= Solution
Take $f(t)=a(t)e^{-i\omega t}$ at a fixed reference point, and define the full-field <temporal optical coherence> function by
$$
\boxed{\Gamma(t,\tau)=C\langle f^*(t)f(t+\tau)\rangle
=C\langle a^*(t)a(t+\tau)\rangle e^{-i\omega\tau},\qquad C=\frac{\epsilon_0c}{2}.}
$$
The angle brackets denote an ensemble <expected value>, or a sufficiently long time average when justified. The factor $C$ puts $\Gamma$ in <light intensity> units for the chosen peak complex-field convention. The normalized <first-order optical coherence> is $g^{(1)}(\tau)=\Gamma(\tau)/\Gamma(0)$ for a stationary source.
“Independent of absolute time” means the second moment depends on the time difference $\tau$ but not on the starting time $t$: shifting both observation times by the same amount leaves it unchanged. Together with a constant mean this is second-order or <weak stationarity>; it does not mean that the random <electric field> is constant in time. Put $I_0=\Gamma(0)=C\langle|a|^2\rangle$. The full-field convention is important: an envelope-only correlation omits $e^{-i\omega\tau}$ and must have that carrier factor restored before deriving optical <interference fringes>.
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