1. Introduction

Surface waves are typified by fields that decay exponentially away from a dielectric surface, with most of its energy contained in or near the dielectrics. They are known to exist in a variety of geometries that involve dielectric interfaces. Given an ideal lossless isotropic homogeneous grounded slab, as shown in Fig. 1(a) , one can determine the TE and TM surface wave modes by [1,2]:

 

Fig. 1 Surface wave propagation on an ideal lossless isotropic homogeneous grounded slab. (a) Configuration of the surface wave guiding structure. The surface wave is propagating along the air-dielectric interface which is depicted by the orange line. In this paper, we consider ${\mu }_{r1}=1$ and ${\mu }_{r2}=1$for all cases studied. (b) Demonstration of the TM₀ mode propagating along the air-dielectric interface of the surface wave guiding structure, at 5 GHz with $d=5\text{mm}$,${\epsilon }_{r1}=3$, and ${\epsilon }_{r2}=1$. (c) Graphical representation of the relationship between the effective refractive index $n_{eff}$of TM₀ mode and substrate material property ${\epsilon }_{r1}$ at 5 GHz with $d=5\text{mm}$ and ${\epsilon }_{r2}=1$. (d) Graphical representation of the relationship between the effective refractive index $n_{eff}$of the TM₀ mode and the relative permittivity of the superstrate ${\epsilon }_{r2}$ at 5 GHz with $d=5\text{mm}$ and ${\epsilon }_{r1}=3$. (e) Graphical representation of the relationship between the effective refractive index $n_{eff}$of the TM₀ mode and the substrate thickness d at 5 GHz with ${\epsilon }_{r1}=3$ and ${\epsilon }_{r2}=1$.

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The wave-number$k_{\rho }$of a surface wave can be interpreted as the product between the free space propagation constant $k_0$ with an equivalent refractive index $n_{eff}.$ As a result, we have

 

Fig. 2 The schematic showing how potentially ${\epsilon }_{r1}$,${\epsilon }_{r2}$ and d can be used to steer the surface wave ray trace. The changing of these three parameters actually leads to a change in the corresponding effective refractive index. The surface wave is assumed to propagate along the interface of ABC, which is depicted in orange. (a) Use ${\epsilon }_{r1}$to change ray trace. (b) Use ${\epsilon }_{r2}$to change ray trace. (c) Used d to change ray trace. (d) Graphical representation of refraction relationship between the surface waves at the boundary B from region AB to region BC with the effective refractive index $n_{effi}$and $n_{effj}$respectively.

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We have demonstrated three methods of controlling the propagation of a surface wave, each based on modification of the effective refractive index. However, a general recipe is still required, in order to offer the design of the permittivity values $\left({\epsilon }_{r1}\text{or}{\epsilon }_{r2}\right)$ and the substrate thickness $\left(d\right),$ such that we are able to manipulate the surface wave in a manner prescribed to our will. Transformation optics provides a practical strategy to control wave propagation [6–13], which connects the geometrical optics and the gradient refractions of electromagnetic materials by [8]

However, in the microwave regime where surface wave modes have longer wavelengths, the dielectric substrate will play an essential role in the surface wave propagation, such that the simplified recipe may not universally function well. In this paper, we investigate two scenarios of the surface wave control at microwaves: One is the surface wave traveling on an uneven interface with a triangular bump along a grounded dielectric slab, and the other is the surface wave circumventing a metallic rhombus-shaped obstacle that is partially buried in a flat dielectric slab. With consideration of the eigenmode properties of the surface wave, we provide a general recipe based on transformation optics to accurately guide the wave propagation. On the one hand, we see that the surface wave is guided nicely on the uneven interface with no scattering into the air as the grounded dielectric slab is flat. On the other hand, we observe that the surface wave is capable of traversing the rhombus obstacle with no shadow left behind, as the rhombus obstacle is cloaked. This technique for the surface wave control should also be valid at the higher frequency ranges, and can easily be extended to encompass other propagating modes.

2. One-dimensional surface wave control

We shall begin with a scenario in which a surface wave travels along an uneven interface ABCDEFGHIJ, as shown in Fig. 3(a) . Similar cases were studied in Refs. [14] and [15]. The substrate chosen here has the permittivity${\epsilon }_{r1}=3$ and the thickness $d=5\text{mm}$ with a triangular perturbation along the grounded dielectric slab. The dielectric perturbation has a height of 16 mm and a length of 144 mm, and is made of the same material with ${\epsilon }_{r1}=3.$Therefore, the thickness of the present structure thus ranges from 5 mm to 21 mm. When the surface wave propagates and impinges upon this perturbation, a large part of the wave will travel into the air, as shown in Fig. 3(b).

 

Fig. 3 Surface wave propagating along an uneven interface ABCDEFGHIJ with a triangular bump on a grounded dielectric slab. The dielectric bump has a dimension of $16\text{mm}\times 144\text{mm}$in the$\widehat{z}$and $\widehat{y}$directions. The substrate is assumed to be infinite in the$\widehat{x}$direction. In this case, the grounded dielectric slab has a thickness ranging from 5 mm to 21 mm, and material with ${\epsilon }_{r1}=3$. (a) Configuration of the considered structure. (b) Surface wave traveling on the uneven air-dielectric interface showing significant energy scattering into the air. (c) Configuration of the considered structure with a carpet cloak device. The cloaking device is the same one as used in Ref. [12], with dimensions of$60\text{mm}\times 137\text{mm}$in the $\widehat{z}$and $\widehat{y}$directions, and make up of 8 dielectric blocks. The unit-cell size is thus$30\text{mm}\times 37.25\text{mm}$. The materials employed in the cloak have ${\epsilon }_{ri}^{\text{'}}=\left[1.17,1.30,1.02,1.46\right]$. (d) Surface wave traveling on the uneven interface with carpet cloak device covering the dielectric bump. However, significant energy is still scattered into the air.

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A transformation based carpet cloaking device can be designed for the surface wave to virtually flatten this dielectric bump, as shown in Fig. 3(c). Here we use the carpet cloak device for the same sized bump in Ref. [12]. Such a technique has proven to be a valid approach for cloaking the bump from surface waves in optical regime, as demonstrated in Refs. [14] and [15]. One would expect this direct recipe from transformation optics to also be valid for our case in the microwave regime. Unfortunately, simply using the values previously prescribed, with ${\epsilon }_{ri}^{\text{'}}=\left[1.17,1.30,1.02,1.46\right]$ as the cloaking cover, gives unsatisfactory results as shown in Fig. 3(d). There is still significant energy scattering into the air.

The simple physics behind this is because the increasing thickness $d\subseteq [5\text{mm},21\text{mm}]$ of the bump will lead to a rising effective refractive index $n_{eff}\subseteq [1.07,1.61],$ as shown in Fig. 1(e), which, in turn, leads to an increase in the wave impedance of the surface wave that propagates along the air-dielectric interface. In the air region, the wave impedance of the TM₀ mode can be expressed as [1,2]

However, the majority of the energy of a surface wave is present in or near the dielectrics and propagating along the interface. In other words, a surface wave can be approximated as a kind of two-dimensional wave in which its propagation is determined by $k_{\rho }.$ The only discontinuity of encountered by the surface wave traveling on the interface ABCDEFGHIJ is thus the varied effective refractive index $n_{eff},$ rather than the distortion of changing altitude in the $\widehat{z}$direction. For the flat grounded dielectric slab with ${\epsilon }_{r2}=1,$${\epsilon }_{r1}=3,$ $d=5\text{mm,}$ the TM₀ mode has $n_{eff}=\mathrm{1.07.}$ The solution to cloak this triangular bump is thus significantly simplified. One only needs to keep the $n_{eff}$of the surface wave propagating on the uneven interface unchanged, and equal to 1.07, such that the wave impedance of the TM₀ mode can be also maintained at a constant.

Through dividing the triangular bump into four parts with an average thickness of $9\text{mm}$ and $17\text{mm}$for the two different blocks, as shown in Fig. 4 , we can retrieve the permittivity of these blocks as ${\epsilon }_{ri\_sub}^{sw}=\left[1.74,1.38\right].$ For example, with a substrate structure of ${\epsilon }_{r2}=1,$$d=9\text{mm,}$ and requiring the guided TM₀ mode to have $n_{eff}=1.07,$from the eigenmode Eqs. (1) and (2) we calculate that the substrate material should have${\epsilon }_{ri\_sub}^{sw}=\mathrm{1.74.}$A clearer view can be viewed in Fig. 1(e) from the graphical representation of the relationship between the $n_{eff}$of the TM₀ mode and substrate thickness $d.$

 

Fig. 4 Surface wave propagating on the uneven interface ABCDEFGHIJ with a self-cloaking scheme by altering the dielectric property of the bump. (a) Configuration of the considered structure with the proposed surface wave cloaking design, where the bump has been divided into four homogeneous blocks. The average thickness of the blocks are $d_i=\left[9\text{mm},17\text{mm}\right]$. (b) Magnified picture of the surface wave self-cloaking dielectric bump with two materials of ${\epsilon }_{ri\_sub}^{sw}=\left[1.74,1.38\right]$ according to the corresponding average thickness of the divided blocks. (c) Normalized E-field distribution of the surface wave TM₀ mode propagating at the interface ABCDEFGHIJ with the self-cloaking dielectric bump. This shows that no scattering is present in the air, and the surface waves propagate well along the uneven interface.

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A full-wave finite-element simulation (Ansoft HFSS v12) is then performed to verify the proposed design. When this self-cloaking device is applied to the substrate of the triangular bump, one may clearly observe that no scattering of the surface wave will exist in the air, and the surface waves propagate nicely along the uneven interface. Such a successful implementation virtually flattens the uneven interface for the surface wave through a simple scheme directly based on the effective refractive index, providing an efficient way to control the surface wave. However, it is noted that such a technique does not provide a universal solution for the surface wave control, but is restricted the present case. Therefore, a more universal method is expected to fulfill the surface wave manipulation once a specified functionality is given.

3. Two-dimensional surface wave control

The second scenario we consider is shown in Fig. 5(a) and Fig. 5(b). In this case, surface waves travel along the air-dielectric interface ABCD, where a metallic rhombus obstacle is partially buried in a flat grounded dielectric slab. Such a conductor has the size of $l_1=32\text{mm,}$$l_2=144\text{mm}\text{.}$The substrate has relative permittivity ${\epsilon }_{r1}=3$ with a thickness $d=5\text{mm}\text{.}$ When the surface wave propagates and impinges upon the rhombus conductor, it splits into two parts and leaves a strong shadow behind the conductor on the interface, as shown in Fig. 5(c).

 

Fig. 5 Surface wave propagating on the air-dielectric interface ABCD with a metallic rhombus obstacle partially buried in a flat grounded dielectric slab. The conductor has a size of $l_1=32\text{mm}$, $l_2=144\text{mm}$. The substrate has the relative permittivity ${\epsilon }_{r1}=3$ with a thickness $d=5\text{mm}$. (a) 3D view of the considered structure. (b) Lateral view of the considered structure. (c) Normalized E-field distribution of the surface wave TM₀ mode propagation on the air-dielectric interface ABCD when propagating and impinging upon the rhombus conductor. Clearly, when the wave impinges upon the metallic rhombus obstacle, the surface wave split into two parts and forms a shadow.

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In order to guide the surface wave in a prescribed way, we propose the following three steps:

For the present case where we want the surface wave to circumvent a metallic rhombus obstacle partially buried in a flat grounded dielectric slab, we basically follow the above three steps. To begin with, we directly employ the free space cloak, which extends 137 mm by 60 mm around the rhombus conductor and consists of 16 different sections. Each dielectric block has a different permittivity value ${\epsilon }_{ri}^{\text{'}}=\left[1.17,1.30,1.02,1.46\right]$ with the corresponding refractive index $n_i^{\text{'}}=\left[1.08,1.14,1.01,1.21\right].$ This device was proven to be a successful cloak in Ref. [12] for the same sized metallic rhombus obstacle. In order to make the cloaking device valid for the surface wave TM₀ mode, the refractive index of transformed device should have the value of $n_i^{sw}=n_{eff}{n}_i^{\text{'}}=\left[1.16,1.22,1.08,1.30\right],$ where $n_{eff}=\mathrm{1.07.}$ Finally, we should note that the relationship between the permittivity and the refractive index can never follow the conventional square root expression for the space wave of TEM mode, but must be retrieved from the dispersion curves, as shown in Fig. 1(c) and Fig. 1(d). Therefore, two schemes of dielectrics with the cloaking function can be achieved.

If the surface wave cloaking device is located in the substrate, as shown in Fig. 6(a) , each dielectric block has the permittivity values of ${\epsilon }_{ri\_sub}^{sw}=\left[4.93,5.95,3.27,6.86\right].$ While if the surface wave cloaking device is located in the air superstrate, as shown in Fig. 6(c), then the dielectric blocks will have the permittivity values of ${\epsilon }_{ri\_top}^{sw}=\left[1.18,1.33,1.02,1.53\right].$ One may notice that ${\epsilon }_{ri\_top}^{sw}$ has almost the same permittivity value${\epsilon }_{ri}^{\text{'}}$ of the free space case and can be directly simplified by the free space cloak. However, such an approximation is valid only when, for the TM₀ on a grounded dielectric slab, $n_{eff}$ is very close to 1. If the substrate material has a larger permittivity value or a much thicker substrate, the approximation will no longer function well. In addition, if we replace the air superstrate by another material ${\epsilon }_{r2},$ this approximation will again fail. Even though we simply modify the permittivity map as ${\epsilon }_{ri\_top}^{sw}={\epsilon }_{r1}{\epsilon }_{ri}^{\text{'}},$ this surface cloaking will not remain valid, due to the nonlinear nature of the surface eigenmode equations.

 

Fig. 6 Two schemes of surface wave cloaking device design with $n_i^{sw}=\left[1.16,1.22,1.08,1.30\right]$for the corresponding blocks in the transformed device. The cloaking device has the dimensions of $120\text{mm}\times 137\text{mm}$in $\widehat{x}$and $\widehat{y}$directions and is composed of 16 dielectric blocks. The intact unit size is thus $30\text{mm}\times 37.25\text{mm}$. (a) Cloaking device embedded in the substrate with the material property of ${\epsilon }_{ri\_sub}^{sw}=\left[4.93,5.95,3.27,6.86\right]$. (b) Normalized E-field distribution of the surface wave TM₀ mode propagating on the interface ABCD with the cloaking device embedded in the substrate, showing surface wave successfully circumventing the rhombus conductor, and with no shadow left. (c) Cloaking device embedded in the superstrate layer with material properties of ${\epsilon }_{ri\_top}^{sw}=\left[1.18,1.33,1.02,1.53\right]$. (d) Normalized E-field distribution of the surface wave TM₀ mode propagation on the interface ABCD with the cloaking device in the air superstrate, showing surface wave successfully circumventing the rhombus conductor.

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A full-wave finite-element simulation (Ansoft HFSS v12) is then performed to verify the proposed design. When we embed the cloaking device around the rhombus conductor in the substrate with ${\epsilon }_{ri\_sub}^{sw}=\left[4.93,5.95,3.27,6.86\right],$or alternatively place it in the air with ${\epsilon }_{ri\_top}^{sw}=\left[1.18,1.33,1.02,1.53\right],$ we clearly observe that surface wavefronts are diffracting around and recomposing on the back of the rhombus conductor, as shown in Fig. 6(b) and Fig. 6(d). Thus, no shadow behind the conductor is left. These investigations validate our design methodology and shows that the surface wave cloaking device functions well to cloak the conductor. The other point we would like to emphasize is that our proposed technique can, in principle, be applied to any other class of propagating modes (TE and TM modes in rectangular waveguide; LSE and LSM modes in H guide; etc.) [1,2] as long as one simply follows the above three steps, thus promises a more controllable field manipulation.

4. Conclusion

In conclusion, we have demonstrated two surface wave control scenarios for cloaking applications. In the first case, a triangular bump located upon the dielectric substrate is cloaked from the surface wave. In the second, a metallic rhombus obstacle, partially buried in the substrate, is cloaked. The success of this manipulation may lead to enhanced performance in applications which are fundamentally based on the surface wave propagation. The proposed technique – based on transformation optics and considering surface wave eigenmode properties – has offered an efficient and accurate way to fulfill such a control. This methodology should also be valid at higher frequency ranges, and easily extended to any other propagating modes.