_{x,\text{av}},$$

$$\Delta E_y=jk{\chi }_{\text{me}}^{xy}{E}_{y,\text{av}}+j\omega \mu {\chi }_{\text{mm}}^{xx}{H}_{x,\text{av}},$$

$$\Delta H_x=(T_1-(1+R_1))H_{x,\text{i}},H_{x,\text{av}}=\frac{1}{2}(T_1+R_1+1)H_{x,\text{i}},$$

$$\Delta E_y=(T_1-(1-R_1))E_{y,\text{i}},E_{y,\text{av}}=\frac{1}{2}(T_1+1-R_1)E_{y,\text{i}}.$$

$$\frac{E_{y,\text{i}}}{H_{x,\text{i}}}=\frac{{\eta }_0}{\cos \theta },$$

$$T_1=\frac{(2+jk{\chi }_{\text{em}}^{yx})(-2j+k{\chi }_{\text{me}}^{xy})-jϵ\mu {\omega }^2{\chi }_{\text{ee}}^{yy}{\chi }_{\text{mm}}^{xx}}{j\frac{\eta }{\cos \theta }2ϵ{\chi }_{\text{ee}}^{yy}\omega +2\mu {\chi }_{\text{mm}}^{xx}\omega \frac{\cos \theta }{\eta }-j(4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}-ϵ\mu {\omega }^2{\chi }_{\text{ee}}^{yy}{\chi }_{\text{mm}}^{xx})},$$

$$R_1=\frac{2(k({\chi }_{\text{me}}^{xy}-{\chi }_{\text{em}}^{yx})+\frac{\eta }{\cos \theta }ϵ{\chi }_{\text{ee}}^{yy}\omega -\mu {\chi }_{\text{mm}}^{xx}\omega \frac{\cos \theta }{\eta })}{j\frac{\eta }{\cos \theta }2ϵ{\chi }_{\text{ee}}^{yy}\omega +2\mu {\chi }_{\text{mm}}^{xx}\omega \frac{\cos \theta }{\eta }-j(4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}-ϵ\mu {\omega }^2{\chi }_{\text{ee}}^{yy}{\chi }_{\text{mm}}^{xx})}.$$

$$T_2=\frac{(-2+jk{\chi }_{\text{em}}^{yx})(2j+k{\chi }_{\text{me}}^{xy})-jϵ\mu {\omega }^2{\chi }_{\text{ee}}^{yy}{\chi }_{\text{mm}}^{xx}}{j\frac{\eta }{\cos \theta }2ϵ{\chi }_{\text{ee}}^{yy}\omega +2\mu {\chi }_{\text{mm}}^{xx}\omega \frac{\cos \theta }{\eta }-j(4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}-ϵ\mu {\omega }^2{\chi }_{\text{ee}}^{yy}{\chi }_{\text{mm}}^{xx})},$$

$$R_2=\frac{2(k({\chi }_{\text{em}}^{yx}-{\chi }_{\text{me}}^{xy})+\frac{\eta }{\cos \theta }ϵ{\chi }_{\text{ee}}^{yy}\omega -\mu {\chi }_{\text{mm}}^{xx}\omega \frac{\cos \theta }{\eta })}{j\frac{\eta }{\cos \theta }2ϵ{\chi }_{\text{ee}}^{yy}\omega +2\mu {\chi }_{\text{mm}}^{xx}\omega \frac{\cos \theta }{\eta }-j(4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}-ϵ\mu {\omega }^2{\chi }_{\text{ee}}^{yy}{\chi }_{\text{mm}}^{xx})}.$$

$$T_1=\frac{(2j-k{\chi }_{\text{em}}^{yx})(-2j+k{\chi }_{\text{me}}^{xy})}{4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}},R_1=\frac{2jk({\chi }_{\text{me}}^{xy}-{\chi }_{\text{em}}^{yx})}{4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}},$$

$$T_2=\frac{(-2j-k{\chi }_{\text{em}}^{yx})(2j+k{\chi }_{\text{me}}^{xy})}{4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}},R_2=\frac{2jk({\chi }_{\text{em}}^{yx}-{\chi }_{\text{me}}^{xy})}{4+k^2{\chi }_{\text{em}}^{yx}{\chi }_{\text{me}}^{xy}}.$$

$$T_1=\frac{-k^2({\chi }_{\text{em}}^{yx}{)}^2+4jk{\chi }_{\text{em}}^{yx}+4}{4+k^2({\chi }_{\text{em}}^{yx}{)}^2},$$

$$T_2=\frac{-k^2({\chi }_{\text{em}}^{yx}{)}^2-4jk{\chi }_{\text{em}}^{yx}+4}{4+k^2({\chi }_{\text{em}}^{yx}{)}^2}.$$

$${\chi }_{\text{me}}^{xy}={\chi }_{\text{em}}^{yx}=\frac{2i}{k}\frac{1-T_{1,\text{spec}}}{1+T_{1,\text{spec}}},$$

$$T_2=\frac{1}{T_{1,\text{spec}}}.$$

$${\chi }_{\text{me}}^{xy}={\chi }_{\text{em}}^{yx}=\frac{2}{k}.$$

Full text

Angle-Independent Nongyrotropic Metasurfaces

G. Lavigne1** and C. Caloz1**

**Abstract   –   We derive a general condition for angle-independent bianisotropic nongyrotropic metasurfaces and present two applications corresponding to particular cases: an angle-independent absorber/amplifier and an angle-independent spatial gyrator. **

• 1Polytechnique Montréal, Department of Electrical Engineering, Blvd. Edouard-Montpetit, H3T 1J4, Montréal, Canada

guillaume.lavigne@polymtl.ca

I.  Introduction

Metasurfaces are generally designed for a given specification, namely the incident, reflected and transmitted fields with specific angles, frequencies and polarizations [1, 2, 3]. Specifically, general bianisotropic metasurfaces have been shown to have intricate and diverse angular scattering properties [4]. It would be particularly desirable for 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: an absorber/amplifier and a spatial gyrator.

II.  Theoretical Derivation

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 sheet transition conditions (GSTCs) and surface susceptibility tensors [5]. The tangential susceptiblity-based GSTCs model of such a metasurface is

[TABLE]

where the symbol $\Delta$ and the subscript ’av’ represent the differences and averages of the tangential electric or magnetic fields at both sides of the metasurface, and $\overline{\overline{\chi}}_{\text{ee}}$, $\overline{\overline{\chi}}_{\text{mm}}$, $\overline{\overline{\chi}}_{\text{em}}$, $\overline{\overline{\chi}}_{\text{me}}$ are the bianisotropic susceptibility tensors characterizing the metasurface.

We seek here a general condition for metasurface angle-independent transformation, with the only restriction of nongyrotropy to avoid excessive complexity at this point. For the sake of conciseness, we limit the investigation to the $s$ polarization, and a similar treatment naturally applies to the $p$ polarization. Under such conditions, Eqs. (1) reduce to

[TABLE]

where the field differences and averages are related to the forward-incidence transmission and reflection coefficients ($T_{1}$ and $R_{1}$) as

[TABLE]

The ratio of the tangential electric and magnetic fields in (3) as a function of the incidence angle $\theta$ can be written as

[TABLE]

where $\eta_{0}$ is the wave impedance in free space. Inserting (3) into (2), respectively dividing the resulting equations by $H_{x,\text{i}}$ and by $E_{y,\text{i}}$ and substituting (4) yields the following expression for the forward transmission and reflection coefficients in terms of the susceptibilities:

[TABLE]

Following the same procedure for the backward case yields

[TABLE]

Inspecting (5) and (6) reveals that a metasurface transformation generally has an angle dependance that is associated with the $\cos\theta$ and $\frac{1}{\cos\theta}$ coefficients in the above expressions of the reflection and transmission coefficients. However, these coefficients only affect the $\chi_{\text{ee}}^{yy}$ and $\chi_{\text{mm}}^{xx}$ susceptibilities. Therefore, imposing the restriction $\chi_{\text{ee}}^{yy}=\chi_{\text{mm}}^{xx}=0$ removes the angular dependance. Practically, even if the $\chi_{\text{ee}}^{yy}$ and $\chi_{\text{mm}}^{xx}$ susceptiblities are not exactly zero but still negligeable compared to the $\chi_{\text{em}}^{yx}$ and $\chi_{\text{me}}^{xy}$ susceptiblities, the response of the metasurface will be essentially independent of the angle, except towards grazing angles $\frac{1}{\cos\theta}\rightarrow\infty$. Inserting this condition into (5) and (6) yields then the angle-independent scattering coefficients

[TABLE]

III.  Angle-Independent Absorber/Amplifier

An extremely useful application of such an angle-independent metasurface would be that of an angle-independent absorber111A bianisotropic metasurface was demonstrated as thin-absorber was theoretical and experimentally demonstrated in [6] and [7], respectively. However, this metasurface absorber is not angle-independent..

Inspecting (7) shows that the reflection at both sides of the metasurface is suppressed by imposing $\chi_{\text{em}}^{yx}=\chi_{\text{me}}^{xy}$ since this leads to $R_{1}=R_{2}=0$. Note that such a condition implies nonreciprocity since as it violates the reciprocity condition $\overline{\overline{\chi}}_{\text{em}}=-\overline{\overline{\chi}}_{\text{me}}^{T}$, which reveals that nonreciprocity is a fundamental condition for metasurface angle-independent absorption. The corresponding foward and backward transmission coefficients reduce then to

[TABLE]

Equations (8) are analysis equations giving the transmission coefficients through a metasurface of susceptibility $\chi_{\text{em}}^{yx}$. For design, we need an inverse, synthesis equation. Such an equations is obtained by solving (8a) for a specific $T_{1}$, $T_{1,\text{spec}}$, which yields

[TABLE]

whose insertion in(8b) yields

[TABLE]

Equation (10) reveals that specifying $T_{1}$ via (9) results in having $T_{2}$ being the opposite of this $T_{1}$. This physically means that an angle-independent absorption corresponding to $T_{1,\text{spec}}$ in the forward excitation implies a gain of the same level in the opposite direction. Therefore, the overall metasurface needs to be active. Such a metasurface may be realized by integrating transistors in the metasurface [6]. The topic of transistor-based metasurface absorber was previously discussed in [7].

IV.  Angle-Independent Spatial Gyrator

Another interesting application would be an angle-independent spatial gyrator. Given the fundamental nature of the gyrator as nonreciprocal component [8], one may easily anticipate that an angle-independent spatial could enable many unique devices. Such a gyrator may be realized by specifying zero reflection and $T_{1}=1e^{i\pi/2}$, so that $T_{2}=1e^{-i\pi/2}$ according to (10), which indeed corresponds to a $\pi$ phase difference between the forward and backward transmissions. The corresponding susceptibilities are found from (9) as

[TABLE]

V.  Conclusion

We have derived a general condition for an angle-independent nongyrotropic metasurface, and theoretically derived two fundamental applications: an angle-independent absorber/amplifier and an angle-independent spatial gyrator.