= Solution
The <annihilation operator> eigenvalue property gives
$$
\boxed{\langle\hat E(r,t)\rangle_\alpha
=\mathcal E\alpha e^{i(kr-\omega t)}+\mathcal E^*\alpha^*e^{-i(kr-\omega t)}
=2|\mathcal E\alpha|\cos(kr-\omega t+\arg(\mathcal E\alpha)).}
$$
This is a classical monochromatic <electric field>: the modulus of $\alpha$ fixes the <wave amplitude>, while its argument fixes the <wave phase> relative to the chosen mode convention. Direct normal ordering also gives $(\Delta E)^2_\alpha=|\mathcal E|^2$, independent of $\alpha$. Hence the fluctuation divided by the peak mean-field amplitude is $1/(2|\alpha|)$, small in the bright <coherent state> limit. A relative fluctuation evaluated exactly at a field zero is not a useful classicality test.
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