Question $4(10\%)$

For the propagation of a Gaussian pulse in a dispersive optical fiber, the Fourier transform of the input pulse is given as:

Ei$($f$)=2$$801020$exp$($$2$f$2801020)$V$/$m$.$E$\_$i$($f$)=2\backslash$sqrt$\{\backslash$frac$\{\backslash$pi$\}\{80\backslash$times $10^\{20\}\}\}\backslash$exp$\backslash$left$(-\backslash$frac$\{\backslash$pi$^2$ f$^2\}\{80\backslash$times $10^\{20\}\}\backslash$right$)\backslash ,\backslash$text$\{$V$/$m$\}.$Ei$($f$)=2801020$$$exp$(801020$$2$f$2)$V$/$m$.$

$($A$)$ Find the spectral width of the optical pulse. The full width at half maximum bandwidth is defined as $2$f$2$f$2$f$,$ where fff is the frequency at which the term exp$\backslash$expexp is equal to $0\mathrm{.}5.$

$($B$)$ For D$=10$ps$/($km$-$nm$)$D $=10\backslash ,\backslash$text$\{$ps$\}/(\backslash$text$\{$km$-$nm$\})$D$=10$ps$/($km$-$nm$),$ find the length of the fiber that leads to a pulse spread of $63\mathrm{.}6$ ps$.$