Low-Frequency Content of THz Emission from Two-Color Femtosecond Filament

Abstract

We experimentally investigate the low-frequency (below 1 THz) spectral content of broadband terahertz (THz) emission from two-color femtosecond filament formed by the 2.7-mJ, 40-fs, $800+400$-nm pulse focused into air. For incoherent detection, we screened the Golay cell by the bandpass filters and measured the THz angular distributions at the selected frequencies $\nu =0.5$, 1, 2 and 3 THz. The measured distributions of THz fluence were integrated over the forward hemisphere taking into account the transmittance of the filters, thus providing the estimation of spectral power at the frequencies studied. The spectral power decreases monotonically with the frequency increasing from 0.5 to 3 THz, thus showing that the maximum of THz spectrum is attained at $\nu \le 0.5$ THz. The THz waveform measured by electro-optical sampling (EOS) based on ZnTe crystal and transformed into the spectral domain shows that there exists the local maximum of the THz spectral power at $\nu \approx 1$ THz. This disagrees with monotonic decrease of THz spectral power obtained from the filter-based measurements. We have introduced the correction to the spectral power reconstructed from EOS measurements. This correction takes into account different focal spot size for different THz frequencies contained in the broadband electromagnetic pulse. The corrected EOS spectral power is in semi-quantitative agreement with the one measured by a set of filters.

1. Introduction

Two-color femtosecond filament [1] with the intensity of ∼100 TW/cm${}^2$ [2,3,4], the beam diameter of 50–100 $\mathsf{\mu }$m [5,6,7], and the plasma density of ${10}^{15}$–${10}^{19}$ cm${}^{-3}$ [8,9] is a terahertz (THz) source with the most broadband spectrum up to 50–100 THz [10,11,12], depending on the pump pulse duration [13]. The high-frequency THz emission from two-color filament has been studied in details both experimentally and theoretically [13,14,15,16,17]. So, the angular distribution in the range of frequencies $\nu \gtrsim 5$ THz is conical [16,18,19,20,21], but the shape of directional diagrams at lower frequencies is still debated [22]. Indeed, the experiments [21,22,23] show the unimodal on-axis angular distribution of THz emission from two-color filament for $\nu \lesssim 3$ THz. However, in other experiments [24,25,26], the conical divergence of THz radiation was observed.

The spectral distribution of the low-frequency THz emission from a two-color filament is still unknown. Even the position of the low-frequency maximum ${\nu }_{\max }$ in the THz spectra measured in the close conditions (pulse energy and duration, geometry of the experiment etc.) can vary by an order of magnitude from one experiment to another. We collected the corresponding data in

Table 1. The frequency of maximum ${\nu }_{\max }$ in the THz spectra measured in different experiments in the conditions close to the ones of our experiment.

Pulse Duration, fs	Pulse Energy, mJ	Focusing Conditions	Detection Method	${\mathit{\nu }}_{\mathbf{max}}$, THz	Ref.
32	∼1	$f=15$ cm	EOS	2.9	[13]
32	0.7	$f=15$ cm	ABCD	4.4	[30]
32	0.7	—	ABCD	∼5	[31]
35	2	$f=25$ cm	EOS	0.4	[27]
35	∼3	$f/15$	Interferometer	13	[28]
40	0.7–2	$f=15$–25 cm	EOS	∼1	[32]
40	∼1	$f=15$ cm	EOS	2.3	[13]
40	2.3	$f=20$ cm	EOS (ZnTe) EOS (GaP) ABCD Interferometer	1.7 2.2 3.4 7	[33]
40	1–6	$f/10$	Interferometer	5–10	[34]
50	0.5	$f=30$ cm	EOS	0.4	[35]
50	1.15	$f/80$	EOS	∼0.5	[36]
50	2.5	$f/40$–$f/4$	EOS	0.5–0.7	[37]
50	1.7	$f=20$ cm	EOS	∼1	[38]
50	∼1	—	ABCD EOS	0.55 1.7	[31]

, which shows that as the pump pulse duration varies in the range of 30–50 fs, the position of the spectral maximum travels in a wide frequency range from 0.4 THz [27] to 13 THz [28]. Self-consistent numerical simulations based on Unidirectional pulse propagation Equation (UPPE [29]) predict the maximum of THz spectrum at the frequency of about a few hundreds of gigahertz [16,17].

This large spreading of experimentally obtained frequencies ${\nu }_{\max }$ can be attributed to the extreme broadness of the spectrum of THz emission from two-color filament. Different components of the THz spectrum have different divergence diagrams. As the result, the THz focal spot size varies significantly across the spectrum. This focal spot size variation affects the collection of THz radiation by coherent detection schemes, in particular, electro-optical sampling EOS [39] and air-biased coherent detection ABCD [40]. Both methods require overlap between the probe optical pulse and the focused THz pulse. The THz spot transverse size is of order of THz wavelength or larger, and, therefore, exceeds the typical ∼1 mm (or even less in case of ABCD detection) diameter of the probe beam [41,42]. For example, at the low THz frequency $\nu \lesssim 0.5$ THz with the corresponding wavelength larger than ∼600 $\mathsf{\mu }$m, the THz beam size in the focus is equal or larger than 1 mm. As a result, a significant fraction of the THz beam does not overlap with the probe, and the coherent time domain sampling (TDS) underestimates the low-frequency THz content [43].

It should be noted that this underestimation does not prevent the time-domain sampling from being an excellent tool in THz spectroscopy [44]. Indeed, through the comparison of the spectrum of THz radiation transmitted through the substance and the one of the reference THz signal, TDS provides the information about both real and imaginary parts of the dielectric susceptibility [39]. The low in the amplitude half-cycle electric field in a standard EOS crystal ZnTe (89 kV/cm for a 1 mm thickness [45]) makes the registration of a weak THz signal possible. Using EOS, Zhong et al. recorded the THz spectra of explosives from the distance of 30 m in the atmosphere [46]. The EOS scheme can be adapted for $2D$ frequency-resolved imaging of organic substances [47]. The ABCD technique was applied to broadband THz spectroscopy [48,49]. The dispersion of TDS sensitivity becomes a problem only if one decides to characterize the THz source rather than the sample.

The incoherent methods of THz spectra registration based on the Michelson interferometer coupled to a THz powermeter (a bolometer or a Golay cell) are aimed at the high-frequency THz content [33]. The THz spectra measured by these devices have the maximum at high THz frequencies ${\nu }_{\max }\ge 5$ THz, see Table 1. The strongly divergent low-frequency THz radiation diffracts out of the aperture of the interferometer mirrors and does not enter the powermeter. Therefore, the low-frequency THz content is underestimated in such interferometric measurements as well.

The purpose of this work is to study experimentally the low-frequency ($\nu \lesssim 1$ THz) spectral content of the broadband THz emission from two-color femtosecond filament, formed in air by 2.7-mJ, 40-fs, $800+400$-nm pulse in the vicinity of 20-cm geometrical focus. To reconstruct the spectral components of THz emission from two-color filament, we measured the angular distributions of the THz radiation by rotation of the Golay cell, the entrance window of which was screened by the bandpass filters centered at 0.5, 1, 2 and 3 THz (see Figure 1a). The spectrum reconstructed from THz angular distributions was compared with the one measured by electro-optical sampling for the same two-color filament (see Figure 1b). The comparison shows relatively low power at the frequency below 1 THz in the case of EOS measurement. We found that the qualitative agreement between the two spectra is achieved if the spectrum obtained from EOS is corrected based on the convolution between the optical probe and THz pump pulse in ZnTe crystal.

2. Materials and Methods

In the experiment we used the laser pulses delivered by Ti:Sapphire laser system Spectra Physics Spitfire Pro XP centred at the wavelength ${\lambda }_1=800$ nm with the energy up to 2.7 mJ, FWHM duration of 40 fs and beam diameter of 12 mm (at $e^{-2}$ level) at 1 kHz repetition rate. To form the two-color filament we inserted a BBO crystal (I-type, 100-$\mathsf{\mu }$m thickness, 10% efficiency) into the pump beam thus providing generation of the second harmonic radiation centered at ${\lambda }_2=400$ nm (Figure 1). The BBO crystal was adjusted to maximize the second harmonic yield. Behind the BBO crystal, polarization of the 400-nm radiation was oriented perpendicularly to the one of the 800-nm pump. Similarly to our work [23], a compensator plate decreased the group delay between the fundamental (${\lambda }_1$) and the second (${\lambda }_2$) harmonic pulses. A dual wavelength phase plate (“${\lambda }_1/2+{\lambda }_2$” in Figure 1) served to align the polarization vectors of the harmonic pulses in order to maximize the THz yield. Thereby, both harmonics had the same vertical linear polarization direction and zero group delay between themselves in the filament. After the dual wavelength phase plate the two-color beam was focused into the atmospheric air by the off-axis parabolic mirror with the focal length $f=20$ cm. In the vicinity of the geometrical focus, the ∼10-mm two-color filament was formed. It was the source of THz radiation, which spectral content was studied in our experiment.

We measured the spectral content of THz emission from two-color laser-induced plasma using two different techniques. In the first case (Figure 1a), we screened the Golay cell by the teflon plate (to block laser radiation) and the bandpass filters (Tydex [50]) centered at the frequencies $\nu =0.5$, 1, 2 and 3 THz. The transmittance spectra provided by the manufacturer for our filters are shown in Figure 2 by the colored areas. We rotated the Golay cell placed at a distance of 20 cm from the geometrical focus around the vertical axis by the angle $\theta$ relative to the laser beam axis with the step $\Delta \theta =2.5^{\circ }$ (Figure 1a). With this setup we recorded the angular distributions of THz fluence $F\left(\theta \right)$ at different frequencies $\nu$. In these measurements the Pockels cell located in the regenerative amplifier provided the modulation of generated THz emission with a frequency of 20 Hz. The resulting THz signal was registered by means of synchronous detection technique using lock-in amplifier. The measured dependencies $F\left(\theta \right)$ were processed according to Ref. [51] to obtain the power S of the spectral components at the set of frequencies studied.

In the second case (Figure 1b), we measured THz waveforms using a standard time-domain system based on the electro-optics sampling technique (EOS [39]). The system consisted of the ZnTe crystal ($10\times 10\times 0.5$ mm${}^3$, $\langle 110\rangle$ cut), the quarterwave plate (“${\lambda }_1/4$” in Figure 1b), the Wollaston prism, and two photodiodes. This setup registered the radiation in the spectral range from ∼0.2 to ∼3 THz. Two 50-mm in diameter teflon lenses with the foci of 6 and 10 cm collimated the THz radiation and focused it onto the ZnTe crystal, respectively (Figure 1b). Optical probe pulse passed through a variable delay line and then was combined with the THz pulse on a lavsan pellicle. So as to decrease a fluctuation of registered signal, we apertured the collimated probe beam down to 1 mm in diameter. By the variation of the delay between the probe and THz pulses, we measured the THz waveforms. The THz spectrum was reconstructed from the measured waveform using the Fourier transform.

3. Results and Discussion

Figure 3a shows the dependence of the THz fluence F on the polar angle $\theta$ measured by the Golay cell at the frequency $\nu =0.5$ THz (red squares), 1 THz (green circles), 2 THz (blue triangles) and 3 THz (violet stars). Angular distributions $F\left(\theta \right)$ get narrower with the frequency increase in reasonable agreement with the well-known law [52] for the divergence angle

$$\delta \propto \sqrt{{\nu }^{-1}}$$

(1)

of electromagnetic waves emitted from an extended source, see Figure 4. Equation (1) is usually used in the studies of THz emission from a single-color filament [53,54,55], however, is valid in the case of two-color filament [18], and even for wire lasers [56].

Bandpass filters centered at different frequencies $\nu$ have different transmittance spectra, see colored areas in Figure 2. So, to compare the amplitudes of THz signal measured with different filters, we recalculated the measured THz fluence F (see Figure 3a) into the THz spectral intensity I (see Figure 3b) according to the equation:

$$I(\nu ,\theta )=\frac{F(\nu ,\theta )}{T\left(\nu \right)\Delta \nu \left(\nu \right)}.$$

(2)

Equation (2) takes into account the transmittance of the filters T, their bandwidth $\Delta \nu$ (see colored areas in Figure 2) and the constant spectral sensitivity of the Golay cell within the range of frequency studied, that is from 0.5 to 3 THz. The maximal transmittance T slightly changes with the central frequency of the bandpass filter $\nu$. At the same time, the filter bandwidth $\Delta \nu$ increases fourfold with the central frequency $\nu$ increase from 0.5 to 3 THz (cf. red and violet colored areas in Figure 2). This results in monotonic decrease in the angularly-resolved THz spectral intensity I with the frequency increase (Figure 3b). In contrast, the measured THz fluence profiles F have almost the same amplitude for all the frequencies studied (Figure 3a).

According to $2D$ measurements of the THz transverse spatial distribution [18,19,57], the latter is axially symmetrical in the case of geometrical focusing into air of the moderate-power pulse [58]. Assuming axial symmetry of THz beam in our experiments, we integrate the directional diagrams $I(\nu ,\theta )$ over the whole forward hemisphere to obtain the spectral power of THz radiation $S\left(\nu \right)$:

$$S\left(\nu \right)=\Delta \theta \sum _{p}I(\nu ,p\Delta \theta )|sin(p\Delta \theta )|,$$

(3)

where $\Delta \theta =2.5^{\circ }$ and the integer index p runs over the angular positions of the Golay cell. The spectral powers $S\left(\nu \right)$ calculated from Equation (3) and divided by $S(\nu =1\mathrm{THz})$ are shown in Figure 2 by symbols corresponding to the ones in Figure 3. The uncertainties are estimated from the asymmetry of THz divergence diagrams, see Figure 3. The dependence $S\left(\nu \right)$ decreases monotonically as the frequency $\nu$ grows. Therefore, the frequency ${\nu }_{\max }$ of the maximum in THz spectrum is below 0.5 THz.

We compare the spectral power S estimated from Equation (3) with the spectrum measured by EOS and shown in Figure 2 by the black curve. Both spectra are normalized to the spectral power at $\nu =1$ THz, at which the spectral maximum in the EOS measurement is achieved. In the range of frequencies $\nu =1$–2 THz the spectral powers obtained by the two methods coincide within an error. At the highest frequency $\nu =3$ THz, the spectral power reconstructed from THz angular distribution exceeds the one measured by EOS by approximately two orders of magnitude due to ZnTe phonon band at $\nu \gtrsim 3$ THz [59].

However, the spectral power measured by EOS at $\nu =0.5$ THz is an order of magnitude lower than the one obtained from integration of the angular distribution (compare the black curve at $\nu =0.5$ THz with the red square in Figure 2). The overall collection angle of the THz collimatimg lens (Figure 1b) is about ${30}^{\circ }$. Therefore, only a little portion of THz emission from the two-color filament escapes the lens and, hence, the ZnTe crystal (Figure 3a). The depletion at $\nu =0.5$ THz in the THz spectrum measured by EOS can be formed due to geometrical focusing of THz radiation into the spot larger than the probe beam transverse size. We developed a model of EOS detection technique, which includes the effect of the broadband THz geometrical focusing onto the ZnTe crystal, see Appendix A. Figure 5a shows the EOS spectral sensitivity $\epsilon \left(\nu \right)$, calculated according to this model in our experimental conditions, i.e., for the probe beam, which was apertured down to 1 mm and afterwards propagated for about 50 cm towards the ZnTe crystal. The diffraction of the apertured probe beam forms the ∼400-$\mathsf{\mu }$m in diameter quasi-Gaussian beamlet surrounded by divergent rings (Figure 5b,c). We corrected the measured by EOS spectrum taking into account the EOS sensitivity $\epsilon \left(\nu \right)$, see Figure 2, dotted curve. The depletion in the corrected EOS spectrum vanished at $\nu <1$ THz in agreement with the spectral power S at $\nu =0.5$ THz (Figure 2, compare black solid and orange dotted curves). This agreement between the corrected EOS spectrum and the spectrum obtained by the integration over the forward hemisphere confirms our result, that the actual spectrum of THz emission from two-color filament has the maximum at few hundreds of gigahertz or even less.

4. Conclusions

In conclusion, the incoherent measurements of the frequency-angular distribution of THz emission from two-color femtosecond filament indicate the monotonic decrease in spectral power from 0.5 to 3 THz and, thus, the absence of spectral maximum in this range. The measured by the rotation of the Golay cell screened by the bandpass filters THz angular distributions were integrated over the forward hemisphere taking into account the filters’ transmittance. As a result, we reconstructed the spectral powers at the filters’ central frequencies. Our technique of the incoherent measurements of THz spectrum does not use the focusing/collimating THz elements and, therefore, allows us to estimate the actual low-frequency content of THz emission from a two-color filament.

In contrast, the spectrum obtained by electro-optical sampling in the same conditions has a maximum at 1 THz. The depletion of low-frequency components in EOS measurements is due to their geometrical focusing into the spot larger than the probe beam diameter and, therefore, weakening of the THz-induced birefrigence of the probe beam. The model of EOS, which accounts for the nonparaxial diffraction and focusing of THz wave, provides a correction factor to EOS spectrum. The corrected spectrum complies with the incoherent measurements.