Poles at s=0

We know from the foregoing that the denominator of the cone reflectance has at least one root at $s=0$. We now investigate the ``dc behavior'' more thoroughly.

• A hasty analysis based on the reflection and transmission filters (see figure) might conclude that the reflectance of the conical cap converges to $-1$ at dc, since $R(0)=-1$ and $T(0)=0$. However, this is incorrect.
• Instead, it is necessary to take the limit as $\omega\to0$ of the complete conical cap reflectance $R_J(s)$:
  $$R_J(s) = \frac{1 - e^{-2s} - 2s e^{-2s}}{2s - 1 + e^{-2s}}$$
  We already discovered a root at $s=0$ in the denominator in the context of the preceding stability proof. However, note that the numerator also goes to zero at $s=0$. This indicates a pole-zero cancellation at dc.

• To find the reflectance at dc, we may use L'Hospital's rule to obtain
  $$R_J(0) = \lim_{s\to0} \frac{N^\prime(s)}{D^\prime(s)} = \lim_{s\to 0}\frac{4s e^{-2s}}{2-2e^{-2s}}$$
  and once again the limit is an indeterminate $0/0$ form.

• We apply L'Hospital's rule again to obtain
  $$R_J(0) = \lim_{s\to0} \frac{N^{\prime\prime}(s)}{D^{\prime\prime}(s)} = \lim_{s\to 0}\frac{(4-8s) e^{-2s}}{4e^{-2s}} = +1$$
  Thus, two poles and zeros cancel at dc, and the dc reflectance is $+1$, not $-1$ as an analysis based only on the scattering filters would indicate.

• From a physical point of view, it makes more sense that the cone should ``look like'' a simple rigid termination of the cylinder at dc, since its length becomes vanishingly small compared with the wavelength in the limit.

• Another method of showing this result is to form a Taylor series expansion of the numerator and denominator:
  $$\begin{eqnarray*} N(s) &=& 2 {s^2} - {{8 {s^3}}\over 3} + 2 {s^4} + \cdots \\ D(s) &=& 2 {s^2} - {{4 {s^3}}\over 3} + {{2 {s^4}}\over 3} + \cdots \end{eqnarray*}$$
  

  Both series begin with the term $2s^2$ which means both the numerator and denominator have two roots at $s=0$. Hence, again the conclusion is two pole-zero cancellations at dc.

• The series for the conical cap reflectance is
  $$R_J(s) = 1 - {{2 s}\over 3} + {{2 {s^2}}\over 9} - {{4 {s^3}}\over {135}} - {{2 {s^4}}\over {405}} + \cdots$$
  which approaches $+1$ as $s\to0$.