Reflectance Magnitude

We have shown that the conical cap reflectance has no poles in the strict right-half plane. For passivity, we also need to show that its magnitude is bounded by unity for all $s$ on the $j\omega$ axis.

We have

$$R_J(j\omega) = \frac{1 - e^{-2j\omega} - 2j\omega e^{-2j\omega}}{2j\omega - 1 + e^{-2j\omega}} = e^{-2j\omega} \frac{e^{2j\omega} - 1 - 2j\omega }{e^{-2j\omega} - 1 + 2j\omega} = e^{-2j\omega} \frac{\overline{D(j\omega)}}{D(j\omega)}$$

so that $\left\vert R_J(j\omega)\right\vert = 1$ , which is exactly lossless, as expected.