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diff --git a/content/know/concept/optical-wave-breaking/index.pdc b/content/know/concept/optical-wave-breaking/index.pdc new file mode 100644 index 0000000..3c509fe --- /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. + |