Fourier-Transform-Based Metrology for Whispering Gallery Mode Spectra in Soft Photonic Microcavities

We present a Fourier-transform (FT)-based framework for quantitative analysis of whispering gallery mode (WGM) spectra in soft photonic microcavities. By treating the WGM spectrum as a quasi-periodic signal, the method enables robust extraction of the optical path length L_opt = λ$_c^2$/Δλ directly in the frequency domain, avoiding explicit peak identification and reducing sensitivity to background and spectral overlap. This quantity is used as a primary measurand within a unified metrological formulation: when the cavity radius R is known, it yields the effective refractive index n_eff = L_opt/(2πR); when the refractive index n is known, it provides an inferred geometric path length l_geo=L_opt/n. Following the Guide to the Expression of Uncertainty in Measurement (GUM), we establish the measurement models and evaluate the uncertainty budget, identifying the FSR determination as the dominant contribution (relative uncertainty 7.7%), with secondary contributions from radius measurement (1.5%) and negligible influence from wavelength calibration. The framework is applied to two representative soft photonic systems as complementary test and consistency cases. For Rhodamine B-doped mesoporous silica microcapsules (R=44 μm), we obtain n_eff=1.164±0.09, corresponding to a porosity of 63.3% via Bruggeman effective medium theory, in close agreement with independent BET measurements (62.8%). For surfactant-stabilized Rhodamine 640-doped benzyl alcohol microdroplets, the method identifies dominant Fourier-domain periodicities and yields inferred geometric path lengths consistent with near-equatorial mode propagation. An additional N=14 droplet analysis gives an FT-inferred radius of 60.78±1.91 μm, in close agreement with the microscopy-estimated radius of approximately 60 μm. By combining Fourier-domain analysis with explicit measurement modeling and uncertainty quantification, this work establishes FT-WGM spectroscopy as a reproducible and generalizable tool for single-particle metrology in complex soft-matter microcavities.