## Abstract

We investigate plasmon-induced transparency (PIT) in a resonator structure consisting of two orthogonally arranged metal-insulator-metal nanocavities. Finite–difference time– domain simulations reveal that when both cavities in this structure resonate at the same frequency, the PIT effect can be used to induce spectral modulation. This spectral modulation depends on the resonance order of the cavity coupled directly to the external field, as it occurs when first-order resonance is exhibited but not with second-order resonance. We confirmed that this behavior is caused by the discrepancies between odd-order and even-order resonances using classical mechanical models analogous to nanocavities. By tuning the resonance frequency and resonance order of the cavities, one can modulate the spectrum of the resonator structure in an order-selective manner. The resonant order-dependent PIT provides insight into the development of metamaterials that function only at specific resonant orders for incident waves of various bands.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

The creation of metamaterials composed of engineered nanoscale resonator structures (also known as meta-atoms) has led to significant advances in optical manipulation [1]. The control of optical properties enabled by metamaterials is one of the foundations for controlling light in free space and of surface plasmon polaritons (SPPs) [2,3]. The unique properties of metamaterials are attributed to the dispersion relations near the resonant frequencies of the meta-atoms. The development of modulation techniques and explanation of the resonant spectrum of meta-atoms are fundamental topics in the study of metamaterials. The resonant wavelengths of many meta-atoms, such as Fabry–Pérot-type resonators, are due to the reflection of free electrons or SPPs at their physical boundaries [4]. When the incident wave has a broad spectral range, resonance spectra exhibit multiple peaks originating from multiple resonance orders [5]. However, as it is difficult to suppress or modulate resonances of a specific order exclusively, the working band of a metamaterial is limited to the resolution of the constituent meta-atoms.

The electromagnetically induced transparency (EIT) effect is a quantum phenomenon that arises from the interference between different excitation pathways in a three-level atomic system. An EIT-like effect known as plasmon-induced transparency (PIT) [6,7] can be used to modulate the spectrum of a plasmonic resonator. As it is a plasmonic analog of EIT, PIT occurs due to the interference between plasmonic structures with electric or magnetic resonances [8–10]. The coupling between the resonator excited directly by external light (the bright resonator) and the indirectly excited resonator (the dark resonator) generates significant modulation in the resonance spectrum [11–13]. Owing to its large spectral modulation effect and high sensitivity, the PIT phenomenon has been applied in many fields, such as slow-light generation [14], sensing [15], and waveguide development [16].

In this study, we investigate the resonance interaction between the two metal-insulator-metal (MIM) cavities in an orthogonally arranged resonator structure to realize spectral modulation depending on the resonance order. Only one of the cavities in this structure is directly excited by SPP wave packets (WPs), which are injected next to the resonator structure on the metal surface. This cavity thus functions as a bright resonator directly coupled to the external field, while the other cavity functions as a dark resonator that is excited indirectly via the bright resonator. A finite–difference time–domain (FDTD) simulation revealed that the spectral modulation due to the PIT phenomenon, which occurs when the two cavities have similar resonance wavelengths, depends on the resonance order of the bright resonator. The mode-splitting characteristic of PIT was observed when the bright resonator exhibited first-order resonance, while the modulation of spectral shape was dependent on the resonance wavelength of the dark resonator. Conversely, no mode splitting was observed when the bright resonator exhibited a second-order resonance, and the effect of the dark resonator on the resonance spectrum was marginal. Analysis of a classical mechanical model analogous to the nanocavities confirms that the difference in the symmetry of the magnetic distribution of the bright resonator induces the order-dependent PIT phenomenon. We believe that explaining the order-selective PIT phenomenon by modeling the effect of spatial symmetry on resonance–order–dependent mode couplingprovides insights into advanced optical control with structures that combine multiple resonators.

Figure 1(a) shows a schematic of the resonator structure studied in this work, as defined in the FDTD simulation. A commercial FDTD software package (FDTD Solution, Lumerical, Inc.) was used for all simulations. Here, we placed a Au block on an Al_{2}O_{3} (thickness *a*: 16 nm)/Au (thickness *t*: 100 nm) layer to form an open-ended horizontal MIM cavity (hereafter called the “open cavity”) [17,18]. The resonant modes of this open cavity are given by the following equation [5,18,19]:

*k*is the vacuum wavenumber,

_{0}*n*is the real component of the effective refractive index of the MIM nanocavity,

_{OC}*N*is the integer defining the order of the resonant mode, and

*ϕ*

_{OC}is the additional phase shift resulting from the opening edge. A thinner

*a*provides a larger

*n*[18]. We used values of

_{OC}*a*and

*L*such that the open cavity exhibited first- and second-order resonances in the near-infrared region.

We defined the second cavity by cutting a narrow slit of width *w* and depth *d* into the Au block. Thus, the Au block had the same geometry as a U-shaped resonator, which is a typical magnetic resonator structure [20,21]. When the slit width of this structure is sufficiently thin, it can be regarded as a half-closed MIM nanocavity (hereafter termed the “closed cavity”) [19,22,23]. The resonant mode for the closed cavity is given by the following equation [19]:

*n*is the real component of the effective refractive index of the closed MIM nanocavity and

_{CC}*ϕ*is the additional phase shift resulting from the opening edge. From Eqs. (1) and (2), the resonance frequency of each cavity is connected either to the cavity length

_{CC}*L*or the slit depth

*d*. Therefore, the resonance wavelengths of the two cavities constituting the resonator can be defined independently.

We placed an Au ridge structure 25 μm from the resonator [Fig. 1(b)], to act as an SPP excitation source. SPP WPs with a broad spectral range were excited by injecting the ridge with ultrashort 1.2 fs pulses with a peak wavelength of 600 nm. These WPs propagate on the metal surface, entering the open cavity from the side. In this configuration, the open cavity can be regarded as the bright resonator, as it is directly excited by the SPP WP, while the closed cavity can be regarded as the dark resonator, as it is excited indirectly via the resonance of the open cavity.

To determine the resonator’s spectral response, we prepared a reference model without meta-atoms that was otherwise identical in construction to our initial model. The time evolution of the vertical component of the electric field (*E _{z}*(

*t*)) was subsequently recorded at point

*P*, located at the right-hand edge of the resonator structure. Then, we evaluated

*R*(

*ω*), the spectral response, as

*F*(

_{res}*ω*) and

*F*(

_{ref}*ω*) are the fast Fourier transforms (FFTs) of the

*E*(

_{z}*t*) waveform recorded at point

*P*in the resonator model and reference model, respectively.

Figure 2(a) depicts the spectra obtained for the resonator structures with *L* fixed at 200 nm and *w* fixed at 2 nm. Peaks can be observed at 1630 and 880 nm in the spectrum for a resonator with no slit in the open cavity (*d* = 0 nm, dashed black line), corresponding to its first and second Fabry–Perot resonant modes, respectively. As the resonance wavelength of the closed cavity is determined by slit depth, this parameter can be adjusted such that both cavities in the structure are resonant at the same wavelength. With *d* = 64 nm, both cavities exhibit first-order resonances at 1630 nm. Because of the resulting mode coupling, splitting can be observed in the spectrum for this structure (solid blue line). In contrast, although both cavities resonate at 880 nm with *d* = 22 nm, in this case the closed cavity exhibits first-order resonance while the open cavity exhibits second-order resonance. Hence, no modulation is observed in the spectrum for this structure (solid red line), which is similar in shape to the spectrum of the open cavity without a slit. The mode coupling strength between open and closed cavities varies with the distance between the two cavities. Upon lowering the height *h* below 100 nm and decreasing the separation of the two cavities, the dip in the resonant spectrum due to the coupling becomes more prominent, as shown in Fig. 2(b).

The magnetic distributions shown in Figs. 2(c) and 2(d) indicate that the line symmetry of the magnetic field of the open cavity with *x* = *L*/2 as an axis depends on the resonance order. The magnetic field of the open cavity is antisymmetric with respect to the closed cavity when the open cavity exhibits even-order resonance. In such cases (e.g., for second-order resonance), the magnetic field in the region of the closed cavity originating from the open cavity is canceled out, and the closed cavity is excited predominantly by diffraction from the Au block. In contrast, when the open cavity exhibits odd-order resonance, such as first-order resonance, the induced magnetic field excites the closed cavity. The mutual inductance of the magnetic fields of the open and closed cavities results in mode coupling [8].

Figures 3(a), 3(b), and 3(c) show color maps of the spectra of resonators with slit widths of 2, 4, and 6 nm, respectively. The color in these images indicates the spectral responses of the resonators, while the slit depth is given by the vertical axis. Here, *h* was fixed at 100 nm and *L* was fixed at 100 nm, such that the open cavity exhibited first-order resonance at a wavelength of 1000 nm. The dashed white lines in each graph indicate the theoretical evaluation of the resonance wavelength of the closed cavity, obtained by setting *ϕ _{CC}* = 0 in Eq. (2), according to the dispersion calculation for an MIM waveguide [5,18]. The three graphs show large modulations in the resonance spectra along the resonance wavelength curves, with the slope of the modulation depending on the slit width. There are shifts between the resonance curves indicated by the white lines and slit depths where the mode modulations are maximized. These shifts in the

*y*direction are explained by the additional phase shift,

*ϕ*, in Eq. (2). Spectral splitting due to the mode coupling of resonators with the same resonance wavelength is a typical feature of PIT, while the calculated peak splitting is up to approximately 470 meV (

_{CC}*w*: 2 nm,

*d*: 100 nm), which is comparable to that seen in previous studies [9,24,25]. In addition, the shift of each split peak with respect to the resonance wavelength of the dark resonator is also consistent with the study by Zhang

*et al.*[26]. In contrast, the resonance spectrum for the structure where

*L*= 240 nm, which exhibited second-order resonance at 1000 nm, was minimally affected by the slit depth of the closed cavity, as shown in Fig. 3(d).

The PIT phenomenon that occurs in this structure does not depend on the resonance order of the closed cavity. Figure 3(a) confirms that there was large PIT-induced mode splitting along the second-order resonance curve at around *d* = 100 nm, whereas there is no mode splitting in Fig. 3(d) at the same *d*. The resonant order of the open cavity affects the line symmetry of the magnetic field with *x* = *L*/2 as an axis. However, the resonant order of the closed cavity does not affect the symmetry in the *x* direction. Therefore, the PIT phenomenon in this configuration of the resonator structure depends only on the resonant order of the open cavity.

To verify the order dependence of the spectral modulation, we constructed numerical models based on a classical mechanical analog to PIT. First, we defined a system consisting of two magnetic dipoles, each representing the open and closed cavities (*P _{O}* and

*P*, respectively), as a model for the case where both exhibit first-order resonance (“first-order model”), as shown in Fig. 4(a). The dynamic equation for this is described by that of a system of linearly-coupled Lorentzian oscillators [6,11,26,27]:

_{C}*c*and

*o*indicate the closed and open cavities, respectively,

*ω*and

_{O}*ω*are resonant frequencies,

_{C}*γ*is the damping constant for a dipole,

*E*is the incident field, and

_{I}*g*is a coupling constant between a dipole and the incident field. Ω is the constant for the coupling between the two dipoles.

In addition, we constructed a second-order model consisting of a single dipole representing the closed cavity (*P _{C}*) and two dipoles representing the open cavity (

*P*,

_{O1}*P*), as shown in Fig. 4(b). Here,

_{O2}*P*and

_{O1}*P*represent the two peaks of the magnetic distribution when the open cavity exhibits second-order resonance, as shown in Fig. 2(c). The dynamic equations for this model can be defined in a similar form to Eq. (4) as follows [28,29]:

_{O2}*g*, the coupling constant between

_{O2}*P*and the incident field, equal to

_{O2}*–g*to represent the π-phase shift between the two magnetic oscillations observed with second-order resonance. To simplify the calculation, we assume that the closed cavity is not coupled to the external field at all. Hence,

_{O1}*g*= 0. In practice, this coupling constant would have a finite value because of diffraction over the top of the Au block. The values of the remaining parameters were defined as follows:

_{C}*ω*=

_{O}*ω*=

_{O1}*ω*= 300 THz,

_{O2}*γ*= 25,

_{C}*γ*= 12.5,

_{O}*Ω*= 75,

*g*= –

_{O1}*g*= 1, and

_{O2}*E*= 1.

_{I}The susceptibility of the dipole response can be obtained from Eqs. (4) and (5), *χ _{O}* =

*P*and

_{O}/E_{I}*χ*=

_{O1}*P*. Figures 4(c) and 4(d) show color maps of the imaginary parts of

_{O1}/E_{I}*χ*and

_{O}*χ*, representing the energy dissipation of the model. Here, the resonance wavelength of the open cavity,

_{O1}*λ*2π

_{O}=*c/ω*, was fixed at 1000 nm, while the vertical axis corresponds to the resonance wavelength of the closed cavity (

_{O}*λ*π

_{C}= 2*c/ω*) as determined by the slit depth. As with the FDTD simulations, mode splitting was observed with the first-order model when the resonance frequencies of the two dipoles coincided (

_{C}*λ*), and the peaks shifted according to

_{C}= λ_{O}*λ*. In contrast, uniform spectra were observed with the second-order model regardless of the value of

_{C}*ω*, as shown in Fig. 4(d). These calculation results are consistent with the FDTD simulation results shown in Fig. 2 and Fig. 3.

_{C}In conclusion, we investigated the spectral modulation of a resonator consisting of an orthogonal arrangement of two types of MIM nanocavity. FDTD simulations revealed mode splitting in the resonance spectra due to the PIT phenomenon, and peak shifts according to the resonance frequency of the cavity acting as the dark resonator. The PIT phenomenon was highly dependent on the resonance order of the cavity acting as a bright resonator. The dependence of the mode splitting and peak shift on the resonance order was confirmed using a classical mechanical model. The shape of the resonance spectrum of the resonator structure used in this study is determined by the cavity geometry, the materials used for each cavity (through the effective refractive index), the alignment of the resonator with the external field, and the distance between the cavities. This demonstrates that resonance spectra design, including the narrowing of line widths, the shifting of peak positions, and the generation of multiple peaks, can be enabled by the fabrication of composite meta-atoms consisting of bright and dark resonators. Changing the number and positions of the slits makes it possible to modulate a specific resonance order. The fabrication of the meta-atoms studied in this study could be achieved by cutting slits in a standard MIM nanocavity fabricated by the EB lithography technique using helium ion etching. In addition, the dependence of mode coupling on the mismatch between even and odd resonance orders offers the potential to modulate a specific resonant order of a meta-atom. Such meta-atoms are the building blocks of metamaterials for broadband incident waves such as femtosecond pulses and supercontinuum waves.

## Funding

Ministry of Education, Culture, Sports, Science and Technology, Q-LEAP ATTO (JPMXS0118068681); Core Research for Evolutional Science and Technology (JPMJCR14F1); Japan Society for the Promotion of Science (JP18967972, JP20J21825).

## Acknowledgments

The authors thank H. T. Miyazaki for advice and valuable discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results may be obtained from the authors upon reasonable request.

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