1. Introduction

The less explored terahertz (THz) gap of the electromagnetic spectrum between the microwave and the far-infrared region has attracted a great deal of attention, particularly from material scientists and security experts. In contrast to x-rays, THz photons have low energy and can be used for non-destructive material research, medical diagnosis, and industrial quality control [1]. In addition, this spectrum range is transparent in many materials, such as wood, paper, plastic, and cloth [2], allowing THz radiation to be used in security applications. Among the wide applications of THz emission, it can also be used in basic physics such as vibrational and rotational spectroscopy of gaseous molecules indicating appreciable transition lines in the sub-millimeter range [3,4]. Over the last two decades, the weakness of generated THz radiation has been a major challenge in the field of nonlinear optics where the efficiency of conversion is inevitably low.

The most popular choice of THz generation is based on the electro-optic methods and two-color filamentation [5–8]. Both methods are capable of producing THz waves with amplitudes in the range of several MV/cm. The plasma radiation spectrum has a suitable width between zero and several tens of THz, and in recent decades, significant efforts have been made to enhance the radiation spectrum of electro-optic methods. For example, Sell et al. [9] successfully produced broadband terahertz radiation with an amplitude of 100 MV/cm up to 70 THz, demonstrating the potential of these techniques for generating high-intensity THz radiation. Other research groups, such as Shalabi and Hauri [10] and Vicario et al. [11], have also reported intriguing experimental results on high-amplitude THz generation using electro-optic methods. While both methods face challenges with focusing optics at high intensities [11], despite the lower efficiency in terahertz generation, plasma is a more suitable option than the electro-optical method due to its temperature insensitivity and its ability to recover in between laser shots [12].

The efficiency of THz generation is approximately 0.01% at a wavelength of 800 nm, which increases to around 7% at longer wavelengths of about 3.2 µm [12,13], but it can handle the intensities in the range of 10²¹ W/cm² [14]. Furthermore, besides the high-power emission, by this method, the received THz has a wide emission band and ultra-short pulses which are in the tens of femtosecond (fs) scale. A typical example is the use of a high power 800 nm laser and its second harmonic to drive a plasma filamentation for THz generation [15–18]. The next effort which was reported by Kim et al. [8,13] was an emission model for a micro-plasma. Their study examined the influence of phase difference between the two-color lasers in the transverse direction of a plasma emission. Moreover, they studied the influence of carrier envelope phase in single few-cycle lasers on the emission of plasma including THz and higher harmonic generation (HHG).

In a recent study, Wang et al. [19], developed a THz near-field microscope that utilizes two femtosecond lasers cross-focused on an air target near the sample. The reflected THz beam was measured during the experiment, and the authors investigated the relationships between the plasma, THz beam, and sample. The THz near-field microscope is based on an air-plasma dynamic aperture, created when the two femtosecond lasers are focused perpendicularly in close proximity to the sample surface. The cross-filament created by the plasma modulates the incident THz beam, and the reflected THz near field signal is measured. Singh and Sharma [20], investigated THz radiation by cross-focusing two collinear Gaussian lasers with a frequency difference in a pre-defined rippled density to reach the phase matching condition. In this process, the applied lasers cause the electrons to oscillate and couple with the rippled density to generate a phase-matched emission. The study optimized various laser and plasma parameters and reported an efficiency of approximately 8 × 10⁻³ for the current scheme. Jafari Milani et al. [21], numerically investigated THz radiation generation by beating two cross-focused high-intensity laser beams in a warm rippled density plasma. The study considered ponderomotive force, ohmic heating, and collisional nonlinearities, and found special ranges of electron temperature and laser intensity with turning points where THz radiation reaches its maximum value. The results showed that increasing the electron density and accounting for the collision frequency enhance THz generation, and the maximum yield of THz radiation occurs when the beat wave frequency approaches the plasma frequency.

The particle in cell (PIC) method has been used by many authors [22–24], as an efficient approach to investigate the various properties of plasma and its emission spectrum inside the THz gap. Another direction of research involves influences of different parameters on the amplitude and spectrum of such THz radiation, which are very susceptible to the chirp of applied laser beam, pulse repetition rate and shape, background gas pressure, and effect of background DC electrical field [25–28]. Using PIC method, Soltani et al. [29] studied the influence of plasma density, laser pulse duration and its intensity on the induced plasma current density and the subsequent effects on the generated THz signal. They showed that induced current density in hydrogen plasma medium is the dominant factor influencing the generation of THz pulse radiation. Also, Nguyen et al. [30] studied THz emission by two-color femtosecond filaments in air using a wide pump wavelengths ranging from 0.8 to 10.6 µm. Their study revealed that produced filament in 10 µm two-color interaction can produce GV/m THz pulses while the efficiency of conversion exceeded the percent level. In an additional effort, Fedorov and Tzortzakis [31] indicated that two-color filamentation of 3.9 µm mid-infrared laser is capable of generating a single-cycle THz pulses with the energy of several milli-joules where the amplitude of generated electrical field reaches GV/cm. They claimed that, the origin of mentioned high efficiency is the enhancement of symmetry breaking by the generated HHG. In an experimental work, Koulouklidis et al. [12] demonstrated that two-color filamentation regime at 3.9 µm allows for THz sub-cycle pulses with the field amplitude above 100 MV/cm. A solution for THz amplification is accelerating hot electrons in a plasma column. In a most recent example represented by Saeed et al. [6] by increasing the input laser intensity to 10¹⁸ W/cm² and breaking the symmetry of the electric field (SBEF), an amplified infrared emission of about three times stronger was obtained.

In the present study, the interaction of a two-color pulses including a fundamental laser and its second-harmonic (SH) entering from the left-hand side (LHS) with plasmatic argon (Ar) gas is numerically simulated using the particle in cell-Monte Carlo collision (PIC-MCC) model to obtain the maximum THz radiation. Three types of interaction are modeled whereas the inputs are chirped and their profiles are changed. In the normal case, the inputs are weakly and highly chirped to check the impact of chirping regime on the THz emission and extent. This is performed by calculating the transverse current density of plasma for several wavelengths of inputs changed from 800 nm to 3.9 µm to consider the asymmetrical feature occurring in the temporal variation of plasma current density. The time-dependent asymmetry and the flowerthorn-like fluctuations observed in the plasma current density are fully discussed and their effect on the THz emission considered.

In the next type, an oppositely-moving pulse entering the Ar slab from the right-hand side (RHS) is involved in the interaction as another laser to induce a modulation in the longitudinal current density of plasma. In this cross-focusing scheme, which is examined in detail, the THz emission is investigated for two chirping regimes of the fundamental and its SH pulses for the LHS laser, as well as for the intensity of the RHS pulse as it is changed. The ultimate result is an appreciable enhancement observed in the emission of plasma. The effect of intensity on the longitudinal current density is studied by increasing the RHS pulse intensity up to 10¹⁵ W/cm² to explore the result of plasma-induced electrons on the emission of THz radiation. In the last step, the delayed cross-focusing interaction is modeled and simulated for which the LHS pulse arrives the target point about 225 fs after the RHS pulse in both highly and weakly chirped regime of LHS laser. The beam profiles of the LHS pulses are also changed from an ordinary Gaussian to a flat-top profile to consider how much the THz amplitude can be affected. Final results confirm that delayed cross-focusing scheme including weakly-chirped LHS one is able to increase the THz emission in terms of both amplitude and bandwidth.

2. Simulation model

According to the photo-current model, plasma emission is the consequence of non-zero current of free electrons, which is generated by ionizing the neutral gas during the propagation of a laser pulse. Ignoring the ponderomotive forces, plasma current density, J, can be written as [8,32]:

 

Fig. 1. Schematic description of the explained steps led to the generation of THz radiation from a plasma column. The indicated algorithm is used for simulation performed based on the PIC-MCC model. The MCC section is a part of the PIC method that models the atomic collisions.

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In a laser-plasma interaction for the input intensity greater than 10¹⁴ W/cm², the photo-ionization is the main mechanism of ionization. The rate of such ionization can be estimated using a theory formulated in Eq. (2). As long as the ionization occurs, the MCC section starts to estimate the number of generated electrons expected in each time step. For the calculations being accurate and stable enough, the required time step, $\Delta $t, used in this simulation is determined by the Courant-Friedrichs-Lewy (CFL) condition, as follows [35,36]:

 

Fig. 2. Simple representation of the two-color laser interaction with a two-dimensional neutral gas of 70 µm long and 140 µm width. The two-color lasers are chirped. The input intensity is I_ω=10¹⁵W/cm² for the fundamental pulse and I_2ω=5 × 10¹⁴W/cm² for its SH. The pulse widths and beam waists of those pulses are τ_FWHM= 40 fs and W₀= 40 µm, respectively. They are collinearly focused on an Argon (Ar) gas slab owning the density of N_AR= 10⁻³ N_c, where N_c =10²⁷m^-3, is the critical density of plasma. The plasma column is enclosed between two 15 µm-long vacuum parts on both sides.

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As Fig. 2 shows, a plasma column is irradiated by two linearly polarized chirped laser fields labeled as fundamental and SH pulses at respective frequencies of ω and 2ω. The intensities of fundamental and SH beams are assumed to be I_ω = 10¹⁵ W/cm² and I_2ω= 5${\times} $10¹⁴ W/cm², respectively, corresponding to 50% of conversion efficiency. It is worth noting that the application of two-color lasers with 50% conversion between the fundamental and the SH waves for plasma emission generation has been the topic of many scientific studies [20]. The simulation region is a 70 µm long Ar gas with a density of N_AR= 10⁻³ N_c. Two 15 µm long vacuum parts on both sides of the plasma slab are embedded for confining the region. Both the fundamental and SH pulses are linearly polarized along the y-axis, and they have an initial phase of zero degrees. Additionally, their carrier-to-envelope phase is also set to zero.

3. Results and discussion

The simulation starts by focusing the two-color femtosecond pulses on the Ar slab based on the scheme discussed in Fig. 2. The chirp parameter is primarily assumed to be b = 10⁻⁴rad/s² and the fundamental wavelength is changed to 800 nm, 1064 nm, 1800nm, 3200 nm and 3900 nm from visible to mid-infrared. Figure 3 presents the simulation results.

 

Fig. 3. Simulated results of the two-color laser-plasma interaction performed for x = 77 µm and y = 70 µm while the fundamental laser wavelength is switched to 800, 1064, 1800, 3200 and 3900 nm while the chirp parameter, b, is 10⁻⁴rad/s². The time variation of plasma current density is plotted from (a) to (e), of the plasma emission amplitude from (f) to (j), and of the plasma emission intensity from (k) to (p). The emission intensity of plasma is plotted with a logarithmic scale and the vertical dashed red line on this set of plots determines the border of the cut-off frequency.

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The time-dependent asymmetry observed in the plasma current density, shown in Fig. 3(a) to (e), is due to the input that consists of two colors with different intensities. This in turn, leads to a similar pattern of asymmetry in the temporal variation of the emission amplitude of plasma plotted from Fig. 3(f) to (j). It can be seen that the magnitude of current density, J_y, increases by changing the fundamental laser wavelength from 800 nm in (a) to 3900 nm in (e). It peaks at about 2.8 unit for 800 nm and at about 11 unit for 3900 nm, indicating a growing trend with fundamental and its SH wavelengths. Furthermore, the chirping effect has caused flowerthorn-like fluctuations in the plots, which are clearly seen in the case of infrared fundamental wavelengths specified at 1.8, 3.2, and 3.9 µm. As a result, two main features can be identified from Fig. 3:

The explained characteristics play a main role in obtaining the THz emission from the plasma column, which will be further studied in the following discussions. The flowerthorn-like fluctuations discussed in this investigation significantly differ from the sawtooth variations frequently discussed in related research, particularly in the context of the influence of input electrical fields, as demonstrated in the study by González et al. [38]. It is noteworthy that these fractures, which are the aforementioned flowerthorn-like fluctuations, are locally observed within the fluctuations of plasma current density.

The variation of the plasma emission amplitude, A_p, indicated in Fig. 3(f) to (j) is calculated using the relation given in Eq. (1). Similar variations seen in A_p and J_y curves are the subsequent of the fact that the electric field of the plasma emission is the first time-dependent derivative of the transverse current density. While the rate of fluctuations decreases from Fig. 3(f) to (j), the imposed distortion on the A_p envelope clearly grows over the time. The intensity of plasma emission is demonstrated in Fig. 3(k) to (p) at 800 nm and 3900 nm of the fundamental wavelength, respectively. Each plot shows an approximately identical spectrum broaden across a few THz to about 2.5 PHz. It also contains of two distinctive peaks that occur at the wavelengths corresponding to their fundamental and SH driving lasers. Such a wide band of plasma emission has been observed in a variety of plasma slabs interacting with sub-ps pulses in a single or two colors scheme [39–41]. However, similar to the work represented here, there has been found a cut-off frequency in the emission spectrum of HHG in the plasma. It has been shown that the HHG emission depends on the intensity of input laser and the type of utilized gas [42]. Since the calculations are performed with the similar intensity, chirp parameter, and gas target, the calculated emission spectrums in Fig. 3(k) to (p) are identical. The noteworthy point to mention is that the emission spectrum of plasma indicates very good agreement with the theoretically obtained results [42], confirming the validity of the present work. To compare the results obtained in this work with the ones conducted by Lee et al. [43], the plasma emission amplitude, A_p, in Fig. 3(f) to (j) is multiplied by the normalization coefficient of 5 × 10⁸ to convert the units to MV/cm. It was observed that the emission amplitude remained approximately consistent across all cases at the intensity of the fundamental laser used in Fig. 3, which is consistent with the findings reported in Ref. [43]. This provides a good validation for the present study. In addition, our analysis demonstrates that weak chirping resulted in an approximately five-fold increase in the field amplitude compared to the un-chirped case reported in the same reference.

To provide a better understanding of THz generation mechanism in the range of the wavelengths under study, the THz range of the plasma radiation spectrum shown in Fig. 3 is depicted in details in Fig. 4.

 

Fig. 4. Detailed THz radiation of plasma emission shown in Fig. 3(k) to (p). The interaction characteristics are the same as explained in Fig. 3. From (a) to (e) fundamental wavelength changed from 800 nm to 3.9 µm. The vertical axis is logarithmic.

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The generated THz spectrum is a wide band radiation extended over 50 THz and it shows a maximum value about plasma frequency below about 15 THz that are regularly observed for all fundamental wavelengths from 800 nm in (a) to 3.9 mm in (e). Although the results represented in Fig. 3 suggests that the asymmetry of current density, J_y, has no effective impact on the overall emission spectrum of plasma, THz region in particular is the case where the described asymmetry plays an important role. This can be considered as one of the important features of laser plasma interaction; whereas the overall spectrum is almost constant, THz region can be amplified by manipulating a set of variables. In Fig. 3(a) and (b) where the current density for the respective wavelengths of 800 and 1064 nm has lower amplitudes, their emission spectrums given in Fig. 4(a) and (b) are approximately peaked at about I_THz= 2.4 × 10⁻⁴. In the case of Fig. 3(d) and (e) simulated for which the fundamental wavelengths are 3.2 and 3.9 µm owning the current densities varied between -6 to 8 and -7.5 to 11 units, respectively, the I_THz corresponding to these two wavelengths indicated in Fig. 4(d) and (e) are 2.5 × 10⁻³ and 3.5 × 10⁻³. This compared to the fundamental wavelengths of 800 and 1064 nm, indicates a growth by one order of magnitude. As a result, a stronger THz emission can be obtained by providing the interaction using longer fundamental wavelengths approaching the MIR region. Moreover, the next advantage of comparison between the plots shown in Fig. 3 and 4 is that the time-dependent asymmetry in the transverse current density, J_y, leads to the THz generation which is obtained for all cases of the fundamental wavelengths. On the other hand, the magnitude asymmetry which is clear mostly in Fig. 3(c), (d) and (e) can be considered as a reason for obtaining a higher amplitude of emission in the corresponding plots of Fig. 4. A closer look at Fig. 3(d) and (e) suggests a new feature of the THz generation in this investigation. Despite a similar pattern and amplitude of J_y observes in either Fig. 3(d) or (e) plots, frequent flowerthorn-like distortions are clearly observed which can be connected to the susceptibility of fundamental wavelengths within the MIR band to the chirping of fundamental inputs. At the first glance, it appears that these fluctuations are a disorder, but on the contrary, they are the main reason for stronger THz emission characterized in Fig. 4(e) than in 4(d). Therefore, the combination of asymmetry and the flowerthorn-like fluctuations in J_y is very important for gaining an efficient THz emission. It is worth noting that in spite of the oscillation seen in other plots in Fig. 3, the lower amplitude of the plasma current density obtained in Fig. 3(a) to (c) is the major reason of THz emission with lower intensity. Approximately, the same fashion of flowerthorn-like fluctuations is raised in the A_p depicted in Fig. 3(i) and 3(j). In conclusion, the results of calculations have confirmed that the THz radiation using the driving wavelength of 3.9 µm and its SH beam reached a value that is about 14 times stronger than the case where the 800 nm and its SH wave are applied to the Ar gas slab. The application of two color MIR laser pulses is reported by Koulouklidis et al. [12] where a wide band emission spectrum from 1 to 40 THz with a maximum at approximately 10 THz was experimentally generated. Hence, the similarity in terms of emitting a wide band THz radiation as well as a plasma emission peak at about 10 THz, between the cited work and the results displayed in Fig. 4(e) again outlines a good validation for the present simulation.

The importance of using the longer wavelength described above is followed by Fig. 5 to examine the effect of chirp coefficient on the THz emission whilst the plasma slab is irradiated by a two color 3.9 µm laser beam.

 

Fig. 5. Variation of the transverse current density of plasma with time. The fundamental wavelength of the two-color laser pulse is 3.9 µm and the input intensities are I_ω=10¹⁵W/cm² and I_2ω=5 × 10¹⁴W/cm². The pulse widths and the beam waists of the fundamental and the SH beam are 40 fs and 40 µm, respectively. The chirp coefficient, b, is changed to three weak values of 10⁻⁴rad/s², 2 × 10⁻⁴rad/s² and 3 × 10⁻⁴rad/s², together in (a) indicated respectively by black, red and blue colors. The highly chirped interaction with greater chirp coefficients of 3 × 10⁻³rad/s², 5 × 10⁻³rad/s²and 9 × 10⁻³rad/s² are plotted in (b), (c) and (d) respectively. Their corresponding THz spectrums are simulated in (e) to (h).

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As Fig. 5 shows, for the three weakly chirped two-color lasers all shown in Fig. 5(a), their time-dependent features and amplitudes are nearly the same and their plasma emission spectrums indicated in part (e) of Fig. 5 are similar. This is differently changed when the chirp is increased with an order of magnitude and reaches a multiple of 10⁻³rad/s² in Fig. 5(b), (c) and (d). It is found that for those cases of higher chirping the amplitude of transverse current density receives an increase of 1.5 times more than that of the weakly chirped ones. Moreover, flowerthorn-like fluctuations in the current density, J_y, are apparent in highly-chirped cases. This is the unique consequence of such interaction when a chirped pulse hits a plasma. It is noteworthy that while the chirp coefficient is increased from 3 × 10⁻³rad/s² in Fig. 5(b) to 9 × 10⁻³rad/s² in Fig. 5(d) the flowerthorn-like fluctuations in the current density, J_y, is more prominent. This becomes clearer if the change of plasma current density inside the red boxes and around the vertical blue dashed line is carefully considered. An increase in the chirp parameter, b, leads to a more prominent flowerthorn-like fluctuations in the local minimum and local maximum of the current density peaks within the delineated boxes and around the indicated dashed line. Figure 5(f) simulated for b = 3 × 10⁻³rad/s² confirms that the magnitude of THz emission maximally reached 4 × 10⁻³. This is while, the maximum value is I_THz is 6 × 10⁻³ for the curve shown in (g) plotted for b = 5 × 10⁻³rad/s², indicating 1.5 times of increase. Interestingly, in the case of b = 9 × 10⁻³rad/s² shown in (h), the THz emission is enhanced to 10⁻². This is very impressive because the plasma emission intensity using the highly chirped 3.9 µm fundamental and its SH pulses is appreciably enhanced by a factor of 40 compared to the case obtained by a two-color 800 nm input pulses. Therefore, it can be claimed that the presence of flowerthorn-like fluctuations in current density leads to the higher amplitude of the plasma emission in the THz region; and, the higher the chirp is, the more efficient THz emission can be achieved. To study the transverse energy distribution of the inputs on the THz intensity characteristics emitted from the induced plasma, a super-Gaussian (SG) profile is assumed for the fundamental and SH pulses. This effect is further investigated for two distinctive chirp parameters of b = 3 × 10⁻⁴rad/s² and b = 9 × 10⁻³rad/s² that have 30 time difference. The electric field of the SG profile is generally written as ${E_j}(y) = {E_{0j}}\exp [ - {(\frac{y}{{{W_0}}})^q}],$ where the subscript j = 1, 2 marks the fundamental and its SH pulses, respectively, and q specifies the degree of SG profile. The simulated results presented in Fig. 6 and Fig. 5(a) indicate that when the q increasingly changed from 2 for an ordinary Gaussian profile to 10 for a flat-top one, the patterns of plasma current density, J_y and the resultant THz emission changed, too.

 

Fig. 6. Variation of the current density of plasma with the change of SG profile for q = 6 and q = 10 represented in (a) and (b), respectively. Their corresponding THz emissions are simulated in (c) and (d). The chirp parameter of the fundamental and its SH beams is weak and fixed at b = 3 × 10⁻⁴rad/s². The y-axis of (c) and (d) plots are provided in a logarithmic form.

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As Fig. 6(a) and (b) results confirm, similar to the case of Fig. 3, the time-dependent asymmetry is seen in both cases of weakly-chirped SG profiles. However, in case of q = 6 the amplitude of density current, J_y, reaches 15 while for q = 10 it increased to 25. In Fig. 6(c) and (d) plots, the THz emission increases from approximately 0.0088 to 0.01 as q changes from 6 to 10, respectively. This results in an approximately 1.13-fold increase in terahertz radiation. The plots in Fig. 6 also show that while the SG profile is changed from 6 in Fig. 6(a) to 10 in Fig. 6(b), the time-dependent asymmetry observed in the magnitude of J_y can be the reason for enhancing the THz generation correspondingly indicated in Fig. 6(c) and (d) plots. Contrarily, in Fig. 3(d) and (c), improving the THz radiation was due to the flowerthorn-like fluctuations. Therefore, in obtaining the highest plasma emission the asymmetry effect induced here by the change of the SG profile is more pronounced than that of the flowerthorn-like fluctuations. Compared to the results discussed in Fig. 5(e), where a normal Gaussian profile with q = 2 was used, the THz emission amplitude characterized in Fig. 6 with the weakly-chirped inputs is significantly improved and reached up to 8 and 8.8 times for q = 6 and 10, respectively. To investigate the combined effect of higher chirp parameters and the change of SG profile, Fig. 7 is provided to indicate the peak variation of plasma emission.

 

Fig. 7. The intensity variation of THz radiation simulated for which the chirp parameter is distinctively changed from weak to high values of 3 × 10⁻⁴rad/s² and 9 × 10⁻³rad/s², respectively, and the SG coefficient is changed in a relatively wide and continuous range from 2 to 10. The intensity of plasma emission is logarithmic. The inset shows the variation of the SG profile.

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As shown, in either case of weakly and highly chirped regimes, the THz amplitude is increased with a similar trend with the raising of SG coefficient, reaching a maximum at about q = 7. This can be interpreted by the fact that, as shown in Fig. 6(b), the use of inputs other than a normal Gaussian form leads to a higher current density. It can be deduced from Fig. 7 that, despite the type of laser profile, highly-chirped inputs result in a THz amplitude of about 0.1, while it approaches 0.01 for weakly-chirped regime. A closer look at the results reveals that the amplitude of THz emission saturates in both regimes of chirping, meaning that the curves are descending at around q = 7. This can be explained in this way that beyond a certain limit of SG number that is q > 7, the flatness of beam profile overcomes the chirping effect to a large extent. In addition, in the weakly-chirped interaction specified by the red curve in Fig. 7, by the increase of SG coefficient from 2 to 4 the emission amplitude is doubled. However, it increases by approximately 1.1 times when the q is changed from 6 to 8. Nearly, the same tendency can be seen for the highly-chirped one with stronger magnitude of emission intensity. The next alternative to further improve the plasma emission is based on the simulation of the cross-focusing scheme. In such interaction the two-color waves containing the fundamental and its SH are chirped and hit the plasma slab from the LHS while the other that is a un-chirped single-color pulse is focused on the medium from the opposite side. The described scheme is displayed in Fig. 8.

 

Fig. 8. Simple schematic display of the cross-focusing interaction where a chirped fundamental and its SH waves enter the Ar gas column from the left and a single-color un-chirped wave hit the medium from the RHS. Except for the SH wave, both pulses in the opposite directions are operating at 3.9 µm. The intensities of the LHS pulses are I_ω= 10¹⁵ W/cm² and I_2ω = 5 × 10¹⁴ W/cm² as before while for the RHS one it is I_ω= 10¹⁴ W/cm². The chirp parameter, b, of the LHS pulses is assumed to be set at the value of 3 × 10⁻⁴rad/s². k_q is the wavenumber of rippled density.

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The point to be noted is that the RHS single-color and un-chirped wave that oppositely enters the column is a pulse with the same properties as the fundamental one but with lower intensity of I_ω = 10¹⁴ W/cm². This pulse is linearly polarized along the y-axis, with an initial phase of 0 degrees, and a carrier-to-envelope phase of zero. The properties of the LHS laser, including its intensity, initial phase, polarization, and carrier-to-envelope phase, are the same as those in Fig. 2 for both the fundamental and SH pulses. The consequent results of this part of investigation are depicted in Fig. 9.

 

Fig. 9. Simulation results of the cross-focusing interaction described in Fig. 8. The plasma density is depicted in (a), the plasma transverse current density, J_y, in (b) and the intensity of THz emission in (c). In (a), k_q, denotes the wavenumber of rippled density. Plasma density is specified by the color bar given in arbitrary unit.

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It is worth mentioning that since the intensity of RHS laser pulse is below the ionization threshold [44], it is not capable of ionizing the Ar gas. What it does is propagating through the gas slab and just modulating the plasma density similar to the vertical striped shape indicated in Fig. 9(a). The longitudinal frequency of this modulation is equal to the wavenumber of the RHS pulsed laser. Besides, the collinear interaction between the modulation frequency and the wave vectors of the applied laser pulses can lead to a new situation which is governed by the phase-matching condition, which can be written as ${k_{out}} = 2k_\omega ^{LHS} - k_{2\omega }^{LHS} + k_\omega ^{RHS} - {k_q}$, where, ${k_{out}} = \frac{{{\omega _{out}}}}{c}[{(1 - \frac{{\omega _p^2}}{{\omega _{out}^2}})^{1/2}} - 1]$ is the wavenumber of the output wave generated due to the phase-matching condition, ω_out, is its frequency, ω_p is the plasma frequency, and k_q is the wavenumber of rippled density. The noteworthy feature of k_out is that it has no emission within the frequency band of investigation and, therefore, no impact on the THz generation. It may result in the generation of a resonance radiation emitting at a frequency that differs from the phase-matched wave vectors with a very narrow bandwidth [45,46].

As a result, because the density modulation is a cause for such radiation outside the THz emission range under investigation, the amplification of THz amplitude is not expected. The phased-matched emission in the presence of laser modulated plasma density is fully discussed and simulated using PIC-MCC method in a previous study [6].

As Fig. 9(b) shows, the described modulation of plasma density with a wave vector of k_q increases plasma current density, Jy, which is approximately three times more than its value that achieved with the non-crossing scheme characterized in Fig. 5(a). The emission spectrum of this scheme simulated in Fig. 9(c), indicates that addition of the RHS pulsed laser has no impact on the THz emission. This is because the frequency of phase-matched emission discussed above is much above the limit we are investigating. The RHS pulse forces the plasma to oscillate at a frequency more than the THz and, therefore, it cannot add to and amplify the radiation at the plasma frequency. To involve the phase-matched radiation in the THz amplification, the plasma emission frequency shall be pushed to a region of higher domain of THz frequencies. One solution is to increase the intensity of the single-color RHS pulsed laser beyond the ionization threshold of Ar gas slab. The consequent results are illustrated in Fig. 10 for the case in which the intensity of the single-color laser is raised up to 10¹⁵ W/cm².

 

Fig. 10. Simulation results performed based on the cross-focusing scheme where the LHS laser is chirped, two-color, and 3.9 µm; and the RHS laser is un-chirped, single-color, and 3.9 µm, oppositely entering the Ar gas slab with the same properties as discussed in Fig. 2. The chirp parameter for the LHS pulses is assumed to be b = 3 × 10⁻⁴rad/s². The intensity of the RHS laser is increased to 10¹⁵ W/cm² while the intensities of the LHS pulses including the fundamental and its SH waves remain constant at I_ω = 10¹⁵ W/cm² and I_2ω = 5 × 10¹⁴ W/cm², respectively. The cross-focusing interaction is simulated at x = 77 µm and y = 70 µm. Color bars indicate the plasma density in arbitrary unit.

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The worthwhile point is that since the intensity of RHS laser is above the ionization threshold, it simultaneously ionizes the Ar gas during the propagation and modulates the plasma density where it interacts with the generated radiations inside the plasma slab. As Fig. 10(a) confirms, the plasma density is modulated more strongly than the case of Fig. 9. However, here the plasma current density shown in Fig. 10(b) is found to be 10 times stronger than the one shown in Fig. 9(b). Meanwhile, part (c) of Fig. 10 indicates a THz emission that is amplified by 4 times. This is because in this case of study at the cross-focusing point placed where x = 77 µm and y = 70 µm, the two-color LHS laser can interact with a plasma induced by the RHS pulses instead of neutral Ar gas. Therefore, it can generate a higher current more effectively as its seen. It should be noted that RHS laser is a single-color laser and cannot be regarded as a THz generation source. Referring to the results previously discussed in Fig. 9, at the first glance it appears that modulating the plasma density enhances the THz spectrum. However, this is against the results which will be discussed in the next section where the rippled density causes the THz emission being degraded.

To better understand the role of density modulation, in the following we will investigate the interaction for which the single-color un-chirped RHS pulse firstly ionizes the plasma and then the two-color chirped LHS pulses interact with such induced plasma. Since the RHS pulse requires to have some delay time for plasma induction in this scheme, it is referred to as delay-cross focusing interaction. Figure 11 displays the simulation results.

 

Fig. 11. Simulation results concerning the plasma current density and the THz emission obtained for the delay-cross focusing scheme. The optical characteristics of the interacting pulses are similar to those explained in the caption of Fig. 9. The simulation is performed at x = 59 µm and y = 70 µm. As characterized, the red and blue boxes in (a) specify the current density produced by the LHS and RHS pulses. The interaction regime is a weakly-chirped one with a chirp parameter of 3 × 10⁻⁴rad/s² and RHS pulse intensity of 10¹⁵ W/cm².

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In the particular position of interaction assumed to be at x = 59 µm and y = 70 µm, the required time for the RHS pulse to firstly ionize the Ar gas and induce the required plasma is about 225 femtoseconds. Soon after this delay the LHS pulses interact with the generated plasma. In this regard, all the energy of LHS laser is consumed to increase the current density of plasma and to accelerate the electrons. As seen and marked in Fig. 11(a), the plasma current density occurred twice, once by the LHS pulses and once again by the RHS laser with a certain delay. However, in part (b) of the Fig. 11, the THz amplitude in the described delay-cross focusing scheme is greatly enhanced and reached about 5.3 times stronger than the cross-focusing interaction obtained in Fig. 9(c). This improvement can be explained by the fact that because LHS laser constantly interacts with the plasma induced by the RHS pulse, it can transfer the maximum asymmetry to the current density instead of consuming its energy for the photo-ionization process. According to Fig. 3, this is what is needed to achieve a stronger THz radiation. A straightforward comparison between the results simulated in Fig. 11(a) and Fig. 10(b) shows that although the amplitude of plasma current density in the case of delay cross-focusing is lower than the cross-focusing one, it contains flowerthorn-like fluctuations in its current. This in turn cause the THz emission to have a higher amplitude in the delay-cross focusing scheme. Figure 11(b) confirms that the received THz is a wideband one. Moreover, compared to the results discussed in Fig. 3(a), here the generated THz is enhanced by two orders of magnitude.

A highly chirped regime of LHS pulse with a coefficient of b = 9 × 10⁻³rad/s² can also be searched if it is enabled to provide a stronger plasma emission. In Fig. 12, a cross-focusing model is simulated for which the LHS pulses are highly chirped and the RHS one is not. The intensity of the RHS pulse is kept at I_ω= 10¹⁴ W/cm² below the threshold of plasma ionization.

 

Fig. 12. Current density and plasma emission spectrum of cross-focusing interaction simulation for the case which two-color LHS pulse is highly chirped with b = 9 × 10⁻³rad/s² while the intensity of the single-color RHS one has increased to I_ω = 10¹⁴ W/cm² below the ionization limit of Ar gas. The intensity characteristics of LHS pulses including a 3.9 µm fundamental wave and its SH wave are similar to those denoted in Fig. 9.

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As it can be clearly seen, the flowerthorn-like fluctuations within the J_y, shown in Fig. 12(a) are quite evident which can cause a plasma radiation with a larger amplitude as discussed in Fig. 5. Even though the RHS intensity is not sufficient to induce a plasma, its presence has made the current density being enhanced. However, comparing Fig. 12(b) and Fig. 5(h) shows that a reduction in the plasma emission has occurred where it decreasingly changed from 10⁻² to 5 × 10⁻³, respectively. This comparison is useful because both Fig. 12(b) and Fig. 5(h) have the same chirp level with the difference that Fig. 5(h) was simulated for a non-cross-focusing scheme. The reason can be explained in a way that similar to the weakly-chirped case simulated in Fig. 8, the RHS pulse forces the generated plasma to oscillate with a frequency far from the THz domain. Similar to the previous case described in Fig. 8, enhancement of the THz emission is seen at a frequency corresponding to the phase-matched frequency. Considering the results of Fig. 12(b) and Fig. 5(h) reveals that in the cross-focusing scheme, the low-intensity RHS pulse reduces the effectiveness of chirping effect. To provide better clarity around the matter, the role of RHS pulse intensity in the cross-focusing interaction has been investigated by increasing the RHS intensity up to one order of magnitude, while the LHS pulses are highly chirped. The consequent results have been brought into Fig. 13.

 

Fig. 13. Simulation of plasma current density (a) and the resultant emission spectrum (b) in which the intensity of un-chirped RHS pulse is increased to 10¹⁵ W/cm². This is while the LHS pulse is a highly-chirped one with a b coefficient of 9 × 10⁻³rad/s². Other specifications of pulses, Ar gas and the xy position of interaction remain unchanged.

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Figure 13 shows that the current density is enhanced but the THz emission is reduced at the plasma frequency. This case is similar to the case of Fig. 12 simulated for highly-chirped LHS fs pulses. This comparison confirms that in highly chirped cross-focusing interactions, the use of RHS laser decreases the emission amplitude but the plasma emission obtains approximately twice the ordinary scheme established without using RHS laser. A deeper understanding of why cross-focusing technique decreases the THz emission is possible by calculating the longitudinal current density for cross-focusing cases discussed so far. The results of this investigation are presented in Fig. 14.

 

Fig. 14. Variation of longitudinal current density with the longitudinal direction, x, while the chirp coefficient is changed from a weak value of 3 × 10⁻⁴rad/s² in (a) to a high value of 9 × 10⁻³rad/s² in (b) both are for the normal two-color interaction without RHS laser. A three-dimensional snapshot of the plasma longitudinal current density given in (c) is for the weakly-chirped cross-focusing scheme including RHS laser with b coefficient of 3 × 10⁻⁴rad/s² for LHS laser. The corresponding two-dimensional sketch of (c) is indicated in (d) at y = 70 µm. The three dimensional sketch of highly-chirped cross-focusing case with b = 9 × 10⁻³rad/s² is shown in (e) and the corresponding two-dimensional plot is displayed in (f) at y = 70 µm. Color bars in (c) and (e) show J_x in the arbitrary unit.

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It is worth noting that in simulating Fig. 14, the intensity of RHS pulse, when present, is kept constant at 10¹⁵ W/cm². In previous studies, utilizing both experimental and analytical methods, have investigated the presence of a longitudinal electric field in the interaction between a chirped laser and plasma [28,47–50]. While these studies have examined the longitudinal field generally, the present paper, has a focus on a sub-picosecond regime, representing a more specific and limited scope of investigation. It is anticipated that the plasma in this regime will exhibit a longitudinal current due to the formation of a longitudinal electric field in chirped laser interactions, as observed in prior studies.

For the normal case of interaction shown in Fig. 14(a) and (b), a gentle and regular change of the longitudinal current density, J_x, is almost seen. In contrast, the variation of J_x indicated in Fig. 14(c) and (d) simulated for weakly-chirped cross-focusing method is very fast and sharp. Such rapid change of J_x reveals that the chirped inputs have been able to accelerate the electrons [36]. This causes the expectation that the longitudinal current density should increase, but it is not the case because the RHS laser traps the electrons and J_x decreases to almost zero. The RHS laser forces the trapped electrons to oscillate by their individual frequencies, leading to a THz emission that is reduced and weak. In a highly-chirped regime of cross-focusing scheme, accelerating plasma-generated electrons is more efficient [37,38]. Therefore, in Fig. 14(e) and (f), the higher density of electrons is accelerated toward the end of plasma column and again are trapped by RHS laser. This can be seen by tracing the current density variation which is approximately between zero and -300. However, in the weakly-chirped case shown in Fig. 14(c) and (d) plots, the maximum variation of J_x is from about zero to about -90. This comparison highlights the effect of chirp on accelerating the electrons and trapping them due to the presence of RHS laser. The consequent effect of the explained trapping is the change of J_x variation form that is shown in Fig. 14(a) and (b) to (d) and (f) plots. The ultimate conclusion is that the trapping mechanism forces the electrons to oscillate at RHS laser frequency differed from THz emission, leading to the decease of THz amplitude by about 50%.

The last investigation is performed to search for a better arrangement that leads to a more efficient generation. One can think of the change of the situation discussed in Fig. 11 from a weakly-chirped to a highly one and increase the RHS intensity to 10¹⁵ W/cm² beyond the Ar ionization. The results of simulation concerning this type of interaction which is called delayed cross-focusing scheme is presented under Fig. 15.

 

Fig. 15. Simulation results of the delayed cross-focusing interaction while the intensity of un-chirped RHS laser is pushed to 10¹⁵ W/cm². The LHS pulses are highly chirped with b coefficient of 9 × 10⁻³rad/s². The other characteristics of RHS pulses are the same as described in the caption of Fig. 11. The results are obtained for x = 59 µm and at y = 70 µm.

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As specified in Fig. 15(a), a delay is observed between the RHS and LHS pulses. This suggests that the single-color un-chirped RHS pulse firstly ionizes Ar gas and after approximately 225 fs the two-color highly-chirped LHS pulse induces a current density. This is the condition that is necessary for THz generation. The results presented in Fig. 15(b) suggest that at a highly-chirped regime of LHS input pulses, the emission amplitude at plasma frequency is enhanced about 1.8 times more than the case simulated in Fig. 5(e) for the ordinary interaction without using the RHS pulse. The ultimate results of all cases discussed in the present study is summarized in Table 1.

Table 1. Summarized results concerning the wide band generation of the THz radiation received based on a model in which a two-color laser, including the fundamental and its SH wave, interacts with a plasma column. The un-chirped RHS pulse is only involved in the cross- and delayed cross-focusing schemes. The weakly- and highly-chirped are denoting the cases of b = 3 × 10⁻⁴rad/s² and 9 × 10⁻³rad/s², respectively. The intensity of fundamental LHS pulse is I_ω= 10¹⁵ W/cm² and remained constant throughout the simulation.

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As Table 1 shows, when the interaction is schemed for cross-focusing and the chirp of the LHS pulses are weak, the THz amplitude is about 43 times higher than the one obtained with the normal interaction at 800 nm simulated in Fig. 3(k). Moreover, by the delayed cross-focusing scheme and weakly chirping the LHS pulses, the THz amplitude is appreciably enhanced by about 233 times the normal interaction. Even though, chirping the inputs in the normal interaction at 800 nm leads to a higher THz by about 33 times at 3.9 µm, the results classified in Table 1 confirms that the use of chirped inputs does not always lead to a desired result. Therefore, the trade-off is that both the type of interaction and the level of chirping should be simultaneously respected.

4. Conclusion

The interaction of a two-color chirped laser pulse is simulated using PIC-MCC method. The target is a slab of neutral Ar gas with the density of N_Ar = 10⁻³ N_c. The emission spectrum of fundamental laser wavelengths of 800, 1064, 1800, 3200 and 3900 nm is reported and their relation between plasma current density and THz amplitude are discussed. It is observed that the emission amplitude enhances more than one order of magnitude for fundamental wavelength of 3.9 µm compared to the 800 nm. The impact of chirped coefficient was studied for the wavelength of 3.9 µm, where the comparison of the weakly-chirped interaction with the highly-chirped one shows that the emission amplitude is enhanced by about 5 times. In both weakly and highly chirped regimes, in addition to the chirping effect on THz emission which is simulated in Fig. 4, it is found that the emission amplitude increases with the increase of SG coefficient from q = 2 for ordinary Gaussian to q = 10 for the flat-top profile. By the combination of chirping the LHS pulses and increasing their SG coefficient shown in Fig. 7, a THz emission of about 10 times higher is obtained for the highly-chirped regime. The next effect is a saturation occurred in the emission amplitude that is seen in both regimes of chirping. For example, in the weakly-chirped interaction it is found that by increasing the SG coefficient, q, from 2 to 4, the THz amplitude can be doubled but when the SG coefficient is changed from 6 to 8, the increase of THz is not significant. A similar effect is also seen for the highly-chirped LHS pulses case. This type of saturation can be connected to the fact that the desired effect of chirping overcomes the sharpness of beam profile.

It is further examined that in the cross-focusing of weakly-chirped two-color laser, adding an un-chirped RHS laser with the intensity of 10¹⁴ W/cm² has no significant impact on THz radiation at plasma frequency but it enhances amplitude of higher THz frequencies more than one order of magnitude. However, it is observed that when the intensity of RHS pulse is raised to 10¹⁵ W/cm², the THz amplitude can be enhanced by 5 times compared to the case in which the RHS pulse in excluded from the interaction; and for the delayed cross-focusing scheme and weakly-chirped regime of LHS laser this enhancement is slightly higher, reaching 5.5 times. In contrast, in the highly-chirped fashion, a destructive effect against the THz growth is found to be the modulation of plasma and trapping the electrons, which are capable of reducing the THz to about 50% at RHS intensity of 10¹⁴ W/cm². However, increasing the RHS intensity to 10¹⁵ W/cm² could not compensate the above decrease. As a result, the delayed cross-focusing interaction operating with a desirable set of input characteristics reported in Table 1 can be assumed as a proper choice for generating a large amplitude and bandwidth of THz emission.