Tunable mantle cloaking utilizing graphene metasurface for terahertz sensing applications

Zahra Hamzavi-Zarghani; Zahra Hamzavi-Zarghani; Alireza Yahaghi; Alireza Yahaghi; Ladislau Matekovits; Ali Farmani; Ali Farmani

doi:10.1364/OE.27.034824

1. Introduction

Cloaking techniques have been demonstrated to lead to highly effective tools that are able to sense electromagnetic radiation, especially at terahertz frequencies with negligible perturbation [1–13]. Different approaches have been proposed for cloaking purpose such as transformation optics [14,15] plasmonic cloaking [16,17] and so on [18–22]. Mantle cloaking is one of the most effective methods showing notable performance and comfortable realization [23,24]. Since in a mantle cloak, the object is covered by a thin metasurface, the proposed cloaking devices have low profile and weight and good flexibility to be formed in a desired shape [25, 26]. At microwave frequencies, patterned metallic sheets have been used as the covering metasurface to achieve the required surface impedance aimed at cancelling the scattered field of the object by producing an anti-phase scattered wave [27,28].

In this regards, several invaluable studies have been presented in literature to make dielectric and conducting cylinders invisible especially for $TM_z$ polarized incident wave propagating in the $x$ direction, i.e, orthogonal incidence. As an example Matekovits et al. [29], considered cloaking of multiple dielectric cylinders. Covering the cylinders by width-modulated microstrip line based mantle cloak has significantly reduced their scattering. Also, Teperik et al. [30], experimentally reported ultra-thin metasurface cloak for hiding a metallic obstacle. In their work, radar cross section (RCS) of conducting cylinders has been remarkably suppressed by coating them with patterned metallic surfaces. Some researches have been devoted to increase cloaking bandwidth. For instance in Granpayeh research group [31,32], metasurfaces based on disks with different sizes have been utilized to cover dielectric cylinders and spheres, respectively in order to create resonances in different frequencies leading to a broad bandwidth. Therefore, metasurfaces have been received remarkable attention from research groups [33–35]. However, noble metasurfaces have relatively high ohmic loss and tunability of their optical properties are challenging [36–39].

To solve the above issue graphene is introduced in [40–46]. Graphene, a two dimensional material, as a allotropes of carbon has attracted significant attention because of its extraordinary properties such as strong light-matter interaction, low ohmic loss, and transparency [47–51]. One of the most important characteristics of graphene is its electrical and optical tunability which results in designing various tunable and reconfigurable devices in electronics and photonics realm [52–55] such as: switches and logic gates [56–58], reconfigurable lenses [59,60], tunable polarization converters [61,62] and tunable absorbers [63,64] and some other applications [65–71]. Tunable mantle cloaking can also be achieved with graphene monolayers or patterned graphene metasurfaces. In [72], scattering of a dielectric cylinder under illumination of $TM_z$ polarized oblique incidence has been reduced using a graphene monolayer. Graphene monolayer is inductive in terahertz range and therefore can not cloak conducting cylinders which needs capacitive surface impedances [73]. To cope with this issue, a nanostructured graphene metasurface with negative reactance has covered a conducting cylinder in order to make it invisible [48]. In the both cases, frequency of cloaking has been tuned by changing the chemical potential of graphene [74]. The majority of researches related to cloaking have been done with consideration of $TM_z$ polarized impinging wave propagating in the $x$ direction. This is because this incident wave produce more pronounced scattered fields compared to $TE$ polarized impinging wave [75]. However, the scattered field from a $TE_z$ illuminated wave is not negligible and in some applications it is necessary to cloak a cylinder under illumination of a $TE_z$ polarized wave. Dual polarized mantle cloak can be a good choice to reduce scattering from a cylinder for both $TE$ and $TM$ polarizations. To achieve mantle cloaking for $TE$ and $TM$ polarizations, an anisotropic metasurface should be designed. With an anisotropic metasurface, one can independently control the surface impedance in each direction, leading to achieve the required values of it for both polarizations in a given frequency [76]. We propose graphene strips as a covering metasurface whose surface impedance tensor has been derived in [77]. In [76], dual polarized cloaking has been achieved in a fixed frequency. However, in our proposed structure, the frequency of dual polarized cloaking can be tuned by adjusting the chemical potential of graphene. This is the highlight feature of the proposed structure.

The rest of this paper is organized as follows. In Section II, the optical and geometrical properties of the proposed sensor is presented and studied. In the same section, the metasurface to achieve the considered goal is designed. Then, in Section III, cloaking behaviour of the structure is investigated. Therefore, numerical results of RCS for uncloaked and cloaked cylinders are illustrated proving scattering reduction for $TE$, $TM$ and circular polarizations. In the same section, the tunable optical properties are considered and cloaking responses further extracted through numerical simulation. Then, in Section IV, finally, we summarize the main conclusions.

2. Designing an anisotropic metasurface based on graphene strips

The proposed structure is shown in Fig. 1, which can exhibit the cloaking behaviour. In this model, a dielectric cylinder under illumination of $TE$ and $TM$ polarized plane waves is used. The $z$ component of electric and magnetic fields for $TM$ and $TE$ polarizations can be written in terms of Bessel and Hankel functions as follows, respectively [78]:

(1)$$E_i = \hat{z}\ E_0\ \sum_{n={-}\infty}^{\infty} j^{{-}n} J_n(\beta_0 r) \ e^{j n \phi}$$

(2)$$E_s = \hat{z}\ E_0\ \sum_{n={-}\infty}^{\infty} j^{{-}n}\ c_{n (TM)}\ H_n^{(2)} (\beta_0 r)\ e^{j n \phi}$$

(3)$$E_{in} = \hat{z}\ E_0\ \sum_{n={-}\infty}^{\infty} j^{{-}n}\ a_{n (TM)}\ J_n(\beta r) \ e^{j n \phi}$$

(4)$$H_i = \hat{z}\ E_0\ \sum_{n={-}\infty}^{\infty} j^{{-}n} J_n(\beta_0 r) \ e^{j n \phi}$$

(5)$$H_s = \hat{z}\ E_0\ \sum_{n={-}\infty}^{\infty} j^{{-}n}\ c_{n (TE)}\ H_n^{(2)} (\beta_0 r)\ e^{j n \phi}$$

(6)$$H_{in} = \hat{z}\ E_0\ \sum_{n={-}\infty}^{\infty} j^{{-}n}\ a_{n (TE)}\ J_n(\beta r) \ e^{j n \phi}$$

where $J_n$ and $H_n^{(2)}$ are Bessel function of the first type and Hankel function of the second type, respectively. $\beta _0$ and $\beta$ are propagation constants in the air (here considered as the background medium) and in the cylinder with defined relative permittivity. Subscripts $i$, $s$ and $in$ represent incident field, scattered field and the field inside the object, respectively.

Fig. 1. Dielectric cylinder under TE and TM polarized incident waves.

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Applying boundary conditions of continuity of tangential electric field and discontinuity of magnetic field as a result of introducing the covering metasurface, and by considering the following relations:

(7)$$H_{\phi(TM)} = \frac{1}{j w \mu } \frac{\partial E_{z(TM)} }{\partial r}$$

(8)$$E_{\phi(TE)} ={-} \frac{1}{j w \epsilon } \frac{\partial H_{z(TE)} }{\partial r}$$

one can achieve scattering coefficients for the two polarizations. By equating the scattering coefficients to zero, the required surface impedances for the both polarizations are obtained.

Monti et al. [76], demonstrated that the required surface impedance to achieve cloaking is different for $TE$ and $TM$ polarizations and invisibility can not be achieved for the two polarizations simultaneously at the same centre frequency with an isotropic metasurface. Therefore, an anisotropic metasurface should be designed whose surface impedance is set in each direction, independently. Graphene strips as the covering metasurface are considered for this purpose. The proposed structure is shown in Fig. 2. For the planar configuration it exhibits a surface impedance tensor as follows [79]:

(9)$$z_{zz} = z_s\ \frac{p}{a}$$

(10)$$z_{xx} = z_s\ \frac{a}{p} + \frac{g}{p\ \sigma_c}$$

(11)$$\sigma_c = \frac{j \omega \epsilon_0 p}{\pi}\ \ln{\csc (\frac{\pi g}{2p})}$$

where $p$ and $a$ are periodicity and size of the strips, respectively, $g=p-a$ is the gap distance between two strips and $z_s$ is the surface impedance of graphene. Surface conductivity of graphene which is revers of surface impedance is modeled by Kubo formula [80,81]. It is the sum of $intra$ and $inter$ conductivity:

(12)$$\sigma_{intra} ={-}j \frac{K_B e^2 T}{\pi \hbar^2 (w-2j \tau^{{-}1})} [\frac{\mu_c}{K_B T}+ 2 \ln (e^{-\frac{\mu_c}{K_B T}}+1)]$$

(13)$$\sigma_{inter} = \frac{j e^2}{4 \pi \hbar} \ln \left ( \frac{2 \vert \mu_c \vert -(w-j \tau^{{-}1}) \hbar }{2 \vert \mu_c \vert +(w-j \tau^{{-}1}) \hbar } \right )$$

where $K_B$ is Boltzmann’s constant, $e$ is the electron charge, $\mu _c$ is the chemical potential, $\tau$ is the relaxation time, $T$ is the temperature and $\hbar$ is the reduced Plank$'$s constant. The chemical potential of graphene can be adjusted by applying different bias voltages resulting in different surface impedances.

Fig. 2. (a) Structure of graphene strips, (b) A dielectric cylinder coated by graphene strips.

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To design the proposed metasurface the following steps have been followed:

1. Obtaining the required surface impedance of the metasurface ($z_{zz}$) for achieving invisibility for $TM_z$ polarization, using the formula (14) from [23] which is useful for infinite cylinder: $(14)$$z_{zz} = \frac{2}{\omega a_1 \epsilon_0 (\epsilon_r -1)}$$$ where $a_1$ and $\epsilon _r$ are radius and relative permittivity of the cylinder.
2. Obtaining the required surface impedance of the metasurface for $TE$ polarization invisibility ($z_{xx}$). There is no closed form expression for cloaking a cylinder under illumination of $TE$ polarization [76]. The required surface impedance can be achieved by optimization.
3. Choosing the characteristics of graphene and obtaining its surface impedance ($z_s$).
4. Achieving the ratio of $p/a$ using eq. (9).
5. By knowing the ratio of $g/p$ from $p/a$, $\sigma _c$ is achieved using eq. (10).
6. The periodicity of the strips ($p$) will be obtained by eq. (11) and by knowing the ratio of $p/a$, the width of srtips ($a$) is also obtained. For give more insight about the design of metasurface, its pseudo code of metasurface design is reported in the following:

3. Results and discussions

To benchmark the performance of the proposed metasurface model, a numerical method is employed to simulate the structure. To sense of terahertz frequencies, the RCS spectra of the structure in regimes of cloaking and uncloaking is calculated. Here we aim to cloak a dielectric cylinder with radius of $a_1=10\mu$m and relative permittivity of $\epsilon _r=4$ for $TE$ and $TM$ polarizations at a reference frequency of $f_r=$ 2.5THz. The required surface reactance to achieve invisibility for $TE$ and $TM$ polarizations are obtained as: 234$\Omega$ and 404$\Omega$, respectively. We choose the characteristic of graphene as: $\mu _c=0.4$eV, $\tau =1$ps and $T=300$K resulting in $z_s=j333.7\Omega$. Following the procedure in the previous section we obtain $a=19.2\mu$m and $p=24\mu$m. It is worth noting that because the number of strips covering the cylinder should be integer and the obtained periodicity and size of the stirps lead to a non integer number of strips, optimization is needed. The optimized values for $z_{xx}$ and $z_{zz}$ is illustrated in Fig. 3 which at 2.5THz are very close to ones which have been achieved analytically. It can be seen that the required surface impedance for TM polarization shown by a green dot is very close to the surface impedance of the metasurface in the z axis. However, the required surface impedance for TE polarization shown by a blue star has a small difference with the surface impedance of the metasurface. The reason is that the surface impedance tensor has been derived for a planer structure while in this research, it has been used for bending structure. This can have low impact on z direction but more impact on x direction. The result is that for TE polarization, more optimization is needed.

Fig. 3. The surface impedance tensor elements $z_{xx}$ and $z_{zz}$ for the optimized parameters of the proposed structure. The determined points with a star and dot correspond to the required surface impedances for $z_{xx}$ and $z_{zz}$, respectively.

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Figure 4 shows RCS of uncloaked and cloaked cylinders for $TM$ and $TE$ polarizations illustrating simultaneous scattering reduction at 2.5THz for the two considered polarizations. It can also be seen that scattering reduction for $TM$ polarization is much higher than that of $TE$ polarization. We refer to [75] which provides an explanation for the reason of this difference. It is illustrated that for $TM$ polarization, the scattering related to the first harmonic is more pronounced than the other harmonics and by canceling the scattering of the first harmonic, significant RCS reduction can be achieved. However for $TE$ polarization, the first three harmonics have very similar amplitude and by canceling one harmonic, the other harmonics still play their role in the scattered field.

Fig. 4. RCS of uncloaked and cloaked cylinders with anisotropic metasurface for (a) $TM$ polarized incident wave and (b) $TE$ polarized incident wave.

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Effect of the relaxation time of graphene on the cloaking performance has been studied. Figure 5 shows the RCS of the cloaked cylinder for different relaxation times of graphene for $TE$ and $TM$ polarizations. It reveals that higher relaxation time results in better cloaking performance and lower RCS of the cylinder for both polarizations. The reason is that higher relaxation time leads to lower loss of graphene.

Fig. 5. RCS of the cloaked cylinders for different amounts of the relaxation time of graphene for (a) $TM$, (b) $TE$ polarizations.

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We exploit the extraordinary property of graphene to achieve tunable invisibility by changing chemical potential of graphene. Figures 6(a) and 6(b) show RCS of the cloaked cylinder with graphene strips for different chemical potentials. The figure indicates a shift in frequency of cloaking to 2.1THz and 2.8THz with chemical potential of 0.25eV and 0.55eV, respectively.

Fig. 6. RCS of uncloaked and cloaked cylinders with anisotropic metasurface for $TE$ and $TM$ polarizations with the chemical potential of (a) 0.25eV and (b) 0.55eV.

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Distribution of the electric field related to $TE$ and $TM$ polarizations for uncloaked and cloaked cylinders is shown in Fig. 7 which depicts that covering the cylinder by the designed graphene strips reduces scattering for both polarizations so that the incident plane waves pass the object with a small perturbation. Figure 8 shows distribution of the electric field at 2.1THz and 2.8THz for cloaked cylinders under illumination of $TE$ and $TM$ polarizations. It indicates that the incident waves can be considered as plane waves after passing the cylinders which means the scattering from the cylinders can be neglected at these frequencies.

Fig. 7. Electric field distribution for the (a) uncloaked and (b) cloaked cylinders for $TM$ polarization and (c) uncloaked and (d) cloaked cylinders for $TE$ polarization

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Fig. 8. Electric field distribution for the cloaked cylinders for (a) and (b) $TM$ polarization, (c) and (d) $TE$ polarization. (a) and (c) at 2.1THz, (b) and (d) at 2.8THz.

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Numerical results obtained by two commercial software CST Microwave Studio and HFSS confirm RCS reduction for $TM$ polarization in $\phi =0^{\circ }$ plane and for $TE$ polarization in plane $\theta =0^{\circ }$ in polar system are shown in Fig. 9. Furthermore, the results from the two software show good agreement.

Fig. 9. Polar plot of RCS related to cloaked and uncloaked cylinders for (a) $TM$ polarized incident wave in $\phi =0^{\circ }$ plane and for (b) $TE$ polarized incident wave in $\theta =0^{\circ }$ plane. Blue: uncloaked, Red: cloaked, Dashed line: CST, Solid line: HFSS

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The designed graphene strips can operate as a covering metasurface for invisibility purpose also for circular polarization. This claim is proved in Fig. 10 which shows scattering reduction for circular polarization at 2.5THz.

Fig. 10. RCS of cloaked and uncloaked cylinders under illumination of circular polarized waves.

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4. Conclusion

A tunable mantle cloaking of a dielectric cylinder under $TE$ and $TM$ polarizations in terahertz range was proposed. By considering the tunable optical properties of graphene metasurface, tunable cloaking behaviour of the proposed structure was investigated. For this purpose, an anisotropic metasurface based on graphene strips has been considered. The proposed covering structure and characteristics of graphene have been designed so that the required surface impedance tensor for achieving invisibility for both polarizations has been obtained. Scattered wave from the cylinder under illumination of circularly polarized wave can also decreases with the designed metasurface. Furthermore, by properly changing the chemical potential of graphene, tunable mantle cloaking has been achieved. To evaluate of the structure and to obtain high reduction in scattering, the numerical software including CST Microwave Studio and HFSS were considered. Numerical results show 2-11 dB reduction in scattering strength relative to the uncloaked configuration for 0.3eV variation of graphene chemical potential. We believe that the proposed model can open up novel avenues for practical applications of high resolution cloaking sensors.

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Tunable mantle cloaking utilizing graphene metasurface for terahertz sensing applications

Abstract

1. Introduction

2. Designing an anisotropic metasurface based on graphene strips

3. Results and discussions

4. Conclusion

References

Cited By

Figures (10)

Equations (14)

Optics Express