Expand knowledge base with material from BSc thesis

18 files changed, 789 insertions, 1 deletions

diff --git a/content/know/category/fiber-optics.md b/content/know/category/fiber-optics.md
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+---
+title: "Fiber optics"
+firstLetter: "F"
+date: 2021-02-26T15:13:26+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/category/nonlinear-dynamics.md b/content/know/category/nonlinear-dynamics.md
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+---
+title: "Nonlinear dynamics"
+firstLetter: "N"
+date: 2021-02-26T20:30:17+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/category/perturbation.md b/content/know/category/perturbation.md
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+---
+title: "Perturbation"
+firstLetter: "P"
+date: 2021-02-26T20:37:33+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/concept/dispersive-broadening/index.pdc b/content/know/concept/dispersive-broadening/index.pdc
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+---
+title: "Dispersive broadening"
+firstLetter: "D"
+publishDate: 2021-02-27
+categories:
+- Physics
+- Optics
+- Fiber optics
+
+date: 2021-02-27T11:48:34+01:00
+draft: false
+markup: pandoc
+---
+
+# Dispersive broadening
+
+In optical fibers, **dispersive broadening** is a (linear) effect
+where group velocity dispersion (GVD) "smears out" a pulse in the time domain
+due to the different group velocities of its frequencies,
+since pulses always have a non-zero width in the $\omega$-domain.
+No new frequencies are created.
+
+A pulse envelope $A(z, t)$ inside a fiber must obey the nonlinear Schrödinger equation,
+where the parameters $\beta_2$ and $\gamma$ respectively
+control dispersion and nonlinearity:
+
+$$\begin{aligned}
+    0
+    = i \pdv{A}{z} - \frac{\beta_2}{2} \pdv[2]{A}{t} + \gamma |A|^2 A
+\end{aligned}$$
+
+We set $\gamma = 0$ to ignore all nonlinear effects,
+and consider a Gaussian initial condition:
+
+$$\begin{aligned}
+    A(0, t)
+    = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
+\end{aligned}$$
+
+By [Fourier transforming](/know/concept/fourier-transform/) in $t$,
+the full analytical solution $A(z, t)$ is found to be as follows,
+where it can be seen that the amplitude
+decreases and the width increases with $z$:
+
+$$\begin{aligned}
+    A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
+    \exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg)
+\end{aligned}$$
+
+To quantify the strength of dispersive effects,
+we define the dispersion length $L_D$
+as the distance over which the half-width at $1/e$ of maximum power
+(initially $T_0$) increases by a factor of $\sqrt{2}$:
+
+$$\begin{aligned}
+    T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2}
+    \qquad \implies \qquad
+    \boxed{
+        L_D = \frac{T_0^2}{|\beta_2|}
+    }
+\end{aligned}$$
+
+This phenomenon is illustrated below for our example of a Gaussian pulse
+with parameter values $T_0 = 1\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
+$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0$:
+
+<img src="pheno-disp.jpg">
+
+The **instantaneous frequency** $\omega_\mathrm{GVD}(z, t)$,
+which describes the dominant angular frequency at a given point in the time domain,
+is found to be as follows for the Gaussian pulse,
+where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$:
+
+$$\begin{aligned}
+    \omega_{\mathrm{GVD}}(z,t)
+    = \pdv{t} \Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big)
+    = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
+\end{aligned}$$
+
+This expression is linear in time, and depending on the sign of $\beta_2$,
+frequencies on one side of the pulse arrive first,
+and those on the other side arrive last.
+The effect is stronger for smaller $T_0$:
+this makes sense, since short pulses are spectrally wider.
+
+The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/)
+leads to many interesting effects,
+such as [modulational instability](/know/concept/modulational-instability/)
+and [optical wave breaking](/know/concept/optical-wave-breaking/).
+Of great importance is the sign of $\beta_2$:
+in the **anomalous dispersion regime** ($\beta_2 < 0$),
+lower frequencies travel more slowly than higher ones,
+and vice versa in the **normal dispersion regime** ($\beta_2 > 0$).
diff --git a/content/know/concept/dispersive-broadening/pheno-disp.jpg b/content/know/concept/dispersive-broadening/pheno-disp.jpg
new file mode 100644
index 0000000..a97312b
--- /dev/null
+++ b/content/know/concept/dispersive-broadening/pheno-disp.jpg
diff --git a/content/know/concept/fourier-transform/index.pdc b/content/know/concept/fourier-transform/index.pdc
index 6d8901a..96653f5 100644
--- a/content/know/concept/fourier-transform/index.pdc
+++ b/content/know/concept/fourier-transform/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-02-22
 categories:
 - Mathematics
 - Physics
+- Optics
 
 date: 2021-02-22T21:35:54+01:00
 draft: false
diff --git a/content/know/concept/modulational-instability/index.pdc b/content/know/concept/modulational-instability/index.pdc
new file mode 100644
index 0000000..d912c04
--- /dev/null
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@@ -0,0 +1,197 @@
+---
+title: "Modulational instability"
+firstLetter: "M"
+publishDate: 2021-02-26
+categories:
+- Physics
+- Fiber optics
+- Optics
+- Perturbation
+- Nonlinear dynamics
+
+date: 2021-02-26T20:36:22+01:00
+draft: false
+markup: pandoc
+---
+
+# Modulational instability
+
+In fiber optics, **modulational instability** (MI)
+is a nonlinear effect that leads to the exponential amplification
+of background noise in certain frequency regions.
+It only occurs in the [anomalous dispersion regime](/know/concept/dispersive-broadening/)
+($\beta_2 < 0$), which we will prove shortly.
+
+Consider the following simple solution to the nonlinear Schrödinger equation:
+a time-invariant constant power $P_0$ at the carrier frequency $\omega_0$,
+which is experiencing [self-phase modulation](/know/concept/self-phase-modulation/):
+
+$$\begin{aligned}
+    A(z,t) = \sqrt{P_0} \exp( i \gamma P_0 z)
+\end{aligned}$$
+
+We add a small perturbation $\varepsilon(z,t)$ to this signal,
+representing background noise:
+
+$$\begin{aligned}
+    A(z,t) = \big(\sqrt{P_0} + \varepsilon(z,t)\big) \exp( i \gamma P_0 z)
+\end{aligned}$$
+
+We insert this into the nonlinear Schrödinger equation to get a perturbation equation,
+which we linearize by assuming that $|\varepsilon|^2$ is negligible compared to $P_0$,
+such that all higher-order terms of $\varepsilon$ can be dropped, yielding:
+
+$$\begin{aligned}
+    0
+    &= - P_0 \sqrt{P_0} \gamma - P_0 \gamma \varepsilon + i \pdv{\varepsilon}{z}
+    - \frac{\beta_2}{2} \pdv[2]{\varepsilon}{t}
+    + \gamma \big(\sqrt{P_0} + \varepsilon\big)^2 \big(\sqrt{P_0} + \varepsilon\big)^*
+    \\
+    &= i \pdv{\varepsilon}{z}
+    - \frac{\beta_2}{2} \pdv[2]{\varepsilon}{t}
+    + \gamma \big( P_0 (\varepsilon + \varepsilon^*) + \sqrt{P_0} |\varepsilon|^2
+    + \sqrt{P_0} \varepsilon (\varepsilon + \varepsilon^*) + \varepsilon |\varepsilon|^2 \big)
+    \\
+    &= i \pdv{\varepsilon}{z} - \frac{\beta_2}{2} \pdv[2]{\varepsilon}{t} + \gamma P_0 (\varepsilon + \varepsilon^*)
+\end{aligned}$$
+
+We split the perturbation into real and imaginary parts
+$\varepsilon(z,t) = \varepsilon_r(z,t) + i \varepsilon_i(z,t)$,
+which we fill in in this equation.
+The point is that $\varepsilon_r$ and $\varepsilon_i$ are real functions:
+
+$$\begin{aligned}
+    0
+    &= i \pdv{\varepsilon_r}{z} - \pdv{\varepsilon_i}{z}
+    - \frac{\beta_2}{2} \pdv[2]{\varepsilon_r}{t} - i \frac{\beta_2}{2} \pdv[2]{\varepsilon_i}{t}
+    + 2 \gamma P_0 \varepsilon_r
+\end{aligned}$$
+
+Splitting this into its real and imaginary parts gives two PDEs
+relating $\varepsilon_r$ and $\varepsilon_i$:
+
+$$\begin{aligned}
+    \pdv{\varepsilon_r}{z} = \frac{\beta_2}{2} \pdv[2]{\varepsilon_i}{t}
+    \qquad \quad
+    \pdv{\varepsilon_i}{z} = - \frac{\beta_2}{2} \pdv[2]{\varepsilon_r}{t} + 2 \gamma P_0 \varepsilon_r
+\end{aligned}$$
+
+We [Fourier transform](/know/concept/fourier-transform/)
+these in $t$ to turn them into ODEs relating
+$\tilde{\varepsilon}_r(z,\omega)$ and $\tilde{\varepsilon}_i(z,\omega)$:
+
+$$\begin{aligned}
+    \pdv{\tilde{\varepsilon}_r}{z} = - \frac{\beta_2}{2} \omega^2 \tilde{\varepsilon}_i
+    \qquad \quad
+    \pdv{\tilde{\varepsilon}_i}{z} = \Big(\frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) \tilde{\varepsilon}_r
+\end{aligned}$$
+
+We are interested in exponential growth, so let us make the following ansatz,
+where $k$ may be a function of $\omega$, as long as it is $z$-invariant:
+
+$$\begin{aligned}
+    \tilde{\varepsilon}_r(z, \omega) = \tilde{\varepsilon}_r(0, \omega) \exp(k z)
+    \qquad \quad
+    \tilde{\varepsilon}_i(z, \omega) = \tilde{\varepsilon}_i(0, \omega) \exp(k z)
+\end{aligned}$$
+
+With this, we can write the system of ODEs for
+$\tilde{\varepsilon}_r(z,\omega)$ and $\tilde{\varepsilon}_i(z,\omega)$
+in matrix form:
+
+$$\begin{aligned}
+    \begin{bmatrix}
+        k & \beta_2 \omega^2 / 2 \\
+        \beta_2 \omega^2 / 2 \!+\! 2 \gamma P_0 & - k
+    \end{bmatrix}
+    \cdot
+    \begin{bmatrix} \tilde{\varepsilon}_r(0, \omega) \\ \tilde{\varepsilon}_i(0, \omega) \end{bmatrix}
+    =
+    \begin{bmatrix} 0 \\ 0 \end{bmatrix}
+\end{aligned}$$
+
+This has non-zero solutions if the system matrix' determinant is zero,
+which is true when:
+
+$$\begin{aligned}
+    k = \pm \sqrt{ - \frac{\beta_2}{2} \omega^2 \Big( \frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) }
+\end{aligned}$$
+
+To get exponential growth, it is essential that $\mathrm{Re}\{k\} > 0$,
+so we discard the negative sign,
+and get the following condition for MI:
+
+$$\begin{aligned}
+    - \frac{\beta_2}{2} \omega^2 \Big( \frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) > 0
+    \quad \implies \quad
+    \boxed{
+        \omega^2 < -\frac{4 \gamma P_0}{\beta_2}
+    }
+\end{aligned}$$
+
+Since $\omega^2$ is positive, $\beta_2$ must be negative,
+so MI can only occur in the ADR.
+It is worth noting that $\beta_2 = \beta_2(\omega_0)$,
+meaning there can only be exponential
+noise growth when the "parent pulse" is in the anomalous dispersion regime,
+but that growth may appear in areas of normal dispersion,
+as long as the above condition is satisfied by the parent.
+
+This result has been derived using perturbation,
+so only holds as long as $|\varepsilon|^2 \ll P_0$.
+Over time, the noise gets amplified so greatly
+that this approximation breaks down.
+
+Next, we define the **gain** $g(\omega)$,
+which expresses how quickly the
+perturbation grows as a function of the frequency offset $\omega$:
+
+$$\begin{aligned}
+    \boxed{
+        g(\omega)
+        = \mathrm{Re}\{k\}
+        = \mathrm{Re} \bigg\{ \sqrt{ - \frac{\beta_2}{2} \omega^2 \Big( \frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) } \bigg\}
+    }
+\end{aligned}$$
+
+The frequencies with maximum gain are then found as extrema of $g(\omega)$,
+which satisfy:
+
+$$\begin{aligned}
+    g'(\omega_\mathrm{max}) = 0
+    \qquad \implies \qquad
+    \boxed{
+        \omega_\mathrm{max} = \pm \sqrt{\frac{2 \gamma P_0}{-\beta_2}}
+    }
+\end{aligned}$$
+
+A simulation of MI is illustrated below.
+The pulse considered was a solition of the following form
+with settings $T_0 = 10\:\mathrm{ps}$, $P_0 = 10\:\mathrm{kW}$,
+$\beta = -10\:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$,
+whose peak is approximately flat, so our derivation is valid there,
+hence it "wrinkles" in the $t$-domain:
+
+$$\begin{aligned}
+    A(0, t)
+    = \sqrt{P_0} \sech\!\Big(\frac{t}{T_0}\Big)
+\end{aligned}$$
+
+<img src="pheno-mi.jpg">
+
+Where $L_\mathrm{NL} = 1/(\gamma P_0)$ is the characteristic length of nonlinear effects.
+Note that no noise was added to the simulation;
+what you are seeing are pure numerical errors getting amplified.
+
+If one of the gain peaks accumulates a lot of energy quickly ($L_\mathrm{NL}$ is small),
+and that peak is in the anomalous dispersion regime,
+then it can in turn also cause MI in its own surroundings,
+leading to a cascade of secondary and tertiary gain areas.
+This is seen above for $z > 30 L_\mathrm{NL}$.
+
+What we described is "pure" MI, but there also exists
+a different type caused by Raman scattering.
+In that case, amplification occurs at the strongest peak of the Raman gain $\tilde{g}_R(\omega)$,
+even when the parent pulse is in the NDR.
+This is an example of stimulated Raman scattering (SRS).
+
diff --git a/content/know/concept/modulational-instability/pheno-mi.jpg b/content/know/concept/modulational-instability/pheno-mi.jpg
new file mode 100644
index 0000000..e45f074
--- /dev/null
+++ b/content/know/concept/modulational-instability/pheno-mi.jpg
diff --git a/content/know/concept/optical-wave-breaking/index.pdc b/content/know/concept/optical-wave-breaking/index.pdc
new file mode 100644
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--- /dev/null
+++ b/content/know/concept/optical-wave-breaking/index.pdc
@@ -0,0 +1,229 @@
+---
+title: "Optical wave breaking"
+firstLetter: "O"
+publishDate: 2021-02-27
+categories:
+- Physics
+- Optics
+- Fiber optics
+- Nonlinear dynamics
+
+date: 2021-02-27T10:09:46+01:00
+draft: false
+markup: pandoc
+---
+
+# Optical wave breaking
+
+In fiber optics, **optical wave breaking** (OWB) is a nonlinear effect
+caused by interaction between
+[group velocity dispersion](/know/concept/dispersive-broadening/) (GVD) and
+[self-phase modulation](/know/concept/self-phase-modulation/) (SPM).
+It only happens in the normal dispersion regime ($\beta_2 > 0$)
+for pulses meeting a certain criterium, as we will see.
+
+SPM creates low frequencies at the front of the pulse, and high ones at the back,
+and if $\beta_2 > 0$, GVD lets low frequencies travel faster than high ones.
+When those effects interact, the pulse gets temporally stretched
+in a surprisingly sophisticated way.
+
+To illustrate this, the instantaneous frequency $\omega_i(z, t) = -\pdv*{\phi}{t}$
+has been plotted below for a theoretical Gaussian input pulse experiencing OWB,
+with settings $T_0 = 100\:\mathrm{fs}$, $P_0 = 5\:\mathrm{kW}$,
+$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$.
+
+In the left panel, we see the typical S-shape caused by SPM,
+and the arrows indicate the direction that GVD is pushing the curve in.
+This leads to steepening at the edges, i.e. the S gradually turns into a Z.
+Shortly before the slope would become infinite,
+small waves start "falling off" the edge of the pulse,
+hence the name *wave breaking*:
+
+<img src="pheno-break-inst.jpg">
+
+Several interesting things happen around this moment.
+To demonstrate this, spectrograms of the same simulation
+have been plotted below, together with pulse profiles
+in both the $t$-domain and $\omega$-domain on an arbitrary linear scale
+(click the image to get a better look).
+
+Initially, the spectrum broadens due to SPM in the usual way,
+but shortly after OWB, this process is stopped by the appearance
+of so-called **sidelobes** in the $\omega$-domain on either side of the pulse.
+In the meantime, in the time domain,
+the pulse steepens at the edges, but flattens at the peak.
+After OWB, a train of small waves falls off the edges,
+which eventually melt together, leading to a trapezoid shape in the $t$-domain.
+Dispersive broadening then continues normally:
+
+<a href="pheno-break-sgram.jpg">
+<img src="pheno-break-sgram.jpg" style="width:80%;display:block;margin:auto;">
+</a>
+
+We call the distance at which the wave breaks $L_\mathrm{WB}$,
+and would like to analytically predict it.
+We do this using the instantaneous frequency $\omega_i$,
+by estimating when the SPM fluctuations overtake their own base,
+as was illustrated earlier.
+
+To get $\omega_i$ of a Gaussian pulse experiencing both GVD and SPM,
+it is a reasonable approximation, for small $z$, to simply add up
+the instantaneous frequencies for these separate effects:
+
+$$\begin{aligned}
+    \omega_i(z,t)
+    &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
+    \\
+%    &= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
+%    + \frac{2\gamma P_0 z}{T_0^2} t \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
+%    \\
+    &= \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
+    + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+\end{aligned}$$
+
+Assuming that $z$ is small enough such that $z^2 \approx 0$, this
+expression can be reduced to:
+
+$$\begin{aligned}
+    \omega_i(z,t)
+    \approx \frac{\beta_2 tz}{T_0^4} \bigg( 1 + 2\frac{\gamma P_0 T_0^2}{\beta_2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+    = \frac{\beta_2 t z}{T_0^4} \bigg( 1 \pm 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+\end{aligned}$$
+
+Where we have assumed $\beta_2 > 0$,
+and $N_\mathrm{sol}$ is the **soliton number**,
+which is defined as:
+
+$$\begin{aligned}
+    N_\mathrm{sol}^2 = \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
+\end{aligned}$$
+
+This quantity is very important in anomalous dispersion,
+but even in normal dispesion, it is still a useful measure of the relative strengths of GVD and SPM.
+As was illustrated earlier, $\omega_i$ overtakes itself at the edges,
+so OWB only occurs when $\omega_i$ is not monotonic,
+which is when its $t$-derivative,
+the **instantaneous chirpyness** $\xi_i$,
+has *two* real roots for $t^2$:
+
+$$\begin{aligned}
+    0
+    = \xi_i(z,t)
+    = \pdv{\omega_i}{t}
+    &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+    = \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
+\end{aligned}$$
+
+Where the function $f(x)$ has been defined for convenience. As it turns
+out, this equation can be solved analytically using the Lambert $W$ function,
+leading to the following exact minimum value $N_\mathrm{min}^2$ for $N_\mathrm{sol}^2$,
+such that OWB can only occur when $N_\mathrm{sol}^2 > N_\mathrm{min}^2$:
+
+$$\begin{aligned}
+    N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
+\end{aligned}$$
+
+Now, consider two times $t_1$ and $t_2$ in the pulse, separated by
+a small initial interval $(t_2 - t_1)$.
+The frequency difference between these points due to $\omega_i$
+will cause them to displace relative to each other
+after a short distance $z$ by some amount $\Delta t$,
+estimated by:
+
+$$\begin{aligned}
+    \Delta t
+    &\approx z \Delta\beta_1
+    \qquad
+    &&\Delta\beta_1 = \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
+    \\
+    &\approx z \beta_2 \Delta\omega_i
+    \qquad
+    &&\Delta\omega_i = \omega_i(z,t_2) - \omega_i(z,t_1)
+    \\
+    &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
+    \qquad \quad
+    &&\Delta\xi_i = \xi_i(z,t_2) - \xi_i(z,t_1)
+\end{aligned}$$
+
+Where $\beta_1(\omega)$ is the inverse of the group velocity.
+OWB takes place when $t_2$ and $t_1$ catch up to each other,
+which is when $-\Delta t = (t_2 - t_1)$.
+The distance where this happens, $z = L_\mathrm{WB}$,
+must therefore satisfy the following condition
+for a particular value of $t$:
+
+$$\begin{aligned}
+    L_\mathrm{WB} \, \beta_2 \, \xi_i(L_\mathrm{WB}, t) = -1
+    \qquad \implies \qquad
+    L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \, f(t^2/T_0^2)}
+\end{aligned}$$
+
+The time $t$ of OWB must be where $\omega_i(t)$ has its steepest slope,
+which is at the minimum value of $\xi_i(t)$, and, by extension $f(x)$.
+This turns out to be $f(3/2)$:
+
+$$\begin{aligned}
+    f_\mathrm{min} = f(3/2)
+    = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2)
+    = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
+\end{aligned}$$
+
+Clearly, $f_\mathrm{min} \ge 0$ when
+$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$, which, when inserted in the
+condition above, confirms that OWB cannot occur in that case. Otherwise,
+if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then:
+
+$$\begin{aligned}
+    L_\mathrm{WB}
+    = - \frac{T_0^2}{\beta_2 \, \sqrt{f_\mathrm{min}}}
+    = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
+    = \frac{L_D}{\sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}}
+\end{aligned}$$
+
+This prediction for $L_\mathrm{WB}$ appears to agree well
+with the OWB observed in the simulation:
+
+<img src="pheno-break.jpg">
+
+Because all spectral broadening up to $L_\mathrm{WB}$ is caused by SPM,
+whose frequency behaviour is known, it is in fact possible to draw
+some analytical conclusions about the achieved bandwidth when OWB sets in.
+Filling $L_\mathrm{WB}$ in into $\omega_\mathrm{SPM}$ gives:
+
+$$\begin{aligned}
+    \omega_{\mathrm{SPM}}(L_\mathrm{WB},t)
+    = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
+\end{aligned}$$
+
+Assuming that $N_\mathrm{sol}^2$ is large in the denominator, this can
+be approximately reduced to:
+
+$$\begin{aligned}
+    \omega_\mathrm{SPM}(L_\mathrm{WB}, t)
+%   = \frac{2 \gamma P_0 t \exp(-t^2 / T_0^2)}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
+    \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
+    = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
+\end{aligned}$$
+
+The expression $x \exp(-x^2)$ has its global extrema
+$\pm 1 / \sqrt{2 e}$ at $x^2 = 1/2$. The maximum SPM frequency shift
+achieved at $L_\mathrm{WB}$ is therefore given by:
+
+$$\begin{aligned}
+    \omega_\mathrm{max} = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
+\end{aligned}$$
+
+Interestingly, this expression does not contain $T_0$ at all,
+so the achieved spectrum when SPM is halted by OWB
+is independent of the pulse width,
+for sufficiently large $N_\mathrm{sol}$.
+
+
+## References
+1.  D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro,
+    [Wave breaking in nonlinear-optical fibers](https://doi.org/10.1364/JOSAB.9.001358),
+    1992, Optical Society of America.
+2.  A.M.  Heidt,  A.  Hartung,  H.  Bartelt,
+    [Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers](https://doi.org/10.1007/978-1-4939-3326-6_6),
+    2016, Springer Media.
+
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diff --git a/content/know/concept/self-phase-modulation/index.pdc b/content/know/concept/self-phase-modulation/index.pdc
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+---
+title: "Self-phase modulation"
+firstLetter: "S"
+publishDate: 2021-02-26
+categories:
+- Physics
+- Optics
+- Fiber optics
+- Nonlinear dynamics
+
+date: 2021-02-27T10:09:32+01:00
+draft: false
+markup: pandoc
+---
+
+# Self-phase modulation
+
+In fiber optics, **self-phase modulation** (SPM) is a nonlinear effect
+that gradually broadens pulses' spectra.
+Unlike dispersion, SPM does create new frequencies: in the $\omega$-domain,
+the pulse steadily spreads out with a distinctive "accordion" peak.
+Lower frequencies are created at the front of the
+pulse and higher ones at the back, giving S-shaped spectrograms.
+
+A pulse envelope $A(z, t)$ inside a fiber must obey the nonlinear Schrödinger equation,
+where the parameters $\beta_2$ and $\gamma$ respectively
+control dispersion and nonlinearity:
+
+$$\begin{aligned}
+    0
+    = i \pdv{A}{z} - \frac{\beta_2}{2} \pdv[2]{A}{t} + \gamma |A|^2 A
+\end{aligned}$$
+
+By setting $\beta_2 = 0$ to neglect dispersion,
+solving this equation becomes trivial.
+For any arbitrary input pulse $A_0(t) = A(0, t)$,
+we arrive at the following analytical solution:
+
+$$\begin{aligned}
+    A(z,t) = A_0 \exp\!\big( i \gamma |A_0|^2 z\big)
+\end{aligned}$$
+
+The intensity $|A|^2$ in the time domain is thus unchanged,
+and only its phase is modified.
+It is also clear that the largest phase increase occurs at the peak of the pulse,
+where the intensity is $P_0$.
+To quantify this, it is useful to define the **nonlinear length** $L_N$,
+which gives the distance after which the phase of the
+peak has increased by exactly 1 radian:
+
+$$\begin{aligned}
+    \gamma P_0 L_N = 1
+    \qquad \implies \qquad
+    \boxed{
+        L_N = \frac{1}{\gamma P_0}
+    }
+\end{aligned}$$
+
+SPM is illustrated below for the following Gaussian initial pulse envelope,
+with parameter values $T_0 = 6\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
+$\beta_2 = 0$, and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$:
+
+$$\begin{aligned}
+    A(0, t)
+    = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
+\end{aligned}$$
+
+From earlier, we then know the analytical solution for the $z$-evolution:
+
+$$\begin{aligned}
+    A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+\end{aligned}$$
+
+<img src="pheno-spm.jpg">
+
+The **instantaneous frequency** $\omega_\mathrm{SPM}(z, t)$,
+which describes the dominant angular frequency at a given point in the time domain,
+is found to be as follows for the Gaussian pulse,
+where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$:
+
+$$\begin{aligned}
+    \omega_{\mathrm{SPM}}(z,t)
+    = - \pdv{\phi}{t}
+    = 2 \gamma z P_0 \frac{t}{T_0^2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
+\end{aligned}$$
+
+This result gives the S-shaped spectrograms seen in the illustration.
+The frequency shift thus not only depends on $L_N$,
+but also on $T_0$: the spectra of narrow pulses broaden much faster.
+
+The interaction between self-phase modulation
+and [dispersion](/know/concept/dispersive-broadening/)
+leads to many interesting effects,
+such as [modulational instability](/know/concept/modulational-instability/)
+and [optical wave breaking](/know/concept/optical-wave-breaking/).
diff --git a/content/know/concept/self-phase-modulation/pheno-spm.jpg b/content/know/concept/self-phase-modulation/pheno-spm.jpg
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diff --git a/content/know/concept/self-steepening/index.pdc b/content/know/concept/self-steepening/index.pdc
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+---
+title: "Self-steepening"
+firstLetter: "S"
+publishDate: 2021-02-26
+categories:
+- Physics
+- Optics
+- Fiber optics
+- Nonlinear dynamics
+
+date: 2021-02-26T15:18:25+01:00
+draft: false
+markup: pandoc
+---
+
+# Self-steepening
+
+For a laser pulse travelling through an optical fiber,
+its intensity is highest at its peak, so the Kerr effect will be strongest there.
+This means that the peak travels slightly slower
+than the rest of the pulse, leading to **self-steepening** of its trailing edge.
+Mathematically, this is described by adding a new term to the
+nonlinear Schrödinger equation:
+
+$$\begin{aligned}
+    0
+    = i\pdv{A}{z} - \frac{\beta_2}{2} \pdv[2]{A}{t} + \gamma \Big(1 + \frac{i}{\omega_0} \pdv{t}\Big) \big(|A|^2 A\big)
+\end{aligned}$$
+
+Where $\omega_0$ is the angular frequency of the pump.
+We will use the following ansatz,
+consisting of an arbitrary power profile $P$ with a phase $\phi$:
+
+$$\begin{aligned}
+    A(z,t) = \sqrt{P(z,t)} \, \exp\big(i \phi(z,t)\big)
+\end{aligned}$$
+
+For a long pulse travelling over a short distance, it is reasonable to
+neglect dispersion ($\beta_2 = 0$).
+Inserting the ansatz then gives the following, where $\varepsilon = \gamma / \omega_0$:
+
+$$\begin{aligned}
+    0 &= i \frac{1}{2} \frac{P_z}{\sqrt{P}} - \sqrt{P} \phi_z + \gamma P \sqrt{P} + i \varepsilon \frac{3}{2} P_t \sqrt{P} - \varepsilon P \sqrt{P} \phi_t
+\end{aligned}$$
+
+This results in two equations, respectively corresponding to the real
+and imaginary parts:
+
+$$\begin{aligned}
+    0 &= - \phi_z - \varepsilon P \phi_t + \gamma P
+    \\
+    0 &= P_z + \varepsilon 3 P_t P
+\end{aligned}$$
+
+The phase $\phi$ is not so interesting, so we focus on the latter equation for $P$.
+As it turns out, it has a general solution of the form below, which shows that
+more intense parts of the pulse will tend to lag behind compared to the rest:
+
+$$\begin{aligned}
+    P(z,t) = f(t - 3 \varepsilon z P)
+\end{aligned}$$
+
+Where $f$ is the initial power profile: $f(t) = P(0,t)$.
+The derivatives $P_t$ and $P_z$ are then given by:
+
+$$\begin{aligned}
+    P_t
+    &= (1 - 3 \varepsilon z P_t) \: f'
+    \qquad \quad \implies \quad
+    P_t
+    = \frac{f'}{1 + 3 \varepsilon z f'}
+    \\
+    P_z
+    &= (-3 \varepsilon P - 3 \varepsilon z P_z) \: f'
+    \quad \implies \quad
+    P_z
+    = \frac{- 3 \varepsilon P f'}{1 + 3 \varepsilon z f'}
+\end{aligned}$$
+
+These derivatives both go to infinity when their denominator is zero,
+which, since $\varepsilon$ is positive, will happen earliest where $f'$
+has its most negative value, called $f_\mathrm{min}'$,
+which is located on the trailing edge of the pulse.
+At the propagation distance where this occurs, $L_\mathrm{shock}$,
+the pulse will "tip over", creating a discontinuous shock:
+
+$$\begin{aligned}
+    \boxed{
+        L_\mathrm{shock} = -\frac{1}{3 \varepsilon f_\mathrm{min}'}
+    }
+\end{aligned}$$
+
+In practice, however, this will never actually happen, because by the time
+$L_\mathrm{shock}$ is reached, the pulse spectrum will have become so
+broad that dispersion can no longer be neglected.
+
+A simulation of self-steepening without dispersion is illustrated below
+for the following Gaussian initial power distribution,
+with $T_0 = 25\:\mathrm{fs}$, $P_0 = 3\:\mathrm{kW}$,
+$\beta_2 = 0$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$:
+
+$$\begin{aligned}
+    f(t) = P(0,t) = P_0 \exp\!\Big(\! -\!\frac{t^2}{T_0^2} \Big)
+\end{aligned}$$
+
+
+Its steepest points are found to be at $2 t^2 = T_0^2$, so
+$f_\mathrm{min}'$ and $L_\mathrm{shock}$ are given by:
+
+$$\begin{aligned}
+    f_\mathrm{min}' = - \frac{\sqrt{2} P_0}{T_0} \exp\!\Big(\!-\!\frac{1}{2}\Big)
+    \quad \implies \quad
+    L_\mathrm{shock} = \frac{T_0}{3 \sqrt{2} \varepsilon P_0} \exp\!\Big(\frac{1}{2}\Big)
+\end{aligned}$$
+
+This example Gaussian pulse therefore has a theoretical
+$L_\mathrm{shock} = 0.847\,\mathrm{m}$,
+which turns out to be accurate,
+although the simulation breaks down due to insufficient resolution:
+
+<img src="pheno-steep.jpg">
+
+Unfortunately, self-steepening cannot be simulated perfectly: as the
+pulse approaches $L_\mathrm{shock}$, its spectrum broadens to infinite