**1. Introduction**

Duality is an indispensable concept in mathematics, physics, and engineering. It relates two seemingly different systems in a nontrivial manner and facilitates deeper insight into the underlying structures behind the relevant physical theories. A duality transformation converts an object to its dual counterpart. Performing duality transformations twice brings a system back to its original state. Advantageously, duality transformation can sometimes convert a difficult problem into a more tractable one. Moreover, if a system is dual to itself, namely self-dual, some special characteristics could be expected. To name a few, self-duality characterizes a critical probability and temperature for percolation on graphs [1,2] and two-dimensional Ising models [3], respectively.

For electromagnetic systems, duality appears in various forms, such as dual circuits [4,5], Keller–Dykhne duality [6–8], electromagnetic duality [9–11], input–impedance duality for antennas [12,13], and Babinet's principle [14–26]. These dualities have been individually developed in various research fields such as electrical engineering, radio-frequency engineering, and photonics because the relevant frequency spectra broadly range from direct-current to the optical regime. Despite the long history of dualities in electromagnetic systems, they have not been sufficiently discussed from a unified perspective. In particular, universal geometrical structures behind the dualities are still mathematically unclear in electromagnetic systems.

Recently, engineered artificial composites called metamaterials have been attracting much attention due to their exotic electromagnetic properties [27]. In particular, two-dimensional

metamaterials are called metasurfaces and are intensively studied as ultrathin functional devices [28–30]. The operating frequency of metamaterials has been extended from the microwave to the optical band, and versatile design principles over extensively wide frequency spectra are in high demand. For example, circuit designs and nano-optics were integrated to establish a universal strategy to design metamaterials [31]. Over the past few decades, duality has been assuming an important role in the research and development of metamaterials. To name a few such cases, duality has been leveraged to design complementary metasurfaces [32–35], self-complementary metasurfaces [36–42], critical metasurfaces [43–52], maximally chiral metamaterials [53], and self-dual metamaterials with zero backscattering [54] or helicity conservation [55–57]. In such situations, the extensive target frequencies of metamaterials require a comprehensive understanding of duality in a unified manner.

In this paper, we establish a unified perspective which synthesizes dualities appearing in electromagnetic systems. The underlying geometrical structures hidden behind dualities are uncovered through Poincaré duality between circuit theory to optics. To this end, we introduce inner and outer orientations of geometrical objects. We stress the importance of these two different kinds of orientations because it is sometimes overlooked in primary electromagnetism. The correspondence relationship among various dualities in electromagnetic systems is thoroughly discussed. Moreover, we comprehensively show that self-duality manifests as frequency-independent effective response, zero backscattering, and criticality in electromagnetic systems. As a whole, we attempted to keep the paper pictorial as much as possible in order to easily grasp the important concepts for duality.

In Section 2, we start with discussions of electrical circuits. Although electrical circuits are simplified and idealized systems, they include the essence of duality. To clearly see duality in circuit theory, we provide an algebraic-topological framework for electrical circuits. Based on the established framework, we strictly formulate circuit duality through Poincaré duality. Furthermore, we explain that the effective response of a self-dual circuit is automatically derived from its self-duality. In Section 3, we discuss zero backscattering for waves in self-dual circuits in terms of impedance matching. Section 4 generalizes circuit duality to Keller–Dykhne duality for a continuous two-dimensional resistive sheet. We show that critical behavior can appear for a self-dual sheet. To extract the essence of the duality in continuous systems, differential forms are utilized. We see that Keller–Dyhkne duality exactly corresponds to circuit duality through discretization. In Section 5, we introduce electromagnetic duality to Maxwell's electromagnetic theory. Electromagnetic duality in a four-dimensional spacetime has a similar structure to Keller–Dykhne duality in a two-dimensional plane. Section 6 introduces duality for wave radiation and scattering by a two-dimensional sheet. Combining electromagnetic duality and mirror symmetry of the sheet, we establish Babinet duality. Babinet duality induces the critical response of a metallic checkerboard-like sheet. Finally, Section 7 summarizes the unified perspective for dualities appearing in electromagnetic systems and gives a conclusion.

## **2. Circuit Theory**

In this section, we establish a comprehensive formulation of circuit theory based on algebraic topology. Within the proposed framework that makes use of Poincaré duality, we unveil the geometrical structure of circuit duality. Next, we discuss the resultant duality in effective response and show that a specific response is automatically guaranteed under self-duality. Compared to the previous algebraic topological formulation of the circuit theory [58–60], we clarify the importance of inner and outer orientations of geometrical objects for circuit duality. Our approach gives a succinct mathematical description of circuit duality, and an amalgamation of circuit theory and algebraic topology yields various benefits in this interdisciplinary area.

#### *2.1. Duality between Current and Voltage*

#### - Circuits as Graphs

A circuit is considered as a network of interconnected components. The interconnection relation can be represented by a *graph* G = (N , E) with a set N = {*ni*|*<sup>i</sup>* = 1, 2, ··· , |N |} expressing the totality of nodes and a set E = {*ei*|*<sup>i</sup>* = 1, 2, ··· , |E|} representing the totality of directed edges (an ordered 2-element subset of N ), where |S| is the number of elements of a set S. Elements in N and E are called 0- and 1-cells, respectively. Circuit elements such as voltage sources or resistors are placed along the graph edges. In this section, we mainly focus on circuits with resistors, voltage, and current sources. Figure 1 shows an example of a circuit and the corresponding graph.

**Figure 1.** Example of (**a**) a circuit and (**b**) its corresponding graph, which represents the circuit interconnection.
