Consider a Michelson interferometer that is used in a Fourier spectroscopy experiment. To obtain high resolution in the computed spectrum, the interferogram must be measured out to large pathlength differences, where the interferogram has fallen to very small values.

(a) Show that under such conditions, the spectrum of the light falling on the detector can be significantly different from the spectrum of the light that entered the interferometer.

(b) If the spectrum of the light entering the interferometer is

\[ \widehat{\mathcal{G}}(v)=\frac{1}{\Delta v} \operatorname{rect} \frac{v-\bar{v}}{\Delta v} \]

calculate the spectrum of the light falling on the detector as a function of pathlength delay. Under what condition will the spectrum of the output be significantly different from the spectrum of the input?