Fabry Perot resonances in a transmission line.One of the most important concepts in optics are Fabry-Perot resonances of certain cavities. Nevertheless, we can understand this resonance phenomenon on the basis of wave propagation in transmission lines! You can also find out how this principle is used in nanophotonics.

Problem Statement

Fabry Perot resonances in a transmission line.A transmission line \(a\) is intercepted at \(x=0\) by another transmission line \(b\) of length \(d\) and with a different characteristic impedance. The voltage reflection coefficient at \(x=0\) shall be given by \(\Gamma_{0}^{+}=\left|\Gamma_{0}^{+}\right|e^{\mathrm{i}\phi_{r}}\). Show that the overall transmission coefficient for this configuration is given by\[\begin{eqnarray*}  T&=&\frac{T_{0}^{+}T_{d}^{+}}{1-\left(\Gamma_{0}^{+}e^{\mathrm{i}k_{b}d}\right)^{2}}\ .  \end{eqnarray*}\]Here, the \(T_{0/d}^{+}\) are the transmission coefficients at \(x=0/d\) and \(k_{b}=k_{b}^{\prime}+\mathrm{i}k_{b}^{\prime\prime}\) is the wavenumber of the transmission line \(b\). For a certain incoming power \(P_{i}\left(x=0\right)\), find the total transmitted power through the system at \(x=d\). Verify that if the intercepting line is lossless, the transmitted power is maximal if the Fabry-Perot resonance condition\[\begin{eqnarray*}  2k_{b}^{\prime}d+2\phi_{r}&=&2\pi n  \end{eqnarray*}\]holds. Derive a modified resonance condition if losses are present, i.e. \(k_{b}^{\prime\prime}>0\).


Background: Fabry-Perot Resonances in Nanoantennas

A circular nanoantenna with its resonant mode as described by a Fabry-Perot model. Illustration by Karsten Verch.Recent advances in fabrication techniques make it possible to fabricate antennas that can work in the near-infrared and optical frequency bands. Such antennas can be made of noble metals and are in the order of just a few hundred nanometers. The description of such nanoantennas, however, cannot be made as in the radio frequency domain - metals are not perfect conductors at such short wavelengths and nanoantennas have to be understood in terms of surface plasmon polaritons, collective electron oscillations coupled to light. Then, the scaling of such antennas can be calculated using the same Fabry-Perot resonance condition we will derive! Only the wavenumber is that of a plasmonic mode and \(\phi_{r}\) is the phase accumulated at its reflection at the termination of the antenna. This approach was used in “Circular Optical Nanoantennas - An Analytical Theory” to understand the characteristics of a kind of antennas that might play an important role in the taylored interaction of light with quantum systems as molecules or quantum dots.

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