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Losses, Sources, and Spatially-varying Coefficients

We can deal with spatially-varying material parameters as well as losses and sources in a manner similar to the (1+1)D case. The full (2+1)D transmission line equations, as originally presented in §3.8, are
$\displaystyle \begin{eqnarray}l\frac{\partial i_{x}}{\partial t}+\frac{\partial... i_{x}}{\partial x}+\frac{\partial i_{y}}{\partial y}+gu+h&=&0 \end{eqnarray}$ (4.70a)

where we have $ r(x,y)\geq 0$ and $ g(x,y)\geq 0$, and $ e$, $ f$ and $ h$ are driving functions of $ x$, $ y$ and $ t$.

The centered difference approximation to (4.58) is

$\displaystyle \begin{eqnarray}I_{x,i+\frac{1}{2},j}(n+{\textstyle \frac{1}{2}})...
...&\Delta\sigma_{U,i,j}\bar{h}_{i,j}(n-{\textstyle \frac{1}{2}})= 0\end{eqnarray}$    


$\displaystyle \rho_{I,k,p}$ $\displaystyle = \frac{2\bar{l}_{k,p}-r_{k,p}T}{2\bar{l}_{k,p}+r_{k,p}T}$ $\displaystyle \sigma_{I,k,p}$ $\displaystyle = \frac{2}{2v_{0}\bar{l}_{k,p}+r_{k,p}\Delta}$ (4.72)
$\displaystyle r_{k,p}$ $\displaystyle = r(k\Delta,p\Delta)$ $\displaystyle \bar{l}_{k,p}$ $\displaystyle = l(k\Delta,p\Delta)+O(\Delta^{2})$ (4.73)

for $ k$, $ p$ half-integer such that $ 2(k+p)$ is odd, and

$\displaystyle \rho_{U,i,j}$ $\displaystyle = \frac{2\bar{c}_{i,j}-g_{i,j}T}{2\bar{c}_{i,j}+g_{i,j}T}$ $\displaystyle \sigma_{U,i,j}$ $\displaystyle =\frac{2}{2v_{0}\bar{c}_{i,j}+g_{i,j}\Delta}$    
$\displaystyle g_{i,j}$ $\displaystyle = g(i\Delta,j\Delta)$ $\displaystyle \bar{c}_{i,j}$ $\displaystyle = c(i\Delta,j\Delta)+O(\Delta^{2})$    

for $ i$, $ j$ integer. For the sources, we have used
$\displaystyle \bar{e}_{i+\frac{1}{2},j}(n)$ $\displaystyle =$ $\displaystyle \frac{1}{2}\left(e_{i+\frac{1}{2},j}(n+{\textstyle \frac{1}{2}})+e_{i+\frac{1}{2},j}(n-{\textstyle \frac{1}{2}})\right)$  
$\displaystyle \bar{f}_{i,j+\frac{1}{2}}(n)$ $\displaystyle =$ $\displaystyle \frac{1}{2}\left(f_{i,j+\frac{1}{2}}(n+{\textstyle \frac{1}{2}})+f_{i,j+\frac{1}{2}}(n-{\textstyle \frac{1}{2}})\right)$  
$\displaystyle \bar{h}_{i,j}(n-{\textstyle \frac{1}{2}})$ $\displaystyle =$ $\displaystyle \frac{1}{2}\Big(h_{i,j}(n)+h_{i,j}(n-1)\Big)$  

$\displaystyle e_{i+\frac{1}{2},j}(n+{\textstyle \frac{1}{2}})$ $\displaystyle =$ $\displaystyle e\left((i+{\textstyle \frac{1}{2}})\Delta,j\Delta,(n+{\textstyle \frac{1}{2}})T\right)$  
$\displaystyle f_{i,j+\frac{1}{2}}(n+{\textstyle \frac{1}{2}})$ $\displaystyle =$ $\displaystyle f\left(i\Delta,(j+{\textstyle \frac{1}{2}})\Delta,(n+{\textstyle \frac{1}{2}})T\right)$  
$\displaystyle h_{i}(n)$ $\displaystyle =$ $\displaystyle h\left(i\Delta,j\Delta,nT\right)$  

Again, we have applied a semi-implicit approximation to the constant-proportional terms of (4.58).

The waveguide network shown in Figure 4.21 is a direct generalization to (2+1)D of Figure 4.15. To the structure of Figure 4.20 we have added an extra port to each scattering junction, series or parallel, which is connected to a self-loop of impedance $ Z_{c}$ and doubled delay length, as well as a port with impedance $ Z_{R}$ to introduce losses and sources. All immittances are indexed by the coordinates of their associated junctions. As before, we set the admittance $ Y$ = $ 1/Z$ for any impedance $ Z$ in the network. In Figure 4.21, the linking admittances of the bidirectional delay lines are indicated only at the parallel junction, since we must have

$\displaystyle Y_{x^{+},i,j} = \frac{1}{Z_{x^{-},i+\frac{1}{2},j}}\hspace{0.3in}...
...\frac{1}{2}}}\hspace{0.3in} Y_{y^{-},i,j} = \frac{1}{Z_{y^{+},i,j-\frac{1}{2}}}$    

The junction admittances and impedances are thus
$\displaystyle Y_{J,i,j}$ $\displaystyle =$ $\displaystyle Y_{x^{-},i,j}+Y_{x^{+},i,j}+Y_{y^{-},i,j}+Y_{y^{+},i,j}+Y_{c,i,j}+Y_{R,i,j}\notag$  
$\displaystyle Z_{J,i+\frac{1}{2},j}$ $\displaystyle =$ $\displaystyle Z_{x^{-},i+\frac{1}{2},j}+Z_{x^{+},i+\frac{1}{2},j}+Z_{c,i+\frac{1}{2},j}+Z_{R,i+\frac{1}{2},j}$  
$\displaystyle Z_{J,i,j+\frac{1}{2}}$ $\displaystyle =$ $\displaystyle Z_{y^{-},i,j+\frac{1}{2}} + Z_{y^{+},i,j+\frac{1}{2}}+Z_{c,i,j+\frac{1}{2}}+Z_{R,i,j+\frac{1}{2}}\notag$  

Beginning from series and parallel junctions, and proceeding through derivations similar to (4.32) yields the difference scheme (4.59) in the junction variables $ U_{J}$, $ I_{xJ}$ and $ I_{yJ}$, provided we set

$\displaystyle Y_{J,i,j}$ $\displaystyle = 2v_{0}\bar{c}_{i,j}+\Delta g_{i,j}$ $\displaystyle Y_{R,i,j}$ $\displaystyle = \Delta g_{i,j}$ $\displaystyle U_{R,i,j}^{+}$ $\displaystyle = -h_{i,j}/2g_{i,j}$    
$\displaystyle Z_{J,i+\frac{1}{2},j}$ $\displaystyle = 2v_{0}\bar{l}_{i+\frac{1}{2},j} + \Delta r_{i+\frac{1}{2},j}$ $\displaystyle Z_{R,i+\frac{1}{2},j}$ $\displaystyle = \Delta r_{i+\frac{1}{2},j}$ $\displaystyle I_{R,i+\frac{1}{2},j}^{+}$ $\displaystyle = -e_{i+\frac{1}{2},j}/2r_{i+\frac{1}{2},j}$ (4.74)
$\displaystyle Z_{J,i,j+\frac{1}{2}}$ $\displaystyle = 2v_{0}\bar{l}_{i,j+\frac{1}{2}}+\Delta r_{i,j+\frac{1}{2}}$ $\displaystyle Z_{R,i,j+\frac{1}{2}}$ $\displaystyle =\Delta r_{i,j+\frac{1}{2}}$ $\displaystyle I_{R,i,j+\frac{1}{2}}^{+}$ $\displaystyle = -e_{i,j+\frac{1}{2}}/2r_{i+\frac{1}{2},j}$    

Figure: (2+1)D waveguide mesh for the varying-coefficient system (4.58), with losses and sources.
% graphpaper(0,0)(480,480...
...ny {$Y_{x^{+}}$}}
\end{picture} \end{center}\par\vspace{-0.2in}

We can again identify three useful ways of setting the immittances:
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Next: Type I: Voltage-centered Mesh Up: The Waveguide Mesh Previous: The Waveguide Mesh
Stefan Bilbao 2002-01-22