A pulse launched into a fibre with group-velocity dispersion has propagated far…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Optics & Photonics · Waves in two and three dimensions, dispersion and wave packets

Modes, Packets and Media, Ch Dispersion and the propagation of wave packets

Problem

A pulse launched into a fibre with group-velocity dispersion

\beta_2 = -21.7\,\mathrm{ps^2\,km^{-1}}

has propagated far enough (

z \gg L_{\rm D}

) that its envelope solution gives a local angular frequency varying linearly across the pulse,

\omega(\tau,z) = \omega_0 + \tau/(\beta_2 z)

, where

\tau

is the local time measured from the pulse centre and

\omega_0

the carrier frequency (this is the time-domain form of the late-time spatial chirp

k_0 + \xi/(\omega''t)

found for the envelope equation, with

x \leftrightarrow t

k \leftrightarrow \omega

\omega'' \leftrightarrow \beta_2

). Its spectrogram at the fibre output shows the instantaneous ordinary frequency

f

(with

\omega=2\pi f

)

8.0\,\mathrm{GHz}

above the carrier at

\tau = -30\,\mathrm{ps}

and

8.0\,\mathrm{GHz}

below the carrier at

\tau = +30\,\mathrm{ps}

. ((a)) Read off the chirp rate

\mathrm{d}\omega/\mathrm{d}\tau

from these two points. ((b)) Use it, with the given

\beta_2

, to find the length

z

of fibre the pulse has passed through.

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