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Abstract
Considering the widespread applications of resonant phenomena in metasurfaces to bend, slow, concentrate, guide and manipulate lights, it is important to gain deep analytical insight into different types of resonances. Fano resonance and its special case electromagnetically induced transparency (EIT) which are realized in coupled resonators, have been the subject of many studies due to their high-quality factor and strong field confinement. In this paper, an efficient approach based on Floquet modal expansion is presented to accurately predict the electromagnetic response of two-dimensional/one-dimensional Fano resonant plasmonic metasurfaces. Unlike the previously reported methods, this method is valid over a wide frequency range for different types of coupled resonators and can be applied to practical structures where the array is placed on one or more dielectric layers. Given that the formulation is written in a comprehensive and flexible way, both metal-based and graphene-based plasmonic metasurfaces under normal/oblique incident waves are investigated, and it is demonstrated that this method can be posed as an accurate tool for the design of diverse practical tunable/untunable metasurfaces.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, metasurfaces or planar versions of metamaterials, have represented a remarkably versatile way to manipulate impinging light at an interface. These artificial interfaces rely on sub-wavelength patterned metallic/dielectric particles to arbitrarily tailor the polarization, phase, amplitude and dispersion of light [1–5]. In these structures, by changing material and designing shapes and sizes of constituents, one can create unique abilities to absorb, concentrate, disperse and guide electromagnetic waves. These engineered sheets are easily fabricated and open the door to a variety of applications that are unattainable with naturally available materials. Owing to these outstanding properties and observation of novel phenomena, metasurfaces have recently been introduced as a promising platform for a diverse range of applications that include beam shaping and steering [6–8], biomedical sensing and imaging [9–11], holography [12,13], nonlinear optics [14,15], optical lenses [16,17], spectrally selective thermal emission [18], polarization converters [19–21] and absorbers [22–24].
The resonant nature of metasurfaces enables phenomena including extraordinary reflection and transmission [25,26], cloaking [27,28], perfect absorption [29,30], bound states in the continuum [31,32] and strong field localizations in the near field [33]. A special type of resonance in optics is the Fano resonance that has received a great deal of attention during the last decades. Fano resonance is a kind of quantum interference phenomenon between a continuum state and a discrete localized state that was originally discovered in atomic physics, and then this concept was introduced to the fields of photonics and metasurfaces [34,35]. From the application’s point of view, sharp reflection/refraction supplied by Fano resonance which is highly sensitive to perturbations in the local environment, paves the way for better biochemical sensors and switching devices in photonics [36–38]. Although ohmic loss that metals experience in high frequencies makes Fano resonances broader than lossless metasurfaces, a variety of metallic metasurfaces were designed to realize the Fano resonances in the past decades due to the high field enhancement and strong confinement of light along plasmonic metasurfaces [39,40].
Providing theoretical methods describing the optical response of metasurfaces gives a deep insight into the treatment of these structures and can become an essential tool for better design. One of the most widely used analytical methods for the analysis of these structures is the dipole method which has been extended by including the effects of higher order multipoles [41–43]. To solve a problem involving a metallic/dielectric layered medium using dipole methods, one must use Dyadic Green’s function, whose equations inevitably become very complicated [44]. On the other hand, including quadrupole and octupole effects in an analytical form is usually restricted to spherical particles, and it is hardly possible to investigate other shapes. In general, the methods of analyzing Fano resonant metasurfaces are limited, and most of the methods only provide general explanations about the phenomenon and are not able to accurately predict their optical response over a wide frequency range.
The variational method is one of the oldest methods that has been employed in the microwave frequency range to analyze waveguides with a discontinuity [45]. In this method, with the help of a current distribution on the patch/antenna or an electric field distribution on the aperture/hole, the equations are written in such a way that scattering parameters are calculated. This method was later applied to periodic structures and analytically justified the phenomenon of extraordinary transmission through periodic arrays of sub-wavelength holes perforated in a metallic film. Over the last two decades, the equivalent circuit model form of the variational method has been used to accurately investigate one- and two-dimensional PEC arrays [46–48]. Recently, the variational method has been extended in a way that, unlike previous models, it could be applied to metal/graphene-based plasmonic metasurfaces and predicts their electromagnetic responses with high accuracy [49]. One of the most important advantages of this method is its high flexibility and ability to be employed for multilayer structures, which is not accurately and easily possible in other methods [46,50].
In this work, we extend the previous equivalent circuit model [51] in a way that precisely analyzes the plane wave scattering by plasmonic metasurfaces comprising coupled resonators. The extended model is definitely much faster than full-wave simulations, in particular, if many frequency points are desired or several simulations are required to reach a design. Unlike its predecessors, this model is not restricted to PEC metasurfaces and can accurately predict the transmission and reflection of plasmonic metasurfaces. In this approach, plasmonic resonators are treated as surface conductivities, and the model is derived by applying the Floquet expansion to the induced currents in such a way that coupling between resonators is well modeled. In addition to including the plasmonic effect, both components of induced currents (x direction and y direction) have been used to drive the model (Ref. [51] considers only the x or y component for each current); in this way, resonators with complex shapes can also be investigated. For validation purposes, we first simulate an asymmetric split-ring resonator under the PEC assumption and also by taking the plasmonic effect into account to see the differences. Next, a symmetric structure exhibiting Fano resonance is investigated, and finally, we analyze the array of graphene ribbons in which the unit cell contains two ribbons with different widths and different bias voltages.
2. Derivation of model
In this section, we first present the derivation of an equivalent circuit model for a single layer of array placed between two different dielectric half-spaces (Fig. 1), then apply it to practical layered metasurfaces. It should be noted that in the following, the time dependence of the form $e^{-j \omega t}$ is implicitly assumed ($\omega$ is the angular frequency). In this paper, we focus on plasmonic metasurfaces; thus in the formulation, surface electric conductivities are considered for resonators to model plasmonic properties at infrared frequencies. In order to keep the formulation more general and applicable to diverse structures, the electrical conductivity of the two resonators is considered different. The surface conductivity of metals, if they have a small thickness, is written as [49]:
Fig. 1. Schematic of a periodic array of asymmetric split-ring resonators that is located between two different semi-infinite dielectrics under a normal incident plane wave. In this structure, the periodicities are $P_{x}$ and $P_{y}$ in $x$ and $y$ directions respectively, the radius of the inner circle is $r_{1}$, the radius of the outer circle is $r_{2}$, the thickness of resonators is $l_{m}$, the longer arc spans $\alpha$, the shorter arc spans $\beta$, and the refractive index of upper and lower environments are $n^{1}$ and $n^{2}$, respectively.
Consider now the array is illuminated by an incident TM/TE polarized plane wave ($TE_{00}$ or $TM_{00}$). The impinging plane wave excites the Floquet modes on both sides of the array (discontinuity) and also the current modes on the resonators. The problem can be posed in terms of a waveguide discontinuity problem when the walls of the unit cell are considered periodic boundary conditions and the Floquet modes alternatively represent waveguide harmonics. In the variational-based method, the equations are written in such a way that the effect of current frequency-dependent coefficients is eliminated in calculation of scattering parameters; thus, the problem is solved only by having the currents’ distributions. In Refs. [47] and [48], types of problems have been solved where only one PEC resonator exists in the unit cell; next, a new model was developed in which the plasmonic effect of metals was also included [49]. In these papers, only one frequency-dependent current was considered, hence this approximation has been found to be valid up to the second excitable resonance of the resonator. Reference [49] also used superposition to account for the effect of high-order induced currents, while in Fano resonance, due to the near-field interaction between the resonators in the unit cell, it is not possible to use superposition. In order to accurately predict the Fano resonance, we must include the current on both resonators simultaneously, so we write the total current as
Fig. 2. Schematic of the equivalent circuit model for the metasurface is depicted in Fig. 1. The expressions $Y_{0}^{2}$ and $Y_{0}^{2}$ are admittances correspond to the incident harmonic in upper and lower environments.
In this circuit model, the equivalent impedance is abtained
where $Z_{11}$ represents the self-effect of the first current, $Z_{22}$ is the self-effect of the second current, as well as $Z_{12}$ and $Z_{21}$ model the interaction between the two resonators. The self and mutual impedances related to two currents is given byAs can be seen from the equations of self and mutual impedances, the frequency-dependent coefficients ($A_{1}(\omega )$ and $A_{2}(\omega )$) have been completely eliminated, thus this is the reason why this method is called Variational. The prime in the series indicates that the incident harmonic is excluded from the summation; instead, the incident wave and its counterpart in the lower environment have appeared as transmission lines in the circuit model. In one-dimensional periodic arrays, the TE/TM incident wave only excites the TE/TM Floquet modes, while in two-dimensional periodic arrays, TE or TM incident waves simultaneously excite TE and TM modes; therefore, the summation should include TM and TE modes. In Eqs. (8) and (9), $\sigma _{1s}$ and $\sigma _{2s}$ are the electrical conductivity of the first and second resonators, respectively; subscript $i$ in $P_{ih}$ refers to the current number. The expression $e_{h}$ is the tangential component of the diffracted Floquet harmonics $h$ ($h$ is associated with a pair of integer numbers $nm$ which refers to the diffraction number) are given by
3. Results and discussion
Due to the resonant nature of metasurfaces, the concept of quality factor associated with field concentration is important for many applications. High-quality factor resonances in metasurfaces can be attained when the structural design of the metasurface is engineered in a way that could support Fano resonance. This paper’s focus is on evaluating the performance of the reported method, not on designing a metasurface for a particular application. Thus, we will introduce three distinct asymmetric, symmetric, and tunable graphene-based metasurfaces and explore the Fano resonance in their electromagnetic response.
3.1 Asymmetric metasurfaces
In recent years, various symmetry-breaking structures exhibiting high-quality Fano resonances have been introduced, and metasurfaces consisting of split-ring resonators are one of these structures [52,53]. In Ref. [52], resonances of symmetric and asymmetric split-ring resonators have been studied, along with evidence that by increasing structural asymmetry, the strength of the asymmetric resonance enhances while its Q factor gradually declines. To validate the model, we initially simulate the array of split-ring resonators illustrated in Fig. 1 when the resonators are PEC and also by taking into account the plasmonic effect of the metal. Suppose the array is placed in free space; we simulate the array using full-wave software to extract the currents on the resonators. Note that we simulate the complete structure instead of just the resonators independently to extract the currents, which means that the coupling effect is taken into account in the current distributions. We have extracted the large arc’s current at the first resonance and the small arc’s current at the second resonance. It should be mentioned that current distributions are not significantly affected by frequency, allowing for the extraction of current distributions at other frequencies. Using the extracted currents in the circuit model, the following results have been achieved for the PEC and plasmonic array:
Under the PEC assumption, impedances $Z_{11}$ and $Z_{22}$ are a combination of inductors and capacitors, which exhibit a capacitive effect at low frequencies. The capacitive and inductive effects cancel each other out when the frequency rises up to a certain point (it has been demonstrated in Ref. [49] that this point is the resonance point where a resonator exists in the unit cell); consequently, the array shows an inductive effect as the frequency increases. Figure 3(a) illustrates the equivalent impedance computed in the PEC scenario using the extracted currents; since the metal is PEC, the real part of the impedance is 0 and is not depicted. We use the calculated equivalent impedance in the circuit model (Fig. 2), and as a result, a good agreement can be seen in the circuit model and full-wave results (Fig. 3(b)). The bandwidth of a resonance is inversely related to the impedance variations around the resonance point. It is obvious that the first resonance’s bandwidth is substantially less than the second one since $Z_{eq}$ varies around it more quickly than it does around the second resonance. Both the impedance and reflection curves show that the first and second resonances are distinct from one another (the first resonance is introduced as the Fano resonance).
Fig. 3. (a) Equivalent impedance calculated using PEC assumption. (b) Comparison between the results of the proposed method and full-wave simulation for the PEC array. (c) Equivalent impedance under the assumption of plasmonic metal (resonators are gold, whose refractive index was taken from Ref. [54]). (d) Comparison between the results of the proposed method and full-wave simulation for the plasmonic array. Geometrical dimensions (see Fig. 1 for definitions): $P_{x}=8\mu m$, $P_{y}=8\mu m$, $r_{1}=2.2\mu m$, $r_{2}=1.7\mu m$, $l_{m}=20nm$, $\alpha =146^{\circ }$, $\beta =120^{\circ }$.
Under the assumption that the metal is plasmonic, an inductive and resistive effect are added to self-impedances ($Z_{11}$ and $Z_{22}$), which leads to a decrease in reflection and a frequency shift in its electromagnetic response. In Fig. 3(c) and Fig. 3(d), the equivalent impedance and the reflection curves are depicted, respectively; it is obvious how the results differ from those obtained using the PEC assumption. It’s noteworthy that the resistivity greatly increases around the Fano resonance but not at the second resonance.
3.2 Symmetric metasurfaces
Fano resonances are often observed and investigated in asymmetric metasurfaces, although these resonances can also be realized in symmetric metasurfaces. In the previous example, an asymmetric metasurface consisting of two resonators in the unit cell was simulated, whereas it can also be used in cases where the number of resonators is more as long as the structural symmetry allows us to reduce the unknown current frequency-dependent coefficients into two.
Consider a periodic structure as shown in Fig. 5(a), due to the symmetry, the currents on both rectangles exhibit a similar frequency dependence, allowing us to express the current in the unit cell as follows:
Fig. 4. (a) Shematic of equivalent circuit model for the structure shown in Fig. 5(a). (b) Definition of the input admittances associated with the $h$-harmonic.
Fig. 5. (a) Schematic of the structure. (b) Equivalent impedance corresponding to the structure. (c) Comparison between the proposed method and full-wave simulation results. Geometrical dimensions: $P_{x}=3.5\mu m$, $P_{y}=3.5\mu m$, $r_{1}=.39\mu m$, $r_{2}=.7\mu m$, $L=1.75\mu m$, $W=.35\mu m$, $l_{s}=.3\mu m$, the gap between the ring and the rectangle is $g=.175\mu m$, and the thickness of resonators is $l_{m}=25nm$.
3.3 Tunable Metasurfaces
Plasmon-induced transparency (PIT), the plasmonic analogue of electromagnetically induced transparency, is a quantum interference effect between radiative (bright) and non-radiative (dark) surface plasmon waves. In the PIT phenomenon, the bright mode, which is easily coupled with the incident wave, interacts with the dark mode in such a way that it creates a transparency window at a frequency near the bright mode resonance [56]. The realization of PIT in different types of coupled resonators has been the subject of many studies during the last few years, and metal/graphene-based metasurfaces have always been proposed as an interesting structure to achieve this phenomenon. From the viewpoint of application, the ability to dynamically control the effect of PIT in plasmonic metasurfaces is very attractive since it enables changing the operating frequency range. Although most of the metal-based metasurfaces have a fixed spectral response and are hardly possible to dynamically tune the PIT responses, graphene-based metasurfaces have provided a fruitful platform for actively controlling the PIT effect [56–59]. Graphene, a two-dimensional material made of carbon atoms, has emerged as a promising platform for plasmonics due to its controllable plasmonic properties, optical transparency, gate-variable optical conductivity, and strong light confinement. The gate-variable conductivity of graphene-based metasurfaces allows us to tune the PIT response by controlling the coupling strength between the resonators.
The array of graphene ribbons depicted in Fig. 6(a) is one of the structures that have been suggested to realize the PIT phenomenon [60]. Figure 6(a) is a one-dimensional array of graphene ribbons in which there are two graphene ribbons with distinct biases /widths in the unit cell. In the case where a TM polarized wave is normally impinging on the metasurface, following the described steps in previous examples, the results shown in Fig. 6 are obtained. As can be seen, the model has high accuracy; however, an error around 9 THz is observed, which is due to the excitation of second-order current mode in the wider ribbon. In one-dimensional structures, changing the incident wave angle does not considerably change the current distributions. Hence, utilizing the extracted currents in normal incident scenario, the transmission spectra as a function of incidence angle and frequency is depicted in Fig. 6(d), which shows appropriate accuracy.
Fig. 6. (a) Schematic of the structure which is illuminated by an oblique TM polarized wave. (b) Comparison between the result of proposed method and full-wave simulation when the structure normally impinged. (c) Transmission mapping as a function of incident angle and frequency computed using circuit model. (d) Transmission mapping as a function of incident angle and frequency obtained using full-wave simulation. Parameters: $P=30\mu m$, $W_{1}=7.5\mu m$, $W_{2}=6\mu m$, $l_{s}=1.5\mu m$, $E_{f1}=.7ev$, $E_{f2}=1ev$, and the relaxation time of both ribbons is $\tau =.45ps$.
4. Conclusion
Herein, a variational-based approach simplified into an equivalent circuit model was introduced to investigate Fano resonant metasurfaces. In this method, exited evanescent Floquet modes were modeled as inductors and capacitors, which are energy storage elements; the incident harmonic was modeled as transmission-lines, and the currents on the plasmonic resonators led to a resistive and an inductive effect. In this study, we found that the equivalent impedance of the array drastically varies around the Fano resonance, and leads to a decrease in the bandwidth and an increase in the loss. Using the proposed model, two 2-D metallic metasurfaces and a 1-D graphene-based metasurface were explored, and we observed that the model accurately predicts the electromagnetic response of the structures.
5. Appendix
To solve the presented problem, electric and magnetic boundary conditions should be applied. At the discontinuity (z=0), the Floquet expansions of the transverse (x, y components) electromagnetic fields in the upper (region 1) and lower (region 2) environments of the array are
The magnetic field boundary condition implies
Using the orthogonality of the Floquet modes in Eq. (25) yields
The currents on the resonators give
Multiply both sides of Eq. (30) by $(\hat {z} \times J_{is}^{*})$ and some mathematical operations result in
Equation (32) can be rewritten as
The system of above equations can be solved to obtain $\frac {P_{10}^{*}}{C_{0}^{*}}$ and $\frac {P_{20}^{*}}{C_{0}^{*}}$
By substituting $\frac {P_{10}^{*}}{C_{0}^{*}}$ and $\frac {P_{20}^{*}}{C_{0}^{*}}$ in Eq. (26), the following equation is obtained, which represents the circuit model of Fig. 2.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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