Expand knowledge base

5 files changed, 714 insertions, 1 deletions

diff --git a/content/know/concept/electromagnetic-wave-equation/index.pdc b/content/know/concept/electromagnetic-wave-equation/index.pdc
index 59e0125..124bcc6 100644
--- a/content/know/concept/electromagnetic-wave-equation/index.pdc
+++ b/content/know/concept/electromagnetic-wave-equation/index.pdc
@@ -122,7 +122,7 @@ In fact, thanks to linearity, these **plane waves** can be treated as
 terms in a Fourier series, meaning that virtually
 *any* function $f(\vb{k} \cdot \vb{r} - \omega t)$ is a valid solution.
 
-Keep in mind that in reality, $\vb{E}$ and $\vb{B}$ are real,
+Keep in mind that in reality $\vb{E}$ and $\vb{B}$ are real,
 so although it is mathematically convenient to use plane waves,
 in the end you will need to take the real part.
 
diff --git a/content/know/concept/salt-equation/index.pdc b/content/know/concept/salt-equation/index.pdc
new file mode 100644
index 0000000..2f2917b
--- /dev/null
+++ b/content/know/concept/salt-equation/index.pdc
@@ -0,0 +1,286 @@
+---
+title: "SALT equation"
+firstLetter: "S"
+publishDate: 2022-02-07
+categories:
+- Physics
+- Optics
+
+date: 2022-01-20T22:01:48+01:00
+draft: false
+markup: pandoc
+---
+
+# SALT equation
+
+The **steady-state *ab initio* laser theory** (SALT) is
+a theoretical description of lasers, whose mode-centric approach
+makes it especially appropriate for microscopically small lasers.
+
+Consider the [Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/),
+governing the complex polarization
+vector $\vb{P}^{+}$ and the scalar population inversion $D$ of a set of
+active atoms (or quantum dots) embedded in a passive linear background
+material with refractive index $c / v$.
+The system is affected by a driving [electric field](/know/concept/electric-field/)
+$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$,
+such that the set of equations is:
+
+$$\begin{aligned}
+    - \mu_0 \pdv[2]{\vb{P}^{+}}{t}
+    &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{1}{v^2} \pdv[2]{\vb{E}^{+}}{t}
+    \\
+    \pdv{\vb{P}^{+}}{t}
+    &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
+    - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D
+    \\
+    \pdv{D}{t}
+    &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
+\end{aligned}$$
+
+Where $\hbar \omega_0$ is the band gap of the active atoms,
+and $\gamma_\perp$ and $\gamma_\parallel$ are relaxation rates
+of the atoms' polarization and population inversion, respectively.
+$D_0$ is the equilibrium inversion, i.e. the value of $D$ if there is no lasing.
+Note that $D_0$ also represents the pump,
+and both $D_0$ and $v$ depend on position $\vb{x}$.
+Finally, the transition dipole matrix elements $\vb{p}_0^{-}$ and $\vb{p}_0^{+}$ are given by:
+
+$$\begin{aligned}
+    \vb{p}_0^{-}
+    \equiv q \matrixel{e}{\vu{x}}{g}
+    \qquad \qquad
+    \vb{p}_0^{+}
+    \equiv q \matrixel{g}{\vu{x}}{e}
+    = (\vb{p}_0^{-})^*
+\end{aligned}$$
+
+With $q < 0$ the electron charge, $\vu{x}$ the quantum position operator,
+and $\ket{g}$ and $\ket{e}$ respectively
+the ground state and first excitation of the active atoms.
+
+We start by assuming that the cavity has $N$ quasinormal modes $\Psi_n$,
+each with a corresponding polarization $\vb{p}_n$ of the active matter.
+Note that this ansatz already suggests
+that the interactions between the modes are limited:
+
+$$\begin{aligned}
+    \vb{E}^{+}(\vb{x}, t)
+    = \sum_{n = 1}^N \Psi_n(\vb{x}) e^{- i \omega_n t}
+    \qquad \qquad
+    \vb{P}^{+}(\vb{x}, t)
+    = \sum_{n = 1}^N \vb{p}_n(\vb{x}) e^{- i \omega_n t}
+\end{aligned}$$
+
+Using the modes' linear independence to treat each term of the summation individually,
+the first two Maxwell-Bloch equations turn into, respectively:
+
+$$\begin{aligned}
+    \mu_0 \omega_n^2 \vb{p}_n
+    &= \nabla \cross \nabla \cross \Psi_n - \frac{1}{v^2} \omega_n^2 \Psi_n
+    \\
+    i \omega_n \vb{p}_n
+    &= \big( i \omega_0 + \gamma_\perp \big) \vb{p}_n
+    + \frac{i}{\hbar} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D
+\end{aligned}$$
+
+With being $\vb{p}_0^{+} \vb{p}_0^{-}$ a dyadic product.
+Isolating the latter equation for $\vb{p}_n$ gives us:
+
+$$\begin{aligned}
+    \vb{p}_n
+    &= \frac{\big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D}{\hbar \big((\omega_n - \omega_0) + i \gamma_\perp\big)}
+    = \frac{\gamma(\omega_n) D}{\hbar \gamma_\perp} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n
+\end{aligned}$$
+
+Where we have defined the Lorentzian gain curve $\gamma(\omega_n)$ as follows,
+which represents the laser's preferred frequencies for amplification:
+
+$$\begin{aligned}
+    \gamma(\omega_n)
+    \equiv \frac{\gamma_\perp}{(\omega_n - \omega_0) + i \gamma_\perp}
+\end{aligned}$$
+
+Inserting this expression for $\vb{p}_n$
+into the first Maxwell-Bloch equation yields
+the prototypical form of the SALT equation,
+where we still need to replace $D$ with known quantities:
+
+$$\begin{aligned}
+    0
+    &= \bigg( \nabla \cross \nabla \cross - \, \omega_n^2 \frac{1}{v^2}
+    - \omega_n^2 \frac{\mu_0 \gamma(\omega_n) D}{\hbar \gamma_\perp} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg) \Psi_n 
+\end{aligned}$$
+
+To rewrite $D$, we turn to its (Maxwell-Bloch) equation of motion,
+making the crucial **stationary inversion approximation** $\pdv*{D}{t} = 0$:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
+\end{aligned}$$
+
+This is the most aggressive approximation we will make:
+it removes all definite phase relations between modes,
+and effectively eliminates time as a variable.
+We insert our ansatz for $\vb{E}^{+}$ and $\vb{P}^{+}$,
+and point out that only excited lasing modes contribute to $D$:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \sum_{\nu, \mu}^\mathrm{active}
+    \bigg( \vb{p}_\nu^* \cdot \Psi_\mu e^{i (\omega_\nu - \omega_\mu) t}
+    - \vb{p}_\nu \cdot \Psi_\mu^* e^{i (\omega_\mu - \omega_\nu) t} \bigg)
+\end{aligned}$$
+
+Here, we make the [rotating wave approximation](/know/concept/rotating-wave-approximation/)
+to neglect all terms where $\nu \neq \mu$
+on the basis that they oscillate too quickly,
+leaving only $\nu = \mu$:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \sum_{\nu}^\mathrm{act.}
+    \bigg( \vb{p}_\nu^* \cdot \Psi_\nu - \vb{p}_\nu \cdot \Psi_\nu^* \bigg)
+\end{aligned}$$
+
+Inserting our earlier equation for $\vb{p}_n$
+and using the fact that $\vb{p}_0^{+} = (\vb{p}_0^{-})^*$ leads us to:
+
+$$\begin{aligned}
+    D
+    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \bigg( \gamma^*(\omega_\nu) \big(\vb{p}_0^{+} \vb{p}_0^{-}\big)^* \!\cdot\! \Psi_\nu^* \cdot \Psi_\nu
+    - \gamma(\omega_\nu) \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \!\cdot\! \Psi_\nu \cdot \Psi_\nu^* \bigg)
+    \\
+    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \bigg( \gamma^*(\omega_\nu) \big(\vb{p}_0^{+} \cdot \Psi_\nu^*\big) \vb{p}_0^{-} \cdot \Psi_\nu
+    - \gamma(\omega_\nu) \big(\vb{p}_0^{-} \cdot \Psi_\nu\big) \vb{p}_0^{+} \cdot \Psi_\nu^* \bigg)
+    \\
+    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \Big( \gamma^*(\omega_\nu) - \gamma(\omega_\nu) \Big) \big|\vb{p}_0^{-} \cdot \Psi_\nu\big|^2
+\end{aligned}$$
+
+By putting the terms on a common denominator, it is easily shown that:
+
+$$\begin{aligned}
+    \gamma^*(\omega_\nu) - \gamma(\omega_\nu)
+    &= \frac{\gamma_\perp ((\omega_\nu - \omega_0) + i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
+    - \frac{\gamma_\perp ((\omega_\nu - \omega_0) - i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
+    \\
+    &= \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
+    = i 2 \big|\gamma(\omega_\nu)\big|^2
+\end{aligned}$$
+
+Inserting this into our equation for $D$ gives the following expression:
+
+$$\begin{aligned}
+    D
+    &= D_0 - \frac{4 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2
+\end{aligned}$$
+
+We then properly isolate this for $D$ to get its final form, namely:
+
+$$\begin{aligned}
+    D
+    &= D_0 \bigg( 1 + \frac{4}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+    \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2 \bigg)^{-1}
+\end{aligned}$$
+
+Substituting this into the prototypical SALT equation from earlier
+yields the most general form of the **SALT equation**,
+upon which the theory is built:
+
+$$\begin{aligned}
+    \boxed{
+        0
+        = \bigg( \nabla \cross \nabla \cross
+        -\,\omega_n^2 \bigg[ \frac{1}{v^2(\vb{x})} + \frac{\mu_0 \gamma(\omega_n)}{\hbar \gamma_\perp}
+        \frac{D_0(\vb{x})}{1 + h(\vb{x})} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg] \bigg) \Psi_n(\vb{x})
+    }
+\end{aligned}$$
+
+Where we have defined **spatial hole burning** function $h(\vb{x})$ like so,
+representing the depletion of the supply of charge
+carriers as they are consumed by the active lasing modes:
+
+$$\begin{aligned}
+    \boxed{
+        h(\vb{x})
+        \equiv \frac{4}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
+        \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu(\vb{x})\Big|^2
+    }
+\end{aligned}$$
+
+Many authors assume that $\vb{p}_0^- \parallel \Psi_n$,
+so that only its amplitude $|g|^2 \equiv \vb{p}_0^{+} \cdot \vb{p}_0^{-}$ matters.
+In that case, they often non-dimensionalize $D$ and $\Psi_n$
+by dividing out the units $d_c$ and $e_c$:
+
+$$\begin{aligned}
+    \tilde{\Psi}_n
+    \equiv \frac{\Psi_n}{e_c}
+    \qquad
+    e_c
+    \equiv \frac{\hbar \sqrt{\gamma_\parallel \gamma_\perp}}{2 |g|}
+    \qquad \qquad
+    \tilde{D}
+    \equiv \frac{D}{d_c}
+    \qquad
+    d_c
+    \equiv \frac{\varepsilon_0 \hbar \gamma_\perp}{|g|^2}
+\end{aligned}$$
+
+And then the SALT equation and hole burning function $h$ are reduced to the following,
+where the vacuum wavenumber $k_n = \omega_n / c$:
+
+$$\begin{aligned}
+    0
+    = \bigg( \nabla \cross \nabla \cross -\,k_n^2 \bigg[ \varepsilon_r
+    + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
+    \qquad
+    h(\vb{x})
+    = \sum_{\nu}^\mathrm{act.} \Big|\gamma(c k_\nu) \tilde{\Psi}_\nu(\vb{x})\Big|^2
+\end{aligned}$$
+
+
+In addition, some papers only consider 1D or 2D *transverse magnetic* (TM) modes,
+in which case the fields are scalars. Using the vector identity
+
+$$\begin{aligned}
+    \nabla \cross \nabla \cross \Psi
+    = \nabla (\nabla \cdot \Psi) - \nabla^2 \Psi
+\end{aligned}$$
+
+Where $\nabla \cdot \Psi = 0$ thanks to [Gauss' law](/know/concept/maxwells-equations/),
+so we get an even further simplified SALT equation:
+
+$$\begin{aligned}
+    0
+    = \bigg( \nabla^2 +\,k_n^2 \bigg[ \varepsilon_r
+    + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
+\end{aligned}$$
+
+The challenge is to solve this equation for a given $\varepsilon_r(\vb{x})$ and $D_0(\vb{x})$,
+with the boundary condition that $\Psi_n$ is a plane wave at infinity,
+i.e. that there is light leaving the cavity.
+
+If $k_n$ has a negative imaginary part, then that mode is behaving as an LED.
+Gradually increasing the pump $D_0$ in a chosen region
+causes the $k_n$'s imaginary parts become less negative,
+until one of them hits the real axis, at which point that mode starts lasing.
+After that, $D_0$ can be increased even further until some other $k_n$ become real.
+
+Below threshold (i.e. before any mode is lasing), the problem is linear in $\Psi_n$,
+but above threshold it is nonlinear, and the amplitude of $\Psi_n$ is adjusted
+such that the corresponding $k_n$ never leaves the real axis.
+When any mode is lasing, hole burning makes it harder for other modes to activate,
+since it effectively reduces the pump $D_0$.
+
+
+## References
+1.  L. Ge, Y.D. Chong, A.D. Stone,
+    [Steady-state *ab initio* laser theory: generalizations and analytic results](http://dx.doi.org/10.1103/PhysRevA.82.063824),
+    2010, American Physical Society.
+
diff --git a/content/know/concept/step-index-fiber/bessel.jpg b/content/know/concept/step-index-fiber/bessel.jpg
new file mode 100644
index 0000000..464a1e7
--- /dev/null
+++ b/content/know/concept/step-index-fiber/bessel.jpg
diff --git a/content/know/concept/step-index-fiber/index.pdc b/content/know/concept/step-index-fiber/index.pdc
new file mode 100644
index 0000000..8847fff
--- /dev/null
+++ b/content/know/concept/step-index-fiber/index.pdc
@@ -0,0 +1,427 @@
+---
+title: "Step-index fiber"
+firstLetter: "S"
+publishDate: 2022-02-11
+categories:
+- Physics
+- Optics
+- Fiber optics
+
+date: 2022-01-31T19:29:33+01:00
+draft: false
+markup: pandoc
+---
+
+# Step-index fiber
+
+As light propagates in the $z$-direction through an optical fiber,
+the transverse profile $F(x,y)$ of the [electric field](/know/concept/electric-field/)
+can be shown to obey the *Helmholtz equation* in 2D:
+
+$$\begin{aligned}
+    \nabla_{\!\perp}^2 F + (n^2 k^2 - \beta^2) F = 0
+\end{aligned}$$
+
+With $n$ being the position-dependent refractive index,
+$k$ the vacuum wavenumber $\omega / c$,
+and $\beta$ the mode's propagation constant, to be determined later.
+In [polar coordinates](/know/concept/cylindrical-polar-coordinates/)
+$(r,\phi)$ this equation can be rewritten as follows:
+
+$$\begin{aligned}
+    \pdv[2]{F}{r} + \frac{1}{r} \pdv{F}{r} + \frac{1}{r^2} \pdv[2]{F}{\phi} + \mu F = 0
+\end{aligned}$$
+
+Where we have defined $\mu \equiv n^2 k^2 \!-\! \beta^2$ for brevity.
+From now on, we only consider choices of $\mu$ that do not depend on $\phi$ or $z$,
+but may vary with $r$.
+
+This Helmholtz equation can be solved by *separation of variables*:
+we assume that there exist two functions $R(r)$ and $\Phi(\phi)$
+such that $F(r,\phi) = R(r) \, \Phi(\phi)$.
+Inserting this ansatz:
+
+$$\begin{aligned}
+    R'' \Phi + \frac{1}{r} R' \Phi + \frac{1}{r^2} R \Phi'' + \mu R \Phi = 0
+\end{aligned}$$
+
+We rearrange this such that each side only depends on one variable,
+by dividing by $R\Phi$ (ignoring the fact that it may be zero),
+and multiplying by $r^2$.
+Since this equation should hold for *all* values of $r$ and $\phi$,
+this means that both sides must equal a constant $\ell^2$:
+
+$$\begin{aligned}
+    r^2 \frac{R''}{R} + r \frac{R'}{R} + \mu r^2
+    = -\frac{\Phi''}{\Phi}
+    = \ell^2
+\end{aligned}$$
+
+This gives an eigenvalue problem for $\Phi$,
+and the well-known *Bessel equation* for $R$:
+
+$$\begin{aligned}
+    \boxed{
+        \Phi'' + \ell^2 \Phi = 0
+    }
+    \qquad \qquad
+    \boxed{
+        r^2 R'' + r R' + (\mu r^2 \!-\! \ell^2) R = 0
+    }
+\end{aligned}$$
+
+We will return to $R$ later; we start with $\Phi$, because it has the
+simplest equation. Since the angle $\phi$ is limited to $[0,2\pi]$,
+$\Phi$ must be $2 \pi$-periodic, so:
+
+$$\begin{aligned}
+    \Phi(0) = \Phi(2\pi)
+    \qquad \qquad
+    \Phi'(0) = \Phi'(2\pi)
+\end{aligned}$$
+
+The above equation for $\Phi$ with these periodic boundary conditions
+is a [Sturm-Liouville problem](/know/concept/sturm-liouville-theory/).
+Consequently, there are infinitely many allowed values of $\ell^2$,
+all real, and one of them is lowest, known as the *ground state*.
+
+To find the eigenvalues $\ell^2$ and their corresponding $\Phi$,
+we in turn assume that $\ell^2 < 0$, $\ell^2 = 0$, or $\ell^2 > 0$,
+and check if we can then arrive at a non-trivial $\Phi$ for each case.
+
+* For $\ell^2 < 0$, solutions have the form $\Phi(\phi) = A \sinh\!(\phi \ell) + B \cosh\!(\phi \ell)$,
+    where $A$ and $B$ are unknown linearity constants.
+    At least one of these constants must be nonzero for $\Phi$ to be non-trivial,
+    but the challenge is to satisfy the boundary conditions:
+
+    $$\begin{alignedat}{3}
+        \Phi(0) &= \Phi(2 \pi)
+        \:\quad &&\implies \quad\:\:
+        0 &&= A \sinh\!(2 \pi \ell) + B \big( \cosh\!(2 \pi \ell) - 1 \big)
+        \\
+        \Phi'(0) &= \Phi'(2 \pi)
+        \: \quad &&\implies \quad \:\:
+        0 &&= A \ell \big( \cosh\!(2 \pi \ell) - 1 \big) + B \ell \sinh\!(2 \pi \ell)
+    \end{alignedat}$$
+
+    This only has non-trivial solutions
+    if the determinant of the system matrix is zero:
+
+    $$\begin{aligned}
+        0
+        &= \mathrm{det}
+        \begin{bmatrix}
+            \sinh\!(2 \pi \ell) & \cosh\!(2 \pi \ell) - 1 \\
+            \cosh\!(2 \pi \ell) - 1 & \sinh\!(2 \pi \ell)
+        \end{bmatrix}
+        = 2 \big( \cosh\!(2 \pi \ell) - 1 \big)
+    \end{aligned}$$
+
+    This can only be zero if $\ell = 0$,
+    which contradicts the premise that $\ell^2 < 0$,
+    so we conclude that $\ell^2$ cannot be negative,
+    because no non-trivial solutions exist here.
+
+* For $\ell^2 = 0$, the solution is $\Phi(\phi) = A \phi + B$.
+    Putting this in the boundary conditions:
+
+    $$\begin{alignedat}{3}
+        \Phi(0) &= \Phi(2 \pi)
+        \qquad &&\implies \qquad
+        A &&= 0
+        \\
+        \Phi'(0) &= \Phi'(2 \pi)
+        \qquad &&\implies \qquad
+        B &&= B
+    \end{alignedat}$$
+
+    $B$ can be nonzero, so this a valid solution.
+    We conclude that $\ell^2 = 0$ is the ground state.
+
+* For $\ell^2 > 0$, all solutions have the form
+    $\Phi(\phi) = A \sin\!(\phi \ell) + B \cos\!(\phi \ell)$, therefore:
+
+    $$\begin{alignedat}{3}
+        \Phi(0) &= \Phi(2 \pi)
+        \quad &&\implies \quad
+        0 &&= A \sin\!(2 \pi \ell) + B \big(\cos\!(2\pi \ell) - 1\big)
+        \\
+        \Phi'(0) &= \Phi'(2 \pi)
+        \quad &&\implies \quad
+        0 &&= A \big(\cos\!(2 \pi \ell) - 1\big) - B \sin\!(2 \pi \ell)
+    \end{alignedat}$$
+
+    This system only has nontrivial solutions
+    if the determinant of its matrix is zero:
+
+    $$\begin{aligned}
+        0
+        &= \mathrm{det}
+        \begin{bmatrix}
+            \sin\!(2 \pi \ell) & \cos\!(2 \pi \ell) - 1 \\
+            \cos\!(2 \pi \ell) - 1 & -\sin\!(2 \pi \ell)
+        \end{bmatrix}
+        = 2 \big(\cos\!(2 \pi \ell) - 1\big)
+    \end{aligned}$$
+
+    Meaning that $\ell$ must be an integer.
+    We revisit the boundary conditions and indeed see:
+
+    $$\begin{alignedat}{3}
+        0 &= A \sin\!(2 \pi \ell) + B \big(\cos\!(2 \pi \ell) - 1\big)
+        \qquad &&\implies \qquad
+        0 &&= 0
+        \\
+        0 &= A \big(\cos\!(2 \pi \ell) - 1\big) - B \sin\!(2 \pi \ell)
+        \qquad &&\implies \qquad
+        0 &&= 0
+    \end{alignedat}$$
+
+    So $A$ and $B$ are *both* unconstrained,
+    and each integer $\ell$ is a doubly-degenerate eigenvalue.
+    The two linearly independent solutions,
+    $\sin\!(\phi \ell)$ and $\cos\!(\phi \ell)$,
+    represent the polarization of light in the mode.
+    For simplicity, we assume that all light is in a single polarization,
+    so only $\cos\!(\phi \ell)$ will be considered from now on.
+
+By combining our result for $\ell^2 = 0$ and $\ell^2 > 0$,
+we get the following for $\ell = 0, 1, 2, ...$:
+
+$$\begin{aligned}
+    \boxed{
+        \Phi_\ell(\phi) = A \cos(\phi \ell)
+    }
+\end{aligned}$$
+
+Here, $\ell$ is called the **primary mode index**.
+We exclude $\ell < 0$ because $\cos\!(x) \propto \cos\!(-x)$
+and $\sin\!(x) \propto \sin\!(-x)$,
+and because $A$ is free to choose thanks to linearity.
+
+Let us now revisit the Bessel equation for the radial function $R(r)$,
+which should be continuous and differentiable throughout the fiber:
+
+$$\begin{aligned}
+    r^2 R'' + r R' + \mu r^2 R - \ell^2 R = 0
+\end{aligned}$$
+
+To continue, we need to specify the refractive index $n(r)$, contained in $\mu(r)$.
+We choose a **step-index fiber**,
+whose cross-section consists of a **core** with radius $a$,
+surrounded by a **cladding** that extends to infinity $r \to \infty$.
+In the core $r < a$, the index $n$ is a constant $n_i$,
+while in the cladding $r > a$ it is another constant $n_o$.
+
+Since $\mu$ is different in the core and cladding,
+we will get different solutions $R_i$ and $R_o$ there,
+so we must demand that the field is continuous at the boundary $r = a$:
+
+$$\begin{aligned}
+    R_i(a) = R_o(a)
+    \qquad \qquad
+    R_i'(a) = R_o'(a)
+\end{aligned}$$
+
+Furthermore, for a physically plausible solution,
+we require that $R_i$ is finite
+and that $R_o$ decays monotonically to zero when $r \to \infty$.
+These constraints will turn out to restrict $\mu$.
+
+Introducing a new coordinate $\rho \equiv r \sqrt{|\mu|}$
+gives the Bessel equation's standard form,
+which has well-known solutions called *Bessel functions*, shown below.
+Let $\pm$ be the sign of $\mu$:
+
+$$\begin{aligned}
+    \begin{cases}
+        \displaystyle
+        0 = \rho^2 \pdv[2]{R}{\rho} + \rho \pdv{R}{\rho} \pm \rho^2 R - \ell^2 R
+        & \mathrm{for}\; \mu \neq 0
+        \\
+        \displaystyle
+        0 = r^2 \pdv[2]{R}{r} + r \pdv{R}{r} - \ell^2 R
+        & \mathrm{for}\; \mu = 0
+    \end{cases}
+\end{aligned}$$
+
+<a href="bessel.jpg">
+<img src="bessel.jpg" style="width:90%;display:block;margin:auto;">
+</a>
+
+Looking at these solutions with our constraints for $R_o$ in mind,
+we see that for $\mu > 0$ none of the solutions decay
+*monotonically* to zero, so we must have $\mu \le 0$ in the cladding.
+Of the remaining candidates, $\ln\!(r)$, $r^\ell$ and $I_\ell(\rho)$ do not decay at all,
+leading to the following $R_o$:
+
+$$\begin{aligned}
+    R_{o,\ell}(r) =
+    \begin{cases}
+        r^{-\ell}
+        & \mathrm{for}\; \mu = 0 \;\mathrm{and}\; \ell = 1,2,3,...
+        \\
+        K_\ell(\rho) = K_\ell(r \sqrt{-\mu})
+        & \mathrm{for}\; \mu < 0 \;\mathrm{and}\; \ell = 0,1,2,...
+    \end{cases}
+\end{aligned}$$
+
+Next, for $R_i$, we see that when $\mu < 0$ all solutions are invalid
+since they diverge at $r = 0$,
+and so do $\ln\!(r)$, $r^{-\ell}$ and $Y_\ell(\rho)$.
+Of the remaining candidates, $r^0$ and $r^\ell$ have a non-negative slope
+at the boundary $r = a$, so they can never be continuous with $R_o'$.
+This leaves $J_\ell(\rho)$ for $\mu > 0$:
+
+$$\begin{aligned}
+    R_{i,\ell}(r) =
+    J_\ell(\rho) = J_\ell(r \sqrt{\mu})
+    \qquad \mathrm{for}\; \mu > 0 \;\mathrm{and}\; \ell = 0,1,2,...
+\end{aligned}$$
+
+Putting this all together, we now know what the full solution for $F$ should look like:
+
+$$\begin{aligned}
+    F_\ell(r, \phi)
+    = R_\ell(r) \, \Phi_\ell(\phi)
+    =
+    \begin{cases}
+        A_\ell \: R_{i,\ell}(r) \, \cos\!(\phi \ell)
+        & \mathrm{for}\; r \le a
+        \\
+        B_\ell \: R_{o,\ell}(r) \, \cos\!(\phi l)
+        & \mathrm{for}\; r \ge a
+    \end{cases}
+\end{aligned}$$
+
+Where $A_\ell$ and $B_\ell$ are constants to be chosen
+based on the light's intensity, and to satisfy the continuity condition at $r = a$.
+
+We found that $\mu \le 0$ in the cladding and $\mu > 0$ in the core.
+Since $\mu \equiv n^2 k^2 \!-\! \beta^2$ by definition,
+this discovery places a constraint on the propagation constant $\beta$:
+
+$$\begin{aligned}
+    n_i^2 k^2 > \beta^2 \ge n_o^2 k^2
+\end{aligned}$$
+
+Therefore, $n_i > n_o$ in a step-index fiber,
+and there is only a limited range of allowed $\beta$-values;
+the fiber is not able to guide the light outside this range.
+
+However, not all $\beta$ in this range are created equal for all $k$.
+To investigate further, let us define the quantities
+$\xi_\mathrm{core}$ and $\xi_\mathrm{clad}$ like so,
+assuming $n_i$ and $n_o$ do not depend on $k$:
+
+$$\begin{aligned}
+    \xi_i(k)
+    \equiv \sqrt{ n_i^2 k^2 - \beta^2(k) }
+    \qquad \qquad
+    \xi_o(k)
+    \equiv \sqrt{ \beta^2(k) - n_o^2 k^2 }
+\end{aligned}$$
+
+It is important to note that the sum of their squares is constant with respect to $\beta$:
+
+$$\begin{aligned}
+    \xi_i^2 + \xi_o^2 = (\mathrm{NA})^2 k^2
+\end{aligned}$$
+
+Where $\mathrm{NA}$ is the so-called **numerical aperture**,
+often mentioned in papers and datasheets as one of a fiber's key parameters.
+It is defined as:
+
+$$\begin{aligned}
+    \boxed{
+        \mathrm{NA}
+        \equiv \sqrt{n_i^2 - n_o^2}
+    }
+\end{aligned}$$
+
+From this, we define a new fiber parameter: the $V$-**number**,
+which is extremely useful:
+
+$$\begin{aligned}
+    \boxed{
+        V
+        \equiv a \sqrt{\xi_i^2 + \xi_o^2}
+        = a k \: \mathrm{NA}
+    }
+\end{aligned}$$
+
+Now, the allowed values of $\beta$ are found
+by fulfilling the boundary conditions (for $\mu \neq 0$):
+
+$$\begin{aligned}
+    A_\ell J_\ell(a \xi_i)
+    &= B_\ell K_\ell(a \xi_o)
+    \\
+    A_\ell \xi_i J_\ell'(a \xi_i)
+    &= B_\ell \xi_o K_\ell'(a \xi_o)
+\end{aligned}$$
+
+To remove $A_\ell$ and $B_\ell$,
+we divide the latter equation by the former,
+meanwhile defining $X \equiv a \xi_i$ and $Y \equiv a \xi_o$
+for convenience, such that $X^2 + Y^2 = V^2$:
+
+$$\begin{aligned}
+    X \frac{J_\ell'(X)}{J_\ell(X)} = Y \frac{K_\ell'(Y)}{K_\ell(Y)}
+\end{aligned}$$
+
+We can turn this result into something a bit nicer
+by using the following identities:
+
+$$\begin{aligned}
+    J_\ell'(x) = -J_{\ell+1}(x) + \ell \frac{J_\ell(x)}{x}
+    \qquad \quad
+    K_\ell'(x) = -K_{\ell+1}(x) + \ell \frac{K_\ell(x)}{x}
+\end{aligned}$$
+
+With this, the transcendental equation for $\beta$
+takes this convenient form:
+
+$$\begin{aligned}
+    \boxed{
+        X \frac{J_{\ell+1}(X)}{J_\ell(X)} = Y \frac{K_{\ell+1}(Y)}{K_\ell(Y)}
+    }
+\end{aligned}$$
+
+All $\beta$ that satisfy this indicate the existence
+of a **linearly polarized** mode.
+These modes are called $\mathrm{LP}_{\ell m}$,
+where $\ell$ is the primary (azimuthal) mode index,
+and $m$ the secondary (radial) mode index,
+which is needed because multiple $\beta$ may exist for a single $\ell$.
+
+An example graphical solution of the transcendental equation
+is illustrated below for a fiber with $V = 5$,
+where red and blue respectively denote the left and right-hand side:
+
+<a href="modes.jpg">
+<img src="modes.jpg" style="width:90%;display:block;margin:auto;">
+</a>
+
+This shows that each $\mathrm{LP}_{\ell m}$ has an associated cut-off $V_{\ell m}$,
+so that if $V > V_{\ell m}$ then $\mathrm{LP}_{lm}$ exists,
+as long as $\beta$ stays in the allowed range.
+The cut-offs of the secondary modes for a given $\ell$
+are found as the $m$th roots of $J_{\ell-1}(V_{\ell m}) = 0$.
+In the above figure, they are $V_{01} = 0$, $V_{11} = 2.405$, and $V_{02} = V_{21} = 3.832$.
+
+All differential equations have been linear,
+so a linear combination of these solutions is also valid.
+Therefore, the fiber modes represent independent "channels" of light.
+However, in practice, they can interact nonlinearly,
+and light can scatter between them, and between polarizations.
+
+
+
+## References
+1.  O. Bang,
+    *Applied mathematics for physicists: lecture notes*, 2019,
+    unpublished.
+2.  B.E.A. Saleh, M.C. Teich,
+    *Fundamentals of photonics*, 1st edition, 1991,
+    Wiley.
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