## Abstract

Plasmonic structures with high symmetry, such as double-identical gap split ring resonators, possess dark eigenmodes. These dark eigenmodes are dominated by magnetic dipole and/or higher-order multi-poles such as electric quadrapoles. Consequently these dark modes interact very weakly with the surrounding environment, and can have very high quality factors (*Q*). In this work, we have studied, experimentally as well as theoretically, these dark eigenmodes in terahertz metamaterials. Theoretical investigations with the help of classical perturbation theory clearly indicate the existence of these dark modes in symmetric plasmonic metamaterials. However, these dark modes can be excited experimentally by breaking the symmetry within the constituting metamaterial resonators cell, resulting in high quality factor resonance mode. The symmetry broken metamaterials with such high quality factor can pave the way in realizing high sensitivity sensors, in addition to other applications.

© 2014 Optical Society of America

## 1. Introduction

Recently, metamaterials have attracted tremendous amount of interest because of their unusual electromagnetic (EM) properties [1–6] including negative refractive index [7–9], perfect focusing [10, 11], and cloaking [12] which are not permissible using the naturally occurring materials. Additionally, metamaterials enable efficient manipulation of EM waves via tuning the fundamental and the higher order resonances [13–15], perfect absorbers [16], thin-film sensors [17, 18] and many more. Typically, metallic split ring resonators (SRRs) are used as the fundamental building blocks of plasmonic metamaterials. Ohmic and radiative losses in the metal-based SRRs are considered a major hindrance to the development of typical plasmonic metamaterials based devices [19]. Recently, asymmetric metamaterials have been proposed as a novel approach to overcome radiative losses, hence improving the quality factor in metamaterials resonances [20–24].

In this work, we report excitation of dark plasmonic resonance modes via breaking the symmetry of the double identical gap split ring resonator, the constituting element of our planar terahertz metamaterial. Although the dark plasmonic modes exist in the symmetric split ring resonator based metamaterials, these modes cannot be excited experimentally with a plane wave with linear polarization. Our theoretical calculations clearly indicate the existence of these modes in metamaterials, however, experimentally these dark plasmonic modes are only accessible by introducing asymmetry at the unit cell level, featuring a sharp resonance in the transmission spectrum. Additionally, these modes offer low radiation losses, therefore they have a high quality factor. We have probed the existence as well as the excitation of these dark modes with the help of a classical perturbation theory. Application of this theory allows us to better understand the physical mechanism in asymmetric metamaterials and plasmonic systems.

In the case of a perfectly symmetric MM, we observe typical dipolar resonance features. With the introduction of an asymmetry in the structure, we observe the appearance of a sharp resonance feature developed at lower frequencies due to the excitation of the dark mode which is inaccessible while the SRR is perfectly symmetric. One of the exciting properties of this evolved resonance is its high quality factor that is measured (simulated) to be 10.2 (35.8) in the current study. Also, this dark plasmonic mode is highly sensitive to the polarization of the incident radiation and can be excited by a particular polarization. The proposed asymmetric metamaterial may enable high sensitivity sensors [25] and strong light mater interaction [26].

## 2. Experiments

Three different sets of metamaterial samples were fabricated on an undoped GaAs wafer as shown in Figs. 1(a)–1(d). All samples were fabricated using photolithography and e-beam deposition of 10 *n*m of titanium, followed by 200 *n*m of gold. Finally, the MM samples were realized through a typical lift off process leading to the formation of the metal SRRs.

Optical images of the fabricated SRRs are shown in Fig. 1. The detailed geometric parameters are indicated in Fig. 1 for all metamaterials samples. The sample MM1 consists of completely symmetric split ring resonators with two identical gaps. In the samples MM2 and MM3, we intentionally introduced asymmetry by extending one side plate of the split gap. The unit cell periodicity in all MM samples was kept constant at *P _{x}* =

*P*= 88

_{y}*μ*m. The fabricated metamaterials samples were characterized using terahertz time domain spectroscopy (THz-TDS) [27,28]. The transmitted THz pulses through samples and reference were measured at room temperature and in a dry atmosphere.

The transmitted THz amplitudes were measured for two orthogonal polarizations: THz electric fields oriented parallel and perpendicular to the gap bearing arms of SRR, as shown in Fig. 2. A piece of bare GaAs identical to the MM sample substrate was measured as a reference. Transmission spectra (*S*_{21}) normalized to the bare GaAs substrate were obtained for all the samples.

Figures 2(a)–2(c) depict the measured spectra for samples MM1, MM2 and MM3, respectively. In the case of the electric field polarization parallel to the gap bearing arms, a resonance appears at 1.174 THz for MM1. With the same excitation polarization, the resonance red shifts to 1.161 THz for MM2 and 1.148 THz for MM3. However, when the polarization of the incident THz is rotated by 90 degree, i.e. the electric field of the THz radiation perpendicular to the gap-bearing arms, we observe the resonance to appear at 0.717 THz for MM1 (Fig. 2(d)). The same resonance is slightly red-shifted to 0.71 THz for MM2, but we also observe an evolution of a resonance feature at 0.486 THz. The evolved resonance feature at low frequency becomes more prominent with greater asymmetry, as shown in Fig. 2(f) for sample MM3. The broad resonances still appear around 0.7 THz for the symmetry broken samples MM2 and MM3. The *Q* factor for the resonances appearing nearing 0.7 THz are measured as ∼ 6 for all three samples. The low frequency dark resonance is experimentally inaccessible in MM1 because of the high structural symmetry. However, by introducing asymmetry one can experimentally access this asymmetric dark resonance mode, which usually offers a high *Q* factor. The measured *Q* factor of the resonance at 0.446 THz for MM3 is ∼ 10.2, much larger than observed for the resonance at 0.7 THz.

## 3. Simulations and discussion

In order to explain our experimental results, we employ the eigenmode analysis developed in [29, 30]. The EM properties of the symmetrical SRR system, MM1, are fully described by the following eigenvalue equation:

where**u**= (

**H**,

**E**,

**P**,

**J**)

^{T}, and

*ℒ*is a non-Hermitian differential operator. As a consequence, the eigenvalues are in general complex, and the corresponding eigenvectors are not orthogonal to each other [31, 32]. However, the eigenvectors are

*bi-orthogonal*to the eigenvectors of the corresponding adjoint equation, From the theory of non-self-adjoint differential equations, one knows that the eigenvalues and eigenvectors of the two mutually adjoint equations can be ordered in such a way that ${\lambda}_{m}={\omega}_{m}^{*}$, and $\u3008{\mathbf{u}}_{m}^{\u2020}|{\mathbf{u}}_{{m}^{\prime}}\u3009={\delta}_{m,{m}^{\prime}}$ (here 〈...〉 ≡ ∫

*d*

^{3}

**r**...). Also, it is possible to associate the same eigenvalue

*ω*with one eigenvector of (1) and with one eigenvector of the adjoint equation (2), i.e.,

_{m}*ω*↔

_{m}**u**

*, ${\mathbf{u}}_{m}^{\u2020*}$. In the literature,*

_{m}**u**

*and ${\mathbf{u}}_{m}^{\u2020*}$ are generally called right- and left- eigenvectors, respectively.*

_{m}For the specific symmetry SRR structure, because it possesses a *y* = 0 mirror symmetry as well as *x* = 0 mirror symmetry, its eigenmodes can be divided into four groups in terms of the *x* component of the electric field. The first group **u*** _{n,ee}* is even in terms of

*x*= 0 mirror plane, and also even in terms of

*y*= 0 mirror plane, such that:

**u**

*is even in terms of*

_{n,eo}*x*= 0 mirror plane, while odd in terms of

*y*= 0 mirror plane, such that:

**u**

*is odd in terms of*

_{n,oe}*x*= 0 mirror plane, while even in terms of

*y*= 0 mirror plane, such that:

**u**

*is odd in terms of*

_{n,oo}*x*= 0 mirror plane, while odd in terms of

*y*= 0 mirror plane, such that:

*x*-polarized incident plane wave can only excite the

**u**

*modes, while*

_{n,ee}*y*-polarized incident plane wave can only excite the

**u**

*modes. The*

_{n,oo}**u**

*and*

_{n,eo}**u**

*can not be excited at all, baring the characters of dark modes [33].*

_{n,oe}Even without numerical simulation, one can note that the current distribution in the first order modes (*n* = 1) should look like:

*eo*mode has the smallest frequency, and (2)

*Re*[

*ω*

_{1,oo}] ≃ 2

*Re*[

*ω*

_{1,eo}],

*Re*[

*ω*

_{1,oe}] ≃ 2

*Re*[

*ω*

_{1,ee}]. Through numerical simulation, three of these four eigenmodes are excited. It is found that the

*ee*mode is around 1.218 THz, the

*oo*mode is around 0.7246 THz and the

*eo*around 0.5739 THz. Furthermore, the corresponding eigenmode pattern are also calculated and plotted in Figs. 3, 4, and 5, and are found to perfectly agree with our analytical prediction.

We now use perturbation theory to study the asymmetric system, whose eigenvalue equation is given by:

with*𝒳*corresponding the additional asymmetric part. For the structure we considered,

*𝒳*(

*x*,

*y*,

*z*) =

*𝒳*(

*x*, −

*y*,

*z*). Consequently, its eigenmodes can be divided into two groups in terms of the

*x*component of the electric field. The first group,

**v**

*, is odd in terms of*

_{n,o}*y*= 0 mirror plane, such that: and the second group,

**v**

*, is even in terms of*

_{n,e}*y*= 0 mirror plane, such that:

To perform the perturbation theory, one need expand **v** in terms of the original eigenmodes **U** such that:

**X**is defined as $\u3008{\mathbf{u}}_{m}^{\u2020}\left|\mathcal{X}\right|{\mathbf{u}}_{n}\u3009$, and the remaining element of the first matrix is zero. Due to the symmetry, the operator

*𝒳*can only couple an

*ee*mode to

*ee*and

*oe*mode, or an

*oo*mode to

*eo*and

*oo*mode. In other words, the new eigenmode can be written as

*will be different to the original*

_{m}*ω*. In other words, the resonance location of the symmetric SRR systems will be changed by the degree of asymmetry.

_{m}When the metamaterial is illuminated by an external EM field, the corresponding Maxwell equations can be written as:

where**S**represents excitation sources or the incident wave. The total field

**V**can now be expanded in terms of the eigenvectors

**v**

*of the source-free problem, namely*

_{m}*are the corresponding complex eigenvalues of the source-free problem. Under a normal*

_{m}*x*-polarized incidence, the total field can be written as

*y*-polarized incidence

We assume that in the interested frequency region, only first-order eigenmodes can be excited. Under *x*-polarized incidence, the asymmetrical structure will present two valleys, around *ω*_{1,ee} and *ω*_{1,oe}. Under a *y*-polarized incidence, the asymmetrical structure also will present two valleys, around *ω*_{1,eo} and *ω*_{1,oo}. Because of the fact that *Re*[*ω*_{1,oe}] ≃ 2*Re*[*ω*_{1,ee}], the first order *oe* mode should be around 2.4 THz. It therefore can be excited but cannot be observed in the frequency regime our measurement system operates. It should be pointed out that the first grating mode of the array, has a frequency of 0.9472 THz, which is much smaller than the frequency of the first order *oe* mode.

To confirm the experimental result, a numerical Maxwell solver using finite-difference-time-domain (FDTD) method is employed [34]. During the simulation, the electric and magnetic fields are temporally separated by a half-time step, and also are spatially interlaced by a half mesh cell. The mesh size is 0.5*μ*m. In order to compare the experiment directly, same geometry parameters are employed. The gold is treated as perfect conductor, which is a reasonable approximation in THz region. Moreover, it is assumed that the GaAs substrate has a semi-infinite thickness, with a permittivity of 12.56. All the MM samples were simulated under both orientations of electric field polarization. The simulated transmission amplitude reproduces the experimental results reasonably well. Experimental and numerical data are compared in Figs. 2 and 3, respectively for both the polarization of excitation beam. Experimental and simulated resonance frequencies are shifted slightly because of fabrication uncertainties. In these simulations we have considered the nominal parameters as used for the sample design. In the case of the electric field of polarization perpendicular to the split gap, the quality factor of the higher frequency resonance appearing near 0.7 THz is simulated to be 6.0 (approximately), matching well with the experimental value. The sharp resonance feature arising at lower frequency possesses much higher quality factor. The simulated quality factors for MM2 and MM3 are 35.8 and 35.2, respectively. The deviations in the amplitudes and line widths between simulations and measurements are due to the longer time scan of the measured and simulated pulses [35]. These deviations lead to the differences in simulated and measured quality factor values for the MM samples.

In order to identify the resonance modes and to compare them with our theoretical outcomes, we numerically simulate the electric field distributions around each resonance (Figs. 3 and 5). The results are consistent with our qualitative picture as described before. For *x* polarization of excitation (electric field parallel to the split gap), *x*- component (**E*** _{x}*) and

*y*- component (

**E**

*) of induced electric fields at the resonances are shown in Fig. 3. At the resonance, the computed electric field distributions for all the samples, asymmetrical as well as the symmetrical structure, demonstrate similar characteristics of the*

_{y}*ee*eigenmodes predicted theoretically (see, Fig. 3(a) and

**u**

_{1,ee}). Under

*y*-polarized excitation (electric field perpendicular to split gap), electric field distributions are computed at the higher-frequency (0.7 THz) resonance. The field distributions for all the MM samples resemble the

*oo*eigenmode (see Fig. 4 and

**u**

_{1,oo}). We have further simulated the

*x*- and

*y*- components of the induced electric field distributions at the theoretically predicted frequencies for

*eo*eigenmodes in Fig. 5. The

*eo*eigenmodes are supposed to be excited at 0.57 THz, 0.51 THz and 0.47 THz for MM1, MM2 and MM3, respectively. We note that for MM2 and MM3, the sharp resonance appears near the predicted frequencies although we do not see any resonance for MM1 in the measurements. Although the

*eo*eigenmode exists for MM1, it cannot be excited with a plane wave excitation. The

*eo*eigenmode for the symmetric structure (MM1) is excited numerically using a few random electric dipoles but bears the same character as predicted theoretically (see Fig. 5(a) and

**u**

_{1,eo}). This mode is generally not possible to excite for symmetric structure irrespective of polarization of probing beam. Interestingly, this

*eo*eigenmode (

**u**

_{1,eo}) is capable of holding the EM energy for longer time period, hence low in radiative resistance. Such low-loss radiation mode is associated with the enhanced quality factor of the

*eo*mode. The measured and simulated quality factors of this mode are observed to be as high as 10.2 and 35.8 in this symmetry broken plasmonic MM. It should be mentioned that this perturbed dark mode has been observed in asymmetrical microwave metamaterial in [20], and was named as trapped-mode resonance. Additionally, one may also interpret it in terms of asymmetric Fano resonance, resulting from the interference between a sharp resonance and a smooth continuum-like spectrum [36, 37].

## 4. Conclusion

In this work, we have investigated dark resonance modes in plasmonic metamaterials. Full-wave eigenmode investigations predicted the existence of these modes in terahertz metamaterials where the unit cell consists of double identical split gap resonators. We have indicated an experimental route to excite these dark plasmonic modes via the introduction of asymmetry by increasing the plate length of one side of the split gap within the SRR. The dark modes are excited in the form of a sharp resonance feature for a particular excitation polarization. The quality factor of these resonance modes can be significantly high because of low radiative loss. Experimentally, we have demonstrated a sharp, yet very clear, resonance feature with a quality factor equal to 10.2. These dark resonance modes are only accessible with the electric field polarization perpendicular to the split gap. The experimental results are further interpreted utilizing perturbation theory, and validated using finite-difference, time-domain numerical simulations. The demonstration of high quality factor resonance modes in such broken-symmetry planar terahertz metamaterials can enable the development of high sensitivity sensors, frequency agile metamaterials, as well as future terahertz devices with versatile functionalities.

## Acknowledgments

This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences Nanoscale Science Research Centre operated jointly by Los Alamos and Sandia National Laboratories. Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396. We gratefully acknowledge the cleanroom facilities of Center for Integrated NanoTechnologies (CINT) located at Sandia National Laboratory for the fabrication of some of the metamaterial samples. The authors also acknowledge the support provided by the State Key Program for Basic Research of China ( 2013CB632705), the National Natural Science Foundation of China ( 11334008), and the Fund of Shanghai Science and Technology Foundation ( 13JC1408800).

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