Stability Proof Outline

• A transfer function $R_J(s) = N(s)/D(s)$ is stable if there are no poles in the right-half $s$ plane. That is, for each zero $s_i$ of $D(s)$, we must have re$\left\{s_i\right\}\leq 0$. If this can be shown, along with $\left\vert R_J(j\omega)\right\vert\leq 1$, then the reflectance $R_J$ is shown to be passive.

• We must also study the system zeros (roots of $N(s)$) in order to determine if there are any pole-zero cancellations (common factors in $D(s)$ and $N(s)$).

• Since re$\left\{s{t_{\xi}}\right\}\geq 0$ if and only if re$\left\{s\right\}\geq 0$, for ${t_{\xi}}>0$, we may set ${t_{\xi}}=1$ without loss of generality. Thus, we need only study the roots of
  $$\begin{eqnarray*} N(s) &=& 1 - e^{-2s} - 2s e^{-2s} \\ D(s) &=& 2s - 1 + e^{-2s} \end{eqnarray*}$$
  

  If this system is stable, we have stability also for all ${t_{\xi}}>0$.

• Since $e^{-2s}$ is not a rational function of $s$, the reflectance $R_J(s)$ may have infinitely many poles and zeros.