# Fundamentals of Lossless, Reciprocal Bianisotropic Metasurface Design

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## Abstract

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## 1. Introduction

## 2. Scattering from Bi-Isotropic Metasurfaces

## 3. Bi-Isotropic Metasurfaces: Bandwidth and Quality Factor

## 4. Scattering from Bianisotropic Metasurfaces

## 5. Design Examples

#### 5.1. Asymmetric Circular Polarizer

#### 5.2. Polarization Rotator

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The geometry of a metasurface between two regions with different material properties. The equivalent surface current densities J (electric) and M (magnetic) describe the interaction of the metasurface with the tangential fields. Under illumination by a normally incident plane wave, each region can contain two plane waves that are denoted by + for a wave propagating toward the surface or − for a wave propagating away from the surface.

**Figure 2.**A metasurface at the interface between air and alumina half-spaces. The metasurface is used to impedance match a normally incident plane wave traveling from the region of air into the alumina.

**Figure 4.**Bi-isotropic metasurface realized using three impedance sheets that are separated by dielectric spacers with thickness d.

**Figure 5.**The quality factor, fractional bandwidth, and the magnitude of the frequency response for metasurfaces that provide impedance matching with six different transmission phases. (

**a**) The quality factor is minimized when the transmission phase is $-68.{5}^{\circ}$ (red circle). (

**b**) The fractional bandwidth is maximized at $-68.{5}^{\circ}$ (red circle). (

**c**) Plots of the transmission amplitude over frequency for several transmission phases and the maximum bandwidth is observed when the transmission phase is $-68.{5}^{\circ}$, as predicted by the quality factor.

**Figure 6.**Plots of the (

**a**) transmission and (

**b**) reflection magnitudes for the interface with the metasurface (${\varphi}_{21}=-68.{5}^{\circ}$), a quarter-wave transformer, and with no impedance matching (bare interface). The simulations of the metasurface were performed in Ansys HFSS. The metasurface has a bandwidth that is comparable to a quarter-wave transformer.

**Figure 7.**(

**a**) The depiction of an inhomogeneous, bi-isotropic metasurface implemented as a three-sheet cascade in free-space. (

**b**) Plots of the sheet reactances for different transmission phases. The solid circles indicate the values used for the linear phase gradient and the empty squares indicate the sheet values used for the perturbed phase gradient. (

**c**) Full-wave simulation results for the real part of the electric field using the metasurface with a linear phase gradient. (

**d**) Full-wave simulation results for the real part of the electric field using the metasurface with a perturbed phase gradient.

**Figure 8.**(

**a**) The quality factor of the metasurface unit cells versus transmission phase. (

**b**) The first derivative of the quality factor with respect to the transmission phase. The solid black circles indicate the values corresponding to the linear phase gradient and the hollow red squares indicate the adjusted values used for the perturbed phase gradient.

**Figure 9.**A comparison of the transmission phases used for the original and perturbed phase gradients. The solid black circles indicate the transmission phases used for the linear phase gradient. The empty red squares indicate the transmission phases that are used for the perturbed phase gradient.

**Figure 10.**An illustration of the wave matrix and the constitutive blocks of the cascaded structure. (

**a**) The definition of a wave matrix. (

**b**) A metasurface interface between two dielectric media. (

**c**) A dielectric spacer.

**Figure 11.**A cascaded metasurface structure consists of three sheets with only electric responses ($\mathbf{\gamma}=\mathbf{\chi}=\mathit{Z}=\mathbf{0}$).

**Figure 12.**An asymmetric circular polarizer. (

**a**) Polarization-converting operation of the metasurface [23]. (

**b**) Unit cell of the asymmetric circular polarizer. (

**c**) Simulated transmission coefficients.

**Figure 13.**A linear polarization rotator. (

**a**) Polarization-converting operation of the metasurface [23]. (

**b**) Unit cell of the polarization rotator. (

**c**) Simulated transmission coefficients.

**Table 1.**The unit cell transmission phases (${\varphi}_{21}$) used in the design of the gradient metasurface for plane wave refraction. The original phase gradient corresponds to the linear phase gradient. The perturbed phase gradient corresponds to the adjusted phases used to improve the performance of the metasurface.

Unit Cell | ${\mathit{\varphi}}_{21}$ (Original) | ${\mathit{\varphi}}_{21}$ (Perturbed) |
---|---|---|

1 | $-{18}^{\circ}$ | $-{31}^{\circ}$ |

2 | $-{54}^{\circ}$ | $-{54}^{\circ}$ |

3 | $-{90}^{\circ}$ | $-{90}^{\circ}$ |

4 | $-{126}^{\circ}$ | $-{126}^{\circ}$ |

5 | $-{162}^{\circ}$ | $-{147}^{\circ}$ |

6 | $-{198}^{\circ}$ | $-{216}^{\circ}$ |

7 | $-{234}^{\circ}$ | $-{234}^{\circ}$ |

8 | $-{270}^{\circ}$ | $-{270}^{\circ}$ |

9 | $-{306}^{\circ}$ | $-{306}^{\circ}$ |

10 | $-{342}^{\circ}$ | $-{330}^{\circ}$ |

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**MDPI and ACS Style**

Szymanski, L.; Raeker, B.O.; Lin, C.-W.; Grbic, A.
Fundamentals of Lossless, Reciprocal Bianisotropic Metasurface Design. *Photonics* **2021**, *8*, 197.
https://doi.org/10.3390/photonics8060197

**AMA Style**

Szymanski L, Raeker BO, Lin C-W, Grbic A.
Fundamentals of Lossless, Reciprocal Bianisotropic Metasurface Design. *Photonics*. 2021; 8(6):197.
https://doi.org/10.3390/photonics8060197

**Chicago/Turabian Style**

Szymanski, Luke, Brian O. Raeker, Chun-Wen Lin, and Anthony Grbic.
2021. "Fundamentals of Lossless, Reciprocal Bianisotropic Metasurface Design" *Photonics* 8, no. 6: 197.
https://doi.org/10.3390/photonics8060197