1. Introduction

In recent years, metamaterials, which have unique electromagnetic properties not readily found in nature, have attracted much attention in the science and engineering community all over the world. In a broad sense, the category covers all man-made materials with periodic or pseudo-periodic unit cells, which get its electromagnetic response mainly from its geometrical structure, rather than its chemical components. In this regard, it includes the well known left-handed materials [1], materials with extreme EM parameters or controllable parameters [2, 3, 4, 5, 6], single negative materials [7], and gradient index materials [8, 9], to name a few.

People’s enthusiasm for metamaterial is further stimulated when the first microwave electromagnetic cloaking device is proposed and immediately verified [10, 11]. Since then, transformation optics, together with the underlying metamaterial technology, has become a standard methodology for novel EM devices design. Marvelous structures are proposed, theoretically investigated, numerically calculated and/or even measured in the laboratory [12, 13, 14, 15, 16, 17, 18, 19, 20], and more novel devices will emerge. One problem associated with optical transformation is that complex materials, which are inhomogeneous, anisotropic and magnetic, are always produced, thus hinder the practical realization. Lots of techniques are put forward to address the problem. One straight forward way is to use TE or TM modes instead of the general one, and this is adopted in almost all the works in this area [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Other methods make use of the freedom in choosing the transformation functions, e.g. nonlinear functions [21]. Variational methods have also been proposed for the optimistic function design, which produce gradient index materials [22, 23, 24, 25].

 

Fig. 1. Schematic of the bend structure (a) and the refractive index distribution for the ideal bend structure (b). Here, A=0.214, r ₁=0.1 m and r ₂=0.148 m.

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Like transformation optics, similar methodology exists in the geometrical optics area. By designing the light rays through the devices properly, one eventually gets the material parameters through Eikonal function in that structure [26]. Unlike its optical transformation counterpart, the resulting materials are nonmagnetic, inhomogeneous and isotropic, i.e. they are gradient-index materials. A good example is the recent work on directional cloak design by Mandatori et al [27]. Leonhardt’s work actually belongs to this category [28]. Jacob et al. took further steps by using geometrical optics in anisotropic and nonmagnetic materials, through which, they successfully designed nonmagnetic cloaking devices and hyperlens [29, 30].

Though gradient index materials have been extensively used in the optical fibers and other optical devices, its combination with metamaterial occurs only a few years ago [8, 9]. In this regard, Liu and coworkers’ experiment proves to be the most successful one, in which, various I-shaped unit cells are utilized for the ground plane carpet cloak [23]. By changing the sizes of the unit cell, they actually obtain artificial metamaterials with gradient index, and hence, the needed cloaking device. After that, devices using controllable refractive index emerge, including beam steering devices, planar microwave lenses, etc.

We have proposed the design of arbitrary waveguide bends using the gradient-index materials [31]. In this paper, the theoretical work is further validated through laboratory measurement. By using the broadband and low loss I-shaped unit cells, we successfully realized the refractive index distribution for the structure. Experimental measurement confirm our previous work. Though the performance is not as good as similar structures in optical fibers [32, 33, 34, 35, 36, 37], the work once again shows the freedom people can have with the metamaterial technology and the potential applications it would produce in the future.

 

Fig. 2. HFSS model of the unit cell (a), its geometry (b) and the relationship curve for the retrieved refractive index with a. The working frequency is 10 GHz.

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Fig. 3. (Color online) The layered structure (a) and its equivalent refractive index distribution(b). Green rectangles represent unit cells, they are elongated for clear demonstration and aligned along the radius direction.

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2. Theoretical Analysis

Our design is a two-dimensional (2D) dielectric bend structure with an angle of π/2, as shown in Fig. 1(a). The light rays are supposed to travel in the circumferential directions (not shown in the figure). Then it is obvious that the linear form Eikonal function is S=Aθ+C, in which A and C are two constants [31]. And the refractive index for the bend structure is n=A/r, which is shown in Fig. 1(b). The constant A gives us more control in the design and simulation of bend devices. In the following part, we will use metamaterials with I-shaped unit cells for the practical implementation.

The geometry of the unit cell is a cube with side length equal to 4 mm, and the model is given in Fig. 2(a) together with the excitation setup. The detailed structure is illustrated in Fig. 2(b). The PCB substrate is F4B, with dielectric permittivity 2.65, loss tangent 0.001 and depth of 0.25 mm; I-shaped copper pattern lies on one side of the PCB. The copper depth is 0.018 mm, and the line width is 0.3 mm. Copper strip length a is variable in the design. The commercial FEM code, Ansoft’s High Frequency Structure Simulator (HFSS, v10), is used to characterize the unit cell. It is illuminated using plane waves under the HFSS environment, through which S parameters are obtained. Using the well developed retrieval method from S parameters [38], we get the relationship between the refractive index and the variable a, which is shown in Fig. 2(c). It is clear that the refractive index increases from 1.0 to 2.5 when a varies from 0.8 to 3.6 mm. Polynomial expression for the curve can be easily got by using curve fitting tools.

In our design, A=0.214, r ₁=0.1 m and r ₂=0.148 m. So the refractive index for the bend lies in a region which can be realized using the above unit cell. The corresponding refractive index distribution is given in Fig. 1(b). For practical implementation, the dielectric bend is divided into 12 layers along the radial direction, each layer is considered as a homogeneous material with refractive index equal to that on its center line, and each one is realized using a strip of unit cells mentioned above. In the design, each layer has the unit cells of the same size since they are supposed to have the same refractive index, and cells from different layers are aligned in the radial direction exactly, as is shown in Fig. 3(a). Intuitively, this arrangement allows for smooth bending effect and confirms to the boundary conditions needed in cubic model (It also leads to problems, as mentioned later). The copper lengths of various cells are determined using simple root finding algorithm based on the result shown in Fig. 2(c). The layered structure and its refractive index distribution is demonstrated in Fig. 3.

 

Fig. 4. (Color online) Photograph of the fabricated waveguide bend (a), the field mapping equipment (b) and two typical strips in the bend (c).

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We remark that the method of geometrical optics always gives the medium refractive index ($n=\sqrt{{\epsilon }_r{\mu }_r}$) but not the medium impedance ($z=\sqrt{{\mu }_r⁄{\epsilon }_r}$). For nonmagnetic materials, this is not a problem since µ_r=1 and the dielectric constant ε_r can be uniquely determined. However, for metamaterials, this can be realized by infinite combinations of material parameters. Under ideal situations, impedance matched parameters, i.e. materials with z=1, are preferable because of zero reflection from the device and this is the case in our theoretical work [31]. For our fabricated device, the characteristic impedance is defined as the equivalent impedance of the constitutive cells, which is also obtained through the standard retrieval process [38].

3. Results and Discussion

The fabricated metamaterial bend structure is 12 mm high and shown in Fig. 4(a), the layout of two typical layers is also given in (c). The PCB strips are fabricated in the factory using lithography technology, and the supporting frame is a handmade foam structure with permittivity nearly to 1. The experimental measurement is carried out using the 2D field mapping equipment similar to that in Schurig’s work [11], and the setup is illustrated in Fig. 4(b). Our setup differs from Schurig and coworkers’ in that we have four probes on the top metal plate, which are connected to the VNA port through a 4 to 1 microwave multiplexer, the scanning area is thus enlarged four times. Quasi-plane waves are excited using a small dipole antenna in a vertically halved X-band rectangular waveguide (not shown in the picture), and propagate between a tapered channel, which is formed by microwave absorbing materials, until they touch the bend structure.

 

Fig. 5. (Color online) Field mapping results without the bend in the 2D field mapping equipment. (a) 10 GHz; (b) 11 GHz.

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Figure 5 shows the field mapping results when the bend structure is absent in the equipment. In both figures, most of the fields are confined in the channel region, where the wave fronts are planar surfaces and perpendicular to the propagation direction. However, outside the channel, the wave radiates into the free space, it diverges into a sector and wave fronts look more like cylindrical surfaces. The higher the frequency, the less divergent the wave is, and the better the directivity. This is in line with the antenna theory. According to our design [31], plane waves or Gaussian waves will have better performance, however, the measurement shows such quasi-plane waves produce similar wave bending results, though the performance is deteriorated a little.

Figure 6 shows the simulation results for electric field distribution with different materials, i.e. perfectly matched materials, nonmagnetic materials and materials for the practical implementation. The simulations are carried out by another commercial FEM code, COMSOL Multi-physics v3.3. The snapshot of the field distribution is shown in (a), (c) and (e) respectively. It is obvious that all three cases give similar wave bending effect. The major difference for them lies in the fact that they give rise to different reflections. This can be clearly identified by observing the norm of the electric field, which is given in subplot (b), (d) and (f) respectively. For the perfectly matched case, no reflection is present and the norm of the field is generally uniform along the wave path; for the other two cases, the norm changes as the wave propagates, which is the result of the superposition of the larger incident field and the relatively smaller reflected field.

In Fig. 7, we give the measured electric field distribution for the bend structure. For comparison, the simulation result is also given. Figure 7(a) shows the simulation result for a 10 GHz Gaussian beam passing through the designed 12-layer bend structure with A=0.196, while the result for A=0.214 has already been given in Fig. 6. The readers are kindly referred to Fig. 6(a) for careful examination. Undoubtedly, the beam is smoothly bent along the circumferential direction according to our prediction. The corresponding measured norm of the electric field and the snapshot of the electric field are given in subplot (b) and (c) respectively. Similar bending effect is clearly observed. Further examination of the related plots shows that field patterns inside the bend are not exactly the same for the two cases. For the simulation in Fig. 6(a), about 11 complete periods can be seen, while only 10 are found in measured data in Fig. 7 (c). The discrepancy is mainly attributed to the cell arrangement in the device. Since cells from different layers are aligned in the radial direction, this means each cell covers a sector having the same central angle but has different arc lengths. As a result, the cube model is approximately satisfied for the inner circles, it changes into a cuboid one for the exterior layers. In other words, the cubic ‘atoms’ are diluted by air gaps between them. This will definitely lead to a decreased equivalent refractive index for the bend structure, and hence, longer wavelength. That explains why we have changed the original A=0.214 to 0.196 in the simulation, By doing so, the measured data agree well with the numerical methods, which firmly support the above mentioned arguments.

 

Fig. 6. (Color online) Simulation results for the bend structure with perfectly matched materials (a), (b), nonmagnetic materials (c), (d) and the practical materials used in the device (e), (f). The snapshot of the electric field distribution is given in (a), (c) and (e) respectively and (b), (d), (f) show the norm of the electric field. f=10 GHz, A=0.214 in all cases.

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The measured distribution for the norm of the electric field in Fig. 7(b) shows clearly that there are reflections in the device. They are produced because of the impedance mismatch. In the work, we focus on the profile of the refractive index, and few efforts are made on the impedance of the final device. Very recently, an improvement over the I-shaped unit cells is proposed by Ma et al. [39], which can provide impedance-matched gradient-index metamaterials. This improvement may lead to possible realization of the ideal case in our theoretical work.

 

Fig. 7. (Color online) Electric field distribution of the dielectric bend structure for the simulation (a), and experimental measurement (b), (c), (d), (e) and (f). Here, A=0.196 for (a) and f=10 GHz for (a), (b) and (c). For subplots (d), (e) and (f), f=9 GHz, 8 GHz and 11 GHz respectively.

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Though small differences can still be observed between subplot (a) and (c), the disparity is largely due to the source of excitation. In the simulation, the source is the Gaussian beam; while in the experiment setup, the incident wave is a quasi-planar wave. Leaked waves from the channel region, especially those on the upper side, interfere with those exit from the bend, and change the output pattern severely. Since the foam structure is handmade, its errors can also lead to performance degradation further.

The wave bending effect is valid for 9 GHz and 8 GHz too, as illustrated in (d) and (e), this is understandable due to the wide bandwidth of I-shaped unit cells. The performance deteriorates for 11 GHz, although the bending effect could still be seen in (f). Actually, as the working frequency goes higher, it approaches the resonant region of the unit cell, which produces different equivalent parameters.

Figure 8 shows the spectral responses of the unit cells in the device. In subplot (a), the real part of refractive index n is given. It is clear that n increases with the frequency for all twelve layers due to frequency dispersion. For outer layers (those with smaller n), it changes very slowly, while for the inner layers (larger n), the change is relatively fast. Besides, the rate of change (or the slope for each curve) is larger at higher frequencies than it at lower frequencies. Since the working frequency is set at 10 GHz, the bending performance will surely be deteriorated if frequency deviates. However, the device works on the relative distribution of the refractive index, not the absolute value of n, this means the deterioration can be alleviated because each layer has similar variation of n when the frequency changes. And this explains why we got efficient bending effect for a more than 2 GHz band although the refractive index changes on the same frequency band.

 

Fig. 8. (Color online) Spectral responses of the unit cells used in the device. (a) The real part of the refractive index. (b) The imaginary part of the refractive index. In both plots, larger values correspond to cells in the inner layers.

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Loss is also a very important factor for practical usage. For this reason, the imaginary part of the refractive index is given in subplot (b). It is shown that below 11 GHz, the loss is very small for all layers. When the frequency is above 11 GHz, the imaginary part of n will increase. For most layers, the loss remains at an acceptable level, but for the two innermost layers, it becomes very large and hence unacceptable. The increase is attributed to the fact that the unit cells approach to their resonant regions as frequency increases, as is mentioned earlier. We thus conclude that the device has very small losses for 8 GHz-11 GHz since it works at non-resonant regions.

Finally, we point out that the fabricated bend device only works for TE polarization. For polarization independent applications, 3D isotropic unit cells must be used to address the problem. A possible solution is to use 3 I-shaped unit cells which are perpendicularly connected at their geometrical centers, and treat them as a composite unit cell. Similar methods have been applied for other unit cell structures. However, this deserves further research because of the possible interactions between each constitutive cell and the difficulty of fabrication..

4. Conclusions

In conclusion, a metamaterial based, 90-degree bend structure is designed and measured using the theory of geometrical optics. Low loss, broadband and I-shaped unit cells are utilized in the design. Those gradient index metamaterials demonstrate how we can control the EM responses at will. When equipped with electrically controllable unit cells, more marvelous devices will emerge. The design can be easily extended to other microwave or optical devices like directional cloaks, beam shifters, beam steering devices and lenses etc.