A metropolitan optical link is of standard single-mode fibre with group-velocity…

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

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

Problem

A metropolitan optical link is 80 km80\,\mathrm{km} of standard single-mode fibre with group-velocity dispersion β2=−21.7 ps2 km−1\beta_2 = -21.7\,\mathrm{ps^2\,km^{-1}} at the operating wavelength. The transmitter launches transform-limited Gaussian pulses of characteristic width T0=5.0 psT_0 = 5.0\,\mathrm{ps}; the pulse duration at distance zz obeys T(z)=T01+(z/LD)2,LD=T02∣β2∣.T(z) = T_0\sqrt{1+(z/L_{\rm D})^2}, \qquad L_{\rm D}=\frac{T_0^2}{|\beta_2|}. ((a)) Find the dispersion length LDL_{\rm D} of this link. ((b)) Find the pulse width on arrival, after the full 80 km80\,\mathrm{km}. ((c)) A length of dispersion-compensating fibre with β2=+120 ps2 km−1\beta_2 = +120\,\mathrm{ps^2\,km^{-1}} is spliced onto the end of the link, exactly cancelling the accumulated β2L\beta_2L product of the whole path. Find the length of compensating fibre required, and state the pulse width after it, idealising both fibres as lossless and ignoring dispersion beyond the quadratic term.

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