Calibrated Plane--Convex Microcavity for Room-Temperature Polaritons with Geometric $g$-Scaling

We present a plane–convex open microcavity that supports room-temperature polariton spectroscopy and offers a simple geometric handle on the coupling rate. The effective length ($L_{\mathrm{e}\mathrm{f}\mathrm{f}}$) is absolutely calibrated from the free-spectral range, and piezo tuning is performed at near-normal incidence ($k_{\parallel }!\approx !0$) to avoid angle-induced degradation. Using spin-coated PEA$2$PbI$4$ quasi-2D perovskites, we observe clear anti-crossings in reflection, with vacuum Rabi splittings up to $71\mathrm{m}\mathrm{e}\mathrm{V}$ (reflection) and $87\mathrm{m}\mathrm{e}\mathrm{V}$ (PL) near $L\mathrm{e}\mathrm{f}\mathrm{f}!\approx !3\mu \mathrm{m}$. A linewidth-corrected analysis converts the apparent splitting into the coherent exciton–photon coupling rate $g$, revealing a robust geometric scaling $g\propto L{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1\mathrm{/}2}$ across multiple longitudinal orders and spatial sites, consistent with the filled-mode thin-film limit where the transverse area cancels in the mode volume. The platform establishes a compact, broadly compatible testbed for room-temperature polaritons and provides a practical design rule: shortening $L_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is a reliable geometric lever to strengthen collective coupling in plane–convex microcavities.