We study angular and frequency-angular distributions of the terahertz (THz) emission of the low-frequency region (0.3–3 THz) from a two-color femtosecond plasma spark experimentally and in three-dimensional numerical simulations. We investigate the dependence of the angular shapes of the THz radiation on focusing conditions and pulse durations by using two laser facilities (pulse durations 35 and 150 fs) for different focusing geometries. Our experiments and simulations show that decrease in the numerical aperture from NA ≈0.2 to NA ≈0.02 results simultaneously in (I) squeezing of the THz angular distribution and (II) formation of the bright conical emission in the THz range. The moderate focusing NA ≈0.05, which forms the relatively narrow unimodal THz angular distribution, is identified as optimal in terms of angular divergence. Numerical simulations with carrier wave resolved show that bright THz ring structures appear at the frequencies ≥2 THz for longer focuses (NA ≈0.02), while for optimal focusing conditions NA ≈0.05 the conical emission develops at THz frequencies higher than 10 THz.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Terahertz (THz) radiation  is widely applied in spectroscopy [2,3], imaging [4–8], medicine [9,10], etc. Femtosecond filament or plasma spark in gases  has the specific features as compared with other THz sources . Gas is self-repairing media after interaction with the high-intensity laser radiation, and THz generation can be realized in the tight focusing conditions. The two-color scheme (usually fundamental and second harmonic of a Ti:Sapphire laser pulse) of THz generation in gases [12–17] allows to achieve THz fields with typical values of 30 kV/cm , and the THz emission from this scheme has the most broadband spectrum among other sources : from ∼100 GHz to 50 – 100 THz or from microwave to the mid infrared range [19–22]. This provides possibilities, for example, for broadband THz time-domain and FTIR spectroscopy . However, the strong angular divergence and the conical structure of the THz emission [14,16,24–31] from the filament have been observed and they reduces its applicability. Thus, it is important to clarify the optimized condition of THz beam generation by the two-color scheme.
Simultaneously with the THz generation the transformation of optical harmonics develops in two-color filament in air [32,33]. This effect originates from both instantaneous electron nonlinear response  and inertial molecular one .
The overall divergence of the THz emission is determined by the filament’s length Lfil mostly and the divergence angle θ is approximatively proportional to , where is THz frequency [24,26,28,36,37]. The conical structure of the THz radiation in case of the two-color filamentation originates from the phase walk-off  and the plasma obstacle [28,30]. The effect of the phase walk-off provides the off-axis synchronism for THz radiation due to its polarity inversion when the phase shift between the optical harmonics increases from 0 to π; this mechanism is important for filaments of ∼2-cm length or longer. The self-induced laser plasma supplants out THz radiation due to strong gradient of the refractive index in- and outside the plasma channel. Experiments and simulations [27,30] show, that the bright ring-like angular structures are formed in the high-frequency THz range ≥3 THz, while the low-frequency THz radiation (∼1 THz) is more uniform [14,27,30]. It means that the changeover from unimodal structure to ring-like structure occurs in the frequency region around 0.3–3 THz and it is important to investigate this frequency region to find the condition for unimodal THz generation. However the angular structure in the spectral range below 3 THz has been measured relatively roughly although the measurements of THz frequency-angular distributions [27,30] were done in the wide frequency range up to 10 – 15 THz. In addition, the spectral lines of different media lie in the spectral region 0.3 – 3 THz , so, it is of special interest due to spectroscopic applications.
In this paper, we investigate angular and frequency-angular distributions of the low-frequency radiation in the range of 0.3 – 3 THz generated by a two-color femtosecond plasma spark in air. Two Ti:Sa femtosecond laser systems with different pulse durations (35 fs and 150 fs) and close energy 2.5–2.7 mJ were used. The spectral width of the THz radiation is inversely proportional to the laser pulse duration : ∼30 THz for 35 fs pules duration and ∼7 THz for 150 fs pulse duration. Using of two laser facilities with significantly different pulse durations allowed us to estimate the influence of the pulse duration on the THz divergence in the range 0.3 – 3 THz in the close focusing conditions. The focal length F was varied from 1 to 8” and corresponding numerical aperture NA was 0.2–0.02 for our beams of ∼1-cm diameter. The length of self-focusing was ∼30 m even for 35-fs 2.7-mJ pulse. Thus, the beam propagation was governed by the geometrical focusing. We found that despite angular distributions are different and are not determined only by the focusing conditions, these distributions have a lot of information in common. Our measurements showed, that increase in the focal length results in the decrease of the angular divergence and enhancement of the conical structure in the low-frequency THz range simultaneously. The optimum focusing was estimated as F ~4” (NA ≈0.05–0.035), corresponding to the relatively low divergence angle without ring structures. Our numerical simulations based on the unidirectional pulse propagation equation  supported this result and show that conical emission in the THz range moves towards the high-frequency range with decrease in the focal length.
2. Experimental setups and numerical model
We built three experimental setups (numbered 1, 2 and 3, see Fig. 1 and Table 1) with similar THz generation scheme, but different detection systems. In all setups part of laser radiation was used to generate second harmonic radiation in a BBO-crystal (type I, 7 × 5 × 0.4 mm3, conversion efficiency 10% for setup 1 and type I, 10 × 10 × 0.2 mm3, conversion efficiency 8% for setups 2 and 3). To produce more powerful THz generation, the polarization states of first and second harmonics were made collinear by a dual wavelength wave plate and the delay between them was compensated by the group velocity delay compensator plate. A two-color pulse was focused into air by parabolic mirrors with different focal lengths F (see Table 1) producing optical-breakdown plasma, which is a source of the THz emission under study. We made two kinds of measurements: the angular dependencies of THz fluence in the range 0.3–3 THz and the THz frequency-angular distributions in the same range.
Angular distributions of the THz fluence in setups 1 and 2 were measured by rotating the Golay cell (Tydex, GC-1P) around the laser-induced plasma spark. To provide measurements laser output was modulated on 20 Hz frequency by means of gating of the electro-optical switches of the regenerative amplifier in setup 2 and was modulated on 15 Hz frequency by means of mechanical chopper in setup 1. For the synchronous detection technique measurements lock-in amplifier (Stanford Research Systems SR830 DSP) was used. The angular step of mesarument in setup 1 was 2.5°; in setup 2 it was 2.5°, 5° and 10° for focal length F = 8”, 4” and 1” (NA ≈0.025, 0.05, 0.2) respectively.
The choice of measurements angles depended on the focal length: the angular distribution is broader for smaller focal lengths of parabolic mirror, further angular steps in setup 2 were selected. The distance between the plasma spark and the Golay cell was 20–25 cm (far zone for the THz radiation emitted from a plasma spark of ∼100-μm in diameter) and did not change during experiments. The aperture with ~7 mm diameter was placed before the Golay cell to separate the radiation propagated into the cone with the opening angle 2.5°. The 2-mm-thick polytetrafluoroethylene (PTFE) screen was placed normally to the source-detector axis at about 5 cm from the plasma spark to block an optical radiation.
Two methods of THz frequency-angular measurements were applied at different setups. In the setup 1 we inserted bandpass filters between the PTFE screen and the Golay cell and provided angular measurements. We used six filters with central frequencies at 0.6, 0.8, 1.0, 1.3, 1.6 and 1.9 THz (see, Fig. 2). These transmission spectra were measured independently by us using the electro-optical sampling with reference source. So, there are 6 frequencies and ∼20 angular points in the reconstructed frequency-angular distributions. To calibrate the angular distributions measured by each filter we measured the on-axis THz spectrum (the orange curve in Fig. 2) by the same electro-optical sampling detector (not shown in Fig. 1) which was used for calibrating of bandpass filters for setup 1. The angular distributions for each filter were normalized to correspond to the measured on-axis spectrum.
Measurements of frequency-angular distributions in setup 3 were done using a standard electro-optical sampling technique  with ZnTe crystal (<110> cut, 10 × 10 × 0.5 mm3), quarter wave plate, Glan-Taylor prism and two photodiodes. The probe laser beam passed a motorized delay line and then propagated through the aperture with diameter ~1 mm to provide a single-point measurement. The signal from the balanced detector was collected by the lock-in amplifier (Stanford Research Systems SR830 DSP). The whole setup was rotated around the plasma spark with the same angular step as in the THz fluence measurement. For each angle a waveform and the amplitude of the signal from one of the photodiodes were collected. To provide the same scaling of the signal from different angles under study, the obtained THz waveforms were normalized to the corresponding amplitude from the photodiode. These “scaled” waveforms were processed by the fast Fourier transform to get THz spectra. To eliminate modulation due to THz reflection inside ZnTe crystal these obtained spectra were smoothened with step of 0.1 THz. All of these measurements were conducted with parabolic mirrors with focal lengths F = 8 and 4” (NA ≈0.025 and 0.05), see Table 1.
We made numerical simulations of THz generation from two-color plasma sources by the unidirectional pulse propagation equation (UPPE)  for the axially-symmetric spatio-temporal Fourier-harmonic12,34] and transient photocurrent [13,34]30] relies on the discrete Hankel transform and is able to resolve the propagation angles up to 45° and frequencies down to 0.05 THz. Important to note that the evanescent modes, which appear when the source term is less than the wavelength of the radiation, are consistently accounted in equation due to the complex longitudinal wavenumber . The initial spatio-temporal distribution of the electrical field in the coordinates (t, r, z) in the numerical simulations was Gaussian in space and time with the spectral composition, energy, diameter and duration corresponding to the experimental ones for the Legend Elite. The beam phase was set in the plane z = 0 by the parabolic dependence corresponding to a focal length F. The parabolic input phase is suitable only for the Gaussian beams with the diameter less than , where is a wave number. The value of is larger than the beam diameter (1 cm) for F = 4, 8” (NA ≈0.05, 0.025), while = 3.3 mm in case of F = 1” (NA ≈0.2). So, the simulations were done only for relatively long focusing F = 4, 8” (NA ≈0.05, 0.025), i.e. we reproduced conditions of the setup 3.
3. Results and discussions
Experimentally measured dependences of the THz fluence on the rotation angle are shown in Fig. 3(a) and 3(b) for setup 1 (measurements without bandpass filters) and 2 respectively. One can see decrease in the THz divergence and increase in the THz fluence with increase of the focal length for both setups. The specific feature of the data for the longest focal length F = 7.5” (NA ≈0.02) in setup 1 is the strong pedestal in angular distributions of THz fluence, while those distributions are uniform for shorter focusing as shown in Fig. 3(a) and 3(b).
In order to clarify more details, we measured frequency-resolved angular distribution of THz radiation with different NA (Figs. 4 and 5). The results for setup 1 (pulse duration 35 fs) are shown in Fig. 4. In the case of F = 2” (NA ≈0.07), the angular distributions are unimodal and rings are not observed as shown in Fig. 4(a). In contrast, two peaks are observed at around ± 7°in the case of F = 4 and 7.5” (NA ≈0.035 and 0.02), see Fig. 4(b) and 4(c), it indicates that ring angular-frequency distribution of THz emission are realized. In the case of smaller NA (Fig. 4(c)), the ring-structure observed more clearly. It means that decreasing NA is important to observe clear ring-structure. Note that, we observe strong on-axis peak in experimental data for F = 4 and 7.5” (NA ≈0.035 and 0.02). This effect is not reproduced in simulations, but it may originate from pulse duration which is significantly shorter than in the numerical simulations.
In the case of setup 2 and 3 (pulse duration: 150 fs), similar dependence of THz field distribution on NA was observed. As shown in Fig. 3(b), angular distributions of the THz fluence was unimodal for shorter focal lengths F = 1 and 4” (NA ≈0.05). This unimodal structure persisted at any THz frequency in the frequency-angular distribution, as shown in Fig. 5(a). In contrast, the angular distribution had the flat top for the longest focal length F = 8” (NA ≈0.025) as shown in Fig. 3(b), and the bright ring structures were observed in the frequency-resolved measurement as shown in Fig. 5(b). Though integrated spectra results shown in Fig. 5(b) (black curve) are close to Golay cell measurements in Fig. 3(b), they have some differences, which can be explained by different response of detectors in this spectral region and relatively high noise level in electro-optical measurements.
In order to confirm the validity of the measurement, we performed numerical UPPE simulation. The simulated frequency-angular distribution at the end of the plasma channel are shown in Fig. 6(a) for F = 4” (NA ≈0.05) and in Fig. 6(b) F = 8” (NA ≈0.025). They demonstrate the unimodal structure with maximum at the beam axis in the low-frequency THz range. The appearance of two peaks on the same angles from the axis in the frequency-angular distribution corresponds to a ring structure. According to numerical simulation for F = 4” (NA ≈0.05), rings do not appear in this frequency region 3 THz available for the experimental measurements. In contrast, bright ring structures are clearly seen for 2 THz in case of F = 8” (NA ≈0.02, Fig. 6(b)–6(c)). Our simulations predict that THz rings can be experimentally observed only in the case of small NA: the longest focusing F = 8”. These results are in good agreement with the measured frequency-angular distributions, which shows bright ring-structure only in the case of large focal length and small NA as shown in Fig. 4(c) and 5(b).
The divergence angle of rings ± 7° in simulations is also in a good agreement with the divergence angle of ring structures in the experiment (Fig. 4(c), 5(b)) and with the width of the flat part of the angular distribution of the THz fluence (Fig. 3). So, both in our experiment and in our simulations rings in the THz frequency-angular distribution shifted towards low frequencies with increase in the focal length. The uniform THz angular-frequency distribution with low angular divergence in the low-frequency THz range (up to ∼10 THz) can be obtained for moderate focusing conditions, which corresponds to the focal length of 4” (NA ≈0.035–0.05) in our simulations and experiments.
The qualitative description of the observed experimentally and reproduced numerically distributions of the THz emission is following. There are two main nonlinear mechanisms of the THz ring formation in case of two-color femtosecond breakdown: phase walk-off between the optical harmonics, which leads to the variation of the polarity of THz pulses , and plasma obstacle for THz radiation [28,30]. Both mechanisms require the long filament for more pronounced THz rings. The length of phase walk-off is about 2 cm; in our experiments the plasma spark length was 7 mm for the focal lengths F = 7.5 and 8” and about 1 mm for lesser ones. Hence, the phase walk-off between the harmonics does not play significant role in our experiment and simulations, and plasma obstacle determined the rings formation.
The plasma spark diameter is ∼100 μm. This obstacle is sub-wavelength for THz radiation with the frequency 3 THz. The low-frequency THz waves bend around such obstacle without significant changes in the far zone of diffraction even in case of the plasma frequency higher than the THz frequency. As a result, the low-frequency THz emission is unimodal without ring structures. Opposite, THz waves with the frequency 3 THz (wavelength shorter than ∼100 μm) diffracts out the plasma spark less efficiently and, therefore, propagates through the plasma as well. This results in a plasma-induced phase shift, which transforms into a ring-like THz fluence distribution in a far zone of diffraction. Longer plasma spark enhances this effect, and it can be observed for the THz waves with lesser frequency.
Thus, our measurements with two different laser facilities show, that the THz emission from the two-color femtosecond plasma spark has relatively narrow unimodal angular distribution with suppressed ring structures (we observe on-axis maximum) in the low-frequency THz range 0.3–3 THz in case of the moderate focusing; this regime corresponds to the focal length of F = 4” (NA ≈0.035–0.05) for both laser systems. In spite of the pulse duration of our lasers differs significantly (35 and 150 fs), we directly found in our experiments, that the shape of the THz radiation in this spectral range is governed by the focusing conditions mostly. According to the observed data, some differences in THz angular distributions for setup 1 and setup 2 are observed. First, one can see a plateau near the optical axis for F = 8” (NA ≈0.025), that was not observed in setup 1 THz angular distribution measurements with the same NA. Moreover, a pedestal existed for all three distributions for setup 1, which is clearly not observed for measurements in setup 2. One of possible reasons of occurred differences is varied dynamics of plasma creation depending on the laser pulse duration. Though the geometrical focusing is realized in the same conditions (same NA), the electron concentration distribution in plasma channel is formed differently: intensities in the beam waist are different and plasma electron concentration strongly depends on the intensity. Plasma channel is more uniform for longer pulse alike in the filamentation case. In case of shorter pulse there is a prominent variation of plasma density along the channel. According to the modified interference model [26,28] plasma could be considered as a source of obstacle and phase walk-off between fundamental and second harmonics that leads to different angular distribution. However, despite considering these effects, angular divergences are similar for setup 1 and 2 measurements.
In conclusion, we measured the angular and frequency-angular distributions of the low-frequency emission in the range 0.3–3 THz from the two-color plasma spark. We used two laser facilities with the significantly different pulse durations of 35 and 150 fs and varied focusing conditions F = 1–8” (NA ≈0.2–0.02). The similarity of the angular shapes of the THz emission in both cases shows, that the THz divergence in the low-frequency range is determined by the focusing conditions mostly; its dependence on the pulse duration is much weaker. In case of the short focal lengths of 1” and 2” (NA ≈0.2, 0.07) the broad unimodal THz angular distribution was observed. Opposite, its distribution for longer focuses F = 7.5 and 8” (NA ≈0.025-0.02) is narrow with bright conical emission. The moderate focal length of 4” (NA ≈0.035–0.05) was found to be the optimal: the THz angular distribution is relatively narrow, and no rings were observed. The numerical simulations support the experimental results: the THz conical emission moves toward the high-frequency THz range with the decrease of the focal length; in case of F = 4” (NA ≈0.035–0.05) it can be seen for frequencies of ∼10 THz, while for F = 8”(NA ≈0.025) ring structures form at ∼2 THz.
Russian Foundation for Basic Research (RFBR) (18-02-00954, 18-52-16020, 18-32-01000); Russian Federation President grant (MK-8562.2016.2); Presidium of the Russian Academy of Sciences Program (I.6); Students and Researchers Exchange Program in Sciences (STEPS); “Basis” Foundation; Scholarship of Russian Federation President (SP-2453.2018.2); The program “UMNIK” of Foundation of assistance to development of small forms of enterprises in scientific-technical sphere (FASIE) (11488GU/2017, 11522GU/2017); The Photon Frontier Network Program of the Ministry of Education, Culture, Sports, Science, and Technology (MEXT).
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