= Solution
<Monochromatic light> has one frequency, or a bandwidth sufficiently narrow for the intended measurement. A lamp can illuminate the experiment through a narrow spectral filter or <monochromator>, followed by a small source aperture at the focus of a collimating <lens>. The aperture supplies <spatial coherence> across the slits; the lens supplies approximately parallel incident wavefronts. Two independently illuminated incoherent slits would not provide a stationary fringe pattern.
In the <Fraunhofer approximation>, the optical path difference at observation angle $\theta$ is $s\sin\theta$. For equal single-slit intensities $\mathcal I_s$, adding the two wave amplitudes gives
$$
\boxed{\mathcal I(\theta)=4\mathcal I_s\cos^2\left(\frac{\pi s\sin\theta}{\lambda}\right),\qquad\sin\theta=\frac{Y}{\sqrt{D^2+Y^2}}.}
$$
The thin-slit idealization suppresses the individual diffraction envelope. If additionally $|Y|\ll D$, then $\sin\theta\simeq Y/D$ and $\mathcal I(Y)\simeq4\mathcal I_s\cos^2(\pi sY/(\lambda D))$. The far-field condition does not by itself impose this small observation angle.
For order $m=10$ at $Y=0.05\,\mathrm m$,
$$
\boxed{\lambda=\frac{s}{10}\frac{Y}{\sqrt{D^2+Y^2}}\simeq5.87\times10^{-7}\,\mathrm m.}
$$
The paraxial estimate is $sY/(10D)=5.88\times10^{-7}\,\mathrm m$, consistent to the precision of the data.
Let the upper slit contain the glass plate and take positive $Y$ towards that slit. The extra <optical path length> is $\Delta=(n_g-n_a)t=5.0\times10^{-5}\,\mathrm m$. Its total upper-minus-lower path difference is $\Delta-s\sin\theta$. The order-zero fringe therefore satisfies
$$
\boxed{\sin\theta_0=\frac\Delta s=\frac12,\qquad\theta_0=30^\circ,\qquad Y_0=D\tan\theta_0=0.491\,\mathrm m.}
$$
It moves \b[towards the covered slit]. Using $Y_0\simeq D\Delta/s=0.425\,\mathrm m$ here is a poor small-angle approximation: $30^\circ$ is not small. The order-zero fringe is being tracked as the optical delay is introduced; because the pattern is periodic in phase, many other bright fringes still lie near the original centre. Neglecting unequal glass transmission changes visibility but not the zero-path-difference condition.
\Image[/past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2016-resit-glass-slit-fringe-shift.png]
{title=Finite-angle fringe shift from a glass plate, with the order-zero fringe at thirty degrees and local fringes near the original centre}
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