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. *Transmission line *b* exhibits a different characteristic impedance than *a*.

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

Here, the \(T_{0/d}^{+}\) are the **transmission coefficients** at *x = 0* and* x = d*, respectively.

\(k_{b}=k_{b}^{\prime}+\mathrm{i}k_{b}^{\prime\prime}\) 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. \(k_{b}^{\prime\prime}>0\).

## Hints

The transmission coefficient can be seen as **sum over all possible transmission scenarios**.

So, which reflection coefficients are needed and how are they related to \(\Gamma_{0}^{+}\)?

The **transmitted power** is given by \(P_{t}=\left|T\right|^{2}P_{i}\).

Ok, let's jump directly into the solution for the Fabry Perot line transmission coefficient!

## Solution

To find the **transmission coefficient** we have to first know understand the reflection inside of transmission line \(b\). After this we will calculate the transmission coefficient by a **summation** over all possible scenarios leading to an overall transmission. Finally, we will see that a maximum transmission is linked to the **Fabry-Perot resonance** condition.

## Calculating the needed Reflection Coefficients

First of all we have to answer how the **reflection coefficient** looks like if we reflect at some point \(x^{\prime}\) and not at the origin. A decomposition in forward and backwards propagating waves in a transmission line \(a\) is given by (see again Impedance Matching of Transmission Lines and Oscillator Circuits for more details)\[\begin{eqnarray*} U\left(x\right)&=&U_{a}^{+}\left(e^{\mathrm{i}k_{a}x}+\Gamma e^{-\mathrm{i}k_{a}x}\right)\ ,\\I\left(x\right)&=&\frac{U_{a}^{+}}{Z_{a}}\left(e^{\mathrm{i}k_{a}x}-\Gamma e^{-\mathrm{i}k_{a}x}\right) \end{eqnarray*}\]where we assume a reflection backwards. In another transmission line, say \(b\), we only consider outwards going waves and find by continuity:\[\begin{eqnarray*} \Gamma_{x^{\prime}}&=&\frac{Z_{b}-Z_{a}}{Z_{b}+Z_{a}}e^{\mathrm{i}2k_{a}x^{\prime}}=\Gamma_{0}^{+}e^{\mathrm{i}2k_{a}x^{\prime}}\ . \end{eqnarray*}\]Here, we may call \(\Gamma_{0}^{+}\) the reflection coefficient in positive direction at the origin. For the propagation in negative direction we have to make the same calculation. Keeping in mind that \(Z_{a}=-U_{a}^{-}/I_{a}^{-}\), we find that there is no sign change for the reflection forwards, \(\Gamma_{x^{\prime}}^{\pm}=\Gamma_{x^{\prime}}\). For our specific problem we need two reflection coefficients, both from medium \(b\) to \(a\):\[\begin{eqnarray*} \Gamma_{d}^{+}&=&\frac{Z_{a}-Z_{b}}{Z_{a}+Z_{b}}e^{\mathrm{i}2k_{b}d}=-\Gamma_{0}^{+}e^{\mathrm{i}2k_{b}d}\ .\\\Gamma_{0}^{-}&=&\frac{Z_{a}-Z_{b}}{Z_{a}+Z_{b}}=-\Gamma_{0}^{+}\ . \end{eqnarray*}\]Now we have all reflection coefficients needed to calculate the transmission coefficient.

## Transmitted Power

To derive the total transmission coefficient we have to consider **all possible situations**: a direct transmission, a transmission with two reflections at the inner boundaries and then transmission etc. Using the geometric sum we find\[\begin{eqnarray*} T&=&T_{0}^{+}T_{d}^{+}+T_{0}^{+}\Gamma_{d}^{+}\Gamma_{0}^{-}T_{d}^{+}+T_{0}^{+}\Gamma_{d}^{+}\Gamma_{0}^{-}\Gamma_{d}^{+}\Gamma_{0}^{-}T_{d}^{+}+\dots\\&=&T_{0}^{+}T_{d}^{+}\sum_{k=0}^{\infty}\left(\Gamma_{0}^{+}e^{\mathrm{i}k_{1}d}\right)^{2}\\&=&\frac{T_{0}^{+}T_{d}^{+}}{1-\left(\Gamma_{0}^{+} e^{\mathrm{i}k_{1}d}\right)^{2}}\ . \end{eqnarray*}\]The **transmitted power** is related to the squared amplitude of the transmission coefficient. Using a decomposition into real and imaginary part of the **wavenumber** \(k_{b}=k_{b}^{\prime}+k_{b}^{\prime\prime}\) and **reflection coefficient** \(\Gamma_{0}^{+}=\left|\Gamma_{0}^{+}\right|e^{\mathrm{i}\phi_{r}}\), we find \[\begin{eqnarray*} P_{t}&=&\left|T\right|^{2}P_{i}=\left|\frac{T_{0}^{+}T_{d}^{+}}{1-\left(\Gamma_{0}^{+}e^{\mathrm{i}k_{b}d}\right)^{2}}\right|^{2}P_{i}\\&=&\frac{\left|T_{0}^{+}T_{d}^{+}\right|^{2}}{1-2\left|\Gamma_{0}^{+}\right|^{2}e^{-2k_{b}^{\prime\prime}d}\cos\left(2k_{b}^{\prime}d+2\phi_{r}\right)+\left|\Gamma_{0}^{+}\right|^{4}e^{-4k_{b}^{\prime\prime}d}}P_{i}\ . \end{eqnarray*}\]

## Fabry-Perot Resonance Condition

We can now ask the question when the transmitted power will be maximal which is called a **resonance of the system**. For the **lossless** case with \(k_{b}^{\prime\prime}=0\) we find\[\begin{eqnarray*} P_{t}&=&\frac{\left|T_{0}^{+}T_{d}^{+}\right|^{2}}{1-2\left|\Gamma_{0}^{+}\right|^{2}\cos\left(2k_{b}^{\prime}d+2\phi_{r}\right)+\left|\Gamma_{0}^{+}\right|^{4}}P_{i}\ . \end{eqnarray*}\]It is obvious that the **denominator** depends on the length of the transmission line. So, to have a maximum output, we may ask when it is minimized. This is obviously the case if the phase inside if the cosine-term is one, hence\[\begin{eqnarray*} \cos\left(2k_{b}^{\prime}d+2\phi_{r}\right)&=&1\ ,\ \mathrm{hence}\\2k_{b}^{\prime}d+2\phi_{r}&=&2\pi n\ . \end{eqnarray*}\]This is the famous **Fabry-Perot resonance condition**. It states that at certain lengths of the intercepting transmission line we will find standing wave solutions possibly with very **strong fields** inside of this transmission line. This is very important to know since **nonlinear effects** might occur. Furthermore, because of the peaked structure of the transmission coefficient, such devices can be used to form extremely sensitive **interferometers**.

Also in this case we can see that \(\left|T_{0}^{+}T_{d}^{+}\right|=\left(1+\left|\Gamma_{0}^{+}\right|^{2}\right)e^{-k_{b}^{\prime\prime}d}\). The first term follows because on resonance, all of the power in the lossless case is transmitted whereas the **exponential damping** corresponds to the well-known **Lambert-Beer** behaviour. So, even though this argumentation is a bit handwaving it enabled us to follow the first law of studies in engineering and the natural sciences: Be lazy and think before you calculate!

Now to the **lossy** case. With respect to the monotoneous decay, the denominator still has to be minimized, so\[\begin{eqnarray*} 1-2\left|\Gamma_{0}^{+}\right|^{2}\cos\left(2k_{b}^{\prime}d+2\phi_{r}\right)e^{-2k_{b}^{\prime\prime}d}+\left|\Gamma_{0}^{+}\right|^{4}e^{-4k_{b}^{\prime\prime}d}&\overset{!}{=}&\mathrm{minimal}\ ,\ \mathrm{so}\\k_{b}^{\prime}\sin\left(2k_{b}^{\prime}d+2\phi_{r}\right)+k_{b}^{\prime\prime}\cos\left(2k_{b}^{\prime}d+2\phi_{r}\right)&=&k_{b}^{\prime\prime}\left|\Gamma_{0}^{+}\right|^{2}e^{-2k_{b}^{\prime\prime}d} \end{eqnarray*}\]using some mad highschool derivation skills.

We can see that if the imaginary part of the propagation constant, \(k^{\prime\prime}\), is much smaller than \(k^{\prime}\), the **Fabry-Perot** resonance condition will still **approximately** hold. If, however, damping is too big, also the resonance lengths will be affected. This is also illustrated in the following figure.

The squared **transmission coefficient** for a transmission line of length \(d\) and wavenumber \(k_{b}=k^{\prime}+\mathrm{i}k^{\prime\prime}\). If the medium is **lossless** (blue line), the transmission coefficient reaches maxima at unity. The maxima are equispaced every \(k^{\prime}\Delta d=\pi\), so every half wavelength since \(k=2\pi / \lambda\). With **losses** the situation changes (magenta and yellow lines). Here, \(k^{\prime\prime}/k^{\prime}=5\%\text{ and }10\%\) have been chosen, respectively. We can see that the **maxima** get not only **lower** but also slightly **shifted**. Note that because of a nonvanishing **phase on reflection** chosen here as \(\phi_{r}=-0.3\neq0\), the first resonances occur already after a very short distance.

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** that we have just derived! Only the wavenumber is that of a **plasmonic mode.** \(\phi_{r}\) 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.

You can see that Fabry-Perot resonances are still very much at the core of scientific activities!

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