*Article* **Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials**

#### **Yosuke Nakata 1,\*,†, Yoshiro Urade 2 and Toshihiro Nakanishi 3**


Received: 31 August 2019; Accepted: 14 October 2019; Published: 28 October 2019

**Abstract:** In electromagnetic systems, duality is manifested in various forms: circuit, Keller–Dykhne, electromagnetic, and Babinet dualities. These dualities have been developed individually in different research fields and frequency regimes, leading to a lack of unified perspective. In this paper, we establish a unified view of these dualities in electromagnetic systems. The underlying geometrical structures behind the dualities are elucidated by using concepts from algebraic topology and differential geometry. Moreover, we show that seemingly disparate phenomena, such as frequency-independent effective response, zero backscattering, and critical response, can be considered to be emergen<sup>t</sup> phenomena of self-duality.

**Keywords:** duality; self-duality; Poincaré duality; circuit duality; Keller–Dykhne duality; electromagnetic duality; Babinet's principle; constant-resistance circuit; zero backscattering; critical response; self-complementary antenna; metamaterials; metasurfaces
