= Solution
The given spatial and carrier factors combine into retarded copies of the same random field:
$$
\mathbf E'(\mathbf r,t)=\frac{f(t-t_1)+f(t-t_2)}{\sqrt2}\widehat{\mathbf z},\qquad
t_j=\frac{\mathbf k_j\cdot\mathbf r}{\omega}.
$$
On the $y$ axis, $t_1=ky/(\sqrt2\omega)$ and $t_2=-ky/(\sqrt2\omega)$. Expand the squared electric-field <wave amplitude>. Each self term contributes $I_0/2$, and the two cross terms together contribute the real part of the full-field correlation. Stationarity and $\Gamma(-\tau)=\Gamma(\tau)^*$ give
$$
\boxed{I(y)=C\langle|\mathbf E'|^2\rangle
=I_0+\operatorname{Re}\Gamma(\tau(y)),\qquad
\tau(y)=\frac{\sqrt2ky}{\omega}.}
$$
In <vacuum>, $k=\omega/c$ and $\tau(y)=\sqrt2y/c$. Reversing the delay convention replaces $\tau$ by $-\tau$ without changing this real-part result. This $I$ is the electric-field <light intensity> convention of the question. For two nonparallel waves it should not automatically be identified with the magnitude of the net <Poynting vector>; part (d) gives that geometric conversion explicitly.
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