Lasers and Laser Light — Question 6

Question 6

Two phase-stable laser frequencies produce a measurable beat signal

Two collinear, equally polarized optical fields have intensities $I_1=I_2=I_0$ and frequencies $\nu_1$ and $\nu_2=\nu_1+80.0\,\mathrm{MHz}$. Their relative phase is stable. The detector averages over optical cycles but resolves the frequency difference.

1. Derive the time-dependent intensity.

2. Find the beat period and the time-averaged intensity if the electronics also average over many beat cycles.

Question 6 - Solution

Two phase-stable laser frequencies produce a measurable beat signal

(a) Method and calculation

$$E=E_a\cos(2\pi\nu_1t)+E_a\cos(2\pi\nu_2t+\phi).$$$$\boxed{I(t)=2I_0[1+\cos(2\pi\Delta\nu t+\phi)]}.$$

(b) Interpretation and further calculation

$$\boxed{T_b=1/\Delta\nu=12.50\,\mathrm{ns}}.$$$$\langle\cos(2\pi\Delta\nu t+\phi)\rangle=0\Rightarrow\boxed{\bar I=2I_0}.$$

Physical check

The intensity can momentarily reach four times one-beam intensity, while its beat-cycle average is the sum of the two beam intensities. Limited detector bandwidth can hide the beat.