-Heaviside Condition

Self-duality can be realized in transmission lines with inductance *Li*+ 12 = *L* ¯ Δ*z*, capacitance *Ci* = *C* ¯ Δ*z*, conductance *Gi* = *G* ¯ Δ*z*, and resistance *Ri*+ 12 = *R* ¯ Δ*z* as shown in Figure 22. The self-dual condition is given by 

$$
\sqrt{\frac{L}{\overline{C}}} = \sqrt{\frac{R}{\overline{G}}}.\tag{21}
$$

This self-dual condition is nothing but a no-distortion signal transmission condition derived by Heaviside [67]. Due to the self-duality, frequency-independent response is realized and backscattering vanishes, while a signal decays as it propagates.

**Figure 22.** LC ladder with resistance and conductance.

#### *3.2. Circuit Model for Huygens' Metasurface*

Two-dimensional artificial structures called metasurfaces have been extensively investigated for controlling the amplitude and phase of transmitted and/or reflected electromagnetic waves [68]. The wavefront control of light can be realized by designing metasurfaces with spatial variations of phase responses [69–72]. The amplitude and phase responses of metasurfaces are generally interdependent. Nevertheless, it is possible to control the phase of the transmitted light with constant power transmission, which could be 100% in ideal conditions without losses, by carefully designing the resonant components of the metasurfaces. Metasurfaces for the arbitrary control of transmission properties, or amplitude and phase control, are called Huygens' metasurfaces, which have been introduced by Pfeiffer and Grbic [73]. In this subsection, we clarify the role of self-duality in Huygens' metasurfaces.

#### - Transmission and Reflection for Huygens' Metasurfaces

It is assumed that monochromatic plane electromagnetic waves with a specific polarization are normally incident on an isotropic metasurface placed at *z* = 0 in a vacuum with a wave impedance of *Z*0. In this paper, the variable with a tilde represents the complex amplitude of a harmonically oscillating quantity *A* = *A* ˜ e j*ω<sup>t</sup>* + c.c., where c.c. denotes the complex conjugate of the preceding term. The complex amplitudes of macroscopic electric and magnetic fields, which are averaged over the typical scale of metasurface elements, are represented by *E* ˜ − and *H* ˜ − (*E* ˜ + and *H* ˜ +) for the input (output) side *z* ≤ 0 (*z* ≥ 0) in the proximity to the surface. Although electric and magnetic fields are represented by vectors, we here focus on scalar amplitudes for a specific polarization. Electromagnetic response of the metasurface is characterized by two parameters: electric sheet admittance *Y*e and magnetic sheet impedance *Z*m. The averaged electric fields *E* ˜ av = (*E*˜− + *E*˜+)/2 induce surface currents *K*˜ = *<sup>Y</sup>*e*<sup>E</sup>*˜av, which demand the boundary condition *H* ˜ − − *H* ˜ + = *K* ˜ on *z* = 0 [74]. In the same way, the magnetic counterpart can be considered, and the averaged magnetic fields *H* ˜ av = (*H*˜ − + *H*˜ +)/2 produce surface magnetic currents *K* ˜ m = *Z*m*H* ˜ av, which require the boundary condition *E* ˜ − − *E* ˜ + = *K* ˜ m. The boundary conditions are summarized as

$$
\hat{H}\_- - \hat{H}\_+ = Y\_\circ \frac{\tilde{E}\_- + \tilde{E}\_+}{2},
\tag{22}
$$

$$
\mathcal{E}\_{-}-\mathcal{E}\_{+} = Z\_{\text{m}}\frac{\hat{H}\_{-}+\hat{H}\_{+}}{2}.\tag{23}
$$

For the incident wave propagating in the +*z* direction with the electric field *E* ˜ in e − j*kz* and the magnetic field *H* ˜ in e − <sup>j</sup>*kz*(= *E*˜in e<sup>−</sup> <sup>j</sup>*kz*/*Z*0), the total electric and magnetic fields (*E*˜, *H*˜ ) are represented as (*E* ˜ in e − j*kz* + *<sup>E</sup>*˜in e j*kz*, *H*˜in e<sup>−</sup> j*kz* − *<sup>H</sup>*˜in e <sup>j</sup>*kz*) in *z* ≤ 0 and (*τ<sup>E</sup>*˜in e<sup>−</sup> j*kz*, *τ<sup>H</sup>*˜in e<sup>−</sup> <sup>j</sup>*kz*) in *z* ≥ 0, where *τ* and are the amplitude transmission and reflection coefficients. By substituting these fields at the metasurface (*z* = 0) into Equations (22) and (23), the amplitude transmission and reflection coefficients are obtained as

$$\begin{array}{rcl} \mathsf{LT} &=& \frac{(Z\_2 - Z\_1)Z\_0}{(Z\_0 + Z\_1)(Z\_0 + Z\_2)} \\ \mathsf{q} &=& \frac{Z\_1 Z\_2 - (Z\_0)^2}{(Z\_0 + Z\_1)(Z\_0 + Z\_2)} \end{array} \tag{24}$$

where *Z*1 = *Z*m/2 and *Z*2 = 2/*Y*e. Hence, the reflection vanishes for *Z*1*Z*2 = (*<sup>Z</sup>*0)2. In addition to the no-reflection condition, if *Z*1 and *Z*2 are purely imaginary impedances, which are expressed as *Z*1 = (*<sup>Z</sup>*0)2/*<sup>Z</sup>*2 = j*bZ*0 with a dimensionless number *b* ∈ R, the transmission coefficient can be written as

$$
\pi = \frac{1 - \text{j}b}{1 + \text{j}b}.\tag{25}
$$

The incident waves are perfectly transmitted through the metasurface due to the fact that |*τ*|<sup>2</sup> = 1 for any *b*, and the transmitted waves acquire a phase of *θ* = −2 arctan *b*. Such metasurfaces, which are typical examples of Huygens' metasurfaces, realize arbitrary phase shift *θ* without losses in an ideal case by tailoring the design of the metasurface structures. Both of the electric and magnetic responses are indispensable for the no-reflection condition.

#### - Circuit Model for Huygens' Metasurfaces

The propagation of electromagnetic waves in a vacuum can be modeled as signal propagation in a transmission line with the wave impedance *Z*0, and the metasurface is represented by circuit elements inserted in the transmission line as shown in Figure 23a. In this model, the electric field *E* ˜ and magnetic field *H* ˜ are replaced with voltage *V* ˜ and current ˜ *I*, respectively; therefore, the circuit model of the metasurface should satisfy the following conditions:

$$
\tilde{I}\_- - \tilde{I}\_+ = \Upsilon\_0 \frac{\tilde{\mathcal{V}}\_- + \tilde{\mathcal{V}}\_+}{2},
\tag{26}
$$

$$
\mathcal{V}\_{-}-\mathcal{V}\_{+}=Z\_{\text{m}}\frac{\tilde{I}\_{-}+\tilde{I}\_{+}}{2}.\tag{27}
$$

A circuit called a lattice circuit as shown in Figure 23b satisfies the above conditions [75]. The electric and magnetic responses are represented by impedances *<sup>Z</sup>*2(= 2/*Y*e) and *<sup>Z</sup>*1(= *<sup>Z</sup>*m/2), respectively. Equations (26) and (27) can be confirmed separately for Figure 23b, considering excitation by waves from both sides of the metasurface. For in-phase excitation *V* ˜ − = *V* ˜ +, all currents are sunk into the bridge circuit, and the currents become antiphase ˜ *I*− = − ˜ *I*+. There is no voltage across *Z*1, and the currents flow only in *Z*2. Hence, we obtain *V* ˜ − = *Z*2 ˜ *I*−, which is identical to Equation (26) for *V* ˜ − = *V* ˜ + and ˜ *I*− = − ˜ *I*+. In the opposite case, *V* ˜ − = −*V* ˜ + and ˜ *I*− = ˜ *I*+, the currents flow only in *Z*1, and *V* ˜ − = *Z*1 ˜ *I*−, so the equation that corresponds to Equation (27) can be derived.

#### - Zero Backscattering Due to Self-Duality

For Figure 23a, *Z*in = *V* ˜ −/ ˜ *I*− provides the input impedance for the metasurface, or lattice circuit, followed by the transmission line in *z* > 0. The uniform semi-infinite transmission line in *z* > 0 can be regarded as a resistor with an impedance of *Z*0 as shown in Figure 21g. As a result, the total system viewed from the input side *z* < 0 is well described by a bridge circuit as shown in Figure 23c, and *Z*in is identical to the impedance of the bridge circuit. As described in Section 2.6, the bridge circuit satisfying *Z*1*Z*2 = (*<sup>Z</sup>*0)<sup>2</sup> is self-dual for the inversion center *Z*0, and the input impedance *Z*in is always *Z*0. The reflection vanishes under this condition, where the wave impedance *Z*0 is impedance-matched to the load represented by the

bridge circuit. As a result, all energy is transmitted to the transmission line in *z* > 0. Thus, the no-reflection condition for Huygens' metasurfaces is interpreted in terms of self-duality.

**Figure 23.** (**a**) circuit model for propagating electromagnetic waves incident on a metasurface; (**b**) circuit model of Huygens' metasurfaces with *Z*1 = *Z*m/2 and *Z*2 = 2/*Y*e; (**c**) lumped circuit model for the metasurface followed by a semi-infinite transmission line.
