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diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md new file mode 100644 index 0000000..64c251a --- /dev/null +++ b/source/know/concept/dispersive-broadening/index.md @@ -0,0 +1,96 @@ +--- +title: "Dispersive broadening" +date: 2021-02-27 +categories: +- Physics +- Optics +- Fiber optics +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. +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 +control dispersion and nonlinearity: + +$$\begin{aligned} + 0 + = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A +\end{aligned}$$ + +We set $\gamma = 0$ to ignore all nonlinear effects, +and consider a Gaussian initial condition: + +$$\begin{aligned} + A(0, t) + = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) +\end{aligned}$$ + +By [Fourier transforming](/know/concept/fourier-transform/) in $t$, +the full analytical solution $A(z, t)$ is found to be as follows, +where it can be seen that the amplitude +decreases and the width increases with $z$: + +$$\begin{aligned} + A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}} + \exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg) +\end{aligned}$$ + +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 $\sqrt{2}$: + +$$\begin{aligned} + T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2} + \qquad \implies \qquad + \boxed{ + L_D = \frac{T_0^2}{|\beta_2|} + } +\end{aligned}$$ + +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 $\gamma = 0$: + +<a href="pheno-disp.jpg"> +<img src="pheno-disp-small.jpg" style="width:100%"> +</a> + +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 $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$: + +$$\begin{aligned} + \omega_{\mathrm{GVD}}(z,t) + = \pdv{}{t}\Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big) + = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2} +\end{aligned}$$ + +This expression is linear in time, and depending on the sign of $\beta_2$, +frequencies on one side of the pulse arrive first, +and those on the other side arrive last. +The effect is stronger for smaller $T_0$: +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** ($\beta_2 < 0$), +lower frequencies travel more slowly than higher ones, +and vice versa in the **normal dispersion regime** ($\beta_2 > 0$). + + + +## References +1. O. Bang, + *Numerical methods in photonics: lecture notes*, 2019, + unpublished. diff --git a/source/know/concept/dispersive-broadening/pheno-disp-small.jpg b/source/know/concept/dispersive-broadening/pheno-disp-small.jpg Binary files differnew file mode 100644 index 0000000..8c70eac --- /dev/null +++ b/source/know/concept/dispersive-broadening/pheno-disp-small.jpg diff --git a/source/know/concept/dispersive-broadening/pheno-disp.jpg b/source/know/concept/dispersive-broadening/pheno-disp.jpg Binary files differnew file mode 100644 index 0000000..a97312b --- /dev/null +++ b/source/know/concept/dispersive-broadening/pheno-disp.jpg |