From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/optical-wave-breaking/index.md | 100 ++++++++++----------- 1 file changed, 50 insertions(+), 50 deletions(-) (limited to 'source/know/concept/optical-wave-breaking/index.md') diff --git a/source/know/concept/optical-wave-breaking/index.md b/source/know/concept/optical-wave-breaking/index.md index aa43d25..42064ff 100644 --- a/source/know/concept/optical-wave-breaking/index.md +++ b/source/know/concept/optical-wave-breaking/index.md @@ -14,18 +14,18 @@ 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 ($\beta_2 > 0$) +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. +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 $\omega_i(z, t) = -\ipdv{\phi}{t}$ +To illustrate this, the instantaneous frequency $$\omega_i(z, t) = -\ipdv{\phi}{t}$$ 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 $\gamma = 0.1/\mathrm{W}/\mathrm{m}$. +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. @@ -41,30 +41,30 @@ hence the name *wave breaking*: 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 +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. +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. +which eventually melt together, leading to a trapezoid shape in the $$t$$-domain. Dispersive broadening then continues normally: -We call the distance at which the wave breaks $L_\mathrm{WB}$, +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 $\omega_i$, +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 +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: $$\begin{aligned} @@ -74,7 +74,7 @@ $$\begin{aligned} + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}$$ -Assuming that $z$ is small enough such that $z^2 \approx 0$, this +Assuming that $$z$$ is small enough such that $$z^2 \approx 0$$, this expression can be reduced to: $$\begin{aligned} @@ -83,8 +83,8 @@ $$\begin{aligned} = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}$$ -Where we have assumed $\beta_2 > 0$, -and $N_\mathrm{sol}$ is the **soliton number**, +Where we have assumed $$\beta_2 > 0$$, +and $$N_\mathrm{sol}$$ is the **soliton number**, which is defined as: $$\begin{aligned} @@ -93,11 +93,11 @@ $$\begin{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 occurs when $\omega_i$ oscillates there, -which starts when its $t$-derivative, -the **instantaneous chirpyness** $\xi_i$, -has *two* real roots for $t^2$: +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, +the **instantaneous chirpyness** $$\xi_i$$, +has *two* real roots for $$t^2$$: $$\begin{aligned} 0 @@ -107,10 +107,10 @@ $$\begin{aligned} = \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big) \end{aligned}$$ -Where the function $f(x)$ has been defined for convenience. As it turns -out, this equation can be solved analytically using the Lambert $W$ function, -leading to the following exact minimum value $N_\mathrm{min}^2$ for $N_\mathrm{sol}^2$, -such that OWB can only occur when $N_\mathrm{sol}^2 > N_\mathrm{min}^2$: +Where the function $$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} \boxed{ @@ -118,15 +118,15 @@ $$\begin{aligned} } \end{aligned}$$ -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. +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 $\omega_i$ +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 $\Delta t$, +after a short distance $$z$$ by some amount $$\Delta t$$, estimated by: $$\begin{aligned} @@ -144,12 +144,12 @@ $$\begin{aligned} &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1) \end{aligned}$$ -Where $\beta_1(\omega)$ is the inverse of the group velocity. -OWB takes place when $t_2$ and $t_1$ catch up to each other, -which is when $-\Delta t = (t_2 - t_1)$. -The distance where this happens first, $z = L_\mathrm{WB}$, +Where $$\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)$$. +The distance where this happens first, $$z = L_\mathrm{WB}$$, must therefore satisfy the following condition -for a particular value of $t$: +for a particular value of $$t$$: $$\begin{aligned} L_\mathrm{WB} \, \beta_2 \, \xi_i(L_\mathrm{WB}, t) = -1 @@ -157,9 +157,9 @@ $$\begin{aligned} L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \, f(t^2/T_0^2)} \end{aligned}$$ -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 $f(3/2)$: +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 $$f(3/2)$$: $$\begin{aligned} f_\mathrm{min} = f(3/2) @@ -167,10 +167,10 @@ $$\begin{aligned} = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2 \end{aligned}$$ -Clearly, $f_\mathrm{min} \ge 0$ when $N_\mathrm{sol}^2 \le N_\mathrm{min}^2$, -which, when inserted above, leads to an imaginary $L_\mathrm{WB}$, +Clearly, $$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}$$, confirming that OWB cannot occur in that case. -Otherwise, if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then: +Otherwise, if $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$, then: $$\begin{aligned} \boxed{ @@ -180,24 +180,24 @@ $$\begin{aligned} } \end{aligned}$$ -This prediction for $L_\mathrm{WB}$ appears to agree well +This prediction for $$L_\mathrm{WB}$$ appears to agree well with the OWB observed in the simulation: -Because all spectral broadening up to $L_\mathrm{WB}$ is caused by SPM, +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: +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 +Assuming that $$N_\mathrm{sol}^2$$ is large in the denominator, this can be approximately reduced to: $$\begin{aligned} @@ -206,18 +206,18 @@ $$\begin{aligned} = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \end{aligned}$$ -The expression $x \exp(-x^2)$ has its global extrema -$\pm 1 / \sqrt{2 e}$ at $x^2 = 1/2$. The maximum SPM frequency shift -achieved at $L_\mathrm{WB}$ is therefore given by: +The expression $$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}$$ -Interestingly, this expression does not contain $T_0$ at all, +Interestingly, this expression does not contain $$T_0$$ at all, so the achieved spectrum when SPM is halted by OWB is independent of the pulse width, -for sufficiently large $N_\mathrm{sol}$. +for sufficiently large $$N_\mathrm{sol}$$. ## References -- cgit v1.2.3