Consider a Mach$-$Zehnder interferometer shown below. The effective index that $1\mathrm{.}5m$ light experiences while traveling inside the interferometer is $3\mathrm{.}5$ when no bias voltage is applied. Due to manufacturing problems, $l_1=100m$ and $l_2=100\mathrm{.}1m$ are not the same.

$($a$)$ What is the output power when the input power is $1$ mW at $1\mathrm{.}5m$ and no bias is applied? Assume $P_{\text{o}\text{u}\text{t}}$ has $co{s}^2$ dependence on $P_{in}.$

We want to use the interferometer as an optical on$/$off switch by applying voltage to the upper arm as shown. The effective index of the upper arm increases $0\mathrm{.}001$ per $1$ volt applied.

$($b$)$ What is the voltage with the smallest absolute value that needs to be applied to make the switch on$?$