# Terahertz time-domain ellipsometry with spintronic emitters: Pauli coefficients as a superior alternative to Jones and Mueller matrices

Pierre Koleják, Jiaming Liu, Robin Silber, Martin Mičica, Ondřej Ballada, Čestmír Barta, Romain Lebrun, Nicolas Tiercelin, Mathias Vanwolleghem, Kamil Postava

► To cite this version:

Pierre Koleják, Jiaming Liu, Robin Silber, Martin Mičica, Ondřej Ballada, et al. Terahertz time-domain ellipsometry with spintronic emitters: Pauli coefficients as a superior alternative to Jones and Mueller matrices. IEEE Transactions on Terahertz Science and Technology, 2025, pp.1-17. 10.1109/TTHZ.2025.3559074. hal-05065593

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https://hal.science/hal-05065593v1

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# Terahertz time-domain ellipsometry with spintronic emitters: Pauli coefficients as a superior alternative to Jones and Mueller matrices

Pierre Koleják, Jiaming Liu, Robin Silber, Martin Mičica, Ondřej Ballada, Čestmír Barta, Romain Lebrun, Nicolas Tiercelin, Mathias Vanwolleghem, and Kamil Postava

**Abstract**—We introduce terahertz complete time-domain spectroscopic ellipsometry (THz-cTDSE), extending traditional ellipsometry by determining full Jones and Mueller matrices for in-depth material characterization. This marks the first application of spintronic terahertz emitters in ellipsometry, achieving pure linear polarization for precise magnetization-based switching. Our complete phase-resolved ellipsometry transforms Jones matrices into an intuitive framework of diattenuations and retardations across all Stokes bases—Pauli exponential coefficients—previously unattainable with incomplete ellipsometry. A novel calibration technique ensures accurate Jones matrix reconstruction without requiring precise alignment of polarizing components, approaching the precision typical of visible-range ellipsometry. We demonstrate THz-cTDSE on quartz and $Hg_2Cl_2$ crystals, advancing the analysis of anisotropic materials.

**Index Terms**—Ellipsometry, Terahertz, Spintronic terahertz emitter, Pauli matrices, Optical anisotropy, Jones matrix, Mueller matrix, Time-domain spectroscopy, Calomel, Calibration

## I. INTRODUCTION

Over the past two decades, terahertz spectroscopic ellipsometry in both frequency-domain (THz-FDSE) and time-domain (THz-TDSE) has significantly advanced material characterization [1]–[4]. Applications include the characterization of thin films [5], [6], magneto-optical ellipsometry for determining carrier dynamics [7], [8], liquids in attenuated-total-reflection configurations [9], in vivo human tissue studies [10], and metasurface analysis [11].

Ellipsometry measures transformations in the polarization state of light [12], [13], where the 2x2 Jones matrix transforms the amplitude and phase of the electric field, while the 4x4 Mueller matrix converts Stokes vectors, capturing the total intensity and the degrees of linear and circular polarization. Due to challenges in generating broadband circular polarization in the terahertz range [14], THz-FDSE is limited to an incomplete 3×3 and 4×3 Mueller matrices that exclude circular polarization effects [15]–[17]. While offering high spectral

resolution and stability, these systems face complex calibration procedures, limited spectral ranges up to 1.5 THz (10 THz with a synchrotron), high costs, and potential inaccuracies due to standing wave formations [18].

Standard ellipsometry, commonly used in time-domain ellipsometry, determines the ratio of complex amplitude coefficients for p- and s-polarized light. The ratio is given as $\tilde{\rho} = \tilde{r}_p/\tilde{r}_s = \tan \Psi \exp(-i\Delta)$, where $\Psi$ denotes the amplitude ratio and $\Delta$ represents the phase difference between p- and s-polarizations [13], [19]. Standard ellipsometry enhances precision by correcting phase errors typically caused by sample-reference displacement in reflective measurements [6], [20], [21]. However, its application is primarily restricted to isotropic or ideally oriented birefringent materials. Polarization control in time-domain ellipsometry is typically achieved using photoconductive antennas with polarizers [22]–[25] or rotating nonlinear crystals [26]–[30]. Several studies have derived Mueller matrices from time-domain measurements to obtain comprehensive ellipsometric information. Nevertheless, these approaches are restricted to intensity-based descriptions that exclude absolute phase information [31]–[33].

Time-domain ellipsometry is a method that captures both amplitude and phase information, enabling the use of the Jones calculus. However, only a single study has developed a method for determining the complete Jones matrix from time-domain measurements [34]. Zhao et al. employ dual-channel detection to measure both polarizations simultaneously. They reconstruct the Jones matrix elements by optimizing polarizer orientations and exploiting trigonometric symmetry. Recent work by Chen et al. extended standard ellipsometry using a multi-channel approach [35]. This method generalizes ellipsometric angles $\Psi$ and $\Delta$ for all polarization conversions, including s-to-p transitions. With precisely aligned components, their system eliminates the need for additional calibration.

The challenge of obtaining complete ellipsometry primarily stems from the need to reconstruct the Jones matrix accurately, which requires advanced calibration techniques, as minor angular deviations can introduce systematic errors. To address this, we propose a novel calibration method that removes the need for precise polarizer alignment. By experimentally determining the actual configurations of polarizing components using reference measurements, our approach ensures accurate ellipsometric analysis and offers a more robust solution compared to methods focusing on precise alignment [21], [36], [37].

P. Koleják, J. Liu, R. Silber, and K. Postava are with the Faculty of Materials Science and Technology, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic (e-mail: kamil.postava@vsb.cz)

P. Koleják, N. Tiercelin, and M. Vanwolleghem are with the University of Lille, CNRS, UMR 8520-IEMN, 59000 Lille, France

M. Mičica is with the Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, F-75005, Paris, France

O. Ballada and Č. Bárta are with the BBT-Materials Processing, s.r.o., Doubická 790/11, 184 00 Praha, Czech Republic

R. Lebrun is with the Laboratoire Albert Fert, CNRS, Thales, Université Paris-Saclay, 1 Avenue Augustin Fresnel, 91767, Palaiseau, France
Achieving pure linear polarization with a high extinction ratio is often challenging to obtain using conventional terahertz polarizers but is crucial for accurate ellipsometric measurements. Equally important is maintaining stable control when switching between generated polarization states. We propose terahertz spintronic emitters (STEs) as a solution for ellipsometry applications. STEs consist of a nonmagnetic/ferromagnetic/nonmagnetic trilayer that emits terahertz radiation through spin-to-charge conversion, allowing for fine control of polarization via magnetic reorientation [38], [39]. They offer ultra-broadband emission up to 30 THz [40] and operate across excitation wavelengths from visible to infrared. While STEs have lower efficiency compared to conventional sources [40], recent advancements aim to enhance performance through efficient pumping [41]–[43], improved spin current generation [44]–[46] and terahertz extraction [47]. We employ a high-performance large-area STE integrated with a photonic crystal, capable of generating strong terahertz fields (1.5 MV/cm with 5 mJ pulses), provided by Rouzegar et al. [48]. This advancement drives progress in time-domain ellipsometry by enabling high-powered signal generation and robust polarization control.

This paper presents terahertz complete time-domain spectroscopic ellipsometry (THz-cTDSE), achieving experimental determination of 2×2 Jones matrices, 4×4 Mueller-Jones matrices, and primary Pauli exponential coefficients. Incorporating these coefficients introduces an innovative framework for data interpretation, simplifying analysis by converting the Jones matrix into diattenuation and retardation coefficients within various Stokes bases. This enables straightforward quantification of polarization phenomena. Although the Pauli linear combination has been theoretically explored [49]–[52], it has not been applied in experimental ellipsometry due to the need for a complete Jones matrix ellipsometer. Our use of the exponential Pauli decomposition, derived from the matrix logarithm of the experimental Jones matrix, is novel in ellipsometric practice.

This article is organized as follows. In Section II, we present the instrumentation of our time-domain terahertz ellipsometer. Using a quartz crystal that exhibits only retardation, we demonstrate how the ellipsometer can precisely determine the amplitude and phase of the Jones matrix. Section III details extracting additional ellipsometric information by converting the Jones matrix into an intensity-based Mueller-Jones matrix and performing a decomposition into Pauli exponential coefficients. We discuss the choice of each approach based on sample properties and demonstrate the advantages of Pauli coefficients using the quartz crystal. Section IV describes the calibration method, detailing how the Jones matrix of a sample is derived from a series of measurements. We estimate the accuracy of our method and present the experimentally determined errors for each element of the Jones matrix. Section V applies our ellipsometric technique to a more complex sample, a Hg<sub>2</sub>Cl<sub>2</sub> crystal exhibiting both diattenuation and retardation. To clarify the fundamental differences between THz-cTDSE and conventional frequency-domain Mueller matrix ellipsometry, we provide a dedicated discussion in Section VI. Finally, Section VII discusses our method's limitations and potential

improvements, comparing various terahertz ellipsometric tech-
niques. The conclusion summarizes and reflects on the findings
of our complete matrix approach.

II. INSTRUMENTATION AND DEMONSTRATION OF
THZ-TDS COMPLETE ELLIPSOMETRY

We introduce the THz complete time-domain spectroscopic ellipsometry (THz-cTDSE) in Figure 1a. The system uses a Ti:sapphire amplifier (Astrella from the company Coherent) to generate a 5 W average power infrared beam at a 1 kHz repetition rate, generating 5 mJ pulses. Pulses are shorter than 35 fs, with a central wavelength of 800 nm. The generated infrared beam is split into probing and pumping beamlines. The pumping beam excites the spintronic emitter, positioned within the magnetic field of a rotatable Halbach array. This configuration ensures a homogeneous and linear magnetic field of 10 mT, allowing precise control over the emitted polarization. As shown in the mapped magnetization distribution (documented in the supplementary material of the referenced study [48]), minor inhomogeneities are symmetric around the magnetization axis and cancel out coherently in time-domain detection. Thus, the detected THz polarization is defined primarily by the net magnetization projection along the emission axis. This intrinsic symmetry enables precise control over the emitted polarization, achieving higher purity than conventional far-infrared polarizers. Consequently, a direct quantitative measurement was not possible, as the available reference polarizers exhibited lower performance than the emitted THz polarization itself.

The spintronic emitter in this setup represents the state-of-the-art terahertz spintronic generation, demonstrating the highest reported performance [48]. It comprises a Pt/CoFeB/W trilayer stack, a one-dimensional SiO<sub>2</sub>/TiO<sub>2</sub> photonic crystal, and a high-resistivity silicon substrate. The photonic crystal increases pump beam absorption in the spintronic layers, thereby boosting terahertz emission [42], [43] The silicon substrate absorbs the residual pump beam, making additional filtering unnecessary. We use this broad-area emitter for collimated excitation with a 25.4 mm beam diameter and 650 mW laser power (0.65 mJ per pulse). In this configuration, the emitter operates well below its damage threshold [48], [53]. Furthermore, the signal can be further improved by increasing the pump power, but the emitter isn't currently utilizing the amplifier's full power of 5 W due to other experiments.

To ensure stable measurements and minimize absorption losses due to atmospheric water vapor, the entire terahertz optical path—including the emitter, sample region, and detection optics—is enclosed in a sealed box purged with dry air. Terahertz radiation is emitted in the form of a collimated beam and is subsequently focused onto a sample using an off-axis parabolic mirror with a focal length of 152.4 mm (6 in). Two wire-grid polarizers are placed after the sample: the analyzer and the fixed polarizer, both on motorized rotation stages. The analyzer selects the output polarization, while the fixed polarizer adjusts it vertically for stable detection by the ZnTe electro-optic crystal. For electro-optic detection, a (110)-cut ZnTe crystal was used, oriented to measure the vertical
<figure><img src="image_2.png" /><figcaption>Fig. 1. THz-cTDSE scheme: a) Overview of the experimental setup and b) detail of the experimental terahertz ellipsometry section. The Ti:sapphire amplifier generates the initial beam, which splits into pump and probe beams. The pump beam excites a spintronic emitter (STE) within a Halbach array that controls the terahertz polarization; the emitted radiation is focused onto the sample; an analyzer selects the desired polarization; a fixed polarizer aligns it vertically for detection by the ZnTe crystal. The probe beam passes through a delay line, a pellicle beamsplitter, a ZnTe crystal, a quarter-wave plate (λ/4), and a Wollaston prism (WP) to balanced photodiodes. A gated integrator averages the signal, and the measurements are then converted into Jones matrices.</figcaption></figure>

component of the THz field with a p-polarized probe beam [29]. The beam is recollimated and then focused onto the ZnTe detector using another parabolic mirror with a 101.6 mm (4 in) focal length, enhancing field intensity while minimizing defocusing and reducing the Gouy phase shift error [54], [55]. The detail of the terahertz ellipsometry arrangement is illustrated in Figure 1b. The probing beam passes through a delay line and is directed onto the 500 µm thick ZnTe crystal using a pellicle beamsplitter. The beam then passes through a λ/4-waveplate and a Wollaston prism polarizer (WP), before being detected by balanced photodiodes. Operating at a 1 kHz repetition rate, we employ a gated integrator that captures all signal harmonics, offering a comprehensive alternative to traditional first harmonic demodulation via a lock-in amplifier.

A. Demonstration of complete Jones matrix ellipsometry

The primary goal of ellipsometry is to determine the transformation matrix that describes how a sample modifies the input polarization into the output polarization. In the time-domain approach, waveform measurements are inherently coherent, enabling the capture of complex field amplitudes. This coherence allows for reconstructing the vector electric field of the terahertz pulse, both with and without the sample, thus enabling the determination of the transformation matrix. The Jones matrix $\tilde{\mathbf{J}}$ characterizes the coherent polarization transformation by converting the input electric field vector $\tilde{\mathbf{E}}^{\text{in}}$ into the output electric field vector $\tilde{\mathbf{E}}^{\text{out}}$ as follows

$$ \left( \begin{matrix} \tilde{\mathbf{E}}_{\mathrm{x}}^{\text{out}} \\ \tilde{\mathbf{E}}_{\mathrm{y}}^{\text{out}} \end{matrix} \right) = \underbrace{\left( \begin{matrix} \tilde{j}_{11} & \tilde{j}_{12} \\ \tilde{j}_{21} & \tilde{j}_{22} \end{matrix} \right)}_{\mathbf{J}} \left( \begin{matrix} \tilde{\mathbf{E}}_{\mathrm{x}}^{\text{in}} \\ \tilde{\mathbf{E}}_{\mathrm{y}}^{\text{in}} \end{matrix} \right). \quad (1) $$

The Jones matrix provides complete and complex information about the polarization properties of the sample. The off-diagonal elements of the matrix indicate the coupling between orthogonal components of the electric field.

We aim to determine the complete Jones matrix, focusing on uniaxial anisotropic crystals as a representative example of the method's broader applicability. We start the demonstration with an A-cut 1.816 mm thick quartz crystal plate, which is

non-absorbing and uniaxial, providing only retardation. We analyze it using time-domain spectroscopy in a transmission configuration, with the optical axis rotated by an angle $\theta$. The polarization components are configured as follows: the spintronic emitter is vertically aligned, the analyzer is set at $\pm 45$ degrees to detect orthogonal modes, and the fixed polarizer is also vertically aligned, as shown in Figure 1b. Initially, we conduct time-domain measurements of the free-space reference without the sample, as shown in Figure 2a. In our case, the reference pulses overlap perfectly, exhibiting no amplitude differences between the polarization states and no time delay that would indicate a phase shift. The quartz sample was then measured under the same conditions, as shown in Figure 2a. The quartz is oriented at $\theta = 45^{\circ}$, so one polarization exhibits the ordinary refractive index while the other displays the extraordinary refractive index. Due to the ideal behavior of reference polarizations, the interpretation of quartz measurements is significantly simplified. The observed temporal delay between the orthogonal pulses highlights the retardation effect. The slight change in pulse shape results from the different dispersions in the ordinary and extraordinary directions. The observed amplitude losses, relative to the references, result from Fresnel reflection at the quartz/air interfaces. Subsequently, we perform Fourier transformation on the time-domain signals to obtain the complex spectra, which include both phase and amplitude. The amplitude spectra are shown in Figure 2b.

Accurate characterization with this elementary polarimetric approach depends on a detailed understanding of the anisotropic sample's orientation and behavior. In contrast, complete ellipsometry provides deeper insights, enabling more precise and comprehensive polarimetric characterization. Figure 2c presents the amplitudes of experimentally determined complete Jones matrix for quartz, compared with the theoretical model, which incorporates the optical constants reconstructed from the ellipsometry data (see Appendix A). Analysis of the amplitude spectrum shows that at $90^{\circ}$ azimuth, the crystal axes align with the ellipsometer-generated polarizations, resulting in minimal off-diagonal elements. At
<figure><img src="image_3.png" /><figcaption>Fig. 2. Comparison of A-cut crystal quartz plate and free-space reference measurements: a) time-domain waveforms, and b) the corresponding spectral intensity $|E|^2$ at orthogonal polarizations. Polarization was selected by setting the analyzer at a ±45° azimuth. Demonstration of the complex Jones matrix for quartz at 15°, 60°, 90°, and 135° azimuths of optical axis: c) Jones amplitude spectra and d) normalized Jones phase spectra. The experimental results are compared to a theoretical model of quartz (gray lines). The quartz crystal functions as a linear retarder, exhibiting half-wave retardation at 1.7 THz. The off-diagonal phases at 90° azimuth were excluded from the plot, as they correspond to zero amplitudes, where the phase is inherently undefined.</figcaption></figure>

135° azimuth, the diagonal components drop to zero signal around 1.7 THz, indicating halfwave plate behavior. Here, the half-wave plate shifts the phase of one polarization component by 180°, effectively rotating the linear polarization direction by 90°. Therefore, the off-diagonal components are maximal at this frequency. Azimuths of 15° and 60° represent intermediate steps between Cartesian and diagonal bases. Measurements at various azimuth angles demonstrate the method's sensitivity to different polarization properties and enable multiple azimuthal-angle fits. To further verify the robustness of the approach, additional measurements were performed over a 180° sample rotation, as detailed in Appendix A. Half-waveplate behavior occurs at a specific frequency $\nu$, where the retardation $\delta$ reaches $\pi$, as $\delta$ is frequency-dependent: $\delta = \frac{2\pi}{c} \nu (n_o - n_e)$, with c being the speed of light. The difference between the ordinary $n_o$ and extraordinary $n_e$ indices is also frequency-dependent due to material dispersion. The analytical form of the Jones matrix for a uniaxial anisotropic crystal is presented in Appendix A for a more detailed understanding.

The complex Jones matrix offers crucial insights into absolute phases in polarization analysis. To enhance clarity in the polarization phase spectra, we normalize the Jones phase matrix, $\arg\{\tilde{\mathbf{J}}\}$, by the phase element $\arg\{\tilde{\mathbf{J}}_{11}\}$. This normalization highlights subtle polarization differences and removes the accumulated phase of hundreds of radians as the wave propagates through the crystal relative to a free-space reference. Figure 2d illustrates the normalized Jones phase matrix compared with the model, where the primary com-

ponent is normalized to zero, and other components display phase differences relative to $\tilde{\mathbf{J}}_{11}$. The normalized phases are linearly proportional to frequency, with slight curvatures in the phase response due to the dispersion between the ordinary and extraordinary refractive indices. A negative phase difference indicates a reduced propagation velocity relative to the $\tilde{\mathbf{J}}_{11}$ element. In a half-wave plate rotated at a 135° azimuth, a phase shift of $\pi$ indicates a reversal in polarization polarity. The figure does not show the phase response in the off-diagonal elements at the 90° azimuth, as these were computed from zero-amplitude signals. Furthermore, as the frequency approaches zero ($\nu \to 0$) for off-diagonal elements, phase offsets are mainly determined by interfaces, thus the phase differences in the complex Fresnel coefficients.

### III. COMPARISON WITH MUELLER AND PAULI TECHNIQUES IN POLARIZATION ANALYSIS

Complete Jones matrices are highly effective in time-domain ellipsometry because they directly describe the transformation of complex electric field components. This method reveals amplitude and phase spectra changes, clearly outlining complex interactions within polarization states through diagonal and off-diagonal elements. The Jones matrix describes the complete polarization response of a coherent non-depolarizing system, therefore, all later discussed descriptions can be directly obtained from it. However, extracting specific effects like retardation and diattenuation from the 2×2 Jones matrix may be challenging, particularly when both effects contribute
to the same matrix elements. The Stokes bases offer an
alternative interpretation approach. This system defines easily
measurable polarization states: horizontal, vertical, diagonal,
anti-diagonal, and left- and right-handed circular polarization.
Traditionally, this approach is applied in intensity descriptions
using Mueller matrices. However, we aim to demonstrate that
this approach is highly robust in the field-resolved approach
when applying Pauli exponential coefficients, ensuring both
accuracy and efficiency in polarization analysis.

A. *Mueller-Jones matrix approach*

Mueller calculus provides a robust intensity-based frame-
work for describing the polarization state of light. However,
this approach results in the loss of total phase information
accumulated during propagation through the sample. Although
it does not directly capture phase, it reveals effects associated
with phase differences. When absolute phase information is
not required, an intensity-based description using the Stokes
vector and 4x4 Mueller matrix offers an efficient polarization
analysis. The transformation from the Jones matrix to the
Mueller matrix proceeds as follows

$$
M = A(J \otimes J^*)A^{-1}, \text{ where } A = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & i & -i & 0 \end{pmatrix}, \quad (2)
$$

enables the conversion of the vectorized coherency matrix
into the Stokes bases [19], [56]. The Mueller matrix derived
from the coherent Jones matrix is inherently non-depolarizing
and commonly called the Mueller-Jones matrix. The Mueller
matrix transforms the Stokes vector components, which quan-
titatively describe polarized intensities as follows

$$
\underbrace{\begin{pmatrix} I'_x + I'_y \\ I'_x - I'_y \\ I'_{+45^\circ} - I'_{-45^\circ} \\ I'_{R} - I'_{L} \end{pmatrix}}_{S_{\text{out}}} = \underbrace{\begin{pmatrix} m_{11} & \cdots & m_{14} \\ \vdots & \ddots & \vdots \\ m_{41} & \cdots & m_{44} \end{pmatrix}}_{M} \underbrace{\begin{pmatrix} I_x + I_y \\ I_x - I_y \\ I_{+45^\circ} - I_{-45^\circ} \\ I_R - I_L \end{pmatrix}}_{S_{\text{in}}}, \quad (3)
$$

where $I_{\xi}$ represents the intensities corresponding to the
$\xi$-polarization of the Stokes bases. Normalizing the Mueller
matrix by the element $m_{11}$ highlights relative changes in
polarization properties. This matrix determines the intensity
ratios for each base relative to the total intensity $I_x + I_y$.
The first row $m_{1,1-4}$ and column $m_{1-4,1}$ are critical for
transforming total intensity and demonstrating diattenuation.
While the submatrix $m_{2-4,2-4}$ plays a crucial role in depicting
retardation by affecting polarization states without altering
total intensity [57].

Figure 3a illustrates the conversion of the Jones matrix of
quartz, shown in Figure 2c,d, into the normalized Mueller-
Jones matrix. The Mueller matrix exhibits the characteristic
form of a linear retarder, in agreement with the analytical
model provided in Appendix B. Importantly, excluding $m_{11}$,
the first row and column are zero, indicating no diattenuation
in the transparent quartz crystal. The remaining submatrix
aligns with the behavior expected of a linear retarder without

diattenuation. At an azimuth of 135°, the Mueller matrix
exhibits a block-diagonal structure. Significantly, the spectrum
$m_{44}$ corresponds to $\cos \delta$ and is independent of the waveplate's
azimuth. The linear retarder behaves as a half-wave plate
at the frequency corresponding to $m_{44} = -1$. The effects
are distributed across the 4×4 matrix elements, simplifying
data interpretation. Yet, this simplification comes at the cost
of losing amplitude and phase information, which are more
intuitive and easier to interpret. Interpreting experimental
Mueller matrices can be challenging without an analytical
model.

B. *Pauli exponential coefficients*

We utilize Stokes bases within a time-resolved frame-
work, providing more detailed information than intensity-
based methods, which is particularly advantageous for time-
domain ellipsometry. A Jones matrix can be constructed by
multiplying elementary Jones matrices J_ξ corresponding to
all ξ-Stokes bases, as shown

$$
\tilde{\mathbf{J}} = \tilde{\mathbf{J}}_0 \tilde{\mathbf{J}}_L \tilde{\mathbf{J}}_D \tilde{\mathbf{J}}_C = \exp(\tilde{p}_0 \sigma_0) \exp(\tilde{p}_L \sigma_L + \tilde{p}_D \sigma_D + \tilde{p}_C \sigma_C), \quad (4)
$$

where each elementary Jones matrix $\tilde{\mathbf{J}}_{\xi} = \exp(\tilde{p}_{\xi}\sigma_{\xi})$ is expressed as a matrix exponential, distinct from element-wise multiplication. The subscripts D, L, C refer to the vertical/horizontal linear, $\pm45^{\circ}$ diagonal, and left/right-handed circular Stokes bases. The matrices $\sigma_{\xi}$, known as the Pauli matrices, correspond to the fundamental Jones matrices used to interpret transformations within the Stokes vector bases. Specifically, $\sigma_L$ describes transitions from horizontal to vertical polarization, $\sigma_D$ describes changes from $+45^{\circ}$ to $-45^{\circ}$ diagonal polarization, and $\sigma_C$ describes transitions between left- and right-handed circular (or linear to circular) polarizations. The Pauli exponential coefficients are expressed as follows

$$
\tilde{p}_{\xi} = \frac{d_{\xi} - i\delta_{\xi}}{2} \qquad (5)
$$

and directly define the attenuation $d_{\xi}$ and phase retardation
$\delta_{\xi}$ for each specific Stokes basis. This formulation presents
the Jones matrix as the product of individual optical effects,
separated into interpretable complex coefficients $\tilde{p}_{\xi}$. Applying
the matrix logarithm $\ln \tilde{\mathbf{J}}$ transforms the Jones matrix into a
linear combination of Pauli matrices $\sigma_{\xi}$, as follows

$$
\begin{align}
\tilde{\mathbf{K}} &= \ln \tilde{\mathbf{J}} = \begin{pmatrix} \tilde{p}_0 + \tilde{p}_L & \tilde{p}_D - i\tilde{p}_C \\ \tilde{p}_D + i\tilde{p}_C & \tilde{p}_0 - \tilde{p}_L \end{pmatrix} = \tag{6} \\
&= \tilde{p}_0 \underbrace{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}_{\sigma_0} + \tilde{p}_L \underbrace{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}}_{\sigma_L} + \tilde{p}_D \underbrace{\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}}_{\sigma_D} + \tilde{p}_C \underbrace{\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}}_{\sigma_C}. \nonumber
\end{align}
$$

Importantly, the Schur decomposition can be used to efficiently
compute the matrix logarithm by transforming the matrix
into a nearly triangular form [58]. This approach is numeri-
cally stable for a broad class of matrices and is extensively
implemented across different computational languages. The
exponential Pauli coefficients are derived from elements $\tilde{k}_{ij}$
of the complex matrix logarithm $\tilde{\mathbf{K}} = \ln \tilde{\mathbf{J}}$ as follows
<figure><img src="image_4.png" /><figcaption>Fig. 3. Ellipsometric data of quartz for azimuth angles of 15°, 60°, 90°, and 135° presented as a) normalized Mueller-Jones matrix; b) exponential Pauli coefficients, where the real part corresponds to diattenuation *d* and the imaginary part to retardation *δ*. The subscript 0 denotes total absorption and phase; subscripts *L*, *D*, and *C* refer to linear polarizations in Cartesian coordinates, ±45° diagonal bases, and circular polarizations, respectively.</figcaption></figure>

$$ \tilde{p}_0 = \frac{\tilde{k}_{11} + \tilde{k}_{22}}{2}, \quad \tilde{p}_L = \frac{\tilde{k}_{11} - \tilde{k}_{22}}{2}, \qquad (7) $$

$$ \tilde{p}_D = \frac{\tilde{k}_{12} + \tilde{k}_{21}}{2}, \quad \tilde{p}_C = \frac{i(\tilde{k}_{12} - \tilde{k}_{21})}{2}. $$

In the case of a linear waveplate, the coefficient $\tilde{p}_0$ takes the form of $\ln(\tilde{e}_o\tilde{e}_e)$, representing the total changes in phase and amplitude of the ordinary and extraordinary polarizations $\tilde{e}_{o,e}$. Conversely, the coefficients $\tilde{p}_L = \cos 2\theta \ln(\tilde{e}_o/\tilde{e}_e)$ and $\tilde{p}_D = \sin 2\theta \ln(\tilde{e}_o/\tilde{e}_e)$ correspond to the linear/diagonal effects of retardation and diattenuation and can readily reveal the presence of linear birefringence and dichroism. To provide a detailed exploration of the polarization response in terms of Pauli exponential coefficients $\tilde{p}_\xi$, the Jones matrix $\tilde{J}$ for A-cut uniaxial achiral anisotropic material from (20) is transformed into its matrix logarithm $\ln \tilde{J}$ and subsequently mapped onto all Stokes bases through equation (8), leading to

$$ \tilde{p}_0 = \frac{d_o - i\delta_o}{2} = \ln(\tilde{e}_o\tilde{e}_e) = \ln|\tilde{t}_o||\tilde{t}_e| + d_o + d_e - i(\arg\tilde{t}_o\tilde{t}_e + \delta_o + \delta_e) \qquad (8) $$

$$ \tilde{p}_L = \frac{d_L - i\delta_L}{2} = \cos 2\theta \ln(\tilde{e}_o/\tilde{e}_e) = \cos 2\theta \left( \ln \frac{|\tilde{t}_o|}{|\tilde{t}_e|} + d - i \left( \arg \frac{\tilde{t}_o}{\tilde{t}_e} + \delta \right) \right) \qquad (9) $$

$$ \tilde{p}_D = \frac{d_D - i\delta_D}{2} = \sin 2\theta \ln(\tilde{e}_o/\tilde{e}_e) = \sin 2\theta \left( \ln \frac{|\tilde{t}_o|}{|\tilde{t}_e|} + d - i \left( \arg \frac{\tilde{t}_o}{\tilde{t}_e} + \delta \right) \right) \qquad (10) $$

$$ \tilde{p}_C = \frac{d_C - i\delta_C}{2} = 0, \qquad (11) $$

where $d_{e,o}$, $\delta_{e,o}$, and $\tilde{t}_{e,o}$ represent the ordinary and extraordinary attenuation coefficients, total phases, and cumulative Fresnel transmission coefficients, respectively. The quantity $d = d_o - d_e$ represents the diattenuation, while the retardation is given by $\delta = \delta_o - \delta_e$.

The Pauli exponential coefficient $\tilde{p}_0$ contains the cumulative phase $\arg \tilde{t}_o \tilde{t}_e + \delta_o + \delta_e$ and amplitude attenuation $\ln|\tilde{t}_o||\tilde{t}_e| + d_o + d_e$. The coefficient $\tilde{p}_0$ remains invariant to the sample's azimuth. In anisotropic materials with linear birefringence and dichroism, the coefficients $\tilde{p}_L$ and $\tilde{p}_D$ undergo a periodic transition between Cartesian and diagonal Stokes bases as a function of the angle $\theta$. The real part, $d_L$ and $d_D$, represents linear diattenuation and is proportional to $\ln(|\tilde{t}_o|/|\tilde{t}_e|) + d$. In contrast, the imaginary part determines retardation, with $\delta_D$ and $\delta_L$ based on $\arg(\tilde{t}_o/\tilde{t}_e) + \delta$. The parameter $\tilde{p}_C$ remains zero, since a uniaxial linearly birefringent sample does not exhibit circular birefringence or dichroism (see Appendix A).

This behavior is evident in the experimentally determined Pauli exponential coefficients in Figure 3b compared with the analytical form. Results demonstrate that the diattenuation coefficients are zero, confirming that quartz exhibits no diattenuation $d_\xi$. Retardation is observed in both the linear and
diagonal bases. In analyzing a uniaxial birefringent crystal oriented at azimuth $\theta$, the linear retardation $\delta_L$ is found to be proportional to $\cos 2\theta$, while the diagonal polarization is proportional to $\sin 2\theta$. As expected, $\delta_C$ is zero, confirming the absence of optical activity and circular birefringence in quartz. Although both Pauli's exponential coefficients and Mueller matrices can be derived from the Jones matrix, the type of information they reveal about the sample differs fundamentally, reflecting distinct aspects of its optical properties. The Pauli exponential coefficients facilitate a detailed polarization analysis in absolute terms, effectively separating amplitude and phase changes for each Stokes basis. This method eliminates the need for the complex modeling typically required in Jones or Mueller's approaches to interpret the individual matrix elements.

Notably, the imaginary part of the Pauli coefficients, expressed as $\sin(2\theta) \cdot \text{phase}$ and $\cos(2\theta) \cdot \text{phase}$, may lead to phase discontinuities if not correctly unwrapped. Therefore, we compute the matrix logarithm by normalizing it as $\ln(\tilde{\mathbf{J}}/\tilde{j}_{11}) + \sigma_0 \ln \tilde{j}_{11}$, which reduces the number of phase discontinuities. Then, we use the azimuth determined from $\tilde{p}_D/\tilde{p}_L = \tan 2\theta$ to scale the phase, unwrap it correctly, and rescale it to ensure continuity in the final result.

## IV. CALIBRATION METHOD

Calibration methods are designed to minimize errors from non-ideal settings or imperfections in optical components. Ellipsometry involves multiple measurements under different polarization conditions, requiring precise combinations to reconstruct the complete Jones matrix accurately. If not properly combined, these spectral matrices can contain systematic errors that may be mistaken for the material's polarization response, mainly when all systems' sensitivity depends on polarization. To overcome these challenges, we demonstrate a calibration method compensating for component misalignments and removing instrumental artifacts. As illustrated in the flowchart in Fig. 4a, the calibration technique involves several key steps: (i) rough alignment of the experimental polarizing components; (ii) conducting free-space reference measurements under various configurations by adjusting the polarization of the STE-generated radiation and the analyzer's azimuth; (iii) calculating the true azimuths of the analyzer and spintronic emitter from the free-space references; (iv) measuring samples in identical configurations; (v) solving a system of linear equations from measurements; and (vi) reconstructing the Jones matrix to determine Pauli exponential coefficients.

### A. Setting of polarizing components

Generating and detecting polarization states that provide sufficient information is crucial for determining the complete Jones matrix. We utilize vertical and horizontal polarization states generated by STE, combined with 45° diagonal and -45° anti-diagonal states at the analyzer, as shown in Figure 1b. Approximate azimuth settings are adequate since the calibration method compensates for misaligned angles, removing the need for precise alignment. This STE and analyzer configuration

ensures strong, non-zero signals in free-space measurements, simplifying the retrieval of phase and amplitude while enhancing sensitivity to subtle polarization effects. Conversely, using crossed polarizers or crossing the analyzer with the STE's polarization for reference measurements is not recommended, as it introduces significant noise from weak signals. Unlike traditional frequency domain ellipsometry, this strategy allows for the complete retrieval of amplitude and phase information using only linear polarizations. This approach removes the need for elliptically or circularly polarized states to detect circular birefringence and dichroism.

### B. Reference and sample measurements

First, we perform four free-space reference measurements $\tilde{t}_{\text{ref}}(\alpha, \beta)$, followed by measurements of the investigated sample $\tilde{t}_{\text{sample}}(\alpha, \beta)$ under the same configurations. Assuming full time-domain coherence, the Jones formalism can be used to model the overall polarization response by multiplying the individual Jones matrices. The free-space reference model $t_{\text{ref}}$ is represented in matrix notation by sequentially multiplying the components as they appear in the experiment. The STE input field $\text{STE}_j$ first passes through the analyzer $\mathcal{P}(\alpha)$, rotated by $\alpha_i$, and then through the fixed polarizer $\mathcal{P}(90^{\circ})$, which is oriented vertically. Since only the vertical component is detected by the ZnTe crystal, the output is multiplied by vertical polarization selection (0 1). For clarity, symbols in bold italic represent experimental variables, while double-struck font denotes theoretical models. Following this conditions, the measured complex amplitude $\tilde{t}_{\text{ref}}(\alpha, \beta)$ should be equal to

$$ \tilde{t}_{\text{ref}}(\alpha_i, \beta_j) \stackrel{!}{=} \tilde{t}_{\text{ref}}(\alpha_i, \beta_j) = \underbrace{(0 \ 1) \mathcal{P}(90^{\circ}) \mathcal{P}(\alpha_i)}_{A_i} \text{STE}_j, \quad (12) $$

where the relation $\stackrel{!}{=}^i$ indicates that the equality is expected between the experimental spectra and the theoretical model. Polarization detection is described by the analyzing vector $A_i$, representing the coherent vertical-field detection after it passes through polarizers. The spintronic terahertz emission $\text{STE}_j$ exhibits perfectly defined linear polarization, which we express using a rotated unit vector $\mathbf{E}_j$ in the following form

$$ \text{STE}_j = \tilde{\mathbf{E}}_{\beta} \mathbf{E}_j, \quad \mathbf{E}_j = \begin{pmatrix} \cos \beta_j \\ \sin \beta_j \end{pmatrix}, \quad (13) $$

where $\tilde{\mathbf{E}}_{\beta}$ refers to the complex amplitude of the emitted terahertz field. The emitted terahertz spectrum may vary with the azimuth $\beta$ depending on the generation mechanism and polarization control. In our case, the spintronic emitter is mounted directly on the rotatable Halbach array, causing it to rotate with the polarization control. This rotation can potentially influence the spectrum.

Assuming an ideal polarizer, its Jones matrix for the azimuth $\alpha_i$ is given by

$$ \mathcal{P}(\alpha_i) = \begin{pmatrix} \cos^2 \alpha_i & \cos \alpha_i \sin \alpha_i \\ \cos \alpha_i \sin \alpha_i & \sin^2 \alpha_i \end{pmatrix}. \quad (14) $$

However, polarizers in the terahertz range can exhibit non-ideal behavior, requiring representation through a Jones matrix
<figure><img src="image_5.png" /><figcaption>Fig. 4. Procedure and stability of the calibration: a)The calibration procedure for generalized time-domain ellipsometry: (1) A rough alignment of the analyzer to ±45° and the spintronic emitter to 0°/90°, corresponding to the vectors $\mathbf{A}_i$ and $\mathbf{E}_j$. (2) Measurement of the free-space reference in all four configurations. (3) Determination of the true azimuths of the analyzer and spintronic emitter. (4) Measurement of the sample in the same four configurations used for the free-space reference. (5) Construction and solution of the complex linear equations for each experimental configuration. (6) Reconstruction of the complex Jones matrix as the solution to the equation system. (7) In the final step, the Jones matrix is converted into Pauli exponential coefficients, directly corresponding to diattenuations and retardations in the linear, diagonal, and circular bases; b) Jones amplitude spectra, compared to a diagonal unit matrix in air, and c) Jones phase spectra, revealing absolute phase errors of free-space sample measurements, demonstrating the stability and accuracy of the generalized time-domain ellipsometer. The analysis compares the effectiveness of using four references versus a single reference in eliminating systematic errors during data reconstruction. The off-diagonal phase elements are excluded, as calculating them from zero signals would produce meaningless results.</figcaption></figure>

of a linear diattenuator or a diattenuating retarder to accurately
capture the extinction ratio or retardation. In our study, this
adjustment had no significant impact and was therefore not
applied. Finally, the sample is placed between the spintronic
emitter and the analyzer for measurement. The expected com-
plex spectrum $\tilde{t}_{\text{sample}}(\alpha_i, \beta_j)$ of the sample, corresponding to
the measured response $\tilde{t}_{\text{sample}}(\alpha_i, \beta_j)$, is expressed as

$$
\tilde{t}_{\text{sample}}(\alpha_i, \beta_j) \stackrel{!}{=} \tilde{t}_{\text{sample}}(\alpha_i, \beta_j) = \mathbf{A}_i \tilde{\mathbf{J}}_{\text{sample}} \text{STE}_j, \quad (15)
$$

where $\tilde{\mathbf{J}}_{\text{sample}}$ describes the Jones matrix of the sample, which
we aim to determine experimentally.

C. Determining the actual orientation of STE and analyzer

The crucial step in the calibration method involves deter-
mining the actual azimuths αᵢ and βⱼ, initially set to detect
± 45° diagonal states for the analyzer, and to generate near-
vertical and horizontal polarization states using the spintronic
emitter. Four free-space reference measurements are used to
calculate these azimuths by analyzing the reference ratios at
identical azimuths of the spintronic emitter. We use a least

squares method to determine the actual azimuths β₁ and α₁
relative to the fixed polarizer, formulated as

$$
\arg \min_{\alpha_1, \beta_1} \left\{ \sum_{\text{Frequency } j=1,2} \sum_{\text{Frequency } j=1,2} \left( \frac{\mathbf{A}_1 \mathbf{E}_j}{\mathbf{A}_2 \mathbf{E}_j} - \left| \frac{\tilde{\mathbf{t}}_{\text{ref}}(\alpha_1, \beta_j)}{\tilde{\mathbf{t}}_{\text{ref}}(\alpha_2, \beta_j)} \right| \right)^2 \right\}. \quad (16)
$$

Since the azimuths α₂ and β₂ are orthogonal to α₁ and β₁, as
ensured by the high precision motorized stages, it follows that
α₂ = α₁ + 90° and β₂ = β₁ + 90°. Additionally, using the
same azimuth ratios for the spintronic emitter (βⱼ) eliminates
the complex spectral amplitude $\tilde{\mathbf{E}}_{\beta}$, which would otherwise
require experimental determination.

D. Reconstruction of complete Jones matrix

The Jones matrix of a sample, $\tilde{\mathbf{J}}_{\text{sample}}$, consists of four
complex unknown spectra, requiring a robust and unique
solution for each frequency. To determine these four com-
plex unknown parameters, at least four distinct measurement
configurations of the sample are necessary. Though a single
reference could theoretically determine the delay relative to
free-space, we define a reference in each sample configuration
to eliminate $\tilde{\mathbf{E}}_{\beta}$'s influence and remove systematic artifacts.
In each configuration, we establish the following equation to
relate the models $\tilde{\mathbf{t}}$ and the measurements $\tilde{\mathbf{t}}$ for both references and samples as

$$ \tilde{\mathbf{t}}_{\text{sample}}(\alpha_i, \beta_j) \stackrel{!}{=} \frac{\tilde{\mathbf{t}}_{\text{sample}}(\alpha_i, \beta_j)}{\tilde{\mathbf{t}}_{\text{ref}}(\alpha_i, \beta_j)} \tilde{\mathbf{t}}_{\text{ref}}(\alpha_i, \beta_j). \quad (17) $$

Applying equations (12) and (15) to each configuration, we obtain

$$ A_i \tilde{\mathbf{J}}_{\text{sample}} E_j \stackrel{!}{=} \tilde{\mathbf{t}}_{ij} A_i E_j, \quad \tilde{\mathbf{t}}_{ij} = \frac{\tilde{\mathbf{t}}_{\text{sample}}(\alpha_i, \beta_j)}{\tilde{\mathbf{t}}_{\text{ref}}(\alpha_i, \beta_j)}, \quad (18) $$

where all experimental configurations are transformed into the complex matrix equation $\mathbf{W} \cdot \tilde{\mathbf{j}}_{\text{sample}} = \tilde{\mathbf{u}}$ in the following form

$$ \begin{pmatrix} A_1 \otimes E_1 & A_1 \otimes E_2 \\ A_2 \otimes E_1 & A_2 \otimes E_2 \end{pmatrix} \begin{pmatrix} \tilde{\mathbf{j}}_{11} \\ \tilde{\mathbf{j}}_{12} \\ \tilde{\mathbf{j}}_{21} \\ \tilde{\mathbf{j}}_{22} \end{pmatrix} = \begin{pmatrix} \tilde{\mathbf{t}}_{11} A_1 E_1 \\ \tilde{\mathbf{t}}_{12} A_1 E_2 \\ \tilde{\mathbf{t}}_{21} A_2 E_1 \\ \tilde{\mathbf{t}}_{22} A_2 E_2 \end{pmatrix}. \quad (19) $$

Symbol $\otimes$ denotes the Kronecker product, and $\tilde{\mathbf{j}}_{\text{sample}}$ is the vectorized complex Jones matrix $\tilde{\mathbf{J}}_{\text{sample}}$. Solving this system of complex linear equations at each frequency yields the complete measured Jones matrix.

The experimental input data are entirely contained within the vector $\tilde{\mathbf{u}}$. Inaccuracies in these data directly impact the Jones matrix solution but do not affect the stability of the equation system. Inaccuracies in the models $A_i$ and $E_j$ within the matrix $\mathbf{W}$ will directly affect the resulting Jones matrix. Because the matrix $\mathbf{W}$ is theoretical, the system's stability depends on our choice of the vectors $A_i$ and $E_j$. These vectors must be linearly independent. The system's stability is evaluated by the condition number of the symmetric positive definite matrix $\mathbf{G} = \mathbf{W}^\dagger \cdot \mathbf{W}$, where $\dagger$ denotes the Hermitian transpose. The condition number $\kappa = \lambda_{\max}(\mathbf{G})/\lambda_{\min}(\mathbf{G})$ is defined by the maximum and minimum eigenvalues of $\mathbf{G}$, $\lambda_{\max}(\mathbf{G})$ and $\lambda_{\min}(\mathbf{G})$, respectively. The chosen Cartesian horizontal and vertical polarizations for $E_j$ and diagonal and anti-diagonal polarizations for $A_i$ result in a condition number $\kappa$ of 1, ensuring ideal conditioning for stability, uniqueness, and robustness. In contrast, selecting horizontal and vertical states for both $A_i$ and $E_j$ leads to $\kappa \to \infty$, indicating a highly unstable solution. This outcome is expected experimentally, as measurements with crossed polarizations lead to zero signals. Thus, selecting $A_i$ and $E_j$ from independent basic polarization bases like Cartesian, diagonal, or circular is recommended for optimal conditioning.

### E. Error analysis of the Jones matrix reconstruction

The precision in determining the Jones matrix relies on the quality of the theoretical descriptions of polarization components $A_i$ and $E_j$, as well as the accuracy of the measured data $\tilde{\mathbf{t}}_{ij}$. Errors in the Jones matrix can arise from both random and systematic artifacts in the data, as well as from inaccuracies in the component descriptions. To assess experimental errors, we follow these steps: (i) Taking four free-space reference measurements; (ii) Collecting an additional four free-space measurements after a one-hour delay, a standard timeframe between reference and sample measurements, this time treating

the second set as the sample; (iii) Determining the Jones matrix, which is expected to be a unit diagonal matrix since both the sample and reference share the same configuration. We specifically selected free-space measurements for the sample because deviations in amplitude and phase from the diagonal matrix directly reflect instrumental errors and instabilities. We reconstructed the resulting Jones matrix in two ways: (a) Using four reference measurements in the corresponding sample configuration, as proposed in our method, to eliminate systematic errors; (b) Using only single reference to determine the Jones matrix, highlighting the advantage of employing four references.

In analyzing the amplitude Jones matrix shown in Figure 4b, the blue dashed line represents the theoretical diagonal unit matrix, while the red dots indicate the Jones matrix derived from four references. The grey lines illustrate the results obtained using only one reference. Relying on only one reference leads to substantial deviations from the theoretical model. These deviations may stem from idealized theoretical models, configuration-specific artifacts, or the azimuthal dependence of $\tilde{\mathbf{E}}_\beta$. Using four references significantly reduces the main diagonal errors to less than 2%. For the off-diagonal elements, the error is more pronounced, around 2-3%, possibly due to the unaccounted terahertz polarizer imperfections in the model.

We present the unnormalized absolute phases of the free-space Jones matrix in Figure 4c to demonstrate the absolute phase errors. Only the main diagonal elements are shown, as the off-diagonal elements have zero amplitude signals, making phase calculation irrelevant. The diagonal elements exhibit a phase shift of tenths of radians, likely caused by temperature or humidity fluctuations in the laboratory due to long beam paths. Notably, a phase error of 0.1 rad corresponds to an optical path length deviation of approximately 5 µm in the air at 1 THz. This deviation arises despite the long beam paths (several meters) required by the 1 kHz amplifier system. However, such millijoule spectroscopic experiments have a high potential for polarization-sensitive material characterization, which requires minimizing such phase errors.

Using only one reference results in systematic errors that deviate significantly from the expected linear trend. In contrast, using four references forms a linear curve, indicating more accurate results. In particular, this represents the absolute phase error, which translates to a relative error of less than one per thousand when measuring a bulk sample with a total phase of hundreds of radians. This substantial reduction in amplitude and phase errors brings the precision of terahertz ellipsometry closer to that of the visible range.

### V. DEMONSTRATION OF COMPLETE ELLIPSOMETRY WITH AN ADVANCED SAMPLE: $Hg_2Cl_2$

Up to now, we have demonstrated THz-cTDSE using the quartz plate, which exhibits only linear retardation. In this section, we explore the more advanced uniaxial anisotropic crystal $Hg_2Cl_2$ (Calomel) with a thickness of 2.15 mm and an in-plane optical axis. Due to its polarization-dependent phonon interactions in the terahertz spectrum, this crystal displays both linear retardation and dichroism. Figure 5a
shows the time-domain signal that exhibits a significant delay and attenuation compared to the free-space reference signals. In one polarization, where strong absorption occurs, the signal follows the behavior of a damped harmonic oscillator. Driven by polarization-dependent phonon interactions, the retardation here is significantly greater than that observed in quartz (see Figure 2a). Figure 5b reveals a corresponding spectrum where phonon-induced cutoff begins at 1 THz, highlighting the diattenuating behavior of the crystal.

### A. Jones analysis

Using our complete ellipsometry, we now investigate linear birefringence and diattenuation using the Jones matrix formalism. The predicted behavior of a birefringent diattenuating sample is detailed in Appendix A for the Jones matrix, while optical constants derived from the oscillator model are out of the scope of this paper and are provided in [59]. The Jones matrix, including phases obtained by THz-cTDSE, enables the reconstruction of both the refractive index spectra and geometrical parameters (thickness, optical axis azimuth) and de-correlate them by data fitting to the model consistent with Kramers-Kronig dispersion relations. To illustrate its complex behavior, we analyze a Calomel crystal at 90° and 135° azimuths of its optical axis, corresponding to Cartesian linear and diagonal Stokes bases. We slightly rotated the azimuth in both orientations to ensure detectable signals in all components and avoid zero off-diagonal elements.

Figure 5c presents the experimentally determined amplitude Jones matrix, compared with the theoretical model. At a 90° azimuth, the generated polarizations travel through the crystal along the ordinary and extraordinary axes, corresponding to the diagonal elements. Consequently, the off-diagonal elements are nearly zero due to the absence of polarization conversion, except for a minor signal caused by the slight rotation in setting the crystal's azimuth. The diagonal amplitudes differ due to reflective losses influenced by different Fresnel coefficients and primarily by distinct phonon absorption. For the element $|\hat{t}_{22}|$, the signal is fully absorbed by phonons starting at 1.4 THz. This notable difference in absorbed amplitude complicates data interpretation compared to quartz, where the amplitude was uniform and dependent on retardation. Nevertheless, the normalized amplitude form is unsuitable, leading to zero dividing at multiple points in the terahertz spectrum.

Setting the azimuth to 135° results in non-zero off-diagonal elements of the Jones matrix, which introduces oscillations in all its elements. As observed in quartz, these oscillations manifest as retardation caused by the difference between the ordinary and extraordinary refractive indices. The Calomel crystal functions as a half-wave plate, rotating the input polarization by 90° at frequencies where the diagonal elements approach zeros. Similarly, this behavior is evident in the amplitude maxima of the off-diagonal elements. The elements exhibit combined signals with ordinary and extraordinary absorption, and the oscillations cease when one polarization component is fully absorbed. It is crucial to recognize that combining these effects in a 2 × 2 matrix can become complex due to

their coupling. Moreover, this sample does not consider more complex mixing, such as those involving circular polarization effects. However, amplitude changes due to the retardation can easily be mistaken for those related to absorption, underscoring the importance of analyzing the phase.

Figure 5d depicts the Jones phase matrix, normalized by the element $t_{11}$. Interpreting the phases is easier because they reflect minimal amplitude effects. The diagonal element $t_{22}$ at 90° azimuth exhibits total retardation reaching hundreds of radians, which is an order of magnitude higher than in quartz. Retardation at a 135° azimuth is largely compensated but not completely nullified due to a slight azimuth deviation. Oscillations in the off-diagonal elements arise from phase jumps of $\pi$ at frequencies corresponding to a half-wave plate caused by polarity changes in the waveform. Although noisy at a 90° azimuth due to minimal diagonal elements, the phase still closely matches the theoretical model. Beyond 1.4 THz, the oscillations cease entirely as one polarization is fully attenuated, thus eliminating retardation.

### B. Mueller matrix analysis

Figure 6a presents the experimentally determined normalized Mueller matrix for the Calomel crystal plate. The spectra of the Mueller matrix elements are in close agreement with the diattenuating retarder model described in Appendix B. The Mueller matrix highlights the retardation submatrix $m_{2-4,2-4}$, characterized by spectra that appear as damped trigonometric functions. Its off-diagonal elements are described by $J \cos 2\theta \sin \delta$ and $J \sin 2\theta \sin \delta$, where $\theta$ is the crystal's azimuth and $\delta$ represents the retardation introduced by the crystal. The coefficient $J$ is a parameter associated with diattenuation. The oscillations in the spectra indicate retardation, whereas the presence of damping reveals diattenuation effects. At the 90° azimuth, the matrix elements $m_{22}$, $m_{34}$, and $m_{43}$ reach their maxima, corresponding to linear retardation in Cartesian coordinates.

At the 135° azimuth, the matrix elements $m_{33}$, $m_{24}$, and $m_{42}$ reach their maxima, corresponding to retardation in diagonal polarizations. The first row ($m_{1,2-4}$) and column ($m_{2-4,1}$), excluding $m_{11}$, directly indicate diattenuation without retardation effects, expressed as $K \cos 2\theta$ and $K \sin 2\theta$, with $K$ associated with diattenuation. Minor diattenuation is observed at low terahertz frequencies. As diattenuation reaches its maximum at 1.4 THz, it indicates complete signal absorption, providing no further information. Similarly, elements $m_{12}$ and $m_{21}$ are associated with diattenuation in Cartesian coordinates, while $m_{13}$ and $m_{31}$ primarily correspond to diattenuation in diagonal coordinates. All retardation phenomena are coupled, complicating the interpretation. The Mueller matrix, comprising 15 elements, is often more comprehensible for complex samples than the Jones matrix. However, it remains less straightforward than analysis based on Pauli exponential coefficients. In contrast to the Jones matrix and Pauli's exponential coefficients, the Mueller matrix does not include total phase information.
<figure><img src="image_6.png" /><figcaption>Fig. 5. Ellipsometric characterization of the Hg<sub>2</sub>Cl<sub>2</sub> (Calomel) crystal at 90° and 135° azimuths, demonstrating significant diattenuation and retardation due to polarization-anisotropic phonons. a) Waveforms obtained via standard time-domain spectroscopy in orthogonal polarizations, compared with a free-space reference using the analyzer at ±45°; and b) the corresponding spectral intensity |E|^2. c) Generalized time-domain ellipsometry: amplitude spectra of the Jones matrix and d) normalized phase spectra of the Jones matrix, compared with a model.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. The ellipsometric analysis of the Calomel crystal at 90° and 135° azimuths presented as a) normalized Mueller-Jones matrix calculated from the corresponding Jones matrix in Figure 5c,d. The first row and column, excluding the first element, correspond to diattenuation effects. The remaining submatrix represents a diattenuating retarder, with oscillations indicating retardation and damping signifying diattenuation. b) Exponential Pauli coefficients are associated with diattenuation represented by *d* and retardation represented by *δ*. The subscript *0* indicates total attenuation and phase, while subscripts *L*, *D*, and *C* correspond to linear polarizations in Cartesian coordinates, ±45° diagonal bases, and circular polarizations, respectively.</figcaption></figure>

### C. *Pauli analysis*

Interpreting the Jones matrix remains challenging despite
the sample exhibiting only linear retardation and diattenu-

ation. By converting to the logarithmic Jones matrix $\ln \tilde{J}$,
we decompose it into Pauli exponential coefficients $\tilde{p}_{\xi}$ for
$\xi = 0, L, D, C$, dividing all effects into retardation $\delta_{\xi}$ and
diattenuation $d_{\xi}$ across the Stokes $\xi$-bases. Equations 11
provide the analytical expressions for diattenuating birefrin-
gent materials regarding Pauli exponential coefficients. The
Calomel crystal's experimentally determined Pauli exponen-
tial coefficients are plotted in Figure 6b. The coefficients'
spectra are shown up to 1.5 THz since the absorption of
one polarization makes determining diattenuation and retar-
dation unattainable. The total absorption coefficient $d_0 =$
$\ln(|\tilde{t}_o||\tilde{t}_e|) + d_o + d_e$ is independent of polarization and sample
rotation. It includes the sum of the ordinary and extraordinary
attenuation coefficients $d_{o,e}$ and Fresnel transmission coeffi-
cients $\tilde{t}_{o,e}$. The total retardation $\delta_0 = \arg(\tilde{t}_o\tilde{t}_e) + \delta_o + \delta_e$
is primarily determined by the sum of the ordinary and
extraordinary phases accumulated during propagation in the
crystal. The coefficients $d_L = \cos 2\theta (\ln|\tilde{t}_o/\tilde{t}_e| + d_o - d_e)$
and $d_D = \sin 2\theta (\ln|\tilde{t}_o/\tilde{t}_e| + d_o - d_e)$ describe diattenuation
effects in the vertical/horizontal and $\pm45^\circ$ diagonal polariza-
tion bases for an uniaxial dichroic crystal. The diattenuation
varies between $\tilde{p}_L$ and $\tilde{p}_D$ due to the crystal's azimuthal
rotation $\theta$. The principal axis of diattenuation can be easily
determined from the ratio $d_D/d_L = \tan\theta$. The primary
onset of diattenuation is observed at 1 THz, although it
slightly diattenuates before this frequency. The retardation
coefficients $\delta_L = \cos 2\theta (\arg(\tilde{t}_o/\tilde{t}_e) + \delta_o - \delta_e)$ and $\delta_D =$
$\sin 2\theta (\arg(\tilde{t}_o/\tilde{t}_e) + \delta_o - \delta_e)$ take an analogous form with
diattenuation coefficients. In contrast, the retardation $\delta_{L,D}$ is
observable across the entire THz spectrum. Finally, the effects
$\delta_C$ and $d_C$ are zero, corresponding to polarization phenomena
in circular bases, which are absent in achiral uniaxial crystals.

VI. COMPARISON OF THZ-CTDSE WITH THE
SPECTRAL-DOMAIN MUELLER MATRIX ELLIPSOMETRY

A. *Signal detection, coherence, and spectral range*

THz-cTDSE can operate within the terahertz and mid-
infrared regions, relying on waveform measurements that
provide amplitude and phase information. The method relies
on fully coherent signals, making it ideal for phase-sensitive
measurements. Spectral-domain Mueller matrix ellipsometry
spans a broader spectral range, from ultraviolet to terahertz,
because it measures signal intensity rather than waveform.
Although this broadens its range of applications, the reliance
on intensity measurements results in the loss of absolute
phase and amplitude information. In visible and ultraviolet
regions, waveform measurements are unattainable to perform,
and Mueller matrix ellipsometry typically operates with par-
tially coherent intensity systems. However, terahertz radiation
remains highly coherent, even in Mueller matrix spectral-
domain ellipsometry, which excludes depolarization analysis
[18].

B. Ellipsometric formulation and inspected phenomena

THz-cTDSE provides more versatility by retrieving the
complete 2 × 2 Jones matrix, allowing detailed polarization
analysis of anisotropic materials. THz-cTDSE also provides
the ability to transform the Jones matrix into the complete

Mueller matrix. A key advantage of THz-cTDSE is its abil-
ity to derive Pauli exponential coefficients, offering a direct
interpretation of ellipsometric data via retardation and diat-
tenuation in Stokes space. THz-cTDSE is the only method
capable of providing such comprehensive polarization insights.
Mueller matrix ellipsometry measures the 4 × 4 Mueller
matrix, providing a detailed characterization of anisotropy and
depolarization. However, depolarization can only be detected
in cases where the system operates with partially coherent
signals.

C. *Experimental requirements*

THz-cTDSE utilizes two generated and two analyzed po-
larization states for matrix reconstruction, enabling detailed
anisotropy analysis while preserving a straightforward experi-
mental approach. Mueller matrix ellipsometry requires sixteen
measurements to determine the Mueller matrix, combining
four generated and four analyzed polarization states. These
states must be independent and span a large area of the
Poincaré sphere, making this method experimentally more
complex than THz-cTDSE, which provides complete infor-
mation with fewer measurements.

VII. PERSPECTIVES OF THZ-CTDSE

This study demonstrates complete ellipsometry in the
time domain, offering significant advantages for analyzing
anisotropic materials. The THz-cTDSE can be particularly
advantageous in the terahertz and even mid-infrared regions, as
it utilizes waveform measurements that provide both amplitude
and phase information [60], [61]. A key advantage of this ap-
proach is that all polarization states are determined exclusively
from linear polarizations, obviating the requirement for cir-
cular polarization-based methods typically needed in Mueller
matrix ellipsometry or circular dichroism measurements [15],
[62]. While many scientific advancements have employed
polarimetry [63]–[65] or Stokes vector analysis [66]–[68],
these techniques provide only limited information about the
output polarization state. In contrast, complete ellipsometry
offers a more comprehensive and intuitive understanding by
capturing retardation and diattenuation of all field-resolved
polarization states.

Although this study focuses on transmission measurements,
straightforward adaptation to reflection configurations are
more prevalent in ellipsometry due to the enhanced polar-
ization sensitivity of reflective Fresnel coefficients at oblique
angles. The reflection ellipsometry approach is crucial for
enhancing sensitivity in thin-film measurements at oblique
incidence angles, where current terahertz techniques face
significant limitations due to the increased precision required
relative to the long wavelengths of terahertz radiation, which
remains a fundamental challenge in this field [69], [70]. A
crucial aspect of adapting the setup for reflection geometry
is optimizing the angle of incidence, ideally near the Brew-
ster angle, where the contrast between s- and p-polarized
reflection is maximized. In the terahertz range, the Brewster
angle typically falls between 60° and 75°, making the use
of a collimated beam impractical due to the increased beam
footprint on the sample. To mitigate this, the beam should be focused to reduce its footprint on the sample. However, excessive focusing broadens the range of incidence angles due to the strong Gaussian nature of the THz beam [54], [55], leading to deviations from the expected Fresnel coefficients. Additionally, precise tilting and positioning of the sample are crucial to ensure the correct alignment of the reflected beam with the detection path. Misalignment may cause the beam to miss the detector or introduce phase errors.

Furthermore, our method could be adapted to reflection by applying similar calibration procedures, such as using an isotropic reference material in reflection to determine the actual azimuth of the generated and analyzed polarization states. The Jones matrix reconstruction will utilize the reflection coefficient ratio $\tilde{\tau}_{ij}$ in the same equation system $A_i \tilde{J}_{\text{sample}} E_j \stackrel{!}{=} \tilde{\tau}_{ij} A_i E_j$, as demonstrated for the transmission. We recommend normalizing the matrix by its first element or calculating the displacement error from the data to avoid phase errors caused by differently positioned references and samples in reflection [71]–[73], as this parameter remains constant across all frequencies and polarization measurements.

Spintronic emitters, demonstrated here as a universal polarization source, offer an additional avenue for enhancement. In our setup, the emitter facilitates straightforward polarization control, eliminating the need for polarizers at the ellipsometer input. Furthermore, by modulating the spintronic emitter’s polarization with an oscillating magnetic field, the sensitivity of the ellipsometer could be improved through polarization demodulation [74], [75]. Integrating spintronic detection via Zeeman torque sampling involves utilizing an external magnetic field to achieve a uniform and extensive polarization-sensitive waveform detection [76]. This breakthrough can potentially create a polarization-sensitive spintronic source and detector within the terahertz region, eliminating the need for polarizers.

Lastly, another potential improvement pertains to the calibration procedure. The calibration method demonstrated in this paper is critical for accurately determining the Jones matrix and compensates for systematic errors, ensuring reliable results. However, experimentally characterizing the vectors $A_i$ and $E_j$, which are assumed to exhibit ideal behavior, could further enhance accuracy and performance. This approach, known as the eigenvalue calibration method, requires multiple known samples to determine the experimental vectors $A_i$ and $E_j$ [77], [78]. This technique is commonly used in Mueller matrix ellipsometry to determine the behavior of all included components. This strategy could prove highly effective in the terahertz region, where accurately describing polarizing components is challenging.

## VIII. CONCLUSION

In this work, we have introduced terahertz time-domain spectroscopic complete ellipsometry (THz-cTDSE), advancing beyond the limitations of standard and Mueller ellipsometry methods. Traditional standard ellipsometry is limited to isotropic samples and lacks comprehensive polarization information. In contrast, Mueller matrix ellipsometry provides

a complete intensity-based characterization, but the resulting data is indirect and interdependent, without explicit separation of phase and amplitude contributions. Our time-domain approach leverages the complex information inherent in terahertz signals, significantly enhancing ellipsometric capabilities by providing a detailed and intuitive description using Pauli exponential coefficients.

We have successfully determined the complete Jones and Mueller matrices, with the Jones matrix approach being unique in capturing the entire complex polarization response. The incorporation of Pauli exponential coefficients, seldom applied in experimental ellipsometry, provides direct and clear insights into diattenuation and retardation across all Stokes vector bases, calculated directly from the Jones matrix. The Pauli approach significantly streamlines ellipsometry analysis compared to the Jones and Mueller matrices, where distinguishing individual polarization phenomena is often complex due to their coupling.

Furthermore, we have developed the calibration method, which is crucial for accurately reconstructing the Jones matrix. Unlike previous terahertz ellipsometry techniques that relied on meticulous alignment, our method not only determines the actual settings of optical components but also compensates for systematic artifacts, enhancing robustness and reliability.

We demonstrated the practical application of spintronic terahertz emitters (STEs) in ellipsometry, taking advantage of their exceptional polarization properties. The STE allows straightforward polarization control of the terahertz source by simply adjusting an external magnetic field, a level of simplicity and precision not attainable with other methods for pulsed mJ-systems.

In conclusion, the THz-cTDSE method substantially advances terahertz ellipsometry, providing complete polarization characterization with improved accuracy and interpretability. This work opens new avenues for investigating complex materials and phenomena in the terahertz regime, with potential applications across complex anisotropic samples, heterostructures, metamaterials, biological specimens, and materials exhibiting magneto-optical effects.

## APPENDIX A

### COHERENT DESCRIPTION OF AN A-CUT UNIAXIAL ANISOTROPIC MATERIAL USING THE JONES MATRIX

In this section, we describe the transmission model for a polarimetric response of an A-cut uniaxial achiral anisotropic material using the Jones matrix $\tilde{J}$ [13]. The transmitted polarizations $\tilde{e}_{o,e}$ propagate along the ordinary and extraordinary directions in such material. We used the rotated Jones matrix for the material’s optical axis rotation by an angle $\theta$ as follows

$$ \tilde{J} = \begin{pmatrix} (\tilde{e}_o \cos^2 \theta + \tilde{e}_e \sin^2 \theta) & (\tilde{e}_o - \tilde{e}_e) \cos \theta \sin \theta \\ (\tilde{e}_o - \tilde{e}_e) \cos \theta \sin \theta & \tilde{e}_o \sin^2 \theta + \tilde{e}_e \cos^2 \theta \end{pmatrix}, \quad (20) $$

We further assume that the material has distinct ordinary and extraordinary complex refractive indices $\tilde{n}_o \neq \tilde{n}_e$, exhibiting
<figure><img src="image_8.png" /><figcaption>Fig. 7. The complex Jones matrix of the A-cut quartz plate as a function of the optical axis azimuth angle. Experimental data at 0.72 THz, 1.69 THz, and 2.35 THz are compared with the model.</figcaption></figure>

both linear birefringence and dichroism in the form

$$ \tilde{e}_o = \frac{4\tilde{n}_o}{(1+\tilde{n}_o)^2} \exp \left[ -i \frac{2\pi\nu}{c} (\tilde{n}_o - 1)\tau \right] = \tilde{t}_o \exp(-d_o) \exp(-i\delta_o), \quad (21) $$

$$ \tilde{e}_e = \frac{4\tilde{n}_e}{(1+\tilde{n}_e)^2} \exp \left[ -i \frac{2\pi\nu}{c} (\tilde{n}_e - 1)\tau \right] = \tilde{t}_e \exp(-d_e) \exp(-i\delta_e), \quad (22) $$

where the amplitude factors $\tilde{t}_{o,e}$ represent the cumulative Fresnel transmission coefficients for a single pass through the air-to-crystal and crystal-to-air interfaces. The exponential term describes the propagation of terahertz radiation through the crystal's thickness $\tau$. Parameter $\nu$ represents the terahertz frequency, and $c$ denotes the speed of light in vacuum. Importantly, this formulation presupposes only direct transmission, excluding subsequent echoes. In this form, the measured waveform should contain only the initial pulse, as time-domain spectroscopy enables the separation of Fabry-Pérot reflections due to their distinct time delays.

Figure 7 shows the azimuthal angle dependence of the Jones matrix for the 1.816 mm thick A-cut crystal quartz plate. The observed 180° periodicity of the matrix components is consistent with the Jones matrix model (20) and demonstrates the reliability of the measurement across different rotations. This behavior arises from the uniaxial symmetry of the crystal and aligns with theoretical predictions. Additionally, phase jumps of $\pi$ correspond to their sign changes, further validating the accuracy of the measurement setup.

Our main interest lies in the differences in phase and absorption between the polarization eigenmodes of the anisotropic crystal, specifically the retardation

$$ \delta = \operatorname{Re}(2\pi\nu/c(\tilde{n}_e - \tilde{n}_o)\tau) = \delta_o - \delta_e \quad (23) $$

and the diattenuation

$$ d = \operatorname{Im}(2\pi\nu/c(\tilde{n}_e - \tilde{n}_o)\tau) = d_o - d_e. \quad (24) $$

We describe the ordinary and extraordinary refractive indices $\tilde{n}_{o,e}$ of quartz crystal using the Sellmeier model

$$ \tilde{n}_{o,e}^2(\lambda) = \varepsilon_{o,e} + \frac{B_{o,e}\lambda^2}{\lambda^2 - C_{o,e}^2} \quad (25) $$

with $\varepsilon_{o,e}$ being the static permittivity, and $B_{o,e}$, $C_{o,e}$ corresponding to the oscillator strength and the resonant wavelength, respectively.

Figure 8 shows a comparison of the refractive index extracted from THz-cTDSE measurements in this work with values from the Palik handbook [79]. The obtained refractive indices and the birefringence $n_e - n_o$ show good agreement with the tabulated data.

<figure><img src="image_9.png" /><figcaption>Fig. 8. The refractive index spectra of crystal quartz obtained by fitting THz-cTDSE measurement data are compared with tabulated values from the Palik handbook [79].</figcaption></figure>

For Hg₂Cl₂ (calomel) crystal plate, we model the refractive indices $\tilde{n}_{o,e}$ using the Lorentz oscillator model

$$ \tilde{n}_{o,e}^2(\omega) = \varepsilon_{o,e} + \sum_{j=1}^{N} \frac{A_j^{o,e}}{\omega_j^{o,e2} - \omega^2 - i\gamma_j^{o,e}\omega} \quad (26) $$

where $A_j^{o,e}$, $\omega_j^{o,e}$, and $\gamma_j^{o,e}$ represent the amplitude, resonance frequencies, and damping factors, respectively. The fitted optical constants are presented in Liu et al. [59]. The extracted complex refractive indexes for Calomel (Hg₂Cl₂) are in good agreement with the optical properties reported in the literature [80].
APPENDIX B

MUELLER MATRIX OF A-CUT UNIAXIAL CHERYSTAL

By transforming the Jones matrix $\tilde{J}$ from (20) into the
Mueller matrix using the conversion (equation (2)), we derive

the following analytical expression [19], [78]:

$$
\mathbf{M}_{\text{rotated}} = \begin{pmatrix} 1 & D \cos 2\alpha & D \sin 2\alpha & 0 \\ D \cos 2\alpha & \cos^2 2\alpha + A \sin^2 2\alpha \cos \delta & A \sin 2\alpha \cos 2\alpha (1 - \cos \delta) & A \sin 2\alpha \sin \delta \\ D \sin 2\alpha & A \sin 2\alpha \cos 2\alpha (1 - \cos \delta) & -A \cos 2\alpha \sin \delta & -A \cos 2\alpha \sin \delta \end{pmatrix}, \quad (27)
$$

where the coefficients *D* and *A* represent the differences
between the ordinary and extraordinary attenuations

$$
A = \frac{2|\tilde{t}_o||\tilde{t}_e|\exp(-d)}{|\tilde{t}_o|^2 + |\tilde{t}_e|^2 \exp(2d)}, \quad D = \frac{|\tilde{t}_o|^2 - |\tilde{t}_e|^2 \exp(2d)}{|\tilde{t}_o|^2 + |\tilde{t}_e|^2 \exp(2d)}. \qquad (28)
$$

The first row and column of the Mueller matrix are scaled by the coefficient *D*, which corresponds directly to diattenuation. The submatrix *m*<sub>2–4,2–4</sub> includes trigonometric functions of retardation *δ*, while also being influenced by diattenuation through the parameter *A*, which diminishes as diattenuation *d* becomes more pronounced.

In the case of quartz, where attenuation is negligible ($d_o = d_e \to 0$) and the transmission coefficients are nearly identical ($\tilde{t}_o \approx \tilde{t}_e$), the values $D \to 0$ and $A \to 1$ yield a Mueller matrix of a wave plate. However, as diattenuation increases ($d \to \infty$), the coefficient $D = -1$, indicating maximal absorption of one polarization, while $A = 0$ signifies the complete attenuation of retardation, as one polarization is fully absorbed.

ACKNOWLEDGMENT

The authors acknowledge the partial support of the follow-
ing projects: Czech Science Foundation 22-33060S; project
No. CZ.02.01.01/00/22 008/0004631- "Materials and Tech-
nologies for Sustainable Development", funded by the Eu-
ropean Union and the state budget of the Czech Republic
within the framework of the Jan Amos Komensky Opera-
tional Program; project REFRESH - Research Excellence For
Region Sustainability and High-tech Industries project num-
ber CZ.10.03.01/00/22 003/0000048 via the Operational Pro-
gramme Just Transition; Student Grant System SP2024/086.
This work was funded by FET project s-NEBULA/Grant No.
863155. We thank Tobias Kampfrath, Tom S. Seifert, and
Alexander Chekhov for providing the large-area spintronic
terahertz emitter, which was essential for the experiments
performed in this study.

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