Tunable narrowband THz waveforms were generated from laser induced gas plasma using shaped optical pulses. Square wave phase patterns were fed to a spatial light modulator. The frequency and amplitude of the square wave phase were used as parameters to tailor the terahertz waveforms. The dependence of THz waveforms on these parameters has been studied in detail. The presence of the ionization thresholds for pulse shaping is also discussed. We have demonstrated the wide and continuous tunability of the central frequency of the narrowband THz waveform from 2.5 to 7.5THz.
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The generation and detection techniques of coherent terahertz (THz) radiation have been advanced extensively in the last few decades. The use of the THz radiation provides unique tools in a wide range of applications from material characterization  and chemical sensing/identification  to label-free genetic analysis  and imaging . The capability of manipulating THz waveforms will expand the flexibility of terahertz radiation for such applications [1–6]. THz pulse shaping techniques have become increasingly important as precise control of the waveform is desirable for many of these applications.
Tunable narrowband THz sources are suitable for the study of phenomena involving relatively narrow bandwidth and are desirable for some applications in spectroscopy, imaging, and coherent control [5,6]. Pulse shaping techniques have been used to generate tunable narrowband THz waves from ZnTe , photoconductive antennas [8,9] and lithium niobate . However, the highest tunable frequencies are limited to approximately 3 THz due to the narrow bandwidth of terahertz waves from these materials. A wider tunable range is desirable for spectroscopic applications.
THz radiation with bandwidth over 20 THz has been generated from laser-induced plasma. Simultaneous high THz electric field and broad bandwidth are the characteristics of THz wave from this type of emitter . After the demonstration of THz wave generation from gas plasma using two-color excitation by Cook et al. , the generation process has been studied extensively [11–19]. A THz field of 400kV/cm was reported by Bartel et al. using 25-femtosecond laser pulses to excite gas plasma . Recently, pulse shaping of the THz wave generated from gas plasma using a spatial light modulator has been demonstrated .
In this letter, we report THz pulse shaping in laser-induced plasma to realize a wider tuning range of narrowband THz waves. We used a spatial light modulator (SLM) to shape optical pump pulses and generated THz waves with the controlled optical pulses. In particular, we used square wave phase patterns in our experiments and controlled the phase pattern by only two parameters, namely, the period and amplitude of the square waves. By using these phase patterns, we can generate multiple optical pulses with different peak intensity and separation. Due to the wide bandwidth of the THz wave generated from plasma, more precise control of the waveform is expected by pulse shaping compared to other solid state materials. THz wave generation from laser-induced plasma also exhibits a threshold effect, which does not exists in other solid state emitters [7–10]. We followed a transient photocurrent model  to simulate the THz waveforms, and a qualitative comparison between experimental results and the simulations was demonstrated. In the end, we showed that a wider tunable range (up to 7.5THz) of narrowband THz waves can be produced using this periodic phase distribution.
2. Theoretical calculations
In this work, we have followed a photocurrent model  with some modifications. According to the model, a non-vanishing transverse plasma current is produced due to the asymmetry of two-color laser field and this photocurrent generates a THz electromagnetic wave. In the simulations, we start with a fundamental pulse at 800 nm.Eq. (1) and E′(ω) is the modified spectrum after the addition of the square-wave phase, φ Square-wave(ω) to E(ω). The inverse Fourier transform of Eq. (3) gives the electric field of shaped optical pulse, E′(t) in the time-domain. The electric field of the second harmonic (400nm) pulse generated from the shaped pulses, E SH (t) is given by,12]. Therefore, we rotate the BBO crystal orientation in such a way that the component of generated second harmonic beam is the maximum along the polarization direction of the fundamental beam . The generated THz field amplitude along the fundamental beam polarization is much higher than the THz amplitude generated along the orthogonal polarization direction and is neglected in our calculation. Hence the total optical excitation field, E Total(t), consisting of components of two-color field along polarization direction of THz wave, can be expressed as,13]. The generated THz power is proportional to cos 2(φrel) . In the calculations, the static tunneling model is used to calculate the ionization rate of nitrogen molecules . The ionization rate (WST) by the total field, was calculated using the static tunneling model, given by,19]. The Ammosov-Delone-Krainov (ADK) model is also often used in literature to calculate the ionization rate . However, it has been reported that both models produce qualitatively similar results in case of the laser power density, used in our experiments .
The velocity of an electron, born at t = t′, is given by the following equation, assuming the initial velocity is zero:21]. These two variables are related as . The decay of electron current was incorporated to include the effects of electron scattering by ions. Rodriguez et al. estimated the electron-ion collision time, τ, to be on the order of 1ps for atmospheric gas density , and we used τ = 1ps for our simulations. The THz electric field from the electron current was calculated using following equation:
3. Experimental setup
The experimental setup included the following: a Ti:sapphire oscillator laser beam with 100fs pulse duration at 76 MHz repetition rate and 795 nm central wavelength was directed towards the programmable pulse shaper setup aligned for transmission mode. The pulse shaper setup consisted of a pair of 1800 lines/mm gratings placed at the focal plane of a pair of cylindrical lenses (focal length, f = 70mm). The programmable liquid crystal-spatial light modulator (LC-SLM) with 128 pixels was placed at a distance of 2f from the gratings. A regenerative femtosecond laser amplifier was seeded by the shaped optical pulses coming out of the SLM. The amplified shaped pulses were used to generate THz waveforms from the laser induced N2 plasma. Furthermore, as the phase-only pulse shaping doesn’t alter the optical spectrum, ideally there should not be any distortion of optical pulse shape due to the amplifier . Realistically, the main effect of amplifier is the broadening of the pulse duration and we observed an increase of pulse duration of transform limited pulse from 95fs to 105fs before and after the amplifier, respectively. Autocorrelation of the generated THz wave was measured using a Michelson interferometer and a pyroelectric detector to accommodate the broad spectral range of the terahertz wave from the laser induced plasma. More details of the experimental setup are given in our previous report .
4. Results and discussions
In our experiments, the amplitude and the frequency of the square wave phase pattern were changed independently and the effects on the generated THz waveforms were studied. Further, we generated narrowband THz pulses with a combination of these two parameters. Generally, when these phase patterns are used for optical pulse shaping, a multiple pulse sequence in time domain is generated, and the peaks are separated temporally by (2π/ωφ) , where ωφ is the frequency of periodic phase and the amplitude of these peaks depends on the amplitude (a) of periodic phase. At a = 0 and 2π, the phase pattern is same as constant phase and the shaped pulse is the replica of the input pulse, i.e. the transform limited pulse. For 0<a<π and π<a<2π, the relative amplitude of the side peaks increases and decreases, respectively, relative to the central peak and the central peak amplitude is zero at a = π, where the temporal separation between the two side peaks become 2 × (2π/ωφ) . On the other hand, for phase-only pulse shaping, the shape of the optical spectrum doesn’t change. Adding phase in the frequency domain generates shaped temporal pulses, keeping the optical spectrum unaltered.
4.1 Square-wave frequency dependence of THz wave generation
Figure 1 shows the frequency dependence of the square wave phase pattern. The THz autocorrelation signal and corresponding spectra are plotted for three different frequencies (1.6, 3.2 and 6.4 THz). The amplitude of the periodic phase was kept fixed at a = 1.0π. The generated THz time-domain pulse has multiple peaks (Fig. 1(a).). The separation between the side peaks in the THz time-domain autocorrelation signal decreases as the frequency of the square wave phase increases [Fig. 1(a).]. The simulation results were plotted for three different frequencies of the periodic phase (1.6, 3.2 and 6.4 THz) in Fig. 2 (a) and (b) .
The amplitude of the periodic phase patterns is fixed at a = π. The simulated spectrum has been corrected for the frequency dependent detector response. The simulation results show that the separation between the peaks in THz time-domain pulse decreases as we increase the frequency of the periodic phase, following the same trend as the experimental results. The largest time separation between peaks corresponds to the minimum frequency of the periodic phase, which is limited by the frequency resolution of the SLM pixels (~0.5 THz). On the other hand, the minimum time separation is limited by the bandwidth of the optical spectrum for both experiment and simulations. The effects of spectral bandwidth and pixel resolution are discussed in detail in Section 4.4. Xie et al. showed enhancement in THz generation, in presence of pre-ionized plasma . They estimated the lifetime of the plasma, to be 185ps.
In our experiment, the consecutive peaks in the shaped optical pulses are separated within a distance of 0.3-1.5ps, which is much less than the plasma lifetime and in the same order of the electron-ion scattering time (~1ps) described in the theoretical model we used (section 2). But in our experimental data, presented here, we did not observe any prominent evidence of enhancement in THz generation, in presence of consecutive peaks in the optical pulse. Our simulation shows, in the case of shaped pulses, the total number of ionized electrons is on the order of 1011 −1012cm−3, where as Xie et al. reported an estimated plasma density of 2 × 1017cm−3. The low plasma density is one of the possible reasons for the absence of any enhancement in THz generation from consecutive peaks in the optical pulse.
4.2 Square-wave amplitude dependence of THz wave generation
Here we discuss the effects of amplitude a of square-wave phase patterns. THz auto-correlation signals (Fig. 3(a) ) and their corresponding spectra (Fig. 3(b)) are shown for three different amplitudes of periodic phases in the range of π<a<2π. The frequency of periodic phase is fixed at 1.6 THz. The results show that the relative intensity of the side peaks to the center peak decreases as the modulation amplitude increases while the peak positions in time domain data remains the same. In addition, the corresponding spectra show the decrease of the depth of the periodic modulation in the THz spectrum (Fig. 3(b)). On the other hand, for
0<a<π, increase in a, shows an increase of the relative intensity of the side peaks to the center peak and an increase in depth of modulation in THz spectra. This is due to the symmetry around a = π. The simulated THz auto-correlation signal and THz spectra for three different amplitudes of periodic phase (a = 1.00π, 1.26π and 1.38π), where π<a<2π, are plotted in Fig. 4 . The simulated data qualitatively reproduce the modulation dependence of the side peak intensity and corresponding modulation in the THz spectra (Fig. 4(b)). In the experiment, to change the modulation depth in the THz spectrum from 90% to 10%, we changed “a” from 1.13π to 1.64π (Δa = 0.51π), whereas in the simulation, to produce similar change in modulation depth we had to change “a” from 1.0π to 1.38π (Δa = 0.38π). This discrepancy may be the result of a slight misalignment in the SLM setup.
4.3 Threshold behavior
One of the distinct characteristics of THz wave generation from gas plasma over other THz wave sources is the existence of the ionization threshold for the optical pump pulse. Figure 5(a) shows autocorrelations of two optical pump pulses with similar pulse shapes except for a small intensity difference of the side peaks at ± 0.37ps and ± 0.73ps. Figure 5(b) shows the THz wave signal generated by these optical pump pulses. The THz wave signal with, a = 0.789π shows two sharp pairs of side peaks, while the signal with a = 0.751π shows one pair of sharp peaks at t = ± 0.73ps and missing a sharp peak around t = ± 0.37ps in Fig. 5(b). This is due to the existence of the ionization threshold, which depends on the ionization potential of the gas molecules. According to the theoretical model, described in section 2, when the amplitude of the periodic phase is increased by a small amount (0.751π to 0.789π) the optical peak electric field, corresponding to the pair of side peaks at t = ± 0.37ps in the optical autocorrelation signal, increased from 1.46 × 1010 to 1.557 × 1010 V/m. The static ionization model predicts that the number of ionized electrons is an exponential function of the optical electric field (Eq. (6)) and increases rapidly around the threshold electric field. Calculation shows that in the electric field range from 1.25 × 1010 to 1.0 × 1011V/m number of ionized electrons increases by 10 orders. The electric field amplitudes (1.46 × 1010 and 1.557 × 1010 V/m) corresponding to the optical autocorrelation signal at t = ± 0.37ps (Fig. 4(a).), are in the range of ionization threshold and produce an increase in the number of ionized electrons by a factor of 10, which gives rise to a sharp middle peak in the THz auto-correlation signal at t = ± 0.37ps.
4.4 Tunable THz wave generation
As we shown in sec 4.1, a periodic pulse train in time domain generally does not produce narrow band spectrum in frequency domain, but produces an intensity-modulated spectrum at the frequency of the inverse of the pulse period. To generate narrowband spectra, combinations of the frequency and amplitude of square-wave phase need to be adjusted to suppress unwanted higher harmonic components. Figure 6 shows the narrowband THz time-domain waveforms and their spectra from the laser-induced plasma obtained by simultaneous controls of square wave amplitude and frequency. Both the THz autocorrelation signals and the THz spectra are normalized. Generated THz narrowband waves produce a signal in the range of 10-15mV in a pyroelectric detector, whereas the THz wave generated from a transform-limited pulse produces a pyroelectric signal of 150mV. Shaped optical pulses have a lower electric field than transform-limited pulses with same energy. Hence, these narrowband THz waves generated from shaped optical pulses have lower energy than the corresponding THz waves generated from transform-limited pulses. The frequencies of periodic square phases for A to E are 15, 13, 10.4, 7.6 and 4.8THz respectively. A narrow band spectrum with no sideband can be obtained only when the multiple peaks in the time-domain THz pulse lie on the Gaussian envelope of the time domain pulse. The peak amplitude of the side pulses with respect to the central peak amplitude can be controlled by the amplitude of the periodic phase. Therefore, a proper choice of periodic phase amplitude ‘a’ suppresses the side-band. The changes in frequency of the periodic phase shift the central frequency of the THz wave and the amplitude of the square wave changes the relative amplitude of each pulse in the time-domain, hence changes side-bands in the frequency domain and a proper choice of amplitude suppresses the side-band. As the separation between time-domain pulses decrease from E to A (Fig. 6(a)), the central frequency of corresponding THz spectrum shifts towards the higher frequency from E to A (Fig. 6(b)). The central frequency was tuned from 2.5 to 7.5 THz. The maximum tunable frequency is limited by the bandwidth of the optical excitation pulses. In our experimental setup, the bandwidth is 12nm and the expected maximum center frequency is 8.0 THz. This value is close to the experimental value of 7.5THz. On the other hand, the lowest tunable frequency is theoretically limited by the width of pixels in the spatial light modulator, and 1 THz is expected in our pulse shaper setup. However, the experimental value of 2.5 THz is limited by the detector response which has low sensitivity in the low frequency region. The use of optical excitation pulses with broader bandwidth is expected to expand the tunable range of narrowband THz pulses. The bandwidth of the THz spectrum changes from 1.0 to 3.0 THz as the center frequency increases from E to A in Fig. 6 (b). The bandwidth of narrowband THz spectrum is governed by the width of the envelope of time domain THz pulse, and this increase is expected from the autocorrelation signals in Fig. 6(a). Therefore, a spectrum with higher central frequency (for example 7.5 THz, the red colored spectrum in Fig. 6(b)) generated using periodic phase with a higher value of ωφ, has larger spectrum width (~3 THz). This feature is specific to the square-wave phase pattern. The use of the higher-energy excitation optical pulses may increase the number of ionizing pulses in the pulse sequence, potentially reducing the bandwidth of the generated narrowband THz wave pulse.
Tunable narrowband THz wave generation from various THz emitters has been reported, such as EO crystal (ZnTe), LiNbO3, and photoconductive switch using shaped optical pulses and the tunable range is from 0.5 to 2.0 THz, 0.8 to 2.5 THz, 0.5 to 3.0 THz, respectively [7,9,10] due to the phonon absorption in solid materials. We have demonstrated a wider tunable range of the narrowband THz spectrum (2.5 - 7.5 THz) using air plasma as a THz radiation source.
The shaped THz wave from gas plasma was generated using square wave phase patterns. Two parameters, the amplitude and frequency of square wave phase, were used to control the THz waveform and the characteristics of the generated THz wave, for each of parameters, were studied. The simulation results agree qualitatively with the experimental results of the amplitude and frequency dependences. The effect of the ionization threshold in pulse shaping was also demonstrated. Finally, tunable narrowband THz waveforms were generated in a broad frequency range from 2.5 to 7.5 THz. Both lower and upper frequency limits are due to experimental reasons, and are not fundamental limitations, unlike the case of solid-state THz emitters where phonon absorption limits the highest available frequency. Further experiments to resolve these limitations are now in progress.
This work is partially supported by the Office of Naval Research.
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