Broadband unidirectional behavior of electromagnetic waves based on transformation optics

High directive antennas are fundamental elements for microwave communication and information processing. Here, inspired by the method of transformation optics, we propose and demonstrate a transformation medium to control the transmission path of a point source, resulting in the unidirectional behavior of electromagnetic waves (directional emitter) without any reflectors. The network of inductor-capacitor transmission lines is designed to experimentally realize the transformation medium. Furthermore, the designed device can work in a broadband frequency range. The unidirectional-manner-based device demonstrated in this work will be an important step forward in developing a new type of directive antennas.

Here, in this paper, anisotropic but homogeneous transformation medium is proposed and implemented to obtain the unidirectional emission of electromagnetic waves without the use of reflectors. Such a transformation medium is fabricated by controlling parameters of inductor-capacitor (L-C) unit cells in a transmission-line network. Our simulated and measured results demonstrate that such a novel device can work over a broad spectral range (about from 15 MHz to 75 MHz). It should be noted that the unidirectional effects proposed in this paper are realized by three kinds of transformation medium, so we just need to design three kinds of L-C unit cells to simulate transformation medium, and fortunately, the position of excitation source is not fixed but flexible. Figure 1(a,b) show the schematic of power flow and wavefront of a point source propagating in free space and a designed transformation medium, respectively. In the free space, as shown in Fig. 1(a), the wavefront of the point source is a circle, and the excited point source radiates into all directions. However, when the point source is located in the transformation medium ( Fig. 1(b)), a Gaussian beam-like wavefront is produced, and the excited point source propagates only along one designated direction, resulting in the unidirectional emission. In order to achieve such a unidirectional behavior, we establish the mapping between Fig. 1(c,d) based on the coordinate transformation. Here, (x, y, z) and (x′ , y′ , z′ ) are coordinates in real space and virtual space, respectively. The regions of OAB and OAC in real space are bended and compressed into the regions of O' A 1 B 1 and O'C 1 B 1 in virtual space, respectively. Meanwhile, the region OBC in real space is stretched into the region O'B 2 C 2 in virtual space. According to the transformation optics, the coordinate transformation between Fig. 1(c,d) can be expressed as follows:

Theoretical Model
In region I: 11 11 11 12 12 12 In region II: In region III: 31  31  31  32  32  32 where, x y x y x y y y y x y x y x y y y y The corresponding transformation medium parameters from Eqs (1-3) can be expressed: In region I: In region II: In region III: It should be noted that the transformation medium in region I, II, and III are anisotropic and homogenous, so it can hardly realize by naturally occurring crystals. It can be realized by using traditional metamaterial technology (such as the SRR (split resonance rings)) and effective theory (i.e., to further reduce transformation medium parameters), but limited to the resonance bandwidth. Although, the GRIN metamaterial technology (see ref. 13) can also be applied to realize the same phenomenon with a broad bandwidth, it cannot be simpler than our method of the L-C circuits.

Results and Discussions
The numerical simulations of unidirectional effects were performed using COMSOL Multiphysics ® . Figure 2(a), shows the electric field distribution of a point source located at (0, 0) with frequency of 3 GHz in free space. The wavefront of the electric field is a circle, which means that the field is radiated isotropically. However, when the same point source is embedded in a properly designed transformation medium (the corresponding parameters is shown in Eq. (7)~Eq. (9)), the radiated field is mostly directed upward as shown in Fig. 2(b), validating the unidirectional emission of the electromagnetic waves. In this case, the wave excited by the point source only propagates in the +y direction. We want to emphasis that the similar phenomenon can also be realized when the excitation source is located at other positions such as (0, 0.015 m) and (0, − 0.01 m) (see Supplementary Fig. S1), indicating that the position of excitation source is no longer fixed at (0, 0) but flexible.
Experimentally, such unidirectional behavior effects based on transformation optics can be realized by using a network of properly designed and implemented L-C transmission lines. A fabricated periodic L-C transmission line network shown in Fig. 3(a) acts as the desired transformation medium (region I, II, and III) and the background (the space outside the region I, II, and III). Figure 3(b) depicts the unit cell of L-C transmission line network. If L x is equal to L y , the L-C transmission line network can be regarded as an isotropic medium with positive permeability and permittivity due to the mapping between the L-C network equations and Maxwell's equations. Therefore, the transformation medium can be precisely designed by selecting proper parameters of inductors and capacitors. When the long wavelength limit is considered, i.e, the model of effective medium, where the dimension (period) of the unit cell is much smaller than the wavelength of the excitation source, the relationship between the real material parameters and L-C transmission line network can be described as follows: where Δ = 5mm is the length of the unit cell (here, the period Δ = 5mm is much smaller than the wavelength of the excited source). Like in refs 39 and 40, (here, in order to simplify the fabrication, the off-diagonal components of the transformation medium (in Eq. (7)~Eq. (8)) are also ignored like in refs 39 and 40, and such an approximation can also ensure that the unidirectional effect is verified in present experiment (see the measured results).) the detailed parameters can be chosen as listed in Table 1. The fabrication and measurement of the device will be discussed in details in Methods.  The unidirectionality for this special medium was studied by both numerical simulations and experimental verifications, as shown in Fig. 4. Figure 4(a1) depicts the simulated voltage distribution of a point source (with frequency of 45 MHz) excited at the node (31, 41), based on the Agilent's advanced design system (ADS). In this configuration, the excited source is coated with the transformation medium. So, almost all of the radiated waves (voltage distribution) from the point source propagate along the + y direction, while the radiation in other directions is significantly suppressed, indicating the unidirectional/directive emission of waves. Meanwhile, we fabricate the corresponding sample ( Fig. 3(a)) with 81 grid nodes along x direction and 101 grid nodes along y direction to demonstrate this phenomenon. Figure 4(b) illustrates the measured voltage distribution of the excited point source located at the node (41, 31) with the frequency of 45 MHz. The excited point source can also just radiate into the + y direction, resulting in unidirectional/directive emission of electromagnetic waves. In comparison with Fig. 4(a,b), both of them have nearly the identical voltage distributions, which demonstrate that the numerical simulation and experimental measurement are matched with each other. In addition, Fig. 4(c) shows voltage distributions of our proposed device at x = 41 (Here, 41 is the node number, and ∈ y [1, 101]) with various excited frequencies. From 15 MHz to 75 MHz, the intensity of voltage distribution at y > 30 is much stronger than that of y < 30, indicating that the designed unidirectional device can be worked in a broadband region (about from 15 MHz to 75 MHz). Now, we discuss the directivity of our designed unidirectional device, as shown in Fig. 5. Figure 5(a) shows the simulated and measured voltage distributions of our designed device at f = 45 MHz. The calculated voltage distributions show agreement with the measured results except for a slight difference because of the non-ideal electric capacitance and electric inductance. Figure 5(b) plots the simulated far-field patterns of the point source in the background medium (blue curve) and the transformation medium (red curve), respectively. The antenna directivity at f = 45 MHz is 14.3 dB. When compared with traditional lens directive antennas (such as a broadband half Maxwell fish-eye lens antenna in ref. 13), the designed device in this paper also presents excellent directivity. In addition, the far-field patterns for the designed device without/with reflectors and just with the bare reflector (without the transformation medium) are shown in the supplementary of Fig. S2.
Next, we illustrate the voltage distributions of all nodes in the transmission line network to further verify the broadband characteristic of unidirectionality. Figure 6(a 1 ,a 6 ) depict the corresponding voltage distributions of device at the design frequencies of 20, 30, 40, 50, 60 and 70 MHz. For different excitation frequencies, our designed device also shows the unidirectional property with antenna directivity of 12.5, 13.5, 13.9, 14.65, 16.85 and 16.95 dB, respectively. Figure 6(b 1 ,b 6 ) present the measured results of such sample at the frequencies of 20, 30, 40, 50, 60 and 70 MHz, which are in agreement with the simulation results, demonstrating that our designed sample can work in a broad frequency range.
Finally, we give a brief discussion on the limitations of the designed device. Although the proposed device shows broadband of unidirectional emitting (which is originated from the non-resonant nature of the transmission line metamaterials), it has more or less limitations. First, the size of the proposed directional emitting  device is still much bigger, which is limited to be applied in the SOC (system of chip). Second, such a conceptual verification of broadband unidirectional emitting can be just realized in RF or microwave region. It can also be realized in THz and visible frequency based on the metamaterial technology such as the SRR, it is limited to the bandwidth due to the resonance property. Therefore, to realize the broadband unidirectional emitting based on TO in high frequency region, is still an urgent problem to be solved. But, we still want to emphasis that it is a new or another method to design broadband directive antennas. On the other hand, if the design is extended to a three-dimensional (3D) case, the permittivity and permeability in Eqs 7~9 are further complicated. Actually, in the 3D case, the corresponding coordinate transformations are between (x, y, z) and (x′ , y′ , z′ ), and each component in permittivity and permeability tensors (Eqs 7~9) is non-zero, which means that the original parameters (in two-dimensional case) are not feasible. In a word, such a TO-based method can be extended into 3D cases, but the parameters are much more complicated, resulting in the limitation in fabrication.
In conclusion, we have proposed a new method to generate unidirectional emitting of electromagnetic waves based on transformation optics. The transformation medium mimicked by L-C transmission line networks was designed to realize the unidirectionality of electromagnetic waves. In particular, it is also demonstrated that such a device can work in a broadband frequency region (from 15 MHz to 75 MHz). Therefore, such a conceptual verification of unidirectional device may pave the way to realize the TO-based high-directive antennas.