- Self-Complementary Antennas

In antenna theory, it is well-known that antennas with self-complementary geometry show a frequency-independent input impedance, which is defined as the ratio of voltage and current at a feeding point of an antenna, and such antennas are called self-complementary antennas [13].

Before moving on to the self-complementary case, we introduce duality for the effective response of antennas. Consider an antenna with an electric sheet admittance *<sup>Y</sup>*e(*<sup>x</sup>*, *y*)(= *<sup>Z</sup>*e(*<sup>x</sup>*, *y*)−<sup>1</sup>) on *z* = 0 as shown in Figure 48a. On a rectangular patch *S*, an external voltage distribution *E* ˜ ext(*<sup>x</sup>*, *y*) is applied. We assume that *E* ˜ ext is directed in the *y*-direction. For the dual setup, we set *<sup>Z</sup>*e satisfying Equation (137) and an external current *K* ˜ ext = 2*Y*0Ω*<sup>σ</sup>ez* × *E* ˜ ext on *S* in Figure 48b. In particular, when antennas are made only of perfect electric conductor (*<sup>Y</sup>*e = <sup>∞</sup>), the corresponding dual antennas have complementary shapes, which are obtained by interchanging metallic regions and hole regions. The input impedances of the original and dual antenna are denoted by *Z*in and *<sup>Z</sup>*in, respectively. From Babinet duality, these input impedances satisfy

$$Z\_{\rm in} Z\_{\rm in}^{\star} = \left(\frac{Z\_0}{2}\right)^2,\tag{148}$$

as derived in Appendix C. This equation means that the input impedance of the dual antenna is related to that of the original antenna.

**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.

Now, we consider self-dual antennas. An example of a self-dual antenna is realized with a self-complementary geometry as shown in Figure 49. From Equation (148) and self-dual condition *Z*in = *<sup>Z</sup>*in, we obtain

$$Z\_{\rm in} = Z\_{\rm in}^{\star} = \frac{Z\_0}{2} \, , \tag{149}$$

and this means that the input impedance of self-complementary antennas is independent of frequency. Thus, the principle of self-complementary antennas called the "Mushiake Principle" [96] plays an important role in designing broadband antennas [13]. Because semi-infinite free spaces in *z* > 0 and *z* < 0 are seen as two parallel transmission lines with the characteristic impedance of *Z*0, Equation (149) shows that self-complementary antennas are perfectly matched to the composite impedance of *Z*0/2.

**Figure 49.** Example of a self-complementary antenna.


Here, we discuss critical behaviors of metallic checkerboard-like metasurfaces from the point of view of self-duality in Babinet duality. We assume the metasurfaces are placed in a vacuum and made of a perfect electric conductor. If we assume the periodicity of the checkerboard-like metasurfaces, we can classify them into three distinct cases as shown in Figure 50: (a) metallic patches are disconnected (disconnected phase), (b) metasurface is self-complementary, i.e., the metallic patches touch each other at ideal point contacts (self-complementary point), and (c) metallic patches are connected (connected phase). Note that structures in (c) are complementary to those in (a) if *w* = *w*; therefore, the two phases are related through Babinet duality. Now, consider the transition from the disconnected phase to the connected phase. Under this transition, the checkerboard-like metasurface passes through the self-complementary point between the two phases, as shown in Figure 50. As we

see below, the electromagnetic responses of checkerboard-like metasurfaces abruptly change at this self-complementary point, and this point is actually a *singular* point [44,45,47].

At first, we explain general transmission properties of checkerboard-like metasurfaces. We consider that a circularly polarized plane wave with wavevector *k* = *k<sup>e</sup>z* is normally incident on the metasurfaces, and we observe the transmission behind the metasurfaces. For the disconnected phase shown in Figure 50a, the power-transmission spectrum is typically like the lower panel of Figure 50a. The metasurfaces in the disconnected phase behave as capacitive filters, which highly transmit lower frequency components, while they resonantly reflect around the higher resonance frequency [97]. Note that the incident frequency axis is clipped at the lowest diffraction frequency *<sup>c</sup>*0/*<sup>a</sup>*, where *a* is the size of the unit cell of the metasurfaces. Above the lowest diffraction frequency, the incident energy is enabled to be transmitted into higher-order diffraction modes with *k* = ±*k<sup>e</sup>z*. In other words, effective loss (mode-conversion loss) appears for the zeroth-order mode. To restrict our discussion to the single mode, we consider frequencies below *c*0/*a* in the following. In addition to this constraint, thanks to the 4-fold rotational symmetry of the metasurfaces, polarization conversion is prohibited (*τ*<sup>⊥</sup> = *ρ*⊥ = 0). Then, we can use Equation (147) to obtain the power-transmission spectrum of the complementary structures: |*τ*| 2 = 1 − |*τ*| 2. In other words, the connected phase shown in Figure 50c exhibits transmission spectra with the upside-down shapes to those of the disconnected phase: it highly reflects lower frequency components, while it resonantly transmits around the higher resonant frequency, as shown in the lower panel of Figure 50c [97].

Now, we derive the criticality of the self-complementary structure. At the self-complementary point shown in Figure 50b, we have

$$
\pi = \pi^\*,
\tag{150}
$$

from the self-duality of the problem. Combining above with Equations (144) and (145), we can derive

$$
\tau = \tau^\star = -\varrho = \frac{1}{2}.\tag{151}
$$

Then, it is concluded that the self-complementary metasurface shows a *finite dissipation A* = 1 − |*τ*| 2 − || 2 = 1/2. The finite dissipation obviously contradicts the assumption that the system is made of a lossless perfect electric conductor. Note that this contradiction occurs below the diffraction frequency *<sup>c</sup>*0/*<sup>a</sup>*. For frequencies above *<sup>c</sup>*0/*<sup>a</sup>*, scattering into the diffraction modes is enabled, and thus *A* is composed of not only dissipation but also mode conversion. In addition to this contradiction, the frequency-independent behavior also contradicts Foster's reactance theorem, which states that the imaginary part of the impedance of a lossless and passive system must increase monotonically with the frequency [98]. Thus, the transmission spectra of such systems cannot be flat. Consequently, we can conclude that there is no physical solution for the ideal self-complementary point and thus the checkerboard-like metasurface at the self-complementary point is singular.

**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*.

Note that the *dissipative* intermediate structure in the transition between the disconnected and connected phases is physically consistent, although the *lossless* intermediate structure is singular. Such a dissipative intermediate structure, which contains dissipative elements (resistive sheets with sheet impedance of *Z*0/2) at the connecting points of the checkerboard-like metasurfaces, can be characterized by the novel frequency-independent response with *τ* = − = 1/2 and dissipation *A* = 1/2 [37], as shown in Figure 51. The frequency-independent response has been experimentally observed in the terahertz frequency region [38]. In addition, introducing randomness into the connectivity of the metallic patches also leads to the similar flat spectrum [42,48]. In this random case, loss *A* results from mode conversion due to the randomness of the metasurface structure.

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

The criticality of metallic checkerboard-like metasurfaces is utilized for dynamical metasurfaces to manipulate electromagnetic waves. By placing photoconductive materials like silicon or phase-change materials like vanadium dioxide (VO2) between the metallic patches of checkerboard-like metasurfaces, researchers have realized optically tunable waveguides [99], capacitive–inductive switchable filters [50], dynamical polarizers [49], and dynamical switching of quarter-wavelength plates [52]. In addition to these experiments, dynamical planar-chirality switching is also theoretically proposed [51]. The advantage of these dynamical checkerboard-like metasurfaces is that we can achieve deep modulation of the electromagnetic characteristics of the metasurfaces because we dynamically induce phase transitions of the checkerboard-like metasurfaces.

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

Here, we consider Babinet duality in light of equivalent circuit models of metasurfaces. As discussed in Section 3, responses of metasurfaces can be described by equivalent circuit models. An electric metasurface in a vacuum can be modeled by an effective shunt impedance *Z*sh inserted between two semi-infinite transmission lines with the characteristic impedance of *Z*0 and the phase velocity of *c*0 as shown in Figure 52. The complex amplitude transmission coefficient *τ* := *V* ˜ t/*V* ˜ in through the metasurface is written as

$$\tau = \frac{Z\_{\rm sh}}{Z\_0/2 + Z\_{\rm sh}}.\tag{152}$$

On the other hand, for the dual problem with the complementary metasurface, the dual transmission coefficient *τ* is written as

$$
\pi^\* = \frac{Z\_{\rm sh}^\*}{Z\_0/2 + Z\_{\rm sh}^\*} \tag{153}
$$

with the effective impedance of the dual metasurface *<sup>Z</sup>*sh.

Requiring the same duality relation with Equation (144):

$$
\pi + \pi^\* = 1,\tag{154}
$$

we obtain

$$Z\_{\rm sh} Z\_{\rm sh}^{\star} = \left(\frac{Z\_0}{2}\right)^2. \tag{155}$$

This equation indicates that the effective impedance of the complementary metasurface is given by the *dual* of the effective impedance of the original metasurface with respect to the reference impedance *Z*0/2. In other words, the response of the complementary metasurface can be described by the dual circuit to the equivalent circuit of the original metasurface.

In particular, when the shunting equivalent circuit is a self-dual circuit like the bridge circuit shown in Figure 19, i.e., *Z*sh = *<sup>Z</sup>*sh = *Z*0/2 (see Section 2.6), we have *τ* = *τ* = 1/2, and the transmission and reflection characteristics are frequency independent. Thus, we can connect the frequency-independent responses of self-dual resistive checkerboard-like metasurfaces to the constant resistance property of self-dual circuits.

**Figure 52.** Transmission-line model for scattering of plane waves by an electric metasurface.
