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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:
+
+$$\begin{aligned}
+    A_m
+    &= (1 - r_L) C_m \exp\!\big( i (\tilde{n}_C \!-\! \tilde{n}_L) \tilde{k}_m L/2 \big)
+    \\
+    D_m
+    &= (1 - r_R) B_m \exp\!\big( i (\tilde{n}_C \!-\! \tilde{n}_R) \tilde{k}_m L/2 \big)
+\end{aligned}$$
+
+
+
+## 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.