= Solution
Let $I_0$ be the <intensity> from either fibre alone at the observation point, assuming equal output <intensities>, matching <polarizations> and negligible drift. The screen samples the relative <phase> $\Phi(X)$ between the two outputs. For an ideal monochromatic <laser>, <first-order optical coherence> is complete and
$$
\boxed{I_{\rm ideal}(X)=2I_0[1+\cos\Phi(X)]},\qquad I_{\min}=0,\quad I_{\max}=4I_0.
$$
The <interference fringes> are approximately equally spaced in the small-angle far field, with a slowly varying <diffraction> envelope. If the output separation is $a$ and the screen distance is $D$, the <interference fringe spacing> is approximately $\lambda_0D/a$. Equal phases at the two \i[input] ends do not ensure a central bright fringe: the extra fibre contributes carrier <phase> $2\pi n_{\rm eff}(l_2-l_1)/\lambda_0$ in addition to the free-space <optical path difference>.
For a finite linewidth, the time-averaged <interference> signal becomes
$$
I(X)=2I_0\{1+|g^{(1)}(\tau)|\cos[\Phi(X)+\arg g^{(1)}(\tau)]\},\qquad V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=|g^{(1)}(\tau)|.
$$
Here $g^{(1)}$ may be defined for the slowly varying field envelope, with the carrier <phase> already included in $\Phi$. A realistic <laser> therefore gives reduced <interferometric visibility>; for $V=1/2$, its extrema are $I_0$ and $3I_0$.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2017-q1-fibre-interference.png]
{title=Ideal and finite-linewidth interference for equal output intensities}
For a <Lorentzian line shape> of <frequency> <full width at half maximum> $\Delta\nu$, <temporal optical coherence> gives
$$
|g^{(1)}(\tau)|=e^{-\pi\Delta\nu|\tau|},\qquad \tau=\frac{n_g(l_2-l_1)}c.
$$
The relevant index is the <group index>. Using the intended nondispersive approximation $n_g\simeq n_{\rm eff}=2$,
$$
\tau=\frac{20\,\mathrm m}{c}=66.71\,\mathrm{ns},\qquad \boxed{\Delta\nu=\frac{\log2}{\pi\tau}=3.31\,\mathrm{MHz}\quad\text{(FWHM)}.}
$$
This <delayed-fibre linewidth measurement> assumes the loss of contrast comes from linewidth. Unequal output <intensities>, polarization mismatch or mechanical <phase> fluctuations also lower <interferometric visibility>. If <dispersion> matters, the stated <phase> index alone does not specify the necessary <group index>.
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