-Maxwell's Equations

Maxwell's equations can be written as

$$
\nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} = -J\_{\mathbf{m}'} \quad \nabla \cdot \mathbf{D} = \rho\_{\mathbf{e}'} \tag{102}
$$

$$
\nabla \times H - \frac{\partial \mathbf{D}}{\partial t} = f\_{\mathbf{o}^\*} \qquad \nabla \cdot \mathbf{B} = \rho\_\mathbf{m} \tag{103}
$$

with electric field *E*, electric displacement field *D*, electric current density *J*e, electric charge density *ρ*e, magnetic field *H*, magnetic flux density *B*, magnetic current density *J*m, and magnetic charge density *ρ*m. While *E*, *D*, and *J* are polar vectors, *H*, *B*, and *J*m are axial vectors. Electric and magnetic charge densities are represented by a scalar and pseudoscalar, respectively. Here, we introduce *J*m and *ρ*m to investigate the duality. Note that *J*m and *ρ*m are fictitious because a magnetic monopole does not exist. Material fields (*<sup>D</sup>*, *H*) are determined from (*<sup>E</sup>*, *B*) through the constitutive equations as described later.

## -Electromagnetic Duality Transformation

Electric and magnetic fields are represented by polar and axial vectors as shown in Figure 37, respectively. To exchange these two different types of vectors, we need to fix the orientation of the space to *σ* and introduce a pseudoscalar Ω*σ* as (Ω*<sup>σ</sup>*)*σ* = +1 and (Ω*<sup>σ</sup>*)−*<sup>σ</sup>* = −1. The pseudoscalar Ω*σ* can be represented by a helix with winding of *σ* as shown in Figure 42. For an axial vector *a*, Ω*σ* converts it to the polar component as Ω*<sup>σ</sup>a* = *aσ*.

**Figure 42.** (**a**) Ω<sup>R</sup> and (**b**) ΩL.

We set a reference resistance *<sup>R</sup>*ref(=: 1/*G*ref). For a spatial orientation *σ*, the electromagnetic duality transformation is given by

$$E^\star = R\_{\rm nsf} \Omega^\sigma H, \qquad \mathbf{D}^\star = G\_{\rm nsf} \Omega^\sigma B,\tag{104}$$

$$H^\star = -G\_{\rm nfl} \Omega^\tau E, \quad \mathcal{B}^\star = -R\_{\rm nfl} \Omega^\tau D. \tag{105}$$

Under the duality transformation, Maxwell's equations are invariant as

$$
\nabla \times E^{\star} + \frac{\partial B^{\star}}{\partial t} = -f\_{\mathbf{m}\prime}^{\star} \quad \nabla \cdot \mathbf{D}^{\star} = \rho\_{\mathbf{v}\prime}^{\star} \tag{106}
$$

$$
\nabla \times H^\star - \frac{\partial \mathbf{D}^\star}{\partial t} = f\_{\alpha \prime}^\star \qquad \nabla \cdot \mathbf{B}^\star = \rho\_{\mathbf{m}}^\star \tag{107}
$$

with

$$
\rho\_o^\* = G\_{\rm ref} \Omega^\sigma \rho\_{\rm m\nu} \qquad \mathcal{J}\_a^\star = G\_{\rm ref} \Omega^\sigma \mathcal{J}\_{\rm m\nu} \tag{108}
$$

$$
\rho\_{\rm m}^{\star} = -R\_{\rm rad} \Omega^{\sigma} \rho\_{\rm o}, \quad \mathcal{J}\_{\rm m}^{\star} = -R\_{\rm rad} \Omega^{\sigma} \mathcal{J}\_{\rm o}. \tag{109}
$$


> Consider the relations called constitutive equations

$$D = \varepsilon E + \zeta H,\tag{110}$$

$$B = \zeta E + \mu H,\tag{111}$$

where *ξ* and *ζ* are twisted. Generally, *ε*, *ξ*, *ζ*, and *μ* are tensors. Under Equations (104) and (105), the constitutive equations are transformed as

$$D^\* = \varepsilon^\* E^\* + \tilde{\zeta}^\* H^\*,\tag{112}$$

$$\mathcal{B}^{\star} = \zeta^{\star} E^{\star} + \mu^{\star} H^{\star} \tag{113}$$

with

$$
\varepsilon^{\star} = (G\_{\rm ref})^2 \mu, \quad \xi^{\star} = -\xi,\tag{114}
$$

$$\zeta^\star = -\xi, \qquad \mu^\star = (R\_{\text{ref}})^2 \mathfrak{e}. \tag{115}$$


Now, we require the set of self-dual conditions *ε* = *ε*, *ξ* = *ξ*, *μ* = *μ*, and *ζ* = *ζ*. Then, the following equations should hold:

$$
\varepsilon = (G\_{\rm rad})^2 \mu, \quad \mathcal{J} = -\mathcal{J}. \tag{116}
$$

Here, the *G*ref satisfying Equation (116) is called the admittance *Y* of the medium. For scalar *ε* and *μ*, we have *Y* = *<sup>ε</sup>*/*μ*. In particular, a vacuum with *ε* = *ε*0, *μ* = *μ*0, *ξ* = *ζ* = 0 is self-dual with respect to the vacuum admittance *Y*0 = *<sup>ε</sup>*0/*μ*0. In circuit theory, a self-dual system did not have backscattering. This statement is also established in electromagnetic systems under certain conditions [54]. In addition, the duality transformation can be extended to continuous one and continuous self-dual symmetry leads to the helicity conservation law [55,83–88].

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

Maxwell's electromagnetic theory in a four-dimensional spacetime has an analogous structure to the sheet problem discussed in Section 4. To see this analogy, we use the differential-form approach to Maxwell's equations [62,89–92]. The wedge product, exterior derivative, integral, and Hodge star are naturally extended in dimensions greater than two.

In a three-dimensional space, an electric field and magnetic field are represented by an untwisted 1-form *E* and twisted 1-form *H*, respectively. On the other hand, a magnetic flux density is denoted by an untwisted 2-form *B*, while an electric displacement is represented by a twisted 2-form *D*. Using these quantities, we define untwisted and twisted 2-forms F and G as

$$\mathsf{F} = \boldsymbol{E} \wedge \mathsf{d}t + \boldsymbol{B}, \tag{117}$$

$$\mathbf{G} = -H \wedge \mathbf{d}t + D\_t \tag{118}$$

respectively. Maxwell's equations without a source are equivalent to

$$\text{df} = 0,\tag{119}$$

$$\text{dG} = 0,\tag{120}$$

with the four-dimensional exterior derivative d. Now, we consider a medium with a scalar permittivity *ε* and permeability *μ*, while we set *ξ* = 0 and *ζ* = 0. The constitutive equation is given by

$$\mathbf{G} = \mathbf{Y} \star \mathbf{F} \tag{121}$$

with the admittance *Y* = *ε*/*μ* (= 1/*Z*) and the four-dimensional Hodge star operator . These equations are summarized in Figure 43. For 2-forms in the four-dimensional space, we have

$$
\star \star = -\mathrm{Id},
\tag{122}
$$

with the identity operator Id. Equation (122) leads to a complex structure.

**Figure 43.** Structure of Maxwell's theory of electromagnetism.

Equations (119)–(121) perfectly correspond to Equations (79)–(81), respectively. When we fix an orientation of the four-dimensional spacetime, the duality transformation can be written as

$$\mathbf{F}^{\star} = -R\_{\text{ref}} \Omega^{\omega} \mathbf{G},\tag{123}$$

$$\mathbf{G}^\* = G\_{\mathrm{ref}} \Omega^\partial \mathsf{F}.\tag{124}$$

Then, the dual admittance is defined as

$$\mathbf{Y}^{\star} = (\mathbf{G}\_{\text{ref}})^2 / \mathbf{Y}. \tag{125}$$

The duality transformation of Equations (123) and (124) interchanges *E* and *H* in the three-dimensional space. If we set *G*ref = *Y*, we obtain *Y* = *Y*, which indicates the system is self-dual. As an example, a vacuum is self-dual with respect to the vacuum admittance *Y*0 = *<sup>ε</sup>*0/*μ*0 (= 1/*Z*0) with vacuum permittivity *ε*0 and vacuum permeability *μ*0. Electromagnetic duality can be considered as a manifestation of Poincaré duality in this spacetime [93].

## **6. Babinet Duality**

*Babinet's principle* known in optics and electromagnetism relates wave-scattering problems of two complementary screens (Figure 44) [94]. Here, we call the duality appearing in Babinet's principle as *Babinet duality*. Babinet duality can be regarded as a high-frequency counterpart of Keller–Dykhne duality, which is discussed in Section 4. At first, we introduce rigorous Babinet's principle for electromagnetic waves. Then, we analyze self-dual systems in terms of Babinet duality, such as the Mushiake principle in antenna theory [13]. Finally, we discuss Babinet duality in the light of circuit duality by using a transmission-line model of metasurfaces.

**Figure 44.** (**<sup>a</sup>**,**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.

#### *6.1. Babinet's Principle for Electromagnetic Waves*

Before moving on to Babinet's principle for electromagnetic waves, we discuss the duality for electromagnetic waves radiated from planar sheets. Next, the formulated duality is utilized to derive Babinet's principle for electromagnetic waves.

#### - Duality for Radiation from Planar Antennas

Here, we formulate the duality for fields radiated from nonuniform planar sheets in a vacuum, while we stress the importance of axial vectors. Consider a sheet on *z* = 0 with an electric sheet impedance *<sup>Z</sup>*e(*<sup>x</sup>*, *y*) and an external electric field *E*˜ ext(*<sup>x</sup>*, *y*) as a voltage source. Generally, *<sup>Z</sup>*e(*<sup>x</sup>*, *y*) is a tensor. Following Maxwell's equations, the induced current distribution radiates electromagnetic waves. The radiated electromagnetic field is represented by (*E*˜ +, *H*˜ +) and (*E*˜ −, *H*˜ −) in *z* ≥ 0 and *z* ≤ 0, respectively. Mirror reflection with respect to *z* = 0 is expressed as M*z* and the considered system is invariant under M*<sup>z</sup>*. First, let us see the symmetry property of electromagnetic fields on *z* = 0. The component of *v* perpendicular to the plane *z* = 0 is obtained by *v*n = (*v* · *<sup>e</sup>z*)*<sup>e</sup>z*. Then, the projection of *v* onto *z* = 0 is given by *v*t = P*v* = *v* − *v*n with P = <sup>−</sup>*ez* × *ez*×. A polar vector *p* and axial vector *a* behave differently for M*z* as

$$\mathcal{M}\_z \mathfrak{p} = \mathfrak{p}\_\mathfrak{t} - \mathfrak{p}\_{\mathfrak{n}'} \tag{126}$$

$$
\mathcal{M}\_z a = -a\_t + a\_n. \tag{127}
$$

These relations are schematically shown in Figure 45. From Equations (126) and (127), we obtain the following symmetry at any point of the plane *z* = 0:

$$\mathcal{P}\mathcal{E}\_{+} = \mathcal{P}\mathcal{E}\_{-},\ \mathbf{e}\_{z}\cdot\mathcal{E}\_{+} = -\mathbf{e}\_{z}\cdot\mathcal{E}\_{-},\tag{128}$$

$$
\mathcal{P}\hat{H}\_{+} = -\mathcal{P}\hat{H}\_{-},\ \mathbf{e}\_{z}\cdot\hat{H}\_{+} = \mathbf{e}\_{z}\cdot\hat{H}\_{-}.\tag{129}
$$

**Figure 45.** Mirror reflection of (**a**) polar and (**b**) axial vectors.

Next, the boundary condition on *z* = 0 is given by

$$\mathcal{P}\mathbb{E}\_{+} = \mathcal{P}\mathbb{E}\_{-},\tag{130}$$

$$\mathbb{E}\_{\text{2D}} = \mathcal{P}(\mathbb{E}\_{+} + \mathbb{E}\_{\text{ext}}) = Z\_{\text{\text{\textdegree}}} \mathbb{K}\_{\text{2D}},\tag{131}$$

with the two-dimensional electric field *E* ˜ 2D on *z* = 0, the sheet current density *K* ˜ 2D = *ez* × (*H*˜ + − *H* ˜ −) = 2*ez* × *H*˜ + which is obtained from Equation (129). Equation (130) represents the continuity of the tangential electric field, while Equation (131) is equivalent to Ohm's law. The boundary conditions for *D* ˜ and *H* ˜ are derived from these boundary conditions [95].

Finally, we consider the duality transformation. We fix a spatial orientation *σ* and introduce a pseudoscalar Ω*σ* satisfying (Ω*<sup>σ</sup>*)*σ* = 1 and (Ω*<sup>σ</sup>*)−*<sup>σ</sup>* = −1. The following duality transformations are considered:

*Symmetry* **2019**, *11*, 1336

1.  $z \ge 0$ :

$$\begin{aligned} \mathsf{\tilde{E}}\_{+}^{\*} &= -\mathsf{Z}\rho \mathsf{I} \mathsf{I}^{\sigma} \mathsf{\tilde{H}}\_{+}, \quad \mathsf{\tilde{H}}\_{+}^{\*} = \mathsf{Y}\rho \mathsf{I} \mathsf{I}^{\sigma} \mathsf{\tilde{E}}\_{+} \end{aligned} \tag{132}$$

− +  +,

 −, 2.  $z \le 0$ :

$$\mathbb{E}\_{-}^{\*} = Z\_{0} \Omega^{\sigma} \mathcal{H}\_{-}, \quad \mathbb{H}\_{-}^{\*} = -\mathbb{Y}\_{0} \Omega^{\sigma} \mathbb{E}\_{-},\tag{133}$$

where *<sup>Z</sup>*0(= 1/*Y*0) is the impedance of a vacuum. The transformed fields (*E*˜ ±, *H*˜ ±) are invariant under M*z* as shown in Figure 46. With this symmetry of Equation (129), we immediately obtain P*E* ˜ + = P*E* ˜ − on *z* = 0. On the other hand, Equation (131) is transformed as

$$\mathbb{K}\_{\text{2D}}^{\star} = \mathbb{Y}\_{\text{q}}^{\star} \mathbb{E}\_{\text{2D}\prime}^{\star} \tag{134}$$

where we defined

$$
\mathbb{K}\_{\text{2D}}^{\star} = 2\mathfrak{e}\_{z} \times \mathbb{H}\_{+}^{\star} + \mathbb{K}\_{\text{ext}}^{\star} \tag{135}
$$

$$
\mathbb{K}\_{\rm ext}^{\star} = 2\mathbb{Y}\_0 \Omega^{\sigma} \mathbf{e}\_z \times \mathbb{E}\_{\rm ext} \tag{136}
$$

with *<sup>Z</sup>*e = (*Y*e )−1. Here, the following general impedance inversion holds:

+

−

$$\|Z\_{\v}^{\star}\|Z\_{\v}\|^{-1} = \left(\frac{Z\_0}{2}\right)^2\tag{137}$$

with

$$J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.$$

Note that the impedance *Z*0 can be replaced with *μ*/*ε* if the screen is placed in an isotropic and homogeneous medium with permeability *μ* and permittivity *ε*. Furthermore, *E* ˜ 2D = P*E* ˜ + and *K* ˜ 2D satisfy

$$\mathbb{E}\_{\text{2D}}^{\star} = \frac{Z\_0}{2} \Omega^{\sigma} \mathfrak{e}\_z \times \mathbb{K}\_{\text{2D}\prime} \tag{138}$$

$$\mathcal{K}\_{\text{2D}}^{\star} = \left(\frac{Z\_0}{2}\right)^{-1} \Omega^r \mathfrak{e}\_z \times \mathcal{E}\_{\text{2D}} \tag{139}$$

on *z* = 0. Equations (137), (138), and (139) perfectly correspond to Equations (49), (43), and (44). For the dual setup with the sheet impedance *<sup>Z</sup>*e and current source *K*˜ ext, the radiated fields are given by (*E*˜ ±, *H*˜ <sup>±</sup>). For a scalar *Z*e, the impedance inversion simplifies to

$$Z\_{\mathbb{P}} Z\_{\mathbb{e}}^{\star} = \left(\frac{Z\_0}{2}\right)^2. \tag{140}$$

Note that Equation (140) includes the duality between the perfect electric conductor (*Z*e = 0) and aperture (*Z*e = ∞). In other words, the sheet-impedance model is a generalization of the binarized case, where only opaque and transparent regions were considered for screens as shown in Figure 44.

**Figure 46.** Duality transformation keeping the mirror symmetry. (**a**) *H*± to *E* ± and (**b**) *E*± to *H* ±.


Here, we derive Babinet's principle by applying the previous discussion to scattering problems. We consider a scattering problem by a screen characterized with a sheet impedance of *<sup>Z</sup>*e(*<sup>x</sup>*, *y*) for an incident electromagnetic field (*E*˜in, *H*˜ in) from *z* < 0, as shown in Figure 47a. Fields scattered by the screen are denoted by (*E*˜ s,±, *H*˜ s,<sup>±</sup>) for *z* ≥ 0 and *z* ≤ 0, respectively. These scattered fields are induced by the external electric field *E*˜ ext = P *<sup>E</sup>*˜in(*<sup>x</sup>*, *y*, 0) on *z* = 0.

**Figure 47.** (**<sup>a</sup>**,**b**) two scattering problems that are dual with each other. The dual screen (**b**) is obtained through the impedance inversion of Equation (137).

Next, we consider the dual wave scattering by *Z* e (*<sup>x</sup>*, *y*) for an incident wave

$$(\mathbf{E}\_{\rm in\prime}^\star, \mathbf{H}\_{\rm in}^\star) = (Z\_0 \Omega^\sigma \mathbf{H}\_{\rm in\prime} - \mathbf{Y}\_0 \Omega^\sigma \mathbf{E}\_{\rm in}),\tag{141}$$

from *z* < 0. In the dual problem, we have to inject an external current rather than apply an electric field. To this end, we virtually consider total reflection by a perfect electric conductor sheet at *z* = 0. The totally reflected field (*E*˜ TR, *H*˜ TR) is obtained by a mirror reflection of (*E*˜ in, *H*˜ in) in *z* > 0 with respect to *z* = 0 and a phase flip:

$$(\mathcal{E}\_{\text{TR}'}^\star, \mathcal{H}\_{\text{TR}}^\star) = -(\mathcal{M}\_z \mathcal{E}\_{\text{in}}^\star, \mathcal{M}\_z \mathcal{H}\_{\text{in}}^\star). \tag{142}$$

Using the total reflection, we can introduce an external current as

$$\mathbb{K}\_{\rm ext}^{\star} = -\mathbf{2} \mathbf{e}\_{\rm z} \times \mathbf{H}\_{\rm ln}^{\star} = \mathbf{2} \mathbf{Y}\_{0} \boldsymbol{\Omega}^{\sigma} \mathbf{e}\_{\rm z} \times \mathbf{E}\_{\rm ext} \tag{143}$$

on *z* = 0. This virtually injected current radiates the electromagnetic field (*E*˜ s,±, *H*˜ s,<sup>±</sup>) as shown in Figure 47b. Here, Equation (143) is identical to Equation (136). Therefore, (*E*˜ s,±, *H*˜ s,<sup>±</sup>) and (*E*˜ s,±, *H*˜ s,<sup>±</sup>) are related through Equations (132) and (133), if *Z* e satisfies Equation (137). Thus, we could relate the scattered fields in the two problems. This duality relationship is the Babinet's principle for vector waves.

Babinet's principle leads to complementary relation on transmission coefficients. Consider a normal incidence of a plane wave to a periodic screen (metasurface) with a sheet impedance *<sup>Z</sup>*e(*<sup>x</sup>*, *y*) on *z* = 0. The complex transmission coefficient to the transmitting mode with the same polarization is denoted by *τ*. In the dual situation, the incident wave of Equation (141) enters the dual metasurface *<sup>Z</sup>*e (*<sup>x</sup>*, *y*) satisfying Equation (137). The complex electric transmission coefficient in the dual setup is represented by *<sup>τ</sup>*. Now, the following dual relation holds:

$$
\tau + \tau^\star = 1.\tag{144}
$$

The detailed derivation of this relation is given in Appendix B.

#### - Babinet's Principle for Power Transmissions

Next, we consider the consequence of Babinet duality on power transmission spectra of periodic screens, i.e., the square of the absolute value of the complex amplitude transmission coefficients. Using the continuity equation 1 + = *τ* with a reflection coefficient , we obtain

$$
\varrho = -\mathfrak{r}^\*.\tag{145}
$$

On the other hand, we have the energy–conservation relation

$$\left|\pi\right|^2 + \left|\varrho\right|^2 = 1,\tag{146}$$

under the following conditions: (i) periodicity of screens is smaller than the wavelength and thus energy scattering into the other diffraction modes except for the zeroth order modes is negligible, and (ii) polarization conversion is negligible (*τ*<sup>⊥</sup> = *ρ*⊥ = 0 in Appendix B). By using Equations (145) and (146), we obtain

$$\left|\pi\right|^2 + \left|\pi^\*\right|^2 = 1.\tag{147}$$

This equation indicates that the power transmission spectrum in the dual problem has the opposite shape of that in the original one.

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

In this subsection, we discuss the manifestation of self-duality in terms of Babinet duality. First, we introduce self-complementary antennas, and then we discuss the criticality of metallic checkerboard-like metasurfaces.
