From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Tue, 20 Dec 2022 20:11:25 +0100 Subject: More improvements to knowledge base --- source/know/concept/optical-wave-breaking/index.md | 51 ++++++++++++++-------- 1 file changed, 32 insertions(+), 19 deletions(-) (limited to 'source/know/concept/optical-wave-breaking') diff --git a/source/know/concept/optical-wave-breaking/index.md b/source/know/concept/optical-wave-breaking/index.md index 882749f..3509bc2 100644 --- a/source/know/concept/optical-wave-breaking/index.md +++ b/source/know/concept/optical-wave-breaking/index.md @@ -54,7 +54,7 @@ Dispersive broadening then continues normally: {% include image.html file="spectrograms-full.png" width="100%" alt="Spectrograms of pulse shape evolution" %} We call the distance at which the wave breaks $$L_\mathrm{WB}$$, -and would like to analytically predict it. +and want to predict it analytically. We do this using the instantaneous frequency $$\omega_i$$, by estimating when the SPM fluctuations overtake their own base, as was illustrated earlier. @@ -84,11 +84,13 @@ and $$N_\mathrm{sol}$$ is the **soliton number**, which is defined as: $$\begin{aligned} - N_\mathrm{sol}^2 \equiv \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|} + N_\mathrm{sol}^2 + \equiv \frac{L_D}{L_N} + = \frac{\gamma P_0 T_0^2}{|\beta_2|} \end{aligned}$$ 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. +but even in normal dispersion, 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 occurs when $$\omega_i$$ oscillates there, which starts when its $$t$$-derivative, @@ -100,17 +102,19 @@ $$\begin{aligned} = \xi_i(z,t) = \pdv{\omega_i}{t} &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) - = \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big) + \equiv \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big) \end{aligned}$$ 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, +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 $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$: $$\begin{aligned} \boxed{ - N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12 + N_\mathrm{min}^2 + = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) + \approx 1.12 } \end{aligned}$$ @@ -129,28 +133,33 @@ $$\begin{aligned} \Delta t &\approx z \Delta\beta_1 \qquad - &&\Delta\beta_1 \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1)) + &&\Delta\beta_1 + \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1)) \\ &\approx z \beta_2 \Delta\omega_i \qquad - &&\Delta\omega_i \equiv \omega_i(z,t_2) - \omega_i(z,t_1) + &&\Delta\omega_i + \equiv \omega_i(z,t_2) - \omega_i(z,t_1) \\ &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1) \qquad \quad - &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1) + &&\Delta\xi_i + \equiv \xi_i(z,t_2) - \xi_i(z,t_1) \end{aligned}$$ 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, +For a certain choice of $$t_1$$ and $$t_2$$, +OWB occurs when they catch up to each other, which is when $$-\Delta t = (t_2 - t_1)$$. -The distance where this happens first, $$z = L_\mathrm{WB}$$, -must therefore satisfy the following condition -for a particular value of $$t$$: +The distance $$L_\mathrm{WB}$$ at which this happens first +must satisfy the following condition for some value of $$t$$: $$\begin{aligned} - L_\mathrm{WB} \, \beta_2 \, \xi_i(L_\mathrm{WB}, t) = -1 + L_\mathrm{WB} \: \beta_2 \: \xi_i(L_\mathrm{WB}, t) + = -1 \qquad \implies \qquad - L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \, f(t^2/T_0^2)} + L_\mathrm{WB}^2 + = - \frac{T_0^4}{\beta_2^2 \: f(t^2/T_0^2)} \end{aligned}$$ The time $$t$$ of OWB must be where $$\omega_i(t)$$ has its steepest slope, @@ -158,7 +167,8 @@ which is at the minimum value of $$\xi_i(t)$$, and by extension $$f(x)$$. This turns out to be $$f(3/2)$$: $$\begin{aligned} - f_\mathrm{min} = f(3/2) + f_\mathrm{min} + = f(3/2) = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2) = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2 \end{aligned}$$ @@ -182,8 +192,9 @@ with the OWB observed in the simulation: {% include image.html file="simulation-full.png" width="100%" alt="Optical wave breaking simulation results" %} 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. +whose $$\omega$$-domain 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: $$\begin{aligned} @@ -205,7 +216,8 @@ $$\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}} + \omega_\mathrm{max} + = \sqrt{\frac{2 \gamma P_0}{e \beta_2}} \end{aligned}$$ Interestingly, this expression does not contain $$T_0$$ at all, @@ -214,6 +226,7 @@ is independent of the pulse width, for sufficiently large $$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), -- cgit v1.2.3