The discrete coordinate transformation (DCT), as a unique technique to control the electromagnetic waves, has been applied for creating all-dielectric devices recently. To extend the applicability of this technique, we propose the concept of multiple discrete coordinate transformation, which serves to deal with more complicated geometries in the transformation domain. As an example, an all-dielectric absorber is created by compressing a pyramidal absorber to a third of its original thickness using the multiple DCT technique. The Finite-Difference Time-Domain (FDTD) method based numerical simulations demonstrate the broadband performance of the transformation absorber from 2 GHz to 20 GHz.
© 2014 Optical Society of America
Indoor measurements can seriously be corrupted by wave scattering from surrounding walls. For this reason, anechoic chambers are in widespread use in order to make accurate antenna measurements from microwave to THz . In order to mimic the free space environment, in anechoic chambers, absorber structures are normally mounted on the walls to attenuate reflections from the chamber confinement. An absorber is usually required to match the impedance of its back surface (e.g., the conducting wall of a chamber) to that of the free space. When propagating inside the absorber, the wave attenuates gradually in the designated medium. As a result, wave reflections from the back wall are efficiently eliminated. The first investigation on practical absorbers was proposed in the 1940s. Resonant absorbers like the λ/4 Salisbury screen  and the “Jaumann Sandwich”  were discussed with the aim of reducing their thicknesses . These resonant absorbers are essentially narrow-band because the operating frequency is partially determined by the dimensions of unit elements. Research on broadband absorbers began in the late 1940s, where gradually-tapered lossy materials were applied. One kind of the broadband absorbers, the urethane pyramidal absorber, has been popularly used in the design of anechoic chambers since the 1960s .
The excellent absorption of the pyramidal absorbers is due to the multiple reflections between neighbouring pyramids. An important property of the pyramidal absorber is, however, to achieve great absorption, the tip-to-base thickness should be no less than one quarter of wavelength. Obviously, it becomes impractical to construct an anechoic chamber at low frequencies as the pyramidal absorber is bulky and heavy. Since 1970s, several kinds of thin absorbers, such as the ferrite tiles  and some metamaterials based absorbers [6–9], have been proposed and successfully applied. However, most of them require magnetic materials with complex permeability to manipulate wave attenuation as well as absorption, but only operate in narrow frequency bands. In this paper, we propose an all-dielectric approach to design a compact broad-band absorber by using a so-called multiple discrete coordinate transformation (DCT) technique.
The coordinate transformation is a unique technique recently developed to control the electromagnetic waves [10, 11] accompanied with its applications from the “cloaks of invisibility”  to other functional devices . However, practically implementing transformation devices is still challenging because the media required in the construction are usually very complex and cannot be found directly in the nature. Instead, metamaterials have been employed for transformation device. They are composed of deep sub-wavelength unit cells and each unit cell is essentially an electric and/or magnetic resonator [14, 15]. With careful design of these resonators, metamaterials can exhibit anisotropic and inhomogeneous distribution of permittivity and permeability that are required by transformation devices. A big challenge is that anisotropic and dispersive metamaterials require meticulous design and sometimes can realise specific permittivity and permeability distribution only in a narrow frequency band. Furthermore, to simultaneously control the permittivity and the permeability, electric and magnetic resonators are both needed in a deep sub-wavelength unit cell, which makes the construction of metamaterials even more complicated, and is likely to further limits the operating bandwidth.
One reliable solution to circumvent this limitation is to apply the discrete coordinate transformation using quasi-conformal mapping . Due to the discretisation of the transformation space, local coordinate systems before and after transformation can be near-orthogonal and isotropic, and therefore, an all-dielectric isotropic solution of a transformation device is achievable.Such devices could be realised using metamaterials composed of only electric resonators. Due to the fact that the structuring is isotropic, and the magnetic resonator is no longer needed in a deep sub-wavelength scale, the all-dielectric isotropic metamaterials are easier to construct and work in wider frequency bands. In addition, another straightforward way to realise the DCT based device is to use an assembly of conventional composites with extremely low loss and nearly constant permittivities over operating frequencies, for example, the polyurethane foam loaded with BiTiO3 . Each block of the composites possesses a specific value of permittivity, and has a size no larger than half the wavelength according to the Nyquist-Shannon sampling theorem . These composites are available in lab or from commercial providers, and special fabricating processes may be demanded and therefore the manufacture process can be sometimes expensive and time consuming.
The DCT technique has been applied to realise several new functional devices, e.g. the carpet cloak [17, 19–21], the planar reflectors [22,23], and the flattened dielectric lenses [24,25]. However, there are restrictions of applying the DCT technique, that, if the transformation space has a boundary with points of singularity, generating near-orthogonal local coordinate systems becomes impossible. As a result, magnetic metamaterials are inevitably needed. Singularity on the boundary has been investigated recently for broadband light harvesting using non-orthogonal conformal mapping . Furthermore, even all boundaries of the transformation space are finite, they should not include geometries with sharp curvature, otherwise, it will degrade the accuracy of the all-dielectric solution. For these reasons, the DCT technique has remained invalid to a variety of applications. To extend the applicability of this technique, in the following text, a multiple discrete coordinate transformation is introduced and analysed.
This paper is organized as follows. First, we will explain from the theory that the method of multiple discrete coordinate transformation can efficiently reduce the inaccuracy which happens in the one-step DCT. Second, an all-dielectric absorber will be created by compressing a pyramidal absorber. Applying the multiple DCT technique, the thickness of the pyramidal absorber can be reduced by 2/3. The compressed absorber requires only lossy dielectrics, hence is easy to fabricate. The Finite-Difference Time-Domain (FDTD) method based numerical simulations will demonstrate the performance of the compressed absorber at the end of the paper.
2. The multiple discrete coordinate transformation
2.1. Orthogonality condition for the discrete coordinate transformation
The discrete coordinate transformation is applied over the conformal mapping between two spaces - the virtual space and the physical space. There exist two conditions when adopting the DCT technique for an all-dielectric and isotropic solution. The first one is that boundaries of the virtual space and the physical space should maintain identical except for slight distortions, in order to guarantee that the region before and after transformation is embedded in the surroundings in the same way. For example, in Fig. 1(a), the solid black line and the blue line (numbered 1) represent the physical space for a carpet cloak, whilst its corresponding virtual space is the square portrayed by the solid black lines and the dashed black line. Mild distortion is observed on the north (N) boundary of the physical space. However, as the north boundaries are perfectly conducting, the virtual space and the physical space is in fact matched to the free space in the same way. A parabolic reflector, as depicted by the green line (numbered 2) in Fig. 1(a), is flattened using the same method. Under the circumstances when the distorted boundary is not conducting, for example, when a convex lens (as depicted by the brown line numbered 3) or a Luneburg lens (as depicted by the red line numbered 4) is compressed, impedance matching layers are needed on the north boundary to compensate the difference before and after transformation. To be noted, the two wings of the transformation region [the grey areas in Fig. 1(a)] are neglectable if permittivities are close to the unity in the two wings. The second condition is that quasi-orthogonal discretising meshes are needed to define local coordinates. Next, we are going to demonstrate the all-dielectric solution when the above two conditions are satisfied.
It has been studied that the electromagnetic parameters in the original (unprimed) and transformed (primed) space have the relationship that Fig. 1(b). The bold reduction from three-dimension (3D) to two-dimension makes the design process much simpler. In practice, because many devices are symmetric, or work at a specified polarization, 3D transformation devices can be obtained by rotating or extending 2D models to an axis.
Different from the analytical coordinate transformation (ACT), local coordinates are used to define the relation between the two spaces. Assuming both spaces are discretised by M × N sub-wavelength cells and in each of them there exists a local coordinate system, e.g., (x, y) in the virtual space and (x′, y′) in the square physical space [see Fig. 1(b)]. As long as the discretisation resolution is fine enough, saying to be sub-wavelength, permittivity and permeability in a cell is considered to be homogeneous. Imaging there is a small dot locating in the (i, j) cell of the original space. It has a random position of (x, y) in the local coordinates and a corresponding position of (X, Y) in the global coordinates. ε and μ at the dot is represented by the permittivity and permeability value at the center of the cell as
It has been proved in  that if , e.g., there is no magnetic dependence, the refractive index n′, which determines the trace of the wave, can be realised by the permittivity alone, leading to an all-dielectric device. This condition is approximately satisfied if we can generate quasi-orthogonal local coordinates in both spaces such thatEq. (4) that
Consequently, as long as the discretising mesh in both the virtual space and the physical space are quasi-orthogonal, and each cell in the grid is quasi-isotropic, one is able to control the propagation of waves using isotropic dielectrics alone. Unfortunately, the DCT technique cannot be straightforwardly applied to all problems as the ACT does. For example, it cannot transform a finite boundary to an infinite boundary because the grid connecting a point and a line can never be near-orthogonal. Even if all boundaries are finite and the discretising mesh is structured, the DCT technique can only deal with “smooth” boundaries. The term of “smooth” is roughly defined as “non-sudden bend by a large angle”. For example, in Fig. 1(b), the brown mesh is acceptably orthogonal because the brown boundary is “smooth”. In contrast, the blue mesh is strongly distorted because the blue boundary is discontinuous, containing a sharp bend of the angle of Φ. An investigation has been conducted, showing that if the bending angle of Φ in Fig. 1(b) is beyond 20°, the orthogonality of the blue mesh is apt to deteriorate significantly. An extreme circumstance happens when Φ approaches 90°. Two corners emerge on the north boundary (shown by the red line) and a near-orthogonal mesh becomes impossible due to singularity on the boundary.
2.2. The multiple discrete coordinate transformation
Here, we propose a so-called multiple DCT technique with the aim to eliminate some aforementioned limitations of the one-step DCT technique. We are going to use the multiple DCT to provide a general way of compressing an arbitrarily shaped dielectric object by a controllable ratio, as illustrated in Fig. 1(c). There are two key features of the multiple DCT technique. Firstly, the space before transformation is always square and the object is embedded in it. Therefore, the boundary of transformation space is no longer dependant on the shape of the object. Secondly, the original space is not necessary to be homogeneous.
Figure 2 shows a simple example to use the multiple DCT technique to compress a space. As mentioned above, one is not able to simultaneously generate M1 × N1 meshes with quasi-orthogonal and isotropic cells for two square spaces unless they are aspect-ratio-locked, and consequently is not able to compress a square space in only one direction using the one-step DCT technique. Instead of executing the DCT directly from the virtual space to the physical space, a transit space is introduced between them. Figures 2(b)–2(d) give the schematic showing of the multiple transformation process. A mild distortion is deliberately added for the transit physical space in Fig. 2(b) and the bending angle is chosen to be 10° – 20°, as mentioned above, to guarantee an acceptable orthogonality in the first step of the DCT between the original virtual space and the transit physical space. Assuming the two spaces are discretised by M1 × N1 cells, and according to Eq. (7), the permittivity and the permeability at a dot that located in the (i1, j1) cell in the transit physical space areFigs. 2(a) and 2(b) respectively.
Next, the left half and the right half of the transit physical space are exchanged, which means the distributions of ε′ and μ′ are re-located in Fig. 2(c) by
The new space in Fig. 2(c) is termed as the “transit virtual space” as it is corresponding to the target physical space in Fig. 2(d). The second step of the DCT is carried out between the transit virtual space and the target physical space. A quasi-orthogonal discretising mesh containing M2 × N2 cells is applied this time. It should be pointed out that discretising meshes in the 1st and the 2nd step are independent at all, and M2 and N2 are unnecessary to be equal to M1 and N1 respectively. Since the bending angle on the boundary of the transit virtual space doesn’t change, the orthogonality condition in Eq. (5) is considered to be satisfied as well and we achieveFigs. 2(c) and 2(d) respectively. After the second-step transformation, the right half and the left half of the space are exchanged back in the same way as described in Eq. (9). Finally we achieve a compressed space and only permittivity distribution is required for the transformation media.
The compressed space exhibits identical electromagnetic properties as the original one. In Fig. 2, the arrowed red lines show that the parallel incidences, as well as the perpendicular ones, propagates in the same way in both the target physical space in Fig. 2(d) and the original virtual space in Fig. 1(a). In addition, an incident wave leaves the transformation area in the same manner and hence an outer detector is not able to distinguish the two spaces. It should be pointed out that the target physical space may not guarantee to be matched to the surroundings as the original space does. This problem should be carefully treated, as we discussed above, by using an conducting boundary or an impedance matching layer.
Based on the two-step DCT, one is able to control the ratio of compression, which is defined asFigs. 2(b) and 2(c). To balance the two aspects, a thick space is discretised into several layers, and a series of two-step DCT are executed in a row. Figure 3(a) illustrates a space to be compressed. As the width of the space is much shorter than the height of it, ΔH cannot be comparable to H and hence the ratio α is limited. To solve this problem, the space is discretised into K layers to make sure the width is much larger than the height of each layer. The two-step DCT is applied to each layer to obtain a Δh = ΔH/K reduction of the height. For example, Figs. 3(b)–3(e) depict how we use the two-step DCT to compress the 1st layer, which is the same as we did in Figs. 2(a)–2(d). Therefore, the permittivity (ε′″1) and the permeability (μ′″1) in Fig. 3(e) can be calculated in the same way using Eqs. (8), (9) and (10).
To be noted, the black north boundary in Figs. 3(b)–3(e) is always maintained and, therefore, the boundary between the 1st layer and the 2nd layer keeps unchanged after transformation. As a result, the compressed 1st layer in Fig. 3(e) could be directly added to the 2nd layer, as depicted in Fig. 3(f), and the permittivity and the permeability distribution becomeFigs. 3(f)–3(i). The same procedure repeats until all of the layers have been compressed. Note that at the beginning of each round of the two-step DCT, the permittivity and the permeability distribution is inherited and updated as
In addition, the proposed multiple DCT technique could be further applied to compress the space in a second direction so as to realise two-dimensional compression. The issue of impedance mismatching on the boundary should be carefully studied in that case.
3. Design example: compression of a pyramidal absorber
Anisotropic complex permittivities and permeabilities are usually required to compress a pyramidal absorber using the technique of analytical coordinate transformation (ACT) . Moreover, such compression becomes a baffling problem to the one-step DCT technique because the pyramidal profile is usually electrically thick and very sharp. As a result, orthogonality and isotropy of local coordinate systems are seriously destroyed and the all-dielectric solution becomes invalid. However, the multiple DCT technique can guarantee the isotropic and dielectric properties of the transformation media, as analysed above. The target physical space serves as a much thinner absorber.
Figure 4 shows an example design of commercial pyramidal absorbers. This microwave absorber is fabricated by the TDK Corporation, and has been tested to provide lower than −20 dB reflectivity from 1 GHz to 100 GHz. This product is mainly made from urethane, and we assume it has a normal permittivity of 2.9ε0 and a loss tangent of 0.5. This kind of material usually has no magnetic response, so we assume the permeability is μ0.
Obviously, because the base of the absorber is much thinner (50 mm) than the pyramids (250 mm), plus the pyramids have very sharp tips, a simple two-step DCT technique is invalid in this case. In Section 2, we discussed the method to compress the thickness of a high space using multi-layered DCT technique. Figure 5 illustrates that the space containing the pyramid is discretised into 10 layers, as marked in Fig. 5 from Layer 0 to Layer 9. Each layer has a thickness of 25 mm. First of all, Layer 0, combined with the base, is compressed from 75 mm thick to 55 mm thick using the same method shown in Fig. 2. Accordingly, the bending angle is about 18°, and hence the boundaries of the transit spaces are considerably “smooth”. Second, Layer 1 is added to the compressed Layer 0, and one more round of the two-step DCT is carried out. After that, the space is compressed from 80 mm (55 mm + 25 mm) thick to 60 mm thick. The same procedure carries on until all ten layers have been compressed, in the same way as described in Section 2. Finally, the compressed space, as well as the target map of permittivity and permeability, is 100 mm in width and 100 mm in thickness, as shown in the right figure in Fig. 5. The total thickness of the pyramidal absorber is therefore reduced by 2/3.
Figure 6 explains the first round of the multiple DCT to compress the base and the first layer (Layer 0 in Fig. 5) in detail as an example. At the beginning, it should be pointed out that, there are two principles during the design. First, in all steps from (a) to (g) in Fig. 6, materials on all boundaries except the perfect electric conductor (PEC) boundary should be the same. In other words, only the PEC boundary is compressed because change of reflection on this boundary only contributes slightly to the total reflection of the transformation absorber. This is key to maintain the reflection to the surroundings before and after transformation, as pointed in Section 2. Second, since the pyramids are symmetric and periodically extended, it is reasonable to flip two halves of the transformation space, like what we do from Fig. 6(c) to Fig. 6(d), and from Fig. 6(f) to Fig. 6(g).
To start with, the base and the first layer of the absorber are located into the transformation space portrayed by the dashed black box in Fig. 6(a). Notice that the top left and top right corners are occupied with the air. Then, the transformation space is compressed from the PEC boundary in Fig. 6(b), similar to what we did in Fig. 3(c). The height of the bump is chosen to be 20 mm, with the bending angle around 18°. It should be clarified that from Fig. 6(b) afterwards, transformation spaces inside the dashed boxes are filled with spatially dispersive media with calculated permittivity distribution. In Fig. 6(c), the transformation space is rotated by 180°, whilst the left half and the right half are exchanged. After this re-arrangement, the second step DCT is executed in Fig. 6(d), similar to what we did in Figs. 3(d)–3(e), compressing the transformation space from the top and consequently reducing the total thickness to 55 mm. In Figs. 6(f) and 6(g), the transformation space is rotated and flipped again to its original arrangement, but the thickness is 20 mm lower than the original value in Fig. 6(a). At the end, the second layer of the discretised space is located above the physical space that achieved in Fig. 6(g), and the same procedure from (a) to (g) will be repeated.
After all the ten layers in Fig. 5 have been compressed, an isotropic all-dielectric transformation absorber is finally achieved. The logarithmic scaled relative permittivity map, including both the real part and the imaginary part, is presented in Fig. 7. The size of one unit pyramid has been reduced from 100 mm × 100 mm × 300 mm to 100 mm × 100 mm × 100 mm. The staircase in Fig. 7 has a resolution of 0.5 mm, which is also the resolution of the FDTD grid during the simulations in the following text. The permittivity value in each 0.5mm-size block ranges from ε0 to about 22 ε0, and the high values only exist in a few blocks close to the bottom boundary. Less-than-unity permittivities are required in very small area, therefore can be ignored. In conclusion, the transformation absorber in Fig. 7 can be constructed using non-resonant metamaterials or common dielectrics. Note that the compressed absorber can be further modified if Δh in Fig. 6(b) is less than 20 mm.
4. Numerical verification
A common way to examine the performance of an absorber is shown in Fig. 8. The absorber array is mounted on the conducting wall (shown as the PEC), while incident waves from different directions are launched towards the wall in order to test the scattering and absorbing properties of the absorber . In the measurement, the location of a transmitter is steered gradually from ϕ = 0° to ϕ = 90° (ϕ defined in Fig. 8), and a receiver is located at the coherent position to measure the reflection. For example, if the transmitter is placed at ϕ = 45°, then the receiver should be placed at ϕ = 135° accordingly.
Before fabricating a prototype and carrying out the measurement, we use numerical simulations to predict the performance of our design. In this case, the FDTD method based simulation is employed. The simulation is in a 2D circumstance, including Hx, Hy and Ez components. The setup of simulation is plotted in Fig. 8, and five pyramidal absorbers and five compressed absorbers are modelled for comparison. Noted that in the measurement, a much larger array is required and the transmitter and the receiver are always located in the far field; but in the simulation, to save computing memory and time, a short source (30 mm long) is originated 500 mm away (which is the radius of the yellow semi-circle in Fig. 8) from the center of the absorber array. This distance is only a few wavelengths below 3 GHz, and hence, the transmitter is not always in the far field. In other words, scattering may happen at all directions from ϕ = 0° to ϕ = 180°. According to this concern, we use a wide-band Gaussian pulse as the source, integrate the electric field along the whole semi-circle at each time step, isolate the reflected signal from the incident signal, and calculate the reflection coefficient in frequency domain by comparing the reflected signal with the incident one. The source has a Gaussian distribution from 2 GHz to 20 GHz in the frequency domain, hence is able to provide broad-band information.
Figure 9 plots the reflection coefficient when the pyramidal absorber and the compressed absorber are mounted on the conducting wall respectively. The incident pulse is located at ϕ = 90° [see Fig. 9(a)], ϕ = 45° [see Fig. 9(b)] and ϕ = 27° [see Fig. 9(c)] for testing (ϕ defined in Fig. 8). At the normal incidence when ϕ = 90°, the red curve has a similar shape as the black one, indicating that the compressed absorber obtains some performance of the pyramidal one, and is able to reduce the reflection from 2 GHz to 20 GHz. However, the reflection coefficient on the red curve is about 10 dB higher than that on the black curve at some frequencies, which means the compressed absorber is not as efficient as the original pyramidal one at a normal incidence. In Fig. 9(b), the red curve possesses a more similar shape as the black one and the difference between them becomes quite small from 11 GHz to 14.5 GHz. When the incidence is from ϕ = 27° in Fig. 9(c), the compressed absorber produces very similar reflection coefficient as the pyramidal one, especially from 7 GHz to 15 GHz. These results have demonstrated that the backwall reflection under an arbitrary incidence can be effectively reduced by the compressed absorber within a broad frequency band.
The degradation of absorbing performance of the transformation absorber, as indicated in Fig. 9, may be explained from different aspects. First of all, it may be due to the insufficient simulation domain in Fig. 8. An absorber array of five elements is not large enough and the absorbing property is not completely observed. Therefore, a large array simulation, or an experimental demonstration is expected for future work. Secondly, it may be mainly due to the reflections. Figures 10(a)–10(c) plots the distribution of the normalised electric field inside the pyramidal absorber, the compressed all-dielectric absorber and the compressed absorber with matched impedance (when the relative permittivity distribution and the relative permeability distribution are identical) respectively when the incidence is from ϕ = 27°. It is found that the compressed all-dielectric absorber in Fig. 10(b) has stronger reflections than the pyramidal absorber in Fig. 10(a), which is mainly caused by the impedance mismatching on the boundary between the compressed absorber and the surroundings. Therefore, one can potentially improve the compressed absorber by discretising the space with more layers and choosing a smaller Δh in Fig. 6(b). In addition, one solution to reduce the reflection is indicated in Fig. 10(c) that if the transformation absorber has matched permittivity and permeability, the reflections to the surroundings are effectively depressed. To demonstrate the assumption, reflection coefficient for the transformation absorber with matched impedance is plotted Fig. 10(d). When the incidence varies from ϕ = 90° to ϕ = 27°, the reflection coefficient substantially maintains unchanged from 2 GHz to 20 GHz. Comparing Fig. 9 and Fig. 10(d), the compressed absorber possesses much lower reflections from 2 GHz to 20 GHz and is less sensitive to the incidence when its permeability is engineered. Therefore, in this work we propose a general approach for compressing the sharp pyramidal absorber, and further optimization may be carried out to hence enhance the absorbing performance. Nevertheless, the compressed all-dielectric absorber shown in Fig. 7 already obtains an acceptable broadband performance, and possesses a much smaller thickness when compared with the original pyramidal absorber.
In this paper, we explained the essential limitation of the one-step DCT technique. The multiple DCT technique was then presented as a solution to this issue. It was analysed from the theory that the multiple transformation can manipulate the propagation of an wave without bringing in anisotropic or magnetic materials. An application of the multiple DCT technique was reported, that the thickness of a large pyramidal absorber can be reduced by 2/3. The achieved transformation absorber only contained lossy dielectrics, which can be easily found or fabricated. Numerical simulation results demonstrated that the transformation absorber can functionally reduce the backwall reflection from 2 GHz to 20 GHz. We envisage that more engineering applications can be developed based on this approach which provides an extension to the technique of coordinate transformation.
Wenxuan Tang’s work is the continuation of her PhD work at Queen Mary University of London, and has been partly supported by the National High Tech (863) Projects under Grant No. 2012AA030402 from China and partly by the National Science Foundation of China under Grant No. 61138001. Rui Yang’s work has been supported by the Fundamental Research Funds for the Central Universities ( K5051202039, K5051302081) from China, Innovation Funds for Excellent Returned Overseas Chinese Talents from Xidian University, and the Newton International Fellowships (follow-on program) from Royal Society, UK.
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