Figure 1 . Schematic set-ups of (a) IR dual-comb polarimetry (IR-DCP) and (b) FTIR polarimetry. The measurement principle is outlined on the right side. Typical measurement parameters are parallel-polarized (ss, pp) and cross-polarized (sp, ps) transmission T (reflection R) and absolute/relative phases Δ in dependence of the azimuthal angle (αi). See text for further methodical details.
The dual-comb system (IRis-F1, IRsweep AG, Switzerland, schematic in Fig. 1(a)) comprises two QCL frequency combs (FC 1 and FC 2). FC 1 probes the sample, whereas FC 2 works as the local oscillator. Polarizers (P1, P2, P3) and beamsplitters (BaF2, Specac, England) are used for controlling incident and output power and polarization states, as well as beam propagation and recombination. P1 regulates the incident power (typically about 0.75 mW at the sample), P2 sets the incident polarization, and P3 acts as an analyzer. Further polarizers (not shown) control the power incident on the reference detector and ensure polarization matching of local oscillator beam and sample beam. The sample was placed between P2 and P3 on a rotational mount. The beam diameter at the sample position was about 3 mm (FWHM, Gauss).
The operating principles of dual-comb spectroscopy have been described previously33-35. Briefly, FC 1 and FC 2 deliver frequency combs with inter-line spacings (frep,1and frep,2 ) of about 10 GHz. When FC 1 and FC 2 are overlaid on a high-bandwidth mercury-cadmium-telluride (MCT) detector, a heterodyne beating corresponding to Δfrep  =frep,1  – frep,2  ≈ 2 MHz can be recorded in the time-domain. Conducting a Fourier transform on this signal allows for the individual frequency contributions to be resolved. The reference detector is used to correct for amplitude and frequency noise of the free-running QCLs. For a given polarizer configuration, measuring the difference between a set-up without and one with a sample gives the complex transmission txy, the angle of which yields the absolute phase. The theoretical time resolution is 1/Δfrep 35,36however, for data-handling and signal-to-noise reasons, the spectrometer is typically operated at a time resolution of 4 µs35,36. In the present study, the integration time was 65 µs. A transmission geometry at 0° incidence angle was used. For universality reasons, and to avoid confusion with the sample azimuthal settings in degrees (0° = horizontal orientation; 90° = vertical orientation), we use the following notation for polarizer/analyzer settings: (i) ss for parallel polarizers in horizontal direction, (ii) pp for parallel polarizers in vertical direction, (iii) sp and ps for crossed polarizers. Note that, for non-depolarizing samples, the ss-, pp-, sp- and ps-polarized measurements are directly related to the Jones matrix and specific combinations of the respective Mueller-matrix elements,37 which enables the analysis of complex dielectric functions (real and imaginary part).38
The FTIR polarimeter (Fig. 1(b)) is coupled to an FTIR IFS 55 spectrometer (BRUKER, Germany) serving as a radiation source. A Jacquinot aperture (A) of 1.85 mm results in a spot size on the sample of about 4.5 mm (FWHM). The sample was placed on a rotational mount. Polarizers (KRS5, Specac, England) P1 and P2 act as polarizer and analyzer, respectively. A liquid-nitrogen-cooled photovoltaic MCT detector (Kolmar Technologies, USA) was used. A retarding element (R) was additionally inserted for sensitive phase measurements. The opening angle was about ±3.5°. Measured experimental quantities are the relative phase Δ = Δpp – Δss and the polarization dependent transmission Tpp = |tpp|2and Tss = |tss|2at 0° incidence angle. |tpp| and |tss| represent the amplitudes of the respective pp- and ss-polarized complex transmission coefficients (for incidence angles larger than 0°, pp is parallel and ss is perpendicular to the optical incidence plane), and Δ is the phase shift between them. Further details are found in Refs.38,39 for the ellipsometric method and in Ref.39 for the employed set-up.