Coherence — Question 10

Question 10

Pulse overlap limits time-integrated interference contrast

Two equal-energy, identically polarized coherent pulses have rectangular field envelopes of duration T. Their relative arrival delay is $\tau$ and their adjustable carrier phase is $\delta$. The detector integrates over both entire pulses. Each isolated pulse deposits energy U, and the field amplitude is constant during its pulse.

1. Derive the integrated energy as a function of phase and delay for $|\tau|<T$.

2. Find the visibility for $\tau=0.30T$ and state what happens for $|\tau|\geq T$.

Question 10 - Solution

Pulse overlap limits time-integrated interference contrast

(a) Method and calculation

$$T_{\mathrm{overlap}}=T-|\tau|,\quad\int E_1E_2\,dt\ \propto\ (T-|\tau|)\cos\delta.$$$$\boxed{\mathcal E=2U\left[1+\left(1-\frac{|\tau|}{T}\right)\cos\delta\right],\quad V=1-|\tau|/T}.$$

(b) Interpretation and further calculation

$$\tau=0.30T\Longrightarrow\boxed{V=0.70}.$$$$|\tau|\geq T:\quad T_{\mathrm{overlap}}=0\Longrightarrow\boxed{\mathcal E=2U,\quad V=0}.$$

Physical check

Mutual carrier coherence alone is insufficient for ordinary intensity interference when the fields never overlap in time at the detector.