A Michelson interferometer is illuminated by a stabilised single-mode laser whose…

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Interference and diffraction · Coherence and statistical optics

What an Instrument Can See, Ch Temporal coherence: correlation, spectrum and the interferometer

Problem

A Michelson interferometer is illuminated by a stabilised single-mode laser whose power spectrum is Lorentzian, of full width at half maximum Δν=2 MHz\Delta\nu = 2\ \mathrm{MHz}, centred at frequency ν0\nu_0. A lossy beamsplitter coating makes the two beams reaching the detector unequal: I1=0.70I0I_1 = 0.70I_0 and I2=0.30I0I_2 = 0.30I_0, where I0I_0 is the irradiance either arm alone would give with no loss. The two beams are exactly overlapped at the detector (spatial coherence is perfect), so only the temporal delay τ\tau between them, set by an optical path difference LL, limits the fringe contrast. (a) Starting from Γ12(τ)=⟨V1(t+τ)V2∗(t)⟩\Gamma_{12}(\tau) = \langle V_1(t+\tau)V_2^*(t)\rangle and the detector field V(t)=V1(t+τ)+V2(t)V(t) = V_1(t+\tau) + V_2(t), derive the interference law I(τ)=I1+I2+2I1I2 Re γ12(τ)I(\tau) = I_1 + I_2 + 2\sqrt{I_1I_2}\,\mathrm{Re}\,\gamma_{12}(\tau) and hence the visibility V(τ)=2I1I2I1+I2∣γ12(τ)∣\mathcal{V}(\tau) = \dfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}(\tau)|. (b) Both beams derive from the same laser, so γ12=γ11\gamma_{12} = \gamma_{11}. Using the Wiener-Khinchin theorem, state γ11(τ)\gamma_{11}(\tau) for this Lorentzian line and compute ∣γ11(τ)∣|\gamma_{11}(\tau)| and V(τ)\mathcal{V}(\tau) at a path difference L=15 mL = 15\ \mathrm{m} (so τ=L/c\tau = L/c).

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