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author | Prefetch | 2022-10-20 18:25:31 +0200 |
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committer | Prefetch | 2022-10-20 18:25:31 +0200 |
commit | 16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch) | |
tree | 76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/dispersive-broadening | |
parent | e5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff) |
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/dispersive-broadening')
-rw-r--r-- | source/know/concept/dispersive-broadening/index.md | 38 |
1 files changed, 19 insertions, 19 deletions
diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md index 816135a..4e4cf82 100644 --- a/source/know/concept/dispersive-broadening/index.md +++ b/source/know/concept/dispersive-broadening/index.md @@ -12,11 +12,11 @@ layout: "concept" In optical fibers, **dispersive broadening** is a (linear) effect where group velocity dispersion (GVD) "smears out" a pulse in the time domain due to the different group velocities of its frequencies, -since pulses always have a non-zero width in the $\omega$-domain. +since pulses always have a non-zero width in the $$\omega$$-domain. No new frequencies are created. -A pulse envelope $A(z, t)$ inside a fiber must obey the nonlinear Schrödinger equation, -where the parameters $\beta_2$ and $\gamma$ respectively +A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation, +where the parameters $$\beta_2$$ and $$\gamma$$ respectively control dispersion and nonlinearity: $$\begin{aligned} @@ -24,7 +24,7 @@ $$\begin{aligned} = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A \end{aligned}$$ -We set $\gamma = 0$ to ignore all nonlinear effects, +We set $$\gamma = 0$$ to ignore all nonlinear effects, and consider a Gaussian initial condition: $$\begin{aligned} @@ -32,10 +32,10 @@ $$\begin{aligned} = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \end{aligned}$$ -By [Fourier transforming](/know/concept/fourier-transform/) in $t$, -the full analytical solution $A(z, t)$ is found to be as follows, +By [Fourier transforming](/know/concept/fourier-transform/) in $$t$$, +the full analytical solution $$A(z, t)$$ is found to be as follows, where it can be seen that the amplitude -decreases and the width increases with $z$: +decreases and the width increases with $$z$$: $$\begin{aligned} A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}} @@ -43,9 +43,9 @@ $$\begin{aligned} \end{aligned}$$ To quantify the strength of dispersive effects, -we define the dispersion length $L_D$ -as the distance over which the half-width at $1/e$ of maximum power -(initially $T_0$) increases by a factor of $\sqrt{2}$: +we define the dispersion length $$L_D$$ +as the distance over which the half-width at $$1/e$$ of maximum power +(initially $$T_0$$) increases by a factor of $$\sqrt{2}$$: $$\begin{aligned} T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2} @@ -56,17 +56,17 @@ $$\begin{aligned} \end{aligned}$$ This phenomenon is illustrated below for our example of a Gaussian pulse -with parameter values $T_0 = 1\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$, -$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0$: +with parameter values $$T_0 = 1\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$, +$$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0$$: <a href="pheno-disp.jpg"> <img src="pheno-disp-small.jpg" style="width:100%"> </a> -The **instantaneous frequency** $\omega_\mathrm{GVD}(z, t)$, +The **instantaneous frequency** $$\omega_\mathrm{GVD}(z, t)$$, which describes the dominant angular frequency at a given point in the time domain, is found to be as follows for the Gaussian pulse, -where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$: +where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$$: $$\begin{aligned} \omega_{\mathrm{GVD}}(z,t) @@ -74,20 +74,20 @@ $$\begin{aligned} = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2} \end{aligned}$$ -This expression is linear in time, and depending on the sign of $\beta_2$, +This expression is linear in time, and depending on the sign of $$\beta_2$$, frequencies on one side of the pulse arrive first, and those on the other side arrive last. -The effect is stronger for smaller $T_0$: +The effect is stronger for smaller $$T_0$$: this makes sense, since short pulses are spectrally wider. The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/) leads to many interesting effects, such as [modulational instability](/know/concept/modulational-instability/) and [optical wave breaking](/know/concept/optical-wave-breaking/). -Of great importance is the sign of $\beta_2$: -in the **anomalous dispersion regime** ($\beta_2 < 0$), +Of great importance is the sign of $$\beta_2$$: +in the **anomalous dispersion regime** ($$\beta_2 < 0$$), lower frequencies travel more slowly than higher ones, -and vice versa in the **normal dispersion regime** ($\beta_2 > 0$). +and vice versa in the **normal dispersion regime** ($$\beta_2 > 0$$). |