The ability to capture terahertz (THz) waveforms has led to the development of many important research tools such as THz time-domain spectroscopy (TDS), material characterization, biomedical imaging, electron beam diagnostics, and scanning tunneling microscopy [1–4]. Electro-optic sampling (EOS) is a commonly used technique that measures out the field-induced birefringence in an electro-optic crystal using a probe infrared laser pulse, allowing direct measurements of the temporal electric field waveform in the THz frequency range [5–8].

Traditional multishot schemes where the time evolution of the birefringence is mapped by scanning the delay of the ultrashort laser pulse over several shots face two main challenges. First, these are not applicable in low repetition rate cases or in setups with large shot-to-shot time-of-arrival fluctuations between the THz and the probe laser. These importantly include most electron-beam-based field sources, such as THz free electron lasers, coherent transition and synchrotron radiation, and near-fields of the beam itself [9–15]. Even when the THz source and probe laser are perfectly synchronized, multishot TDS systems still fail when analyzing samples that are dynamically changing with poor repeatability [16–22], highlighting the necessity of single-shot detection schemes. Secondly, in its typical implementation, multishot EOS measures the electric field along one axis, not allowing arbitrary polarization states to be measured without reorienting the optical setup and performing a second independent scan.

The first challenge can be addressed using single-shot setups where the waveform is encoded either spatially or spectrally in the IR pulse. Spatial encoding requires non-collinear propagation in the crystal and typically can be useful only over short time windows, limiting the application range. In spectral encoding approaches, the probe pulses are stretched in time by adding a frequency chirp so that the THz waveform information can be collected over very long (100s of ps) time windows in a single shot by simply recording the probe pulse spectrum [23–25]. However, this technique has been severely limited by distortions that appear when trying to capture wide frequency ranges over very long time windows [26]. With the new advancement of diversity electro-optic sampling (DEOS) by Roussel et al. [27], these limitations can be overcome and accurate waveform reconstruction of long THz waveforms with broad spectral ranges is now achievable. The approach essentially relies on the use of a two-channel encoding scheme with a shifted frequency response so that combining the information from the two channels makes it possible to reconstruct the full waveform.

The second challenge related to the THz field polarization reconstruction has been addressed in multishot EOS schemes by mechanically rotating the optical components of the readout [28,29]. For single-shot polarimetry, a general formalism based on probe pulse stretching was developed in the visible range, avoiding the need for mechanical intervention in the setup [30]. This concept was recently adapted to the THz range [31] allowing the capture of a THz waveform with polarization information in a single-shot EOS scheme, but without addressing the limitations associated with spectral-temporal encoding [26,27].

In this paper, we will demonstrate how the DEOS formalism can be extended to arbitrary polarization states, allowing THz polarization measurements while still addressing the waveform distortions that appear in the spectral-temporal encoding scheme. We present an analytical model for the measured signals and a reconstruction method for THz waveforms that are polarized at an arbitrary angle or ellipticity. We also highlight additional effects in DEOS measurements that alter the reconstruction results. Finally, using an optical-rectification-based THz source, we demonstrate the use of a stretched pulse EOS setup to measure the THz wave polarization state, allowing THz time-domain ellipsometry to be performed in a single shot over a wide frequency range.

We start the discussion with a quick review of the EOS technique and its phase-diversity implementation to establish the formalism, which we will later use to retrieve the polarization of the THz electric field. For an electro-optic crystal cut in the (110) plane, a THz field incident at angle $\alpha$ with respect to the $[-110]$ axis results in a birefringent index-ellipsoid with slow and fast axes in the (110) plane [see Fig. 1(a)] having refractive index [27,32] $$n_{\mathrm{1,2}}(\alpha )=n_0+{\eta }_{\mathrm{1,2}}(\alpha )E_{\mathrm{THz}},$$(1)where ${\eta }_{\mathrm{1,2}}(\alpha )=\frac{n_0^3{r}_{41}}{4}(\sin \alpha \pm \sqrt{1+3{\cos }^2\alpha })$, $n_0$ is the refractive index in the absence of an electric field, $r_{41}$ is the EO coefficient of the crystal, and $E_{\mathrm{THz}}$ is the applied THz electric field. The orientation of the index ellipsoid is determined by the angle $\mathrm{\Psi }$ between the slow axis and the $[-110]$ axis, related to $\alpha$ by $\tan 2\mathrm{\Psi }=2\cot \alpha$ [32] and portrayed in Fig. 1. The birefringence $\mathrm{\Delta }n(\alpha )=n_1(\alpha )-n_2(\alpha )$ is maximized for $\alpha =0$ as in the standard configuration for EOS [see Fig. 1(b)].

The induced birefringence can be read out measuring the polarization state of the probe pulse, which can be modeled using Jones-matrix formalism [33]. In the crystal frame, the EO-induced matrix is diagonal and can be written as $$M_{\mathrm{Xtal}}=\left[\begin{array}{cc}{e}^{i{\varphi }_1(\alpha ,E_{\mathrm{THz}})}& 0\\ 0& e^{i{\varphi }_2(\alpha ,E_{\mathrm{THz}})}\end{array}\right],$$(2)where ${\varphi }_{\mathrm{1,2}}(\alpha ,E_{\mathrm{THz}})=k_{0}L(n_0+{\eta }_{\mathrm{1,2}}(\alpha )E_{\mathrm{THz}})$, where $L$ is the length of the crystal and $k_0$ is the wavenumber of the probe pulse. Typically, after the crystal an EOS setup includes a quarter-waveplate (QWP, $\eta =\pi /2$) followed by a half-waveplate (HWP, $\eta =\pi$) with corresponding Jones matrices $$W(\eta ,\theta )=e^{i\eta /2}\left[\begin{array}{cc}{\cos }^2\theta +e^{i\eta }{\sin }^2\theta & (1-e^{i\eta })\sin \theta \cos \theta \\ (1-e^{i\eta })\sin \theta \cos \theta & {\sin }^2\theta +e^{i\eta }{\cos }^2\theta \end{array}\right],$$(3)so that the full change in polarization state of the probe pulse is described by the single matrix $$M=W(\pi ,{\theta }_H)W(\pi /2,{\theta }_Q)R^{-1}(\mathrm{\Psi })M_{\mathrm{Xtal}}R(\mathrm{\Psi }),$$(4)in which $R$ is the rotation matrix and ${\theta }_H$, ${\theta }_Q$ are the angles of the waveplates’ fast axes with respect to the $[-110]$ axis of the EO crystal [33]. The Jones transport operates on the incoming probe electric field vector resulting in an output signal: $$E_s=M{E}_0\left(\begin{array}{c}\cos {\theta }_{\mathrm{IR}}\\ \sin {\theta }_{\mathrm{IR}}\end{array}\right),$$(5)where the incident field has magnitude $E_0$ and is polarized at angle ${\theta }_{\mathrm{IR}}$ with respect to the $[-110]$ axis. For the standard configuration we set $\alpha =0$ [i.e., $\mathrm{\Psi }=\pi /4$; see Fig. 1(b)] and use a linearly polarized IR probe pulse with ${\theta }_{\mathrm{IR}}=0$, a QWP at ${\theta }_Q=\pi /4$, and an HWP at ${\theta }_H=0$, so that the modulated probe pulse becomes $$\left(\begin{array}{c}{E}_{s,1}\\ E_{s,2}\end{array}\right)=\frac{E_0}{\sqrt{2}}\left(\begin{array}{c}\cos \left(\frac{\mathrm{\Gamma }}{2}\right)-\sin \left(\frac{\mathrm{\Gamma }}{2}\right)\\ -i(\cos \left(\frac{\mathrm{\Gamma }}{2}\right)+\sin \left(\frac{\mathrm{\Gamma }}{2}\right))\end{array}\right)\approx \frac{1}{\sqrt{2}}\left(\begin{array}{c}1-\frac{\mathrm{\Gamma }}{2}\\ -i(1+\frac{\mathrm{\Gamma }}{2})\end{array}\right)$$(6)for small phase shifts. $\mathrm{\Gamma }$ is related to the applied THz field by $$\mathrm{\Gamma }={\varphi }_1-{\varphi }_2=\underset{=\text{:}\beta }{\underbrace{k_{0}L{n}_0^3{r}_{41}{\mathit{T}}_{\mathrm{EO}}}}{E}_{\mathrm{THz}},$$(7)where ${\mathit{T}}_{\mathrm{EO}}$ is a correction term that accounts for Fresnel losses, absorption, and dephasing effects in the crystal [34,35]. We define $\beta$ as the calibration factor between the electric field and the induced phase shift. By separately measuring the intensity of the two polarizations $I_{s,i}\propto {|E_{s,i}|}^2$, in the limit of small $\mathrm{\Gamma }$, we can retrieve the phase shift as $$\mathrm{\Gamma }=\frac{I_{s,2}-I_{s,1}}{I_{s,2}+I_{s,1}}$$(8)and hence the applied electric field by inverting Eq. (7). The standard configuration results in the strongest modulation of the probe pulse [32,34], but as discussed below, in a single-shot spectrally encoded scheme where the probe pulse is stretched, temporal distortions appear and other configurations might be favored [27].

To better understand the bandwidth limitations in spectrally encoded setups as well as the phase diversity solution, we review the principles of photonic time stretching in more detail [36]. A chirped probe pulse can be modeled as $$E_0(t)=A(t)\exp (-i\frac{C}{2}{t}^2-i{\omega }_{0}t),$$(9)with $A(t)$ being the pulse envelope, $C=\partial \omega /\partial t$ the chirp parameter, and ${\omega }_0$ the center frequency. For a linear chirp, the spectral-temporal correlation is expressed as $t=-(\omega -{\omega }_0)/C$. The corresponding spectral amplitude is obtained by Fourier transform for a slowly varying $A(t)$ as [27] $${\tilde{E}}_0(\omega )\approx \sqrt{-\frac{i2\pi }{C}}A(-\frac{\omega -{\omega }_0}{C})\exp \left(i\frac{{(\omega -{\omega }_0)}^2}{2C}\right).$$(10)

The modulation of the probe pulse by the EO effect in the crystal can be expressed as $E_s(t)=T(t)E_0(t)$, with $T(t)$ being the THz-induced modulation function, which is written in a general form for small phase retardation $\mathrm{\Gamma }(t)$ as $$T(t)=\mathcal{T}(1+m{e}^{i\psi }\mathrm{\Gamma }(t)).$$(11)

For example, for the two polarizations in the standard configuration case the transfer function coefficients can be extracted from Eq. (6) as ${\mathcal{T}}_1=1/\sqrt{2},{\mathcal{T}}_2=-i/\sqrt{2},m_{\mathrm{1,2}}=\mp 1/2,{\psi }_{\mathrm{1,2}}=0$. The signal spectrum is given by the convolution integral ${\tilde{E}}_s(\omega )=\tilde{T}(\omega )*{\tilde{E}}_0(\omega )$ and after background subtraction and normalization the recorded signal can be written as $$Y(\omega )=\frac{|{\tilde{E}}_s(\omega {)|}^2-|\mathcal{T}{\tilde{E}}_0(\omega {)|}^2}{|\mathcal{T}{\tilde{E}}_0(\omega {)|}^2}=\frac{2\Re \mathfrak{e}[{\tilde{E}}_0^{*}(\omega )m{e}^{i\psi }\tilde{\mathrm{\Gamma }}\ast {\tilde{E}}_0(\omega )]}{|{\tilde{E}}_0(\omega {)|}^2}.$$(12)

Using the expression from Eq. (10) and assuming that the slowly varying spectral amplitude $A$ can be carried out of the integral and simplified away by the normalization (see Appendix A), we have $$Y(\omega )=2\Re \mathfrak{e}{\int }_{-\infty }^{\infty }\tilde{\mathrm{\Gamma }}(\mathrm{\Omega })m{e}^{i{\mathrm{\Omega }}^2/C+\psi }{e}^{-i\mathrm{\Omega }(\omega -{\omega }_0)/C}\mathrm{d}\mathrm{\Omega }\approx \mathcal{F}(H(\mathrm{\Omega })\tilde{\mathrm{\Gamma }}(\mathrm{\Omega })),$$(13)where $\tilde{\mathrm{\Gamma }}(\mathrm{\Omega })$ is the Fourier transform of the applied THz field and the Fourier transform $\mathcal{F}$ is intended to be taken with respect to the stretched time $t$, correlated to $\omega$ by the frequency chirp. In this picture, $H(\mathrm{\Omega })$ represents the frequency response transfer function of the measurement technique. In fact, if $H(\mathrm{\Omega })=1$, then the measured spectrum signal would be an exact copy of $\mathrm{\Gamma }(t)$ and we would have perfect spectral-temporal encoding. In practice, the chirp $C$ affects our ability to retrieve the signal $\mathrm{\Gamma }$ as, after using the Hermitian property of $\mathrm{\Gamma }(\mathrm{\Omega })$ to simplify the real part operator, we have $$H(\mathrm{\Omega })=2m\cos (\frac{{\mathrm{\Omega }}^2}{2C}+\psi ),$$(14)and all spectral information is lost for frequencies in which $H(\mathrm{\Omega })=0$. A detailed derivation of the transfer function is presented in Appendix A.1. In the standard EOS configuration $H_{\mathrm{1,2}}(\mathrm{\Omega })=\mp \cos (\frac{{\mathrm{\Omega }}^2}{2C})$ and the signals retrieved from each probe pulse polarization have the same zeros so that no information can be retrieved for those frequencies. As we increase the measurable time window by stretching the probe pulse, we decrease $C$, causing the zeros of the transfer function to move to lower frequencies as seen in Fig. 2 where we plot the absolute value of the transfer function $H(\mathrm{\Omega })$ for different chirp parameters $C$. As an example, for $C=1$, we would have no way to retrieve electromagnetic field information at 282 GHz, 488 GHz, 631 GHz, and so on.

Phase diversity EOS makes use of the parameter $\psi$ in Eq. (14) to enable full spectral reconstruction. By adjusting the polarization readout away from the standard configuration, we can tune ${\psi }_1$ and ${\psi }_2$ for the two separate polarizations $E_{s,1}$, $E_{s,1}$ so that the zeros of the transfer functions $H_1(\mathrm{\Omega })$ and $H_2(\mathrm{\Omega })$ are offset from each other. For example, by orienting the THz field to be perpendicular to the [001] axis (i.e., $\alpha =\pi /2$, $\mathrm{\Psi }=0$) and then setting our incoming probe pulse polarization to ${\theta }_{\mathrm{IR}}=\pi /4$ and adjusting ${\theta }_Q=0$ and ${\theta }_H=\pi /8$ as in Fig. 6(c) (shown later), it can be shown that the two transfer functions take on the form of [27] $$H_{\mathrm{1,2}}^{\mathrm{PD}}=\mp \frac{1}{\sqrt{2}}\cos (\frac{{\mathrm{\Omega }}^2}{2C}\mp \frac{\pi }{4}).$$(15)(Our definition of $\mathrm{\Gamma }$ leads to a factor of two difference in the DEOS transfer function compared to that of Roussel et al.). This polarization readout is referred to as the DEOS configuration and allows the full waveform to be recovered, as $Y_1$ and $Y_2$ now contain complementary information. The spectral profile of the THz signal is then recovered via $$\tilde{\mathrm{\Gamma }}(\mathrm{\Omega })=\frac{H_1(\mathrm{\Omega }){\tilde{Y}}_1(\mathrm{\Omega })+H_2(\mathrm{\Omega }){\tilde{Y}}_2(\mathrm{\Omega })}{H_1^2(\mathrm{\Omega })+H_2^2(\mathrm{\Omega })}.$$(16)

The temporal profile is then obtained with an inverse Fourier transform and has been shown to result in accurate waveform reconstruction in tabletop experiments where multishot and single-shot EOS measurements can be easily compared [27].

It is important to note that the spectral intensity of the unmodulated probe pulse $|{\tilde{E}}_0(\omega {)|}^2$ in the DEOS case cannot be recorded as easily as in the standard configuration where $|{\tilde{E}}_0(\omega {)|}^2=|{\tilde{E}}_{s,1}^{\mathrm{std}}(\omega {)|}^2+|{\tilde{E}}_{s,2}^{\mathrm{std}}(\omega {)|}^2$. For DEOS, a reference line is necessary to capture this normalization spectrum and retrieve $Y_1,Y_2$ separately. Another drawback of the proposed DEOS polarization readout is that the induced response is weaker, and thus signal-to-noise levels are lower.

Let us now investigate the cases of arbitrary THz polarization, studying how adding an independent orthogonal polarization affects the measured EOS signal. We assume that the fields in two perpendicular directions independently contribute to the induced birefringence. This can be modeled by introducing a secondary phase shift ${\mathrm{\Gamma }}_y=\beta E_{\mathrm{THz}}^y$, where $\beta$ is the calibration factor from Eq. (7). Referring to the Jones calculus described in Eq. (4), the second field polarization can be taken into account by applying the transport matrix $M$ after an additional transport element $R(-\mathrm{\Psi }({\alpha }_y))M_{\mathrm{Xtal}}({\alpha }_y)R(\mathrm{\Psi }({\alpha }_y))$, where ${\alpha }_y=\alpha +\pi /2$. In the standard EOS configuration, when assuming small phase shifts ${\mathrm{\Gamma }}_x,{\mathrm{\Gamma }}_y\ll 1$, the modulated probe pulse is then described by (see Appendix A.2 for full expression) $$E_s^{\mathrm{std}}\approx \frac{E_0}{\sqrt{2}}\left(\begin{array}{c}1-\frac{{\mathrm{\Gamma }}_x}{2}+i\frac{{\mathrm{\Gamma }}_y}{2}\\ i(1+\frac{{\mathrm{\Gamma }}_x}{2}+i\frac{{\mathrm{\Gamma }}_y}{2})\end{array}\right),$$(17)and using the same reasoning that led to Eqs. (13) and (14), the Fourier transforms of the observed signals result in $${\tilde{Y}}_{\mathrm{1,2}}^{\mathrm{std}}=\mp \underset{=\text{:}{H}_x}{\underbrace{\cos \left(\frac{{\mathrm{\Omega }}^2}{2C}\right)}}{\tilde{\mathrm{\Gamma }}}_x-\underset{=\text{:}{H}_y}{\underbrace{\sin \left(\frac{{\mathrm{\Omega }}^2}{2C}\right)}}{\tilde{\mathrm{\Gamma }}}_y.$$(18)

It is then immediate to note that in the standard EOS analysis using Eq. (8) the contribution of the $y$-polarization is canceled and Eq. (7) only allows us to retrieve ${\mathrm{\Gamma }}_x$. Second, a spectral signature of the $y$-polarization is only present in the signal in the case of a chirped pulse, as in the limit of a short (or compressed) pulse the transfer functions $H_x(\mathrm{\Omega })\stackrel{C\to \infty }{\to }1$ and $H_y(\mathrm{\Omega })\stackrel{C\to \infty }{\to }0$.

In the case of linear polarization incident at an angle of 45° with respect to the $x$-axis ($\alpha =\frac{\pi }{4}$), ${\mathrm{\Gamma }}_x={\mathrm{\Gamma }}_y=\frac{1}{\sqrt{2}}\mathrm{\Gamma }$ and we note that we can rewrite ${\tilde{Y}}_{\mathrm{1,2}}^{\mathrm{std}}$ as $\cos (\frac{{\mathrm{\Omega }}^2}{2C}\pm \frac{\pi }{4})\tilde{\mathrm{\Gamma }}$ as we would expect when computing the transfer functions in the standard configuration for $\alpha =\pi /4$ instead of $\alpha =0$. This proves that our extended model is in agreement with the DEOS formalism, and this particular case shows that diversity arises in the standard configuration when rotating the THz polarization from $\alpha =0$ to $\alpha =\pi /4$.

In a DEOS setup, in the presence of arbitrary THz polarization, we calculate the modulated probe pulse to be (see Appendix A.2 for full expression) $$E_s^{\mathrm{PD}}\approx \frac{E_0}{\sqrt{2}}\left(\begin{array}{c}1-(1-i)\frac{{\mathrm{\Gamma }}_x}{4}+i\frac{{\mathrm{\Gamma }}_y}{2}\\ i(1+(1+i)\frac{{\mathrm{\Gamma }}_x}{4}+i\frac{{\mathrm{\Gamma }}_y}{2})\end{array}\right),$$(19)which results in the relationship $${\tilde{Y}}_{\mathrm{1,2}}^{\mathrm{PD}}=\mp \frac{1}{\sqrt{2}}\cos (\frac{{\mathrm{\Omega }}^2}{2C}\mp \frac{\pi }{4}){\tilde{\mathrm{\Gamma }}}_x-\sin \left(\frac{{\mathrm{\Omega }}^2}{2C}\right){\tilde{\mathrm{\Gamma }}}_y.$$(20)

It is worth noting here that, in the presence of ${\mathrm{\Gamma }}_y\propto E_y^{\mathrm{THz}}$, the reconstruction from Eq. (16) fails, as it is contaminated by the presence of ${\mathrm{\Gamma }}_y$ that requires separate extraction. In fact, in the presence of a vertical polarization field $E_y^{\mathrm{THz}}$, DEOS-based reconstruction of ${\mathrm{\Gamma }}_x$ is not possible without prior knowledge of ${\mathrm{\Gamma }}_y$.

This can be seen in general by writing the relationship between the applied THz field and the measured signals as $$\left(\begin{array}{c}{\tilde{Y}}_1\\ {\tilde{Y}}_2\end{array}\right)=\left(\begin{array}{cc}{H}_{1x}& H_{1y}\\ H_{2x}& H_{2y}\end{array}\right)\left(\begin{array}{c}{\tilde{\mathrm{\Gamma }}}_x\\ {\tilde{\mathrm{\Gamma }}}_y\end{array}\right).$$(21)

In order to extract ${\tilde{\mathrm{\Gamma }}}_x$ and ${\tilde{\mathrm{\Gamma }}}_y$ from the observed signals ${\tilde{Y}}_1$, ${\tilde{Y}}_2$, the matrix needs to be invertible, so this can be done only for those frequencies within the observed signal range for which the matrix determinant is non-zero. Note that in both the standard and DEOS configurations, this is not the case and the retrieval of the two fields is not possible. In fact, full reconstruction of both ${\mathrm{\Gamma }}_x$ and ${\mathrm{\Gamma }}_y$ simultaneously and independently is an open challenge, as there is no known polarization readout in which the transfer function matrix of Eq. (21) has a non-zero determinant over the entire spectral range.

In case the polarization state of the incoming THz wave is known, full waveform reconstruction is possible. The incident field in this case can be described without loss of generality by ${\tilde{\mathrm{\Gamma }}}_x=r\tilde{\mathrm{\Gamma }}$ and ${\tilde{\mathrm{\Gamma }}}_y=\sqrt{1-r^2}{e}^{i\varphi }\tilde{\mathrm{\Gamma }}$, with $r\in [\mathrm{0,1}]$ and $\varphi \in [-\pi ,\pi )$. For example, if $r=1$, we retrieve linear polarization along $x$ and for $r=\frac{1}{\sqrt{2}}$, $\varphi =\pm \frac{\pi }{2}$, we have a left- or right-handed circular polarization.

In the standard EOS configuration, the measured observables take on the form $${\tilde{Y}}_1^{\mathrm{std}}=\underset{=\text{:}{H}_{1,m}^{\mathrm{std}}}{\underbrace{(-r\cos \left(\frac{{\mathrm{\Omega }}^2}{2C}\right)-\sqrt{1-r^2}{e}^{i\varphi }\sin \left(\frac{{\mathrm{\Omega }}^2}{2C}\right))}}\tilde{\mathrm{\Gamma }},$$(22)$${\tilde{Y}}_2^{\mathrm{std}}=\underset{=\text{:}{H}_{2,m}^{\mathrm{std}}}{\underbrace{(+r\cos (\frac{{\mathrm{\Omega }}^2}{2C})-\sqrt{1-r^2}{e}^{i\varphi }\sin (\frac{{\mathrm{\Omega }}^2}{2C}\left)\right)}}\tilde{\mathrm{\Gamma }}.$$(23)

Since these newly defined transfer functions $H_{1,m},H_{2,m}$ have offset zeros, we can reconstruct the unknown spectral amplitude $\mathrm{\Gamma }$ using an equivalent of Eq. (16) by substituting $H_1,H_2$ with $H_{1,m},H_{2,m}$. Note that if some ellipticity is present (i.e., $r\ne \mathrm{0,}1$ and $\varphi \ne 0,\pi$), we obtain complex-valued transfer functions with $|H_{1,m}|,|H_{2,m}|\ne 0$ over the entire spectral range, and the waveform can be retrieved independently from each channel simply by dividing out the transfer functions. The reconstruction would work in a similar way in the DEOS configuration after substituting the correct transfer functions from Eq. (20) in $H_{1,m}^{\mathrm{PD}}$ and $H_{2,m}^{\mathrm{PD}}$. Interestingly, this suggests the possibility of reconstructing the full THz waveform in a single shot using spectral-temporal encoding simply by using a polarizer and a properly oriented well-characterized THz waveplate on the THz path.

A numerical example is presented in Fig. 3 in which we demonstrate reconstruction with different configuration readouts. We start from the spectrum of a single-cycle THz pulse ${\tilde{\mathrm{\Gamma }}}_{\text{act.}}$ with a bandwidth extending to 1.5 THz. The probe pulse is an 800 nm, 40 fs pulse, linearly stretched to 18 ps (FWHM), resulting in a chirp parameter $C=3.413{\mathrm{ps}}^{-2}$. We simulate the modulation of the probe pulse using Jones matrices and obtain the measured signals ${\tilde{Y}}_1,{\tilde{Y}}_2$ from Eq. (13). The original ${\tilde{\mathrm{\Gamma }}}_{\text{act.}}$ is then reconstructed using the appropriate choice of transfer functions (i.e., standard or phase diversity) depending on the EOS configuration readout. In Fig. 3 top left, we show the simulation of a standard configuration polarization readout with the THz polarization adjusted by 45° with respect to the $[-110]$ axis ($\alpha =\pi /4$) so that both ${\tilde{\mathrm{\Gamma }}}_{x,y}$ are present. In the bottom left, we show the simulation of the DEOS polarization readout with the THz polarization perpendicular to the $[-110]$ ($\alpha =\pi /2$) as prescribed for the DEOS setup. The Fourier transforms of the measured signals ${\tilde{Y}}_1,{\tilde{Y}}_2$ are plotted together with the corresponding transfer function $H_1,H_2$ to highlight the lack of response in the measurement at certain frequencies. Note that the curves in the standard 45° and DEOS 0° cases have the same zeros (highlighted by dashed vertical lines), but differ in amplitude, allowing higher signal-to-noise ratios to be achieved in the standard 45° configuration. In both cases, complementary information is present in the two channels and it is possible to fully reconstruct the original signal. On the right, we present a simulated dataset in the DEOS polarization readout with the THz polarization rotated 45° away from the expected THz polarization, causing both ${\mathrm{\Gamma }}_x$ and ${\mathrm{\Gamma }}_y$ to be present. The zeros of the two transfer functions are closer together, which is undesirable for reconstruction, as information is lost over wider regions of the spectrum. We also show the accurately reconstructed signal (enabled by prior knowledge of the THz polarization), but highlight the reconstruction error that could occur by neglecting the additional polarization component ${\mathrm{\Gamma }}_y$ and applying the classical DEOS transfer functions from Eq. (15) for reconstruction.

Let us now study in detail the case in which the polarization state is unknown and needs to be retrieved as in the case for THz-ellipsometry application. For simplicity, we assume a birefringent sample of thickness $d$ with a complex index ellipsoid defined by $n_x^{*}$, $n_y^{*}$ in which a THz pulse propagates at 45° between the two main axes, so that the incident THz pulse is described by ${\tilde{\mathrm{\Gamma }}}_{x,y}^{\mathrm{in}}(\mathrm{\Omega })=\frac{\tilde{\mathrm{\Gamma }}(\mathrm{\Omega })}{\sqrt{2}}$. The transmitted THz pulse is attenuated due to the absorption in the material as well as Fresnel losses at the dielectric interfaces and acquires a phase shift due to the refractive indices of the sample, resulting in an output EOS waveform described by $$\left(\begin{array}{c}{\tilde{\mathrm{\Gamma }}}_x(\mathrm{\Omega })\\ {\tilde{\mathrm{\Gamma }}}_y(\mathrm{\Omega })\end{array}\right)=\frac{\tilde{\mathrm{\Gamma }}}{\sqrt{2}}\left(\begin{array}{c}{\tau }_x\exp (-i{\mathrm{\Phi }}_x(\mathrm{\Omega }))\\ {\tau }_y\exp (-i{\mathrm{\Phi }}_y(\mathrm{\Omega }))\end{array}\right).$$(24)

Note that in general, the radiation pulse could experience multiple reflections in the sample, resulting in a somewhat complicated dependency of the transmission coefficient from the real $n_{x,y}$ and imaginary ${\kappa }_{x,y}$ parts of the index of refraction. In these cases, the reconstruction of the dielectric response becomes a more involved task [37]. Often in single-shot EOS setups, the reflections are not even recorded as the induced time delay can be larger than the temporal window of the measurement. Here we just assume that absorption is small and neglect the Fabry-Perot effect so that we can write ${\tau }_{x,y}=\frac{4n_{x,y}}{{(n_{x,y}+1)}^2}{e}^{-\mathrm{\Omega }{\kappa }_{x,y}d/c}$ as the transmission coefficient and $${\mathrm{\Phi }}_{x,y}(\mathrm{\Omega })=\frac{\mathrm{\Omega }}{c}(n_{x,y}-1)d.$$(25)

By comparing the measured polarization components before and after inserting the sample we can retrieve the induced attenuation and phase shifts on the two polarizations and hence the sample birefringence. In practice, we need to be careful of the transfer functions in this process.

In the case of standard configuration single-shot EOS, we rearrange the measured waveforms $H_{x,y}{\tilde{\mathrm{\Gamma }}}_{x,y}=({\tilde{Y}}_2\mp {\tilde{Y}}_1)/2$ to separate the orthogonal polarization contributions with $H_{x,y}$ being the transfer functions defined in Eq. (18). Then multiplying by the respective filter functions $H_{x,y}$, we can leverage $H_{x,y}^2\ge 0$ to reliably extract the phase $${\mathrm{\Phi }}_{x,y}(\mathrm{\Omega })=-\arg \left(\frac{H_{x,y}({\tilde{Y}}_1\mp {\tilde{Y}}_2)}{\sqrt{2}\tilde{\mathrm{\Gamma }}}\right)=-\arg \left(\frac{\sqrt{2}{H}_{x,y}^2{\tilde{\mathrm{\Gamma }}}_{x,y}}{\tilde{\mathrm{\Gamma }}}\right)$$(26)after normalizing by the spectrum measured without the sample. This works as long as $H_{x,y}(\mathrm{\Omega })\ne 0$, which defines the range of frequencies over which we can perform sample ellipsometry. It is important to note that $\mathrm{\Phi }\in (-\pi ,\pi )$ due to the definition of the arg-function on the complex plane. Therefore, in order to retrieve $n_{x,y}$ by inverting Eq. (25) we need to first unwrap the phase so that it is a continuous function across the entire spectrum. Numerically, unwrapping the phase for a given dataset is a delicate task, as experimental noise can cause unwanted extrapolations.

Once the real part of the index of refraction is recovered, we can use $${\kappa }_{x,y}(\mathrm{\Omega })=\frac{c}{\mathrm{\Omega }d}\mathrm{Re}(\ln \frac{H_{x,y}({\tilde{Y}}_1\mp {\tilde{Y}}_2)}{\sqrt{2}\tilde{\mathrm{\Gamma }}}-\ln H_{x,y}^2-\ln \frac{4n_{x,y}}{{(n_{x,y}+1)}^2})$$(27)to retrieve the absorption coefficient.

For common materials that have a steady and continuous refractive index in the spectral region of interest, it is possible to extrapolate the refractive index in the regions where the spectral response of the measurement is minimal. Instead of extracting the refractive indices analytically, in fact, the problem can be approached numerically, modeling $n_{x,y}$ as a second-order polynomial in $\mathrm{\Omega }$ and ${\kappa }_{x,y}$ as a constant in the frequency domain. The unknown model parameter (the polynomial coefficients of the real index of refraction and its imaginary part in $x$ and $y$, respectively) can be found by minimizing the difference between the computed and measured spectral profiles.

A simulated dataset is presented in Fig. 4 for a single-cycle THz waveform $\mathrm{\Gamma }=\frac{t}{{\sigma }_t}\exp (\frac{t^2}{{\sigma }_t^2})$, with ${\sigma }_t=0.5\mathrm{ps}$. The EOS signals are collected after numerically propagating the waveform through a birefringent crystal that has real and complex indices of refraction different in $x$ and $y$ and with a parabolic frequency dependence, as shown by the solid lines in Fig. 5. For this particular example, we used $n_x=5-0.3f-1.3f^2+0.7f^3$, $n_x=6-0.7f+1.3f^2-0.7f^3$, ${\kappa }_x=0.1+0.01f+0.03f^2-0.02f^3$, ${\kappa }_y=0.05+0.01f+0.003f^2$. The simulated EOS waveforms are obtained with Jones-matrix calculations. The resulting ${\tilde{\mathrm{\Gamma }}}_{x,y}$ are modulated by the standard configuration transfer functions $H_{x,y}$ from Eq. (18) as shown in Fig. 4. We follow the procedures outlined above to extract $n_{x,y}$ and ${\kappa }_{x,y}$ both by analytical inversion and numerical optimization overlaying the results in Fig. 5. The real part of the index ellipsoid is retrieved with great accuracy, only showing minor deviations from the actual values close to the zeros of the transfer function up to the upper limit of the spectral bandwidth of the input THz pulse. The imaginary part responsible for absorption within the sample is less accurate and shows greater deviations, due to the small values extracted by a logarithmic function [38]. The results are portrayed in Fig. 5 and show some significant advantages compared to the analytical extraction. The modeled refractive index is forced to be continuous and extrapolates well in the regions where the transfer functions are zero, especially when retrieving the imaginary part responsible for absorption.

To validate reconstruction and polarization detection techniques, we demonstrate single-shot EOS measurements using an optical-rectification-based THz source presented in Fig. 6 in which a Mg-doped ${\text{LiNbO}}_3$ crystal is pumped with 40 fs 0.6 mJ 800 nm pulses with a spectral bandwidth $\mathrm{\Delta }\lambda =25\mathrm{nm}$. The strong mismatch of refractive indices of IR and THz in $\mathrm{Mg}\text{:}{\text{LiNbO}}_3$ requires the use of tilted pulse fronts and a grating-lens optical system to achieve phase matching and efficient conversion from IR to THz [39–41].

In our setup, the compressor stage of the pump laser system is adjusted so that stretched pulses of 10–20 ps can be generated. To recover maximum efficiency in the THz optical rectification process, we utilize a two-lens optical system ($f_1=500\mathrm{mm}$ and $f_2=75\mathrm{mm}$) from the grating to the LN crystal and adjust the second lens position away from the perfect imaging condition. HWPs before and after the grating are used to maximize the THz generation efficiency. One of the main benefits of this arrangement is that it is possible to perform stretched pulse single-shot EOS diagnostics at different pulse lengths (or chirp parameters $C$) with minimal variation ($<10\%$) in conversion efficiency and without losing time synchronization between pump and probe.

The generated THz is collected by a Teflon lens and refocused with a TPX lens down to the EO crystal. In between the two THz lenses we can insert a wiregrid polarizer or/and a birefringent sample. For EOS diagnostics, a small portion of the pump (8%) is split off by a pellicle beamsplitter, propagated through a delay stage to a second pellicle beamsplitter that overlaps THz and IR with minimal loss in THz transmission. The IR transmitted through the pellicle is used as a reference line, bypassing the THz-induced modulation in the EO crystal. The overlapped probe and THz are propagated through a 0.5 mm thick (110)-cut ZnTe crystal and then through a QWP and an HWP before being split into vertical and horizontal polarization with a Wollaston prism. A third pellicle beamsplitter is then used to re-overlap the reference line with the two split polarizations, which are then analyzed by a spectrometer. The spectrometer is home-built and consists of a $1200\text{\hspace{0.17em}lines}/\mathrm{mm}$ grating, a 100 mm lens, and a CMOS sensor with $1440\times 1080$ resolution and $3.45\mathrm{\mu m}\times 3.45\mathrm{\mu m}$ pixel size. It is set up in a $1f-1f$ configuration, which means that the distance between the grating and the lens, as well as the lens and the CMOS, is 100 mm. This results in collimated rays in $x$ and focused in $y$, forming three spectral lineouts on the CMOS sensor. Two of them are the modulated signals $|{\tilde{E}}_{1,s}(\omega {)|}^2$ and $|{\tilde{E}}_{2,s}(\omega {)|}^2$, while the third is from the reference line.

To measure $Y_1,Y_2$, we must obtain $|{\mathcal{T}}_1{\tilde{E}}_0(\omega {)|}^2$, $|{\mathcal{T}}_2{\tilde{E}}_0(\omega {)|}^2$ for each shot. This is performed by measuring all three spectral lines without a THz signal and determining the intensity ratio between the two split polarizations and the reference line. $|{\mathcal{T}}_{\mathrm{1,2}}{\tilde{E}}_0(\omega {)|}^2$ are then obtained for each individual shot from the calibrated reference line, allowing us to determine $Y_1,Y_2$ using Eq. (13).

The time calibration as well as the spectral-temporal calibration factor $C$ is obtained by adjusting the delay stage by a known amount and measuring the signal position on the spectrometer. A linear fit of the signal position with respect to time then yields the time calibration as $\frac{\partial t}{\partial px}=(2.25\pm 0.04)\frac{\mathrm{fs}}{\text{pixel}}$. Higher order terms are also considered in the fit but contribute around 1% over the range of the spectrometer and are therefore neglected. With the known spectral bandwidth of our laser pulse, we obtain the chirp parameter as $C=\frac{\partial \omega }{\partial px}{(\frac{\partial t}{\partial px})}^{-1}=3.5{\mathrm{ps}}^{-2}$. These parameters are important as they inform the transfer functions to be used in the reconstruction of the temporal profiles.

We first investigate the case in which the THz polarization is reoriented by a wiregrid polarizer. The source emits THz polarized in $\widehat{y}$. The detected polarization state transmitted through the polarizer at angle $\theta$ in respect to the $y$-axis is described as $$E_{\mathrm{THz}}^{\theta }(t)=\left(\begin{array}{c}\sin \theta \cos \theta \\ {\cos }^2\theta \end{array}\right)E_{\mathrm{THz}}(t).$$(28)

The wiregrid is then rotated from 0° to 90° in increments of 15° and measurements are performed at each polarizer angle with the captured single-shot temporal and spectral profiles for ${\mathrm{\Gamma }}_x$ and ${\mathrm{\Gamma }}_y$ presented in Fig. 3. Although these temporal profiles are representative of the THz fields $E_x$ and $E_y$, it is important to note that the signal is modulated spectrally by the transfer functions, as shown in the spectrum in Fig. 3.

Given that the THz polarization that illuminates the crystal is known, we can reconstruct the temporal profile accurately, as discussed earlier. In the case of the wiregrid polarizer, the relation between ${\mathrm{\Gamma }}_x$ and ${\mathrm{\Gamma }}_y$ is given as $r=\tan \theta$. Using the transfer functions in Eqs. (22) and (23) we can then reconstruct the temporal profile with Eq. (16). The reconstructed temporal profile and spectrum are presented in Fig. 3 up to 45° with a waveform captured and reconstructed with DEOS for reference.

To demonstrate THz ellipsometry, we place a $y$-cut ${\text{LiNbO}}_3$ crystal between the wiregrid polarizer and the TPX lens. This material has shown strong birefringence in the THz range and is well characterized, making it a well-suited sample to demonstrate single-shot-based ellipsometry [42]. First, we perform two separate measurements in which the sample’s ordinary and extraordinary axes are parallel to the THz polarization and the transmitted THz waveform is captured with DEOS-based reconstruction methods. This provides us with a benchmark measurement of the sample index ellipsoid without distortions from the transfer functions and establishes reference values to be compared to and noted as Ref. in Fig. 8.

The main axes of the sample are then oriented 45° to the incident THz polarization, and the refractive indices are retrieved through analytical extraction and numerical fit. Both are in good agreement with the reference and literature values [43]; however, the analytical extraction deviates slightly at the zeros of the transfer functions. The numerical model is constrained to a third-order polynomial, and therefore does not present the same deviations and is in better agreement with the reference. At frequencies where the spectral amplitude is small, both the analytical extraction and the numerical fit stray more from the reference values. This demonstrates that in a single-shot measurement, information on the entire index ellipsoid can be retrieved over a broad spectral bandwidth with minor limitations due to the transfer functions. This allows THz ellipsometry measurements to be performed on samples that vary in time or where the THz source has significant shot-to-shot fluctuations or low repetition rates.

In summary, in this paper we discuss effects that arise for arbitrary THz polarization in single-shot, spectral-temporal encoding EOS schemes. The transfer-function-based formalism established in DEOS is modified to consider an additional polarization component, allowing complex polarized THz waveforms to be captured and characterized. We highlight the importance of knowing the THz polarization state when performing DEOS with arbitrary polarization and present the necessary modifications to the reconstruction method to avoid inaccurate results. In this framework, we have presented methods to extract THz polarization information over a large spectral range, allowing THz ellipsometry to be performed on an unknown sample. These methods are first demonstrated numerically and then experimentally supported with a table-top optical-rectification-based THz source.

Using the THz polarization sensitive single-shot methods introduced in this work, we envision the possibility to analyze and study samples that exhibit dynamically changing birefringence in the THz range or exhibit poor repeatability in a single shot and without the need to mechanically intervene. Furthermore, THz sources with significant shot-to-shot jitter can be characterized not only temporally and spectrally but also in their time-dependent polarization state, which could be of particular importance for the characterization of electron-beam-based THz sources such as THz-FELs using helical undulator geometries.