Quartz as an Accurate High-Field Low-Cost THz Helicity Detector

The advent of high-field THz sources has opened the field of nonlinear THz physics and unlocked access to fundamental low energy excitations for ultrafast material control. Recent advances towards controlling and employing chiral excitations, or generally angular momentum of light, not only rely on the measurement of undistorted intense THz fields, but also on the precise knowledge about sophisticated THz helicity states. A recently reported and promising detector material is $\alpha$-quartz. However, its electrooptic response function and contributing nonlinear effects have remained elusive. Here, we establish z-cut $\alpha$-quartz as a precise electrooptic THz detector for full amplitude, phase and polarization measurement of intense THz fields, all at a fraction of costs of conventional THz detectors. We experimentally determine its complex detector response function, which is in good agreement with our model based on predominantly known literature values. It also explains previously observed thickness-dependent waveforms. These insights allow us to develop a swift and reliable protocol to precisely measure arbitrary THz polarization and helicity states. This two-dimensional electrooptic sampling (2D-EOS) in $\alpha$-quartz fosters rapid and cost-efficient THz time-domain ellipsometry, and enables the characterization of polarization-tailored fields for driving chiral or other helicity-sensitive quasiparticles and topologies.


Introduction
THz sources with peak field strengths in the ~1 MV/cm regime, employing optical rectification in LiNbO3 1 , difference frequency generation 2 , large-area spintronic emitters 3 and acceleratorbased facilities 4 , are becoming more widely accessible.This development has enabled the selective drive of low-energy excitations such as phonons 5,6 , magnons 7 or other quasiparticles, thereby allowing for ultrafast control over material properties and non-equilibrium material design towards light-induced superconductivity 8 , ferroelectricity 9 , ferromagnetism 10 and spindynamics 7,11 .However, despite large improvements in THz generation, the detection of intense single-cycle THz fields without distortions has remained challenging 12,13 .
Field-resolved THz detection provides precise frequency resolution of amplitude and phase of the light field.This feature is crucial for, e.g., THz time-domain spectroscopy (THz-TDS) 14 , THz emission spectroscopy 13 and state-of-the-art experiments involving THz high-harmonic generation in topological insulators 15 , graphene 16 or superconducting cuprates 17 .Moreover, emerging field-driven effects, e.g., for ultrafast control of topological 18 or chiral 19-21 material properties, are inherently sensitive to the carrier-envelope phase (CEP) and polarization (e.g., helicity) of the driving THz pulse.Full vectorial THz-field characterization is required for the precise detection of arbitrary THz polarization states.This information constitutes the basis for THz time-domain ellipsometry, which allows for the characterization of tensorial dielectric properties in opaque 22 , anisotropic materials 23 , and transient metamaterials 24 , where traditional THz-TDS faces limitations.Another application is THz circular-dichroism spectroscopy, which has been applied in chiral nanostructures and molecular assemblies 25 , thermoelectric solids 26 , or bio-relevant systems such as DNA 27 , and living cancer cells 28 .However, partly due to the difficulty of precise polarization-resolved THz detection, THz time-domain ellipsometry and circular-dichroism spectroscopy have not been widely adopted yet.
The common technique to detect phase-stable THz fields is electrooptic sampling (EOS) 29 .
Here, the incident THz pulse induces a change in birefringence proportional to the THz electric field in a nonlinear crystal like ZnTe 30 or GaP 31 , which can be stroboscopically sampled by a visible (VIS) or near-infrared (NIR) sampling pulse as a function of time delay .However, the measured instantaneous signal () is, in general, not simply proportional to the instantaneous field () because of noninstantaneous features such as phonon resonances and velocity mismatch of the THz and sampling pulse.Within linear response theory and in frequency space, a response function ℎ connects  and  at THz frequencies Ω via (Ω) = ℎ(Ω)(Ω) and captures the frequency dependence of the nonlinear susceptibility  (2) , which can be strongly modulated by phonons 29 , and non-local effects, such as phase mismatch between THz-and sampling pulse 32,33 .
For (110)-oriented zincblende-type electrooptic crystals such as ZnTe (110), resolving the polarization state of THz pulses typically requires rotation of the detector crystal and sampling pulse polarization 34 .Unfortunately, such measurements can be easily polluted by inhomogeneities of the detector crystal, birefringence effects, or inaccurate rotation axes.On the other hand, (111)-oriented zincblende crystals enable polarization state retrieval by simply modulating the sampling pulse polarization by using, e.g., a photoelastic modulator 35 or employing a dual detection scheme based on two balanced detections 36 .Nonetheless, the specific detector requirements and additional experimental effort have limited the application of polarization-resolved EOS so far.
Extending these concepts to highly intense THz fields poses extra challenges, since they can lead to distorted signals in conventional EOS crystals, such as ZnTe or GaP, which include over-rotation 12 or higher-order nonlinearities such as the THz Kerr effect 37,38 .This aspect means that the amplitude and phase of intense THz fields cannot be reliably extracted within the linear response.However, attenuating the THz fields by using, e.g., wiregrid polarizers or filters might induce additional spectral distortions 39 .
Here, we focus on z-cut α-quartz, which is a widely used substrate material for THz-TDS due to its high THz transparency 14 and in-plane optical isotropy.It recently attracted attention as a promising nonlinear THz material 40 , i.e., as broadband THz emitter via optical rectification 41 or as THz detector via EOS 42 .In fact, its electrooptic coefficient,  11 = 0.1 − 0.3 pm/V 43 , is about an order of magnitude smaller than  41 =4 pm/V of ZnTe 30 , thereby moving nonlinear EOS responses to much higher THz field amplitudes.Its large bandgap and optical transparency allow for a broad dynamic range and high damage threshold.Moreover, α-quartz is widely available at 2 orders of magnitude lower cost than typical EOS crystals.However, there are significant drawbacks that prevented the reliable use of quartz for THz detection so far.In particular, the response function ℎ has been unknown, and its peculiar thickness dependence lead to the open question regarding bulk versus surface  (2) contributions 42 .Likewise, the polarization-sensitivity has remained mostly unexplored.
In this work, we experimentally measure the quartz response function and model it predominantly based on known literature values.We show that arbitrary THz polarization states can be measured by a simple and time-efficient method utilizing only two EOS measurements with different sampling pulse polarizations.The latter is achieved by a simple rotation of a half-waveplate (HWP) in the VIS spectral range.As a textbook example for timedomain ellipsometry, we determine the birefringence of y-cut quartz as commonly used for commercial THz waveplates.We find that the transmitted single-cycle pulses exhibit complex polarization states in the highly polychromatic regime 44 , which cannot be described by a single polarization ellipse, Jones vector or set of Stokes parameters.Our study establishes z-cut α-quartz as a reference detector for amplitude, phase, and arbitrary polarization states of THz fields exceeding 100 kV/cm, fostering cost-efficient high-field THz time-domain ellipsometry and tailoring helical THz driving fields for ultrafast material control.

Experimental Setup
Intense single-cycle THz fields (1.3 THz center frequency, 1.5 THz full width at half maximum (FWHM)) with peak fields exceeding 1 MV/cm are generated by tilted-pulse-front optical rectification in LiNbO3 (Ref.1).The THz field strengths or its linear polarization angle  relative to the vertical direction in the lab frame are altered using a THz polarizer pair, P1 and P2 in Fig. 1a.The THz field-induced birefringence in the EOS crystal is probed by synchronized VIS sampling pulses (800 nm center wavelength, ~20 fs duration) using a balanced detection scheme.The sampling pulse's incident linear polarization is set to arbitrary angles  by a broadband VIS HWP.We measure EOS in a ZnTe (110) crystal (10 µm thickness) and various z-cut α-quartz plates with thicknesses of 35, 50, 70, and 150 µm as a function of sampling pulse polarization , THz polarization , and the crystal's azimuthal angle  at normal incidence (see Fig. 1a).Finally, we also trace the THz field after collimated transmission through highly birefringent y-cut α-quartz (700 µm thickness), which corresponds to a commercial quarter-wave plate (QWP) for 2.2 THz.

Electrooptic Response Function
We first confirm the linear response function relation.Fig. 1b shows the measured THzinduced birefringence signals ()/ max in 50 µm quartz for different THz peak fields.The induced birefringence scales linearly with the THz electric field strength (see inset of Fig. 1b), confirming a linear electrooptic effect as recently observed by Balos et al. 42 .The normalized time-and frequency-domain shapes (see Fig. 1c) do not change substantially for different THz fluences, ruling out over-rotation effects and demonstrating that the higher-order nonlinearities 40 (e.g.<1.5 THz) are small for THz fields on the order of 1 MV/cm.This finding confirms that quartz can reliably sample THz electric fields ≥0.1 MV/cm within the linearresponse regime.
To experimentally extract the linear response function of 50 µm quartz, we compare the quartz EOS signal  Q with the signal  ZnTe from 10 µm ZnTe, whose response function ℎ ZnTe is known 32

(see Fig 2a).
To avoid nonlinear distortions, the THz power for ZnTe was attenuated by the THz polarizer pair by a factor of ~40.We Fourier transform these traces and extract the quartz response using ℎ Q =ℎ ZnTe ( Q / ZnTe ) in the frequency-domain.The amplitude and phase of ℎ Q are shown as blue dots in Figs.2b and 2d, respectively, demonstrating that the quartz response covers the full 0.1-4 THz bandwidth of the LiNbO3 source without gaps.
However, it contains a substantial frequency dependence in the form of modulations with a frequency spacing of ~1.4 THz as well as an enhancement at low frequencies <0.9 THz and at around 3.9 THz.

Modelling
To understand the experimental response function ℎ Q,exp (Ω) of 50 µm quartz, we model the response ℎ calc as function of THz frequency Ω by extending the formalism of Ref. 32 and use: Fresnel transmission coefficients for propagating from air into quartz and quartz into air, respectively. eff (2) (Ω) =  eff (2) ( c ,Ω) is the effective nonlinear susceptibility of the detection crystal under the assumption that  eff (2)  To calculate ℎ Q , we use the known quartz refractive indices in the THz 14 and optical region 45 .
However, the nonlinear susceptibility  (2) (Ω) is not known and we therefore model it by: where   (2) is the pure electronic susceptibility.The last term corresponds to the ionic contribution with  TO being the frequency, and Γ being the damping of the respective transverse-optical (TO) phonon, while the Faust-Henry coefficient  defines the ratio between the lattice-induced and electronic contributions 29,46 .We take the phonon parameters Ω TO /(2) = 3.9 THz, and Γ/(2) = 0.09 THz from Davies et al. 14 and find  = 0.15 to provide good agreement with our experimental values (see red curves in Figs.2b,d).We assume that the striking low-frequency enhancement of ℎ Q (Ω) (see Fig. 2b) arises from  (2) and model it by a phenomenological Debye-type relaxation contribution  with characteristic time scale  D (second term in Eq. ( 2)).Choosing  = 0.7 and  D = 0.5 ps provides nearly perfect agreement with the 0.1-0.9THz range in ℎ Q,exp .We will discuss possible physical origins of such a contribution below.Thus, by analytic modeling, we find dominating contributions by the phase matching factor , the field transmission coeffient  F , and the nonlinear susceptibility  (2) , disentangled in Fig. 2c,e.
We apply the response function to calculate the exact THz electric field (red) from the quantitative EOS signal in 50 µm quartz (blue) in Figs.3a and 3b in the time-and frequencydomain, respectively.To determine the absolute field strength, we use the measured THz pulse energy and focal size (see Supplementary Note 1).We obtain a peak field strength of 1.04 MV/cm.We can therefore estimate the effective electrooptic coefficient  eff , which equals the  11 tensor component, of z-cut quartz to be 0.1 pm/V (see Supplementary Note 2).This value agrees well with previous reports of  11 at optical frequencies ranging between 0.1 and 0.3 pm/V in z-cut quartz 43,47 .

Thickness Dependence and Nonlinear Origin
The response function also depends on the crystal thickness, which typically presents a tradeoff between sensitivity and bandwidth.The first report of EOS in quartz suggested a strong surface  (2) contribution 42 .Indeed, the surface and bulk  (2) have a similar order of magnitude 48 .As the surface contribution originates from a depth of ~1 nm (Ref.48), its contribution will be small in comparison to the bulk contribution for a quartz crystal with a thickness >10 µm.The response functions presented here (Figs.2b,d and 3f) strongly indicate a pure bulk  (2) effect and provide a reasonable estimate of  11 , both sufficient to explain the experimental observations.
We suggest the low-frequency (0.1-0.9 THz) enhancement in  (2) to be caused by disorder.In fact, the frequency region 0.1-1.2THz of fused silica and other glasses is often associated with the so-called Boson-peak behavior corresponding to low frequency vibrational modes 49,50 .Its nature and origin remain debated, but it is known to affect the Raman, neutron, and linear dielectric responses of quartz and related glasses [49][50][51] .Our finding, thus, motivates further research into the nonlinear susceptibility in the sub-0.9THz region.In addition, there is considerable variability of the reported values for the 3.9 THz phonon damping parameter Γ/2 between 0.09 THz (Ref.14) and 0.39 THz (Ref.51).This variation indicates that the  (2) model The peak at 3.9 THz stems from the phonon contribution to  (2) .
parameters are highly sensitive to the sample quality and may be fine-tuned for better agreement.

Polarization-Resolved EOS
So far, we have treated both ℎ Q and  THz as scalars and only considered the specific case in which the THz-pulse and sampling-pulse polarizations are parallel and the quartz azimuthal angle is optimized for maximum (), i.e., oriented parallel to one of the in-plane crystalline axes.However, the THz electric field is a vectorial observable and can have an arbitrary (and thus even helical) polarization state and ℎ Q is generally dependent on the azimuthal angle  and sampling pulse polarization .Nonetheless, we can assume the same frequencydependence of the allowed  (2) tensor elements and any corresponding linear combination of them, because of the in-plane symmetry of the 3.9 THz phonon.Since the other quantities in Eq. ( 1), such as  F or , refer to linear optical properties, they are also in-plane isotropic in zcut quartz.We can, therefore, assume the same frequency evolution of the response function for all  and , but the absolute sensitivity will be rescaled by the global symmetry of   (2) (, ), which ultimately allows for polarization-sensitive THz EOS.
To explore the sensitivity of 50 µm quartz to different THz field polarization components, Fig. 4a shows the measured peak EOS signal  (blue dots) as a function of quartz azimuthal angle  for three different probe polarizations  = 0°, 45°, 90° with respect to the THz field ( = 0°, linearly polarized along the y-axis).Each azimuthal dependence () exhibits a perfect 3- fold symmetry in agreement with the first reported quartz EOS 42 .We therefore calculate the expected dependence of (, ) for a THz field  THz linearly polarized at an arbitrary angle , and sampling field  s linearly polarized at angle  in the x-y plane (see Fig. 1a).We use the 2 nd order nonlinear polarization   (2) =  0   (2)   THz   s , which we can rewrite using the nonlinear susceptibility tensor in contracted notation   with only non-zero  11 and  14 terms due to quartz's  3 point group, evaluated for the z-cut plane 52 (see Methods).The blue line in Fig. 4a shows the expected sensitivity for a vertically polarized THz field   THz (i.e. =0°), in perfect agreement with the measured azimuthal dependence.The expected peak signal for a horizontally polarized THz field   THz (i.e. = 90°), shown as a red line, features the same 3fold symmetry but shifted by 30°.These opposite EOS sensitivities for the x-and y-projections of the THz field allow for a full THz polarization determination by simply measuring EOS for two different sampling pulse polarizations, e.g. =0° for obtaining   THz and  = 45° for obtaining   THz at azimuth  =0 (see square markers in Fig. 4a).
To prove this concept, we rotate the linear polarization of the THz pulse by setting polarizer P1 to 45° and scanning P2 by angle .Next, we measure () for sampling pulse polarization  = 0° ( 0 ) and 45° ( 45 ) for a set of THz polarizer angles .Fig. 4b shows that arctan( 45 / 0 ) is identical to the THz polarizer angle  and, thus, precisely measures the THz polarization by only two EOS measurements at different sampling pulse polarizations.After applying the calculated response function ℎ Q to  0 and  45 , the full vectorial THz field  THz () can be extracted as shown in the 2D EOS traces for selected  between 0° and 90° in Fig. 4c.We note that the perfect 3-fold symmetry is not found in the common ZnTe (110) or GaP (110) EOS crystals, where this convenient procedure cannot be used 36 .

Broadband THz Helicity Measurement
For driving chiral or, generally, helicity-dependent excitations, e.g., for ultrafast control of phonon angular momentum 20,21,53 or topology modulation 18 , CEP-stable table-top THz sources are beneficial due to their inherent synchronization with sub-cycle probing pulses.
Nevertheless, to reach the required peak fields, the energy has to be squeezed into few-or single cycle pulses at low repetition rates.Therefore, the lack of broadband THz waveplates leads to complicated polarization states when aiming for THz pulses with specific helicities.In contrast to conventional multi-cycle optical light, helical few-or single-cycle THz pulses are highly polychromatic and, generally, cannot be described by a single polarization state, i.e., neither by a pair of ellipticity angles (, ) nor by one fixed Jones or Stokes vector 44 .Instead, the polarization state must be generally described as an evolution in frequency space or, equivalently, by the full temporal trajectory of the light's electric field vector  THz ().
To demonstrate the complete detection of arbitrary polarization states in quartz, in particular for complicated helical fields, we characterize the polarization state of single-cycle THz pulses following collimated traversal of the textbook birefringent y-cut quartz (see Supplementary Fig. S1), which is nearly identical to commercially available THz waveplates.Fig. 5a shows the transmitted electric field of a collimated THz beam ( = 45°) through 0.7 mm crystalline ycut quartz for three different crystal orientations, which is detected in 50 µm z-cut quartz.The transmitted THz polarizations for 0° and 90° orientations appear highly elliptical, which is when the incident THz pulse polarization is at 45° to the in-plane crystal axes and therefore experiences maximum birefringence.This form of time-domain ellipsometry permits the direct measurement of the birefringence Δ(Ω) using arg (  THz )−arg (  THz )=ΔΩ/ as shown in  5c,e).In contrast, the incident THz pulse acquires a small ellipticity for the 45° orientation (see Fig. 5d) only at higher frequencies, which are more sensitive to a small Δ.
Usually, broadband QWPs create opposite helicities for ±45° rotation.This behavior is evidently not the case here, as the two () trajectories in Fig. 5f are not perfectly opposite.
We project the polarization state from a linear into a circular basis to resolve the frequencydependent helicity (see Methods).Figs.5g,h depict the full frequency-dependent righthanded ( RCP ) and left-handed ( LCP ) circularly polarized intensity components for the 0° (red) and 90° (blue) orientations, normalized for every frequency component (see Fig. 5g) and as absolute intensity spectra (see Fig. 5h).Fig. 5g highlights that the helicity changes quite drastically across the single THz pulse spectrum and that a circular polarization is achieved at slightly different frequencies for opposite QWP angles (0° vs. 90°), in agreement with the ellipticity parameters (Ω) in Figs.5c,e.The latter can be related to a slightly tilted axis of rotation with respect to the quartz plate's y-axis, which highlights the challenges of helicitydependent measurements in the THz spectral range.

Discussion
We now discuss the detector performance of α-quartz in more detail.As we find a pure bulk  (2) effect, the phase-matching term  governs the trade-off between detection sensitivity and bandwidth.The effective detector bandwidth is, thus, limited by the first zero in  (Fig. 3f), giving a cut-off frequency  cuttoff = /  THz −  s (g)  = 1/(GVM ⋅).The group-velocity mismatch (GVM) in quartz is about 1.8 ps/mm (assuming  THz (1 THz) = 2.09 and group index  s (g) (800 nm) = 1.55), which is only slightly inferior to ZnTe (GVM = 1.1 ps/mm) 30 .A full comparison of  eff and GVM between quartz and the widely used EOS crystals ZnTe and GaP is shown in Supplementary Table S1.Therefore, to sample the whole THz spectrum of typical high-field THz sources based on LiNbO3 (~0.1-4THz), the quartz thickness should not exceed 130 µm.Sampling of higher THz frequencies poses limitations due to substantial dispersion of the linear THz refractive index and nonlinear susceptibility  (2) due to the 3.9, 8, 12, and 13.5 THz TO phonons of quartz 55 .This fact is especially relevant for more broadband high-field sources such as large-area spintronic emitters 3,42 .
The polarization sensitivity of EOS in quartz generally permits time-domain ellipsometry, allows for the direct measurement of complex and even non-equilibrium 24 tensorial material properties in anisotropic media 22,23 and optical activity of chiral phonons 25,26,56 , as well as THz circulardichroism spectroscopy 26,27 , or decoding high-harmonic THz emission of complex quantum materials [15][16][17] .The ability to detect intense THz fields in amplitude and phase without distortions is well suited for any ultrafast spectroscopy based on strong THz-field excitation 13 , e.g., for understanding nonlinear THz polarization responses 38 or driving phase transitions [8][9][10]18 , where an accurate characterization of the driving field is crucial. Moeover, the demonstrated precise helicity characterization of intense THz driving fields is urgently needed for the emerging field of chiral (or circular) phononics.In this field, lattice modes are driven on chiral or circular trajectories with phonon angular momentum 53 leading to magnetization switching 57 , transient multiferroicity 21 , large magnetic fields 20 or other yet unexplored spin-lattice-coupled phenomena.These first explorations in the uncharted territory of phonon-angular-momentum control highlight the challenges for THz helicity differential detection, i.e., extracting signals proportional to ( RCP )−( LCP ), which must be employed to isolate helicity-dependent effects.Using quartz as a reliable high-field THz helicity detector will help to clarify and support these novel types of measurements and will foster further studies of chiral or helicity-selective phenomena in the THz spectral region.
As the demonstrated 2D-EOS protocol only relies on a single HWP rotation, it enables a rapid measurements and, therefore, keeps the phase error due to temporal drifts between adjacent EOS scans minimal.Accordingly, the scheme is also easy to implement in commercial time-domain spectrometer systems as it only relies on the addition of low-cost and widely available thin quartz wafers and standard HWPs in the VIS or NIR spectral range.As another benefit, quartz is well suited for measuring THz fields and their polarization states in systems, where space constraints often prohibit the use of motorized rotation mounts for the detection crystal, in particular in cryostats at cryogenic temperatures.Supplementary Fig. S2 shows quartz EOS at 80 K, demonstrating that the THz field can still be reliably sampled at low temperatures, although the response function is modified due to the enhanced phonon contribution to  (2) (Ref.14).Conveniently, our work may also allow for all-optical synchronization of THz pump and optical probe pulses via THz slicing 58 or in-situ field and polarization characterization in already installed z-cut quartz windows at free-electron-laser facilities, where even a noncollinear THz-and sampling-beam geometry is feasible (see Supplementary Fig. S4).
In conclusion, z-cut α-quartz can reliably sample intense THz fields of the order of 1 MV/cm without over-rotation and with negligible higher-order nonlinearities.We measured and modeled the frequency-dependent electrooptic response function, consistent with a pure bulk  (2) effect dominated by Fabry-Perot resonances, phonon modulations in the Faust-Henry formalism, phase matching effects, and a low frequency Debye-like contribution.We determined the electrooptic coefficient to the order of 0.1 pm/V and proved a perfect 3-fold symmetry of the electrooptic response.Based on this knowledge, we developed an easily implementable protocol to measure the full vectorial THz polarization state by simply toggling between 0° and 45° sampling pulse polarizations.With this approach, we establish quartz as a powerful detector for full amplitude, phase and polarization state of highly intense THz radiation at a fraction of the cost of conventional detection crystals.This work will accordingly foster rapid and cost-efficient high-field THz spectroscopy 5,6,13,15 , THz time-domain ellipsometry 23 , THz circular-dichroism spectroscopy 26,27 and will enable broadband THz helicity characterization of polarization-tailored pulses for driving angular-momentum phonons 25,26,56 or other helicity-dependent excitations [18][19][20][21] in the future.

Generation and electrooptic sampling of intense THz pulses
Intense THz pulses (1.3 THz center frequency, 1.5 THz FWHM) with peak fields exceeding 1 MV/cm are generated using optical rectification in LiNbO3 using the titled pulse front technique 1 .For this, the LiNbO3 crystal is pumped with laser pulses from an amplified Ti:sapphire laser system (central wavelength 800 nm, pulse duration 35 fs FWHM, pulse energy 5 mJ, repetition rate 1 kHz).The THz field strengths are altered by rotating THz polarizer P1 (Fig. 1a), while keeping P2 fixed at  =0°.In this way, the peak fields of the transmitted THz pulses are proportional to the cosine squared of the polarizer P1 angle.
Similarly, the THz polarization  can be set to an arbitrary angle by keeping P1 fixed at 45°a nd P2 at .The sampling pulses are provided by a synchronized Ti:sapphire oscillator (central wavelength 800 nm, repetition rate 80 MHz) and are collinearly aligned and temporarily delayed with respect to the THz pulse.The sampling pulse polarization is set to specific angles by using a half-waveplate (HWP) before the EOS crystal.
The THz pulse induces a change in birefringence (electrooptic effect or Pockels effect) in the EOS crystal.This birefringence causes the sampling pulse to acquire a phase difference between its polarization components parallel and perpendicular to the THz pulse polarization.
This phase difference is detected in a balanced detection scheme consisting of a quarter-and half-waveplate (/4 , /2) followed by a Wollaston prism (WP) to spatially separate the perpendicular polarization components.The intensity of the two resulting beams is detected by two photodiodes ( 1 and  2 ), which leads to the EOS signal  =( 1 −  2 )/( 1 +  2 ), that is equal to twice the THz-induced phase difference (see Supplementary Note 2).

Further details of the electrooptic response function model
The electrooptic response is modeled using Eq. ( 1) and (2) in the main text.In this equation, the field transmission coefficient  F (Ω) accounts for the transmitted THz field, includes multiple reflections of the field inside the crystal, and can be expressed using: where (Ω)= 21 (Ω) 23 (Ω), and θ=2Ω(Ω)/.Here,  12 (Ω) and  12 (Ω) are the Fresnel transmission and reflection coefficients at THz frequencies at the respective interfaces (1,3air; 2 -quartz), respectively.

EOS response for arbitrary quartz azimuthal angles and sampling pulse polarizations
To compute the full quartz EOS dependence for the crystalline azimuthal angle , sampling pulse polarization , and THz polarization angle , we consider the second-order nonlinear polarization  (2) , which for z-cut α-quartz (D3 point group 52 ) can be written using contracted notation in the matrix form: where  11 = 0.3 pm/V and  14 = 0.008 pm/V (Ref.52).For our experimental configuration, the probe and THz polarizations are in the x-y plane, and the z components of these fields are zero.Equation (4) thus implies that only the  11 component affects quartz EOS in our geometry.The balanced-detection signal is proportional to the difference of the intensities of the orthogonally polarized x-and y-components of the total electric field at the detector, separated by the Wollaston prism and projected on the two photodiodes.Thus, the signal can be calculated as: where  (2) is the electric field emitted by the nonlinear polarization  (2) as described by the inhomogeneous wave equation.The sampling-pulse polarization angle is defined by  = atan2(  s ,   s ) and the THz polarization angle by  = atan2   s ,   s , where atan2 corresponds to the four-quadrant arctan function.
A convenient way to numerically simulate the nonlinear polarization  (2) obtained with an azimuthal rotation of the sample in the x-y plane by an angle  is to apply two-dimensional rotation matrices () to the  s and  THz fields while using an unchanged form of   in Eq. ( 4) and then rotate the calculated nonlinear polarization  (2) ′ by - back into the original lab frame.

Polarization state representations of polychromatic THz fields
The polarization state of a THz field (Ω) can be described using a polarization ellipse representation (see inset Fig. 5a), where the orientation (Ω) is given by: Another useful way to describe (Ω), which is typically measured in a linear basis ( ,  ), is the circular basis ( ,  ̂).In this representation, the right-and left-hand circular polarized field components,  RCP (Ω) and  LCP (Ω) respectively, are given by the projection: LCP (Ω) = 1  where  o and  e are the ordinary and extraordinary refractive indices of quartz respectively.
In this new coordinate system, the index ellipsoid becomes:

Fig. 1 |
Fig. 1 | Electrooptic sampling in quartz and its THz fluence dependence.a Experimental setup: THz pulses are generated via optical rectification (OR) in LiNbO3.The THz pulse induces a refractive index change in quartz, leading the sampling pulse to acquire ellipticity.This ellipticity is read out as signal () as a function of time delay  in a balanced detection scheme. is related to the incident THz field  THz via the complex detector response function ℎ Q .b EOS in quartz (z-cut, 50 µm thickness) for different THz fluences, normalized to the  =0 peak EOS values.Inset: Linear dependence of peak () on peak  THz .c (Ω) amplitude spectrum via Fourier transform of EOS signals () in (b) normalized to spectral peak amplitude.

Fig. 2 |
Fig. 2 | Experimentally measured and calculated detector response.a Normalized EOS signal () in quartz (50 µm thickness) and ZnTe (10 µm thickness).b, d Complex quartz response function ℎ Q for 50 µm is experimentally extracted using known ZnTe response (blue) and modeled (red) in amplitude and phase.c, e Calculated (2)  , transmitted field coefficient  F , and phase matching factor  in amplitude and phase, showing how these factors contribute to the quartz response function.
Fig.3cshows the measured dependence of the maximum EOS signal on the quartz crystal thickness between 35 and 150 µm (blue dots), which clearly deviates from an ideal phase-matched behavior, i.e., a linear scaling with the crystal thickness.We also observe a noticeable thickness dependence of the time-domain EOS shapes in Fig.3d, even clearer in the spectral bandwidth in Fig.3e.Fig.3fdisplays the calculated response function for each thickness in amplitude (red) and phase (grey), which explains the measured features.For instance, the effective bandwidth is significantly lower for 150 µm quartz due to the zero in the phase-matching factor (Ω, ), while the thicknessdependent frequency spacing of the modulations generally arise from Fabry-Perot fringes in the field transmission coefficient  F (Ω).The calculated response function, thus, also explains the experimentally observed EOS thickness dependence in Fig.3c(red line), mainly by the phase mismatch (Ω, ) of THz and sampling pulse.

Fig. 3 |
Fig. 3 | Thickness dependence and extracted THz electric fields.a Absolute THz electric field  THz extracted by applying the response function ℎ Q to the measured EOS signal  with 50 µm quartz and b corresponding Fourier amplitude spectrum.c Maximum EOS signal () as a function of quartz thickness (blue markers) and calculated quartz response (red curve).d EOS signal for four different quartz thicknesses below 150 µm with e respective Fourier amplitude spectra.f Modulus and phase of calculated detector response ℎ Q of quartz for the respective thicknesses.The small oscillatory variations below 4 THz are Fabry-Perot resonances.The zero in (e) for 150 µm is dictated by the phase matching factor .The peak at 3.9 THz stems from the phonon contribution to (2)  .

Fig. 4 |
Fig. 4 | Polarization and azimuthal angle dependence for 2D-EOS.a Measured azimuthal angle  dependence of maximum quartz () for different sampling pulse polarizations () with THz pulse polarized along y (blue dots).Blue and red lines are the calculated azimuthal angle dependence for the respective sampling pulse polarizations and THz polarized along y (blue line) and x (red line).b The arctan of the peak EOS signals measured at  = 45° ( 45 ) and  = 0° ( 0 ) perfectly matches the THz polarizer angle , demonstrating that the full THz polarization state can be extracted by measuring  0 () and  45 ().c 2D-EOS:   THz () and   THz () for selected  between 0° and 90°, which were extracted from  45 () and  0 () by applying the quartz response function ℎ Q .

Fig. 5 |
Fig. 5 | Detection of arbitrary THz polarization states and their helicity.a 2D-EOS of the THz electric field transmitted through a 0.7 mm y-cut quartz plate for three different y-cut quartz orientations, detected in 50 µm z-cut quartz .The y-cut quartz plate was aligned with one of its facets parallel to the y-axis (corresponds to 0°).The incident THz field was linearly polarized at 45°. b Extracted birefringence for the three different y-cut quartz azimuthal angles, demonstrating that the THz field experiences the largest birefringence for 0° and 90° quartz-plate orientations.c, d, e Corresponding frequency-resolved THz polarization states expressed in polarization ellipse rotation (Ω) and ellipticity (Ω) for 0°, 45° and 90° y-cut plate azimuthal angle, respectively.f Projection of   THz () and   THz () into the (  THz ,   THz ) plane for 0° and 90° y-cut plate azimuthal angles, unveiling the different, but not exactly opposite helicity states.g Corresponding LCP and RCP intensity spectra normalized for every frequency Ω to | THz (Ω)| 2 and h corresponding absolute intensities.

√2
[  (Ω) −   (Ω)].cut α-quartz, belonging to the D3 point group and 32 symmetry class1,2  , and oriented along the principle axes, the induced change to the index ellipsoid by a THz electric field  THz may then be written as:where   is the electrooptic tensor.For our experimental configuration, the sampling pulse field's and THz pulse field's polarizations are in the x-y plane and the z components of these fields are zero.We now consider the case, where the THz field is polarized along the y-axis:  THz,2 =  12  THz ;  THz,1 =  THz,3 =0.Here,  THz,2 is already the field inside the sample and related to the incident THz field  THz via the Fresnel transmission coefficient,  12 = (2/(1 +  THz )), if we neglect multiple internal reflections and absorption.Note that the nonlinear susceptibility tensor   (methods section of main text) and   are related via   =−4  /

Fig. S3 |
Fig. S3 | Calculated quartz EOS dependence on azimuthal angle , sampling pulse polarization  and linear THz polarization . a Calculation for all possible  and  angles for fixed THz electric field x-and y-components,  = 90° and  =0°, respectively.It is evident that the EOS signal obeys a 3-fold symmetry in , and a 2-fold symmetry in .b Calculated EOS signal for all possible  and  angles at the fixed sampling pulse polarization  = 0°.

Fig. S4 |
Fig. S4 | Calculated quartz EOS azimuthal  dependence as a function of sampling pulse angle of incidence  for the two geometries relevant for 2D-EOS.a Calculation for all possible  angles for fixed THz electric field y-component and sampling pulse polarization  =0° as a function of sampling pulse angle of incidence  between 0° and 45°.This plot illustrates that the THz ycomponent can be extracted (blue square) even at large angles of incidence with proper adjustment of the quartz response function.b Corresponding calculation for fixed THz electric field x-component and sampling pulse polarization at  = 45°, highlighting that the THz x-component may also still be extracted (red square) with the proper calibration of the EOS sensitivity.