21. January 2026

MA: Modeling of microspectroscopic reflectance and transmittance measurements for the determination of the refractive index

Johannes Bauer –

This work deals with the modeling of microspectroscopic measurements. The aim is to determine a material’s complex refractive index from its reflectance and transmittance spectra for the first time. The small spotsize enabled by microspectroscopy allows the investigation of samples with lateral dimensions in the micrometer scale, which makes it an interesting characterisation technique for microelectronics. Starting from already known reflectance modeling, a custom method for simulating both the reflectance and the transmittance of a multilayer stack is developed. It incorporates several correction mechanisms to account for different effects evoked from the type of measurement system used for microspectroscopy, namenly a microscope coupled spectrometer. Objective specific corrections are applied for consistent results across different numerical apertures. Additionally, incomplete detection of the reflected and transmitted light due to lateral displacement of higher reflecion orders is taken into account. Owing to the limited spectral resolution of the measurement system, the interference vanishes for thick substrate layers. This effect is modeled by averaging different phase shifts applied to the coherent simulation. The new model is validated to give correct results both for isotropic and uniaxially anisotropic layers by comparison with established transfer-matrix calculations. It is also able to calculate the optical response for biaxial materials along their in-plane axes. As this modeling has never been done before for transmission, the approach is thorougly discussed and its necessity is demonstrated using reflectance and transmittance measurements. Spectra were recorded for optical glasses, sapphire, strontium titanate, and silicon carbide substrates as well as silicon carbide and silicon nitride thin films. Additionally, anisotropic 2D-materials were studied such as uniaxial graphite (HOPG), biaxial molybdenum trioxide (MoO3), and rhenium disulfide (ReS2). Layer thicknesses were successfully extracted from both spectra with high conformity to complementary measurements. Refractive index determination is shown to work very well for thick, incoherent substrates. For HOPG too the refractive index as well as the extinction coefficient could be extracted successfully. Thin samples present a challenge because instead of a singular curve an ambiguous space of possible solutions is obtained. This ambiguity can be mitigated by measuring multiple samples of the same material with varying thicknesses.

—