1. Introduction

Control and measurement of terahertz (THz) pulses allow scientists to unravel ultra-fast phenomena in plasmas, semi-, and superconductors. The method of laser-based electro-optic sampling (EOS) [1, 2] has proven a powerful technique for characterizing broad-bandwidth THz radiation. EOS also serves as a single-shot temporal diagnostic for femtosecond accelerator-produced electron beams [3–5]. Conventional EOS-based techniques rely on a temporal cross-correlation of the THz field profile with an optical probe pulse [3, 6, 7]. Here the covered spectral bandwidth is intrinsically limited by the probe laser bandwidth. Although few-fs, few-cycle probe pulses have been applied in multi-shot EOS configurations [8], single-shot diagnostics have been limited to longer >30 fs probe beams and <10 THz bandwidth coverage. Furthermore, the multiple probe beams, secondary nonlinear effects and femtoscond laser control make for a challenging single-shot diagnostic.

Here we experimentally demonstrate and discuss an EOS configuration operating directly in the spectral domain, by measuring the spectral sidebands created outside the probe bandwidth (bandwidth coverage now only limited by the EO crystal), as depicted in Fig. 1. This technique has single-shot capabilities and is compatible with fiber integration, offering strong practical advantages. While Jamison et al. [9] recently observed EO-induced spectral broadening of a narrow-bandwidth probe by ultra-short (broad-bandwidth) electron beams, here in this paper the sideband generation concept is specifically studied as a stand-alone broad-bandwidth diagnostic for THz pulses or electron beams. By developing a narrow-bandwidth optical probe and a laser-driven broad-bandwidth THz source rich in spectral features (0–8 THz bandwidth coverage), conditions are realized to gain insight into the diagnostic validity, advantages, and potential challenges. The emphasis lies on study of the relevant diagnostic parameters, where the retrieved spectral resolution, sideband amplitude, and dependency on probe delay are experimentally characterized. THz field waveforms longer than the probe pulse are found to have a more complex correlation to the optical sidebands, while the diagnostic for short THz pulses not only proves to be valid but also excels in setup simplicity and bandwidth coverage (no longer limited to the probe bandwidth). Also, data is presented where the diagnostic is verified with a traditional scanning EOS technique. These novel results provide a guide to utilizing EO-induced sideband generation as a practical single-shot diagnostic.

 

Fig. 1 Through interaction of a THz pulse and a narrow-bandwidth optical probe pulse (f₀=375 THz) in an EO crystal, the THz pulse spectrum appears in the optical domain as spectral sidebands.

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2. Electro-optic spectral sidebands

EOS can be understood [2] as nonlinear mixing in an EO crystal of the THz pulse frequency components at ν with the optical probe field components at f to produce new optical components at the sum and difference frequencies f ± ν. In conventional EOS techniques the optical spectrum of the probe pulse is typically broadband, extending well past the EO-induced components f ± ν, such that the new and existing optical contributions add up coherently and appear as an amplitude or phase shift. Measuring these shifts then allows for THz field retrieval. Note that relying on a broadband probe laser limits the THz spectral coverage not only to the crystal properties but also to the probe laser bandwidth (for single-shot applications <10 THz). The latter limit can be circumvented by concentrating on the optical sidebands created outside the probe laser bandwidth.

In the sideband generation diagnostic, an optical probe and a THz pulse are both being focused onto an EO crystal. For simplicity, the probe field is defined as a Gaussian E_opt(t) ∼ exp[−t²/τ²]cos(2πf₀t), with τ the pulse duration, f₀ = c/λ₀, c the speed of light, and λ₀ the carrier optical wavelength. The spectrum can be written as $E_{\text{opt}}\left(f\right)=E_0\text{exp}\left[-{\left(f-f_0\right)}^2/{\sigma }_f^2\right]$, with σ_f = 1/πτ the spectral bandwidth and E₀ the spectral amplitude at f₀. The field of the THz pulse is labeled as E_THz(t), or as the complex function E_THz(ν) in the spectral domain. Note that E_THz(ν) can also represent the self-fields of a femtosecond electron bunch through the Fourier transformation E_THz(ν) ∼ ∫ dtQ(t)exp(−ı2πνt), with Q(t) the temporal charge profile. Following the approach by Jamison et al. [2, 9, 10], frequency mixing inside the EO crystal leads to sum and difference frequency generation (labeled as E_sum). In a coordinate system where the induced two-photon polarization P and input optical field E_opt are parallel, the nonlinear polarization can be expressed as $P={\varepsilon }_0{\chi }_{\text{eff}}^{\left(2\right)}{E}_{\text{THz}}{E}_{\text{opt}}$, with ${\chi }_{\text{eff}}^{\left(2\right)}$ the effective electro-optic parameter. Applying the slowly varying envelope approximation to the optical field envelopes [2], the solution to the wave equation can be rewritten as

A further simplification can be made if, within the range σ_f around any given THz frequency ν, the complex profile E_THz(ν) is non-evolving. This occurs when the probe laser is much longer then the duration T_THz of the THz pulse (T_THz ≪ τ). Under this condition, the integral in Eq. (1) is only nonzero around ν ≃ |f – f₀| and Eq. (1) can be simplified as

Equations (1) and (2) indicate that the crystal transfer function T_cr(ν) needs to be considered [11]. T_cr(ν) incorporates the frequency dependent EO parameter ${\chi }_{\text{eff}}^{\left(2\right)}\left(\nu \right)$ (see [12] for the relationship between ${\chi }_{\text{eff}}^{\left(2\right)}$ and the EO coefficient r), the effect of Fabry-Perot reflections of the THz pulse inside the crystal, and the phase-matching function [2] which incorporates THz absorption, dispersion, and velocity walk-off between optical and THz fields. T_cr(ν) can either be measured or retrieved from the literature. Note that for ultra-intense THz fields and thick EO crystals [for example E_THz ≳ 0.1 MV/cm and L ≳ 1 mm], the expressions provided here and in Ref. [2] need to be adjusted for non-negligible higher-order spectral contributions at f₀ ± 2ν, f₀ ± 3ν, etc., as described in Ref. [10]. This regime is outside the scope of this paper.

3. Electro-optic sidebands from a laser-driven THz source

In order to experimentally study EO-induced optical sidebands, and to evaluate the complexities in Eq. (1), a setup was developed (see Fig. 2) consisting of a narrow-bandwidth probe pulse (τ = 2.9 ps) and a broad-bandwidth THz source rich in spectral features. The EO crystals used were either 200-μm-thick ZnTe (zinc telluride) or GaP (gallium phosphide), which defined the spectral domain of study to 0–3.5 THz or 0–8 THz, respectively. The laser-based THz generation arm was derived from a 1-kHz titanium-sapphire laser system (λ₀ = 804 nm, f₀=373 THz). The beamsplitter transmitted 92% of the laser light towards the THz arm. The beam diameter in this arm was 3.5 mm (defined by an iris), the pulse duration 45 fs [intensity full-width-at-half-maximum (FWHM)], and the pulse energy after the iris 100 μJ. The laser was propagated through a 100 mm focal length lens (lens 1), where 15 mm before focus a 100-μm-thick frequency doubling BBO crystal (type-I β-barium borate) was placed to produce both fundamental f and frequency-doubled 2 f radiation at the focus in air. This configuration has been demonstrated (Refs. [13, 14] and citations therein) to yield broad-bandwidth THz radiation due to the creation of directional currents of the electrons in the air-based plasma. The THz radiation was collected by a 50-mm-diameter silicon lens (7 mm thick, focal length of 75 mm), and focused through a thin nitro-cellulose pellicle onto the EO crystal. The entire THz line was in air. It is important to emphasize that this study was not geared towards optimization or understanding of the THz source itself, but solely focused on the sideband generation concept.

 

Fig. 2 Experimental setup for the THz-induced optical sideband diagnostic. A laser beam is split by a beamsplitter into a pump arm for THz generation and a probe arm for EO detection. The THz pulse is focused by a Silicon lens through a pellicle onto an EO crystal. The probe arm is reflected off a grating-lens-slit combination to produce a 0.11-THz-bandwidth probe pulse onto the EO crystal, after which an imaging spectrometer records the EO-modulated optical spectrum.

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The narrow-bandwidth probe beam E_opt was derived from the same laser line as the THz generation arm to guarantee temporal synchronization. The size of the probe beam was 5 mm (FWHM). As illustrated in Fig. 2, the reflection off the beam splitter (∼ 15 μJ/pulse) was sent through a delay stage onto a 600 lines/mm grating to accomplish spectral dispersion. A lens placed after the grating (lens 2, focal length 150 mm) produced a horizontal line focus, with each color focused to a different position (spatial chirp). A slit was placed in the focal plane of the lens, therefore only transmitting a narrow controllable part of the optical bandwidth. Lens 3 (focal length of 100 mm) imaged the line focus onto the EO crystal, with a vertical size of 15 μm FWHM and a horizontal size as controlled by the grating slit. The reflection off the pellicle was circa 9%. The polarizer and EO crystal z-axis were rotated such that the EO-generated optical beam E_sum was polarized orthogonally to E_opt [15]. The analyzer was rotated for maximum transmission of E_sum. Due to imperfect extinction and intrinsic crystal birefringence, ∼0.15% of the probe energy still made it through the analyzer (∼0.017% in the case of GaP). The EO-modulated probe beam E_sum (plus remnant E_opt) was then imaged at ×4.2 demagnification on the entrance slit of an imaging optical spectrometer (300 grooves/mm grating). The CCD camera in the spectrometer was a 1392×1032 pixel 10-bit (effective) camera, with a 0.047 nm/pixel calibration around 800 nm. Since the spectrometer slit was transmitting only a controllable portion (of order 2%) of the horizontal line focus at the EO crystal, the grating slit (see Fig. 2) served no benefit and was eventually removed from the setup.

An image of the measured optical spectrum, measured with the 200-μm-thick ZnTe crystal is shown in Fig. 3(a). For this measurement the spectrometer slit was set at 10 μm, which corresponds to a horizontal acceptance at the EO crystal of 45 μm. While the sideband technique has single-shot capabilities, the weak THz field (∼ 1 kV/cm, as will be determined later) and the weak probe beam energy (losses from low beamsplitter reflectivity, low pellicle reflectivity, and strong bandwidth reduction) required the CCD shutter to be opened for 5 seconds (5000 laser shots), after which 8 consecutive images were averaged. Although both sidebands are quantitatively symmetric (as predicted), the small asymmetry in amplitude is most likely attributed to spectrally-dependent optics transmission and reflection. The vertical size of the optical sidebands was measured to be 15 μm (FWHM), consistent with the size of the probe beam at the EO crystal. Note that remnant unmodulated probe radiation |E_opt(f)|² frustrated THz retrieval around λ₀, hence the black bar at 803 nm.

 

Fig. 3 (a) Spectrometer image obtained with 200-μm-thick ZnTe as the EO crystal (spectral range of 0–3.5 THz). (b) Lineouts of the spectral images for ZnTe (red curve) and GaP (blue curve, multiplied by 5). The spectral range of 200-μm-thick GaP is 0–8 THz. The spectral intensity was normalized to $E_0^2$. The spectral features, a result from absorption in the plasma and in air-based water vapor, line up for both measurements and were found to be as sharp as 0.15 THz (FWHM). The black curve in (b) shows the air-based water absorption coefficient, where the frequencies of strongest absorption line up with the minima in the retrieved THz spectrum.

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In a subsequent experiment, the EO crystal was switched to 200-μm-thick GaP (acquisition time increased to 6 seconds, with 34 consecutive images averaged). Based on the spectrometer images for ZnTe and GaP, the retrieved spectral line-outs are shown in Fig. 3(b), where the wavelength axis was converted to a frequency axis ν = c/λ – f₀. One can see that the spectra are consistent with the expected smaller 0–3.5 THz bandwidth for ZnTe and larger 0–8 THz bandwidth for GaP. The GaP-induced sideband spectrum was a factor of 4 weaker than ZnTe. Note that since the THz source relies on a strong nonlinearity at focus in air, one can expect drifts in THz peak field and spectral content over time. Another observation in Fig. 3(b) is the many sharp spectral features in the spectra (which match for the ZnTe and GaP observations, see dashed lines). Such spectra were also observed with a scanning Michelson interferometer by Kim et al. [13], and can be attributed to absorption of the THz radiation in the generation plasma and in air-based water vapor. The black curve in Fig. 3(b) shows the air-based water absorption coefficient, as measured at lower spectral resolution with a Michelson interferometer in an independent study, where the frequencies of strongest absorption indeed line up with several of the minima in the THz spectrum. Spectral features as sharp as 0.15 THz (FWHM) were observed. This observation can be compared to the expected spectral resolution, which is determined by the convolution of the laser probe bandwidth σ_f and the intrinsic spectrometer resolution. The probe bandwidth was σ_f = 0.11 THz (equivalent to 0.13 THz intensity FWHM), while the intrinsic spectrometer resolution was measured with a calibration line-source lamp to be 0.11 THz (FWHM). This yields a convolution of 0.17 THz, close to the measured 0.15 THz. Sideband spectra were also recorded (not shown) at larger spectrometer slit widths (and hence poorer spectrometer resolution) of 25 μm, 50 μm, 100 μm, and 150 μm, and the spectral features indeed were washed out.

The presence of narrow spectral lines in the THz spectra of Fig. 3 was attributed to absorption. While large parts of the THz pulse spectrum fall outside of the absorption bands, those spectral components closer to the bands experience absorption and dispersion, both leading to long time-domain field structure. The pulse lengthening can result in the condition T_THz ≳ τ such that Eq. (2) is no longer valid, leading to a more complex interplay between the probe and THz pulse (including a dependency of the relative sideband profile on the temporal delay). This concept was experimentally verified by recording the retrieved THz spectra as a function of probe delay, both for ZnTe and GaP crystals. For these measurements the spectrometer slit was opened to 50 μm, at an acquisition time of 6 seconds (no averaging). The data for ZnTe and GaP is shown in Figs. 4(a) and 4(b), respectively. Each horizontal line in the two-dimensional images represents the sideband spectrum at a certain probe delay. The effect of the probe delay is consistent in both ZnTe and GaP, although the larger bandwidth of GaP highlights the delay dependence on the spectrum most clearly. Figure 4(c) shows the sideband amplitude (normalized to unity) versus delay for several spectral bands. Those spectral components of the THz pulse not affected by absorption and dispersion are all optimized around the same delay (≃ 7 ps). Since the THz pulse was created by a short 45-fs laser pulse, and the unaffected THz frequencies coherently add up to a short THz burst, for those frequencies the condition T_THz ≪ τ is met. Hence, the temporal width of the bottom curves in Fig. 4(c) represents a measurement of the probe pulse duration, and a Gaussian fit |E_opt(t)|² ∼ exp[−2t²/τ²] yielded τ =2.9 ps. However for other spectral bands (such as around 2.7, 3.9, and 6.1 THz), absorption and dispersion resulted in stretching of the field profile, leading to a later optimum timing for the probe pulse. This phenomenon is clearly evident in Figs. 4(a)–(c).

 

Fig. 4 Each horizontal line in (a) for ZnTe, or (b) for GaP, represents a sideband spectrum for a given delay between the probe and THz pulse. By scanning the probe delay and stacking the sideband spectra, the two-dimensional images are formed. Due to strong THz absorption and dispersion at specific spectral bands (such as 2.7, 3.9 and 6.1 THz), frequency-dependent THz pulse lengthening yields a dependency of the sideband amplitude on probe delay as shown in (c). Those spectral components not affected by absorption (such as 2.0, 3.4 and 4.8 THz) are optimized around the same delay (≃ 7 ps), where a Gaussian fit yields a probe duration τ = 2.9 ps.

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4. Scanning electro-optic sampling as a diagnostic validation

In order to verify the retrieved THz pulse spectrum, the setup of Fig. 2 was modified to a scanning two-dimensional EOS (2D-EOS) configuration [16] with the same THz source and ZnTe as the EO crystal. The grating was replaced by a flat mirror, all slits were removed, and the lenses 2 and 3 were positioned such that the probe beam at the EO crystal was now spatially overfilling the THz spot. The transverse probe profile at the EO crystal was imaged by a conventional CCD camera. The probe beam had a duration of 55 fs (FWHM). By scanning the delay between the probe beam and the THz pulse, the varying spatial imprint of the THz pulse was recorded. The intrinsic birefringence in the ZnTe crystal (and the fact that the THz pulse was weak) allowed for sign-resolved EOS [17]. By comparing the transverse probe profiles to the reference profile at full analyzer transmission, the spatial distribution of the THz-induced phase retardation Γ_THz(x, y, t) was derived [17]. The relation between Γ_THz and the field amplitude E_THz can be approximated [15] as ${\Gamma }_{\text{THz}}=\left(2\pi /{\lambda }_0\right)L{n}_0^3{r}_{41}{E}_{\text{THz}}$, with n₀ = 2.85, L = 200 μm, and r₄₁ = 4 × 10⁻¹² m/V [18].

Figure 5(a) shows the retrieved field waveform E_THz(t), averaged over a 65 × 65 μm² area within the THz spot. Due to fluctuations in laser and THz parameters during the scan, the low frequency components ≲1 THz in the scanning 2D-EOS spectrum contained artifacts and were omitted. The peak field was found to be 1 kV/cm (a peak phase retardation Γ_THz of 15 mrad). By multiplying the retrieved profile E_THz(t) by a Gaussian with duration τ = 2.9 ps, we imposed an artificial temporal window which allowed for comparison of the retrieved THz spectrum from the two methods [scanning 2D-EOS versus the sideband diagnostic results from Fig. 4(a)]. Figure 5(b) shows both spectra side by side (in arbitrary units). The two spectra were retrieved several hours apart from one other, and due to the nonlinear nature of the THz source some drift in THz parameters is likely to occur. Nevertheless, the scanning 2D-EOS retrieved THz waveform provided the approximate field strength of the THz pulse at the EO crystal, and confirmed our previous insight into its temporal structure (a short THz burst at a delay of 7 ps followed by longer field oscillations). Also, the two spectra in Fig. 5(b) show quantitative agreement, with the spectral features in both measurements lining up.

 

Fig. 5 (a) Field waveform of the THz pulse (in units of kV/cm), retrieved with a conventional scanning 2D-EOS configuration (with ZnTe). One can observe a short THz pulse followed by long-lasting field oscillations. (b) Comparison of the retrieved spectrum from scanning EOS to the sideband technique yields quantitative agreement, with the sharp spectral features lining up.

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For the data in Fig. 3 circa 10⁵ shots were accumulated for a sideband spectrum, mainly due to the weak probe beam (∼ 1.5 nJ at the EO crystal) and weak THz field (∼1 kV/cm). As Eq. (2) indicates, the sideband energy scales linearly with the THz intensity |E_THz|² and the probe intensity |E₀|². Thus, for ×10⁵ more available probe energy on the EO crystal (∼150 μJ) a similar spectrum is expected to be obtained in a single-shot manner. Also, for stronger THz fields the necessary probe energy for the same signal-to-noise ratio is reduced (for example, a reduction of ×100 for E_THz =10 kV/cm).

5. Modeled spectral sidebands: effect of temporal overlap

As confirmed by the scanning 2D-EOS retrieval, the optical sidebands of Fig. 4 identified the two regimes of the sideband diagnostic, characterized by T_THz ≪ τ and T_THz ≳ τ. Similar insight can be obtained theoretically by modeling the retrieval using Eqs. (1) and (2). To illustrate this, an example THz pulse is modeled as the sum of two Gaussians, namely $E_{\text{THz}}\left(t\right)/\alpha =\left(1/\sqrt{\pi }{T}_1\right)\text{exp}\left[-t^2/T_1^2\right]-\left(1/\sqrt{\pi }{T}_2\right)\text{exp}\left[-t^2/T_2^2\right]$, where α is chosen such that the peak field is 1 kV/cm. Note that ∫dtE_THz(t) = 0. Values of T₁=40 fs, T₂=80 fs, and τ = 1.2 ps for the probe field are used. Other parameters, based on typical EO crystals, are n_opt = 2.85, L=200 μm, and ${\chi }_{\text{eff}}^{\left(2\right)}=2.6\times {10}^{-10}\text{m}/\text{V}$. The THz pulse is delayed by either τ₁ =0 ps [see blue curve in Fig. 6(a)] or τ₂ =1 ps [red curve in Fig. 6(a)]. In this paper (including the experimental part) only the absolute spectrum of the sidebands |E_sum(f)| is studied, since this represents an optical spectrometer measurement. The optical spectra |E_sum(f)| for both delays are shown as open circles in Fig. 6(b). In these examples, the THz pulse duration T_THz is much shorter than the probe duration τ, such that Eq. (2) applies. As Fig. 6(b) demonstrates, the sideband spectra are indeed identical to the input THz pulse spectrum shifted by f₀ = 375 THz (see black curves in normalized units).

 

Fig. 6 (a) Example of time domain field envelope for the probe field E_env,opt(t) = exp[−t²/τ²] (green curve) and the THz field E_THz(t) at two different delays τ₁ = 0 ps (blue curve) and τ₂=1 ps (red curve). (b) The normalized optical sideband spectra |E_sum(f)/E₀| for the two different delays, with the black curves representing the input THz pulse spectrum shifted by f₀ = 375 THz. Note that the THz pulse duration is much shorter than the probe pulse, and hence the sideband spectra match the input THz pulse spectrum. For the examples in (c), with τ₁ = −0.5 ps and τ₂=0.5 ps, the same THz pulse spectrum is now chirped to longer duration, with the sideband spectra plotted in (d). One can see that the sidebands are now a function of delay and no longer match the input THz spectrum.

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The situation changes when the THz pulse (same spectrum |E_THz(ν)|) is now chirped to a longer duration, see Fig. 6(c), at two delays τ₁=−0.5 ps (blue curve) or τ₂=0.5 ps (red curve). The input THz spectrum [black curves in Fig. 6(d)] is no longer a match to the retrieved sideband spectra (open circles), and even depends on the temporal delay between both pulses. The measured sideband spectra are then only meaningful in experiments where the absolute spectrum is not relevant (for example, in sample-in versus sample-out transmission measurements). Note that the sideband spectra in Figs. 6(b) and (d) are normalized to the probe spectral amplitude |E_opt(f₀)| = E₀, allowing for insight into the sideband strength. In this example, the maximum sideband spectral intensity |E_sum(f)/E₀|² was 1.6 × 10⁻⁵.

6. Conclusion

In conclusion, the electro-optic generation of optical sidebands on a narrow-bandwidth probe by broad-bandwidth THz field pulses was presented as a THz diagnostic. This technique, with single-shot capabilities and operating directly in the frequency domain is not limited by the laser bandwidth and excels in simplicity since the optical setup avoids complexities from conventional EOS cross-correlation with broad-bandwidth optical pulses. The diagnostic was experimentally studied with a τ = 2.9 ps probe pulse (0.11 THz bandwidth) and a broad-bandwidth THz source rich in spectral features. Features as sharp as 0.15 THz (FWHM) were retrieved with bandwidth coverage up to 8 THz (as limited by the EO crystal). Spectrally-dependent absorption and dispersion of the THz pulse in the generation plasma and in air-based water vapor led to lengthening of the THz pulse. The data revealed spectral bands unaffected by absorption such that for this short THz burst the condition T_THz ≪ τ is valid, where the diagnostic is simple and intuitive. However, the data also revealed other spectral bands participating in pulse stretching beyond τ so that the sideband diagnostic becomes more complex, with stronger spectral sidebands at later delays. By operating the setup in a scanning EOS geometry, quantitative agreement between both EOS-based techniques was obtained. Note that the analysis and experiments in this paper focused solely on the absolute sideband spectra |E_sum(f)|², leaving the spectral phase retrieval of the THz pulse for future work.