After we have applied the APML boundary conditions on our basic Maxwell’s equations, in this section, we implement the two-step APML model in our fundamental equations. For this purpose, the discretization of Equations (3.3)-(3.6) is essential, in this regards, the “Yee” staggered-grid Finite-Difference Time Domain (FDTD) discretization in space-time domain is commonly used. Among various FDTD discretization orders, we started with the second-order FDTD discretization. In this context, using the exponential time-stepping ( ${\text{e}}^{-{\sigma }_{E_x}},{\text{e}}^{-{\sigma }_{E_y}},{\text{e}}^{-{\sigma }_{B_x}},{\text{e}}^{-{\sigma }_{B_y}}$) for the electric and magnetic filed, with the assumption that the coefficients ${\bar{\sigma }}_{E_x}={\bar{\sigma }}_{E_y}={\bar{\sigma }}_{B_x}={\bar{\sigma }}_{B_y}=0$, Equations (3.3)-(3.6) in the standard second-order FDTD discretization are written as:

${B_{zx}|}_{i+1/2,j+1/2}^{n+1/2}={\text{e}}^{-{\sigma }_{B_x}\Delta t}{B_{zx}|}_{i+1/2,j+1/2}^{n-1/2}-\frac{1-{\text{e}}^{-{\sigma }_{B_x}\Delta t}}{{\sigma }_{B_x}\Delta x}\frac{C_{B_x}}{c}\left({E_y|}_{i+1/2,k+1/2}^{n+1}-{E_y|}_{j,k+1/2}^{n+1}\right),$ (3.7)

${B_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={\text{e}}^{-{\sigma }_{B_y}\Delta t}{B_{zy}|}_{i+1/2,j+1/2}^{n-1/2}-\frac{1-{\text{e}}^{-{\sigma }_{B_y}\Delta t}}{{\sigma }_{B_y}\Delta y}\frac{C_{B_y}}{c}\left({E_x|}_{i+1/2,j+1}^{n+1}-{E_x|}_{i+1/2,j}^{n+1}\right),$ (3.8)

${E_x|}_{i+1/2,j}^{n+1}={\text{e}}^{-{\sigma }_{E_y}\Delta t}{E_x|}_{i+1/2,j}^n+\frac{1-{\text{e}}^{-{\sigma }_{E_y}\Delta t}}{{\sigma }_{E_y}\Delta y}\frac{C_{E_y}}{c}\left({B_z|}_{i+1/2,j+1/2}^{n+1/2}-{B_z|}_{i+1/2,j-1/2}^{n+1/2}\right),$ (3.9)

${E_y|}_{i,j+1/2}^{n+1}={\text{e}}^{-{\sigma }_{E_x}\Delta t}{E_y|}_{i,j+1/2}^n-\frac{1-{\text{e}}^{-{\sigma }_{E_x}\Delta t}}{{\sigma }_{E_x}\Delta y}\frac{C_{E_x}}{c}\left({B_z|}_{i+1/2,j+1/2}^{n+1/2}-{B_z|}_{i-1/2,j+1/2}^{n+1/2}\right).$ (3.10)

where, $\Delta x\mathrm{,}\Delta y$ are the spatial grid size along $x$ and $y$ direction, respectively, $\Delta t$ is the time step, $i\mathrm{,}j$ are the spatial notation at the center position of the relevant cell, and $n$ is the time step.

Equations (3.7)-(3.10) are the two-step APML formulation in the second-order FDTD algorithm for Equations (3.3)-(3.6).

With the following special choices for the $C_{B_x}$ and $C_{B_y}$ coefficients:

$C_{B_x}=c{\text{e}}^{-{\sigma }_{B_x}\Delta t}\frac{{\sigma }_{B_x}\Delta t}{1-{\text{e}}^{-{\sigma }_{B_x}\Delta t}},$

$C_{B_y}=c{\text{e}}^{-{\sigma }_{B_y}\Delta t}\frac{{\sigma }_{B_y}\Delta t}{1-{\text{e}}^{-{\sigma }_{B_y}\Delta t}}.$

Equations (3.3)-(3.7) are re-written as:

${B_{zx}|}_{i+1/2,j+1/2}^{n+1/2}={\text{e}}^{-{\sigma }_{B_x}\Delta t}\left[{B_{zx}|}_{i+1/2,j+1/2}^{n-1/2}-\frac{\Delta t}{\Delta x}\left({E_y|}_{i+1,j+1/2}^n-{E_y|}_{j,j+1/2}^n\right)\right],$ (3.11)

${B_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={\text{e}}^{-{\sigma }_{B_y}\Delta t}\left[{B_{zy}|}_{i+1/2,j+1/2}^{n-1/2}+\frac{\Delta t}{\Delta y}\left({E_x|}_{i+1/2,j+1}^n-{E_x|}_{i+1/2,j}^n\right)\right],$ (3.12)

${E_x|}_{i+1/2,j}^{n+1}={\text{e}}^{-{\sigma }_{E_y}\Delta t}\left[{E_x|}_{i+1/2,j}^n+\frac{\Delta t}{\Delta y}\left({B_z|}_{i+1/2,j+1/2}^{n+1/2}-{B_z|}_{i+1/2,j-1/2}^{n+1/2}\right)\right],$ (3.13)

${E_y|}_{i,j+1/2}^{n+1}={\text{e}}^{-{\sigma }_{E_x}\Delta t}\left[{E_y|}_{i,j+1/2}^n-\frac{\Delta t}{\Delta x}\left({B_z|}_{i+1/2,j+1/2}^{n+1/2}-{B_z|}_{i-1/2,j+1/2}^{n+1/2}\right)\right].$ (3.14)

Equations (3.11)-(3.14) are a novel two-step APML formulation in the second-order FDTD algorithm for Equations (3.3)-(3.6).

When we push the electric and magnetic field into the vacuum and then reshaping the electric and magnetic field by the exponential coefficients ${\text{e}}^{-{\sigma }_{E_u}\Delta t}$ and ${\text{e}}^{-{\sigma }_{b_u}\Delta t}$ along $u$-direction, respectively, the following modified formulation of Equations (3.11)-(3.14) are explored:

${{\tilde{B}}_{zx}|}_{i+1/2,j+1/2}^{n+1/2}={B_{zx}|}_{i+1/2,j+1/2}^{n-1/2}-\frac{\Delta t}{\Delta x}\left({E_y|}_{i+1,j+1/2}^n-{E_y|}_{i,j+1/2}^n\right),$ (3.15)

${{\tilde{B}}_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={B_{zy}|}_{i+1/2,j+1/2}^{n-1/2}+\frac{\Delta t}{\Delta y}\left({E_x|}_{i+1/2,j+1}^n-{E_x|}_{i+1/2,j}^n\right),$ (3.16)

${B_{zx}|}_{i+1/2,j+1/2}^{n+1/2}={\text{e}}^{-{\sigma }_{B_x}\Delta t}{{\tilde{B}}_{zx}|}_{i+1/2,j+1/2}^{n+1/2},$ (3.17)

${B_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={\text{e}}^{-{\sigma }_{B_y}\Delta t}{{\tilde{B}}_{zy}|}_{i+1/2,j+1/2}^{n+1/2},$ (3.18)

${{\tilde{E}}_x|}_{i+1/2,j}^{n+1}={E_x|}_{i+1/2,j}^n+\frac{\Delta t}{\Delta y}\left({{\tilde{B}}_z|}_{i+1/2,j+1/2}^{n+1/2}-{{\tilde{B}}_z|}_{i+1/2,j-1/2}^{n+1/2}\right),$ (3.19)

${{\tilde{E}}_y|}_{i,j+1/2}^{n+1}={E_y|}_{i,j+1/2}^n-\frac{\Delta t}{\Delta x}\left({{\tilde{B}}_z|}_{i+1/2,j+1/2}^{n+1/2}-{{\tilde{B}}_z|}_{i-1/2,j+1/2}^{n+1/2}\right),$ (3.20)

${E_x|}_{i+1/2,j}^{n+1}={\text{e}}^{-{\sigma }_{E_y}\Delta t}{{\tilde{E}}_x|}_{i+1/2,j}^{n+1},$ (3.21)

${E_y|}_{i,j+1/2}^{n+1}={\text{e}}^{-{\sigma }_{E_x}\Delta t}{{\tilde{E}}_y|}_{i,j+1/2}^{n+1}.$ (3.22)

Equations (3.15)-(3.22) are the two-step time-staggered APML (APML-2SS) formulation in the second-order FDTD algorithm for Equations (3.3)-(3.6).

The two-step time-centered APML (APML-2SC) is an alternative formulation of the (APML-2SS), where the field components are time-centered pushing into the vacuum. In this formulation, Maxwell’s equations are solved in one-time step and the field components are updated by multiplying each of this component by its corresponding damping coefficient ${\text{e}}^{-{\sigma }_n\Delta t}$. Under these considerations, this alternative formulation becomes:

${{\tilde{B}}_{zx}|}_{i+1/2,j+1/2}^{n+1/2}={B_{zx}|}_{i+1/2,j+1/2}^n-0.5\frac{\Delta t}{\Delta x}\left({E_y|}_{i+1,j+1/2}^n-{E_y|}_{i,j+1/2}^n\right),$ (3.23)

${{\tilde{B}}_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={B_{zy}|}_{i+1/2,j+1/2}^n-0.5\frac{\Delta t}{\Delta y}\left({E_x|}_{i+1/2,j+1}^n-{E_x|}_{i+1/2,j}^n\right),$ (3.24)

${{\tilde{E}}_x|}_{i+1/2,j}^{n+1}={E_x|}_{i+1/2,j}^n+\frac{\Delta t}{\Delta y}\left({{\tilde{B}}_z|}_{i+1/2,j+1/2}^{n+1/2}-{{\tilde{B}}_z|}_{i+1/2,j-1/2}^{n+1/2}\right),$ (3.25)

${{\tilde{E}}_y|}_{i,j+1/2}^{n+1}={E_y|}_{i,j+1/2}^n-\frac{\Delta t}{\Delta x}\left({{\tilde{B}}_z|}_{i+1/2,j+1/2}^{n+1/2}-{{\tilde{B}}_z|}_{i-1/2,j+1/2}^{n+1/2}\right),$ (3.26)

${{\tilde{B}}_{zx}|}_{i+1/2,j+1/2}^{n+1}={{\tilde{B}}_{zx}|}_{i+1/2,j+1/2}^{n+1/2}-0.5\frac{\Delta t}{\Delta x}\left({{\tilde{E}}_y|}_{i+1/2,j+1/2}^{n+1}-{{\tilde{E}}_y|}_{i,j+1/2}^n\right),$ (3.27)

${{\tilde{B}}_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={{\tilde{B}}_{zy}|}_{i+1/2,j+1/2}^{n+1/2}+0.5\frac{\Delta t}{\Delta y}\left({{\tilde{E}}_x|}_{i+1/2,j+1}^n-{{\tilde{E}}_x|}_{i+1/2,j}^n\right),$ (3.28)

${B_{zx}|}_{i+1/2,j+1/2}^{n+1}={\text{e}}^{-{\sigma }_{B_x}\Delta t}{{\tilde{B}}_{zx}|}_{i+1/2,j+1/2}^{n+1},$ (3.29)

${B_{zy}|}_{i+1/2,j+1/2}^{n+1}={\text{e}}^{-{\sigma }_{B_y}\Delta t}{{\tilde{B}}_{zy}|}_{i+1/2,j+1/2}^{n+1},$ (3.30)

${E_x|}_{i+1/2,j}^{n+1}={\text{e}}^{-{\sigma }_{E_y}\Delta t}{{\tilde{E}}_x|}_{i+1/2,j}^{n+1},$ (3.31)

${E_y|}_{i,j+1/2}^{n+1}={\text{e}}^{-{\sigma }_{E_x}\Delta t}{{\tilde{E}}_y|}_{i,j+1/2}^{n+1}.$ (3.32)

Equations (3.23)-(3.32) are the two-step time-centered APML (APML-2SC) formulation of the second-order FDTD algorithm for Equations (3.3)-(3.6).

The two-step APML High-order Spatial FDTD (APML-HOFDTD) is an extension formulation for the APML-2SS and APML-2SC where the spatial derivatives of our fundamental Maxwell’s equations are highly-order Finite-Difference Time Domain (FDTD) discretized in the “Yee” staggered spacial domain. Thus, the two-step APML formulation of the high-order FDTD algorithm (APML-HOFDTD) for Equations (3.3)-(3.6) are given by

${B_{zx}|}_{i+1/2,j+1/2}^{n+1/2}={B_{zx}|}_{i+1/2,j+1/2}^{n-1/2}-\frac{\Delta t}{\Delta x}{\displaystyle \underset{l=1}{\overset{p/2}{\sum }}}{c}_l^p\left({E_y|}_{i+l,j+1/2}^n-{E_y|}_{i-\left(l-1\right),j+1/2}^n\right),$ (3.33)

${B_{zy}|}_{i+1/2,j+1/2}^{n+1/2}={B_{zy}|}_{i+1/2,j+1/2}^{n-1/2}+\frac{\Delta t}{\Delta y}{\displaystyle \underset{l=1}{\overset{p/2}{\sum }}}{c}_l^p\left({E_x|}_{i+1/2,j+1+l}^n-{E_x|}_{i+1/2,j-\left(l-1\right)}^n\right),$ (3.34)

${E_x|}_{i+1/2,j}^{n+1}={E_x|}_{i+1/2,j}^n+\frac{\Delta t}{\Delta y}{\displaystyle \underset{l=1}{\overset{p/2}{\sum }}}{c}_l^p\left({B_z|}_{i+1/2+\left(l-1\right),j+1/2}^{n+1/2}-{B_z|}_{i+1/2-l,j-1/2}^{n+1/2}\right),$ (3.35)

${E_y|}_{i,j+1/2}^{n+1}={E_y|}_{i,j+1/2}^n-\frac{\Delta t}{\Delta x}{\displaystyle \underset{l=1}{\overset{p/2}{\sum }}}{c}_l^p\left({B_z|}_{i+1/2,j+1/2+\left(l-1\right)}^{n+1/2}-{B_z|}_{i-1/2,j+1/2-l}^{n+1/2}\right).$ (3.36)

where $c_{\ell }^p\mathrm{,}\ell =\mathrm{1,2,}\cdots ,p/2$ are the Fornberg coefficients [25] for any arbitrary discretization order $p$ of the splited Maxwell’s equations in vacuum. These coefficients are calculated by the Taylor expansion-based closed-form formulas [26] , and given on the staggered grid by [27]

$c_{\ell }^p=\frac{{\left(-1\right)}^{\ell +1}{16}^{1-p/2}\left(p-1\right){\mathrm{<mo>\; !\; </mo>}}^2}{{\left(2\ell -1\right)}^2\left(p/2+\ell -1\right)\mathrm{<mo>\; !\; </mo>}\left(p/2-\ell \right)\mathrm{<mo>\; !\; </mo>}\left(p/2-\ell \right){\mathrm{<mo>\; !\; </mo>}}^2}\mathrm{,}\ell =\mathrm{1,2,}\cdots ,p/2\mathrm{.}$

As clearly listed in the initial beam profile given in Equation (4.1), in whole, the fundamental frequency ${\omega }_0$, the initial input intensity $I_0$, the initial pulse duration ${\tau }_0$, the fractional energy factor $\zeta$, and the relative phase $\varphi$ are the key parameters of this profile. For an efficient THz production, suitable initial values have to be given for these parameters. In this regards, ${\omega }_0=800\text{nm}$ value is necessary given for best coupling between the FH and the SH pulse [29] , and $I_0\approx {10}^{14}~{10}^{17}\text{W}/{\text{cm}}^{\text{2}}$ period is normally selected, since this period is above the threshold intensity of the air gases ionization and below the relativistic intensity to avoid the relativistic dynamics, and ${\tau }_0=40\text{fs}$ is the pulse duration value where the yield and the efficiency of the produced THz radiation is enhancing [20] in the presence of the pulse chirping effects. With respect to $\zeta$ and $\varphi$, in actual fact, the suitable values of these two parameters can’t be given, but it should be optimized in accordance with the conditions of the problem of our interest. In our previous studies [30] and other similar researches [31] [32] , $\zeta =0.5$ value is acknowledged to avoid the energy exchange between the two fs beams and arrest the Modulation Instability (MI) effects [33] , and $\varphi =\pi /2$ is identified for an efficient THz radiation within the selected input intensity period.

To examine the performance of the derived formulations, in this section, we re-optimized the suitable fractional energy factor $\zeta$ and the relative phase $\varphi$ value for an efficient THz production. In this respect, we simulated the excitation energy ${\mu }_{THz}$ of produced THz radiation as function of $\zeta$ and $\varphi$, the possible energy fraction factor $\zeta =0\text{-}1$ and relative phase $\varphi =0\text{-}\pi$ values are considered in this simulation, and an initial input intensity value withing the selected period $I_0={10}^{15}\text{W}/{\text{cm}}^{\text{2}}$ is given, the result of this simulation is displayed in Figure 1 .

As clearly noted in the ${\mu }_{THz}$ spectrum demonstrated in Figure 1 , the maximum THz radiation is approximately obtained at $\zeta \approx \pi /2$ and $\varphi \approx \pi /2$ for the three formulations. It should be emphasized that, although the same acknowledged $\zeta$ and identified $\varphi$ value are also obtained for the three formulations in this figure, the APML-HOFDTD spectrum around these values is more refined than APML-2SS and APML-2SC formulation.

The optical-to-THz radiation conversion efficiency ${\eta }_{THz}$ is the ratio between the intensity of the produced THz radiation ${\left|E_{THz}\right|}^2$ and the input beam intensity ${\left|E_0\left(r\mathrm{,}t\right)\right|}^2$ [34] ,

${\eta }_{THz}={\displaystyle {\int }_0^{THz}}{\left|E_{THz}\left(\omega \right)\right|}^2\text{d}\omega /{\displaystyle {\int }_0^t}{\left|E_0\left(r\mathrm{,}t\right)\right|}^2\text{d}t\mathrm{.}$

For an efficient THz radiation production, a considerable conversion rate is crucially required. For this purpose, numerous researches have been carried out [29] [35] in order to calculate the conversion efficiency and analyze its behavior at different input beam parameters and air plasma structures. ${\eta }_{THz}={10}^{-3}\text{-}{10}^{-4}$ conversion rate is achieved in these researches, which is recognized as standard conversion efficiency value in THz radiation production topic.

In order the check the accuracy of the three derived formulations, herein we carried out a numerical comparison between the conversion efficiency as function of the selected input intensity period obtained by theses formulations and the standard ${\eta }_{THz}$ conversion [36] [37] . The optimum energy fraction factor $\eta =0.5$ and relative phase $\varphi =\pi /2$, in addition to the suitable initial pulse duration ${\tau }_0=40\text{fs}$ are considered in this comparison, the result of this comparison is shown in Figure 2 .

As clearly seen in Figure 2 , at low input intensity range, the three formulations have almost exact behavior to the standard conversion efficiency, moreover the maximum efficiency value obtained by APML-HOFDTD is approximately identical with the standard value, while the maximum values obtained by the APML-2SS and APML-2SC are little lower. At the high input intensity range, a discrepancy between the three formulations behavior and the standard one is emerged, although of that, the APML-HOFDTD has an insignificant discrepancy, while the APML-2SS and APML-2SC have a growing discrepancy with increasing the input intensity in this range.

The Self-Phase Modulation (SPM) is the variation of the pulse phase as it is propagating [38] . Due to this variation, an addition time-dependent phase is added ( $\Delta {\varphi }_{SPM}$). In nonlinear optical medium, the SPM is originated due to the Kerr-nonlinearity effect [38] where intensity is temporarily varying by the own pulse and due to the plasma induced SPM effect [20] where time-varying electron density $n\left(t\right)$ is evolved due the ionization. In both effects, and time dependent refractive index $n\left(r\mathrm{,}t\right)=n_0+\Delta n\left(t\right)$ is established. The presence of the SPM leads to spectral broadening of the pulse due to the generation of extra frequencies $\Delta \omega$ that are added to the central frequency ( ${\omega }_0$) of the pulse $\omega \left(t\right)={\omega }_0+\Delta \omega \left(t\right)$. In this part, we checked the reliability of the three applied formulations in monitoring this broadening. In this context, we presented in Figure 3 a time-evolution for the produced THz radiation envelope where the three formulations are applied. Physical assumptions and considerations are assumed where a laser-envelope evolution which has a linear phase modulation with a symmetric frequency variation from the central frequency of the pulse is proceeding.

As illustrated in Figure 3 , in a comparison of the time-evolution among the three envelopes, it clearly noted a small spectral broadening is ongoing with the APML-2SS and APML-2SC, while a clear spectral broadening is proceeding with the APML-HOFDTD formulation.