Boundary Conditions as Perturbations

To study the effect of boundary conditions on the state transition matrices $\mathbf{A}_W$ and $\mathbf{A}_K$ , it is convenient to write the terminated transition matrix as the sum of the ``left-clamped'' case $\mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$_W$ (for which $g_l=-1$ ) plus a series of one or more rank-one perturbations. For example, introducing a right termination with reflectance $g_r$ can be written

$$\mbox{$\stackrel{{\scriptscriptstyle \vdash\!\!\dashv}}{\mathbf{A}}$} _W= \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$} _W+ g_r{\mathbf d}_{8,7} = \mathbf{A}_W- {\mathbf d}_{1,2} + g_r{\mathbf d}_{8,7}, \protect$$ (E.39)

where ${\mathbf d}_{ij}$ is the $M\times M$ matrix containing a 1 in its $(i,j)$ th entry, and zero elsewhere. (Following established convention, rows and columns in matrices are numbered from 1.)

In general, when $i+j$ is odd, adding ${\mathbf d}_{ij}$ to $\mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$_W$ corresponds to a connection from left-going waves to right-going waves, or vice versa (see Fig.E.2). When $i$ is odd and $j$ is even, the connection flows from the right-going to the left-going signal path, thus providing a termination (or partial termination) on the right. Left terminations flow from the bottom to the top rail in Fig.E.2, and in such connections $i$ is even and $j$ is odd. The spatial sample numbers involved in the connection are $2\lfloor (i-1)/2\rfloor$ and $2\lfloor (j-1)/2\rfloor$ , where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .

The rank-one perturbation of the DW transition matrix Eq.(E.39) corresponds to the following rank-one perturbation of the FDTD transition matrix $\mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$_K$ :

   $$\mbox{$\stackrel{{\scriptscriptstyle \vdash\!\!\dashv}}{\mathbf{A}}$}$$$$_K\;\isdef \;$$   $$\mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$$$_K+ g{\bm \Delta}_{8,7}$$

where

$${\bm \Delta}_{8,7} \isdef \mathbf{T}{\mathbf d}_{8,7}\mathbf{T}^{-1} = \left[\! \begin{array}{rrrrrrrrrrr} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \end{array}\!\right]. \protect$$ (E.40)

In general, we have

$${\bm \Delta}_{ij} = \sum_{\kappa=j}^M (-1)^{\kappa-j} \left({\mathbf d}_{i\kappa}+{\mathbf d}_{i-1,\kappa}\right). \protect$$ (E.41)

Thus, the general rule is that ${\mathbf d}_{ij}$ transforms to a matrix ${\bm \Delta}_{ij}$ which is zero in all but two rows (or all but one row when $i=1$ ). The nonzero rows are numbered $i$ and $i-1$ (or just $i$ when $i=1$ ), and they are identical, being zero in columns $1:j-1$ , and containing $[1,-1,1,-1,\ldots]$ starting in column $j$ .