Bandwidth Limited Pulse Calculator
Computes the bandwidth limited duration of an optical pulse from its spectrum 🛈 \begin{equation} a(t) = \left | a(t) \right | e^{i \phi(t)} \xrightarrow{\mathfrak{F}} A(\omega) = \left | A(\omega) \right | e^{i \psi(\omega)} \end{equation} Under bandwidth limited conditions; \begin{align} \psi(\omega) &= 0 \\ \Rightarrow a(t) &= \mathfrak{F}^{-1} \left [ \sqrt{\left | A(\omega) \right |^2} \right ] \end{align}
Pure client-side JavaScript. Source code openly available on GitHub
| Numerical 🛈 Full width at half maximum of the numerical data | Fit 🛈 Full width at half maximum of the fit to the numerical data | |||
|---|---|---|---|---|
| \( \Delta \nu \) [PHz] | ||||
| \( \Delta t \) [fs] | Gauss 🛈 \begin{align} a(t) &= e^{-(t/t_p)^2} \\ \left | A(\omega) \right | ^2 &= e^{-(\omega^2 t^2_p / 2)} \\ \Delta t &= 1.177 t_p \\ \Delta \nu \Delta t &= 2 \ln{(2)} / \pi \\ &= 0.441 \end{align} | Sech 🛈 \begin{align} \newcommand{\sech}{\mathop{\rm sech}\nolimits} a(t) &= \sech \left( \frac{t}{t_p} \right ) \\ \left | A(\omega) \right | ^2 &= \sech^2 \left( \frac{\pi \omega t_p}{2} \right ) \\ \Delta t &= 1.763 t_p \\ \Delta \nu \Delta t &= 0.315 \end{align} | ||
| \( \Delta \nu \Delta t \) [-] | 0.441 | 0.315 |