Two-line coherence beating ($|g^{(1)}(\tau )|=|\cos (\Delta \omega \tau /2)|$)

Two equally intense monochromatic spectral components have normalized first-order optical coherenceThe visibility zeros occur at $\tau =(2j+1)\pi /|\Delta \omega |$, while its revivals repeat every $2\pi /|\Delta \omega |=1/|\Delta \nu |$. This envelope describes self-interference of each spectral component after a delay, not a time-resolved measurement of inter-frequency beating at one fixed point. Unequal line intensities prevent complete cancellation at the minima.

For two equal-intensity mutually incoherent narrow spectral lines with wavenumbers $k_1,k_2$, a Michelson interferometer with optical path difference $x$ has normalized intensity $I/I_0=1+\cos (\bar{k}x)\cos (\Delta kx/2)$, where $\bar{k}=(k_1+k_2)/2$ and $\Delta k=|k_1-k_2|$. With mirror displacement $d=x/2$, the fast period is $\pi /\bar{k}\simeq \bar{\lambda }/2$; successive interference visibility revivals are separated by $\pi /\Delta k\simeq {\bar{\lambda }}^2/(2\Delta \lambda )$. The first visibility zero is halfway to the first revival. A signed envelope has twice the period of its absolute value. Unequal line strengths prevent perfect zeros.