**7. Conclusions**

A space can be discretized with primary and dual meshes. Topological properties of the primary and dual meshes are related through Poincaré duality with each other. Especially in a 2*n*-dimensional space such as a plane (*n* = 1) and spacetime (*n* = 2), the dual counterpart of an inner-oriented *n*-chain is given by the outer-oriented *n*-chain. Thus, in an even-dimensional space, duality transformation does not alter the dimension of an *n*-chain; rather, it changes the characteristics of orientation for the chain. This special feature of even-dimensional spaces induces the complex structure for *n*-chains. Inner-oriented and outer-oriented objects work as real and imaginary parts of a complex number, respectively. Due to this complex structure, the direction of an object is reversed when duality transformations are applied twice for the object. Electromagnetic duality is induced by a four-dimensional complex structure which appears from Poincaré duality. For a planar structure in the three-dimensional structure, Babinet duality holds due to the combination of electromagnetic duality and mirror symmetry. Keller–Dykhne duality is considered as an appearance of the Babinet duality at lower frequencies of direct current and alternating current. Circuit duality is interpreted as a discretized version of Keller–Dykhne duality.

Moreover, we can consider an additional invariance under duality transformation. When the system has such internal symmetry, it is called self-dual. The effective response of a self-dual system is automatically determined by the self-duality regardless of its components. This consequence of self-duality can lead to a frequency-independent input impedance and zero backscattering. Furthermore, critical response can even be predicted from self-duality, and it is leveraged to manipulate the spectra and polarization of electromagnetic waves.

In conclusion, we have unveiled the underlying geometrical structures behind various dualities in electromagnetic systems. Now, various dualities in electromagnetic systems emerge from the correspondence between quantities with two different kinds of orientations through Poincaré duality. The manifestations of self-duality in electromagnetic systems were consistently confirmed in a broad frequency range.

**Author Contributions:** Conceptualization, Y.N.; investigation, all authors; writing—original draft preparation, Y.N. (Sections 1, 2, 4, 5 and 7), Y.U. (Section 6), T.N. (Section 3); review and editing, all; visualization, all.; project administration, Y.N.; funding acquisition, Y.N. and T.N.

**Funding:** This research was funded by a gran<sup>t</sup> from the Murata Science Foundation, a gran<sup>t</sup> from Shimadzu Science Foundation, and JSPS KAKENHI Grant Nos. 17K17777 and 17K05075.

**Acknowledgments:** The authors thank Shuhei Tamate for his interpretation of the algebraic treatment of circuits, Shogo Tanimura and Masao Kitano for their fruitful discussions on foundation of electromagnetism. Shin-itiro Goto and Jacob Koenig carefully read the manuscript and gave us valuable comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

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

The inner-orientation representation for outer orientation is generalized for any *m*-dimensional space. We use the fact that an inner orientation of a vector space *U* can be specified by a totally antisymmetric tensor *ω* : *U* × *U* ×···× *U* → R, which is linear for all input slots and satisfies *<sup>ω</sup>*(*<sup>u</sup>*1, *u*2, ··· , *ui*, ··· , *uj*, ··· , *um*) = − *<sup>ω</sup>*(*<sup>u</sup>*1, *u*2, ··· , *uj*, ··· , *ui*, ··· , *um*) for all possible *i* = *j*. For an inner-oriented vector space *U*, we select a positively-oriented basis (*<sup>e</sup>*1,*e*2, ··· ,*em*) and set *ω* satisfying *<sup>ω</sup>*(*<sup>e</sup>*1,*e*2, ··· ,*em*) = 1. If *ω*(*e* 1,*e* 2, ··· ,*e m*) > 0 is satisfied, a basis (*e* 1,*e* 2, ··· ,*e m*) has the positive orientation. Reversely, for given *ω* = 0, we can specify a basis (*<sup>e</sup>*1,*e*2, ··· ,*em*) with positive orientation as *<sup>ω</sup>*(*<sup>e</sup>*1,*e*2, ··· ,*em*) > 0. Now, consider an outer-oriented linear subspace *W* in *U* with *l* := dim *W* < *m* := dim *U*. The surjective projection is denoted by *u* ∈ *U* → [*u*] ∈ *U*/*W*. Assume that an orientation of ambient space *U* is specified by a totally antisymmetric tensor with *m* slots *ωU* and take a basis ([*<sup>u</sup>*1], [*<sup>u</sup>*2], ··· , [*um*−*<sup>l</sup>*]) which has positive orientation with respect to *U*/*W*. Then, we can define a totally antisymmetric tensor *ω W* over *W* as

$$
\omega\_W(w\_1, w\_2, \dots, w\_l) := \omega\_{\mathcal{U}}(u\_1, u\_2, \dots, u\_{m-l}, w\_1, w\_2, \dots, w\_l), \tag{A1}
$$

where *w*1, *w*2, ··· , *wl* ∈ *W* and the first (*m* − *l*)-slots for the orientation tensor *ωU* are contracted by an outer orientation. Thus, *ω W* determines the inner orientation of *W* depending on the ambient-space orientation *U*. It is easily shown that the defined orientation of *W* does not depend on a specific choice of *u*1, *u*2, ··· , *um*−*l*. The above discussion is applied to all tangent spaces smoothly, and then we can represent an outer-oriented cell as two inner-oriented cells.

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

We derive Babinet's principle for transmission and reflection coefficients for periodic screens (metasurfaces) in order to ge<sup>t</sup> insight into Babinet's principle under practical situations. We assume that the complex amplitude of the electric field of the incident plane wave in Figure 47a may be written as

$$\mathcal{E}\_{\rm in} = \mathcal{E}\_{\rm in} \mathbf{e}\_{\rm in} \exp(-\mathbf{j}kz), \tag{A2}$$

where *e*in is a unit polarization vector perpendicular to *ez*. Here, *e*in can be a linear polarization like *ex*, *ey* or circular polarizations *e*± := (*<sup>e</sup>x* ± j*<sup>e</sup>y*)/√2. Then, the scattered electric field can be expanded as

$$\mathcal{E}\_{\mathbf{s},\pm} = \left(\mathcal{E}\_{\mathbf{s}}\mathbf{e}\_{\mathrm{in}} + \mathcal{E}\_{\mathbf{s}}^{\perp}\mathbf{e}\_{\mathrm{in}}^{\perp}\right) \exp\left(\mp \mathrm{j} k z\right) \tag{A3}$$

in the far-field region, where *e*⊥in is the unit polarization vector orthogonal to *e*in. While in general all diffracted waves caused by the periodic screens should also be considered, we here focus on the zeroth order modes with the wavevector ±*k<sup>e</sup>z* for simplicity (for a more general formulation, see [37]). Thus, the complex amplitude transmission coefficients for the parallelly and orthogonally polarized modes *τ* and *τ*<sup>⊥</sup> are given by

$$
\pi := \frac{E\_{\rm in} + E\_{\rm s}}{E\_{\rm in}},
\tag{A4}
$$

$$
\pi^{\perp} := \frac{E\_{\text{s}}^{\perp}}{\tilde{E}\_{\text{in}}} \, \text{s} \tag{A5}
$$

respectively. Similarly, we can write the complex amplitude reflection coefficients for the parallelly and orthogonally polarized modes and ⊥ as

$$
\varrho := \frac{\mathcal{E}\_{\mathfrak{s}}}{\mathcal{E}\_{\mathrm{in}}} = \tau - 1,\tag{A6}
$$

$$
\varrho^{\perp} := \frac{E\_{\mathbf{s}}^{\perp}}{\tilde{E}\_{\text{in}}} = \tau^{\perp},
\tag{A7}
$$

respectively.

Next, we consider the dual problem shown in Figure 47b. Here, we fix the spatial orientation as *σ* = R for simplicity, where R represents the right-hand system. As shown in Equation (141), the complex amplitude of the electric field of the dual incident wave is given by

$$\begin{split} \mathsf{E}\_{\mathrm{in}}^{\*} &= Z\_{0} (\varprojlim)\_{\mathbb{R}} \\ &= \tilde{E}\_{\mathrm{in}} (\mathsf{e}\_{z} \times \mathsf{e}\_{\mathrm{in}})\_{\mathbb{R}} \exp(-\mathsf{j}kz) \\ &= \tilde{E}\_{\mathrm{in}} \mathsf{e}\_{\mathrm{in}}^{\*} \exp(-\mathsf{j}kz), \end{split} \tag{A8}$$

where *<sup>e</sup>*in := (*<sup>e</sup>z* × *<sup>e</sup>*in)R is the dual incident polarization vector and *H*˜ in = *Y*0*ez* × *E*˜in is used, which can be derived from Faraday's law. According to Equation (142), the totally reflected field is written as *E* ˜ TR = −*E* ˜ in*e*in exp(j*kz*). From Babinet's principle, the dual scattered electric field is given by

$$\begin{split} \mathbf{E}\_{\mathbf{s},\pm}^{\star} &= \mp \mathbf{Z}\_{0} (\hat{\mathbf{H}}\_{\mathbf{s},\pm})\_{\mathbb{R}} \\ &= \mp \mathbf{Z}\_{0} (\pm \mathbf{Y}\_{0} \mathbf{e}\_{\mathbf{z}} \times \mathbf{E}\_{\mathbf{s},\pm})\_{\mathbb{R}} \\ &= -\left[ \mathbb{E}\_{\mathbf{s}} (\mathbf{e}\_{\mathbf{z}} \times \mathbf{e}\_{\mathbf{in}})\_{\mathbb{R}} + \mathbb{E}\_{\mathbf{s}}^{\perp} (\mathbf{e}\_{\mathbf{z}} \times \mathbf{e}\_{\mathbf{in}}^{\perp})\_{\mathbb{R}} \right] \exp(\mp \mathbf{j} kz) \\ &= -\left( \vec{E}\_{\mathbf{s}} \mathbf{e}\_{\mathbf{in}}^{\star} + \vec{E}\_{\mathbf{s}}^{\perp} \mathbf{e}\_{\mathbf{in}}^{\perp,\star} \right) \exp(\mp \mathbf{j} kz), \end{split} \tag{A9}$$

where *e*⊥, in := (*<sup>e</sup>z* × *<sup>e</sup>*⊥in)<sup>R</sup> is the dual orthogonal polarization vector. Finally, the dual complex transmission and reflection coefficients can be expressed as

$$
\tau^\star := -\frac{E\_\mathbf{s}}{\tilde{E}\_{\text{in}}} \, \tag{A10}
$$

$$
\tau^{\perp,\star} := -\frac{\tilde{E}\_{\rm s}^{\perp}}{\tilde{E}\_{\rm in}},
\tag{A11}
$$

$$
\rho^\star := -\frac{E\_{\rm in} + E\_{\rm s}}{\tilde{E}\_{\rm in}} = \tau^\star - 1,\tag{A12}
$$

$$
\xi^{\perp,\star} := -\frac{\mathcal{E}\_{\mathbf{s}}^{\perp}}{\mathcal{E}\_{\mathrm{in}}} = \tau^{\perp,\star}.\tag{A13}
$$

Note that these coefficients are defined over the dual polarization basis *<sup>e</sup>*in and *e*⊥, in . If *e*in is linearly polarized, then *<sup>e</sup>*in corresponds to the orthogonal linear polarization. On the other hand, if *e*in is circularly polarized, then *<sup>e</sup>*in = ± j*<sup>e</sup>*in and the dual polarization state is the same as the original one up to a phase factor ± j, where + (−) corresponds to the left (right) circular polarization. Here, we define handedness of circularly polarized plane waves from the receivers' side. This convention is commonly used in optics.By comparing Equations (A10) and (A11) with Equations (A4) and (A5), respectively, we finally obtain

$$
\tau + \tau^\star = 1,\tag{A14}
$$

$$
\tau^{\perp} + \tau^{\perp, \star} = 0.\tag{A15}
$$

#### **Appendix C. Duality for Input Impedances of Antennas**

We derive a dual relation for input impedances of antennas shown in Figure A1a,b. To simplify the discussion, we fix the spatial orientation as *σ* = R, where R represents the right hand. Two curves *c*1 and *c*2 on *S* are defined in Figure A1c. The total current along *c*1 is calculated as

$$I = -2\int\_{c\_2} (\mathbf{\tilde{H}}\_+)\_{\mathbb{R}} \cdot \mathbf{dr}.\tag{A16}$$

The electromotive force along *c*1 is given by

$$
\mathcal{V} = \int\_{\mathcal{C}\_1} \mathbf{\mathcal{E}}\_{\text{ext}} \cdot \mathbf{dr}.\tag{A17}
$$

For the dual antenna, we have a current along the *c*2 direction as

$$I^\star = \int\_{\varepsilon\_1} \mathbb{K}\_{\text{ext}}^\star \cdot (\mathfrak{e}\_z \times \mathrm{d}\mathbf{r})\_\mathbb{R} = 2 \int\_{\varepsilon\_1} \mathbb{Y}\_0 \mathbb{E}\_{\text{ext}} \cdot \mathrm{d}\mathbf{r} = 2\mathbb{Y}\_0 \mathcal{V}. \tag{A18}$$

On the other hand, the electromotive force along *c*2 is given by

$$\mathcal{V}^{\star} = \int\_{\mathcal{C}\_{2}} \mathbf{E}\_{+}^{\star} \cdot \mathbf{dr} = -Z\_{0} \int\_{\mathcal{C}\_{2}} (\mathbf{H}\_{+})\_{\mathbb{R}} \cdot \mathbf{dr} = \frac{Z\_{0}}{2}I. \tag{A19}$$

Here, we introduce input impedances of antennas *Z*in = *V* ˜ / ˜ *I* and *<sup>Z</sup>*in = *V* ˜ / ˜ *I*, where the real part of an antenna input impedance represents electromagnetic radiation as a loss. These input impedances satisfy

$$Z\_{\rm ln} Z\_{\rm ln}^{\star} = \left(\frac{Z\_0}{2}\right)^2. \tag{A20}$$

**Figure A1.** (**a**) antenna with a sheet admittance *<sup>Y</sup>*s(*<sup>x</sup>*, *y*) =: *<sup>Z</sup>*s(*<sup>x</sup>*, *y*)−<sup>1</sup> on *z* = 0 is connected to voltage source *E* ˜ ext(*<sup>x</sup>*, *y*), which is assumed to be in the *y* direction, on *S*; (**b**) dual antenna with a sheet admittance *<sup>Y</sup>*s (*<sup>x</sup>*, *y*) (=: *<sup>Z</sup>*s (*<sup>x</sup>*, *y*)−<sup>1</sup>) satisfying Equation (137) on *z* = 0 is connected to current source *K* ˜ ext(*<sup>x</sup>*, *y*) = 2*Y*0Ω*<sup>σ</sup>ez* × *E*˜ ext on *S*; (**c**) definition of two curves on *S*.
