= Solution
An optically <isotropic> film changes the phase of both transverse polarization components equally, so crossed <polarizers> still extinguish the transmitted light. The etched rod-and-gap composite has \b[<form birefringence>]: its effective <permittivity> depends on the direction of the <electric field>, even when the constituent material is intrinsically <isotropic>. In a simple subwavelength lamellar model,
$$
\epsilon_\parallel=f\epsilon_r+(1-f)\epsilon_g,\qquad
\epsilon_\perp=\left(\frac f{\epsilon_r}+\frac{1-f}{\epsilon_g}\right)^{-1}.
$$
The precise effective-medium formula changes with rod geometry, but the <anisotropy> remains. Resolving incident light along the two principal directions produces a <phase retardation>. An ideal lossless specimen between crossed <polarizers> transmits
$$
I=I_P\sin^2(2\alpha)\sin^2(\pi\Gamma/\lambda),\qquad\Gamma=(n_{\rm slow}-n_{\rm fast})t,
$$
where $I_P$ is the intensity after the first polarizer. Extinction occurs when a principal direction follows either polarizer; transmission is strongest near $45^\circ$.
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