Manipulating surface plasmon polaritons with infinitely anisotropic metamaterials

Zheping Shao; Yihao Yang; Zuojia Wang; Muhiddeen Yahaya; Bin Zheng; Shahram Dehdashti; Huaping Wang; Hongsheng Chen

doi:10.1364/OE.25.010515

1. Introductions

Surface plasmon polaritons (SPPs) are confined surface electromagnetic (EM) waves that propagate along the interface between metals and dielectrics, which arise from the interaction between free electrons and light [1–5]. The ability of SPPs to confine light at a deep-subwavelength scale inspires many applications in photonic devices, such as broadband light concentration [6], bio-sensing [7], photonic circuits [8, 9], photovoltaics [10] and lasers [11]. The concept of SPPs has also been extended to low frequencies, such as far-infrared, terahertz and microwave regimes, using structured metal surfaces to mimic the noble metals at optical frequencies, namely, the so-called designer SPPs [12–15]. The SPPs and designer SPPs remain non-radiating on a metal surface as long as the surface is translationally invariant in the direction of propagation [16]. However, when the translation invariance is broken, such as a bump on the surface, a taper structure, or a sharp corner, the SPPs will radiate into the background or reflect backward, suffering from severe losses.

To suppress both reflection and radiation losses of the surface states, topological photonics, an analog of topological electronics, have been introduced in the optical system [17–21]. The topological protected photonic surface states, in analogy with the electronic chiral edge states in quantum Hall systems and topological insulators, are robust against the disorders and sharp corners [22, 23]. However, in the optical systems, such protected surface states so far are found in geometrically complex structures or magnetic-biased structures, and hence, suffer from narrow operational bandwidths [20, 24]. Another way to guide the surface waves through the non-uniform metal surfaces is based on the transformation optics (TO) [25, 26]. The magic power of TO is the space transformation between varied objects, for example, a point to a spherical surface (spherical cloaks) [25], a line to a surface (cylindrical cloaks) [27], a surface to another surface (carpet cloaks) [28–30]. A curved space can also be transform to a straight one, without disturbing the translation invariance of the curved physical space. The so-called surface wave cloaks are based on this theory [16, 31–33]. Due to the nature of TO, both amplitude and phase are well preserved in the surface wave cloaks, however, when the shapes of cloaks are irregular, the constitutive parameters are usually too complicated to implement.

In this paper, we report an alternative method to guide the surface waves through slopes, bumps and sharp corners with arbitrary shapes, based on infinitely anisotropic metamaterials (IAMs). In our method, the SPPs are coupled into and out of the bulk modes in the IAMs with high efficiencies. These bulk modes are efficiently routed in the IAMs by altering their principle axis, therefore successfully guided through the slopes, bumps, or sharp corners with almost total transmissions. These high transmissions can be explained from both microscopic and macroscopic perspectives. Several functional SPP devices, including adapter, cloak, and sharp bending waveguide, are presented in the simulations. Two proof-of-concept devices are experimentally demonstrated, where the SPPs are mimicked by the designer SPPs at microwave frequency.

2. Theories

In the following, we will apply the IAMs to guide the SPPs through non-uniform metal surfaces. We focus our studies on manipulating the energy flows in the dielectric layer. The ratio of energy in the dielectric to that in the metal is $η = Re {ε_{m}^{2} / ε_{d}^{2}}$ , where $ε_{m}$ and $ε_{d}$ correspond to the permittivity of the metal and the dielectric, respectively. In most cases, $| ε_{m} | ≫ | ε_{d} |$ , therefore, most of SPPs are travelling in the dielectric [16].

The IAMs can be implemented in practice by stacking multilayers of plasmonic and dielectrics materials at optical frequencies [34], all-dielectric m or stacked PEC-dielectric layers at low frequencies, as shown in Fig. 1. According to Ref [35], the effective constitutive parameters of the PEC-dielectric metamaterial are given by

ε_{⊥} = ε_{d} p / w, ε_{/ /} = - \infty, u_{z} = w / p,

where

ε_{⊥}

(x direction in Fig. 1(a)) and

ε_{/ /}

(y direction in Fig. 1(a)) are the transverse and longitudinal permittivities of the IAM, respectively,

ε_{d}

is the relative permittivity of the dielectric, p is the period, and w is the width of the dielectric. As an example, we simulate a TM polarized Gaussian beam at 0.6 THz incident from air onto a homogenous IAM with

ε_{⊥} = 2

,

ε_{/ /} = - \infty

,

u_{z} = 0.5

(Fig. 1(c)) and an effective IAM (Fig. 1(d)) at 60°, where

w = 25 ​ ​ ​ μ m

, and

p = 50 μ m

. It is obvious that both of simulated results match each other very well.

Fig. 1 (a) IAM structure. The IAM is composed of PEC (gray)-dielectric/air (green) layers, where w is the thickness of the dielectric, and p is the period. (b) Isofrequency contours of the background medium ( $ε_{b} = 1$ ) and the IAM. (c) Hz field distributions when a Gaussian beam incident from air onto a homogenous IAM with incident angle of 60°. (d) Hz field distributions when a Gaussian beam incident from air onto an effective IAM with incident angle of 60°.

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One of the most important properties of the IAMs is that the EM waves propagate in them with a diffraction-free manner [34]. This can be explained by the dispersion relation in the IAMs, i.e., $k_{/ /}^{2} / ε_{⊥}^{2} = k_{0}^{2}$ , which means that the EM waves with different $k_{⊥}$ will propagate with the same $k_{/ /}$ , preventing the beam diffraction. Here, $k_{⊥}$ and $k_{/ /}$ are the transverse (x direction in Fig. 1(b)) and longitudinal (y direction in Fig. 1(b)) wave vectors, respectively. This property is very useful for designing surface wave devices. Considering lossless materials, when surface waves travel from one medium to another, two main losses may be induced. One is the reflection loss which results from the impedance mismatch, while the other is the radiation loss. Usually, when the surface waves cross the interface between two mediums, they will diffract, where one part is still coupled to the surface waves, and the other part is coupled to the scattering mode, leading to the radiation into the background. This is a fundamental problem for surface wave devices [36]. However, when the surface waves propagate from a medium (dielectric or IAM) to the IAMs, or vice versa, because the IAMs are diffraction-free, no radiation loss will be caused. Therefore, only the reflection loss involves, which simplifies the design of surface wave devices. We should notice that all-dielectric structures can also achieve non-diffracting properties, therefore, they may be used to design surface wave devices with low losses [37].

When designing SPP devices with the IAMs, two types of interfaces should be considered: IAM-isotropic dielectric medium (IDM) interface and IAM-IAM interface, as shown in Fig. 2. According to the electromagnetic wave theory, we can obtain the transmissivity and reflectivity [33].

T_{1} = \frac{4 \sqrt{μ_{0} ε_{b} μ_{z} ε_{⊥}} \cos θ_{i} \cos θ_{1}}{{(\sqrt{ε_{b} μ_{z}} \cos θ_{1} + \sqrt{ε_{⊥} μ_{0}} \cos θ_{i})}^{2}}, R_{1} = \frac{{(\sqrt{ε_{b} μ_{z}} \cos θ_{1} - \sqrt{ε_{⊥} μ_{0}} \cos θ_{i})}^{2}}{{(\sqrt{ε_{b} μ_{z}} \cos θ_{1} + \sqrt{ε_{⊥} μ_{0}} \cos θ_{i})}^{2}}

for the IAM-IDM interface, where

μ_{z}

,

ε_{⊥}

and

θ_{1}

are the relative permeability, the relative transverse permittivity, and the rotational angle of the principal axis of the IAM, respectively.

ε_{b}

is the relative permittivity of the dielectric background, and

θ_{i}

is the incident angle;

T_{2} = \frac{4 \sqrt{μ_{z 1} ε_{⊥ 1} μ_{z 2} ε_{⊥ 2}} \cos θ_{1} \cos θ_{2}}{{(\sqrt{ε_{⊥ 1} μ_{z 2}} \cos θ_{2} + \sqrt{ε_{⊥ 2} μ_{z 1}} \cos θ_{1})}^{2}}, R_{2} = \frac{{(\sqrt{ε_{⊥ 1} μ_{z 2}} \cos θ_{2} - \sqrt{ε_{⊥ 2} μ_{z 1}} \cos θ_{1})}^{2}}{{(\sqrt{ε_{⊥ 1} μ_{z 2}} \cos θ_{2} + \sqrt{ε_{⊥ 2} μ_{z 1}} \cos θ_{1})}^{2}}

for the IAM-IAM interface, where

μ_{z 1}

(

μ_{z 2}

),

ε_{⊥ 1}

(

ε_{⊥ 2}

) and

θ_{1}

(

θ_{2}

) are the relative permeability, the relative transverse permittivity, and the rotational angle of the optical axis of the IAM1 (IAM2), respectively [34]. It is interesting to see that, for the IAM-IAM interface, the transmissivity and reflectivity are independent on the incident angle, which is very different from regular Fresnel formulae. Considering the impedance-matched cases, i.e., zero reflection, we can easily get the total transmissivity condition

\sqrt{μ_{0} / ε_{b}} \cos θ_{i} = \sqrt{μ_{z} / ε_{⊥}} \cos θ_{1},

for the IAM-IDM interface, and

\sqrt{μ_{z 1} / ε_{⊥ 1}} \cos θ_{1} = \sqrt{μ_{z 2} / ε_{⊥ 2}} \cos θ_{2},

for the IAM-IAM interface.

Fig. 2 (a) Interface between an IAM and an IDM. The inset is the corresponding metamaterial structure. Here, $μ_{z}$ , $ε_{⊥}$ , and $θ_{1}$ are the relative permeability, the relative transverse permittivity, and the rotational angle of the optical axis of the IAM, respectively. $ε_{b}$ is the relative permittivity of the dielectric background, and $θ_{i}$ is the incident angle. (b) Interface between an IAM (IAM1) and another IAM (IAM 2). The inset is the corresponding metamaterial structure. Here, $μ_{z 1}$ ( $μ_{z 2}$ ), $ε_{⊥ 1}$ ( $ε_{⊥ 2}$ ), and $θ_{1}$ ( $θ_{2}$ ) are the relative permeability, the relative transverse permittivity, and the rotational angle of the optical axis of the IAM1 (IAM2), respectively.

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For the IAM-IDM interface, by setting $ε_{b} = 3$ , $ε_{⊥} = 2$ , $μ_{z} = 0.5$ , and $θ_{1} = 0 °$ , we can get the total transmissivity incidence angle of $θ_{i} = 60 °$ , which has been demonstrated in Figs. 1(c) and 1(d).

Considering the IAM-IAM interface, both IAMs can be realized with the PEC-dielectric layered structure in practice. Substituting Eq. (1) to Eq. (5), we can get:

\sqrt{(w_{1} / p_{1}) / (ε_{1} p_{1} / w_{1})} \cos θ_{1} = \sqrt{(w_{2} / p_{2}) / (ε_{2} p_{2} / w_{2})} \cos θ_{2} .

Suppose that,

p_{1} / \cos θ_{1} = p_{2} / \cos θ_{2}

and

ε_{1} = ε_{2}

, it’s easy to get that

w_{1} = w_{2} .

Here, Eq. (7) means that the projected periods of both IAMs along the interface are the same. Therefore, the PEC and dielectric layers are continuous across the interface. Equation (8) is very interesting, which manifests that as long as the widths of the dielectric layer are the same, the transmission always stay unity. The underlying physical mechanism is very simple. The PEC-dielectric IAMs consist of the arrays of deep-subwavelength waveguides. A fascinating property of these waveguides is their ability to guide the EM waves through arbitrary sharp corners with total transmissions [38]. Therefore, adopting the same width is an efficient approach towards impedance match for the system.

3. Simulations

Based on the above theories, we design several useful SPP devices. Simulations performed in the COMSOL Multiphyiscs are used to validate our proposal. In the simulations, we choose InSb as the substrate metal for supporting SPPs, which is a semiconductor with a Drude-type dielectric function of conductivity at terahertz frequency $ε_{i n s b} (ω) = ε_{\infty} - ω_{p}^{2} / (ω^{2} + i ω γ)$ , where $ε_{\infty} = 15.6$ , $ω_{p} = 46 \times 10^{12}$ rad/s, and $γ = 0.3 \times 10^{12}$ rad/s [39]. The IAM is composed of gold and dielectric (air) layers, where the permittivity of the gold is defined by a Drude model with $ε_{\infty} = 9.1$ , $ω_{p} = 1.2 \times 10^{16}$ rad/s, and $γ = 1.2 \times 10^{14}$ rad/s. The first device is SPP adapter, which is used to couple the SPPs on metal slabs with different widths (Fig. 3(a)). In our example, the permittivity of the background dielectric is $ε_{b} = 2$ and the tilt angle of the taper is $θ = 45 °$ . According to Eq. (4), we get the total transmission condition $\sqrt{μ_{0} / ε_{b}} \cos θ = \sqrt{μ_{z} / ε_{⊥}} \cos θ_{1}$ . Without loss of generality, we choose $θ_{1} = 0 °$ , $μ_{z} = 0.5$ , and $ε_{⊥} = 2$ in the simulation. It is obvious that without the IAM taper, most of the SPPs radiate into the background due to sudden change of the metal surface (Fig. 3(c)). But with the IAM taper, the SPPs are successfully guided through the non-uniform metal films with preserved mode and considerably high efficiency (Fig. 3(a)). In the effective medium, the IAM is composed of gold and air, where $p = 50 μ m$ , and $w = 25 μ m$ . From the simulation results, one can see that the gold-air layers behave as the IAMs, and the SPPs are successfully coupled into the IAM and out of it with a preserved mode and amplitude. The layered structure works as a sampler or an imaging device, which samples the phase and amplitude of one boundary, then without loss, transfers this information to another boundary. The resolution of the homogenous IAM is infinitely high, while that of the effective IAM is still at a deep-subwavelength scale, namely $p$ . With this resolution, it serves as a superlens [40, 41] with a high resolution, and can image SPPs at a deep-subwavelength scale. Another important consideration is that, the adapter based on the gold-air layered IAMs shows a broadband property (Fig. 3(d)).

Fig. 3 (a) Hz field distributions with the homogenous IAMs at 0.57 THz. Here, $ε_{b} = 2$ , $μ_{z} = 0.5$ , $ε_{⊥} = 2$ , and the tilt angle of the slope is $45 °$ . (b) Hz field distributions with the effective IAMs at 0.57 THz. The IAMs are composed of gold and air, where $p = 50 μ m$ , and $w = 25 μ m$ . (c) Hz field distributions without the IAMs at 0.57 THz. (d) Normalized transmission with the practical IAMs (red dots), with the homogenous IAMs (black line), and without the IAMs (blue line), respectively.

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We also apply the IAMs to design an arbitrary-shaped SPP cloak. Without loss of generality, we design a sin-shaped cloak, as shown in (Fig. 4(a)). The parameters of this cloak seem very complicated, because both $ε_{⊥}$ and $μ_{z}$ should vary with the slope of the cloak. However, in practice, according to Eq. (8), we keep the widths of the subwavelength waveguides unchanged and the effective impedance $\sqrt{μ_{z} / ε_{⊥}} \cos θ_{1}$ stays the same. Therefore, the cloak shows a perfect performance. In the design, we further relax the restrictions and just keep the impedance the same, i.e., w/p is a constant. In doing so, the cloak still works with a nearly perfect performance. On the one hand, the transmissivity is still over 89%, provided that the slop is less than 60°. On the other hand, the effective impedance $\sqrt{μ_{z} / ε_{⊥}} \cos θ_{1}$ slowly vary with the slope of the cloak, which reduces the reflection further. We know that a SPP cloak should keep both amplitude and phase of the SPPs the same with that of the SPP propagating on a flat metal surface. In the design, we can properly choose the refraction index of the cloak, i.e., $\sqrt{ε_{d}}$ , to match the phase condition and carefully alter w/p to match the impedance condition. In the simulation, we set $ε_{b} = 2.82$ , $μ_{z} = 0.6$ , and $ε_{⊥} = 1.667$ . In Figs. 4(a) and 4(b), it is clear that with the IAM cloak, the SPPs are smoothly guided through the bump with a preserved amplitude and phase, while without the cloak, most of the SPPs are reflected and scattered. In the effective IAM, where $ε_{d} = 1$ , $p = 50 μ m$ , and $w = 30 μ m$ , the cloaking performance is still quite good (Fig. 4(c)) in a broad bandwidth (Fig. 4(d)).

Fig. 4 (a) z-oriented magnetic field distributions with the homogenous IAM cloak at 0.57 THz. Here, $ε_{b} = 2.82$ , $μ_{z} = 0.6$ , $ε_{⊥} = 1.67$ , and the bump is in a shape of sin function. (b) z-oriented magnetic field distributions with the effective IAM cloak at 0.57 THz. The IAM are composed of gold and air, where $p = 50 μ m$ , and $w = 30 μ m$ . (c) z-oriented magnetic field distributions without the IAM cloak at 0.57 THz. (d) Normalized transmission with the practical IAMs (red dots), with the homogenous IAMs (black line), and without the IAMs (blue line), respectively.

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The last example is a sharp SPP bending waveguide. The SPPs are first coupled into the bulk modes in the IAMs, which propagate through the optical axis of the IAMs and couple back to the SPPs, with all of coupling efficiencies nearly 100%. In the simulation, we set $ε_{b} = 4$ , and the angle of the sharp corner is 60°. The interfaces between two IAMs are along the angle bisector, therefore, according to Eq. (6), the constitutive parameters in both of IAMs are the same, i.e., $μ_{z} = 0.5$ , and $ε_{⊥} = 2$ . From Figs. 5(a)-5(d), we can clearly observe that the IAM sharp bending waveguide also shows an almost perfect performance in a broad bandwidth.

Fig. 5 (a) Hz field distributions with the homogenous IAMs at 0.57 THz. Here, $ε_{b} = 4$ , $μ_{z} = 0.5$ , $ε_{⊥} = 2$ , and the interfaces between two IAMs are along the angle bisector. (b) Hz field distributions with the effective IAMs at 0.57 THz. The IAMs are composed of gold and air, where p = 50 um, and w = 25 um. (c) Hz field distributions without the IAMs at 0.57 THz. (d) Normalized transmission with the practical IAMs (red dots), with the homogenous IAMs (black line), and without the IAMs (blue line), respectively.

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As a practical IAM at optical regime may involves plasmonic materials, intrinsic absorption can’t be avoided, which will limit the effectiveness of the surface wave devices. To investigate the role of loss in the IAM based surface wave devices, we introduce tangent loss of $δ = i m (ε_{⊥}) / r e (ε_{⊥})$ into the transverse permittivity of the IAM. We take the SPP adapter as an example. We set $ε_{⊥} = 2 + i * 2 δ$ , and the other parameters stay the same as that in Fig. 3. As expected, when increasing $δ$ from 0 to 0.2, the amplitude of the SPPs reduces dramatically. However, the presence of the lossy term in the constitutive parameters of IAM won’t introduce a noticeable impedance mismatch, and the modes preserve well, as shown in Fig. 6.

Fig. 6 The SPP energy transmitted through the SPP adapter with different losses in the IAM at 0.57 THz. The inset is the Hz field distribution near the SPP adapter when $δ = 0.1$ .

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4.Experiments

In the experiments, we mimic the SPPs with the designer-SPPs, which are supported by corrugated metal surfaces (CMSs) as shown in Fig. 7(a). We first measure the transmission of an irregular-shaped IAM cloak. Its outline consists of straight lines and arc curves. The IAM is composed of iron sheets with a thickness of 0.05 mm and air layers (polymer foam cylinders, 1.06 mm Rohacell 71HF with a relative permittivity of 1.1 and $tg δ < 0.0016$ at 10.0 GHz, placed between neighboring iron sheets), as shown in Figs. 8(a) and 8(b). The period of the iron sheets along the y direction is 6 mm. The effective parameters for the iron sheet arrays are $ε_{⊥} = 1$ , $ε_{/ /} = \infty$ and $μ = 1$ at their local coordinate systems. To measure the transmission of the designer-SPPs, an electric dipole antenna is placed at one side of the CMS to excite the designer SPPs and another one is placed at the other side of the CMS as a receiver. Both antennas connect with the vector network analyzer (VNA) to get the transmission spectrum. The simulated and measured transmissions with and without the IAM adapter are shown in Fig. 8(e). It is very clear that with the IAM adapter, the transmission improves by almost 10.0 dB from 5.0 GHz to 13.0 GHz. Though the effective impedance $\sqrt{μ_{z} / ε_{⊥}} \cos θ_{1}$ is not constant in the experiment, the cloaking performance is still nearly perfect. On the one hand, the transmissivity is over 97%, provided that the slope varies within small ranges of $θ$ between 0 $°$ - 45 $°$ . On the other hand, the effective impedance slowly varies with the slope of the cloak, which reduces the reflection further. Note that in the cloak, we abandon the phase preserved condition. This is because, we only focus on the transmission and mode of the surface wave. In our cloak, both transmission and surface wave mode are well preserved (Figs. 8(c) and 8(d)).

Fig. 7 (a) Schematic diagram of the metallic grooved structure, where d = 2 mm, w = 4 mm, and p = 8 mm. (b) Dispersion relation of designer SPPs on the grooved structure.

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Fig. 8 (a)Fabricated sample. (b) Experimental setup of the designer SPP cloak based on the IAM. Two electric dipole antennas are place at each side of the CMS, working as a receiver and a transmitter, respectively. These two antennas are connected to the VNA to get the transmission spectrum. Inset is the front view of the cloak and the geometry parameters of the iron sheets (black line). The thickness of the iron sheet is 0.05 mm. All of the dimensions are in millimeters. (c) Simulated magnetic energy distributions with the IAM cloak at 9.0 GHz. (d) Simulated magnetic energy distributions without the IAM cloak at 9.0 GHz. (e) Simulated and measured transmissions with and without the IAM cloak, respectively. (f) Simulated transmissivity of different optical axis angle of the IAM at normal incidence.

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We also experimentally demonstrate a designer-SPP adapter to couple the designer SPPs on CMS slabs with different thicknesses. Due to mirror symmetrical nature of the slabs, we only fabricate half of them in the experiment (Figs. 9(a) and 9(b)). The tilt angle of the slope is 45°, the thickness of the iron sheet is 0.05 mm, and the period is 6 mm. The effective parameters for the iron sheet arrays at their own local coordinate systems are extracted as, $ε_{⊥} = 1$ , $ε_{/ /} = \infty$ and $μ = 1$ . When the coupling efficiency at each interface exceeds 97%, the performance of the adapter is very efficient. The experimental setup is the same as that of the designer SPP cloak (Fig. 9(b)). Both simulated and measured results match very well and the transmission with the adapter is improved by more than 10.0 dB from 5.0 GHz to 13.0 GHz, compared with its counterpart without the adapter (Fig. 9(e)). Magnetic energy distributions at 9.0 GHz confirm that most of energies on the thin CMS slab are successfully coupled to the thick CMS slab with a high efficiency (Figs. 9(c) and 9(d)).

Fig. 9 (a) Fabricated sample. (b) Experimental setup of the designer SPP adapter based on the IAM. Inset is the front view of the adapter and the geometry parameters of the iron sheets (black line). The thickness of the iron sheet is 0.05 mm. All of the dimensions are in millimeters. (c) Simulated magnetic energy distributions with the IAM adapter at 9.0 GHz. (d) Simulated magnetic energy distributions without the IAM adapter at 9.0 GHz. (e) Simulated and measured transmissions with and without the IAM adapter, respectively.

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5. Conclusions

As a summary, we propose and experimentally demonstrate a method to guide surface waves through slopes, bumps and sharp corners with arbitrary shapes, on the basis of the IAMs. In this method, the SPPs are coupled into and out of the bulk modes in the IAMs with high efficiencies. These bulk modes are perfectly routed in the IAMs by altering their optical axis, therefore, successfully guided through these arbitrary shapes with almost total transmissions over broad bandwidth. These high transmission efficiencies are guaranteed by both microscopic and macroscopic theories. Several functional SPP devices, including adapter, cloak, and sharp bending waveguide, are presented in the simulations. Two proof-of-concept experiments are demonstrated, where the SPPs are mimicked with the designer SPPs at microwave frequency. Our method can find broad applications in manipulating surface waves and photonic circuits.

Funding

National Natural Science Foundation of China (NSFC) (61625502, 61574127 and 61601408); the ZJNSF (LY17F010008); the Postdoctoral Science Foundation of China (2015M581930); the Top-Notch Young Talents Program of China; the Fundamental Research Funds for the Central Universities; the Innovation Joint Research Center for Cyber-Physical-Society System.

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Manipulating surface plasmon polaritons with infinitely anisotropic metamaterials

Abstract

1. Introductions

2. Theories

3. Simulations

4.Experiments

5. Conclusions

Funding

References

Cited By

Figures (9)

Equations (8)

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