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---
title: "Fabry-Pérot cavity"
firstLetter: "F"
publishDate: 2021-09-18
categories:
- Physics
- Optics
date: 2021-09-18T00:42:59+02:00
draft: false
markup: pandoc
---
# Fabry-Pérot cavity
In its simplest form, a **Fabry-Pérot cavity**
is a region of light-transmitting medium
surrounded by two mirrors,
which may transmit some of the incoming light.
Such a setup can be used as e.g. an interferometer or a laser cavity.
## Modes of macroscopic cavity
Consider a Fabry-Pérot cavity large enough
that we can neglect the mirrors' thicknesses,
which have reflection coefficients $r_L$ and $r_R$.
Let $\tilde{n}_C$ be the complex refractive index inside,
and $\tilde{n}_L$ and $\tilde{n}_R$ be the indices outside.
The cavity has length $L$, centered on $x = 0$.
To find the quasinormal modes,
we make the following ansatz, with mode number $m$:
$$\begin{aligned}
E_m(x)
=
\begin{cases}
A_m \exp\!(-i \tilde{n}_L \tilde{k}_m x) & \mathrm{if}\; x < -L/2 \\
B_m \exp\!(i \tilde{n}_C \tilde{k}_m x) + C_m \exp\!(-i \tilde{n}_C \tilde{k}_m x) & \mathrm{if}\; -\!L/2 < x < L/2 \\
D_m \exp\!(i \tilde{n}_R \tilde{k}_m x) & \mathrm{if}\; L/2 < x
\end{cases}
\end{aligned}$$
On the left, $B_m$ is the reflection of $C_m$,
and on the right, $C_m$ is the reflection of $B_m$,
where the reflected amplitude is determined
by the coefficients $r_L$ and $r_L$, respectively:
$$\begin{aligned}
B_m \exp\!(-i \tilde{n}_C \tilde{k}_m L/2)
&= r_L C_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2)
\\
C_m \exp\!(-i \tilde{n}_C \tilde{k}_m L/2)
&= r_R B_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2)
\end{aligned}$$
These equations might seem to contradict each other.
We recast them into matrix form:
$$\begin{aligned}
\begin{bmatrix}
1 & - r_L \exp\!(i \tilde{n}_C \tilde{k}_m L) \\
- r_R \exp\!(i \tilde{n}_C \tilde{k}_m L) & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
B_m \\ C_m
\end{bmatrix}
=
\begin{bmatrix}
0 \\ 0
\end{bmatrix}
\end{aligned}$$
Now, we do not want to be able to find values for $B_m$ and $C_m$
that satisfy this for a given $\tilde{k}_m$.
Instead, we only want specific values of $\tilde{k}_m$ to be allowed,
corresponding to the cavity's modes.
We thus demand that the determinant to zero:
$$\begin{aligned}
0
&= 1 - r_L r_R \exp\!(i 2 \tilde{n}_C \tilde{k}_m L)
\end{aligned}$$
Isolating this for $\tilde{k}_m$ yields the following modes,
where $m$ is an arbitrary integer:
$$\begin{aligned}
\boxed{
\tilde{k}_m
= - \frac{\ln\!(r_L r_R) + i 2 \pi m}{i 2 \tilde{n}_C L}
}
\end{aligned}$$
These $\tilde{k}_m$ satisfy the matrix equation above.
Thanks to linearity, we can choose one of $B_m$ or $C_m$,
and then the other is determined by the corresponding equation.
Finally, we look at the light transmitted through the mirrors,
according to $1 \!-\! r_L$ and $1 \!-\! r_R$:
$$\begin{aligned}
A_m \exp\!(i \tilde{n}_L \tilde{k}_m L/2)
&= (1 - r_L) C_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2)
\\
D_m \exp\!(i \tilde{n}_R \tilde{k}_m L/2)
&= (1 - r_R) B_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2)
\end{aligned}$$
We simply isolate for $A_m$ and $D_m$ respectively,
yielding the following amplitudes:
$$\begin{aligned}
A_m
&= (1 - r_L) C_m \exp\!\big( i (\tilde{n}_C \!-\! \tilde{n}_L) \tilde{k}_m L/2 \big)
\\
D_m
&= (1 - r_R) B_m \exp\!\big( i (\tilde{n}_C \!-\! \tilde{n}_R) \tilde{k}_m L/2 \big)
\end{aligned}$$
Note that we have not demanded continuity of the electric field.
This is because the mirrors are infinitely thin "magic" planes;
if we had instead used a full physical mirror structure,
then the we would have demanded continuity,
as you might have expected.
## References
1. P.T. Kristensen, K. Herrmann, F. Intravaia, K. Busch,
[Modeling electromagnetic resonators using quasinormal modes](https://doi.org/10.1364/AOP.377940),
2020, Optical Society of America.
|
