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Diffstat (limited to 'source/know/concept/fabry-perot-cavity/index.md')
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1 files changed, 23 insertions, 23 deletions
diff --git a/source/know/concept/fabry-perot-cavity/index.md b/source/know/concept/fabry-perot-cavity/index.md index 3f47c3e..980fa54 100644 --- a/source/know/concept/fabry-perot-cavity/index.md +++ b/source/know/concept/fabry-perot-cavity/index.md @@ -15,9 +15,9 @@ which may transmit some of the incoming light. Such a setup can be used as e.g. an interferometer or a laser cavity. Below, we calculate its quasinormal modes in 1D. -We divide the $x$-axis into three domains: left $L$, center $C$, and right $R$. -The cavity $C$ has length $\ell$ and is centered on $x = 0$. -Let $n_L$, $n_C$ and $n_R$ be the respective domains' refractive indices: +We divide the $$x$$-axis into three domains: left $$L$$, center $$C$$, and right $$R$$. +The cavity $$C$$ has length $$\ell$$ and is centered on $$x = 0$$. +Let $$n_L$$, $$n_C$$ and $$n_R$$ be the respective domains' refractive indices: <a href="cavity.png"> <img src="cavity.png" style="width:70%"> @@ -28,8 +28,8 @@ Let $n_L$, $n_C$ and $n_R$ be the respective domains' refractive indices: In its simplest "microscopic" form, the reflection at the boundaries is simply caused by the index differences there. -Consider this ansatz for the [electric field](/know/concept/electric-field/) $E_m(x)$, -where $m$ is the mode: +Consider this ansatz for the [electric field](/know/concept/electric-field/) $$E_m(x)$$, +where $$m$$ is the mode: $$\begin{aligned} E_m(x) @@ -40,9 +40,9 @@ $$\begin{aligned} \end{cases} \end{aligned}$$ -The goal is to find the modes' wavenumbers $k_m$. -First, we demand that $E_m$ and its derivative $\idv{E_m}{x}$ -are continuous at the boundaries $x = \pm \ell/2$: +The goal is to find the modes' wavenumbers $$k_m$$. +First, we demand that $$E_m$$ and its derivative $$\idv{E_m}{x}$$ +are continuous at the boundaries $$x = \pm \ell/2$$: $$\begin{aligned} A_1 e^{i k_m n_L \ell/2} @@ -95,8 +95,8 @@ $$\begin{aligned} \end{aligned}$$ We want non-trivial solutions, where we -cannot simply satisfy the system by setting $A_1$, $A_2$, $A_3$ and -$A_4$; this constraint will give us an equation for $k_m$. Therefore, we +cannot simply satisfy the system by setting $$A_1$$, $$A_2$$, $$A_3$$ and +$$A_4$$; this constraint will give us an equation for $$k_m$$. Therefore, we demand that the system matrix is singular, i.e. its determinant is zero: $$\begin{aligned} @@ -106,7 +106,7 @@ $$\begin{aligned} &+ (n_C^2 + n_L n_R) \big( e^{i k_m (2 n_C - n_L - n_R) \ell/2} - e^{- i k_m (2 n_C + n_L + n_R) \ell/2} \big) \end{aligned}$$ -We multiply by $e^{i k_m (n_L + n_R) \ell / 2}$ and +We multiply by $$e^{i k_m (n_L + n_R) \ell / 2}$$ and decompose the exponentials into sines and cosines: $$\begin{aligned} @@ -125,8 +125,8 @@ $$\begin{aligned} \end{aligned}$$ Thanks to linearity, we can choose one of the amplitudes -$A_1$, $A_2$, $A_3$ or $A_4$ freely, -and then the others are determined by $k_m$ and the field's continuity. +$$A_1$$, $$A_2$$, $$A_3$$ or $$A_4$$ freely, +and then the others are determined by $$k_m$$ and the field's continuity. ## Macroscopic cavity @@ -134,7 +134,7 @@ and then the others are determined by $k_m$ and the field's continuity. Next, consider a "macroscopic" Fabry-Pérot cavity with complex mirror structures at boundaries, e.g. Bragg reflectors. If the cavity is large enough, we can neglect the mirrors' thicknesses, -and just use their reflection coefficients $r_L$ and $r_R$. +and just use their reflection coefficients $$r_L$$ and $$r_R$$. We use the same ansatz: $$\begin{aligned} @@ -147,10 +147,10 @@ $$\begin{aligned} \end{cases} \end{aligned}$$ -On the left, $A_3$ is the reflection of $A_2$, -and on the right, $A_2$ is the reflection of $A_3$, +On the left, $$A_3$$ is the reflection of $$A_2$$, +and on the right, $$A_2$$ is the reflection of $$A_3$$, where the reflected amplitudes are determined -by the coefficients $r_L$ and $r_R$, respectively: +by the coefficients $$r_L$$ and $$r_R$$, respectively: $$\begin{aligned} A_3 e^{- i k_m n_C \ell/2} @@ -185,8 +185,8 @@ $$\begin{aligned} &= 1 - r_L r_R e^{i 2 k_m n_C \ell} \end{aligned}$$ -Isolating this for $k_m$ yields the following modes, -where $m$ is an arbitrary integer: +Isolating this for $$k_m$$ yields the following modes, +where $$m$$ is an arbitrary integer: $$\begin{aligned} \boxed{ @@ -195,12 +195,12 @@ $$\begin{aligned} } \end{aligned}$$ -These $k_m$ satisfy the matrix equation above. -Thanks to linearity, we can choose one of $A_2$ or $A_3$, +These $$k_m$$ satisfy the matrix equation above. +Thanks to linearity, we can choose one of $$A_2$$ or $$A_3$$, and then the other is determined by the corresponding reflection equation. Finally, we look at the light transmitted through the mirrors, -according to $1 \!-\! r_L$ and $1 \!-\! r_R$: +according to $$1 \!-\! r_L$$ and $$1 \!-\! r_R$$: $$\begin{aligned} A_1 e^{i k_m n_L \ell/2} @@ -210,7 +210,7 @@ $$\begin{aligned} &= (1 - r_R) A_3 e^{i k_m n_C \ell/2} \end{aligned}$$ -We simply isolate for $A_1$ and $A_4$ respectively, +We simply isolate for $$A_1$$ and $$A_4$$ respectively, yielding the following amplitudes: $$\begin{aligned} |