The Waveguide Mesh

Consider the original form (2+1)D waveguide network, or mesh [198], operating on a rectilinear grid. Each scattering junction (parallel) is connected to four neighbors by unit sample bidirectional delay lines. The spacing of the junctions is $\Delta$ (in either the $x$ or $y$ direction) and the time delay is $T$ in the delay lines (see Figure 4.19). We now index a junction (and all its associated voltages and currents and wave quantities) at coordinates $(i\Delta,j\Delta)$ by the pair $(i,j)$.

Figure 4.19: (2+1)D waveguide mesh and a representative scattering junction.

As in the (1+1)D case, at each parallel junction at location $(i,j)$, we have voltages at every port, given by

$$U_{x^{+},i,j} = \notag$$

$$U_{x^{-},i,j} = \notag$$

$$U_{y^{+},i,j} = \notag$$

$$U_{y^{-},i,j} = \notag$$

and current flows

$$I_{x^{+},i,j} = \notag$$

$$I_{x^{-},i,j} = \notag$$

$$I_{y^{+},i,j} = \notag$$

$$I_{y^{-},i,j} = \notag$$

as well as wave quantities

$$U_{q,i,j} = U_{q,i,j}^{+}+U_{q,i,j}^{-}$$

$$I_{q,i,j} = I_{q,i,j}^{+}+I_{q,i,j}^{-}$$

where $q$ is any of $x^{+}$, $x^{-}$, $y^{+}$ or $y^{-}$. The variables superscripted with a $+$ refer to the incoming waves, and those marked $-$ to outgoing waves. The voltage and current waves are related by

$$I_{q,i,j}^{+} = Y_{q,i,j}U_{q,i,j}^{+}\hspace{1.5in}I_{q,i,j}^{-} = -Y_{q,i,j}U_{q,i,j}^{-}$$ (4.67)

where $Y_{q,i,j}$ is the admittance of the waveguide connected to the junction with coordinates $(i\Delta,j\Delta)$ in direction $q$. The junction admittance is then

$$Y_{J,i,j} \triangleq Y_{x^{-},i,j}+Y_{x^{+},i,j}+Y_{y^{-},i,j}+Y_{y^{+},i,j}$$ (4.68)

and the scattering equation, for voltage waves, will be, from (4.15),

$$U_{r,i,j}^{-} = -U_{r,i,j}^{+} + \frac{2}{Y_{J,i,j}}\left(Y_{x^{-... ...},i,j}^{+}+Y_{y^{+},i,j}U_{y^{+},i,j}^{+}+Y_{y^{-},i,j}U_{y^{-},i,j}^{+}\right)$$ (4.69)

where $r$ is any of $x^{+}$, $x^{-}$, $y^{+}$, or $y^{-}$. Voltage waves are propagated by:

$$U_{x^{+},i,j}^{+}(n) = U_{x^{-},i+1,j}^{-}(n-1) U_{x^{-},i,j}^{+}(n) = U_{x^{+},i-1,j}^{-}(n-1)$$

$$U_{y^{+},i,j}^{+}(n) = U_{y^{-},i,j+1}^{-}(n-1) U_{y^{-},i,j}^{+}(n) = U_{y^{+},i,j-1}^{-}(n-1)$$

The case of flow waves is similar except for a sign inversion. The complete picture is shown in Figure 4.19.

Similarly to the (1+1)D case, it is possible to obtain a finite difference scheme purely in terms of the junction voltages $U_{J,i,j}$, under the assumption that the admittances of all the waveguides in the network are identical, and equal to some positive constant $Y$. Thus, from (4.56), $Y_{J,i,j} = 4Y$. We have, for the junction at location $x=i\Delta$, $y=j\Delta$,

$$U_{J,i,j}(n+1)\hspace{0.1in} =\hspace{0.1in} \frac{2}{Y_{J,i,j}}\sum_{q}Y_{q,i,j}U_{q,i,j}^{+}(n+1),\hspace{0.5in}q=\{x^{-},x^{+},y^{-},y^{+}\}\notag$$

$$= \frac{1}{2}\left(U_{x^{-},i+1,j}^{-}(n)+U_{x^{+},i-1,j}^{-}(n)+U_{y^{-},i,j+1}^{-}(n) + U_{y^{+},i,j-1}^{-}(n)\right)\notag$$

$$= \frac{1}{2}\left( U_{J,i+1,j}(n)+U_{J,i-1,j}(n)+U_{J,i,j+1}(n) + U_{J,i,j-1}(n)\right)\notag$$

$$-\hspace{0.1in}\frac{1}{2}\left( U_{x^{-},i+1,j}^{+}(n)+U_{x^{+},i-1,j}^{+}(n)+U_{y^{-},i,j+1}^{+}(n) + U_{y^{+},i,j-1}^{+}(n)\right)\notag$$

$$= \frac{1}{2}\left( U_{J,i+1,j}(n)+U_{J,i-1,j}(n)+U_{J,i,j+1}(n) + U_{J,i,j-1}(n)\right)\notag$$

$$-\hspace{0.1in}\frac{2}{Y_{J,i,j}}\sum_{q}Y_{q,i,j}U_{q,i,j}^{-}(n-1)\notag$$

$$= \frac{1}{2}\left( U_{J,i+1,j}(n)+U_{J,i-1,j}(n)+U_{J,i,j+1}(n) + U_{J,i,j-1}(n)\right)-U_{J,i,j}(n-1)$$

This is identical to $\ref{magicdiff2d}$ if we replace $U_{J}$ by $U$.

If we now replace all the bidirectional delay lines in Figure 4.19 by the same split pair of lines shown in Figure 4.11, then we get the arrangement in Figure 4.20.

Figure 4.20: (2+1)D interleaved waveguide mesh.

We have placed the split lines such that the branches containing sign inversions are adjacent to the western and southern ports of the parallel junctions. We also introduce new junction variables $I_{xJ}$ at the series junctions between two horizontal half-sample waveguides, and $I_{yJ}$ at the series junctions between two vertical delay lines, as well as all the associated wave quantities at the ports of the new series junctions. It is straightforward to show that upon identifying $I_{xJ}$, $I_{yJ}$ and $U_{J}$ with $I_{x}$ and $I_{y}$ and $U_{J}$, the mesh will calculating according to scheme (4.52) with constant coefficients, if we choose

$$v_{0} = \sqrt{\frac{2}{lc}}\hspace{1.0in}Z = \sqrt{\frac{2l}{c}}$$

where $Z$ is the impedance in all the delay lines. We are again at the magic time step, but the impedance has been set to be larger than the characteristic impedance of the medium. Also, notice that the speed of propagation along the delay lines is not the wave speed of the medium, which is $\gamma = 1/\sqrt{lc}$. Such a mesh is called a slow-wave structure [90] in the TLM literature. At this point, it is useful to compare Figures 4.20 and 4.18.