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

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. \(k_{b}^{\prime\prime}>0\).

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Background: Fabry-Perot Resonances in Nanoantennas

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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