---
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.