Broadband transmission-line illusions based on transformation electromagnetic

. We demonstrate broadband transmission-line illusions based on transformation electromagnetics at microwave frequencies by using the distributed full-tensor anisotropic medium. Due to an intrinsic nature of the non-resonantunitcellofthemedium,theillusionsoperatefromDCtoanupperlimitfrequencywherethehomogeneous medium approximation holds. Two-dimensional groove and bump illusion media mimicking scattered waves by an original groove and a bump are designed. Their broadband and incident angle independent operations are con ﬁ rmed by circuit simulations. The groove illusion medium is implemented on a dielectric substrate with microstrip-line technology, and it is con ﬁ rmed experimentally by near- ﬁ eld measurements that the illusion medium well mimics scattered waves by the original groove in the broadband frequency range from 2.60GHz to 4.65GHz.


Introduction
The distributed full-tensor anisotropic medium [1] has been proposed. The medium consists of a two-dimensional (2-D) array of unit cells each of which is a distributed transmission-line network yielding equivalent full-tensor material parameters [1]. According to the equivalent circuit analysis of the medium, it is directly shown that the diagonal components of the permeability tensor correspond to inductances per unit length in the x-and y-branches and the off-diagonal components correspond to the mutual inductance per unit length between the x-and y-branches. With these intuitive properties, the medium has full controllability of the material tensor parameters and is suitable for implementations of cloaks of invisibility [1][2][3][4][5][6][7][8][9][10][11][12][13][14] based on the concept of transformation electromagnetics [15][16][17]. In addition, the medium is advantageous in wideband operations from DC to an upper limit frequency where the homogeneous medium approximation holds due to an intrinsic nature of the non-resonant unit cells [1]. With the full-tensor anisotropic medium, a carpet cloak of invisibility [1] and coordinate transformation resonators [18] have been implemented and their unique operations have been demonstrated.
Due to the flexibility of the design, the medium can be used not only for suppressing scattered waves to hide an object or mimicking a shape of other resonator but also for mimicking scattered waves from a non-existent object to realize electromagnetic/optical illusions [19,20]. In this paper, we demonstrate broadband transmission-line illusions by using the distributed full-tensor anisotropic medium. In Section 2, we show the concept of groove and bump illusions and recall the theory of the transmission-line anisotropic media. In Section 3, the groove and bump illusion media are designed and transmission-line parameters are obtained. In Section 4, SPICE circuit simulations are carried out to confirm operations of the designed groove and bump illusions. In Section 5, the groove illusion medium is implemented on a dielectric substrate and the illusion operation is demonstrated by near-field measurements at microwave frequencies.
2 Illusions with transmission-line anisotropic media

Groove and bump illusions
Here, we consider two types of illusions: a groove illusion and a bump illusion. The groove illusion mimicking scattering waves by a 2-D groove on a flat surface whose cross section is shown in Figure 1a. The illusion medium of Figure 1b is designed to generate identical scattered waves by the groove. In addition, the bump illusion mimicking the scattering waves by a 2-D bump on a flat surface is designed with the same manner (see Fig. 2).
For both cases, the illusions can be realized with inhomogeneous anisotropic media equivalent to a corresponding coordinate transformation from an original Cartesian coordinate system (x, y) into a non-conformal coordinate system (x 0 , y 0 ). According to the material interpretation of the coordinate transformation [15][16][17], the material tensor parameters are given by where g i 0 j 0 is the metric for the coordinate transformation.

Broadband transmission-line illusion media
The groove and bump illusions are realized by using the distributed transmission-line full-tensor anisotropic medium shown in Figure 3 [1]. The characteristic impedances and electrical lengths of the transmission-line sections in the unit cell are directly determined from the material tensor parameters of equation (1) by [1]: where Dd is the unit cell length and bl = b x l x = b y l y are the electrical lengths of the transmission-lines in the x-and y-branches in Figure 3. It is worth mentioning that the medium essentially has extremely broadband characteristics. The broadband nature of this medium can be seen from the fact that, as the frequency approaches to zero, the material parameters become frequency independent [1] as: ; ð8Þ where (L 0 0x , C 0 0x ), (L 0 0y , C 0 0y ), (L 0 0M , C 0 0M ) are the inductances per-unit-length and the capacitances perunit-length of the corresponding the transmission-line sections with the characteristic impedances of Z 0x , Z 0y , and Z 0M , respectively. Figure 4 shows the coordinate transformation (x, y) ↦ (x 0 , y 0 ) used in the groove illusion of Figure 1. The   coordinate transformation is given by

Groove illusion
where 2p is the width of the cloak and A is the depth of the groove. The original and transformed regions are given with −p x p, ÀA{1-(x/p) 2 } 2 y h, −p x 0 p, and 0 y 0 h. Based on equation (10), the characteristic impedances and electrical lengths for each of the discretized unit cells are designed by solving equations (2) through (5) simultaneously with the designed material tensor parameters given by equation (1). In this design, we choose the depth and the width of the groove A = 0.3h and 2p = 2h, respectively, where h is the height of the groove illusion medium. The absolute refractive index n = 2.14 and the wave impedance h = 63.6 V of the original medium are also chosen in the following design. Figure 5 shows the calculated characteristic impedances and electrical lengths Z 0x , Z 0y , b x l x = b y l y bl, and b M l M . Here, we choose Z 0M = 1.5h where h is the wave impedance in the original medium. Figure 6 shows the coordinate transformation (x, y) ↦ (x 0 , y 0 ) used in the bump illusion of Figure 2. In this case, we choose the coordinate transformation:

Bump illusion
where 2p is the width of the illusion medium and A is the height of the bump. In this case, the original and transformed regions are given with Àp x p, A{1-(x/p) 2 } 2 y h, Àp x 0 p, and 0 y 0 h. Based on equation (11), the characteristic impedances and electrical lengths for each of the discretized unit cells are designed with the same manner as in the groove illusion. For the bump geometry with A = 0.3h and 2p = 2h, where h is the height of the illusion medium, the obtained characteristic impedances and electrical lengths Z 0x , Z 0y , b x l x = b y l y bl, and b M l M are shown in Figure 7. The values n, h, and Z 0M are chosen as the same values for the case with the groove illusion.    In order to show the equivalence of scattered waves by the groove and bump illusion media to those by the original obstacles as well as their broadband characteristics, wave scattering simulations are carried out using a SPICE simulator. Figure 8 shows the configuration of the circuit simulations. An illusion medium with the anisotropic transmission-line parameters of either Figure 5 or Figure 7 is discretized with 20 Â 10 unit cells and is arranged at the bottom center of an isotropic medium of 300 Â 150 unit cells [1]. The nodes on the bottom boundaries are short-circuited including the illusion area, and the nodes on the right-hand side boundary are terminated by resistors. Voltage sources are connected at the nodes on the other boundaries to illuminate the illusion medium with a normal or oblique Gaussian beams whose beam waist is 10 Dd. Complex voltage distributions at the center nodes of the unit cells are calculated, and the results are compared to those with and without an original obstacle.

Simulation setup
Here, the wave impedance and the refractive index in the isotropic area are chosen as h = 63.6 V and n = 2.14, respectively, considering feasibility of practical implementations. Sixty-two Ohm resistances are used for the terminations to simulate a practical situation used in the experiments. The internal impedance of each voltage source is also set to be 62 V⋅ As seen in Figure 9a, the incident wave is directly reflected back toward the source by the flat surface, whereas the incident wave is strongly scattered by the original groove in several directions as in Figure 9b. For the case with the groove illusion medium as in Figure 9c, the incident wave is scattered as if scattered by the original groove almost completely.

Groove illusion
In order to quantitively evaluate the performance, the radar cross sections (RCSs) for the groove illusion medium are calculated and compared with those with the original groove. Figure 10 shows those calculated results. It is seen from the figure that the RCSs agree well with each other and the scattered waves are in the directions of u scat = 0, ±27, ±43 deg, etc., where u scat is the reflection angle from the normal direction, which is consistent with the voltage distribution simulations in Figure 9. Therefore, it can be concluded that the illusion medium well mimics scattered waves by the original groove.
As mentioned in Section 2.2, the illusion medium intrinsically operates in an extremely broad frequency band down to DC. For instance, Figure 11a-c shows results for the case with the doubled wavelength of l g = 24 Dd. In this case, the beam waist of the source is also doubled with 20 Dd to form a directive beam. It is seen from the figure that scattered waves by the designed illusion medium in   Moreover, simulations with oblique incident waves are carried out. Figure 12a-c shows the results for the incident wave with the angle of (a) 30 deg, (b) 45 deg, and (c) 60 deg, respectively. The wavelength and the beam waist of the incident wave are chosen as 12 Dd and 10 Dd, respectively. It is seen from the results that the groove illusion medium perfectly operates with these oblique incident angles. Thus, the incident angle independent operations of the groove illusion medium are confirmed. Figure 13 shows similar calculated results with a normal incident wave for the cases with (a) an original bump, and (b) the bump illusion medium. The wavelength and the beam waist of the incident wave are chosen to be the same as in the previous simulations, i.e., 12 Dd and 10 Dd, respectively. It is seen from the figure that the bump illusion medium well mimics the scattered waves by the original bump. Figure 14 shows calculated RCSs for the bump illusion medium and the original bump. It is seen from the figure that both the RCSs agree well with each other. Figure 15 also shows results for the cases with the oblique incident waves with (a) 30 deg, (b) 45 deg, and (c) 60 deg, respectively. It is seen from the figure that the bump illusion medium mimics the original bump well regardless of the oblique incident angles as in the case for the groove illusion.

Groove illusion experiments
The groove illusion medium designed in Section 3 is implemented with microstrip-line technology and its operation is confirmed by near-field measurements.

Implementation
The groove illusion medium is implemented by using anisotropic unit cells of Figure 16 [1]. The unit cells are direct implementations of Figure 3 consisting of microstripline sections with line widths, W x , W y , and W M , and line lengths, l x , l y , and l M (see Fig. 16) to satisfy the characteristic impedances and the electrical lengths given by equations (2)- (5). A dielectric substrate ARLON DiClad880 with the relative permittivity e r = 2.17, the thickness t = 0.254 mm, and the dielectric loss tand = 0.00085 is used. The width and the height of the illusion area are chosen as 2p = 100 mm and h = 50 mm, respectively, and the unit element size is also chosen as Dd = 5 mm. Figure 17 shows the line widths W x and W y , and the line lengths l x = l y = l and l M calculated by Keysight ADS assuming the effective phase constant in the substrate b = b x = b y = b M = 1.316k 0 , where k 0 is the wave number in vacuum. The width W M is discretionally determined as 0.231 mm to give the value of Z 0M = 1.5h. Figure 18a shows a photograph of the fabricated illusion medium. For comparison, an original medium with the groove shown in Figure 18b is also fabricated. Figure 18c    T. Nagayama and A. Sanada: EPJ Appl. Metamat. 6, 23 (2019) other boundaries are terminated by chip resistors with 62 V thorough metallic lands with 0.8 Â 0.5 mm 2 to prevent reflections.

Field distributions with a source at the top center
First, a coaxial cable for excitation is connected at the top center of the sample (Position A in Fig. 19), and near-field distributions 0.5 mm above the two samples are measured by a vector network analyzer using an electric probe with an automated xyz-stage at the sampling points with the period of Dd/4 = 1.25 mm (see the insets in Fig. 19 for the electric probe and the excitation method). The length and radius of tip of the electric probe are 6.0 mm and 0.14 mm, respectively. Figure 20a and b shows measured complex electric near-field distributions for the cases with the groove illusion medium and with the original groove, respectively, at the highest frequency of 4.65 GHz (l g = 6.02 Dd) where illusion operations are observed in the experiments. It is seen from the figure that the groove illusion medium mimics the original groove well. For comparison, Figure 21a and b shows corresponding simulated complex voltage distributions for the cases with the simulated results with groove illusion medium and with the original groove, respectively, at the same frequency. By comparing Figure 20 with 21, it is seen that the measured field distributions agree well with the simulated predictions. Note that the major scattered waves are in ±29 deg   according to RCS calculation, which is consistent with the measured field distributions in Figure 20. Figure 22a and b shows measured complex electric near-field distributions at a much lower frequency of 2.60 GHz (l g = 10.8 Dd) where a good impedance matching is obtained at the excitation source. It is also seen from     T. Nagayama and A. Sanada: EPJ Appl. Metamat. 6, 23 (2019) the figure that scattered waves by the groove are well mimicked by the illusion medium. Figure 23a and b shows corresponding simulated results with the groove illusion medium and with the original groove, respectively, at the same frequency. By comparing Figure 22 with 23, it is seen that the measured field distributions agree well with the simulated results. According to RCS calculation, major scattered waves are in 0 and ±48 deg in this case. From these results, the broadband characteristics of the illusion medium from 2.60 GHz to 4.65 GHz are confirmed.

Field distributions with a source at the top corner
Next, the excitation position is moved to the top corner (Position B in Fig. 19) and similar near-field measurements are carried out. Figure 24a and b shows measured complex electric near-field distributions at 4.65 GHz for the cases with the groove illusion medium and with the original groove, respectively. It is seen from Figure 24 that the field distributions agree well with each other as well. Figure 25 shows corresponding simulated results to Figure 24.    According to RCS calculation, simulated major scattered waves are in +35, +12, and À9 deg in this case, which is also consistent with the measured field distributions in Figure 24. Figure 26a and b shows measured complex electric near-field distributions at 2.60 GHz for the cases with the groove illusion medium and with the original groove, respectively. It is seen from Figure 26 that the field distributions agree well with each other also in this case. Figure 27 shows corresponding simulated results to Figure 26. According to RCS calculation, simulated major scattered waves are in +32 and À7 deg, which is also consistent with the measured field distributions in Figure 26. Thus, it is concluded that the angle independent and broadband operations of the groove illusion medium are confirmed experimentally.

Conclusions
Broadband transmission-line illusions based on transformation electromagnetics have been demonstrated by using the distributed full-tensor anisotropic medium. First, the concepts of a groove and a bump illusion media mimicking scattered waves by a groove and a bump have been presented based on transformation electromagnetics, and the design formulas of the distributed full-tensor aniso-    T. Nagayama and A. Sanada: EPJ Appl. Metamat. 6, 23 (2019) tropic medium have been shown. Then, the groove and bump illusion media have been designed with the transmission-line anisotropic medium. The broadband and angle independent operations have been confirmed by calculating the scattered field distributions and the RCSs with SPICE circuit simulations and the validity of the design theory has been shown. In addition, the groove illusion medium has been implemented on a dielectric substrate with microstrip-line technology. Near-field measurements have been carried out for different two source positions and it has been confirmed that the groove illusion medium well mimics scattered waves by the original groove in the broadband frequency range as wide as from 2.60 GHz to 4.65 GHz. It is noted that the groove illusion medium operates fundamentally at any lower frequencies down to DC.
Incidentally, we can implement the illusion medium for a plane wave illumination by stacking multiple transmission-line structures. In this case, some design modifications are required for the plane wave to transmission-line mode conversion and the impedance matching with a free space.