Fig. 1. Transmission line schematic representation of metatronic loaded network to solve PDEs. (a) Equivalent circuit representation of a single node of the proposed analogue processor. Each node is connected to four adjacent nodes via a T-circuit. In this representation, the out-of-plane Hz-field at the center of each junction (Ha, where a indicates the junction number) is represented by the current flowing around that junction counterclockwise (Ia). (b) (top) Waveguide-based metatronic structure that can emulate a T-circuit from (a). It consists of three thin dielectric slabs separated by a distance λ0/4. (bottom) Equivalent T-circuit model. (c) Full-wave numerical simulation results of a 3×3 network of waveguide-based metatronic structures emulating T-circuits with Zp=2.1522iZ0 and Zs=−0.9311iZ0 corresponding to h=0.4646 and k=2.8313. (left) Nodal representation of the simulation setup with a single monochromatic (10 GHz) incident signal excited from the top-left waveguide junction (labeled 1). (middle) Simulation results of the out-of-plane Hz-field (amplitude and phase) extracted at the center of each waveguide junction (labeled as 1–9). These results are normalized such that the input signal at junction 1 is unity. (right) Numerical (black crosses) and theoretical (red hollow circles) results of the phasor values of Hz-field recorded at junctions 2, 4, 5, 6, and 8.

Fig. 2. Scaling of PDE results. (a) Nodal representation of an arbitrarily sized network of interconnected metatronic elements. (b) (Top view) 3×3 sections of a larger waveguide network extending in all directions. The perfect electric conductor (PEC) regions for the waveguides are represented as gray blocks. The red and yellow slabs represent the elements that enable us to emulate the parallel and series metatronic elements, respectively. (c)–(e) Full-wave numerical results of the Hz-field distribution of a 25×25 waveguide network. As in Fig. 1(c), a 10 GHz monochromatic input signal is excited from the left waveguide of the top-left junction. These results have been normalized so that the out-of-plane Hz-field seen at the top left junction is unity. Here, λ1 is the wavelength of the PDE in the simulation space, while λ0 is the wavelength of the incident signal in free space. (c) Numerical results for the case where there are no dielectric slabs present within the connecting waveguides; see inset. (d), (e) Analytical (left), theoretical (middle), and numerical (right) results of the same setup from (c) (same color scale applies here) but now when the waveguides are loaded with the dielectric slabs emulating metatronic elements with effective impedances Zp=2.498iZ0, Zs=−0.9003iZ0 and Zp=5.001iZ0, Zs=−0.4501iZ0, respectively. These values correspond to PDEs with parameters h=0.4003, k=2.999(λ1=2.095), and h≈0.2, k=3.001(λ1=2.094), respectively. The EM and geometrical parameters after optimization are L1=L4=7.394 mm (∼0.2465λ0), L2=L3=7.307mm (∼0.2436λ0), ws=0.2111mm (∼7.0366×10−3λ0), wp=0.1741 mm (∼5.8033×10−3λ0), εs=21.5, and εp=12 for the results presented in panel (d), where L1, L2, L3, and L4 are the lengths of the impedance transforming waveguides from left to right, as shown in Fig. 1(b). ws, wp, εs, and εp are the widths and permittivity values of the slabs representing the series and parallel elements, respectively. These parameters are L1=L4=7.390mm (∼0.2463λ0), L2=L3=7.294mm (∼0.2431λ0), ws=0.2201mm (∼7.3367×10−3λ0), wp=0.1911mm (∼6.37×10−3λ0), εs=10.80, and εp=6.000 for the results presented in panel (e). The line plots in the rightmost panels of (d), (e) show the numerical (green triangles), analytical (red squares), and theoretical (gray circles) results of the magnitude of the Hz-field taken along a straight line from the top-left to bottom-right corners of the simulation space, respectively.

Fig. 3. Solving Dirichlet boundary value problems. (a), (c) Schematic representations of a 25×25 network of junctions of waveguide-based metatronic circuits excited with different Dirichlet boundary conditions. The metatronic elements are chosen considering Zp=5.001iZ0 and Zs=−0.4501iZ0 (see Fig. 2 caption for EM and geometrical parameters), corresponding to h≈0.2 and k=3.001 (λ1=2.094). (a) The left-hand boundary is set to g=1 while the top, right, and bottom boundaries are g=0. (c) The boundary conditions are such that the magnitude at each boundary junction is 1 but the phase along the boundary spatially varies from 0 to 2π, counterclockwise. (b), (d) Analytical, theoretical, and numerical results of the scenarios from (a) and (c), respectively. The top panels from (b) and (d) represent the normalized instantaneous Hz-field values calculated at each of the junctions inside the network. The bottom panels from (b) and (d) represent the magnitude of the calculated analytical (red squares), theoretical (gray circles), and numerical (green triangles) Hz-field values along the dashed lines from the top panels.

Fig. 4. Lensing and particle scattering. (a), (d) Schematic representations of a 50×50 subnetwork of waveguide-based metatronic circuits used to solve open boundary value problems. The 50×50 grid is constructed with Zp=5.001iZ0 and Zs=−0.4501iZ0, corresponding to h≈0.2 and k=3.001 (λ1=2.094, where λ1 is the wavelength seen inside the simulation space; see Fig. 2 caption for EM and geometrical parameters). (a) A 10 GHz monochromatic wave is excited at each of the boundary waveguides connecting to the left-hand boundary junctions of the subnetwork. The amplitude and phase of these signals is selected such that the Hz values at the boundary junctions resemble the output signal from a lens designed to produce a focus at x=1.432λ1 and y=2.387λ1 (15 and 25 junctions in the x and y directions, respectively), represented by a gray spot. (d) As in panel (a), a 10 GHz monochromatic signal is excited at the left-hand boundary waveguides, now with amplitude and phase selected such that the Hz values at the boundary junctions resemble a planewave. A 0.9549×0.9549λ1 (10×10 junctions) g=0 insert is placed at the center of the simulation space, represented by a white block. (b), (e) Theoretical (left) and analytical (right), power distribution for the scenarios presented in panels (a) and (b), respectively. In panels (b) and (e), results are normalized with respect to the power distribution at the focus and the maximum standing wave, respectively. (c), (f) Theoretical (gray circles) and analytical (red squares) values of power distribution along the vertical (top) and horizontal (bottom) lines drawn in panels (b) and (e), respectively.