= Solution
The <wave impedance> is the travelling-wave field ratio $Z=E/H$; for the stated nonmagnetic media it is $Z_0/n$. A <reflection coefficient> is a reflected-to-incident <complex amplitude> ratio with an explicitly chosen field orientation. With this <reflection-amplitude sign convention>, using the question's formula for incidence inside silicon gives
$$
\boxed{r=\frac{Z_0/3.5-Z_0}{Z_0/3.5+Z_0}=-\frac59},\qquad R=|r|^2=\frac{25}{81}.
$$
With a common tangential electric-field direction, the usual electric-field <Fresnel equations> instead give $r_E=(Z_2-Z_1)/(Z_2+Z_1)=+5/9$ for this incidence. The printed formula corresponds to the opposite reflected-field sign convention, for example the magnetic amplitude convention. It gives the same <reflectance> and the same round-trip factor $r^2$.
Each extra internal round trip multiplies the transmitted amplitude by $r^2e^{i\delta}$, with $\delta=4\pi nd/\lambda$. A <geometric series> therefore gives
$$
A_{\rm out}=A_{\rm first}\sum_{j=0}^\infty(r^2e^{i\delta})^j=\frac{A_{\rm first}}{1-r^2e^{i\delta}},\qquad
\boxed{I\propto\frac1{1+R^2-2R\cos\delta}.}
$$
This is <Fabry–Pérot transmission from multiple reflections>. Assuming lossless silicon with wavelength-independent $n$, the full <intensity> transmittance between air regions is
$$
T=\frac{(1-R)^2}{(1-R)^2+4R\sin^2(\delta/2)}.
$$
Maxima have $2nd/\lambda=j$, minima $2nd/\lambda=j+\tfrac12$. Here $2nd=7000\,\mathrm{nm}$, so within the specified tuning range the only maximum is \b[$1400\,\mathrm{nm}$], and the two minima are \b[$1555.6\,\mathrm{nm}$] and \b[$1272.7\,\mathrm{nm}$]. The contrast is
$$
\boxed{\frac{I_{\max}}{I_{\min}}=\left(\frac{1+R}{1-R}\right)^2=\left(\frac{53}{28}\right)^2=3.583.}
$$
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2011/paper-1-etalon.png]
{title=Transmission contrast and resonance narrowing with reflectivity}
{description=Original ideal lossless symmetric-etalon curves. The coating example uses chosen R=0.90, unchanged optical thickness and zero additional round-trip phase.}
Identical highly reflecting coatings increase the <finesse>: transmission maxima become narrower and the <intensity> between them falls. Ideal symmetric lossless coatings retain $T_{\max}=1$; absorption or unequal mirrors reduce it. The illustrated maxima stay fixed only because additional reflection <wave phases> are held unchanged. Real coating <wave phases> can also shift the <resonances>.
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