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Physics - Optics @@ -10,223 +10,470 @@ categories: layout: "concept" --- -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). +In fiber optics, **optical wave breaking (OWB)** is an effect +that can occur in light pulse envelopes $$A(z, t)$$ governed by +the [nonlinear Schrödinger equation](/know/concept/nonlinear-schrodinger-equation/): + +$$\begin{aligned} + 0 + &= i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma_0 |A|^2 A +\end{aligned}$$ + +OWB is caused by an interaction between +the [group velocity dispersion (GVD)](/know/concept/dispersive-broadening/) +caused by the $$\beta_2$$-term, +and the [self-phase modulation (SPM)](/know/concept/self-phase-modulation/) +caused by the $$\gamma_0$$ term. It only happens in the normal dispersion regime ($$\beta_2 > 0$$) -for pulses meeting a certain criterium, as we will see. +for pulses meeting certain criteria, as we shall 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. +In short, SPM creates low frequencies at the front of the pulse +and high ones at the back, and for $$\beta_2 > 0$$, +GVD makes 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 $$\omega_i(z, t) = -\ipdv{\phi}{t}$$ -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*: - -{% include image.html file="frequency-full.png" width="100%" - alt="Instantaneous frequency profile evolution" %} - -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: +To illustrate the resulting dynamics, +the simulated power $$|A|^2$$ of a Gaussian pulse with settings +$$T_0 = 100\:\mathrm{fs}$$, $$P_0 = 5\:\mathrm{kW}$$, +$$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$ +is plotted below as a function of $$z$$, +with the time domain on the left +and the frequency domain on the right: + +$$\begin{aligned} + A(0, t) + &= \sqrt{P_0} \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) +\end{aligned}$$ + +{% include image.html file="simulation-full.png" width="100%" + alt="Plot of optical wave breaking simulation results" %} + +OWB occurs at a distance called $$L_\mathrm{WB}$$, +and until that point things look relatively normal, +with SPM causing spectral broadening +and GVD causing subtle internal deformation in the time domain. +After $$L_\mathrm{WB}$$, the pulse suddenly explodes due to GVD, +and complicated so-called **sidelobes** appear in the frequency domain, +which seem to block any further SPM. +To investigate, we plot a series of spectrograms of the same simulation: {% include image.html file="spectrograms-full.png" width="100%" - alt="Spectrograms of pulse shape evolution" %} + alt="Spectrograms of simulated pulse shape evolution" %} + +At first, we see the appearance of SPM's typical "S" shape, +which quickly starts turning into a "Z" due to GVD. +When the transition to "Z" is complete, +there are many overlapping frequencies at the edges of the pulse. +This causes a complicated interaction +that generates the sidelobes, +and causes a train of small waves to "fall off" +the near-vertical pulse edges in the time domain, +hence the name *wave breaking*. +Eventually, those small waves melt together, +leaving behind a curious trapezoid shape +that gets stretched by GVD as usual. + +We would like to theoretically predict +the distance $$L_\mathrm{WB}$$ at which the wave breaks. +First we show the general principle, +and then we apply it to a couple of example pulses. + + + +## General method -We call the distance at which the wave breaks $$L_\mathrm{WB}$$, -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. +We make the following ansatz for the complex envelope $$A(z, t)$$, +without loss of generality: -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: +$$\begin{aligned} + A(z, t) + = \psi(z, t) \exp\!\big(i \phi(z, t)\big) +\end{aligned}$$ + +Inserting this into the NLS equation and dividing out $$e^{i \phi}$$ yields: $$\begin{aligned} - \omega_i(z,t) - &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t) - = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} - + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) + 0 + &= i \psi_z - \psi \phi_z - \frac{\beta_2}{2} (\psi_{tt} + 2 i \psi_t \phi_t + i \psi \phi_{tt} - \psi \phi_t^2) + \gamma_0 \psi^3 \end{aligned}$$ -Assuming that $$z$$ is small enough such that $$z^2 \approx 0$$, this -expression can be reduced to: +Since $$\psi$$ and $$\phi$$ are real by definition, +we can split this into its real and imaginary parts: $$\begin{aligned} - \omega_i(z,t) - \approx \frac{\beta_2 tz}{T_0^4} \bigg( 1 + 2\frac{\gamma P_0 T_0^2}{\beta_2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) - = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) + 0 + &= \psi_z - \frac{\beta_2}{2} (2 \psi_t \phi_t + \psi \phi_{tt}) + \\ + 0 + &= - \psi \phi_z - \frac{\beta_2}{2} (\psi_{tt} - \psi \phi_t^2) + \gamma_0 \psi^3 \end{aligned}$$ -Where we have assumed $$\beta_2 > 0$$, -and $$N_\mathrm{sol}$$ is the **soliton number**, -which is defined as: +For our purposes, the second equation is enough. +We divide it by $$\psi$$ to get an expression for $$\phi_z$$: $$\begin{aligned} - N_\mathrm{sol}^2 - \equiv \frac{L_D}{L_N} - = \frac{\gamma P_0 T_0^2}{|\beta_2|} + \phi_z + &= - \frac{\beta_2}{2} \frac{\psi_{tt}}{\psi} + \frac{\beta_2}{2} \Omega_i^2 + \gamma_0 \psi^2 \end{aligned}$$ -This quantity is very important in anomalous dispersion, -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, -the **instantaneous chirpyness** $$\xi_i$$, -has *two* real roots for $$t^2$$: +Where $$\Omega_i \equiv -\phi_t$$ is the **instantaneous frequency**, +also called the **frequency-chirp variation**, +which describes the dominant frequency component at a given point $$(z, t)$$; +basically the center of the spectrograms shown earlier. +For small $$z$$, this gives us a linear approximation of $$\phi$$: $$\begin{aligned} - 0 - = \xi_i(z,t) - = \pdv{\omega_i}{t} - &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) - \equiv \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big) + \phi(z, t) + &\approx \bigg( \!-\! \frac{\beta_2}{2} \frac{\psi_{tt}}{\psi} + + \frac{\beta_2}{2} \Omega_i^2 + \gamma_0 \psi^2 \bigg)\bigg|_{z = 0} z + + \phi(0, t) \end{aligned}$$ -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 $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$: +And therefore $$\Omega_i$$ is as follows, +assuming no initial chirp variation $$\Omega_i(0, t) = 0$$: $$\begin{aligned} \boxed{ - N_\mathrm{min}^2 - = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) - \approx 1.12 + \Omega_i(z, t) + = -\pdv{\phi}{t} + \approx \bigg( \frac{\beta_2}{2} \frac{\psi_{ttt}}{\psi} + - \frac{\beta_2}{2} \frac{\psi_{tt} \psi_t}{\psi^2} + - 2 \gamma_0 \psi \psi_t \bigg) \bigg|_{z = 0} z } \end{aligned}$$ -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. -GVD is unable to keep up with SPM, so OWB will not occur. +Once we have $$\Omega_i$$ for a known input pulse, +we can check whether OWB is even possible under the given circumstances: +$$\Omega_i$$ must be non-monotonic, +i.e. $$\ipdv{\Omega_i}{t} = 0$$ must have a solution. +In other words, there must be a sufficiently prominent "bump" in $$\Omega_i$$ +that gets pulled away by GVD faster than its surroundings, +until those more-off-center frequencies overtake +less-off-center ones and lead to the overlap +that generates the sidelobes and other OWB phenomena. + +Let us assume that OWB will occur. +Consider two parts of the pulse, located $$t_1$$ and $$t_2$$ for $$z = 0$$, +so separated by a small initial interval $$\Delta{t} \equiv t_2 - t_1$$. +Due to $$\Omega_i$$ there is a frequency difference between these points, +causing $$\Delta{t}$$ to change by an amount $$\tau$$ +after the pulse has propagated a short distance $$z$$, +estimated as follows: + +$$\begin{alignedat}{2} + \tau + &\approx z \Delta\beta_1 + \approx z \pdv{\beta_1}{\Omega} \Delta{\Omega_i} + = z \beta_2 \Delta\Omega_i + \approx z \beta_2 \pdv{\Omega_i}{t} \Delta{t} +\end{alignedat}$$ + +Where $$\Delta\Omega_i \equiv \Omega_i(z,t_2) - \Omega_i(z,t_1)$$, +and $$\Delta{\beta_1}$$ is the difference in inverse group velocity $$\beta_1(\Omega)$$ +between $$t_2$$ and $$t_1$$, specifically +$$\Delta\beta_1 \equiv \beta_1(\Omega_i(z,t_2)) - \beta_1(\Omega_i(z,t_1))$$. +OWB takes place when $$t_1$$ and $$t_2$$ catch up to each other, +which is when $$\tau = -\Delta{t}$$. +In that case, we have: + +$$\begin{aligned} + z + = - \frac{1}{\beta_2 \displaystyle\pdv{\Omega_i}{t}} +\end{aligned}$$ -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 $$\Delta t$$, -estimated by: +Assuming $$\beta_2 > 0$$, +this implies that the wave starts breaking first +at the $$t$$-values where $$\Omega_i$$ has its most negative slope +(note that for a symmetric input pulse, +$$\ipdv{\Omega_i}{t}$$ is also symmetric, +so OWB will occur simultaneous on both sides). +We can therefore write an equation for $$L_\mathrm{WB}$$ like so, +valid for any input pulse shape +for which we know $$\Omega_i(z, t)$$: $$\begin{aligned} - \Delta t - &\approx z \Delta\beta_1 - \qquad - &&\Delta\beta_1 - \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1)) + \boxed{ + L_\mathrm{WB} + = - \frac{1}{\beta_2 \: \mathrm{min}_t\bigg\{ \displaystyle\pdv{\Omega_i}{t} \Big|_{z = L_\mathrm{WB}} \bigg\}} + } +\end{aligned}$$ + +Let us apply this method to a few specific examples: +a Gaussian input pulse, and a soliton-shaped one +(keeping in mind that true [bright solitons](/know/concept/optical-soliton/) +do not exist for $$\beta_2 > 0$$). + + + +## Gaussian pulse + +For a Guassian input, the amplitude $$\psi$$ is as follows +in our ansatz $$A = \psi e^{i \phi}$$: + +$$\begin{aligned} + \psi(0, t) + &= \sqrt{P_0} \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) +\end{aligned}$$ + +For reference, its relevant $$t$$-derivatives are given by: + +$$\begin{aligned} + \psi_t(0, t) + &= - \frac{\sqrt{P_0}}{T_0^2} t \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \\ - &\approx z \beta_2 \Delta\omega_i - \qquad - &&\Delta\omega_i - \equiv \omega_i(z,t_2) - \omega_i(z,t_1) + \psi_{tt}(0, t) + &= \frac{\sqrt{P_0}}{T_0^2} \bigg( \frac{t^2}{T_0^2} - 1 \bigg) \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \\ - &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1) - \qquad \quad - &&\Delta\xi_i - \equiv \xi_i(z,t_2) - \xi_i(z,t_1) + \psi_{ttt}(0, t) + &= \frac{\sqrt{P_0}}{T_0^4} \bigg( 3 - \frac{t^2}{T_0^2} \bigg) t \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \end{aligned}$$ -Where $$\beta_1(\omega)$$ is the inverse of the group velocity. -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 $$L_\mathrm{WB}$$ at which this happens first -must satisfy the following condition for some value of $$t$$: +Substituting these into our general linear approximation +of $$\Omega_i$$ leads us to: $$\begin{aligned} - L_\mathrm{WB} \: \beta_2 \: \xi_i(L_\mathrm{WB}, t) - = -1 - \qquad \implies \qquad - L_\mathrm{WB}^2 - = - \frac{T_0^4}{\beta_2^2 \: f(t^2/T_0^2)} + \Omega_i(z, t) + &= z \frac{\beta_2 t}{T_0^4} \bigg( 1 + 2 \frac{\gamma_0 P_0 T_0^2}{\beta_2} \exp\!\Big( \!-\!\frac{t^2}{T_0^2} \Big) \bigg) +\end{aligned}$$ + +Since we are in the normal dispersion regime, $$\beta_2 > 0$$, +so we can recognize the **soliton number** $$N_\mathrm{sol}$$ here, +which is a useful measure of the relative strengths of GVD and SPM: + +$$\begin{aligned} + N_\mathrm{sol}^2 + \equiv \frac{\gamma_0 P_0 T_0^2}{|\beta_2|} + = \frac{L_D}{L_N} +\end{aligned}$$ + +We thus have the following expression for $$\Omega_i$$, +sketched below for several values of $$N_\mathrm{sol}$$: + +$$\begin{aligned} + \Omega_i(z, t) + &= z \frac{\beta_2 t}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) +\end{aligned}$$ + +{% include image.html file="gauss-omega-full.png" width="75%" + alt="Sketch of instantaneous frequency of Gaussian pulse" %} + +At a certain value of $$N_\mathrm{sol}$$, which we call $$N_\mathrm{min}$$, +we see that $$\Omega_i$$ transitions from having no extrema, +to having a local minimum and maximum with respect to $$t^2$$. +Those "bumps" get pulled outward by GVD as indicated by the arrows, +steepening the outer edges until the slope becomes infinite, +at which point OWB occurs. +However, for $$N_\mathrm{sol} < N_\mathrm{min}$$, +the bumps are not prominent enough: +the peaks cannot catch up to the outer edges, +so OWB can never happen. + +We would like to find $$N_\mathrm{min}$$. +To do so, we demand that $$\Omega_i$$ has local extrema +where the derivative $$\ipdv{\Omega_i}{t}$$ vanishes, as illustrated below. +Abbreviating $$f(x) \equiv (1 - 2x) e^{-x}$$: + +$$\begin{aligned} + 0 + = \pdv{\Omega_i}{t} + &= z \frac{\beta_2}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) + \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) + \\ + &= z \frac{\beta_2}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \: f\Big(\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}$$ -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 $$f(3/2)$$: +{% include image.html file="gauss-domegadt-full.png" width="75%" + alt="Sketch of derivative of instantaneous frequency of Gaussian pulse" %} + +Here we see that as $$N_\mathrm{sol}$$ increases, +it pulls down the minimum of $$f(x)$$ until it hits the horizontal axis +when $$N_\mathrm{sol} = N_\mathrm{min}$$. +We should therefore find the location $$x_\mathrm{min}$$ of this minimum: + +$$\begin{aligned} + 0 + = f'(x) + = (2 x - 3) e^{-x} + \qquad\implies\qquad + x_\mathrm{min} + = \frac{3}{2} +\end{aligned}$$ + +So the corresponding minimum value of $$f(x)$$ is given by: $$\begin{aligned} f_\mathrm{min} - = f(3/2) - = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2) - = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2 + = f(x_\mathrm{min}) + = -2 e^{-3/2} +\end{aligned}$$ + +Inserting this into our demand that $$\ipdv{\Omega_i}{t} = 0$$ +yields a simple expression for $$N_\mathrm{min}$$: + +$$\begin{aligned} + 0 + = 1 + 2 N_\mathrm{min}^2 \: f_\mathrm{min} + \qquad\implies\qquad + \boxed{ + N_\mathrm{min}^2 + = \frac{e^{3/2}}{4} + \approx 1.12 + } +\end{aligned}$$ + +If $$N_\mathrm{sol}^2 < N_\mathrm{min}^2$$, +then our demand cannot be satisfied: +$$\Omega_i$$ cannot overtake itself, +GVD is unable to keep up with SPM, and OWB cannot occur. +From now on, we assume $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$. + +We now have everything we need to calculate the OWB distance $$L_\mathrm{WB}$$ +using its general recipe. +Inserting $$\ipdv{\Omega_i}{t}$$, +whose minimum we already know, we get: + +$$\begin{aligned} + L_\mathrm{WB}^2 + = - \frac{T_0^4}{\beta_2^2 (1 + 2 N_\mathrm{sol}^2 f_\mathrm{min})} + = \frac{T_0^4}{\beta_2^2 (N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1)} \end{aligned}$$ -Clearly, $$f_\mathrm{min} \ge 0$$ when $$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$$, -which, when inserted above, leads to an imaginary $$L_\mathrm{WB}$$, -confirming that OWB cannot occur in that case. -Otherwise, if $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$, then: +Leading to the following prediction for $$L_\mathrm{WB}$$, +which appears to agree well with the OWB +observed in the simulation shown earlier. +Note that if $$N_\mathrm{sol} < N_\mathrm{min}$$ +then $$L_\mathrm{WB}$$ is imaginary, +confirming that OWB is not possible in that situation: $$\begin{aligned} \boxed{ L_\mathrm{WB} - = \frac{T_0^2}{\beta_2 \, \sqrt{- f_\mathrm{min}}} - = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}} + = \frac{T_0^2}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}} } \end{aligned}$$ -This prediction for $$L_\mathrm{WB}$$ appears to agree well -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 $$\omega$$-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: +## Soliton-shaped pulse + +Although solitons do not exist in the normal dispersion regime, +we can still create pulses with the same shape, given by: $$\begin{aligned} - \omega_{\mathrm{SPM}}(L_\mathrm{WB},t) - = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) + \psi(0, t) + &= \sqrt{P_0} \sech\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$ -Assuming that $$N_\mathrm{sol}^2$$ is large in the denominator, this can -be approximately reduced to: +For reference, we also calculate its relevant $$t$$-derivatives: $$\begin{aligned} - \omega_\mathrm{SPM}(L_\mathrm{WB}, t) - \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) - = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) + \psi_t(0, t) + &= - \frac{\sqrt{P_0}}{T_0} \tanh\!\Big( \frac{t}{T_0} \Big) \sech\!\Big( \frac{t}{T_0} \Big) + \\ + \psi_{tt}(0, t) + &= \frac{\sqrt{P_0}}{T_0^2} \bigg( \tanh^2\!\Big( \frac{t}{T_0} \Big) - \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg) + \sech\!\Big( \frac{t}{T_0} \Big) + \\ + \psi_{ttt}(0, t) + &= \frac{\sqrt{P_0}}{T_0^3} \bigg( 5 \sech^2\!\Big( \frac{t}{T_0} \Big) - \tanh^2\!\Big( \frac{t}{T_0} \Big) \bigg) + \tanh\!\Big( \frac{t}{T_0} \Big) \sech\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$ -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: +Substituting these into our general linear approximation of $$\Omega_i$$, +and once again recognizing the soliton number $$N_\mathrm{sol}$$, +leads us to the following function, sketched below: $$\begin{aligned} - \omega_\mathrm{max} - = \sqrt{\frac{2 \gamma P_0}{e \beta_2}} + \Omega_i(z, t) + &= z \frac{2 \beta_2}{T_0^3} \big( 1 + N_\mathrm{sol}^2 \big) + \sech^2\!\Big( \frac{t}{T_0} \Big) \tanh\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$ -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 $$N_\mathrm{sol}$$. +{% include image.html file="sech-omega-full.png" width="75%" + alt="Sketch of instantaneous frequency of soliton-shaped pulse" %} + +Curiously, this $$\Omega_i$$ is non-monotonic for all $$N_\mathrm{sol}$$, +so OWB occurs even in the linear limit $$N_\mathrm{sol} \to 0$$. +This suggests that OWB is not an inherently nonlinear effect, +instead happening as long as there are bumps in $$\Omega_i$$, +regardless of their origin (SPM or simply the pulse shape). + +We do not care where those local extrema are, only that they exist, +so we move on immediately to finding where $$\Omega_i$$ +has its most negative slope, +which is at some (but not all) solutions of: + +$$\begin{aligned} + 0 + &= \pdvn{2}{\Omega_i}{t} + \\ + &= z \frac{8 \beta_2}{T_0^5} \big( 1 + N_\mathrm{sol}^2 \big) + \bigg( \tanh^2\!\Big( \frac{t}{T_0} \Big) - 2 \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg) + \sech^2\!\Big( \frac{t}{T_0} \Big) \tanh\!\Big( \frac{t}{T_0} \Big) +\end{aligned}$$ + +One solution is clearly $$t = 0$$ because $$\tanh(0) = 0$$, +but from the plot we can see that $$\Omega_i$$'s slope is positive there, +so we must continue our search. +The next candidate is: + +$$\begin{aligned} + 0 + &= \tanh^2(x) - 2 \sech^2(x) + \\ + &= 3 \tanh^2(x) - 2 +\end{aligned}$$ + +Where we have used the standard identity $$\sech^2(x) + \tanh^2(x) = 1$$. +Isolating for $$x$$ and writing out $$\tanh^{-1}(x)$$ as a logarithm yields: + +$$\begin{aligned} + x + &= \tanh^{-1}\!\bigg( \!\pm\!\sqrt{\frac{2}{3}}\bigg) + \\ + &= \frac{1}{2} \ln\!\bigg( \frac{1 \pm \sqrt{2/3}}{1 \mp \sqrt{2/3}} \bigg) + \\ + &= \frac{1}{2} \ln\!\bigg( \frac{\sqrt{3} \pm \sqrt{2}}{\sqrt{3} \mp \sqrt{2}} \bigg) + \\ + &= \frac{1}{2} \ln\!\bigg( \frac{(\sqrt{3} \pm \sqrt{2})^2}{(\sqrt{3} \mp \sqrt{2}) (\sqrt{3} \pm \sqrt{2})} \bigg) + \\ + &= \frac{1}{2} \ln(5 \pm 2 \sqrt{6}) +\end{aligned}$$ + +Note that $$\ln(5 \!+\! 2 \sqrt{6}) = - \ln(5 \!-\! 2 \sqrt{6}) \equiv 2 x_0$$. +The values of $$\sech$$ and $$\tanh$$ are given by: + +$$\begin{aligned} + \sech(\pm x_0) + = \frac{1}{\sqrt{3}} + \qquad\qquad + \tanh(\pm x_0) + = \pm \sqrt{\frac{2}{3}} +\end{aligned}$$ + +The minimum value of the slope $$\ipdv{\Omega_i}{t}$$ is therefore as follows: + +$$\begin{aligned} + \mathrm{min}_t\bigg\{ \displaystyle\pdv{\Omega_i}{t} \bigg\} + &= z \frac{2 \beta_2}{T_0^4} (1 + N_\mathrm{sol}^2) + \bigg( \sech^2\!\Big( \frac{t}{T_0} \Big) - 2 \tanh^2\!\Big( \frac{t}{T_0} \Big) \bigg) \sech^2\!\Big( \frac{t}{T_0} \Big) + \bigg|_{t = x_0 T_0} + \\ + &= - z \frac{2 \beta_2}{3 T_0^4} \big( 1 + N_\mathrm{sol}^2 \big) +\end{aligned}$$ + +Inserting this into $$L_\mathrm{WB}$$'s general equation, +we find that OWB occurs at a distance with a similar +$$T_0^2 / \beta_2$$-dependence as for the Gaussian pulse, +confirming that OWB is mostly linear: + +$$\begin{aligned} + \boxed{ + L_\mathrm{WB} + = \frac{\sqrt{3} T_0^2}{\beta_2 \sqrt{2 + 2 N_\mathrm{sol}^2}} + } +\end{aligned}$$ @@ -237,4 +484,3 @@ for sufficiently large $$N_\mathrm{sol}$$. 2. A.M. Heidt, A. Hartung, H. 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