Generalized equivalent circuit model for analysis of graphene/metal-based plasmonic metasurfaces using Floquet expansion

Mohammad Pasdari-Kia; Mohammad Memarian; Amin Khavasi

doi:10.1364/OE.471558

1. Introduction

Metasurfaces, two-dimensional or flat versions of metamaterials, have received a great deal of attention due to their unique ability to manipulate different characteristics of impinging light. These engineered structures are easily fabricated and open the door to a wide range of applications that are unattainable with naturally available materials. These structures are essential in terms of industrial applications, and the observation of novel phenomena has increased the importance of metasurfaces [1–4]. One of the unique characteristics of these structures is that by designing shapes and sizes of sub-wavelength constituents of the unit cell, one can control wave propagations on the surface and scattering into the surrounding space, with versatility. In addition to geometry, materials used in the structure can significantly impact performance and even enable new functionalities, which is why a large variety of dielectrics and metals or newer materials such as graphene and phase-change materials have been used in metasurface design in the past decade [5–7].

In plasmonic metasurfaces, patches are made of metals experiencing plasmonic effects in the frequency of interest, such as gold, silver and aluminum. Use of graphene and its various interesting features in metasurface patches, has more recently gained significant attention. Graphene is an allotrope of carbon consisting of a single layer of atoms tightly bound in a hexagonal honeycomb lattice. Graphene is chiefly noted for its high electrical conductivity, gate-variable optical conductivity, and controllable plasmonic properties. Plasmonic metasurfaces have various applications, including sensors [8–10], absorbers [11,12], plasmonic switches [13], polarizers [14], etc. Due to the high importance of these structures, providing an analytical method describing their exact optical response can significantly reduce the simulation time and improve the design and optimization process. The transmission-line model is considered in this paper due to its appropriate accuracy and high flexibility, which can be easily generalized to layered structures.

One of the analytical methods for investigating plasmonic metasurfaces is the eigenvalue problem method. In the previous works, using the eigenvalue problem method, the one-dimensional arrays of ribbons [15] as well as the two-dimensional arrays of disks with surface conductivity were investigated [16]. In this method, first, the problem is written as an integral equation, and then the integral equation is solved using the eigenvalue method to find the current modes on the patches. By solving the eigenvalue problem, the electric current modes on the patches and the values of the modes are obtained analytically when the array of patches is illuminated by a normal incident plane-wave and finally converted into a circuit model. One of the main disadvantages of this method is that layered metasurfaces (array of patches embedded in a layered medium) can not be examined with reasonable accuracy, especially when the thickness of the dielectric layers is low. This problem has been solved in Refs. [17] and [18] for a special graphene-insulator-metal metasurface in which the metal used is modeled as PEC, which is not a suitable approximation in the high-frequency range. In addition, due to the approximations used in solving the eigenvalue problem, the accuracy of the method decreases with increase in the surface conductivity of the patches.

Another analytical method for investigating metasurfaces is using multi-modal equivalent circuit models, which have been used frequently in the microwave frequency range [19–21]. These circuit models are created using Floquet expansion along with current distribution on the patches (the current profile can be obtained from full-wave simulations at a single frequency, or a closed-form expression can be used as the current on the patches). With the help of these methods, one-dimensional arrays of PEC strips [22,23] as well as two-dimensional arrays of PEC patches have been examined [19,20]. Unlike the eigenvalue problem method, the value of the electric current on the patches is not important, and the equations are written in such a way that only the spatial profile of the current affects the values of the circuit elements. This method can predict the behaviors of layered metasurfaces with very high accuracy and is also readily applicable to different shapes of patches, which is why it has received so much attention.

In the previous multi-modal circuit models, the patches were considered PEC that can not be used at high frequencies due to the plasmonic properties of metals; furthermore, graphene-based metasurfaces can not be modeled in this way. Another disadvantage of these models is that at high frequencies, the changes in the current profile in terms of frequency are significant, which can not be modeled by considering only one current mode. In addition to the stated analytical methods, semi-analytical methods for plasmonic metasurfaces have also been proposed [24–26]. In such methods, first, the array is simulated at all desired frequencies; subsequently, the circuit/environmental parameters are extracted, then obtained parameters are used for layered metasurfaces. This procedure also does not model the effect of the added dielectric layers well, and the error in layered metasurfaces is also significant.

This article presents a circuit model based on Floquet expansion that solves the shortcomings of the state-of-the-art methods mentioned above. Our proposed method is very flexible and unlike its predecessors, it is extended in such a way that it can predict the behavior of plasmonic metasurfaces. The previous versions of this method were used in microwave frequencies thus only dealing with PEC structures. Moreover, they usually only needed one current mode, but for plasmonic metasurfaces, especially graphene-based ones, we need to consider several current modes. Unlike semi-analytical methods that require simulating the structure at all frequencies, this circuit model only requires finding the current modes of the patches. This means that once the current modes are extracted with the help of a full-wave simulator, the model is then able to handle multilayer structures with any number of layers. In addition to the stated advantages, it is shown that this method has a much better performance compared to the eigenvalue method. This paper uses a simplified form of generalized boundary conditions to investigate metal-based plasmonic metasurfaces. In this paper, With the help of these boundary conditions, metals are modeled as surface conductivity similar to a graphene monolayer and applying these boundary conditions along with the proposed circuit model leads to high accuracy in modeling metal-based plasmonic metasurfaces. The two most commonly used layered metasurfaces studied in the literature are also investigated using this new method, and the results of full-wave simulation with high accuracy confirm the results of the new circuit model.

2. Proposed circuit model

In this section, we present an accurate circuit model for one-dimensional and two-dimensional arrays of metallic/graphene patches. In the proposed circuit model, surface electric conductivity is considered for patches to model plasmonic properties at infrared frequencies. It should be noted that in the following, a time dependence of the form exp($-j\omega t$) is implicitly assumed. A graphene monolayer is modeled as surface electric conductivity accounting for intra-band and inter-band conductivity [27]:

(1)$$\sigma_{s}(\omega)=\frac{2e^{2}k_{B}T}{\pi \hbar^2} \frac{j}{j\tau^{{-}1} + \omega} \log[2\cosh(\frac{E_{f}}{2k_{B}T})] + \frac{e^2}{4\hbar}[H(\omega/2) + \frac{4j\omega}{\pi}\int_{0}^{\infty} \frac{H(\varepsilon) - H(\omega/2)}{\omega^2 - 4\varepsilon^2} d\varepsilon]$$

where $e$ is the electron charge, $E_F$ is the Fermi energy, $\hbar$ is the reduced Plank constant, $k_{B}$ is the Boltzmann constant, $\omega$ is the angular frequency, $T$ is the temperature, $\tau$ is the relaxation time, and function $H(\varepsilon )$ is given by

(2)$$H(\varepsilon)=\frac{\sinh(\frac{\hbar\varepsilon}{k_{B}T})}{\cosh(\frac{E_{f}}{k_{B}T})+\cosh(\frac{\hbar\varepsilon}{k_{B}T})}$$

For metals, if they are very small in thickness, we can use generalized boundary conditions to model metals as a surface conductivity like graphene monolayer [28–30]. In these generalized boundary conditions, in addition to electric current, the magnetic current is also considered; therefore, both electric and magnetic resistivities are used. In the special case, if the thickness of the metal is very small (the thickness should not be much greater than the penetration depth), in the far-infrared and mid-infrared range, we can ignore the magnetic current and with this assumption, similar to the graphene monolayer, we can consider a surface electric conductivity in the form of Eq. (4), which is the inverse of the surface electric resistivity. In these equations, $\eta ^{0}$ is the intrinsic impedance of the vacuum, $n_{m}$ is the refractive index of the metal, $L$ is the thickness of the metal, and $k_{0}$ is the free space wavenumber. Furthermore, $E_{t}^{1}$ and $E_{t}^{2}$ are the tangential electric fields on both sides of the metal; in the same way, $H_{t}^{1}$ and $H_{t}^{2}$ are the tangential magnetic fields, and $Z_{se}$ is surface electric resistivity of metal.

(3)$$\begin{cases} E_{t}^{1}=E_{t}^{2} \\ E_{t}^{1}={-}Z_{se}\hat{z} \times (H_{t}^{1}- H_{t}^{2}) \end{cases}$$

with

(4)$$Z_{se}=\frac{1}{\sigma_{s}}=\frac{j\eta^{0}}{2n_{m}}\cot(n_{m}k_{0} \frac{L}{2})$$

Consider now the periodic structure of Fig. 1 which is illuminated by an incident TM/TE polarized wave. The impinging plane-wave excites the Floquet modes on both sides of the array and also current modes on the patches. The Floquet expansions of the transverse (x, y components) electromagnetic fields at the discontinuity ($z=0$) are

(5)$$\begin{cases} E^{1}(x,y)=(1 + R)e_{0}^{1}(x,y) + \sum_{h}^{'}V_{h}^{1} e_{h}^{1}(x,y) \\ H^{1}(x,y)=(1 - R)Y_{0}^{1}(\hat z \times e_{0}^{1}(x,y)) - \sum_{h}^{'}V_{h}^{1} Y_{h}^{1}(\hat z \times e_{h}^{1}(x,y)) \end{cases}$$

(6)$$\begin{cases} E^{2}(x,y)=Te_{0}^{2}(x,y) + \sum_{h}^{'}V_{h}^{2} e_{h}^{2}(x,y) \\ H^{2}(x,y)=TY_{0}^{2}(\hat z \times e_{0}^{2}(x,y)) + \sum_{h}^{'}V_{h}^{2} Y_{h}^{2}(\hat z \times e_{h}^{2}(x,y)) \end{cases}$$

where the expression $e_ {0}^{1}(x,y)$ is the tangential component of the incident wave whose reflection is R. The incident wave can be TE or TM polarized, and the prime in the series indicates that the incident wave is excluded from the summation. The superscript refers to the medium (we name the upper environment of the array as region I ($n_{1}$) and the lower environment as region II ($n_{2}$)). $e_ {0} ^ {2}(x,y)$ is the tangential component of the corresponding incident wave in region II, which is the transmission of the incident wave. The expressions $e_ {h} ^ {1}(x,y)$ and $e_ {h} ^ {2} (x,y)$ are the tangential components of the diffracted harmonics $h$ in the regions I and II, respectively.

Fig. 1. A square array of gold disks. In this structure, the thickness of disks is $L_{d}$, the diameter of disks is D, the period is P in both directions and the refractive index of upper and lower environments is $n_{1}$ and $n_{2}$, respectively.

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In the case of a one-dimensional periodic array, the TE incident wave only excites the TE Floquet modes; likewise, the TM incident wave only stimulates TM modes, but in two-dimensional periodic arrays, TE or TM incident waves simultaneously excite TE and TM modes; therefore in Floquet expansion both TE and TM harmonics should be considered. In two-dimensional periodic arrays, two parameters (m,n) are used to determine the harmonics, but in the above expansion, only the $h$ is used to specify the harmonics, which is only to reduce the complexity of the appearance of equations. For a two-dimensional periodic array, the Floquet harmonics can be written as [19,20]:

(7)$$e_{h}^{1}(x,y)=\frac{e^{{-}jK_{th}^{1}.\hat{\rho}}}{PP} \hat{e_{h}}^{1} \qquad \hat{\rho}=x\hat{x} + y\hat{y}$$

(8)$$k_{th}^{1}=k_{xm}^{1}\hat{x} + k_{yn}^{1}\hat{y} = (k_{m}^{1} + k_{x0}^{1} )\hat{x} + (k_{n}^{1} + k_{y0}^{1})\hat{y}$$

(9)$$k_{x0}^{1}=k^{1}\sin(\theta)\cos(\phi) \qquad k_{y0}^{1}=k^{1}\sin(\theta)\sin(\phi)$$

(10)$$k_{m}^{1}=\frac{2\pi m}{P} \qquad k_{n}^{1}=\frac{2\pi n}{P}$$

(11)$$\hat{e_{h}}^{1}= \begin{cases} \hat{k_{th}}^{1} \hspace{.5cm} & TM \\ (\hat{k_{th}}^{1} \times \hat{z} ) \hspace{.5cm} & TE \end{cases} \qquad \hat{k_{th}}^{1}=\frac{k_{th}^{1}}{\mid k_{th}^{1} \mid} $$

The modal admittances $Y_{h}^{1}$ are given by

(12)$$Y_{h}^{1}=\frac{1}{\eta^{1}} \begin{cases} \frac{k^{1}}{k_{zh}^{1}} \hspace{.5cm} & TM \\ \frac{k_{zh}^{1}}{k^{1}} \hspace{.5cm} & TE \end{cases} \hspace{.5cm} k_{zh}^{1}=\sqrt{(k^{1})^{2} -{\mid} k_{th}^{1} \mid ^{2}}$$

with

(13)$$k^{1}=n_{1}k_{0} \qquad \eta^{1}=\frac{\eta^{0}}{n_{1}}$$

where $k_{0}$ is the vacuum wavenumber and $\eta ^{0}$ is the intrinsic impedance of free space. These equations are for any incident angle; however, we focus on normal incident waves in this paper for simplicity in presentation. In the case of a normal incident wave ($\theta = 0, \phi = 0$), we can use modes of a waveguide with PEC and PMC walls instead of using the written Floquet modes which decrease computational volume significantly [31,32].

As in any electromagnetic problem, we must apply boundary conditions at the discontinuity to solve this problem, but the number of unknown variables is equal to the excited Floquet modes which we have considered. Assuming we have the spatial profile of the excited current mode on the patches, we can reduce the number of these unknown variables to one, and by applying the boundary conditions at the discontinuity (z=0), the only unknown variable can be computed (in the desired solution, we use the excited current modes separately in the equations and finally sum the effect of all of them.). With the help of this procedure, we can easily calculate any of these unknown variables ($V_{h}$ or $R$), but we are only going to compute the reflection ($R$). To calculate the reflection, the equations can be written in the form of a circuit model, the details of which are given in the appendix (Fig. 2 is the obtained equivalent circuit model).

Fig. 2. Generalized circuit model for plasmonic periodic arrays like Fig. 1

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In the presented circuit model, $Z_{eq}$ is the equivalent impedance of plasmonic array and obtained as:

(14)$$Z_{eq}=\frac{C_{0}^{*}\iint\limits_p (J_{s}\times (\hat{z} \times J_{s}^*)).\hat{z} ds}{\sigma_{s} P_{0}P_{0}^{*}} + \sum_{h}\phantom{}^{'} \hspace{.1cm}\frac{P_{h}P_{h}^{*}C_{0}^{*}}{C_{h}^{*}P_{0}P_{0}^{*}}\frac{1}{Y_{h}^{1} + Y_{h}^{2}}$$

where

(15)$$P_{h}= \iint\limits_{p} (e_{h} \times (\hat{z} \times J_{s}^{*})).\hat{z} ds$$

(16)$$C_{h}=\iint\limits_{c} (e_{h} \times (\hat{z}\times e_{h})^{*}).\hat{z} ds$$

For computing $P_{h}$, integration is on the patch (p), and in the calculation of $C_{h}$, integration is on the unit cell (c), as noted in the appendix. As mentioned, to reduce the number of unknown variables, we considered a spatial profile for the patch current. If we consider the current as $J_{s}=A(\omega )j_{s}(x,y)$, by placing this current in the equation of $Z_{eq}$, the frequency dependence of the current ($A(\omega )$) will be ineffective, and so we may say that $Z_{eq}$ depends only on the spatial profile of current and the current value has no effect on it.

As it turns out, the computed equivalent impedance consists of two parts, the first part of $Z_{eq}$ depends on the surface conductivity of the patches, and the modal admittances have no effect on it. Unlike the first part, the second part of $Z_{eq}$ depends only on the admittances of excited modes. The fact that the effect of modal admittances, as well as the effect of surface conductivity is obtained separately makes the model more flexible, and also because the effect of intrinsic loss of metal or graphene is obtained separately, it can improve the design process especially for metasurface absorbers [11,33]. It should be noted that the evanescent modes are equivalent to the energy storage elements, so the TE modes are equivalent to the inductors, and the TM modes are equivalent to the capacitors.

First, let us examine an array of gold disks shown in Fig. 1 with the help of the provided circuit model. For validation purposes, we have simulated the structure by full-wave software in two ways (for gold, we have used the refractive index in Ref. [34]). We have considered the thickness $L_{d}$ for the disks and simulated the array and also with the help of the surface conductivity obtained for the metals (Eq. (4)), we have re-simulated the structure. We have extracted the current spatial profile on the disk at the resonance frequency from full-wave software and use this current profile to compute $Z_{eq}$. Two parts of $Z_{eq}$ are depicted separately in Fig. 3; we have assumed that the effect of metal’s surface conductivity is $Z_{P1}$ and the effect of excited Floquet modes is $Z_{P2}$.

Fig. 3. Figures (a) and (b) show the first and second parts of $Z_{eq}$ respectively ($n_{1}=1,n_{2}=1$, $P=4$ (um), $D=2$ (um)).

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As shown in the Fig. 3(a), the surface conductivity creates an inductive effect that increases with increasing frequency; in addition, a resistive effect can also be seen in Fig. 3(a), which is the result of the ohmic loss of metal at high frequencies. The second part of $Z_{eq}$, which is drawn in Fig. 3(b) and is the result of the excited modes, does not have a resistive effect because all the excited Floquet modes are evanescent at the investigated frequencies. But as it can be seen, the imaginary part shows a capacitive effect at first, which turns into an inductive effect as the frequency increases. The reflection reaches its maximum value when the imaginary part of $Z_{eq}$ becomes zero, in which the inductive effects and capacitive effects neutralize each other. If we use the computed equivalent impedance in the circuit model of Fig. 2, the reflection of incident wave is obtained as Fig. 4(b).

Fig. 4. The figure on the left shows the absolute electric current profile on the disk extracted from a full-wave simulation at the resonance frequency (the resonance point is shown in the figure on the right), and the figure on the right shows the result of the circuit model obtained using the extracted current profile ($n_{1}=1,n_{2}=1$, $P=4$ (um), $D=2$ (um)).

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As can be seen in Fig. 4, the circuit model has a very high degree of adaptation to the full-wave simulation result; however, the model has a slight error compared to the simulation with thickness due to the supposed surface conductivity and depth of penetration. This is while previous methods such [16] completely fail to capture the optical response as shown.

In metal-based metasurfaces, at a thickness of about 5 nm and greater, it is typically sufficient to consider only one current mode (it is better to extract the current mode in the resonance frequency, but in a metal-based plasmonic metasurface, the profile in other frequencies also gives accurate results). In general, current spatial profile changes with respect to frequency depend on the surface conductivity and shapes of the patches. The higher the surface conductivity of the patches, the less changes in the current profile vs frequency, such that in PEC patches, the changes are very small. It should be noted that using the first current mode is sufficient in many applications, but in graphene-based metasurfaces, we usually need to consider more current modes. For example let us consider the structure of Fig. 5. Separately, three modes have been used in the presented circuit model, and the results of the circuit model and full-wave simulations are depicted in Fig. 6.

Fig. 5. The structure of the cross-shaped graphene array with dimensions $W_{1}$ and $W_{2}$. In this structure, the period is $P$ in two directions, the upper environment has a refractive index $n_{1}$, and the lower environment has a refractive index $n_{2}$.

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Fig. 6. Figures (a), (b), and (c) are the profiles of the current modes that were extracted from the full-wave simulator at the 3 resonances (the resonance points are shown in the figures (d), (e) and (f)). Figures (d), (e) and (f) show the results of the circuit model computed using the current modes ("R" refers to reflection and the subscript of "R" refers to the resonance number). The geometrical parameters of the structure are $W_ {1}=10$ (um), $W_ {2}=3$ (um) and $P=20$ (um). The upper and lower environments are the vacuum, and the characteristics of the graphene are considered: $\tau = 1ps$ and $E_ {f} = 1eV$.

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As mentioned, in eigenvalue-problem methods, current modes on the patch and their value are computed, but for a cross-shaped patch, the current modes on the cross cannot be obtained analytically. It can be said with great accuracy, in different resonances, one of the current modes has a dominant effect, so we can use the current profile extracted from the full-wave simulation at different resonances as cross-shaped patch current modes [15,16]. As it can be seen in Fig. 6, using a current mode produces a good result in a frequency range where that mode has a dominant effect. Since different current modes excite different Floquet modes, the second part of equivalent impedance ($Z_{eq}$) corresponding to each current mode is different from the others. Moreover, the first part of $Z_{eq}$ for three current modes is different due to the difference in the spatial profile of current modes (Supplement 1), and for this reason, their reflections are also different. If we sum up the effect of three current modes, the circuit model will gives the result depicted in Fig. 7, as it is clear that the results of the full-wave simulation confirm the results of the circuit model in a wide frequency range.

Fig. 7. The sum of circuit model results for three current modes

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3. Multi-layer structure

In practical applications, dielectric and metallic layers may be used in metasurfaces to create new features, greater strength, and adhesion of layers, so the analysis of arrays of plasmonic patches embedded in a layered medium is very important. In Ref. [18], a PEC–backed array of graphene ribbons has been studied using the eigenvalue method, and with the addition of the dielectric and PEC layers, the formulation has been rewritten based on the new geometry. Reference [18] has modeled the end metal layer as PEC, which would make a considerable error in the mid-infrared frequency range, and as mentioned, there is an increase in error with increasing surface conductivity of ribbons in this model. One of the advantages of the proposed method is that the effect of adding different layers of any thickness to the structure can be easily applied in the circuit model, which makes this method highly flexible. In this section, we examine two commonly used transmittive/reflective plasmonic metasurfaces in which an array of plasmonic patches is placed on a layered medium.

A type of reflective plasmonic metasurface that uses Meta-Insulator-Metal as a basic building block, is constructed by adding a dielectric spacer and a metallic mirror layer beneath the array of plasmonic patches. In such structures, the dielectric spacer is thin to confine the electromagnetic field and increase the near-field coupling between the array and metallic mirror [35]. An example of such MIM structure is shown in Fig. 8(a). In the circuit model for this structure, transmission-lines related to the metallic mirror layer and dielectric spacer layer are added. In addition, in Eq. (14), modal admittances should then be replaced by the input admittances to the corresponding cascade of transmission-lines seen from the array to the up and down (in the simulations, the refractive index of gold and $SiO_{2}$ are used from Ref. [34] and [36], respectively). The result of reflections show excellent agreement between theory and full-wave simulation results.

(17)$$Z_{eq}=\frac{C_{0}^{*}\iint\limits_p (J_{s}\times (\hat{z} \times J_{s}^{*})).\hat{z} ds}{\sigma_{s} P_{0}P_{0}^{*}} + \sum_{h}\phantom{}^{'} \hspace{.1cm}\frac{P_{h}P_{h}^{*}C_{0}^{*}}{C_{h}^{*}P_{0}P_{0}^{*}}\frac{1}{Y_{in,h}^{D} + Y_{in,h}^{U}}$$

with

(18)$$Y_{in,h}^{D}=Y_{h}^{2} \frac{(Y_{h}^{3} - jY_{h}^{2}\tan(K_{zh}^{2}L_{SiO_{2}}))}{(Y_{h}^{2} - jY_{h}^{3}\tan(K_{zh}^{2}L_{SiO_{2}}))} \qquad Y_{in,h}^{U}=Y_{h}^{1}$$

Fig. 8. Figure (a) shows a Meta-Insulator-Metal metasurface which is used as an absorber sensor. The material of the disks and the end mirror layer is gold, and the middle dielectric material is $SiO_ {2}$ ($P = 4$ (um), $D = 2$ (um), $L_ {d} = 10$ (nm), $L_ {SiO_{2}} = 350$ (nm)). Figure (b) shows the circuit model of the absorber where the parameter $Z_{eq}$ related to the structure is written in Eq. (17) (corresponding $Z_{eq}$ is depicted in Supplement 1). Figure (c) is a comparison of the full-wave simulation and the circuit model results (it should be noted that in the full-wave simulation, the disks are considered with thickness).

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Transmittive metasurfaces are another type of metasurfaces in which the metallic mirror layer does not exist beneath the array and metasurfaces are composed of only dielectric layers. In recent years, examples of them have been fabricated for applications such as sensors and switches [8,9,13]. Such structure used as a sensor is shown in Fig. 9(a), and the equivalent circuit model for this structure, which consists of cascading of transmission-lines associated with the layers, is also shown in Fig. 9(b). By increasing the number of layers to obtain the amount of transmitted and reflected power as well as input admittances associated with the $h$-harmonic, the transfer matrix method can be used for its analysis [37].

Fig. 9. Figure (a) is a metasurface consisting of an array of graphene ribbons ($E_{f}=1.4eV,\tau =.7ps$ ), a layer of $SiO_ {2}$, and a layer of $Si$. In the simulation, the refractive index of $SiO_ {2}$ and $Si$ are approximately 2 and 3.4, respectively. Figure (b) is a circuit model related to the structure. Geometrical parameters of the structure are considered as $P = 30$ (um), $W = 15$ (um), $L_ {Si} = 10$ (um), $L_ {SiO_{2}} = 1$ (um).

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If the incident wave is TM polarized, the results of scattered waves are as depicted in Fig. 10. TM polarized incident wave has been investigated in Ref. [15], as it is clear that the model has a large error in predicting the reflection and transmission relative to our proposal, when compared to full-wave simulation results.

Fig. 10. Figures (a) and (b) show the transmission and reflection of the TM incident wave (corresponding $Z_{eq}$ is depicted in Supplement 1).

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If the incident wave is TE polarized, the results are shown in Fig. 11. In this circuit model, TE and TM polarized incident waves can be easily simulated without the need to write different formulations. For the TE and TM polarization, the excited current modes are different, and by applying this change in the equations, the results are easily obtained. An excellent match between theory and full-wave simulations can be observed.

Fig. 11. Figures (a) and (b) show the transmission and reflection of the TE incident wave (corresponding $Z_{eq}$ is depicted in Supplement 1).

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4. Conclusion

Herein, we developed an equivalent circuit model using Floquet expansion along with spatial profiles of current modes of plasmonic patches to investigate plasmonic metasurfaces. In this circuit model, the array of plasmonic patches was modeled as an equivalent impedance, which consists of the modal admittances and the surface conductivity of the patches. First, an array of gold disks and also an array of graphene crosses were investigated and it was shown that the proposed method results have excellent agreement with full-wave simulation results. In the presented results, we saw that sometimes several current modes may be needed; therefore we added the results of several current modes together for proper results. Next, the circuit model for two layered structures was examined and we observed excellent performance. Unlike the previous models, the presented circuit model does not need to change the formulation for adding layers and changes in incident wave polarizations, and this issue is considered the strength of this model.

5. Appendix

First, we apply the electric field boundary condition at the discontinuity ($z=0$) and the following equations are obtained. As it turns out, the Floquet modes coefficients are obtained the same in regions I and II.

(19)$$\begin{cases} 1+R=T \\ V_{h}^{1}=V_{h}^{2}=V_{h} \end{cases}$$

We need to use the spatial profile of excited current mode on the patch to create a circuit model. The magnetic field boundary condition implies

(20)$$\begin{aligned} -\hat z \times((1 - R)Y_{0}^{1}(\hat z \times e_{0}(x,y)) - & \sum_{h}\phantom{}^{'}V_{h} Y_{h}^{1}(\hat z \times e_{h}(x,y)) - TY_{0}^{2}(\hat z \times e_{0}(x,y)) \\ - & \sum_{h}\phantom{}^{'}V_{h} Y_{h}^{2}(\hat z \times e_{h}(x,y)) )=J_{s} \end{aligned}$$

Using the orthogonality of the modes gives

(21)$$((1 - R^{*})(Y_{0}^{1})^{*} - (1 + R^{*})(Y_{0}^{2})^{*})=\frac{\iint\limits_{p} (e_{0} \times (\hat{z} \times J_{s}^{*})).\hat{z} ds}{\iint\limits_{c} (e_{0} \times (\hat{z}\times e_{0})^{*}).\hat{z} ds}=\frac{P_{0}}{C_{0}}$$

(22)$$-((Y_{h}^{1})^{*} + (Y_{h}^{2})^{*})V_{h}^*=\frac{\iint\limits_{p} (e_{h} \times (\hat{z} \times J_{s}^{*})).\hat{z} ds}{\iint\limits_{c} (e_{h} \times (\hat{z}\times e_{h})^{*}).\hat{z} ds}=\frac{P_{h}}{C_{h}}$$

To compute $V_{h}$, we divide Eq. (22) and Eq. (21).

(23)$$V_{h}=\frac{P_{h}^{*}C_{0}^{*}}{C_{h}^{*}P_{0}^{*}}\frac{ (1 + R)Y_{0}^{2} - (1 - R)Y_{0}^{1}}{Y_{h}^{1} + Y_{h}^{2}}$$

In addition to the written boundary conditions, the current on the patch can be written in terms of the electric field on the patches.

(24)$$(1 + R)e_{0}(x,y) + \sum_{h}\phantom{}^{'}V_{h}e_{h}(x,y)=\frac{J_{s}}{\sigma_{s}}$$

Multiplying both sides of Eq. (24) in $(\hat {z} \times J_{s}^{*})$ and some mathematical operations give

(25)$$(1 + R)P_{0} + \sum_{h}\phantom{}^{'}V_{h}P_{h}=\iint\limits_p (J_{s} \times (\hat{z} \times J_{s}^{*})).\hat{z} ds$$

Placing the $V_ {h}$ (Eq. (23)) in the above equation and a little mathematical operation, the following equation can be obtained, which represents a circuit model.

(26)$$\frac{1 + R}{(1 + R)Y_{0}^{2} - (1 - R)Y_{0}^{1}}={-}\sum_{h}\phantom{}^{'} \hspace{.1cm}\frac{P_{h}P_{h}^{*}C_{0}^{*}}{C_{h}^{*}P_{0}P_{0}^{*}}\frac{1}{Y_{h}^{1} + Y_{h}^{2}} - \frac{C_{0}^{*}\iint\limits_p (J_{s} \times (\hat{z} \times J_{s}^{*})).\hat{z} ds}{\sigma_{s} P_{0}P_{0}^{*}}$$

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.

Supplemental document

See Supplement 1 for supporting content.

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Generalized equivalent circuit model for analysis of graphene/metal-based plasmonic metasurfaces using Floquet expansion

Abstract

1. Introduction

2. Proposed circuit model

3. Multi-layer structure

4. Conclusion

5. Appendix

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (11)

Equations (26)

Optics Express