G-0K8J8ZR168
검색
검색 팝업 닫기

Ex) Article Title, Author, Keywords

## Article

Curr. Opt. Photon. 2023; 7(1): 97-103

Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.97

## Babinet-principle-inspired Metasurfaces for Resonant Enhancement of Local Magnetic Fields

Seojoo Lee1, Ji-Hun Kang2,3,4

1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA
2Department of Optical Engineering, Kongju National University, Cheonan 31080, Korea
3Department of Future Convergence Engineering, Kongju National University, Cheonan 31080, Korea
4Institute of Application and Fusion for Light, Kongju National University, Cheonan 31080, Korea

Corresponding author: *jihunkang@kongju.ac.kr, ORCID 0000-0002-2201-1689

Received: November 29, 2022; Revised: January 1, 2023; Accepted: January 1, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we propose Babinet-principle-inspired metasurfaces for strong resonant enhancement of local magnetic fields. The metasurfaces are designed as complementary structures of original meta-surfaces supporting the local enhancement of electric fields. We show numerically that the complementary structures can support spoof magnetic surface plasmons that induce strong local magnetic fields without sacrificing the deep sub-wavelength-thick nature of the metasurface. By introducing a periodic array of metallic rods in the proximity of the metasurfaces, we demonstrate that a resonant enhancement of the local magnetic fields, more than 80 times the amplitude of an incident magnetic field, can emerge from a resonance of the spoof magnetic surface plasmons.

Keywords: Babinet's principle, Complementary structure, Magnetic resonance, Metamaterials, Metasurfaces

OCIS codes: (160.3918) Metamaterials; (240.6680) Surface plasmons; (260.5740) Resonance

### I. INTRODUCTION

Manipulations of electromagnetic (EM) waves by using structured media require a deep understanding of light interaction with the media and consequent EM field distributions in both near and far fields [17]. A prime example is the designing of optical metamaterials and their two-dimensional equivalent, metasurfaces. Engineering of far-field transmission and reflection coefficients using a metasurface allows us to acquire unprecedented effective electric permittivities and magnetic permeabilities [810], while control of the diffracted EM waves in the proximity of a metasurface has enabled metamaterial-based sub-wavelength optical resonators [3, 11].

Although the near and far fields are connected only in terms of the effective description (i.e. the equivalence of the far field to the spatial averaging of the near field), their phenomenological relationships can be found from the successful excitation of the surface-bound EM waves, so-called spoof surface plasmons (SSPs), by a metasurface with effective negative permittivity [9, 1113]. This fact has removed an ambiguity of relating the effective indices with optical responses in a near-field, so that various plasmonic metasurfaces have been designed based on the negative effective permittivity that makes use of a resonant response of the electric field [14, 15]. Since the effective description works very well for the SSPs, it is quite reasonable to expect the excitation of their counterpart, spoof magnetic surface plasmons (SMSPs), by a metasurface with negative effective permeability. However, while a resonant electric response and negative effective permittivity are quite feasible with a metasurface, the deep sub-wavelength-thick nature of the metasurface has been assumed to be a main obstacle for the resonant magnetic response and negative permeability because the polarization of the magnetic field is considered to be out-of-plane of ring-type unit resonators in the metasurface to have induced loop currents. A cylinder corrugated structure made of a perfect electric conductor (PEC) [1618] and a simple grating structure patterned on a perfect magnetic conductor [19] were introduced to excite SMSPs in a sample with sub-wavelength thickness. As a strong local magnetic response plays a crucial role in the spectral regimes of terahertz and higher frequencies [20], resonant excitation of SMSPs in a thin structure has great importance for the further spatial localization of strong magnetic fields.

Here, we demonstrate numerically that Babinet’s principle can be an alternate way to design metasurfaces for the strong local magnetic resonance without sacrificing the deep sub-wavelength-thick property. Our metasurfaces are conceived as Babinet’s complementary structures of their counterparts, plasmonic metasurfaces with negative effective permittivity. We show that the complementary structures can support SMSPs that induce strong local magnetic fields. By introducing a periodic array of metallic rods in the proximity of the complementary metasurfaces, we demonstrate that a resonant enhancement of the local magnetic fields can emerge from a resonance of the SMSPs.

### 2.1. Babinet’s Principle and Metasurfaces

Babinet’s principle states that the diffraction patterns from the original diffracting structure made of PEC are the same as those emerging from the Babinet-inverted complementary structure with exchanged polarizations of the incident electric and magnetic fields [21]. It should be noted that a rigorous statement of Babinet’s principle requires the diffractive structure to be infinitesimally thin. However, it has been demonstrated that the spirit of Babinet’s principle is also valid for a structure with sub-wavelength but finite thickness to predict qualitatively, not quantitatively, the diffraction patterns from the counterpart structure [22].

In terms of Babinet inversion, metasurfaces are one of the most suitable structures. As shown in Fig. 1, it is not clear whether Babinet’s principle can be applied in the case of excitation of surface plasmons in real metal with negative permittivity. However, a plasmonic metasurface with negative effective permittivity, operating in a terahertz or microwave spectral regime where most metals can be considered a PEC, is fully applicable to Babinet’s principle. Here, our main idea comes from the fact that SSPs supported by the plasmonic metasurface are a phenomenological description of surface-bound waves excited by the diffraction of incident EM waves. Therefore, if Babinet’s principle can be applied to the plasmonic metasurface, SSPs would appear in the form of SMSPs.

Figure 1.The idea of Babinet inversion and the complementary structure of a metasurface. Applying Babinet inversion is not clear for a homogeneous metal plate with finite negative permittivity. In all cases, we assume that the thicknesses of the structures are at a deep sub-wavelength scale.

For negative effective permittivity, let us first consider an exemplary metasurface consisting of rectangular metallic ring resonators as shown in Fig. 2(a). Because Babinet inversion will be applied later, we designed the metasurface to operate in the microwave spectral regime. In our previous study, it was shown that this structure can exhibit negative effective permittivity as a result of a competition between two light channels in the metasurface, and that the effective permittivity in the x-direction can be written as

Figure 2.Metasurfaces for spoof electric and magnetic surface plasmons. (a) A metasurface of negative effective permittivity, made of PEC ring resonators. The red dot contour denotes the unit lattice of the metasurface. Here, we set a = 3 mm, b = 42 mm, Px = 6 mm, Py = 45 mm. The thickness of the metasurface is set to be 1 mm. For the resonators, the width of the rim is 1 mm. The distance between neighboring unit resonators in both x and y directions is given as 1 mm. (b) The complementary structure of (a). PEC, perfect electric conductor.

εeffα1ωt2ω2

when the unit resonators are tightly coupled to each other [14, 23]. Here, α and ωt are a positive coefficient and the transition frequency, respectively: both are defined by the geometrical parameters of a metasurface. Since it is able to have a negative effective permittivity, this type of metasurface can support SSPs with an x-polarized incident light when ω > ωt. However, for the excitation of the SSPs, it should be noted that momentum matching between the SSPs and the incident light is required, for example by locating another diffractive structure near the metasurface or just by truncating the metasurface; the latter has been demonstrated in our previous study [11]. We also note that the aspect ratio of the two sides of the ring resonator, a and b, are set to be large (a << b) to provide a sufficient structural resolution of the surface to both SSPs and SMSPs.

Now, one can readily apply Babinet inversion to the metasurface. For 0 ≤ zh, the metallic resonators and free space in the original metasurface become free space and metal in the inverted metasurface, respectively, as shown in Fig. 2(b). The polarization of the incident changes to the z-direction.

### 2.2. Excitation of SMSPs with Babinet-inverted Metasurfaces

With the Babinet-inverted metasurfaces, we numerically calculated the near-field spectra of the x-component of the magnetic field (Hx) and far-field transmission coefficients by using the finite-difference time-domain (FDTD) method. We considered a TE-polarized incident (Hx, Ey) light that is impinging normally upon the metasurfaces. Shown in Fig. 3 are the spectra of local enhancement of Hx that are taken near the metasurface at (z = 0.2 mm), and the far-field (zero-th order) transmission. The spatial distribution of EM waves in the near-field is primarily determined by the structural detail of the metasurface. We can see a resonant local enhancement of the magnetic field around 2.8 GHz. The enhancement factor, normalized by the incident magnetic field, is shown to be more than 8. We note that this peak corresponds to the resonant transmission of EM waves, exhibiting 100% light passing through the metasurface, as shown by blue dashed line. This means that the local enhancement is due to the resonant funneling of the incident EM waves [24, 25]: all of the incident energy is spatially concentrated while passing through the narrow light channels in the metasurface.

Figure 3.Finite-difference time-domain (FDTD)-calculated spectra of near-field enhancement of Hx (black solid curve), and far-field transmission (blue dashed line). The near field is taken at the center of the core perfect electric conductor (PEC) plate of the unit resonator [Fig. 2(b)], with the measuring height 0.2 mm from the Babinet-inverted metasurface. The enhancement factor is defined as |Hx|loc / |Hx|inc, where |Hx|loc and |Hx|inc are amplitudes of the local magnetic field and the incident, respectively. For transmission, 1.0 means a 100% transmission.

We have seen that the Babinet-inverted metasurface itself already allows local enhancement of the magnetic field without introducing surface-bound waves like SMSPs. In order to incorporate the SMSPs into the magnetic field enhancement, we located PEC rods in front of the metasurface as shown in Fig. 4(a). The rods are infinitely long in the y-direction, and located periodically in the x-direction with three times the periodicity of the metasurface lattice (3px). The role of the rods is to diffract the incident waves, splitting some portion of the incident into EM continuum of planewaves with −∞ ≤ kx ≤ ∞ where kx is the momentum of a planewave fraction in the x-direction. This eventually allows the excitation of the SMSPs via momentum matching of SMSPs with diffracted incident waves. Shown in Fig. 4(b) are near-field spectra of Hx taken at two points p1 (black solid curve) and p2 (red dashed curve) that are located at (x, y, z) = (0, 0, −0.2 mm) and (3px/4, 0, −0.2 mm), respectively. In both cases, we can see very sharp spectral peaks at 3.079 GHz. The enhancement factor reaches around 80, which is about 10-fold greater than that by the bare metasurface without PEC rods. However, we also can see that a broad peak at 2.907 GHz, where the enhancement factor is around 20, appears only for the spectrum taken at p2, while there is no apparent broad resonant behavior in the spectrum taken at p1.

Figure 4.Local enhancement of Hx by the excitation of spoof magnetic surface plasmons. (a) Schematic of the system of metasurface with diffractive perfect electric conductor (PEC) rods. The structural parameters for the metasurface are the same as those in Fig. 3. The width and thickness of the PEC rods are 2.5 mm and 1 mm, respectively, and the rods are located 1 mm from the metasurface. The yellow dot in the top view indicates the origin of the x-y plane. p1 and p2 are 0.2 mm from the metasurface. The periodic boundary condition (PBC) is applied to the system, resulting in an infinitely periodic system in both x- and y-directions. (b) Finite-difference time-domain (FDTD)-calculated spectra of Hx taken at p1 (black solid curve) and p2 (red dashed curve). All spectra are normalized by the amplitude of the incident.

To see more details of the resonant behavior, we increased the periodicity of the PEC rods to five times the lattice of the metasurface (5px) as shown in Fig. 5(a). We took near-field spectra at points p1 and p2, which are located at (x, y, z) = (0, 0, −0.2 mm) and (5px/4, 0, −0.2 mm), respectively. In Fig. 5(b), we can again see that a broad peak is only noticeable for p1 at 2.884 GHz. However, compared to the previous case in which only a single sharp spectral peak was observable, we now have two sharp resonant peaks at 3.044 GHz and 3.090 GHz. This spectral dependency on the periodicity of the rods can be explained in terms of the resonant cavity mode of SMSPs, where the cavity boundaries are defined by the PEC rods. Wider rod periodicity of the PEC rods allows SMSPs to be a cavity mode with a longer wavelength, or a higher-order cavity mode with a wavelength not significantly changed. Also, it is quite reasonable to assume that the shortest possible wavelength of SMSPs is limited by the finite lattice size of the metasurface in the x-direction. We note that this is why we set a << b (see Fig. 2) to accommodate more SMSP modes in a narrower cavity.

Figure 5.Higher order resonance of spoof magnetic surface plasmons. (a) Schematic of the system of metasurface and PEC rods with increased periodicity of the rods. All structural parameters, except for the periodicity of the PEC rods are exactly the same as those used in Fig. 4. In the side-view, p1 and p2 are 0.2 mm from the metasurface. (b) FDTD-calculated spectra of Hx taken at p. PEC, perfect electric conductor; FDTD, finite-difference time-domain.

A direct way to confirm this model based on the cavity mode of SMSPs is to see how the cavity modes, corresponding to the spectral peaks, form the near-field. Shown in Figs. 6(a) and 6(b) are numerically calculated x-y maps of near-field profiles of |Hx| for the two spectral peaks at 2.907 GHz and 3.079 GHz, respectively. The maps are obtained at three different distances from the metasurface (z = −0.1 mm, z = −1 mm, and z = −5 mm). At z = −0.1 mm, the field maps show structural details of the metasurface so that it is not clear to determine the cavity mode. However, those surface details in the near-field maps get less pronounced as the distance from the metasurface increases to 1 mm and 5 mm. In Fig. 6(a) for the broader peak shown in Fig. 4(b), the field amplitude is minimized at the center of the map on which the PEC rod is located, while in Fig. 6(b), for the sharp peak, the center is maximized. Specifically, the spatial configuration of the field map in Fig. 6(b) forms a whole wavelength of the mode inside the cavity, which supports our aforementioned cavity model. Therefore, from Figs. 6(a) and 6(b), we can have a preliminary conclusion that the broader peak in Fig. 4(b) is related to the direct transmission of the incident EM wave which is discussed in Fig. 3, and that the sharp peak is resonant cavity mode of surface-bound waves called SMSPs.

Figure 6.FDTD-calculated near-field x-y maps of |Hx| for the rods/metasurface system where the periodicity of the rods is 3px (Fig. 4). The maps are taken at three different distances from the metasurface as denoted by m1, m2, and m3 in (a). (a) Maps for 2.907 GHz and (b) 3.079 GHz incident frequencies. FDTD, finite-difference time-domain.

It should be noted that the cavity should be able to support higher modes, in principle. However, as we have discussed previously, the finite surface resolution of the metasurface limits the excitation of SMSPs with a shorter wavelength, so that what we see in Figs. 4 and 6 is only a single cavity mode. In this sense, we have assumed that the additional sharp spectral peak in Fig. 5(b) at 3.090 GHz could be the higher cavity mode, and this is confirmed by Fig. 7. Shown in Figs. 7(a) and 7(b) are near-field maps of |Hx| for the two spectral peaks at 3.044 GHz and 3.090 GHz in Fig. 5, respectively. The field maps are taken at z = −0.1 mm and z = −5 mm. Again, for the field maps at z = −0.1 mm, the structural details of the metasurface are too clear to see the cavity mode, and increasing the distance from the metasurface helps us to distinguish the cavity mode. As shown in Fig. 7(a), the field map at 3.044 GHz shows that one wavelength is roughly formed inside the cavity. This means that the peak at 3.044 GHz in Fig. 5(b) corresponds to that at 3.079 GHz in Fig. 4(b). Also, in Fig. 7(b), we can see that an apparent two full wavelengths are formed inside the cavity, meaning that the peak at 3.090 in Fig. 5(b) is the second cavity mode, which is not observable in Fig. 4(b) due to the limited spatial resolution of the metasurface.

Figure 7.FDTD-calculated near-field x-y maps of |Hx| for the rods/metasurface system where the periodicity of the rods is 5px (Fig. 5). The maps are taken at two different distances from the metasurface as denoted by m1, and m3 in (a). (a) Maps for 3.044 GHz and (b) 3.090 GHz incident frequencies. FDTD, finite-difference time-domain.

### III. Discussion

We have applied Babinet’s principle to a plasmonic metasurface of effective negative permittivity supporting SSPs, and have successfully demonstrated the excitation of SMSPs in the Babinet-inverted metamaterial systems. We note that the excitation of the SMSPs by the spirit of Babinet’s principle is a phenomenological interpretation of light diffraction by metasurfaces in a near field, and is not strictly related to the effective medium description of the metasurfaces. Specifically, a Babinet inversion of a metasurface with negative effective permittivity does not mean that the complementary metasurface must possess negative permeability. A prime example of this can be found in [10], demonstrating that a Babinet inversion of a plasmonic structure with negative effective permittivity can result in a complementary structure with a positive effective permittivity with a high effective index of refraction.

### IV. CONCLUSION

In this paper, we have demonstrated a Babinet-principle-inspired metasurface for resonant enhancement of magnetic fields. By applying Babinet’s principle to a plasmonic metasurface supporting SSPs in the form of surface-bound waves, we have shown that the complementary structures can support SMSPs that induce strong local magnetic fields. For excitation of the spoof magnetic plasmons and resonant enhancement of the magnetic fields, we introduced a periodic array of diffractive rods in the proximity of the metasurface. The resonant enhancement of the local magnetic fields has been shown to emerge from the resonance of the SMSPs inside a cavity defined by the periodically located rods. We believe that our scheme provides an intuitive way to realize magnetic resonance in an ultrathin structure, and that the proposed metasurface system could play important roles in various disciplines where strongly enhanced magnetic fields are required.

### DISCLOSURES

The authors declare no conflicts of interests.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

Research grant from Kongju National University in 2020; National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (Grant No. NRF-2021R1A2C2012617, NRF-2020R1C1C1012138).

### References

1. K. Yao and Y. Liu, “Plasmonic metamaterials,” Nanotechnol. Rev. 3, 177-210 (2014).
2. S. Yoo, S. Lee, J.-H. Choe, and Q.-H. Park, “Causal homogenization of metamaterials,” Nanophotonics 8, 1063-1069 (2019).
3. J.-H. Kang and Q.-H. Park, “Local enhancement of terahertz waves in structured metals,” IEEE Trans. Terahertz Sci. Technol. 6, 371-381 (2016).
4. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152-156 (2009).
5. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728-731 (2008).
6. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
7. J. H. Kang, D. S. Kim, and Q.-H. Park, “Local capacitor model for plasmonic electric field enhancement,” Phys. Rev. Lett. 102, 093906 (2009).
8. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369-373 (2011).
9. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
10. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
11. J.-H. Kang and Q.-H. Park, “Fractional tunnelling resonance in plasmonic media,” Sci. Rep. 3, 2423 (2013).
12. A. I. Fernández-Dománguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE J. Sel. Top. Quantum Electron. 14, 1515-1521 (2008).
13. C. Ropers, G. Stibenz, G. Steinmeyer, R. Müller, D. J. Park, K. G. Lee, J. E. Kihm, J. Kim, Q. H. Park, D. S. Kim, and C. Lienau, “Ultrafast dynamics of surface plasmon polaritons in plasmonic metamaterials,” Appl. Phys. B 84, 183-189 (2006).
14. J.-H. Kang, S.-J. Lee, B. J. Kang, W. T. Kim, F. Rotermund, and Q.-H. Park, “Anomalous wavelength scaling of tightly-coupled terahertz metasurfaces,” ACS Appl. Mater. Interfaces 10, 19331-19335 (2018).
15. S. Lee, W. T. Kim, J.-H. Kang, B. J. Kang, F. Rotermund, and Q.-H. Park, “Single-layer metasurfaces as spectrally tunable terahertz half-and quarter-waveplates,” ACS Appl. Mater. Interfaces 11, 7655-7660 (2019).
16. P. A. Huidobro, X. Shen, J. Cuerda, E. Moreno, L. Martin-Moreno, F. J. Garcia-Vidal, T. J. Cui, and J. B. Pendry, “Magnetic localized surface plasmons,” Phys. Rev. X 4, 021003 (2014).
17. F. J. Garcia-Vidal, A. I. Fernández-Domínguez, L. Martin-Moreno, H. C. Zhang, W. Tang, R. Peng, and T. J. Cui, “Spoof surface plasmon photonics,” Rev. Mod. Phys. 94, 025004 (2022).
18. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351-1353 (2004).
19. L.-L. Liu, Z. Li, C.-Q. Gu, P.-P. Ning, B.-Z. Xu, Z.-Y. Niu, and Y.-J. Zhao, “A corrugated perfect magnetic conductor surface supporting spoof surface magnon polaritons,” Opt. Express 22, 10675-10681 (2014).
20. C. Sirtori, “Bridge for the terahertz gap,” Nature 417, 132-133 (2002).
21. J. R. Jiménez and E. Hita, “Babinet’s principle in scalar theory of diffraction,” Opt. Rev. 8, 495-497 (2001).
22. S. Koo, M. S. Kumar, J. Shin, D. Kim, and N. Park, “Extraordinary magnetic field enhancement with metallic nanowire: Role of surface impedance in Babinet’s principle for sub-skin-depth regime,” Phys. Rev. Lett. 103, 263901 (2009).
23. S. Lee and J.-H. Kang, “Gap-size-dependent effective phase transition in metasurfaces of closed-ring resonators,” Crystals 11, 684 (2021).
24. J.-H. Choe, J.-H. Kang, D.-S. Kim, and Q.-H. Park, “Slot antenna as a bound charge oscillator,” Opt. Express 20, 6521-6526 (2012).
25. P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Hadar, and J. L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).

### Article

#### Article

Curr. Opt. Photon. 2023; 7(1): 97-103

Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.97

## Babinet-principle-inspired Metasurfaces for Resonant Enhancement of Local Magnetic Fields

Seojoo Lee1, Ji-Hun Kang2,3,4

1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA
2Department of Optical Engineering, Kongju National University, Cheonan 31080, Korea
3Department of Future Convergence Engineering, Kongju National University, Cheonan 31080, Korea
4Institute of Application and Fusion for Light, Kongju National University, Cheonan 31080, Korea

Correspondence to:*jihunkang@kongju.ac.kr, ORCID 0000-0002-2201-1689

Received: November 29, 2022; Revised: January 1, 2023; Accepted: January 1, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we propose Babinet-principle-inspired metasurfaces for strong resonant enhancement of local magnetic fields. The metasurfaces are designed as complementary structures of original meta-surfaces supporting the local enhancement of electric fields. We show numerically that the complementary structures can support spoof magnetic surface plasmons that induce strong local magnetic fields without sacrificing the deep sub-wavelength-thick nature of the metasurface. By introducing a periodic array of metallic rods in the proximity of the metasurfaces, we demonstrate that a resonant enhancement of the local magnetic fields, more than 80 times the amplitude of an incident magnetic field, can emerge from a resonance of the spoof magnetic surface plasmons.

Keywords: Babinet's principle, Complementary structure, Magnetic resonance, Metamaterials, Metasurfaces

### I. INTRODUCTION

Manipulations of electromagnetic (EM) waves by using structured media require a deep understanding of light interaction with the media and consequent EM field distributions in both near and far fields [17]. A prime example is the designing of optical metamaterials and their two-dimensional equivalent, metasurfaces. Engineering of far-field transmission and reflection coefficients using a metasurface allows us to acquire unprecedented effective electric permittivities and magnetic permeabilities [810], while control of the diffracted EM waves in the proximity of a metasurface has enabled metamaterial-based sub-wavelength optical resonators [3, 11].

Although the near and far fields are connected only in terms of the effective description (i.e. the equivalence of the far field to the spatial averaging of the near field), their phenomenological relationships can be found from the successful excitation of the surface-bound EM waves, so-called spoof surface plasmons (SSPs), by a metasurface with effective negative permittivity [9, 1113]. This fact has removed an ambiguity of relating the effective indices with optical responses in a near-field, so that various plasmonic metasurfaces have been designed based on the negative effective permittivity that makes use of a resonant response of the electric field [14, 15]. Since the effective description works very well for the SSPs, it is quite reasonable to expect the excitation of their counterpart, spoof magnetic surface plasmons (SMSPs), by a metasurface with negative effective permeability. However, while a resonant electric response and negative effective permittivity are quite feasible with a metasurface, the deep sub-wavelength-thick nature of the metasurface has been assumed to be a main obstacle for the resonant magnetic response and negative permeability because the polarization of the magnetic field is considered to be out-of-plane of ring-type unit resonators in the metasurface to have induced loop currents. A cylinder corrugated structure made of a perfect electric conductor (PEC) [1618] and a simple grating structure patterned on a perfect magnetic conductor [19] were introduced to excite SMSPs in a sample with sub-wavelength thickness. As a strong local magnetic response plays a crucial role in the spectral regimes of terahertz and higher frequencies [20], resonant excitation of SMSPs in a thin structure has great importance for the further spatial localization of strong magnetic fields.

Here, we demonstrate numerically that Babinet’s principle can be an alternate way to design metasurfaces for the strong local magnetic resonance without sacrificing the deep sub-wavelength-thick property. Our metasurfaces are conceived as Babinet’s complementary structures of their counterparts, plasmonic metasurfaces with negative effective permittivity. We show that the complementary structures can support SMSPs that induce strong local magnetic fields. By introducing a periodic array of metallic rods in the proximity of the complementary metasurfaces, we demonstrate that a resonant enhancement of the local magnetic fields can emerge from a resonance of the SMSPs.

### 2.1. Babinet’s Principle and Metasurfaces

Babinet’s principle states that the diffraction patterns from the original diffracting structure made of PEC are the same as those emerging from the Babinet-inverted complementary structure with exchanged polarizations of the incident electric and magnetic fields [21]. It should be noted that a rigorous statement of Babinet’s principle requires the diffractive structure to be infinitesimally thin. However, it has been demonstrated that the spirit of Babinet’s principle is also valid for a structure with sub-wavelength but finite thickness to predict qualitatively, not quantitatively, the diffraction patterns from the counterpart structure [22].

In terms of Babinet inversion, metasurfaces are one of the most suitable structures. As shown in Fig. 1, it is not clear whether Babinet’s principle can be applied in the case of excitation of surface plasmons in real metal with negative permittivity. However, a plasmonic metasurface with negative effective permittivity, operating in a terahertz or microwave spectral regime where most metals can be considered a PEC, is fully applicable to Babinet’s principle. Here, our main idea comes from the fact that SSPs supported by the plasmonic metasurface are a phenomenological description of surface-bound waves excited by the diffraction of incident EM waves. Therefore, if Babinet’s principle can be applied to the plasmonic metasurface, SSPs would appear in the form of SMSPs.

Figure 1. The idea of Babinet inversion and the complementary structure of a metasurface. Applying Babinet inversion is not clear for a homogeneous metal plate with finite negative permittivity. In all cases, we assume that the thicknesses of the structures are at a deep sub-wavelength scale.

For negative effective permittivity, let us first consider an exemplary metasurface consisting of rectangular metallic ring resonators as shown in Fig. 2(a). Because Babinet inversion will be applied later, we designed the metasurface to operate in the microwave spectral regime. In our previous study, it was shown that this structure can exhibit negative effective permittivity as a result of a competition between two light channels in the metasurface, and that the effective permittivity in the x-direction can be written as

Figure 2. Metasurfaces for spoof electric and magnetic surface plasmons. (a) A metasurface of negative effective permittivity, made of PEC ring resonators. The red dot contour denotes the unit lattice of the metasurface. Here, we set a = 3 mm, b = 42 mm, Px = 6 mm, Py = 45 mm. The thickness of the metasurface is set to be 1 mm. For the resonators, the width of the rim is 1 mm. The distance between neighboring unit resonators in both x and y directions is given as 1 mm. (b) The complementary structure of (a). PEC, perfect electric conductor.

$εeff≈α1ωt2−ω2$

when the unit resonators are tightly coupled to each other [14, 23]. Here, α and ωt are a positive coefficient and the transition frequency, respectively: both are defined by the geometrical parameters of a metasurface. Since it is able to have a negative effective permittivity, this type of metasurface can support SSPs with an x-polarized incident light when ω > ωt. However, for the excitation of the SSPs, it should be noted that momentum matching between the SSPs and the incident light is required, for example by locating another diffractive structure near the metasurface or just by truncating the metasurface; the latter has been demonstrated in our previous study [11]. We also note that the aspect ratio of the two sides of the ring resonator, a and b, are set to be large (a << b) to provide a sufficient structural resolution of the surface to both SSPs and SMSPs.

Now, one can readily apply Babinet inversion to the metasurface. For 0 ≤ zh, the metallic resonators and free space in the original metasurface become free space and metal in the inverted metasurface, respectively, as shown in Fig. 2(b). The polarization of the incident changes to the z-direction.

### 2.2. Excitation of SMSPs with Babinet-inverted Metasurfaces

With the Babinet-inverted metasurfaces, we numerically calculated the near-field spectra of the x-component of the magnetic field (Hx) and far-field transmission coefficients by using the finite-difference time-domain (FDTD) method. We considered a TE-polarized incident (Hx, Ey) light that is impinging normally upon the metasurfaces. Shown in Fig. 3 are the spectra of local enhancement of Hx that are taken near the metasurface at (z = 0.2 mm), and the far-field (zero-th order) transmission. The spatial distribution of EM waves in the near-field is primarily determined by the structural detail of the metasurface. We can see a resonant local enhancement of the magnetic field around 2.8 GHz. The enhancement factor, normalized by the incident magnetic field, is shown to be more than 8. We note that this peak corresponds to the resonant transmission of EM waves, exhibiting 100% light passing through the metasurface, as shown by blue dashed line. This means that the local enhancement is due to the resonant funneling of the incident EM waves [24, 25]: all of the incident energy is spatially concentrated while passing through the narrow light channels in the metasurface.

Figure 3. Finite-difference time-domain (FDTD)-calculated spectra of near-field enhancement of Hx (black solid curve), and far-field transmission (blue dashed line). The near field is taken at the center of the core perfect electric conductor (PEC) plate of the unit resonator [Fig. 2(b)], with the measuring height 0.2 mm from the Babinet-inverted metasurface. The enhancement factor is defined as |Hx|loc / |Hx|inc, where |Hx|loc and |Hx|inc are amplitudes of the local magnetic field and the incident, respectively. For transmission, 1.0 means a 100% transmission.

We have seen that the Babinet-inverted metasurface itself already allows local enhancement of the magnetic field without introducing surface-bound waves like SMSPs. In order to incorporate the SMSPs into the magnetic field enhancement, we located PEC rods in front of the metasurface as shown in Fig. 4(a). The rods are infinitely long in the y-direction, and located periodically in the x-direction with three times the periodicity of the metasurface lattice (3px). The role of the rods is to diffract the incident waves, splitting some portion of the incident into EM continuum of planewaves with −∞ ≤ kx ≤ ∞ where kx is the momentum of a planewave fraction in the x-direction. This eventually allows the excitation of the SMSPs via momentum matching of SMSPs with diffracted incident waves. Shown in Fig. 4(b) are near-field spectra of Hx taken at two points p1 (black solid curve) and p2 (red dashed curve) that are located at (x, y, z) = (0, 0, −0.2 mm) and (3px/4, 0, −0.2 mm), respectively. In both cases, we can see very sharp spectral peaks at 3.079 GHz. The enhancement factor reaches around 80, which is about 10-fold greater than that by the bare metasurface without PEC rods. However, we also can see that a broad peak at 2.907 GHz, where the enhancement factor is around 20, appears only for the spectrum taken at p2, while there is no apparent broad resonant behavior in the spectrum taken at p1.

Figure 4. Local enhancement of Hx by the excitation of spoof magnetic surface plasmons. (a) Schematic of the system of metasurface with diffractive perfect electric conductor (PEC) rods. The structural parameters for the metasurface are the same as those in Fig. 3. The width and thickness of the PEC rods are 2.5 mm and 1 mm, respectively, and the rods are located 1 mm from the metasurface. The yellow dot in the top view indicates the origin of the x-y plane. p1 and p2 are 0.2 mm from the metasurface. The periodic boundary condition (PBC) is applied to the system, resulting in an infinitely periodic system in both x- and y-directions. (b) Finite-difference time-domain (FDTD)-calculated spectra of Hx taken at p1 (black solid curve) and p2 (red dashed curve). All spectra are normalized by the amplitude of the incident.

To see more details of the resonant behavior, we increased the periodicity of the PEC rods to five times the lattice of the metasurface (5px) as shown in Fig. 5(a). We took near-field spectra at points p1 and p2, which are located at (x, y, z) = (0, 0, −0.2 mm) and (5px/4, 0, −0.2 mm), respectively. In Fig. 5(b), we can again see that a broad peak is only noticeable for p1 at 2.884 GHz. However, compared to the previous case in which only a single sharp spectral peak was observable, we now have two sharp resonant peaks at 3.044 GHz and 3.090 GHz. This spectral dependency on the periodicity of the rods can be explained in terms of the resonant cavity mode of SMSPs, where the cavity boundaries are defined by the PEC rods. Wider rod periodicity of the PEC rods allows SMSPs to be a cavity mode with a longer wavelength, or a higher-order cavity mode with a wavelength not significantly changed. Also, it is quite reasonable to assume that the shortest possible wavelength of SMSPs is limited by the finite lattice size of the metasurface in the x-direction. We note that this is why we set a << b (see Fig. 2) to accommodate more SMSP modes in a narrower cavity.

Figure 5. Higher order resonance of spoof magnetic surface plasmons. (a) Schematic of the system of metasurface and PEC rods with increased periodicity of the rods. All structural parameters, except for the periodicity of the PEC rods are exactly the same as those used in Fig. 4. In the side-view, p1 and p2 are 0.2 mm from the metasurface. (b) FDTD-calculated spectra of Hx taken at p. PEC, perfect electric conductor; FDTD, finite-difference time-domain.

A direct way to confirm this model based on the cavity mode of SMSPs is to see how the cavity modes, corresponding to the spectral peaks, form the near-field. Shown in Figs. 6(a) and 6(b) are numerically calculated x-y maps of near-field profiles of |Hx| for the two spectral peaks at 2.907 GHz and 3.079 GHz, respectively. The maps are obtained at three different distances from the metasurface (z = −0.1 mm, z = −1 mm, and z = −5 mm). At z = −0.1 mm, the field maps show structural details of the metasurface so that it is not clear to determine the cavity mode. However, those surface details in the near-field maps get less pronounced as the distance from the metasurface increases to 1 mm and 5 mm. In Fig. 6(a) for the broader peak shown in Fig. 4(b), the field amplitude is minimized at the center of the map on which the PEC rod is located, while in Fig. 6(b), for the sharp peak, the center is maximized. Specifically, the spatial configuration of the field map in Fig. 6(b) forms a whole wavelength of the mode inside the cavity, which supports our aforementioned cavity model. Therefore, from Figs. 6(a) and 6(b), we can have a preliminary conclusion that the broader peak in Fig. 4(b) is related to the direct transmission of the incident EM wave which is discussed in Fig. 3, and that the sharp peak is resonant cavity mode of surface-bound waves called SMSPs.

Figure 6. FDTD-calculated near-field x-y maps of |Hx| for the rods/metasurface system where the periodicity of the rods is 3px (Fig. 4). The maps are taken at three different distances from the metasurface as denoted by m1, m2, and m3 in (a). (a) Maps for 2.907 GHz and (b) 3.079 GHz incident frequencies. FDTD, finite-difference time-domain.

It should be noted that the cavity should be able to support higher modes, in principle. However, as we have discussed previously, the finite surface resolution of the metasurface limits the excitation of SMSPs with a shorter wavelength, so that what we see in Figs. 4 and 6 is only a single cavity mode. In this sense, we have assumed that the additional sharp spectral peak in Fig. 5(b) at 3.090 GHz could be the higher cavity mode, and this is confirmed by Fig. 7. Shown in Figs. 7(a) and 7(b) are near-field maps of |Hx| for the two spectral peaks at 3.044 GHz and 3.090 GHz in Fig. 5, respectively. The field maps are taken at z = −0.1 mm and z = −5 mm. Again, for the field maps at z = −0.1 mm, the structural details of the metasurface are too clear to see the cavity mode, and increasing the distance from the metasurface helps us to distinguish the cavity mode. As shown in Fig. 7(a), the field map at 3.044 GHz shows that one wavelength is roughly formed inside the cavity. This means that the peak at 3.044 GHz in Fig. 5(b) corresponds to that at 3.079 GHz in Fig. 4(b). Also, in Fig. 7(b), we can see that an apparent two full wavelengths are formed inside the cavity, meaning that the peak at 3.090 in Fig. 5(b) is the second cavity mode, which is not observable in Fig. 4(b) due to the limited spatial resolution of the metasurface.

Figure 7. FDTD-calculated near-field x-y maps of |Hx| for the rods/metasurface system where the periodicity of the rods is 5px (Fig. 5). The maps are taken at two different distances from the metasurface as denoted by m1, and m3 in (a). (a) Maps for 3.044 GHz and (b) 3.090 GHz incident frequencies. FDTD, finite-difference time-domain.

### III. Discussion

We have applied Babinet’s principle to a plasmonic metasurface of effective negative permittivity supporting SSPs, and have successfully demonstrated the excitation of SMSPs in the Babinet-inverted metamaterial systems. We note that the excitation of the SMSPs by the spirit of Babinet’s principle is a phenomenological interpretation of light diffraction by metasurfaces in a near field, and is not strictly related to the effective medium description of the metasurfaces. Specifically, a Babinet inversion of a metasurface with negative effective permittivity does not mean that the complementary metasurface must possess negative permeability. A prime example of this can be found in [10], demonstrating that a Babinet inversion of a plasmonic structure with negative effective permittivity can result in a complementary structure with a positive effective permittivity with a high effective index of refraction.

### IV. CONCLUSION

In this paper, we have demonstrated a Babinet-principle-inspired metasurface for resonant enhancement of magnetic fields. By applying Babinet’s principle to a plasmonic metasurface supporting SSPs in the form of surface-bound waves, we have shown that the complementary structures can support SMSPs that induce strong local magnetic fields. For excitation of the spoof magnetic plasmons and resonant enhancement of the magnetic fields, we introduced a periodic array of diffractive rods in the proximity of the metasurface. The resonant enhancement of the local magnetic fields has been shown to emerge from the resonance of the SMSPs inside a cavity defined by the periodically located rods. We believe that our scheme provides an intuitive way to realize magnetic resonance in an ultrathin structure, and that the proposed metasurface system could play important roles in various disciplines where strongly enhanced magnetic fields are required.

### DISCLOSURES

The authors declare no conflicts of interests.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

### FUNDING

Research grant from Kongju National University in 2020; National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (Grant No. NRF-2021R1A2C2012617, NRF-2020R1C1C1012138).

### Fig 1.

Figure 1.The idea of Babinet inversion and the complementary structure of a metasurface. Applying Babinet inversion is not clear for a homogeneous metal plate with finite negative permittivity. In all cases, we assume that the thicknesses of the structures are at a deep sub-wavelength scale.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### Fig 2.

Figure 2.Metasurfaces for spoof electric and magnetic surface plasmons. (a) A metasurface of negative effective permittivity, made of PEC ring resonators. The red dot contour denotes the unit lattice of the metasurface. Here, we set a = 3 mm, b = 42 mm, Px = 6 mm, Py = 45 mm. The thickness of the metasurface is set to be 1 mm. For the resonators, the width of the rim is 1 mm. The distance between neighboring unit resonators in both x and y directions is given as 1 mm. (b) The complementary structure of (a). PEC, perfect electric conductor.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### Fig 3.

Figure 3.Finite-difference time-domain (FDTD)-calculated spectra of near-field enhancement of Hx (black solid curve), and far-field transmission (blue dashed line). The near field is taken at the center of the core perfect electric conductor (PEC) plate of the unit resonator [Fig. 2(b)], with the measuring height 0.2 mm from the Babinet-inverted metasurface. The enhancement factor is defined as |Hx|loc / |Hx|inc, where |Hx|loc and |Hx|inc are amplitudes of the local magnetic field and the incident, respectively. For transmission, 1.0 means a 100% transmission.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### Fig 4.

Figure 4.Local enhancement of Hx by the excitation of spoof magnetic surface plasmons. (a) Schematic of the system of metasurface with diffractive perfect electric conductor (PEC) rods. The structural parameters for the metasurface are the same as those in Fig. 3. The width and thickness of the PEC rods are 2.5 mm and 1 mm, respectively, and the rods are located 1 mm from the metasurface. The yellow dot in the top view indicates the origin of the x-y plane. p1 and p2 are 0.2 mm from the metasurface. The periodic boundary condition (PBC) is applied to the system, resulting in an infinitely periodic system in both x- and y-directions. (b) Finite-difference time-domain (FDTD)-calculated spectra of Hx taken at p1 (black solid curve) and p2 (red dashed curve). All spectra are normalized by the amplitude of the incident.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### Fig 5.

Figure 5.Higher order resonance of spoof magnetic surface plasmons. (a) Schematic of the system of metasurface and PEC rods with increased periodicity of the rods. All structural parameters, except for the periodicity of the PEC rods are exactly the same as those used in Fig. 4. In the side-view, p1 and p2 are 0.2 mm from the metasurface. (b) FDTD-calculated spectra of Hx taken at p. PEC, perfect electric conductor; FDTD, finite-difference time-domain.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### Fig 6.

Figure 6.FDTD-calculated near-field x-y maps of |Hx| for the rods/metasurface system where the periodicity of the rods is 3px (Fig. 4). The maps are taken at three different distances from the metasurface as denoted by m1, m2, and m3 in (a). (a) Maps for 2.907 GHz and (b) 3.079 GHz incident frequencies. FDTD, finite-difference time-domain.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### Fig 7.

Figure 7.FDTD-calculated near-field x-y maps of |Hx| for the rods/metasurface system where the periodicity of the rods is 5px (Fig. 5). The maps are taken at two different distances from the metasurface as denoted by m1, and m3 in (a). (a) Maps for 3.044 GHz and (b) 3.090 GHz incident frequencies. FDTD, finite-difference time-domain.
Current Optics and Photonics 2023; 7: 97-103https://doi.org/10.3807/COPP.2023.7.1.97

### References

1. K. Yao and Y. Liu, “Plasmonic metamaterials,” Nanotechnol. Rev. 3, 177-210 (2014).
2. S. Yoo, S. Lee, J.-H. Choe, and Q.-H. Park, “Causal homogenization of metamaterials,” Nanophotonics 8, 1063-1069 (2019).
3. J.-H. Kang and Q.-H. Park, “Local enhancement of terahertz waves in structured metals,” IEEE Trans. Terahertz Sci. Technol. 6, 371-381 (2016).
4. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152-156 (2009).
5. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728-731 (2008).
6. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
7. J. H. Kang, D. S. Kim, and Q.-H. Park, “Local capacitor model for plasmonic electric field enhancement,” Phys. Rev. Lett. 102, 093906 (2009).
8. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369-373 (2011).
9. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847-848 (2004).
10. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
11. J.-H. Kang and Q.-H. Park, “Fractional tunnelling resonance in plasmonic media,” Sci. Rep. 3, 2423 (2013).
12. A. I. Fernández-Dománguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE J. Sel. Top. Quantum Electron. 14, 1515-1521 (2008).
13. C. Ropers, G. Stibenz, G. Steinmeyer, R. Müller, D. J. Park, K. G. Lee, J. E. Kihm, J. Kim, Q. H. Park, D. S. Kim, and C. Lienau, “Ultrafast dynamics of surface plasmon polaritons in plasmonic metamaterials,” Appl. Phys. B 84, 183-189 (2006).
14. J.-H. Kang, S.-J. Lee, B. J. Kang, W. T. Kim, F. Rotermund, and Q.-H. Park, “Anomalous wavelength scaling of tightly-coupled terahertz metasurfaces,” ACS Appl. Mater. Interfaces 10, 19331-19335 (2018).
15. S. Lee, W. T. Kim, J.-H. Kang, B. J. Kang, F. Rotermund, and Q.-H. Park, “Single-layer metasurfaces as spectrally tunable terahertz half-and quarter-waveplates,” ACS Appl. Mater. Interfaces 11, 7655-7660 (2019).
16. P. A. Huidobro, X. Shen, J. Cuerda, E. Moreno, L. Martin-Moreno, F. J. Garcia-Vidal, T. J. Cui, and J. B. Pendry, “Magnetic localized surface plasmons,” Phys. Rev. X 4, 021003 (2014).
17. F. J. Garcia-Vidal, A. I. Fernández-Domínguez, L. Martin-Moreno, H. C. Zhang, W. Tang, R. Peng, and T. J. Cui, “Spoof surface plasmon photonics,” Rev. Mod. Phys. 94, 025004 (2022).
18. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351-1353 (2004).
19. L.-L. Liu, Z. Li, C.-Q. Gu, P.-P. Ning, B.-Z. Xu, Z.-Y. Niu, and Y.-J. Zhao, “A corrugated perfect magnetic conductor surface supporting spoof surface magnon polaritons,” Opt. Express 22, 10675-10681 (2014).
20. C. Sirtori, “Bridge for the terahertz gap,” Nature 417, 132-133 (2002).
21. J. R. Jiménez and E. Hita, “Babinet’s principle in scalar theory of diffraction,” Opt. Rev. 8, 495-497 (2001).
22. S. Koo, M. S. Kumar, J. Shin, D. Kim, and N. Park, “Extraordinary magnetic field enhancement with metallic nanowire: Role of surface impedance in Babinet’s principle for sub-skin-depth regime,” Phys. Rev. Lett. 103, 263901 (2009).
23. S. Lee and J.-H. Kang, “Gap-size-dependent effective phase transition in metasurfaces of closed-ring resonators,” Crystals 11, 684 (2021).
24. J.-H. Choe, J.-H. Kang, D.-S. Kim, and Q.-H. Park, “Slot antenna as a bound charge oscillator,” Opt. Express 20, 6521-6526 (2012).
25. P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Hadar, and J. L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).

Wonshik Choi,
Editor-in-chief