1 Introduction

The terahertz (THz) radiation generation has created a lot of interest in sciences due to its potential applications in biological imaging, remote sensing, chemical and security identification, outer space communication, and spectroscopic identifications of complex molecules [1]. Conventional THz sources were usually obtained by semiconductor photoconductive antenna and optical rectification in nonlinear crystals, while the THz field intensity by these two schemes is relatively low and its spectral range is narrow [2, 3]. Examination of several other mechanisms indicated both longitudinal and radial emission of THz radiation from plasma and provided high-power generation of broadband THz pulses [4,5,6,7,8]. Yet another approach, proposed by Kim et al., elucidates THz generation by combining the fundamental and second harmonic of the laser pulse interacting in a gas medium. The generation mechanism is based on creation of photocurrents, induced by the free electrons, tunnel-ionizing gas molecules [9, 10]. In this model, tunnel ionization and subsequent electron motion will form a directional current density, which is responsible for the intense generation of THz radiation. This model has been verified by experiments and implemented theoretically through simulations [11, 12]. Compared to other methods of THz radiation generation, two-color laser wave mixing through plasma current mechanism provides intense, short, and broadband THz pulses with moderate conversion efficiency [13]. As for the THz waves generated from two-color laser-induced plasma, several means for increasing the THz energy yield emitted by photocurrents have been explored. Despite the proven effectiveness of these methods, there exists yet a direct approach to improve the THz generation in plasma current mechanism, which involves changing of the polarization state of the first and second harmonic components [14]. The polarization state of THz radiation can be altered by varying the relative phase of the driving two-color laser field pulses.

Kosareva et al. studied polarization control of THz radiation from two-color femtosecond gas breakdown plasma [15]. They controlled polarization of laser pulses, linear-to-elliptical, and detected a threshold-like appearance of THz ellipticity at around 85° angle between the fundamental and second harmonics of laser pulse field polarization directions that confirmed the abrupt change of THz polarization. Chenhui Lu et al. investigated effect of two-color laser pulse duration on intense THz generation at different laser intensities [16]. They showed that the photocurrent mechanism plays a crucial role in THz radiation and confirmed that the generation of THz emission depends on the laser pulse duration and the intensity of the two-color laser field. Furthermore, they realized that the THz amplitude increases with enhancement of the laser pulse duration at low laser intensity. Tailliez et al. considered generation of THz pulse by two-color laser fields with circular polarization [17]. They considered four laser configurations of particular polarization and showed when the pump pulses have same helicity, the increase in the THz yield is associated with longer ionization sequences and higher electron transverse momenta obtained in the driving field. Chao Meng et al. carried out enhancement of THz radiation using circularly polarized two-color laser fields [18]. They demonstrated through experiments that the efficiency of THz generation of the circularly polarized laser fields with the same helicity is five times higher than that of linearly polarized two-color femtosecond pulses in high laser intensities. Kuk et al. investigated generation of scalable THz radiation from cylindrically focused two-color laser pulses in air [19]. They focused terawatt, two-color laser pulses in air with spherical and cylindrical lenses and demonstrated that expanding the plasma source into a two-dimensional sheet, leads to scalable THz generation.

Control of wave polarization is a crucial concern in THz radiation. Among the variety of applications of THz waves, elliptically or circularly polarized THz waves are often required in the imaging of birefringent materials, gas molecules and chiral molecules that rotate or vibrate in the THz region. In this survey, a newfound configuration based on a three-pulse design in two-dimensional, ordinary and extraordinary modes, is proposed. The method relies on controlling the pulse duration, increasing the number of colors and optimum tuning of the intensity in plasma. With adjustable time delays between two pulses and phase difference between the ordinary and extraordinary components, various polarizations of THz radiation from gaseous plasma can be obtained. The numerical findings of present study show that the addition of the third harmonic effectively intensifies the THz electric field and the conversion efficiency significantly compared to two-color scheme. The asymmetric three-color femtosecond laser fields can be used to control the polarization state, the required radiation power and the directivity of the THz radiation. Furthermore, variation of electric field of generated THz waves for different laser pulse shapes in two-dimensional photocurrent mechanism has been evaluated. The results indicated that the generated THz radiation can be tuned by regulating the index number and the skew (decentered) parameter for different laser pulse profiles. The organization of this paper is as follows: In Sect. 2, the photocurrent mechanism of two-dimensional THz waves by multi-color femtosecond laser pulses is considered. Discussion on results and properties of THz emission are presented in Sect. 3. Conclusions are drawn in Sect. 4.

1.1 THz radiation mechanism

Consider a scheme of two laser pulse profiles with frequencies \({\omega }_{0}\) and \({s\omega }_{0}\) propagating along the \(z\)-axis. The parameter \(s\) is the harmonic order of the laser pulse. In this scheme, the fundamental laser pulse passes through a birefringent medium such as BBO crystal and its amplitude decomposes into two orthogonal waves with their polarizations lying along the ordinary \(\left(\widehat{o}\right)\) and extraordinary \(\left(\widehat{e}\right)\) axes (Fig. 1). As for the fundamental \({\omega }_{0}\) pulse, suitable time delays and intensity ratio are included in its two orthogonal components in the \(\widehat{e}-\widehat{o}\) plane. This is achieved by splitting off the orthogonally polarized higher-order harmonics of laser pulse with a half-wave plate and rotating its polarization to be parallel to the fundamental pulse polarization before pulses combine again for THz generation. It should be noted that when a fundamental laser pulse passes through a birefringent BBO crystal, the fundamental pulse becomes elliptically polarized after the passage, whereas the higher-order harmonics of laser pulse, produced by phase-matching scheme, are polarized along the \(\widehat{e}\) axis. This setting invokes two-dimensional transverse electron current for THz generation. Therefore, the laser pulse with frequency \({\omega }_{0}\) will co-propagate with its higher harmonics in a plasma medium to ionize plasma molecules. The spatial laser profiles after passing through the BBO Crystal are invariant. However, the temporal profiles of laser pulses alter through the adjustable time delays between the two orthogonal components of the electric fields of the fundamental and higher harmonics. To describe the role of various laser pulse profiles in their interaction with plasma, a spatial laser profile function \(P=\mathrm{Cosh}\left(\frac{yb}{{y}_{0}}\right)\mathrm{exp}\left[-{\left(\frac{y}{{y}_{0}}\right)}^{q}\right]\) is defined, where \(q\) is the index number specifying different laser profiles, \(b\) is the skew (decentered) parameter of laser pulse and \({y}_{0}\) is the laser pulse beam width. Different laser pulse profiles will be accessible through the change of these parameters (e.g., \(b=0,q=2\) refers to Gaussian profile; \(b=0,q=\mathrm{4,6}\) displays super-Gaussian; \(b=1,q=2\) represents cosh-Gaussian; \(b=5,q=2\) indicates the hollow-Gaussian profile and so on). Figure 2 shows the pattern of normalized electric field of laser pulse as a function of normalized distance over laser pulse beam width for different skew and index parameters. It can be noticed from the Figure that the flatness of the top increases with rise in \(q\) index for super-Gaussian and a dip in the pattern peak occurs with increase of the skew parameter \(b\). Consideration of both parameters makes the problem cumbersome; however, one can make a generalized treatment of the laser profile for THz radiation generation and can convert bifocal into single focal radiation or vice versa. Different combinations of these two parameters will be treated as critical constraints of the laser pulses in the present analyses. The electric field of the laser pulse can be written as:

$${\overrightarrow{E}}_{\mathrm{L}}\left(y,z,t\right)=\left[{\overrightarrow{E}}_{1}\left(z,t\right)+\sum_{s\ge 2}{\overrightarrow{E}}_{s}\left(z,t\right)\right]\mathrm{Cosh}\left(yb/{y}_{0}\right){e}^{-{\left(y/{y}_{0}\right)}^{q}},$$
(1)

where \({\overrightarrow{E}}_{1}\left(z,t\right)\) and \({\overrightarrow{E}}_{s}\left(z,t\right)\) are the electric fields of fundamental and higher harmonics of laser pulse, respectively. The three-pulse configuration can then be presented as:

$${\overrightarrow{E}}_{1}\left(z,t\right)={E}_{0}\mathrm{Cos}\left[{k}_{1}z-{\omega }_{0}\left(t-{t}_{\mathrm{de}}\right)\right]{e}^{-2Ln2\frac{{\left(t-{t}_{\mathrm{de}}\right)}^{2}}{{\tau }_{L}^{2}}}\widehat{e}+{E}_{0}\mathrm{Cos}\left[{k}_{1}z-{\omega }_{0}\left(t-{t}_{\mathrm{do}}\right)-{\xi }_{e0}\right]{e}^{-2Ln2\frac{{\left(t-{t}_{\mathrm{do}}\right)}^{2}}{{\tau }_{L}^{2}}}\widehat{o},$$
(2)
$${\overrightarrow{E}}_{s}\left(z,t\right)={E}_{0}\mathrm{Cos}\left[{k}_{s}z-{s\omega }_{0}t-{\delta }_{s}\right]{e}^{-2Ln2\frac{{t}^{2}}{{\left(\sqrt{s}{\tau }_{L}\right)}^{2}}}\widehat{e,}$$
(3)

where \({E}_{0}\) is the amplitude of laser pulse, \({k}_{1}=1/c\sqrt{{\omega }_{0}^{2}-{\omega }_{p}^{2}}\) and \({k}_{s}=1/c\sqrt{{{s}^{2}\omega }_{0}^{2}-{\omega }_{p}^{2}}\) are wave numbers of pulses, \({\omega }_{p}\) is plasma frequency, \({t}_{\mathrm{do}}\) and \({t}_{\mathrm{de}}\) are the adjustable time delays between the two orthogonal components of the electric fields of the fundamental and higher harmonics of laser pulse, \({\delta }_{s}=\sum_{s\ge }{\delta }_{0}+s{\omega }_{o}{t}_{\mathrm{de}}\) is the phase retardation between the \({\omega }_{0}\) and \(s{\omega }_{0}\) harmonics along \(\widehat{e}\) axis, \(\xi ={\xi }_{eo}+{\omega }_{0}\left({t}_{\mathrm{de}}-{t}_{\mathrm{do}}\right)\) is the phase difference between the ordinary and extraordinary components of electric field of fundamental laser pulse, \({\xi }_{eo}\) and \({\delta }_{0}\) are the initial phases, and \({\tau }_{L}\) is pulse duration.

Fig. 1
figure 1

a Schematic of laser plasma interaction for THz generation in multi-color scheme. b Vectors diagram for fundamental (\({\overrightarrow{E}}_{1}\left(z,t\right)\)) and its harmonics (\({\overrightarrow{E}}_{s}\left(z,t\right)\)) along the ordinary (\(\widehat{o}\)) and extraordinary (\(\widehat{e}\)) axes of BBO crystal

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Fig. 2
figure 2

Pattern of normalized electric field of laser pulse as a function of normalized distance over beam width of the laser pulse for different laser profiles

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The THz radiation generation, under the three-pulse scheme, is evaluated based on the transient plasma current model. The tunneling-ionized electrons result in a two-dimensional directional current density \({\overrightarrow{J}}_{\mathrm{THz}}=-e{n}_{e}\left(t\right)\overrightarrow{v}\left(t\right)\). Using the electron cold fluid equations, the current density induced by the laser pulse fields is given by:

$$\frac{\partial {\overrightarrow{J}}_{\mathrm{THz}}}{\partial t}+{\nu }_{e}{\overrightarrow{J}}_{\mathrm{THz}}=\frac{{e}^{2}{n}_{e}\left(t\right)}{{m}_{e}}{\overrightarrow{E}}_{\mathrm{L}}+{\overrightarrow{F}}_{p},$$
(4)

where \({\nu }_{e}\) is the electron–ion collision frequency, \(\overrightarrow{v}\) is the oscillation velocity in the laser field and \({\overrightarrow{F}}_{p}\) represents the ponderomotive force due to the laser pulse and the generated nonlinear driving current density. When the laser spot size is large compared to the plasma filament spot size, the ponderomotive force can be approximated as [20]:

$${\overrightarrow{F}}_{\mathrm{pz}}\approx -\frac{e}{16\pi {m}_{e}}\frac{{\omega }_{p}^{2}\left(t\right)}{{\omega }_{0}^{2}}\overrightarrow{\nabla }{\left|{E}_{\mathrm{L}}\left(y,z,t\right)\right|}^{2},$$
(5)

where \({\omega }_{p}^{2}\left(t\right)=4\pi {e}^{2}{n}_{e}\left(t\right)/{m}_{e}\) is time-dependent local plasma frequency due to ionized electron density. The density \({n}_{e}\left(t\right)\) is time-dependent plasma electron density, which is formed by photoionization as the three-pulse configuration of laser fields is focused on plasma. Besides optical nonlinearities, the plasma response of electron density is modeled by the electron source equation [20, 21]:

$$\frac{\partial {n}_{e}\left(t\right)}{\partial t}=W\left(E\right)\left[{n}_{g}-{n}_{e}\left(t\right)\right]-{\beta }_{r}{\left({n}_{e}\left(t\right)\right)}^{2}-{\beta }_{a}{n}_{e}\left(t\right){n}_{g,}^{2}$$
(6)

where \({n}_{g}\) is the initial gas density, \({\beta }_{r}\) is the recombination coefficient, \({\beta }_{a}\) is the attachment coefficient, and \(W\left(E\right)\) is the ionization rate that is dependent upon the magnitude of the electric field of laser pulse \(\left(\left|{E}_{L}\left(y,z,t\right)\right|\right)\). The last two terms in Eq. (6) were neglected due to the recombination and attachment time scales that are much longer than the pulse duration (\(50 \mathrm{fs}\)). The instantaneous ionization rate is provided by Ammosov–Delone–Krainov (ADK) theory. For hydrogen-like atoms, the rate is reduced to the well-known quasi-static tunneling (QST) rate [21, 22]:

$$\mathrm{W}\left(\mathrm{E}\right)=\frac{{\alpha }_{\mathrm{ST}}{E}_{a.u}}{\left|{E}_{L}\left(y,z,t\right)\right|}\mathrm{exp}\left[-\frac{{\beta }_{\mathrm{ST}}{E}_{a.u}}{\left|{E}_{L}\left(y,z,t\right)\right|}\right],$$
(7)

where \({E}_{a.u}=5.14\times {10}^{9} \mathrm{V}/\mathrm{cm}\) is the electric field in atomic unit, \({\alpha }_{\mathrm{ST}}=4{\omega }_{a.u}{r}_{H}^{5/2}\), \({\beta }_{\mathrm{ST}}=\left(2/3\right){r}_{H}^{3}\), \({\omega }_{a.u}=4.134\times {10}^{16} \mathrm{rad}.{\mathrm{s}}^{- 1}\) is the atomic frequency unit and \({r}_{H}={U}_{\mathrm{ion}}/{U}_{\mathrm{ion}}^{H}\) is the ionization potential of the plasma gas relative to hydrogen atom. In this study, \({U}_{\mathrm{ion}}=15.6\mathrm{ eV}\) (for \({N}_{2}\) gas) and \({U}_{\mathrm{ion}}^{H}=13.6\mathrm{ eV}\). When the first harmonic of laser pulse, in the three-pulse configuration, is co-propagating with other harmonics in plasma, the symmetry of the laser fields is broken through the second and third harmonics. This leads to nonzero current density in the direction of laser fields along the ordinary and extraordinary axes. The free electrons due to the laser electric force, which is a linear force, gain a linear drift velocity (\({v}_{L}\)) and as a result, the transverse current density is linear (\({J}_{L}\)). Therefore, the total two-dimensional linear current density of plasma electrons can be written as:

$${\overrightarrow{J}}_{L}\left(t\right)={J}_{e}\widehat{e}+{J}_{o}\widehat{o},$$
(8)

where \({J}_{e}\) and \({J}_{o}\) are the current density components directed along the \(\widehat{e}\) and \(\widehat{o}\) axes, and can be, respectively, expressed by:

$${J}_{e,o}=-e{\int }_{{t}_{0}}^{t}{v}_{L}\left(t,{t}^{^{\prime}}\right)d{n}_{e}\left({t}^{^{\prime}}\right),$$
(9)

where \({v}_{L}\left(t\right)=-\frac{e}{{m}_{e}}{\int }_{{t}_{0}}^{t}{E}_{L}\left({t}^{^{\prime}}\right)d{t}^{^{\prime}}\). The electrons accelerated at this velocity and subjected to the laser fields, not only oscillate at laser frequencies but also experience a drift velocity. Furthermore, the released electrons gain a nonlinear velocity, as well as a nonlinear current density due to the ponderomotive force. The nonlinear motion of plasma electrons leads to a nonlinear current density:

$${J}_{\mathrm{NL}}=-e{\int }_{{t}_{0}}^{t}{v}_{\mathrm{NL}}\left(t,{t}^{^{\prime}}\right)d{n}_{e}\left({t}^{^{\prime}}\right),$$
(10)

where \({v}_{\mathrm{NL}}\left(t\right)=\frac{1}{{m}_{e}}{\int }_{{t}_{0}}^{t}{F}_{\mathrm{pz}}\left({z,t}^{^{\prime}}\right)d{t}^{^{\prime}}\). Therefore, the total current density due to linear and nonlinear current density can be written as:

$${\overrightarrow{J}}_{\mathrm{THz}}\left(t\right)={\overrightarrow{J}}_{L}\left(t\right)+{\overrightarrow{J}}_{\mathrm{NL}}\left(t\right).$$
(11)

The time-evolving electron current density generates the THz electric field, which is proportional to the derivative of electron current density and can be expressed as:

$${E}_{\mathrm{THz}}\propto \frac{\partial {J}_{\mathrm{THz}}}{\partial t}=e\left[{v}_{L}\left(t\right)+{v}_{\mathrm{NL}}\left(t\right)\right]\frac{\partial {n}_{e}\left(t\right)}{\partial t}.$$
(12)

Thus, the spectrum intensity of THz radiation can be evaluated as \({I}_{\mathrm{THz}}={\left|{\overrightarrow{E}}_{\mathrm{THz}}(t)\right|}^{2}\).

2 Results and discussion

Damagesei threshold and absorbed THz electromagnetic waves in nonlinear crystals and other materials are major problems when strong laser pulses are used. However, these issues are of less concern for THz radiation generated in plasma medium. Furthermore, use of plasma leads to broadband THz production which is suitable for spectroscopy applications. The advent of sub-mJ and ultrafast mid-IR laser pulses made it possible to increase the THz conversion efficiency. There are several means to enhance THz energy yield emitted by photocurrents. These methods include variation of pump pulse duration, reducing the plasma dimensions and modifying the frequency ratio between the two colors. However, the efficiency is still low and alternative methods must be investigated. In present study, THz generation in multi-color laser-induced plasma has been evaluated via the two-dimensional photocurrent mechanism. A novel and effective technique for adaption of different polarization based on three-pulse configuration in ordinary and extraordinary modes with adjustable time delays in different laser profiles is presented.

To evaluate the resulting THz wave radiated spectrum, numerical simulations were considered, which included the tunneling ionization process and the diffused electrons motion under the laser fields. However, inspecting the Keldysh parameter (\(\gamma =\sqrt{{E}_{B}/2{U}_{p}}=\omega \sqrt{2{E}_{B}/I}\)), indicates that the tunneling ionization is the prominent process [23]. Here, \({E}_{B}\) is the field-free binding energy of electron in atom, \({U}_{p}\) is the laser ponderomotive potential energy and \(\omega\) is the frequency of the ionizing field of intensity \(I\). For the intensity regime of interest (\({10}^{14}-{10}^{15} \mathrm{W}/{\mathrm{cm}}^{2}\)), the tunneling ionization becomes the dominant ionization route. Therefore, Ammosov–Delone–Krainov tunneling ionization rate was considered in the present work. The collisional processes during the laser pulse application were also considered, as a field-ionized gas can undergo further ionization through electron–ion and electron–neutral collisional processes. The electrons recombination to ions and electrons attachment to neutral molecules have relatively long lifetimes, hundreds of picoseconds, which are much longer than the pulse length of \(50 \mathrm{fs}\) used in the simulations. Thus, these two processes are excluded in present study. Moreover, the contribution from electrons displacement is not considered since the maximum displacement during the interaction is much smaller than the radiated THz wavelength. In the present simulation, the initial laser intensity was chosen \(3\times {10}^{15} \mathrm{W}/{\mathrm{cm}}^{2}\) at \(\lambda =800 \mathrm{nm}\) and other parameters are the same as those listed in Ref [18]. The relativistic effects were ignored.

In Fig. 3, normalized THz electric field as a function of normalized retarded time over laser pulse duration for two-color, as well as three-color scheme for a Gaussian laser profile in extraordinary and ordinary mode is presented. According to the Figure, the three-color setup provides higher THz radiation electric field amplitude in both extraordinary and ordinary modes. This indicates that addition of third harmonic of laser pulse leads to an increase in THz output and the conversion efficiency is significantly improved. This is due to the asymmetry of the laser electric field. The asymmetry of the laser electric field in three-color scheme compared to two-color setup is more noticeable. The THz radiation generation is initiated by laser-induced ionization of molecules followed by transverse movement of electrons driven by the optical field. The transverse velocity of ionized electrons at the extrema of the laser pulses pump electric fields periodically modifies. As a result, the ionization process and the waveform of the three-color laser fields experience quite different path compared to two-color fields. In each ionization period, due to the combined impact of ionization time and asymmetry in incident waveform, the net orientation, as well as the number and the movement of electrons in the forward direction is more pronounced. Therefore, under equal intensity and laser pulse energy condition, more free electrons will be available using a three-color scheme rather than a two-color field. A larger transverse net current along laser electric field in the forward direction can be generated in three-color laser scheme compared to analogous two-color configuration, which leads to higher THz radiation efficiency. The results of electric field of THz radiation in interaction of three-color laser pulses with plasma, compared to two-color scheme, are in close agreement with the experimental findings of Vaičaitis et al. and Danni Ma et al. [24, 25].

Fig. 3
figure 3

Variations of normalized THz electric field as a function of normalized retarded time over laser pulse duration for two- and three-color schemes for Gaussian laser profile in extraordinary and ordinary modes

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During the process of THz generation, laser parameters including the phase between components, laser profile, and the wavelength have major impact on radiation output and must be selected carefully to achieve the desired efficiency. Plots of ionization rate as a function of normalized retarded time over laser pulse duration for a three-color scheme having different laser profiles and wavelengths are shown in Fig. 4. With modification of laser profile, the ionization rate is changed and for hollow-Gaussian profile higher ionization rate is achieved. Theoretical simulation for the three-color laser scheme indicates that the ionization rate can be increased by selecting the appropriate available laser wavelength, which leads to larger THz output and conversion efficiency. This is due to the fact that more plasma electrons are induced and the electron drift velocity increases as the skew and index parameters along with laser wavelength rise. The normalized electron density as a function of normalized retarded time over laser pulse duration for the three-color scheme is presented in Fig. 5. Variation of electron density for different laser profiles was considered, in which different polarization states of THz wave are included. Due to ionization, occurring near the maxima of electric fields, the electron densities experience a rather stepped increase at microscopic scale and exponentially in macroscopic state. The macroscopic behavior reaches saturation due to accumulated plasma defocusing action. This process is repeated in each period and leads to generation of ionization current which is responsible for THz radiation. According to Fig. 5, for all laser profiles, the ionization rate and electron drift velocity, is maximized for circular polarization. The linear polarization experiences the lowest values of variation of electron density and saturates faster. These can be linked to the mutual effect of extraordinary and ordinary modes. Once the phase difference between the electric fields of two modes becomes closer to that of circular polarization, the drift velocity of electrons increases and more electrons align in forward direction; therefore, the net nonlinear current density increases and larger THz radiation power is generated. Furthermore, the laser fields for larger \(b\) and \(q\) parameters of the laser beams enhance the amplitude of THz field and the efficiency of the mechanism. The index parameter \(q\) and skew parameter \(b\) resolve the laser field profile; thus, play an important role in determination of intensity of THz radiation. The amplitude and direction of THz radiation can be controlled by proper selection of \(q\) and \(b\) parameters for desired application.

Fig. 4
figure 4

Variations of ionization rate as a function of normalized retarded time over laser pulse duration for a different laser profiles and b different wavelengths

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Fig. 5
figure 5

Plot of normalized electron density as a function of normalized retarded time over laser pulse duration for three-color scheme, with different polarization states and laser profiles

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In Figs.6, 7, 8, 9, normalized THz electric field of ordinary mode as a function of normalized retarded time over laser pulse duration for different values of polarization is presented. In these Figures, four different laser profiles including Gaussian, super-Gaussian, cosh-Gaussian, and hollow-Gaussian are considered. It has been demonstrated that the polarization of emitted THz waves can be varied by adjusting the field amplitudes and relative phase difference of two-color laser pulses [26, 27]. The theoretical and experimental results show that, depending on polarization of laser pulses, the polarization of THz radiation is either linear or elliptical. In present study, due to different time delays, the formation of the pulse array of three-color configuration is beyond the conventional two-color scheme. Changing the time delays \({t}_{de}\) and \({t}_{do}\), as well as the phase difference between the ordinary and extraordinary components of fields, different polarizations of THz radiation can be achieved. For any arbitrary \(\xi\), the phase retardation (\({\delta }_{s}\)) between \({\omega }_{0}\) and \({s\omega }_{0}\) along the extraordinary axis can be changed. With suitable time delays and amplitude ratios, any THz radiation polarization can be considered, where a continuous variation of time delays of three field components leads to continuous tuning of polarized THz wave. The two characteristics, the amplitude ratio and the phase difference between the ordinary and extraordinary components, can be used to generate circularly polarized pulses. When the time delays of ordinary and extraordinary modes are zero, i.e., \({t}_{\mathrm{de}}=0\) and \({t}_{\mathrm{do}}=0\), a linearly polarized THz wave is generated. Therefore, the polarization state can be tuned from circular to linear by adjusting the time delays. Plots of normalized THz electric field for different laser beam profiles (Figs.6, 7, 8, 9), indicate that the maximum THz radiation belongs to circular polarization. As mentioned before, if the phase difference between the electric fields of two modes is closer to that of circular polarization, ionization rate increases and not only more electrons are induced but also electrons drift velocity increases. This leads to enhancement of electric field amplitude of THz radiation. Furthermore, the increase of skew and index parameters also enhances the amplitude of electric field. This is due to the ponderomotive force exerted by the laser pulses and the enhanced intensity gradient on electrons. Since the larger nonlinear current density is produced by stronger ponderomotive force, it is clear that the higher amplitude THz field will be created by the use of intense laser pulses. The skew and index parameters of laser beams, through which ponderomotive force can be modified, play a crucial role in determination of laser intensity gradient.

Fig. 6
figure 6

Variations of normalized THz electric field as a function of normalized retarded time over laser pulse duration for Gaussian laser profile in ordinary and extraordinary modes and different polarization states

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Fig. 7
figure 7

Pattern of variations of normalized THz electric field as a function of normalized retarded time over laser pulse duration for super-Gaussian laser profile in ordinary and extraordinary modes and different polarization states

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Fig. 8
figure 8

Variations of normalized THz electric field as a function of normalized retarded time over laser pulse duration for cosh-Gaussian laser profile in ordinary and extraordinary modes and different polarization states

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Fig. 9
figure 9

Pattern of variations of normalized THz electric field as a function of normalized retarded time over laser pulse duration for hollow-Gaussian laser profile in ordinary and extraordinary modes and different polarization states

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Plots of normalized THz electric field as a function of normalized retarded time over laser pulse duration for extraordinary mode and different polarization states are also illustrated in Figs.6, 7, 8, 9. Again, different laser profiles including Gaussian, super-Gaussian, cosh-Gaussian and hollow-Gaussian are considered. As the Figures show, the amplitude of electric fields, for the same condition as the ordinary mode, meaning same intensity and laser pulse energy, are larger for all laser profiles and polarization states, which is due to the enhanced transverse velocity. For the extraordinary mode of the three-color configuration, electrons experience larger ponderomotive force. The larger drift velocity of electrons in forward direction leads to higher nonlinear current density, which enhances the THz wave emission. The numerical results indicate that the extraordinary component of laser pulse electric field propagates in plasma electrons with less energy loss, leading to more efficient production of THz radiation. The plot of normalized THz intensity as a function of normalized retarded time over laser pulse duration for a three-color scheme having different laser profiles and circular polarization state is shown in Fig. 10. From another perspective, these plots demonstrate the efficiency of interaction of three-color laser pulses with a collisional plasma. A significant enhancement in conversion efficiency is observed with appropriate assignment of proper values to the parameters \(q\) and \(b\), while leaving the wave width unchanged. It is clear that the increase of the skew and index parameters of the laser profile leads to an upsurge of the laser field and the ponderomotive force, while the side lobes start to dampen and disappear gradually. As the amplitude of the main peak of the THz electric field enhances, it produces larger wave intensity in forward direction. When the third harmonic of the laser field is present and the phase difference between electric field components along the ordinary axis is such that both can be considered circular, the realization of a large THz ellipticity is still possible. In this case, the Hollow-Gaussian laser pulse has the most effective normalized THz intensity compared to other profiles. Therefore, the three-pulse configuration in ordinary and extraordinary modes is an effective method to increase the efficiency of THz radiation in laser-induced plasma. The efficiency of this mechanism is optimized and controlled by the laser pulses and plasma parameters.

Fig. 10
figure 10

Pattern of variations of normalized THz intensity as a function of normalized retarded time over laser pulse duration for circular polarization and different laser profiles in three-color scheme

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3 Conclusions

A two-dimensional three-color laser pulse scheme for the generation of broadband THz radiation is presented. The method encompasses a novel optical coherent control configuration of polarization states of emitted THz waves using a three-color laser pulse scheme with adjustable time delay in ordinary and extraordinary modes. The corresponding numerical results are compared to experimental findings to confirm the validity of the mechanism. In this method, larger quantities of electrons with faster drift velocity are regarded. Variations of electron density for different laser profiles indicated that ionization typically occurs near the maxima of electric fields in each period. The electron density profiles experience a stepped increase at the microscopic state and vary exponentially in the macroscopic state before reaching saturation. The simulation results of the present scheme supported by experimental observation of different polarization of THz generation reveal the possibility of using asymmetric three-color laser field as a means to control the polarization of THz waves while enhancing conversion efficiency. The component of the extraordinary mode of the generated THz electric field has a larger amplitude compared to ordinary mode due to the presence of third harmonic. The present numerical findings indicated that the profile of the laser pulse, as well as the skew and index parameters of the laser pulse, through which ponderomotive force can be controlled, play an important role in changing the magnitude of laser intensity for the generation of THz radiation.