$3.$ Optical Resonator Sensor: Consider a Fabry$-$Perot resonator with a round$-$trip distance $(\backslash (2$ l $\backslash$cos $\backslash$theta $\backslash )$ in class$)$ of $\backslash ($ L$=1\mathrm{.}5\backslash$mathrm$\{$~cm$\}\backslash ).$ The Fabry$-$Perot resonator has a mirror reflectivity of $\backslash ($ R$=\backslash )0\mathrm{.}95,$ and assume the medium inside the resonator is lossless air $(\backslash ($ n$=1\backslash )).$ In this problem, we design a sensor based on this Fabry$-$Perot resonator using light at vacuum wavelengths near $1550$ nm $.(35$ points$)$

a$.$ We know this resonator should have a number of equally spaced $($in frequency$)$ resonances. We are particularly interested in the resonance that is positioned near $1550$ nm wavelength. Calculate the resonant wavelength which is closest to $1550$ nm $.(5$ points$)$

b$.$ Estimate the linewidth $($FWHM$,$ in terms of wavelength$),$ Finesse and $\backslash ($ Q $\backslash )$ factor of the resonator. $(5$ points$)$

c$.$ Estimate the $\backslash ($ Q $\backslash )$ factor due to loss from imperfect mirror reflection $\backslash ($ R$\backslash$left$($Q$\_\{\backslash$mathrm$\{$r$\}\}\backslash$right$)\backslash )$ and the $\backslash ($ Q $\backslash )$ factor due to propagation absorption $\backslash (\backslash$left$($Q$\_\{\backslash$mathrm$\{$a$\}\}\backslash$right$)\backslash ).(5$ points$)$

d$.$ Upon a global refractive index $($inside the entire resonator$)$ change, you would expect the resonant wavelength of the resonator to shift accordingly, therefore acting as a refractive index sensor$.$ The sensitivity of this optical sensor is defined as the resonant wavelength shift per refractive index change, i$.$e$.\backslash ($ S$=\backslash$Delta $\backslash$lambda $/\backslash$Delta n $\backslash ).$ Calculate the sensitivity of this Fabry$-$Perot resonator in terms of nm$/$RIU $($nanometer per refractive index unit$).(5$ points$)$

e$.$ Now we add an object $($to be sensed$)$ inside the air$-$filled Fabry$-$Perot resonator. The object has a length of $\backslash (\backslash$Delta L $\backslash ),$ a refractive index of $\backslash (1+\backslash$Delta n $\backslash ),$ and a propagation $($power$)$ loss rate of $\backslash (\backslash$alpha $\backslash ).$ Qualitatively describe how $\backslash (\backslash$Delta n $\backslash )$ and $\backslash (\backslash$alpha $\backslash )$ will affect the resonance transmission characteristic, respectively, and provide physical explanation for the effects. $(5$ points$)$

f$.$ The detection limit of this sensor is defined as the smallest object size $($in terms of length$)$ that can be detected. Here we only use the resonant wavelength shift information for sensing$,$ and assume the smallest detectable resonance wavelength shift is equal to the linewidth $($FWHM$),\backslash (\backslash$Delta n$=0\mathrm{.}5\backslash )$ and $\backslash (\backslash$alpha$=10\backslash$mathrm$\{$~cm$\}^\{-1\}\backslash ),$ calculate the detection limit of the sensor$.(5$ points$)$

Note: light passes through this object twice in one round trip.

g$.$ Assume we have inserted an object with the size of the detection limit$,\backslash (\backslash$Delta n$=0\mathrm{.}5\backslash )$ and $\backslash (\backslash$alpha $\backslash )\backslash (=10\backslash$mathrm$\{$~cm$\}^\{-1\}\backslash ),$ calculate the new $\backslash ($ Q $\backslash )$ factor of the resonator. $(5$ points$)$