1. Introduction

The availability of continuum pulses simultaneously covering both the THz and IR spectral ranges opens the way for new spectroscopies with pumping [1,2] and/or probing [3,4] of low-energy excitations, e.g. intra- and inter-band conductivity, phonons and other quanta in complex solids with femtosecond (fs) time resolution. The generation of such pulses can employ nonlinear mixing of the frequency components of fs near-IR (NIR) optical pulses in either ${\chi ^{(2)}}$- or ${\chi ^{(3)}}$-media. For ${\chi ^{(2)}}$-crystals, the use of ultra-broadband NIR pulses from two independent OPAs and mixing in suitable crystals (e.g. GaSe) can yield bandwidths reaching 100 THz [5], and very high peak fields (reaching ${\sim }10~{\,\mathrm {MVcm}^{-1}}$ [6] or even ${\sim }100~{\,\mathrm {MVcm}^{-1}}$ for narrow-band mid-IR pulses [7]), although the requirement of a crystalline medium introduces issues due to phonon absorption, phase-matching, damage threshold, and interference/losses due to surface reflections. In contrast, pulse generation in a gas plasma with an ultra-broadband, two-color (${\omega {-}2\omega }$) NIR-UV pump [8–12] can reach bandwidths exceeding 100 THz, also with a stable carrier envelope phase, without any inherent damage threshold or low-frequency cut-off in the conventional THz range, and field strengths readily exceeding 1 ${\,\mathrm {MVcm}^{-1}}$ for sufficiently high pump energy [13–20].

Simultaneous coherent detection of the full THz-IR bandwidth is non-trivial. Time-domain electro-optic (EO) field sampling in ultra-thin ${\chi ^{(2)}}$-crystals [5–7] with extremely short optical detection pulses can provide a direct determination of the temporal field (provided the EO response is carefully taken into account [21]), albeit with similar issues for ${\chi ^{(2)}}$-emitters mentioned above. Moreover, sampling the intra-cycle field for bandwidths exceeding 100 THz places severe requirements on the pulse duration – as shown in Fig. 1(a), this essentially requires an optical pulse duration below 10${\,{\mathrm {fs}}}$. One can also use ${\chi ^{(3)}}$-media as the non-linear medium: in particular a combination of terahertz- and electric-field-induced second harmonic generation (TFISH, EFISH) in a gas medium comprises the method of air‐biased coherent detection (ABCD) detection [22,23], now widely employed for conventional THz time-domain detection. While the higher-order nonlinearity requires higher optical intensities, similar detection efficiencies can be achieved with amplifier-laser pulses. The ABCD method was applied to THz-IR transients [20], although with limited sensitivity due to a strong frequency roll-off. One attractive feature of ABCD detection is that the DC electric field $\cal {E}_{\textrm {b}}$ used for EFISH, in principle, provides an absolute field strength with which to calibrate the field of the THz pulse (as opposed to deducing the field from the crystal ${\chi ^{(2)}}$-value and optical retardation in EO detection). As discussed by others [22,24–26] (and in this report for the full THz-IR range), one must correctly account for the propagation geometries for both EFISH and TFISH to correctly use the EFISH signal for such calibration (as opposed to directly using the signal ratios [27]).

 

Fig. 1. (a) Response function for electro-optic time-domain sampling for different optical detection pulse duration, $T_{\mathrm {FWHM}}$ ($S(\nu )=\exp (-\tfrac {\pi ^2}{4\ln {2}}\nu ^2{T_{\mathrm {FWHM}}}^2)$). (b) Schematic of frequency conversion processes for detection of THz-IR pulses, including FW-DFG (equivalent to TFISH for THz frequencies) used here in air. (c) Setup used for spectrogram/ABCD experiments. (d) Example spectrogram using detection pulses with ${T_{\mathrm {FWHM}}}=150{\,{\mathrm {fs}}}$.

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If one forgoes field detection with sub-cycle resolution, an alternative is to spectrally resolve the signal from mixing the THz-IR and optical-detection pulses. Indeed, such frequency conversion techniques date well back in the history of time-resolved detection of femtosecond pulses. Again, one can employ ${\chi ^{(2)}}$- or ${\chi ^{(3)}}$-media, as depicted in Fig. 1(b). The mixing of the THz-IR and NIR pulses generates sum-frequency (SFG) and difference-frequency (DFG) sidebands which can be detected with an optical spectrometer, and as a function of delay to yield a spectrogram. While the classical method is to employ SFG/DFG in ${\chi ^{(2)}}$-media (which we also employed previously to detect THz-IR pulses [28–30]), this has again the issues of using crystalline media, and the SF-/DF-sidebands also overlap with the input optical spectrum for THz frequencies. However, it was realized in recent years [17,19,31,32] that one can extend the TFISH approach to cover the whole THz-IR range – the resulting four-wave (FW) processes between the fundamental optical ($E_1$) and THz-IR (${\mathcal {E}}$) fields also referred to as FW-SFG/DFG – with detection of the signal ($E_2$) in the UV-vis range. As established for TFISH/ABCD [23–25,33] and discussed in detail in this paper, the FW-SFG process suffers destructive interference during co-propagation through the focal region [32] due to the relative Gouy-phase evolution of the nonlinear signal polarization $P_{\mathrm{SF}}^{(3)} \propto E_1^2 \mathcal{E}$ and generated field $E_2$ (also well-known for optical third-harmonic generation [34]), although the FW-DFG signal with $P_{\mathrm{DF}}^{(3)} \propto E_1^2 \mathcal{E}^*$ survives.

The resulting FW-DFG spectra can be acquired as a function of delay $\tau$ between $E_1$ and ${\mathcal {E}}$ to build up a spectrogram, which also contains temporal information (relative phase) about ${\mathcal {E}}$, being essentially an XFROG (cross‐correlation frequency resolved optical gating) spectrogram of ${\mathcal {E}}(t)$ with a gate pulse $a(t)=E_1^2(t)$ [19] (similar to that used in the infrared‐UV range [35,36], only here the THz‐IR spectrum extends down to zero‐frequency). Here several modalities have been demonstrated to recover the THz-IR pulse, including the use of chirped optical detection pulses $E_1$ [17,37] (which facilitates single-shot detection and can achieve a finer frequency resolution than the bandwidth of the gate pulse, but requires subsequent phase correction for the retrieved data) or conventional XFROG deconvolution with short detection pulses (e.g. with 30-fs duration [19]). The ability to recover the sub-cycle field profile (i.e. to determine the absolute, or carrier-envelope, phase), as achieved with EO time-domain sampling [5–7], is highly desirable, especially for non-linear THz-IR spectroscopy. Here an elegant approach was demonstrated [19], whereby an auxiliary ABCD measurement in the same detection scheme delivers the field profile $\mathcal{E}_{\mathrm{THz}}(t)$ of the low-frequency range of ${\mathcal {E}}$, hence providing the absolute phase offset for the latter determined from the FW-DFG spectrogram.

In this paper, we combine and adapt these approaches to estimate the absolute temporal electric field profile ${\mathcal {E}}(t)$ of THz-IR pulses from a two-color air plasma. Here we measure FW-DFG spectrograms with relatively long (150-fs) detection pulses, which we show can still be used to extract the relative spectral intensity and phase of the THz-IR transients (with a duration which can reach below 10 fs) even without explicit XFROG deconvolution, while the absolute phase offset is provided by an auxiliary ABCD measurement of $\cal {E}_{\textrm {THz}}$ as per [19]. We introduce for the calibration of the absolute field a novel approach where we also analyze the relative EFISH and FW-DFG signal intensities and take into account propagation effects (i.e the amplitude- and phase-development of the on-axis input- and signal-fields propagating through the detection focal region, including Gouy- and dispersive phase mismatch) to determine the frequency-dependent FW-DFG response relative to the EFISH response. As this requires determination of the focal behavior of the THz-IR field through the focus [26], we performed additional measurements of the FW-DFG signal vs. radial truncation of the THz-IR beam before focusing, which we show can provide sufficient information to quantify the propagation effects. This method also has potential for absolute field calibration of ABCD measurements of conventional-bandwidth THz pulses.

2. Experimental details

A schematic of the experimental setup is shown in Fig. 1(c), based on a 1-kHz Ti:Al$_2$O$_3$ amplifier laser (Clark-MXR CPA-2101, $\lambda _0=775$ nm, ${T_{\mathrm {FWHM}}}=150{\,{\mathrm {fs}}}$, pulse energy $900{\,\mathrm{\mu}}\textrm {J}$). As in [16,30], we spectrally broaden these pulses in an Ar-gas capillary (2.1 bar) to achieve optical bandwidth corresponding to ${T_{\mathrm {FWHM}}}<20{\,{\mathrm {fs}}}$ and pulse energy ${\sim }400{\,\mathrm{\mu}} \textrm {J} $ with subsequent compression with negative-dispersion mirrors. These pulses are focused in ambient air (${f_{\mathrm {p}}}=200$ mm) via a 150-$\mathrm{\mu}$m-thick $\beta$-BBO crystal ($32^\circ$-cut for type-I second-harmonic generation, SHG) to achieve ${\omega {-}2\omega }$ plasma emission. Note that the orientation angles (and position) of the SHG crystal are carefully optimized to yield the resultant THz-IR bandwidth, which, as we previously reported [16], depends strongly on the resultant frequency detuning of the $\omega$- and $2\omega$-waves. The emitted THz-IR beam passes a small axial stop (to block the $\omega$ and $2\omega$ pump beams), and is collimated by an off-axis paraboloidal mirror (OAPM, protected Al, eff. focal length $76.2$ mm). The beam is focused through a position for various spectroscopy measurements and re-collimated (protected Au, eff. focal lengths 101.6 mm), and then focused for coherent detection (OAPM, $f=76.2$ mm) with a variably delayed co-propagating 150-fs optical pulse ($1/e^2$-beam diameter 3.6 mm, energy $40{\,\mathrm{\mu}} \textrm {J}$, focused by a lens with $f_1=125$ mm through a hole in the last OAPM). Note that the indium-tin-oxide mirror is used to introduce on-axis optical pump beams for other experiments, while a 50-$\mathrm{\mu}$m-thick high-resistivity Si wafer in the collimated beam path suppresses residual $\omega$ and $2\omega$ plasma pump light.

The focal region is equipped with cylindrical electrodes (diameter 2.5 mm with flat end surfaces, spacing 0.45 mm) to apply a 1-kHz/500-Hz sinusoidal electric field with peak amplitude $\cal {E}_{\textrm {b}}$ = 25.8 kV cm^-1 used for the 1-kHz EFISH calibration measurements (as confirmed with high-voltage probes directly on the electrodes, remaining below air breakdown) to allow the generation of an additional ABCD/EFISH detection signal. Following the detection focus, the signal beam passes dichroic filters to suppress the input optical beam $E_1$, is recollimated and coupled with variable split ratio to a compact UV-vis spectrometer (Avantes Avaspec-HSC1024x58TEC-EVO, 335-510 nm, resolution <0.4 nm) for FW-DFG/EFISH spectral measurements, and a photomultiplier tube (Hamamatsu H9656-04) for ABCD measurements. The spectral response of the spectrometer is flat over the FW-DFG spectral range, and the spectra are used without further wavelength correction. The complete setup is purged with dry air to minimize THz/IR absorption from water vapor.

3. Results

3.1 Extraction of THz-IR spectrum and chirp

For our approach, we employ a relatively long (150-fs), transform-limited optical detection pulse to measure the FW-DFG spectrogram $S_T(\nu,\tau )$, in contrast to previous reports (e.g. 30 fs in [19]). There are several motivations for this: (i) the resultant spectral convolution of the THz-IR spectrum $I(\nu )$ with that of the detection gate pulse still yields a reasonable spectral resolution without deconvolution; (ii) for pump-probe experiments, one can use a constant detection delay $\tau =\tau _0$ (as the rather long gate pulse can maintain temporal overlap with the THz-IR pulse) and only scan the pump-probe delay ${{{\tau }_{\mathrm {ex}}}}$ to acquire $\Delta I(\nu ;{{{\tau }_{\mathrm {ex}}}})$ via $S_T(\nu,\tau _0;{{{\tau }_{\mathrm {ex}}}})$ [30]; (iii) one can more readily treat propagation effects in the FW-DFG detection process (Sec. 3.3) when the optical gate is reasonably narrow-band. Nevertheless, given that the THz-IR pulses have a duration on the order of 10 fs, one could question whether any robust retrieval of temporal information is possible with such relatively long detection pulses. We demonstrate that this is the case in this section.

A measured reference FW-DFG spectrogram $S_T(\nu ',\tau )$ is shown in Fig. 1(d), where $\nu '=2\nu _0-\nu _S$ is the DF offset (i.e. THz-IR) frequency, $\nu _S=c/\lambda$ the measured signal frequency, and $\nu _0=c/\lambda _0$ ($\lambda _0$ the center wavelength of the Ti:Al$_2$O$_3$ amplifier laser). A small portion of the FW-SFG range ($\nu ' < 0$) is also shown, where one sees that the signal is strongly suppressed (as mentioned above and discussed in detail below). Hereon in this section, we will drop the dash and write the DF frequency as $\nu$ to improve readability, although in later sections (beginning in Sec. 3.3 to the end) $\nu =\nu _S-2\nu _0$ will carry a sign to describe both FW-SFG ($\nu >0$) and FW-DFG ($\nu =-\nu '<0$). In order to extract spectral and temporal information from the spectrogram, we employ standard spectrogram analysis methods [30], where we first neglect propagation effects (treated below) and assume an ideal spectrogram signal (Appendix 5.1). To estimate the THz-IR intensity spectrum $I(\nu )$, we calculate the spectral marginal $M(\nu )=\smallint \mathrm {d}\tau \, S_T(\nu,\tau )$ (integration over $(-\infty,\infty )$ implicit), which as derived in Eq. (9) of Appendix 5.1 yields $M(\nu )\propto I(\nu )\ast G(\nu )$, i.e. the spectrum convolved with the detection gate spectrum $G(\nu )=|a(\nu )|^2$ where $a=E_1^2$ (here $G(\nu )$ has a FWHM of 2.6 ${\mathrm {THz}}$, as determined from the EFISH spectrum in Sec. 3.2 below). The spectral marginal $M(\nu )$ is shown in Fig. 1(d) (to the left of the spectrogram), and demonstrates a reasonably continuous THz-IR spectrum with a 10%-bandwidth of ${\sim }100\;{\mathrm {THz}}$, reaching the noise floor near 150 ${\mathrm {THz}}$. The corresponding temporal marginal $M(\tau )=\smallint \mathrm {d}\nu \, S_T(\nu,\tau )$ is also shown below the spectrogram, and as expected reflects only the temporal profile of the optical detection pulse $E_1$. However, as employed previously [30], we can also extract the temporal first moment ${\bar \tau }(\nu )=\smallint \mathrm {d}\tau \, \tau \cdot S_T(\nu,\tau )$ from the spectrogram, which one can show (Eq. (11)) yields ${\bar \tau }(\nu ) = \langle {T_{\mathrm {g}}}(\nu )\rangle$, i.e. the spectral group delay ${T_{\mathrm {g}}}(\Omega )=-\partial _\Omega \varphi (\Omega )$ ($\Omega =2\pi \nu$) convolved with $G$. In Fig. 1(d), we plot ${\bar \tau }$ superposed on the spectrogram, which is seen to vary by less than 20 ${\mathrm {fs}}$ across the full bandwidth, with small dispersive deviations due to residual H$_2$O/CO$_2$ absorption lines, mainly the H$_2$O bending mode at 45 THz (1500 cm$^{-1}$) and CO$_2$ asymmetric stretch mode at 70 THz (2300 cm$^{-1}$), also seen in $M(\nu )$. This weak dispersion is mainly due to the thin (50-$\mathrm{\mu}$m) Si wafer in the beam (Sec. 2), where one can calculate a group dispersive delay of 180 fs mm$^{-1}$ [38] over the range 0-100 ${\mathrm {THz}}$ (despite the absence of dispersion from allowed-phonon contributions – note that Ge has an even higher dispersion).

In order to demonstrate the sensitivity of ${\bar \tau }$, we performed transmission measurements on a thin (15-$\mathrm{\mu}$m) polypropylene (PP) membrane at the sample focus: the reference and sample spectrograms are shown in Fig. 2(a) and (b), respectively. The relative intensity transmission $T(\nu )=M(\nu )/{M_{\mathrm {ref}}}(\nu )$ and group delay $\Delta {T_{\mathrm {g}}}(\nu )={\bar \tau }(\nu )-{{\bar \tau }_{\mathrm {ref}}}(\nu )$ extracted from the spectrograms are shown in Fig. 2(c), along with model results calculated using literature data for the PP vibrational absorption lines [39] (specifically, CH$_{2,3}$ bending modes at ${\sim }40$-$45\;{\mathrm {THz}}$ and stretching modes at ${\sim }85$-$90{\,{\mathrm {THz}}}$) and accounting for the 2.6-THz frequency resolution. As can be seen, the spectra estimated from the spectrogram analysis are quite faithful to the model data. In particular, the expected etalon interference effects are clearly resolved in both $T(\nu )$ and $\Delta {T_{\mathrm {g}}}(\nu )$, the latter demonstrating a relative sensitivity below 10 fs. Note that the model calculations are based on a simple Lorentzian-band model of the main absorption lines with a constant background refractive index $n_{\infty }=1.45$ of PP, such that some of the residual deviations seen in Fig. 2(c) are due to neglecting weaker bands (moreover, the truncation of $\Delta {T_{\mathrm {g}}}$ about the stretching modes arises due to the finite $t$-scan range of the spectrograms).

 

Fig. 2. (a) Reference spectrogram, and (b) spectrogram with 15-$\mu$m-thick polypropylene film in sample position. Additional chirp for reference/sample (compared to Fig. 1(d)) due to the inclusion of an additional 525-$\mu$m-thick high-resistivity silicon wafer in THz-IR beam path for these measurements. (c) Corresponding intensity transmission and group delay of polypropylene sample extracted from $M(\nu )$ and ${\bar \tau }(\nu )$ of the reference/sample spectrograms (solid curves), and model data using literature IR absorption line data [39].

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Having established that ${\bar \tau }$ provides a reliable estimator of ${T_{\mathrm {g}}}$, and hence also the relative spectral phase via $\varphi (\Omega )=-\smallint ^{\Omega }\mathrm {d}\Omega '\,{T_{\mathrm {g}}}(\Omega ')$, a determination of the low-frequency absolute phase is sufficient to the absolute phase $\Phi (\nu )$ for the full THz-IR range [19].

3.2 Field and phase calibration using EFISH/ABCD

We now proceed to use auxiliary EFISH/ABCD measurements to determine the absolute phase and temporal field of the THz-IR pulses, at first without the correction for propagation effects which will be treated in the next section. While the EFISH signals are typically only measured in a spectrally integrated way via their interferometric contribution to an ABCD signal in conventional THz time-domain detection (with alternating bias field polarity $\pm \cal {E}_{\textrm {b}}$), one can also couple the EFISH signal to the spectrometer as per the FW-DFG signal. This is demonstrated in Fig. 3, which shows spectrograms for (a) FW-DFG alone ($S_T(\nu,\tau )$) and (b) with the addition of the EFISH signal ($S_{TE}(\nu,\tau )$). For the latter, we use a 1-kHz frequency for the sinusoidal bias field, such that $\cal {E}_{\textrm {b}}$ is unipolar for each laser shot, and the optical phase of the EFISH signal is constant (and not alternating, as for ABCD). As expected from the principle of ABCD, given that the two TFISH/EFISH signal fields, $b_T$ and $b_E$ respectively, are coherent (see Appendix 5.1 for their mathematical form), they exhibit interference depending on their relative phase, as can be scrutinized by forming the residual $\Delta S_{TE}=S_{TE} - S_{T} - S_{E}$ ($S_{E}$ clearly being independent of $\tau$). The resulting spectrogram $\Delta S_{TE}(\nu,\tau )$ is shown in Fig. 3(d) (with corresponding marginals included in Fig. 3(c), where we scale $M_E = N_\tau \cdot S_{E}$ (where $N_\tau$ is the number of delays in each spectrogram), which shows that the interference is constructive for the lowest THz frequencies (for the laboratory direction of EFISH bias chosen). In principle, this measurement is sufficient to deduce the absolute phase $\Phi$ of the low-frequency range of the THz-IR pulses, seeing as (neglecting the spectral convolution with the optical detection gate)

However, to circumvent some of the issues with the spectral convolution, we adopt here the same approach as in [19], and use an auxiliary ABCD measurement of $\cal {E}_{\textrm {THz}}$, as shown in Fig. 3(e), with the corresponding spectral intensity and phase in (f), which remain above the noise floor out to ${\sim }5{\,{\mathrm {THz}}}$. Here we employ the extreme frequency range of the ABCD spectral phase for matching to that extracted from the FW-DFG spectrogram (as shown in Fig. 3(f)), to minimize the impact of spectral convolution for the latter, where satisfactory phase overlap is achieved.

 

Fig. 3. Measured spectrograms with (a) FW-DFG alone ($S_T(\nu,\tau )$), and (b) both FW-DFG and EFISH $S_{TE}(\nu,\tau )$ (the latter with uni-polar bias $\cal {E}_{\textrm {b}}$), focusing on low-frequency range (see Fig. 1(d) for full spectral range for reference pulse). (c) Corresponding frequency marginals and that from (d) interferometric residual $\Delta S=S_{TE}-S_T-S_E$. (e) Low-frequency ABCD time-domain field for the same pulses and (f) corresponding spectral intensity and phase. Also included in (f) is the phase obtained from integrating the group delay ${\bar \tau }(\nu )$ extracted from the spectrogram (see Sec. 5.1) after adjusting to the absolute ABCD phase in the interval shown.

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Having fixed the absolute phase of the THz-IR pulses, it only remains to calibrate the absolute field strength. As detailed in Appendix 5.1, if one simply considers only the on-axis fields on the focal plane (e.g. assumed in [27]), one arrives at the intuitive normalization condition for the recovered field (Eq. (8)):

3.3 Accounting for beam profile and propagation effects

The role of spatial effects for the co-propagating THz and optical fields on the resultant TFISH and EFISH signals has been considered in detail in previous reports, e.g. [24–26,33] for conventional THz bandwidths. Notably, the effect of the Gouy phase on the axial evolution of the nonlinear signal source ${P^{(3)}}(\nu )\propto a(\nu -\nu '){\mathcal {E}}(\nu ')$ ($a=E_1$) and FISH signal $E_2(\nu )$ leads to a suppression of the FW-SFG signal ($\nu >0$) and saturation of the FW-DFG signal ($\nu <0$). While the Gouy-phase effect is not explicitly considered in many studies with ABCD detection (as it is assumed that the relative time-domain signal is not affected), one sees this effect immediately when spectrally resolving the TFISH signal. We build here on those previous treatments, extended to apply to the present case with an ultra-broadband THz-IR pulse (where quasi-DC TFISH treatments no longer hold). However, the propagation effects are in general more complex than the basic result (integrating the axial fields along $z$ with the Gouy phase variation [33]) due to (i) a deviation of the THz-IR beam from a Gaussian spatial form and its frequency dependence [29], and (ii) effects of dispersive phase mismatch, which were considered when assessing different gas media and pressures [24,33] and were also more recently shown to be significant for low-frequency TFISH detection [40].

The theoretical basis is covered in detail in Appendix 5.2, which shows that the “raw” THz-IR field determined from the spectrogram and applying the EFISH calibration (Eq. (1)) must still be corrected for both the EFISH electrode geometry [25,33] (Eq. (18)) and the evolution of the on-axis optical and THz-IR fields throughout the focal region [26], via the relative responsivity factor $H(\nu )$ (Eq. (20)). The latter would appear to necessitate measurements of the THz-IR beam profile throughout the focal region [41], although to perform this with spectral resolution over the full THz-IR bandwidth would indeed be a demanding task.

However, here we adopt a novel, alternate approach, by measuring the FW-DFG spectra $S_T(\nu,\tau =0, R)$ with a variable aperture (radius $R$) in the collimated beam before focusing (see scheme in Fig. 4). As derived in Appendix 5.2 (Eq. (23)), under the assumption of a radially symmetric beam intensity and linear polarization (reasonable assumptions for our THz-IR source [29]), one can deduce the responsivity correction $P(\nu )$ for the FW-DFG signal due to propagation. This highly useful result is facilitated by the Fourier optics relation between the collimated beam field ${\mathcal {E}}_C(r,\nu )$ (a distance $f$ before the focusing OAPM whose focal length is also $f$), the focal-plane field ${\mathcal {E}}(r,z=0,\nu )$, and the axial field ${\mathcal {E}}(r=0,z,\nu )$ in the relevant vicinity of the focal plane. That such an analysis allows one to connect the FW-DFG signal to the collimated beam field in closed-form is not immediately obvious, given that the variable, radial beam truncation simultaneously changes the numerical aperture and circular portion of the transmitted THz-IR beam being focused, but one can show that these two effects essentially factor into two terms in the integral transformation (Eq. (23)).

 

Fig. 4. Beam profile results from measurements with variable aperture in collimated THz-IR beam path. (a) Experimental and (b) fitted FW-DFG spectra $S_T(\nu,\tau =0,R)$ vs. iris radius $R$, the latter based on the collimated radial beam profile $I_C(r,\nu )$ in (c) (plotted here normalized for each frequency) – see Eq. (2) for model function for ${\mathcal {E}}_C(\nu,R)$ and Appendix 5.2 (Eqs. (23) and (24)) for the integral transformation between ${\mathcal {E}}_C(\nu,R)$ and $S_T(\nu,R)$. Schematic of experimental geometry shown left (${\mathcal {E}}(z,\nu )$ denotes the axial THz-IR field about the focal point, $b_T(z,\nu )$ the FISH signal field, and ${\mathcal {E}}_C(r,\nu )$ the field of the collimated THz/IR beam).

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The experimental results are shown in Fig. 4(a). While the approach described in Appendix 5.2 can in principle be applied to an arbitrary beam profile ${\mathcal {E}}_C(r,\nu )$, we find the data can be modeled well assuming a conical beam

Besides the beam-shape propagation effects treated above (which include the influence of the Gouy phase), as apparent in Eqs. (18,19,23), the dispersive wavevector mismatch ${\Delta k}$ (Eq. (15)) in air also adds an additional factor to the FISH responsivity functions [25,33,40]. While often this is neglected, due to the notion that at ambient pressure the weak dispersion of air leads to NIR-UV phase walk-off lengths on the few-cm scale (compared to mm-scale Rayleigh ranges), we will see that at least for the THz range, it is not negligible for our THz-IR beam focal parameters. That such effects must be considered was again emphasized in a recent report with a weakly focused THz beam, which showed severe effects due to ${\Delta k}$, even well below ambient pressure [40]. The calculated values ${\Delta k}$ (for $\lambda _0=775$ nm here) are shown in Fig. 5(a) for ambient air, and both dry and humid air (using the full set of THz-IR absorption bands for H$_2$O and CO$_2$ [43] and electronic response of N$_2$, O$_2$, Ar and H$_2$O from [44]). One sees that the mismatch is dominated by the residual electronic dispersion, and is asymmetric between FW-DFG and FW-SFG. The latter is as expected, as the DFG process involves wavelengths on the red side of $\lambda _2=\lambda _0/2$ which are closer to $\lambda _0$ such that ${\Delta k}$ is progressively smaller for increasing $\nu '=-\nu$ (one notes that this is a fortuitous situation for FISH detection, given that only FW-DFG survives the Gouy-phase effects). To consider the magnitudes more readily, the right scale in Fig. 5(a) shows the corresponding coherence length $L_c=1/{\Delta k}$ (note this is defined differently than the typical expression $L_c=\pi /{\Delta k}$ for plane-wave nonlinear generation, as its combination here with the beam $z$-dependence yields an exponential (and not sinc-function) dependence, see e.g. Eq. (17)). One sees that $L_c \approx 7$ mm for THz frequencies, growing to $L_c \approx 13$ mm at $\nu '=100{\,{\mathrm {THz}}}$. We can compare these values with the predicted axial-field $z$-dependence obtained from the THz-IR beam analysis above, as shown in Fig. 5(b). Here one sees that especially in the range below $10{\,{\mathrm {THz}}}$, the tails of the axial fields do persist on the many-mm scale and one should evaluate phase-matching factors to inspect how significantly they affect the FW-DFG signal evolution. Note that the non-monotonic development of the curves in Fig. 5(b) around 10 THz is linked to the characteristic shape of $I_C(r,\nu )$ shown in Fig. 4(c), where the intensity of the collimated radiation is found to be concentrated closer to the beam axis than at lower and higher frequencies. We also note that axial fields actually decay here more rapidly with $|z|$ than for a Gaussian beam due to the conical beam shape, which actually reduces the impact of phase mismatch.

 

Fig. 5. (a) Calculated phase mismatch ${\Delta k}$ vs. frequency, due to dispersion of air (for two values of relative humidity ${c_{\mathrm {H_2O}}}$); corresponding $1/e$-coherence length scale at right. (b) Normalized on-axis field amplitude vs distance about focal plane based on the fitted THz-IR beam parameters from Fig. 4. (c) Corresponding field responsivity factor $H(\nu ')$ for FW-DFG (blue solid curve), relative to EFISH for geometry used here. Dashed curves: responsivity factors due to dispersive phase mismatch alone. (d) Recovered temporal electric field in focal plane: raw result using simple focal-plane treatment (dashed) and after correcting for relative FW-DFG responsivity in (c) due to propagation effects.

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In Fig. 5(c), we plot the total FW-DFG field responsivity spectrum $H(\nu ')$ (including both beam propagation and mismatch), relative to the EFISH responsivity (from Eq. (18), plotted as a horizontal line). As can be seen, this exceeds unity, reflecting the fact that the effective interaction length for FW-DFG is larger than for EFISH with our experimental beam parameters. Also plotted are the relative contributions due to phase mismatch alone (dashed curves), calculated by taking the ratio of $H(\nu )$ with ${\Delta k}\neq 0$ and ${\Delta k}=0$, respectively. One sees here that ${\Delta k}$ does significantly degrade the responsivity below ${\sim }10{\,{\mathrm {THz}}}$, falling to ${\sim }0.5$ for $\nu '\rightarrow 0$ (while for our electrode geometry and optical focusing, ${\Delta k}$ only reduces the relative EFISH field responsivity by a small factor 0.05).

We can now apply this relative responsivity function to the THz-IR spectral field determined from the spectrogram/ABCD analysis in the last section. The corresponding temporal electric fields ${\mathcal {E}}(t)$ (from both the focal-plane calibration and after responsivity correction) are shown in Fig. 5(d), where one sees an essentially single-cycle pulse (blue curve) with a small residual positive chirp. While the duration of the main half-cycle is $9.5{\,{\mathrm {fs}}}$ (reflecting the mean frequency of ${\sim }50{\,{\mathrm {THz}}}$), a more objective measure of the pulse duration is the FWHM of the field envelope ($\hat {{\mathcal {E}}}(t)$, also included in the figure) which is $17.0{\,{\mathrm {fs}}}$, while the corresponding intensity FWHM based on $I(t)=|\hat {{\mathcal {E}}}(t)|^2$ is $10.1{\,{\mathrm {fs}}}$. Importantly, as expected from the responsivity spectrum, the corrected peak field is ${\mathcal {E}}_0 = 1.2{\,\mathrm {MVcm}^{-1}}$, compared to $2.1{\,\mathrm {MVcm}^{-1}}$ from the simple focal-plane calibration. Hence for our present geometry, one would significantly overestimate the field using the simplified approach. Note that ${\mathcal {E}}_0$ is not corrected for the loss/dispersion of the thin Si wafer in the beam (field transmission $0.7$), such that higher fields are potentially available at the sample location. Moreover, from the THz-IR beam results in Fig. 4(c), we see that the numerical aperture (and hence ${\mathcal {E}}_0$) could be increased significantly, by (i) increasing the collimated beam diameter (which is well below 10 mm across most of the THz-IR spectrum of Fig. 4(c)) using a longer focal length for the initial collimation-OAPM after the plasma, and (ii) using a tighter re-focusing than used here ($f=76.2$ mm). Hence, for this THz-IR source, peak fields approaching 10${\,\mathrm {MVcm}^{-1}}$ could be achieved.

To further elucidate the above results, it is instructive to inspect how the propagation effects are predicted to develop vs. distance through the focal region. In Fig. 6(a), we plot the FW-DFG/SFG intensity spectrum for a set of $z$-positions, using the experimental spectrum and beam parameters from above, i.e. based on Eq. (14), where we also include the results for ${\Delta k}=0$. One sees that for ${\Delta k}=0$ (dashed curves), while FW-DFG and SFG initially grow symmetrically before reaching the focal plane ($z<0$), for the FW-SFG process the destructive Gouy-phase effects already set in upon reaching the focal plane ($z=0$), with the signal vanishing with increasing $z>0$. The phase mismatch for FW-SFG exacerbates these effects, explaining why the FW-SFG signal is so effectively suppressed experimentally, even if one considers a loss of focal symmetry in practice, e.g. due to non-ideal beam shape/wavefront and collinearity. For the FW-DFG process, one sees again that the phase mismatch is only significant at lower DF frequencies $\nu '=-\nu$, which further emphasizes how the situation here differs from conventional plane-wave THz phase-matching (where the THz dispersion plays the dominant role in ${\Delta k}$ and typically increases for higher frequencies due to the THz-wavelength scaling, ${\Delta k}\propto \nu$, and typical THz dispersion).

 

Fig. 6. (a) Simulated evolution of the FW-DFG/SFG signal intensity spectrum for selected positions $z$ relative to focal plane. Results are based on the experimental THz-IR spectrum and beam parameters determined above, and include both results with (solid curves, ${\Delta k}\neq 0$) and without (dotted curves, ${\Delta k}=0$) dispersive phase mismatch. (b,c) Focal spot diameter (top), Rayleigh range (middle) and relative field responsivity (bottom, as per Fig. 5(c)) for two experimental scenarios with a Gaussian THz-IR beam and frequency-independent collimated beam diameter $2w_C$: (b) $2w_C=25$ mm, $f=76.2$ mm, $f_1=125$ mm; (c) $2w_C=15$ mm, $f=152.4$ mm, $f_1=250$ mm.

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Finally, we consider some auxiliary model predictions for the propagation effects for other beam/focal scenarios in Fig. 6(b,c). For simplicity, we assume a Gaussian (not conical) collimated/focal beam profile, with a frequency-independent collimated beam diameter $2w_C$ (as one would obtain e.g. with a large-area THz emitter [45]). In each case, we plot the focal spot diameters $2w_{0,1}$ for the THz-IR and optical beams, respectively, the corresponding Rayleigh ranges $z_{0,1}$ and resultant field responsivity (as per Fig. 5(c), normalized to the EFISH signal, and including the relative contribution due to dispersive mismatch alone). We assume the same geometry for the EFISH electrodes as above. The first scenario in Fig. 6(b) employs the same focal lengths $f,f_1$ and input optical beam diameter as the experiments presented above, but with a larger THz-IR beam with $2w_C=25$ mm. One sees that while the FW-DFG responsivity below 10 THz rises above unity, for higher frequencies it falls significantly below this. One can see why this is the case by inspecting the Rayleigh range $z_0(\nu ')$, which becomes increasingly shorter for larger $\nu '$, such that the effective interaction length becomes much less than the EFISH electrode length. Clearly, in such a scenario, the focal-plane calibration would seriously underestimate the IR spectral range. For our experimental results above, such a trend is countered by the fact that the collimated beam diameter decreases with increasing frequency (Fig. 4(c)), due to the physical emission mechanisms in the plasma [42]. In Fig. 6(c), we inspect a scenario with much weaker focusing, in order to be close to the experimental conditions in [40], i.e. both $f,f_1$ a factor 2 higher and a moderate collimated THz beam size $2w_C=15$ mm. We also adopt the larger input optical beam used in [40] with a 10-mm diameter, such that $z_1$ is actually slightly smaller than the last scenario. We see that the THz responsivity is now below unity, while the IR responsivity is above for the whole range shown. However, our main interest in this scenario is due to the result in [40] that dispersive mismatch seriously degrades the TFISH conversion efficiency (for a low-bandwidth LiNbO$_3$ THz source). As seen, the responsivity in this range (<3 THz) falls to very low values, due to the long Rayleigh range of the THz beam whereby the phase mismatch effects become highly destructive (which was studied and circumvented in [40] by terminating the THz co-propagation with an iris in the focal region). As demonstrated above, the dispersive mismatch effects are much less significant for our THz-IR source and focal parameters (Fig. 5(c)), although clearly one should bear dispersive mismatch effects in mind when designing the focal geometry.

4. Conclusion

We have demonstrated an approach to reconstruct the calibrated, temporal electric field of ultra-broadband THz-IR pulses using FISH in ambient air. Here we have used longer, transform-limited optical detection pulses than in previous reports [19] and extracted the intensity spectrum and relative phase directly from the FW-DFG spectrogram (without deconvolution), which has some practical advantages in certain contexts, e.g. allowing simple single-shot detection of the intensity spectrum via $S_T(\nu ',\tau =0)$ with the same configuration, albeit with a reduced spectral resolution (although in many cases this resolution is sufficient). While not employed here, deconvolution of the full spectrograms would still be possible to recover superior resolution [28]. Extending on the THz-range studies of propagation effects for the relative FW-DFG and EFISH signals, we have shown that one must account for these (in our case, to avoid overestimating the field strength), and we have presented a formalism and technique for estimating these with auxiliary measurements with variable radial truncation of the collimated beam. This technique could also be adapted to account for propagation effects in ABCD measurements on conventional THz sources, in order to correctly calibrate the peak temporal fields.

The approach here again emphasizes how FISH detection in ambient gases is admittedly prone to an “uncontrolled” effective interaction length, and in this sense detection in thin sensor materials (such as EO time-domain field sampling with careful calibration of the EO response [21]) allows a more direct determination of the focal electric field. However, as discussed in the Introduction, for such high THz-IR bandwidths this requires very short optical detection pulses, which may not be available. Nevertheless, it is evident that the field calibration would be simplified if the ${\chi ^{(3)}}$-interaction would be confined to a thin sensor medium. One could consider a narrow gas jet [24], although as shown here, any vacuum cell windows would have to be chosen carefully not to cause severe dispersion/absorption effects. One very interesting approach would be to use FISH in solid-state ${\chi ^{(3)}}$-media [46,47], although again, even for very thin media, dispersion/absorption effects will need to be addressed. Hence, the use of a truly surface-based ${\chi ^{(3)}}$-effect (integrating an optional bias field for the EFISH signal, and using tight focusing) may have the most potential in future approaches.