Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 1 / c / Solution

Let $d=\delta t$. At the port that is bright for zero delay, the ideal 50:50 Michelson interferometer givesEach arm has field amplitude one half after two beam-splitter interactions. For $|d|\ll \tau$, approximate both real Gaussian envelopes by $E_0{e}^{-a{t}^2}$ while retaining their rapidly varying phases. Put $A={\omega }_{0}t+b{t}^2$ and $B={\omega }_0(t-d)+b(t-d{)}^2$. The identity $(e^{-iA}+e^{-iB})/2=e^{-i(A+B)/2}\cos [(A-B)/2]$ giveswhereA delay-dependent constant overall phase can be shifted by a port convention, but the expression above follows directly from the specified bright-port sum.

The output intensity is a Gaussian envelope modulated by a squared cosine:The modulation maxima are separated byThis is the spacing of resolved adjacent pulse peaks when the envelope changes little over a modulation period, requiring $b|d|\tau \gg 1$. The source's assumption $|d|\ll \tau$ alone does not ensure several visible peaks: a weakly chirped pulse may show only one broad maximum. At $d=0$ the output is the original pulse and the modulation-spacing formula has no finite-peak interpretation. The figure compares the exact bright-port field sum with the small-delay approximation.