1. Introduction

Femtosecond gas-based plasma is a broadband source of terahertz (THz) radiation [1–3] with the spectrum spanning from a few GHz [4] to tens of THz [5–8]. The wavelength of the emitted radiation exceeds or is of order of the transverse size of the plasma source. The divergence of THz radiation is therefore quite large and strongly different across the broadband THz spectrum. As a femtosecond filament forms continuous and longitudinally extended plasma channel [9–13] (except for the case of the microplasma [14–16]), the THz radiation divergence angle $\Delta \theta$ [17–19] obeys in general the law well-known from radio antennas [20]:

There are three commonly used schemes for THz generation in a femtosecond gas-based plasma. (i) A femtosecond single-color plasma channel, further also referred to as unbiased filament, emits radially polarized [17] THz wave into a cone due to the plasma column oscillations. These oscillations occur along the filament axis $z$ [15,18] due to the light pressure force. The dipole moment forms along $z$-axis and produces the corresponding dipole local source of THz radiation. Besides, the filament plasma column can undergo axially symmetrical expansion-contraction in the plane perpendicular to the laser pulse propagation direction. The suggested reason for these oscillations is the ponderomotive force yielding quadrupole moment and creating the corresponding quadrupole local source of THz radiation [19,21].

In a femtosecond plasma channel transverse dipole moment cannot be induced due to isotropy of gaseous media. The isotropy break is implemented by introducing either (ii) external electric field bias [3] or (iii) the seed pulse at another wavelength [2,22]. The resulting plasma oscillations appear orthogonally to the propagation axis $z$, and the local THz source is the transverse dipole producing linearly polarized THz emission [23,24]. Out of these three schemes, the two-color filamentation (iii) is the most efficient in terms of THz energy yield [25]. Symmetry breaking by multi-frequency mixing is well elaborated for different pump wavelength [22,26–29]. The coherent THz photocurrent, driven by the overlapped optical pulses, is sensitive to the intra-pulse ionization dynamics [30]. On the contrary, THz emission of single-color unbiased (i) and DC-biased (ii) filament is determined mostly by the plasma dynamics after the pulse is gone [18]. Experimentally, the advantage of THz generation from a single-color filament is the absence of the second harmonic pulse and the demanding adjustment of overlap between harmonics [31,32]. If a single-color filament is biased by a $\sim$10-kV/cm field, the THz yield is higher than in the unbiased case by an order of magnitude at least [3,33].

The far-field distribution of THz radiation changes from the conical one for the unbiased filament [17] to the on-axis one if a DC bias is applied [23]. The low static field of 0.5 kV/cm resulted in the angular distribution asymmetry followed from the interference of radially polarized emission of unbiased filament and linearly polarized dipole emission due to DC bias. The last signature of asymmetry was observed at 3 kV/cm. For higher field the dipole source dominated, and THz angular distribution became symmetrical. Similar interplay between the two mechanisms of THz emission from a filament was observed in $2D$ transverse profile measurements of THz beam under the static electric field of 1.75 kV/cm [34].

Competition between contributions to THz generation from unbiased and DC-biased plasma channels was studied in Ref. [35] by the time-domain measurements. The THz waveforms were recorded from the filament of 1.55-$\mu$m, 50-fs, 0.5-mJ laser pulse biased by electrostatic field varied in the range of 0–10 kV/cm. Without DC bias a single spectral maximum at 1.55 THz was observed. The DC bias led to the appearance of additional, lower-frequency spectral maximum at 0.64 THz. The spectral intensities of the two separate spectral maxima became equal at the electic field of 0.3 kV/cm. This value was interpreted in [35] as the one providing the balance of emission from THz sources in unbiased and DC-biased femtosecond filaments.

In this work, we examine the question of the emission balance from THz sources in a single-color unbiased and DC-biased filament based on both spatially and spectrally resolved measurements of THz radiation. For a set of electrostatic field values and THz bandpass filters, we measured angular distribution of THz emission and decomposed it into the sum of quadrupole and dipole contributions from the plasma column oscillations in the same filament. THz emission from the two sources becomes equal at the field of $E_{eq}=(3.2 \pm 0.8)$ kV/cm.

2. Experiment

In the experiment [see the sketch of the setup in Fig. 1(a)] we used 10-Hz femtosecond laser system (Avesta ltd.) with the central wavelength of 744 nm, pulse duration 90 fs (FWHM) and energy $\sim$1 mJ. The beam with the diameter $\sim$3 mm at $e^{-1}$ fluence level was focused by a lens with the focal length of 20 cm into a $d=4.5$ mm gap between 15-mm-long electrodes biased by the voltage $U_{DC}$ varied from zero to 1500 V. A femtosecond filament was formed between the electrodes. We registered the THz radiation from the filament plasma channel by superconducting hot-electron NbN bolometer (Scontel), sensitive to the radiation in the frequency range 0.1–3 THz. To obtain the angular distributions of THz intensity $I(\theta )$ at different frequencies $\nu$, we rotated the bolometer in the plane of the electrodes by the angle $\theta$ relative to the laser beam axis and screened the bolometer entrance window by bandpass filters centered at $\nu =0.1$, 0.3, 0.5 and 1 THz (the same as in Ref. [36]). The sensitivity of the bolometer at the frequencies studied as well as the spectral width at the level of 1/2 and the peak transmittance of the filters are shown in Table 1. The distance between the electrodes and the bolometer entrance window was 50 cm.

 

Fig. 1. (a) Schematic representation of experimental geometry. The DC bias $E_{DC}$ is parallel to the $y$-axis. Bolometer (blue detector) was rotated by the angle $\theta$ relatively to the $z$-axis in the plane of the electrodes $(x,z)$. $3D$ directional diagrams of (b) the dipole THz local source in the filament biased by external electrostatic field $E_{DC}$, (c) the quadrupole THz local source and (d) cross sections of the diagrams (b) and (c) in the observation plane $(x,z)$. Solid and dashed black semicircles in (d) represent the dipole directional diagrams when the dipole contribution to THz local source dominates or falls below the quadrupole one. The coordinate system $(x,y,z)$ is the same for panels (a)–(c).

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Table 1. Sensitivity of bolometer $\alpha$; maximal transmittance $T$ and FWHM width $\Delta \nu$ of bandpass filters.

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The measured angular distributions of THz emission are shown by symbols in Fig. 2. Experimental data in Fig. 2 are corrected for the bolometer sensitivity $\alpha$ as well as the transmittance of the filters $T$ and their spectral width $\Delta \nu$ (Table 1). In each row of Fig. 2 the THz bandpass filter is the same, while the applied voltage $U_{DC}$ increases from left, where $U_{DC}=500$ V, to right, where $U_{DC}=1500$ V. The corresponding electric field $E_{DC}=U_{DC}/d$ varies from 1.1 to 3.3 kV/cm. For any applied voltages and any frequencies, the variation in the polarity of the external bias did not influence the THz angular distributions significantly.

 

Fig. 2. Experimental (symbols) and simulated (solid curves) angular distributions of THz intensity for different external field (a–c) and bandpass filters (1, 0.5, 0.3, and 0.1 THz, from top to bottom row, respectively). The simulated angular distributions (solid curves) fit the experimentally measured ones (symbols) according to Eq. (3) with the filament length preserved at $L = 5.73$ mm and the dipole contribution $\alpha$ varied. Simulated curves are renormalized to the maximal value of the experimentally obtained angular distributions. The two grey horizontal lines correspond to the maximal value of the measured distributions and one half of this maximal value.

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The frequency of THz radiation increases from bottom to top in each column from (a) to (c) of Fig. 2. Since the pulse duration, energy and central wavelength as well as the lens and its position in the beam remained the same throughout the experiment, the filament and its plasma channel were fully reproduced for each position of the bolometer and each bandpass filter studied. In such conditions, the plasma channel length was considered as the constant one. The clearly observable effect of THz angular distributions $I(\theta )$ getting narrower with increasing frequency was estimated using Eq. (1) with a constant plasma length $L$. As a bandpass filter frequency increases by a factor of 10 from 0.1 to 1 THz, the full angular width at half maximum $\Delta \theta$ of THz distributions in Fig. 2 decreases by a factor of 3, from $\Delta \theta \approx 100^{\circ }$ to $\Delta \theta \approx 30^{\circ }$, respectively.

In addition to the decrease in the angular width, the THz signal also decreases monotonically with increasing bandpass filter frequency from $\nu =0.1$ to 1 THz (Fig. 2). Here we note that several earlier observations of THz spectrum from single-color filament using electrooptics sampling technique (EOS, see Ref. [37]) reported the local maximum in THz spectrum within the frequency range 0.5–2 THz [3,35,38,39] including the experiments, where the filament geometry is essentially different from ours [14,40]. We associate this disagreement between the monotonic spectral intensity decrease in our experiment, where the bandpass filters were used, and the appearance of the local maximum in several mentioned above experiments [3,14,35,38–40] with specific properties of EOS detection. The suggested explanation is that in order to obtain the THz waveform by EOS, one has to focus the THz beam onto the electrooptics crystal, like ZnTe, and overlap it with the probe femtosecond pulse with $d_{EOS}\approx 1$ mm diameter [16,41]. The different spectral components of the broadband THz pulse are focused into the spots with different diameter [42,43], which is of order of a THz wavelength $\lambda$ in the case of high-aperture collection. For $\lambda \gtrsim d_{EOS}$ (i. e. $\nu \lesssim 0.3$ THz), a significant fraction of the THz beam area, of about $1-d_{EOS}^{2}/\lambda ^{2}$, does not overlap with the probe femtosecond beam [41] and, therefore, does not induce birefringence of the probe. As the result, the THz registration technique based on EOS underestimates the low-frequency THz components. Consequently, formation of the local spectral intensity maximum at 0.5–2 THz becomes possible.

In the case of low applied voltage $U_{DC}=500$ V the pronounced ring-like THz angular distributions were observed at all the frequencies studied [Fig. 2(a)]. Such distributions are typical for conical forward THz emission from a single-color unbiased or weakly biased filament [17,34]. For the selected bandpass filter, i. e. if the frequency $\nu$ is kept constant, the increase in the applied voltage $U_{DC}$ from 500 to 1500 V results in the decrease of the ring contrast (see Fig. 2). For the largest voltage $U_{DC}=1500$ V applied to the electrodes in our experiment, the THz conical emission was transformed into on-axis forward emission. This can be followed in Fig. 2(c), where the local minimum at zero angle disappeared at $\nu =0.3$, 0.5, 1.0 THz. For the lowest frequency studied $\nu =0.1$ THz the ring did not vanish completely, however, the relative on-axis intensity increased by a factor of four (Fig. 2, the bottom row).

The experimentally observed transformation of conical THz emission to the on-axis one with the applied voltage growing in the range of 500–1500 V enables us to search for the electrostatic field value, which balances the THz source in the unbiased single-color filament if the external voltage is increasing.

3. Balance of contributions from quadrupole and dipole THz sources

Let the external electric field $E_{DC}$ be directed along the $y$-axis [Fig. 1(a)]. The laser pulse propagated along the $z$-axis and formed the plasma filament between the electrodes. The detector was rotated in the plane $(x,z)$ orthogonal to the external electric field $E_{DC}$ [Fig. 1(a)].

Whether an external DC bias is applied or not, a plasma channel of a single-color filament emits THz component which is radially polarized and propagates into a cone [17]. The circular distribution of THz radiation in the far field can be modulated as the azimuth angle changes [44]. The suggested physical mechanisms defining the local source of THz emission from an unbiased plasma channel might be either dipole oscillations along the laser beam propagation direction [18,45] or radial quadrupole oscillations [19,21]. These two mechanisms are indistinguishable in the far zone for the moderate focusing conditions. Both longitudinal dipole and quadrupole local sources were used in the simulations to fit the measured THz angular distributions in Ref. [15].

In our experiment the numerical aperture was moderate, about 0.01. The transition from conical to unimodal angular distribution with increasing external electric field [Fig. 2] was observed in the far zone of THz diffraction (about 50 cm from the filament). In our search for the balancing electric field we assume that each short part of the plasma channel $\mathrm {d}z$ produces radial quadrupole oscillations independent of the presence or absence of the biasing field. Quadrupole local source emits the field proportional to $\sin 2\theta$, see $3D$ directional diagram in Fig. 1(c) and its cross section in the $(x,z)$ plane in Fig. 1(d), green curve. Let us assume the relative contribution of quadrupole emission into the overall THz emission from the biased filament equal to unity.

The rotation of the bolometer takes place in the $(x,z)$ plane, in which the radially polarized quadrupole field has an $x$-component only. At the same time, the electrostatic field $E_{DC}$ is parallel to the $y$-axis, so the dipole emission from a local source $\mathrm {d}z$ does not depend on the angle $\theta$ and is $y$-polarized [Fig. 1(b) and black curves in Fig. 1(d)]. Therefore, a vector superposition of these sources should be considered [23,44]. According to this consideration, the THz field $\mathbf {E}(\theta )$ emitted by a small region of the filament $\mathrm {d}z$ can be represented by the equation:

Since the THz waves originating from quadrupole and dipole mechanisms are orthogonally polarized [red vectors in Fig. 1(b), 1(c)], their intensities are added on the detector without the interference cross-term. We verified that the reverse in the DC bias polarity in our experiment did not influence the THz angular distributions. The DC-bias induced asymmetry in the THz divergence diagrams was observed in Ref. [23] due to the rotation of the detector in $(y,z)$ plane in the notations of Fig. 1. The geometry of the experiment [23] corresponds to the local THz source $\mathbf {E}(\theta ) \propto \mathbf {e}_y \sin {2\theta } + \mathbf {e}_y \alpha (E_{DC}, \nu )\cos {\theta }$, which provides the interference between the fields from DC-biased and unbiased local sources, and, therefore, the asymmetry in THz divergence diagrams.

To obtain the THz angular distribution in the far zone of diffraction, we used the interference model [19]. An interference model is a common tool in simulations of THz emission from femtosecond filament [18,19,46,47]. Any of such models simplifies the plasma distribution in the focal volume and assumes it to be a set of identical local THz emitters. The self-consistent (i. e. based on nonparaxial Unidirectional pulse propagation equation [48]) simulations of THz emission are elaborated for the case of two-color filamentation [29,49–53]. Self-consistent simulations of THz generation in the filament biased by the relatively high $\sim$10-kV/cm field are possible as well [54]. In the works [29,49–54], the unbiased source of THz radiation is negligible. In our present work it cannot be neglected leading to the axial symmetry and polarization uniformity breakup (see Appendix A). For this reason, the self-consistent simulations were not applied yet to the THz generation from unbiased or weakly-biased single-color filament.

In the interference models, the plasma density is assumed to be constant over the effective length $L$ of the macroscopic THz source. The THz distribution in the far zone is given by [19]:

We fit the measured THz angular distributions $I(\theta )$ by Eq. (3) using the least square method. First, we found the effective filament length $L$ by fitting the distribution $I(\theta )$ in the absence of the external DC field ($U_{DC}=0$, $E_{DC}=0$). The measured THz signal systematically decreased with the frequency increase for any applied voltage, including $U_{DC}=0$ (see Fig. 2 and Fig. 3). To provide the most reliable fit, we used the measured distribution $I(\theta )$ with the highest signal (and thus the highest signal-to-noise ratio), i.e. the one at $\nu =0.1$ THz [Fig. 3(a), circles]. Since $E_{DC}=0$, we assumed $\alpha =0$ in Eq. (2) and calculated the integral [Eq. 3] in the range of $L$ from 1 to 20 mm with the step of 10 $\mu$m. The length $L=5.73$ mm provides the minimum of the approximation error, see the best fit dependence $I(\theta )$ in Fig. 3(a). Such filament length $L$ is in agreement with the plasma length measured by the capacity probe in the same experimental setup [55]. The length $L=5.73$ mm provides an excellent agreement between the experiment and interference integral [Eq. 3] for the frequency $\nu =0.3$ THz, see Fig. 3(b). At higher frequencies, the THz signal from unbiased filament was barely detectable and too low for quantitative analysis.

 

Fig. 3. Angular distribution of THz radiation at (a) $\nu =0.1$ THz (red circles) and (b) $\nu =0.3$ THz (green squares) measured in the experiment without DC bias and their fits (solid curves) according to Eq. (3) with a fixed dipole contribution $\alpha =0$.

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From here on, the plasma channel length is fixed as $L=5.73$ mm for further analysis. The right-hand side of Eq. (3) with the locally emitted THz field [Eq. 2] is scaled linearly with regard to $\alpha ^{2}$:

As the on-axis intensity increases, the dipole contribution $\alpha$ to the THz source term increases as well. For all the angular distributions measured in the experiment and fit by Eq. (3), the dependence of $\alpha$ on the electric field $E_{DC}$ is summarized in Fig. 4, where the symbols correspond to the same frequencies $\nu$ as those in Fig. 2. The knowledge of the dipole and quadrupole contributions into the THz source allows us to reconstruct the $2D$ distribution of the THz polarization in the far zone, see Appendix A.

 

Fig. 4. The dipole contribution $\alpha$ for different frequencies shown by symbols corresponding to the ones in Fig. 2 in the dependence on the applied electric field strength $E_{DC}$ and its linear fit (grey line).

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The sensitivity of the dipole contribution $\alpha$ to the variation of THz frequency $\nu$ is weakly pronounced. In contrast, the linear trend in the dependence of $\alpha$ on the biasing field $E_{DC}$ is clearly revealed (Fig. 4, grey solid line). We fit the dependence of the dipole contribution $\alpha$ on the DC bias $E_{DC}$ by the linear function $\alpha (E_{DC})\propto E_{DC}$ over all the frequencies $\nu$ using the least square method. The intercept $\alpha (0)=0$ was fixed since in the absence of external DC field ($E_{DC}=0$) the local source is quadrupole only [see Eq. (2)]. The fitting line obeys the equation

This value agrees with the estimates that one can perform based on the data published in Refs. [23,34]. Indeed, the interplay between on-axis and conical THz emission was observed at 3 kV/cm [23] and 1.75 kV/cm [34]. At 5 kV/cm the symmetrical on-axis emission dominates [23]. So, the balance field is in the range of 1.75–5 kV/cm. The value of the balancing electrostatic field found in Ref. [35], where THz waveforms were measured by EOS, is an order of magnitude lower than our result of $E_{eq}$. We note, that time-domain EOS measurements are sensitive to the geometry of THz collection (Section 2 of this manuscript). For a narrow-cone on-axis collection (even $10^{\circ }$ can be considered as narrow for $\sim$1 THz) the balancing electrostatic field will be shifted towards zero due the absence of on-axis emission from the unbiased filament, from which the conical emission dominates [17]. For that reason the measurements performed in Ref. [35] can underestimate the balancing electrostatic field.

4. Conclusion

By rotating the bolometer with the input window screened by the bandpass filters, we measured the set of angular distributions of THz emission from the filament biased by the external DC field. Our experimental data are obtained for three varying quantities: 0.1–1 THz frequency range, $-60^{\circ }$ to $60^{\circ }$ rotation angle and 0–3.3 kV/cm external electric field (DC bias). The increase in the DC bias from 0 to 3.3 kV/cm results in the THz distribution transformation from the conical to the one with the on-axis maximum and without the rings in the far zone of diffraction. Each of the measured THz angular distributions was decomposed into the radially polarized quadrupole contribution and the linearly polarized dipole one. The contribution from the dipole radiation was found to be proportional to the DC bias and almost insensitive to the THz frequency. Based on the accumulated experimental data we quantitatively estimated the balancing electrostatic field as $(3.2\pm 0.8)$ kV/cm, at which the local THz source in the biased plasma channel contributes equally with THz source from the unbiased channel to the far-field THz radiation distribution.

Appendix A. Reconstruction of THz polarization in $2D$ angular domain

The polarization of THz radiation generated by a single-color unbiased filament was found to be radial [17]. With the DC bias increase up to $\sim$5 kV/cm the radial polarization is transformed to the linear one as it was shown in the experiment [57]. In the transition regime, which takes place at the external field values between 0 and 1 kV/cm, the emitted from a single-color plasma channel THz radiation exhibits elliptical polarization [57]. We attribute elliptical polarization of the THz emission from the unbiased or weakly biased filament with the high-aperture collection of THz radiation used in the experiment [57]. The polarization state of THz field varied across the beam, and the average polarization state was registered as the elliptical one.

The measurements of $2D$ transverse distribution of the laser beam polarization [58] were performed using the method described in Ref. [59]. In the THz part of spectrum, the registration of the polarization distribution across the beam was done by EOS technique [60,61]. The propagation of the optical or THz radiation in the experiments [58,60,61] was collimated. In our experiments, the THz radiation was emitted from several-millimeter long, $\sim$100-$\mu$m wide plasma channel. The divergence of THz radiation from such a source was up to $\Delta \theta \approx 100^{\circ }$ at low frequencies [see Fig. 2, Fig. 3]. THz field polarization measurement would produce a large error in this geometry. Therefore, we did not perform it. At the same time, the interference integral [Eq. 3] allows us to reproduce the THz field polarization distribution if one accounts both polar $\theta$ and azimuthal $\varphi$ angles in the source term [Eq. 2].

Figure 5 represents the transformation of $2D$ distribution of THz field (green arrows) with the increase in voltage $U_{DC}$ from 0 to 1500 V (corresponding to the DC bias ranging from 0 to 3.3 kV/cm) at the frequency of 0.3 THz. The THz intensity is shown by the grey color. The increase in voltage $U_{DC}$ results in the breakup of THz beam axial symmetry [cf. Fig. 5(a) to (c)] in agreement with the experiments [23,34]. For $U_{DC}=0$, the radial polarization of THz emission is clearly seen in Fig. 5(a). In the most intense part of THz beam emitted from the filament biased by the largest voltage $U_{DC}=1500$ V, the polarization tends to be the linear one [Fig. 5(c)]. This reproduces qualitatively the experimental results of Chen et al. [57].

 

Fig. 5. Simulated two-dimensional distributions of THz field (green arrows) and THz intensity (grey shading in the background) for the frequency of 0.3 THz and three values of $\alpha$ corresponding to the best fits in Fig. 2, third row. For visual purposes, only the main lobe is drawn. Side panels to (c) show the polarization (arrows) and intensity (red lines) that would be observed in vertical and horizontal sections through the center of two-dimensional diagram.

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Funding

Russian Science Foundation (21-49-00023); National Natural Science Foundation of China (12061131010).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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