From aab299218975a8e775cda26ce256ffb1fe36c863 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 30 May 2021 15:54:40 +0200 Subject: Expand knowledge base --- .../know/concept/optical-wave-breaking/index.pdc | 54 ++++++++++++---------- 1 file changed, 30 insertions(+), 24 deletions(-) (limited to 'content/know/concept/optical-wave-breaking/index.pdc') diff --git a/content/know/concept/optical-wave-breaking/index.pdc b/content/know/concept/optical-wave-breaking/index.pdc index 757a633..2ab3ff1 100644 --- a/content/know/concept/optical-wave-breaking/index.pdc +++ b/content/know/concept/optical-wave-breaking/index.pdc @@ -75,11 +75,10 @@ the instantaneous frequencies for these separate effects: $$\begin{aligned} \omega_i(z,t) &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t) - \\ % &= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2} % + \frac{2\gamma P_0 z}{T_0^2} t \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big) % \\ - &= \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} + = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}$$ @@ -97,14 +96,14 @@ and $N_\mathrm{sol}$ is the **soliton number**, which is defined as: $$\begin{aligned} - N_\mathrm{sol}^2 = \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|} + N_\mathrm{sol}^2 \equiv \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|} \end{aligned}$$ 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, +so OWB occurs when $\omega_i$ oscillates there, +which starts when its $t$-derivative, the **instantaneous chirpyness** $\xi_i$, has *two* real roots for $t^2$: @@ -122,11 +121,17 @@ leading to the following exact minimum value $N_\mathrm{min}^2$ for $N_\mathrm{s such that OWB can only occur when $N_\mathrm{sol}^2 > N_\mathrm{min}^2$: $$\begin{aligned} - N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12 + \boxed{ + N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12 + } \end{aligned}$$ -Now, consider two times $t_1$ and $t_2$ in the pulse, separated by -a small initial interval $(t_2 - t_1)$. +If this condition $N_\mathrm{sol}^2 > N_\mathrm{min}^2$ is not satisfied, +$\xi_i$ cannot have two roots for $t^2$, meaning $\omega_i$ cannot overtake itself. +GVD is unable to keep up with SPM, so OWB will not occur. + +Next, consider two points at $t_1$ 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 $\Delta t$, @@ -136,21 +141,21 @@ $$\begin{aligned} \Delta t &\approx z \Delta\beta_1 \qquad - &&\Delta\beta_1 = \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1)) + &&\Delta\beta_1 \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1)) \\ &\approx z \beta_2 \Delta\omega_i \qquad - &&\Delta\omega_i = \omega_i(z,t_2) - \omega_i(z,t_1) + &&\Delta\omega_i \equiv \omega_i(z,t_2) - \omega_i(z,t_1) \\ &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1) \qquad \quad - &&\Delta\xi_i = \xi_i(z,t_2) - \xi_i(z,t_1) + &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1) \end{aligned}$$ 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, $z = L_\mathrm{WB}$, +The distance where this happens first, $z = L_\mathrm{WB}$, must therefore satisfy the following condition for a particular value of $t$: @@ -161,7 +166,7 @@ $$\begin{aligned} \end{aligned}$$ 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 $f(x)$. +which is at the minimum value of $\xi_i(t)$, and by extension $f(x)$. This turns out to be $f(3/2)$: $$\begin{aligned} @@ -170,16 +175,17 @@ $$\begin{aligned} = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2 \end{aligned}$$ -Clearly, $f_\mathrm{min} \ge 0$ when -$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$, which, when inserted in the -condition above, confirms that OWB cannot occur in that case. Otherwise, -if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then: +Clearly, $f_\mathrm{min} \ge 0$ when $N_\mathrm{sol}^2 \le N_\mathrm{min}^2$, +which, when inserted above, leads to an imaginary $L_\mathrm{WB}$, +confirming that OWB cannot occur in that case. +Otherwise, if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then: $$\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}} + \boxed{ + 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}} + } \end{aligned}$$ This prediction for $L_\mathrm{WB}$ appears to agree well @@ -196,7 +202,7 @@ Filling $L_\mathrm{WB}$ in into $\omega_\mathrm{SPM}$ gives: $$\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) + = \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}$$ Assuming that $N_\mathrm{sol}^2$ is large in the denominator, this can @@ -205,8 +211,8 @@ be approximately reduced to: $$\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) + \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 -- cgit v1.2.3