1. Introduction

Light-matter interactions within the THz frequency range (0.1-10 THz) attract a growing number of studies aiming for a better fundamental understanding of physical processes and making a basis for future technologies [1]. A substantial part of this research is concentrated on field-resolved pump-probe experiments measuring both amplitude and phase characteristics of the studied processes. Such experiments typically require CEP stable light sources. Existing CEP-stable high-field THz sources are based on superradiant emission by linear accelerators [2] and laser pulse down-conversion [3,4]. They are actively being used in various recent field-resolved studies on THz control of matter [5,6], nonlinear THz photonics in Dirac materials [7–11], Higgs spectroscopy of superconductors [12,13], electron acceleration [14–16], and many other phenomena [17]. In those studies, the high-field THz pulse induces a nonlinear response in a sample, which is typically probed by a synchronized femtosecond laser pulse. Using a delay line, the laser pulses are overlapped in time with different parts of the CEP-stable THz waveform. At each time delay, the induced response is determined by an instantaneous THz field value and collected within a certain number of shots, giving information on the coherent dynamics of the studied ultrafast process on a sub-cycle time scale. The CEP stability is crucial for such studies because the response at each corresponding probe delay depends on the CEP of the THz pump pulse. If the CEP is not pulse-to-pulse stable, the readout field-resolved response vanishes when averaged over a sequence of probe laser pulses, even when the probe laser pulse and THz pulse envelopes are perfectly synchronized.

Infrared FEL (IR-FEL) oscillators are high-power photon sources offering unique radiation characteristics in the THz range. IR-FELs provide a combination of extremely high repetition rates, high spectral brightness, and wide tunability of the radiation wavelength. This allows for efficient excitation of matter by the pumping of the electronic, lattice or spin system and the subsequent probing with high data acquisition speeds [18–21]. A large number of materials exhibit coherent modes with resonance frequencies located in the THz/mid-IR range, and IR-FELs enable efficient selective control of on-resonance and off-resonance processes in these materials [22–24]. However, because FEL oscillators are not CEP-stable, it has not been possible so far to perform field-resolved measurements with such sources and only THz-envelope driven temporal dynamics could be revealed.

The novel metrology technique presented here overcomes these limitations to enable field-resolved research using IR-FEL oscillators and other CEP-unstable THz sources [25]. The method is based on parallel measurements of the THz-induced signal of interest in conjunction with monitoring of the THz CEP for each pulse, allowing one to retrieve a phase-resolved signal after a proper sorting process. The “measure-and-sort” concept has demonstrated its extreme usefulness for time-resolved pump-probe studies using femtosecond x-ray pulses which exhibit pulse-envelope timing jitter [26–29]. Here, we demonstrate for the first time a means for realizing a similar scenario for THz pulses originating from sources without CEP stability. The approach is schematically shown in Fig. 1. The THz beam is split into two parts. One part goes to a pump- probe experiment, and the other is used for the CEP monitoring. Trains of probe laser pulses in both arms are synchronized to the repetition frequency of THz pulses or to an integer harmonic of it. To make the CEP monitoring feasible at high repetition rates, we implemented a specially designed electro-optical (EO) setup that minimizes the required amount of collected data and resources. This “CEP-detection” setup (Fig. 1) utilizes EO-induced ellipticity and rotational changes in the polarization state of the femtosecond laser pulse from overlapping it with the THz wave. The induced ellipticity and rotation variations are caused by the phase and amplitude changes, respectively, of the complex laser pulse amplitude and are maximal when the probe laser pulses overlap with the THz wave at different wave phases shifted by π/2 relative to each other [30]. By measuring and analyzing the ratio between variations of the laser polarization ellipticity and the orientation induced by each THz pulse, we determine the shot-to-shot THz electric field amplitude and phase values. To the best of our knowledge, this is the first demonstration, where both amplitude and phase of the THz electric field are EO encoded in the polarization state of a single femtosecond laser pulse. The measured phase of each THz pulse can be correlated with the corresponding THz-pump laser-probe experiment, thus allowing THz pump pulses to be sorted by their CEP value. Subsequently, the recorded signals for different CEP values can either be considered as independent types of response from THz pulses with different CEP values, or can be combined after a certain correction to increase the sensitivity of the measurements. Based on the suggested concept, we demonstrate field-resolved time-domain spectroscopy with a CEP unstable IR-FEL working at a continuous 13 MHz repetition rate. The experiments have been conducted at a user end-station of the FELBE IR-FEL located at the ELBE Center for High Power Radiation Sources of Helmholtz-Zentrum Dresden-Rossendorf.

 

Fig. 1. Schematic of the presented approach consisting of the CEP detection and THz-pump optical-probe parts. The electric field of the THz waveform is: $E({{t_0}} )= A({{t_0}} )\cdot sin({\varOmega {t_0} + \varphi } )$, where t₀ determines the time overlap between the laser probe pulse and the THz wave in the EO crystal, A(t₀) is the amplitude envelope of the FEL pulse, Ω is the THz carrier frequency, φ is the CEP; a and b are the main axes of the laser polarization ellipsoid induced by the THz wave, and θ is the angle of the THz-induced rotation of the ellipsoid. For each THz pulse i, the φ_i can be determined from the measured ratio (a/b)_i / θ_i. Afterwards, each THz pulse in the THz-pump optical-probe experiment can be sorted by this parameter enabling field-resolved measurements.

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2. Methods

The experimental scheme is depicted in Fig. 2. The THz beam is split into two parts by a Si plate to deliver THz pulses with matching CEP to the CEP detection unit and the Measurement unit. Each branch of the THz beam is focused on one of the EO crystals – 2 mm and 1 mm thick (110) ZnTe crystals – in the CEP detection and Measurement units, respectively. Probe laser pulses are generated by a Ti:Sapphire femtosecond oscillator with 100 fs pulse duration, 800 nm central wavelength and 78 MHz repetition rate. They are synchronized with THz pulses having a six times lower repetition rate (i.e. only one in six laser pulses interacts with the THz field). The CEP detection unit aims to determine the shot-to-shot CEP of the FEL radiation, while the Measurement unit measures the THz waveform using a standard scanning EO detection technique with a mechanical delay line (delay 1) [31,32]. A separate delay line (delay 2) is used to set the temporal overlap between the THz pulse and the probe laser pulse in the CEP detection unit. The absolute THz phase value determined in the CEP detection unit depends on the delay position, laser-THz velocities mismatch in the EO crystal and position of the crystal along the beam-waist, so that the value of the THz CEP in the Measurement unit is known with respect to a certain value determined by the setup performance, which is constant during measurements. A detailed description of the detection methods is presented below.

 

Fig. 2. Experimental setup: Si – silicon plate beam splitter, ITO – glass plate covered by indium-tin-oxide, WP – Wollaston prism, QWP and HWP – quarter- and half-wave plates, HPF and LPF – high- and low-pass optical filters.

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The detection schemes used in the CEP detection and Measurement units are based on EO effects induced during laser pulse and THz wave interaction [30,33–35]. It was demonstrated [30,33,34], that THz wave induce both phase (Δφ_ω) and amplitude (ΔA_ω) changes of the laser pulse complex amplitude at frequency ω. One of the differences between the changes of Δφ_ω and ΔA_ω is their dependence on the THz wave phase φ_Ω (or CEP). As it is shown (Ref. [30], formulas below Eq. (1)), the dependence of the EO modulation Δφ_ω [${\sim} u({\varOmega ,{\beta_\varOmega },{\varphi_\varOmega }} )]$ on the THz phase φ_Ω shifts by π/2 relative to the dependence ΔA_ω [${\sim} \partial /\partial {\varphi _\varOmega }u({\varOmega ,{\beta_\varOmega },{\varphi_\varOmega }} )]$; where $u({\varOmega ,{\beta_\varOmega },{\varphi_\varOmega }} )\equiv a({{\beta_\varOmega },\varOmega ,L} )sin{\varphi _\varOmega } + b({{\beta_\varOmega },\varOmega ,L} )cos{\varphi _\varOmega }$, ${\beta _\varOmega }$ is the absorption coefficient of the crystal at THz wave frequency $\varOmega $, L is the crystal length, and the functions $a({{\beta_\varOmega },\varOmega ,L} )$ and $b({{\beta_\varOmega },\varOmega ,L} )$ are introduced to denote the phase matching and absorption parameters [33].

To make use of the discussed EO modulations, Δφ_ω and ΔA_ω, the detection schemes presented in Fig. 2 for Measurement and CEP detection units are implemented. All the schemes consist of an EO crystal where the optical-terahertz interaction occurs under the same conditions: laser and THz pulses co-propagate along the $\left\langle {110} \right\rangle$ axis in a $({110} )$ cut EO crystal and are polarized along the $\left\langle {1\bar 10} \right\rangle $ crystal axis. The standard EO detection scheme^31,32,35 is used in the Measurement unit for THz electric field detection. It is based on the Pockels effect (the EO induced ellipticity of laser pulse polarization) caused by opposite signs of phase modulation Δφ_ω of the orthogonal polarization components of the laser pulse. The optical scheme consists of a quarter-wave plate (QWP) and a Wollaston prism (WP) that transform the ellipticity modulation to energy changes of separate beams past the WP with opposite signs (without THz impact the QWP transforms the probe laser pulse polarization from the linear to the circular state, which results in a zero difference signal after the balanced circuit; when the laser polarization in the EO crystal becomes elliptical, it misbalances the probe pulse energies arriving at the separate channels).

The CEP detection unit consists of two schemes with separate balanced detectors (Fig. 2). The laser beam is transmitted through a high-pass filter (HPF), the EO crystal, a low-pass filter (LPF), and after that, is split into two parts. These two parts of the laser beam transmit separately through two different sets of optical elements: (i) a quarter-wave plate (QWP), a Wollaston prism (WP), and (ii) a half-wave plate (HWP), a WP; and then each part is measured in separate balanced photodetector circuits. The only difference between parts (i) and (ii) is the replacement of the QWP with the HWP. In part (i), the relative phase change of the orthogonal polarization components of the laser pulse, induced by the THz wave, is measured. This scheme is similar to the standard one introduced in the Measurement unit. In addition to the standard scheme, two edge filters are added. One of the filters cuts the side-band of the laser pulse spectrum before the EO crystal, the other one placed after the EO crystal transmits only the optical waves with frequencies corresponding to the slope of the first filter providing a residual background without THz impact [30]. The transmission of the filters in the edge region had an average reduction/increase approximately 18 times per 1 THz, and the detected edge width was 3.5 THz. The cut-on and cut-off frequencies of those interference filters were adjusted by changing the angle of incidence (by rotating the filter) in order to optimize signals coming from the balanced circuits [30,36]. This scheme permits measuring the THz induced Pockels effect for a laser pulse with edge-cut spectrum at frequencies corresponding to its slope, similar to the standard EO detection scheme [31,32,35]. Additional LPF and HPF elements increase the sensitivity of part (i) at high THz frequencies [30,33,36].

The second part of the CEP detection scheme (consisting of the HWP) is sensitive to the amplitude changes of the orthogonal polarization components of the laser pulse also induced via the EO effect [30,33,34,36–38]. Without THz impact, the HWP rotates the laser polarization resulting in zero difference signal after the balanced circuit; when the THz electric field rotates the laser polarization in the EO crystal, it misbalances the probe pulse energies arriving at the separate channels. The scheme is intended for measuring the amplitude modulation ΔA_ω (instead of Δφ_ω) of the same orthogonal polarization components as in the previous schemes at frequencies corresponding to the slope of the laser pulse spectrum. These polarization components are chosen to maximize the EO modulation by utilizing the same components of the crystal’s nonlinear susceptibility tensor as is usually done in the standard detection scheme. A half-wave plate (HWP) is used to rotate the orientation of the optical polarization by 45^°. In this case, the WP splits the optical waves with orthogonal polarization components into separate horizontally propagated beams. Actually, it is possible to make a scheme without HWP just by rotating the orientation of the WP by 45^°, but in this case, the position of the photo-detectors should be adjusted. We used the additional, HWP element just for more convenient geometry. Due to the different signs of the nonlinear susceptibility tensor coefficients related to the orthogonal laser polarization components, the amplitude changes lead to a rotation of the laser polarization detected by the optical scheme of part (ii) [30].

In principle, this CEP detection unit can also work without the additional optical spectral filters. In this case, the response of the detection part with the HWP is proportional to the instantaneous time derivative of the THz field overlapped with the probe laser pulse in the EO crystal, whereas the response of the QWP part is proportional to the instantaneous amplitude of the THz field [33]. Thus, the phase of the HWP detection branch is automatically shifted by π/2 compared to the response of the QWP branch for all detected THz wavelengths. However, the response in this scheme is much weaker compared to the response of the standard technique using the QWP. To enhance it, additional spectral filtering can be used [30,34,38]. As was shown earlier [30], when both the standard technique utilizing the “phase changes” (QWP branch) and the technique based on the “amplitude changes” (HWP branch) are used with additional spectral filtering, the sensitivities of both techniques are very close to each other. However, their respective responses in the time-domain differ by π/2 relative to each other [$\varDelta {R_1}\sim u({\varOmega ,{\beta_\varOmega },{\varphi_\varOmega }} )$ and $\varDelta {R_2}\sim \partial /\partial {\varphi _\varOmega }u({\varOmega ,{\beta_\varOmega },{\varphi_\varOmega }} )]$ [30]. As a result, the CEP of each THz pulse can be determined from the ratio between signals from the separate balanced circuits related to these two parts of this scheme. The spectral filtering leads to certain changes in the laser pulse temporal profile. However, this relates equally to both HWP and QWP branches, which are being compared for the CEP detection, and therefore does not lead to any additional inaccuracy.

The THz-induced signals S1 and S2 from the QWP and HWP branches, respectively are given by $S1\sim {E_\varOmega } = {A_\varOmega } \cdot sin({\varOmega {t_0} + {\varphi_\varOmega }} )$, where ${E_\varOmega }$ is the electric field of the THz wave at frequency Ω, and according to Eqs. (2)b-2c: $S2\sim {A_\varOmega } \cdot \partial /\partial {\varphi _\varOmega }sin({\varOmega {t_0} + {\varphi_\varOmega }} )= {A_\varOmega }\; \cdot cos({\varOmega {t_0} + {\varphi_\varOmega }} )$. Then, $\alpha \cdot S1/S2 = {\rm{tan}}({\varOmega {t_0} + {\varphi_\varOmega }} )$, and the phase $\varphi \equiv \varOmega {t_0} + {\varphi _\varOmega }$ can be found from the expression φ=tan^-1(α·S1/S2); where α is an additional coefficient determined by the slightly different laser pulse energies in schemes (i) and (ii), the sensitivities of the balanced circuits, and the detection methods. The $S1\;$ and $S2$ signals are proportional to THz-induced ellipticity and rotation of the laser polarization, respectively, which is schematically depicted in Fig. 1 as the ratio (a/b) of the ellipsoid main axes and the rotation angle θ resulting in the similar formula φ= tan^-1[(a/b)/θ], in the case when $\alpha = 1$. As the CEP (or φ) of the THz radiation emitted by the FEL fluctuates in time, the signal mapped on to the S_x/S_y plane, where S_x=α·S1 and S_y= S2 (Fig. 3(a)) shows a ring-shaped distribution of data points. This is in close analogy to the phase diffusion of a laser field when operating above threshold [39]. The instant intensities of the THz pulses are encoded in the radius of each point and can be extracted in arbitrary units as I = (S_x)²+ (S_y)².

 

Fig. 3. Phase-resolved measurement of a CEP-unstable 1.5 THz waveform. (a) CEP detection for 3999 shots, (b) THz intensity trace measured with CEP detection unit, (c) THz time traces with various CEPs obtained with the Measurement unit and (d) their Fourier transform power spectra.

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THz pulses in the Measurement and CEP detection units represent two parts of the same incoming beam, thus the CEPs of the THz pulses within these two beams uniquely relate to each other. For the Measurement unit, we attribute each incoming THz pulse to its subsequent number relating to the same pulse number in the CEP detection unit and to the position of the variable delay 1 (Fig. 2). In this unit, only one balanced circuit is used in a standard EO setup [31,32,35]. The resulting signal in this setup is proportional to the THz electric field value overlapping in space and time with the femtosecond laser pulse in the EO crystal. Using the variable delay 1, the laser pulse is temporally overlapped with different parts of the THz envelope and THz field values for all the shots are collected with respect to the shot number and the delay position. After that, we perform sorting by the CEP values related to the pulse numbers and then, for each delay position, we average the collected THz field values within different CEP bins. As a result, time-domain traces of the THz electric field related to different CEP values are produced. Alternatively, instead of the EOS set-up implemented in the Measurement unit, any other THz-pump optical-probe scheme can be used.

In the measurements presented here, the repetition rate of the laser we use is six times larger (78 MHz) than the repetition rate of the FEL (13 MHz). As a result, only one of six sequential laser pulses is modulated by the THz radiation. All of the electrical signals were digitized using an Analog to Digital Converter (ADC) with a sampling rate of 1 GSa/s and then evaluated by a specially designed software. No Lock-in amplifier has been used. To increase the signal-to-noise (SNR) ratio, the signal from the balanced circuits corresponding to one of the laser pulses which was not modulated by the THz pulse is subtracted from the modulated signal. The procedure proceeds as follows: using a trigger derived from the 13 MHz reference of the FEL, the signal from the differential output of the balanced detector is acquired for approximately 30 ns, such that the signal from one laser pulse overlapped with the THz pulse and the following unmodulated laser pulse are both captured. Two short equal size segments of the ADC signal centered at each laser pulse are summed to provide two single values representing the magnitude of the response for each of the two laser pulses. For the acquisition segment corresponding to the laser pulse without THz, the summation voltage from these points is close to zero. For the acquisition segment of the laser pulse overlapped with the THz pulse, the summation varies according to the instantaneous THz-induced response. This procedure of acquiring, windowing, and summing can easily be performed in real-time, allowing for measurement of every FEL pulse at 13 MHz. This procedure relates to both CEP and Measurement units.

3. Results

The FELBE FEL operates at a continuous 13 MHz repetition rate and was set up to produce THz pulses with energies of 20 nJ at a central frequency of 1.5 THz. Figure 3(a) represents a set of single-shot measurements of THz amplitudes and phases obtained with the CEP detection unit. Note, no delay line is moved during this measurement, the CEP delay line 2 position is fixed overlapping in time the maximum of the THz envelope with the probe laser pulses. S_x and S_y are proportional to the signal values received from two separate balanced circuits as shown in the Fig. 2. The data represent a circle, and each point relates to one THz shot. The CEP of each pulse is determined as arctan(S_y/S_x) and the instantaneous intensity is determined by the expression I∼(S_x)²+ (S_y)² (see the Methods section).

It is seen that the phase of the collected THz pulses varies over the whole 2π range. Shot-to-shot intensity fluctuations are encoded in the point-to-point fluctuations of the radius-vector squared magnitude (Fig. 3(a), black arrow). These values relate to instant intensities of the THz pulses overlapping with the probe laser pulses and their changes are determined both by the THz pulse energy fluctuations and by the instabilities of the laser-THz timing synchronization resulted in the laser pulse overlapping with different parts of the THz waveform. The standard deviation of the shot-to-shot radius-vector squared magnitude presented in the Fig. 3(a) is approximately 6%, which matches well the typical few percent level of the FELBE power fluctuations.

The minimal THz power requirements for the CEP monitoring technique are determined by the ratio of the EO modulation depth and the noise level. The latter strongly depends on the laser probe power fluctuations, electrical noises and might vary significantly from one setup to another. The EO modulation is determined by a combination of parameters like the THz wave carrier frequency, its pulse duration and EO crystal of choice. Additionally, in comparison with the standard EO detection, the THz-induced modulation can be several times higher in the schemes with spectral filtration [30]. The modulation depth in the presented experiment was below 10% level, to ensure that the electro-optic measurement was performed under linear operation conditions. In the presented approach, the CEP detection unit provides only information for the procedure for sorting THz pulses by their CEP. The possible errors in this procedure might be caused by certain deviations of the EO response from linear behavior leading to a summation of the responses from the Measurement unit having different phases, which will reduce the Measurement setup sensitivity. For example, strong THz pulses can induce large EO modulations causing distortions from the linear regime. In this case, the problem can be easily solved by attenuation of the incoming THz beam. In the presented experiments, the sorted time traces clearly confirm the correct operation of the CEP monitor (please, see below the inset of Fig. 3(c)).

Using the variable delay 2, the laser pulse can be overlapped with different parts of the THz waveform. As a result, the envelope shape of the THz pulse can be obtained [Fig. 3(b) – the CEP detection unit delay line is moved overlapping in time the probe laser pulses with different parts of the THz envelope, the graph represents an average intensity time profile ($S_x^2 + S_y^2$) of the arrived THz pulses to the CEP detection unit, only the data from this unit are collected]. Additionally, if the probe laser pulse is positioned at the slope of the THz intensity profile, the information on the pulse-to-pulse intensity value could be used for the determination of the timing jitter of the whole pulse envelope if needed. The latter feature has not been studied in the present work.

In order to validate the concept, we performed shot-to-shot THz electric field measurements with respect to monotonically changed variable delay 1. Each measured THz field value was attributed to the time delay 1 position and to the CEP determined in the parallel CEP measurement. Figure 3(c) shows the THz electric fields obtained in the Measurement unit versus various time delays for different CEP values of the THz pulses. The procedure for this measurement is performed as follows: the Measurement unit delay line 1 is moved, overlapping the probe laser pulses in time with different parts of the THz envelope; the CEP delay line position is fixed to overlap the probe laser pulses in time with the maximum of the THz envelope; the data received from both units are collected simultaneously, with each single-shot THz electric field registered in the Measurement unit attributed to the CEP determined by the CEP detection unit; all of the data in the Measurement unit are sorted by the CEP. Each point at a certain Measurement delay time in Fig. 3(c) represents an average THz electric field value for measurements sorted by CEP phases within 10 degree bins. The resulting time traces clearly show quasi-monochromatic pulses with various CEPs determined by the CEP detection unit (inset in Fig. 3(c)). The amplitude Fourier spectrum of the time traces for different CEP phases is centered at 1.5 THz and agrees well with the spectrum measured by a commercial Fourier-spectrometer. In Fig. 3(c), one can also see the overall response without phase sorting (green curve), which practically vanishes due to averaging of the signals over the constantly varying CEP phases (only a rectified signal driven by the THz envelope is observed).

After proper phase correction, the responses of all measurements can be averaged increasing the SNR of the power spectrum (Fig. 3(d), cyan curve). With the increased SNR, an additional signal at the second harmonic becomes observable. The intensity of the second harmonic is about four orders of magnitude smaller than the signal at the fundamental frequency. We suppose that this is caused by some imperfections in our measurement setup, resulting in a small nonlinearity of its response giving a peak in the spectrogram around zero frequency (which is the same for all measurements including those without phase sorting) and at the second harmonic frequency. This is well seen for the averaged curve (cyan), and to a smaller extent on the phase-sorted-only curves at selected CEP-bins. Another possible source of the harmonic signal is the off-axis radiation of the FEL itself. It is well known that a byproduct of FEL operation is the generation of coherent odd harmonics. The undulator radiation spectrum also contains even harmonics at non-zero emission angles, and FEL oscillator operation at second harmonics was demonstrated experimentally [40]. In our case, additional measurements are necessary to understand the origin of the second harmonic signal. Nevertheless, we shall note that the demonstrated high dynamic range of the measurement system shows the additional potential for being applicable for FEL physics studies. Note that during time-domain measurements (Figs. 3(c), 3(d)), we collected data points within only an 800 µs window at each delay stage position and remarkably obtained a SNR of about 7 orders on the power spectral graph.

One of the benefits of using intense radiation sources is the ability to study nonlinear excitations of matter, e.g., by probing its transient optical properties with short laser pulses. To verify this capability within the suggested approach, we modified the Measurement unit to the cross-polarization geometry: the QWP was rotated in the position close to the linear polarization transmission and additional attenuation was made for one of the channels of the balanced circuit [41]. In this case, an additional quadratic response proportional to THz power appears simulating an arbitrary physical process with nonlinear response on the THz field. This can be understood as follows: the THz-induced probe intensity modulation transmitted to a photodetector through two crossed polarizers, an arbitrary oriented QWP and EO crystal in-between is determined by the equation [42]: $\varDelta {\rm{I}}({{\mathrm{\varGamma }},{\rm{\beta }}} )= {\rm{I}}({{\mathrm{\varGamma }},{\rm{\beta }}} )- {\rm{I}}({0,{\rm{\beta }}} )\sim {{\;}}[{\rm{si}}{{\rm{n}}^2}({{\mathrm{\varGamma }}/2} )\cdot{\rm{co}}{{\rm{s}}^2}({2{\rm{\beta }}} )+ 0.5{\rm{sin}}({2{\mathrm{\beta }}} )\cdot{\rm{sin}\mathrm{\varGamma }}]$, where ${\mathrm{\varGamma }}$ is the phase retardation induced by the THz field in the EO crystal, ${\mathrm{\beta }}$ is the angle of the QWP with respect to the laser polarization axis. It consists of a quadratic-response (${\sim} {\rm{si}}{{\rm{n}}^2}({{\mathrm{\varGamma }}/2} )$) and a linear-response (∼sin${\mathrm{\varGamma }}$) terms. The ratio between linear- and quadratic-response terms changes with β and the relative impact of the quadratic-response term increases when beta becomes close to zero (linear polarization transmission). This effect is effectively simulating a process with non-linear responsivity with respect to the THz electric field with the aim to verify the feasibility of detecting non-linear responses with the presented approach. In Fig. 4, the measured time trace and its Fourier power spectrum are presented in comparison with linear EO sampling. Approximately two orders increase in power for rectified and second harmonics components at zero and double frequencies (Fig. 4(b)) and also strong asymmetry for the plus-minus components in the time trace (Fig. 4(b)) are clearly observed proving the applicability of the investigated technique for non-linear studies.

 

Fig. 4. Phase-resolved measurement in cross-polarization geometry. (a) THz time trace after proper correction and averaging of all traces with various CEPs obtained with the Measurement unit and (b) its Fourier-transform power spectrum – violet line and Fourier transform power spectra from Fig. 3(d) – cyan line.

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To further validate the technique, we have performed time-domain measurements at a higher frequency, namely by tuning the FEL radiation to 4.9 THz, with 20 nJ pulse energy. For this carrier frequency, 100 µm and 400 µm GaP crystals were employed in the CEP detection and Measurement units, respectively. The measured waveform and its Fourier spectrum, demonstrating a quasi-monochromatic signal, are presented in Fig. 5. The SNR ratio is one and a half orders of magnitude smaller than in the case of pulses with 1.5 THz carrier frequency due to lower detection sensitivity (caused by a rather long probe pulse duration relative to this carrier frequency and the smaller EO coefficient and thickness of the GaP crystal). There are also some pronounced modulations observed in the time trace, which can be attributed to THz beam reflections in the Si beamsplitter and the GaP detection crystal. For the 1.5 THz measurements we do not see such modulation due to much longer THz pulse duration and the thicker ZnTe detection crystal.

 

Fig. 5. Phase-resolved measurement of a CEP-unstable 4.9 THz waveform. (a) THz time trace after proper correction and averaging of all traces with various CEPs obtained with the Measurement unit and (b) its Fourier-transform power spectrum.

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4. Summary and discussion

We have demonstrated a “measure-and-sort” approach allowing THz-pump laser-probe measurements with CEP-unstable THz sources. Using this method, the time-domain electric field traces of a CEP-unstable FEL operating at 1.5 and 4.9 THz carrier frequencies at a 13 MHz repetition rate have been measured using an EO detection scheme. The technique developed here can work over a very broad frequency range. Depending on the laser pulse duration, its wavelength and the detection crystal of choice, the EO sampling (utilized by the technique proposed here) can work across the entire THz frequency range (0.1-10 THz) and beyond [43,44]. Compared to incoherent detection techniques, the phase-resolved THz detection has several benefits: (i) it detects only highly coherent radiation and it is not sensitive to parasitic thermal background radiation, which is often rather strong in the THz range; (ii) moreover, so far incoherent THz detection with a reasonable SNR can only be achieved by using bulky helium-cooled bolometers, whereas EO sampling easily works at room temperature; (iii) the dynamic range of EOS is much higher than that of a bolometer since the former measures the electric field and the latter the intensity [45].

In place of the sequential EO sampling demonstrated here, any other THz-pump laser-probe scheme can be used in combination with the same CEP detection technique providing access to charge carriers motions, spin dynamics, rotations of molecules, vibrations of crystal lattices etc., depending on the researchers’ needs. The presented approach should be particularly useful in studying processes driven not only by the THz intensity envelope, but by the THz electric and magnetic fields. Among such potential studies, accessible with IR-FELs using the presented approach, are THz-induced harmonic generation, magneto-optical Faraday and Kerr effects, phonon dynamics and others. For example, replacement of the EO sampling unit by a standard Faraday rotation setup will provide direct access to precessional motions in magnetic materials, which is hardly achievable at CEP instable sources without a sorting procedure. Without this procedure the induced rotations by THz pulses with different CEP will destructively compensate each other during averaging (as in the case of the green curve, Fig. 3(c)). However, when the incoming THz pulses are sorted by their CEP it is possible to resolve these coherent dynamics.

There is also a certain compromise to be made between the width of each CEP bin (higher CEP accuracy) and the number of THz pulses within the same bin (more averaging and higher SNR within a given CEP bin). The proper CEP bin size should be defined under the particular experimental conditions. In most cases, it is possible to use all detected pulses after a proper phase correction, as we did for EO sampling, achieving the highest possible SNR. The CEP detection accuracy does not depend on the THz and laser power fluctuations, as far as it is determined by the ratio of the signals S_y/S_x (CEP = arctan(S_y/S_x)), both of which are proportional to THz and laser amplitude and power, respectively. We expect that the main contribution to the CEP inaccuracy is the electronic noise as far as it can independently influence the S_y and S_x. Another source of inaccuracies in the CEP monitoring might be the non-ideal performance of the optical beam-splitter placed after the LPF (Fig. 2). If it adds some changes in ellipticity or rotation of the laser polarization this can lead to distortions of the circular phase diagram (Fig. 3(a)) due to mixing of the effects measured in the QWP and HWP arms of the CEP detection unit (Fig. 2).

As an alternative to the approach presented in this work, single-shot detection techniques based on spectral decoding [2,46–52] can be potentially considered within two scenarios: i) as a CEP detection unit and ii) as a Measurement unit eliminating the need of the additional CEP monitoring. However, at high repetition rates their use in both cases demands sophisticated equipment and an evaluation of extremely large amounts of data, which is challenging to implement (e.g. for a timing-jitter detection unit realized at TELBE facility [2,49] working at up to 100 kHz repetition rates, tens of TB of data are generated per day). While the latter configuration, ii), might promise certain simplification due to absence of the CEP monitoring, its application for pump-probe experiments with relatively narrow bandwidth pulses comprised of many tens of cycles requires highly sophisticated procedures and equipment to achieve the required combination of a large time/spectral window and fine time/spectral resolution. Further, single-shot techniques are typically used for EO detection and their direct application for non-linear studies has not been sufficiently explored yet.

The approach we present in this work is simple, robust, and utilizes a very similar geometry to common pump-probe studies conducted with CEP-stable sources, thus providing a straightforward way of studying the same effects but with CEP-unstable sources. Aside from the synchronization system, there are no special requirements on the probe laser system. In the measurements presented here, we have used a rather standard Ti:Sa oscillator (SpectraPhysics Tsunami), which has been in routine operation at the FELBE end-station for almost 20 years. Such an approach easily provides a measurement window of hundreds of ps, with a time resolution determined by the laser pulse duration and is workable at MHz repetition rates.

For the method presented here, instead of using long linearly chirped laser pulses and imprinting the FEL field distribution on them, we implemented a specially designed CEP monitoring EO setup minimizing the information necessary to extract from each THz pulse. In this case, both the instantaneous amplitude and phase of the THz pulse are encoded in the polarization state of a single femtosecond laser pulse, which after decoding gives direct access to these parameters. Thus, measurement of the CEP for each THz pulse with this technique requires only two measured values from each balanced detector related to the modulation of the laser pulses by the instantaneous THz field. For an FEL repetition rate of 13 MHz this requires to a synchronous ADC sampling rate per channel of only 26 MSa/s. Based on a high-end ADC, this means that the proposed technique can be used even at GHz repetition rates, which is of great demand, for example, at synchrotron IR sources, where the repetition rates can be as high as 500 MHz and the pulses are not CEP stable.

In conclusion, the demonstrated concept is robust, can be easily implemented, and enables phase-resolved coherent THz pump-laser probe experiments with CEP-unstable sources. In particular, pump-probe FEL experiments with sub-cycle fs time-resolution are enabled by the method presented here, thus substantially expanding the palette of experiments that can be performed with such a versatile light source. Based on the suggested concept, field resolved time-domain spectroscopy with a CEP-unstable IR-FEL has been demonstrated for the first time.