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

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## Abstract

**:**

## 1. Introduction

## 2. Circuit Theory

#### 2.1. Duality between Current and Voltage

#### 2.2. Planar Graph as Cellular Paving

#### 2.3. Inner and Outer Orientations

#### 2.4. Essence of Poincaré Duality

#### 2.5. Dual Circuits

#### 2.6. Self-Dual Circuit

## 3. Zero Backscattering from Self-Duality

#### 3.1. Self-Dual Transmission Lines

#### 3.2. Circuit Model for Huygens’ Metasurface

## 4. Keller–Dykhne Duality

#### 4.1. Two-Dimensional Resistive Sheets

#### 4.2. Duality in Laplace Equation

#### 4.3. Generalized Duality

#### 4.4. Effective Response and Duality

#### 4.5. Self-Duality and Singularity

#### 4.6. Differential-Form Approach for Duality

#### 4.7. Summary of Basic Equations in Differential-Form Approach

#### 4.8. Keller–Dykhne Duality with Differential Forms

#### 4.9. Discretization

- (1)
- For I and V discretized from K and E, we can consider the dual circuit with current ${I}^{\u2605}:={G}_{\mathrm{ref}}{({\u2605}^{1})}^{-1}(V)$ and voltage ${V}^{\u2605}:={R}_{\mathrm{ref}}{\u2605}_{1}(I)$ distributions for a circuit on ${\mathcal{K}}^{\u2605}$:$$\begin{array}{cc}\hfill {I}^{\u2605}& ={G}_{ref}{\displaystyle \sum _{i=1}^{\left|\mathcal{E}\right|}}{V}_{i}{({\u2605}^{1})}^{-1}({e}^{i})={G}_{ref}{\displaystyle \sum _{i=1}^{\left|\mathcal{E}\right|}}{V}_{i}{\stackrel{\u02c7}{e}}_{i},\hfill \end{array}$$$$\begin{array}{cc}\hfill {V}^{\u2605}& ={R}_{ref}{\displaystyle \sum _{i=1}^{\left|\mathcal{E}\right|}}{I}^{i}{\u2605}_{1}({e}_{i})={R}_{ref}{\displaystyle \sum _{i=1}^{\left|\mathcal{E}\right|}}{I}^{i}{\stackrel{\u02c7}{e}}^{i}.\hfill \end{array}$$
- (2)
- On the other hand, we discretize ${E}^{\u2605}={R}_{\mathrm{ref}}{K}_{\varpi}$ and ${K}^{\u2605}=-{G}_{\mathrm{ref}}{\mathsf{\Omega}}^{\varpi}E$ in a dual mesh ${\mathcal{K}}^{\u2605}$. We need to choose a specific orientation $\varpi $ of the plane, and ${({\stackrel{\u02c7}{e}}_{i})}_{\varpi}$ is regarded as an inner-oriented edge in ${\mathcal{K}}^{\u2605}$. Here, we introduce the ☆-conjugate operation to give an outer-oriented dual edge as ${{e}_{i}}^{\u2605}={\stackrel{\u02c7}{e}}_{i}$. The dual edge ${{({\stackrel{\u02c7}{e}}_{i})}_{\varpi}}^{\u2605}$ is outer-oriented, and represented as ${\left({{({\stackrel{\u02c7}{e}}_{i})}_{\varpi}}^{\u2605}\right)}_{\varpi}=-{e}_{i}$ (Figure 36), which reflects the complex algebraic structure of the plane. Then, discretized ${E}^{\u2605}$ and ${K}^{\u2605}$ are given as$$\left({\int}_{{({\stackrel{\u02c7}{e}}_{i})}_{\varpi}}{E}^{\u2605}\right){({\stackrel{\u02c7}{e}}^{i})}_{\varpi}={R}_{\mathrm{ref}}\left({\int}_{{\stackrel{\u02c7}{e}}_{i}}K\right){({\stackrel{\u02c7}{e}}^{i})}_{\varpi}={({V}^{\u2605})}_{\varpi},$$$$\left({\int}_{{{({\stackrel{\u02c7}{e}}_{i})}_{\varpi}}^{\u2605}}{K}^{\u2605}\right){({\stackrel{\u02c7}{e}}_{i})}_{\varpi}=\left({\int}_{-{e}_{i}}{({K}^{\u2605})}_{\varpi}\right){({\stackrel{\u02c7}{e}}_{i})}_{\varpi}={G}_{\mathrm{ref}}\left({\int}_{{e}_{i}}E\right){({\stackrel{\u02c7}{e}}_{i})}_{\varpi}={({I}^{\u2605})}_{\varpi}.$$

## 5. Electromagnetic Duality

#### 5.1. Preliminary

**A**and

**B**. Clearly,

**A**and

**B**are invariant under a mirror reflection with respect to the plane spanned by

**A**and

**B**. Therefore, it is natural to consider $\mathit{A}\times \mathit{B}$ as an axial vector as shown in Figure 40a. Then, the mirror symmetry is kept. The representation by polar vectors is also depicted in Figure 40b. The vector product between a polar vector and an axial vector is also defined to give a polar vector.

**B**is an axial vector. Therefore, a magnetic charge should be a pseudoscalar if it exists. The difference between electric and magnetic charges is illustrated in Figure 41.

#### 5.2. Formulation of Electromagnetic Duality

**E**, electric displacement field

**D**, electric current density ${\mathit{J}}_{\mathrm{e}}$, electric charge density ${\rho}_{\mathrm{e}}$, magnetic field

**H**, magnetic flux density

**B**, magnetic current density ${\mathit{J}}_{\mathrm{m}}$, and magnetic charge density ${\rho}_{\mathrm{m}}$. While

**E**,

**D**, and

**J**are polar vectors,

**H**,

**B**, and ${\mathit{J}}_{\mathrm{m}}$ are axial vectors. Electric and magnetic charge densities are represented by a scalar and pseudoscalar, respectively. Here, we introduce ${\mathit{J}}_{\mathrm{m}}$ and ${\rho}_{\mathrm{m}}$ to investigate the duality. Note that ${\mathit{J}}_{\mathrm{m}}$ and ${\rho}_{\mathrm{m}}$ are fictitious because a magnetic monopole does not exist. Material fields $(\mathit{D},\mathit{H})$ are determined from $(\mathit{E},\mathit{B})$ through the constitutive equations as described later.

#### 5.3. Analogy between Keller–Dykhne Duality and Electromagnetic Duality

## 6. Babinet Duality

#### 6.1. Babinet’s Principle for Electromagnetic Waves

- $z\ge 0$:$${\tilde{\mathit{E}}}_{+}^{\u2605}=-{Z}_{0}{\mathsf{\Omega}}^{\sigma}{\tilde{\mathit{H}}}_{+},\phantom{\rule{1.em}{0ex}}{\tilde{\mathit{H}}}_{+}^{\u2605}={Y}_{0}{\mathsf{\Omega}}^{\sigma}{\tilde{\mathit{E}}}_{+},$$
- $z\le 0$:$${\tilde{\mathit{E}}}_{-}^{\u2605}={Z}_{0}{\mathsf{\Omega}}^{\sigma}{\tilde{\mathit{H}}}_{-},\phantom{\rule{1.em}{0ex}}{\tilde{\mathit{H}}}_{-}^{\u2605}=-{Y}_{0}{\mathsf{\Omega}}^{\sigma}{\tilde{\mathit{E}}}_{-},$$

#### 6.2. Self-Dual Systems in Terms of Babinet Duality

#### 6.3. Babinet Duality in Transmission-Line Models

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. General Inner-Orientation Representation for Outer Orientation

## Appendix B. Babinet’s Principle for Transmission and Reflection Coefficients

## Appendix C. Duality for Input Impedances of Antennas

**Figure A1.**(

**a**) antenna with a sheet admittance ${Y}_{s}(x,y)\phantom{\rule{0.166667em}{0ex}}\left(=:{{Z}_{s}(x,y)}^{-1}\right)$ on $z=0$ is connected to voltage source ${\tilde{E}}_{ext}(x,y)$, which is assumed to be in the y direction, on S; (

**b**) dual antenna with a sheet admittance ${Y}_{s}^{\u2605}(x,y)\phantom{\rule{0.166667em}{0ex}}(=:{{Z}_{s}^{\u2605}(x,y)}^{-1})$ satisfying Equation (137) on $z=0$ is connected to current source ${\tilde{K}}_{ext}^{\u2605}(x,y)=2{Y}_{0}{\mathsf{\Omega}}^{\sigma}{e}_{z}\times {\tilde{E}}_{ext}$ on S; (

**c**) definition of two curves on S.

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**Figure 1.**Example of (

**a**) a circuit and (

**b**) its corresponding graph, which represents the circuit interconnection.

**Figure 5.**(

**a**) series and (

**b**) parallel resistors with composite resistances R and ${R}^{\prime}$, respectively. The duality between resistance and conductance appears as $R={R}_{1}+{R}_{2}$ and $1/{R}^{\prime}=1/{R}_{1}^{\prime}+/{R}_{2}^{\prime}$.

**Figure 7.**Two types of orientation: (

**a**) inner and (

**b**) outer orientation of a surface in a three-dimensional space.

**Figure 8.**(

**a**) tangent space ${\mathrm{T}}_{P}M$ at a point P on a surface M; (

**b**) inner-oriented 2-cell with continuously oriented tangent spaces.

**Figure 9.**Outer-oriented 2-cell S in a three-dimensional space is continuously outer-oriented in all tangent spaces.

**Figure 11.**Inner-oriented components of an outer-oriented edge in a two-dimensional plane. The plane orientations are given by the subscripts ($\u2940,\u2941$).

**Figure 13.**Boundary operation for an outer-oriented 1-cell in a two-dimensional plane is defined through the inner-oriented component.

**Figure 14.**Boundary operation for an outer-oriented (

**a**) edge and (

**b**) face in a two-dimensional plane.

**Figure 15.**Correspondence between a cellular paving $\mathcal{K}$ and its dual paving ${\mathcal{K}}^{\u2605}$. (

**a**) a face ${f}_{i}$ in $\mathcal{K}$↔ a point ${\stackrel{\u02c7}{n}}_{i}$ in ${\mathcal{K}}^{\u2605}$; (

**b**) a point ${n}_{j}$ in $\mathcal{K}$↔ a face ${\stackrel{\u02c7}{f}}_{j}$ in ${\mathcal{K}}^{\u2605}$. In both figures, an edge ${e}_{j}$ in $\mathcal{K}$ corresponds to an edge ${\stackrel{\u02c7}{e}}_{j}$ in ${\mathcal{K}}^{\u2605}$.

**Figure 21.**(

**a**) LC ladder circuit and (

**b**) its dual circuit; (

**c**) capacitors in the circuit (

**a**) are shifted; (

**d**) LC ladder circuit excited by a voltage source and (

**e**) its dual circuit; (

**f**) capacitors in the circuit (

**d**) are shifted; (

**g**) input impedance of the LC ladder circuit.

**Figure 23.**(

**a**) circuit model for propagating electromagnetic waves incident on a metasurface; (

**b**) circuit model of Huygens’ metasurfaces with ${Z}_{1}={Z}_{\mathrm{m}}/2$ and ${Z}_{2}=2/{Y}_{e}$; (

**c**) lumped circuit model for the metasurface followed by a semi-infinite transmission line.

**Figure 24.**(

**a**) two-dimensional resistive sheet with boundary conditions; (

**b**) solution for a sheet with a constant sheet conductance $G(x,y)={G}_{\mathrm{ref}}$. Black lines represent the current flow. The potential is shown as a color map with isopotential gray contours.

**Figure 25.**(

**a**) holomorphic function $w(z)=u(z)+\phantom{\rule{0.166667em}{0ex}}\mathrm{j}v(z)$ defines an orthogonal coordinate around a point with $\mathrm{d}w/\mathrm{d}z\ne 0$; (

**b**) harmonic potential $\phi $ and the lines of force $-\nabla \phi $; (

**c**) harmonic conjugate $\psi $ for $\phi $ and the lines of force $-\nabla \psi $.

**Figure 26.**Current and potential distributions for (

**a**) original and (

**b**) its dual resistive sheets with a uniform conductance.

**Figure 27.**(

**a**) resistive sheet with a sheet conductance $G(x,y)$ and two terminals; (

**b**) corresponding counterpart with ${G}^{\u2605}={({G}_{\mathrm{ref}})}^{2}J{G}^{-1}{J}^{-1}$. Unit normal vectors are denoted by ${n}_{1}$ and ${n}_{2}$ for curves ${c}_{1}$ and ${c}_{2}$, respectively.

**Figure 28.**(

**a**) ideal checkerboard sheet with sheet admittances ${Y}_{1}$ and ${Y}_{2}$; (

**b**) domain of definition for the effective admittance ${Y}_{eff}({Y}_{1})=\sqrt{{Y}_{1}{Y}_{2}}$. The branch cut along the positive imaginary axis is indicated by a wavy line.

**Figure 29.**(

**a**) illustration of a covector $\alpha $. The signed number of lines that a vector v pierces is given by $v\phantom{\rule{0.166667em}{0ex}}\alpha $. The positive direction of $\alpha $ is depicted by carets; (

**b**) covector field (1-form); (

**c**) discretized curve c with displaced vectors $\Delta {r}_{i}$ to define the integral of a 1-form $\alpha $ along c.

**Figure 30.**(

**a**) $\mathrm{d}x\wedge \mathrm{d}y$ as directed circles; (

**b**) example of a 2-form; (

**c**) discretization for integration.

**Figure 31.**(

**a**) interior product between an outer-oriented vector $\stackrel{\u02c7}{v}$ and an outer-oriented covector ${\stackrel{\u02c7}{\alpha}}_{P}$ at a point P; (

**b**) integration of a twisted 1-form $\stackrel{\u02c7}{\alpha}$ along an outer-oriented 1-chain $\stackrel{\u02c7}{c}$.

**Figure 38.**Representation of an axial vector by polar vectors depending on the space orientation. A spatial orientation is entered in the subscript position and its output is a polar vector which depends on the spatial orientation.

**Figure 40.**(

**a**) axial vector obtained from a vector product of polar vectors; (

**b**) its polar-vector representation.

**Figure 44.**(

**a**,

**b**) two scattering problems that are dual with each other. Opaque and transparent regions are interchanged under the duality transformation. The screen in (

**b**) is called the complementary screen of that in (

**a**), and vice versa.

**Figure 46.**Duality transformation keeping the mirror symmetry. (

**a**) ${\mathit{H}}_{\pm}$ to ${E}_{\pm}^{\u2605}$ and (

**b**) ${E}_{\pm}$ to ${\mathit{H}}_{\pm}^{\u2605}$.

**Figure 47.**(

**a**,

**b**) two scattering problems that are dual with each other. The dual screen (

**b**) is obtained through the impedance inversion of Equation (137).

**Figure 48.**(

**a**) antenna on $z=0$ with voltage source on S; (

**b**) dual antenna with current source on S. Note that metallic and vacant regions are interchanged through the impedance inversion.

**Figure 50.**Metallic checkerboard-like metasurfaces and their typical power-transmission spectra. (

**a**) disconnected phase; (

**b**) self-complementary point; and (

**c**) connected phase. The size of the unit cell is denoted by a.

**Figure 51.**Resistive checkerboard-like metasurface, which is self-dual in terms of Babinet duality, and its power transmission, reflection, and absorption spectra.

Circuit | Dual Circuit |
---|---|

Current distribution I | Voltage distribution ${V}^{\u2605}={R}_{\mathrm{ref}}{\u2605}_{1}(I)$ |

Voltage distribution V | Current distribution ${I}^{\u2605}={G}_{\mathrm{ref}}{({\u2605}^{1})}^{-1}(V)$ |

Face current F | Potential ${\phi}^{\u2605}=-{R}_{\mathrm{ref}}{\u2605}_{2}(F)$ |

Potential $\phi $ | Face current ${F}^{\u2605}={G}_{\mathrm{ref}}{({\u2605}^{0})}^{-1}(\phi )$ |

Voltage source ${V}_{s}$ | Current source ${{I}_{s}}^{\u2605}={G}_{\mathrm{ref}}{V}_{s}$ |

Current source ${I}_{s}$ | Voltage source ${{V}_{s}}^{\u2605}={R}_{\mathrm{ref}}{I}_{s}$ |

Resistance R | Conductance ${G}^{\u2605}={({G}_{ref})}^{2}R$ |

Conductance G | Resistance ${R}^{\u2605}={({R}_{ref})}^{2}G$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nakata, Y.; Urade, Y.; Nakanishi, T.
Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials. *Symmetry* **2019**, *11*, 1336.
https://doi.org/10.3390/sym11111336

**AMA Style**

Nakata Y, Urade Y, Nakanishi T.
Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials. *Symmetry*. 2019; 11(11):1336.
https://doi.org/10.3390/sym11111336

**Chicago/Turabian Style**

Nakata, Yosuke, Yoshiro Urade, and Toshihiro Nakanishi.
2019. "Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials" *Symmetry* 11, no. 11: 1336.
https://doi.org/10.3390/sym11111336