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We know from the above that the denominator of the cone reflectance
has at least one root at
. In this subsection we investigate
this ``dc behavior'' of the cone more thoroughly.
A hasty analysis based on the reflection and transmission filters in
Equations (C.175) and (C.176) might conclude that the reflectance
of the conical cap converges to
at dc, since
and
.
However, this would be incorrect. Instead, it is necessary to take the
limit as
of the complete conical cap reflectance
:
![$\displaystyle R_J(s) = \frac{1 - e^{-2s} - 2s e^{-2s}}{2s - 1 + e^{-2s}}$](img4502.png) |
(C.186) |
We already discovered a root at
in the denominator in the context of
the preceding stability proof. However, note that the numerator also goes
to zero at
. This indicates a pole-zero cancellation at dc. To find
the reflectance at dc, we may use L'Hospital's rule to obtain
![$\displaystyle R_J(0) = \lim_{s\to0} \frac{N^\prime(s)}{D^\prime(s)} = \lim_{s\to 0}\frac{4s e^{-2s}}{2-2e^{-2s}}$](img4503.png) |
(C.187) |
and once again the limit is an indeterminate
form.
We therefore apply L'Hospital's rule again to obtain
![$\displaystyle R_J(0) = \lim_{s\to0} \frac{N^{\prime\prime}(s)}{D^{\prime\prime}(s)} = \lim_{s\to 0}\frac{(4-8s) e^{-2s}}{4e^{-2s}} = +1$](img4505.png) |
(C.188) |
Thus, two poles and zeros cancel at dc, and the dc reflectance is
, not
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 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:
Both series begin with the term
which means both the numerator
and denominator have two roots at
. Hence, again the conclusion
is two pole-zero cancellations at dc. The series for the conical cap
reflectance can be shown to be
![$\displaystyle R_J(s) = 1 - \frac{2 s}{3} + \frac{2 s^2}{9} - \frac{4 s^3}{135} - \frac{2 s^4}{405} + \cdots$](img4511.png) |
(C.191) |
which approaches
as
.
An alternative analysis of this issue is given by Benade in [37].
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