3.1. Harmonic generation in O-mode configuration $({\boldsymbol{E}}_l \parallel {\boldsymbol{B}}_0)$

We first consider the case when the frequency of the incident laser pulse was chosen to be $0.4 \omega _{pe}$ and the polarization of the laser fields was considered to be in O-mode configuration, i.e. ${\boldsymbol {E}}_l \parallel {\boldsymbol {B}}_0$ (in $\hat z$ direction). Here, $\boldsymbol {E}_l$ is the laser electric field and $\boldsymbol {B}_0$ is the externally applied magnetic field. The transverse $\hat y$ and $\hat z$ components of the magnetic field, $B_y$ and $B_z$, are shown by red solid lines for $B_0 = 2.5$ at a particular instant of time $t = 1000$ in figures 2(a) and 2(b), respectively. It is to be noted that at time $t = 0$, a laser pulse with EM fields $B_y$ and $E_z$ was set to propagate along the positive $\hat x$ direction from the location $x = 950$. Thus, the structure in $B_y$ present in the vacuum region ($x \approx 200$) at $t = 1000$, as seen from 2(a), is associated with the reflected part of the incident laser propagating along the $-\hat x$ direction. A small fraction of $B_y$ is also present inside the bulk plasma, as depicted in the zoomed scale in 2(a1). We will identify this structure as the third-harmonic radiation (and the higher odd harmonics) in the following section. In 2(b), it is seen that the $\hat z$ component of the oscillating magnetic field $B_z$ which is not present before the laser beam impinges on the plasma surface, has been produced at a later time and exists in both vacuum and bulk plasma. There are two types of disturbances observed inside the bulk plasma, as can be seen from 2(b). One is the large-scale disturbance near the plasma surface, which will be identified as the magnetosonic perturbation, and the other is a disturbance moving with the faster group velocity. In the following discussion, we will show that this is the second-harmonic radiation (even harmonics) generated due to the interaction of laser pulse with plasma particles. The simulation observations for $B_0 = 0$ are also depicted in figure 2 by blue dotted lines. It is seen that in this case, also the $\hat y$ component of the magnetic field, i.e. $B_y$, is present in both reflected and transmitted radiations. However, no $\hat z$ component of the magnetic field, i.e. $B_z$, is generated in this case, as can be seen in 2(b).

Figure 2. Transverse time-varying magnetic fields (a) $B_y$ and (b) $B_z$ with respect to $x$ are shown at a particular instant of time $t = 1000$ (when the laser beam already gets reflected back from the system). In (a1), $B_y$, which exists inside the plasma, is shown on a different scale. Here, the black dotted line at $x = 1000$ represents the plasma surface. It is to be noted that the EM fields $\boldsymbol {B_l}$ and $\boldsymbol {E_l}$ of the incident laser pulse are along the $\hat y$ and $\hat z$ directions, respectively. Red lines represent $B_0 = 2.5$ with $E_l \parallel B_0$ and blue dotted lines $B_0 = 0$.

The time fast Fourier transform (FFT) of these reflected and transmitted radiations is shown in figure 3. The FFTs of $E_z$ and $B_y$ at the location $x = 500$ (vacuum) show two distinct peaks at frequencies $\omega \approx 0.4\omega _{pe}$ and $\omega \approx 1.2\omega _{pe}$ (figure 3a). It is important to recall that at $t = 0$, the incident laser pulse was located between $x= 750$ and $950$. The first peak with higher power at the location $\omega \approx 0.4\omega _{pe}$ is essentially the original laser pulse which has been reflected from the plasma surface, as the plasma is overdense. The second peak located at $\omega \approx 1.2\omega _{pe}$ is the third-harmonic radiation. The third harmonic is also present inside the bulk plasma and has been demonstrated by carrying out the FFT in time for the $E_z$ and $B_y$ signals at the location $x = 2000$ (plasma), as shown in figure 3(b). Thus, it is now clear that the small disturbance in $B_y$ present inside the plasma, as shown in figure 2(a1), is essentially associated with the third-harmonic radiation. The FFTs of $E_y$ and $B_z$ in time at the locations $x = 500$ and $2000$ are shown in figures 3(c) and 3(d), respectively. The FFT at the location $x = 2000$ (plasma), in 3(d), has been evaluated within a time window $t = 1000$ to $3000$. This is to eliminate the slowly moving magnetosonic disturbance as shown in figure 2(b). It is seen that the frequency spectrum of $E_y$ and $B_z$ has a distinct peak at the location $\omega \approx 0.8\omega _{pe}$ in both cases (vacuum and plasma). This ensures that the second-harmonic radiation has been generated and travels in both vacuum and plasma mediums. The observation of second and third harmonics in the reflected radiation in the presence of a transverse external magnetic field had been reported in an earlier study by Mu et al. (Reference Mu, Li, Sheng and Zhang2016). However, in our study, we have observed the harmonic radiations in both reflected and transmitted radiations, as shown in figure 3. In the case without an external magnetic field, the third harmonic (odd harmonics) shows up with polarization ($E_z$, $B_y$) in both reflected and transmitted radiations, as shown in figure 3(e,f). It is to be noticed that, unlike a magnetized case, no harmonics with polarization ($E_y$, $B_z$) are generated in the absence of an external magnetic field, as shown in figure 2(b). In the case with $B_0 = 0$, the existence of so-called selection rules for the polarization of harmonic generation reflected from an overdense plasma surface was predicted by Lichters et al. (Reference Lichters, Meyer-ter Vehn and Pukhov1996). Their study shows that for a normal incident linearly polarized laser, only the odd harmonics with linear polarization appear in the reflected radiation. The analogy of third-harmonic generation reported for the present study is the same as reported by Lichters et al. (Reference Lichters, Meyer-ter Vehn and Pukhov1996). However, here we have observed and characterized them in both reflected and transmitted radiations. It is interesting to note that we have shown the FFT of transverse electric and magnetic field components in each subplot as a pair. This is done to show the EM nature of harmonic radiations. It is also important to observe that the polarization of third-harmonic radiation is the same as that of the incident laser pulse. In contrast, the polarization of the second-harmonic radiation is different from that of the incident laser pulse. In § 3.3, we discuss the reason behind this.

Figure 3. Fourier transform of EM fields with time after the laser beam is reflected from the plasma surface. The FFT of $E_z$ and $B_y$ with time in (a) vacuum ($x = 500$) and (b) the bulk plasma ($x = 2000$). (c,d) The same for the fields ($E_y$, $B_z$). In (a)–(d), the external magnetic field ($B_0$) is considered to be $2.5$. These FFTs clearly indicate that higher harmonics have been generated, and they are present in both vacuum and the bulk plasma. The time FFT of $E_z$ and $B_y$ without any external magnetic field ($B_0 = 0$) in (e) vacuum and (f) plasma.

The conversion efficiencies of second and third harmonics are provided in table 2 for different values of $a_0$ ($=eE_l/m\omega _lc$) and laser frequency $\omega _l$. The conversion efficiency has been calculated by taking the ratio of spatially integrated EM field energy of the harmonic to that of the incident laser pulse. We have observed that the conversion efficiency solely depends on the strength of the laser fields for a given magnetic field. Keeping $a_0$ constant when we increase $\omega _l$, field strength increases, and so do the efficiencies of the harmonics. On the other hand, as we keep the value of $(a_0\omega _l)$ constant for a different combination of $a_0$ and $\omega _l$, the field strength of the incident laser pulse remains the same, and so does the conversion efficiency. This is clearly shown in table 2.

Table 2. Conversion efficiencies of harmonics in O-mode configuration of incident laser pulse for $B_0=2.5$.

3.2. Harmonic generation in X-mode configuration $(\boldsymbol{E}_l \perp \boldsymbol{B}_0)$

The higher harmonics can also be observed for the case when the polarization of the incident laser pulse is chosen to be in the X-mode configuration, i.e. ${\boldsymbol {E}}_l \perp {\boldsymbol {B}}_0$. This is clearly illustrated in figure 4. A laser pulse with frequency $0.4 \omega _{pe}$ was set up initially ($t = 0$) to propagate in the $\hat x$ direction from the location $x = 950$. In this case also, the external magnetic field $B_0$ has been chosen to be $2.5$ and applied in the $\hat z$ direction. It is to be noticed that for our chosen values of system parameters, the frequency of the incident laser pulse lies between the left-hand cutoff ($\omega _L = 0.376$) and upper hybrid frequency ($\omega _{UH} = 2.7$). Thus, the incident laser pulse with polarization in the X-mode configuration will match the passband of the plasma X-mode dispersion curve. As a result, a significant part of the incident laser pulse penetrates inside the bulk plasma. This is clearly illustrated in figure 4(a) where we show the $z$ component of the magnetic field $B_z$ as a function of $x$ at a particular instant of time $t = 1000$. It is also seen that a part of the incident pulse gets reflected from the vacuum–plasma interface and propagates in the $-\hat x$ direction in the vacuum. The other part of the incident laser radiation penetrates the plasma surface and propagates through the medium. It can be observed that a small disturbance, as highlighted by the dotted rectangular box in figure 4(a), is also present, which moves with a higher group velocity in the plasma medium. These are essentially the higher harmonics generated by the plasma. The FFTs in time for this signal of the transverse fields $E_y$ and $B_z$ in figure 4(b,c) corroborate this. The FFTs of $E_y$ and $B_z$ inside the plasma ($x = 2500$), as shown in figure 4(c), are evaluated within a time window $t = 200$ to $2200$. This choice eliminates the originally transmitted laser pulse with frequency $0.4\omega _{pe}$ (having higher power) from the frequency spectrum. Two distinct peaks observed at $\omega \approx 0.8\omega _{pe}$ and $1.2\omega _{pe}$ in figure 4(b,c) correspond to the second and third harmonics, respectively. It is interesting to notice that unlike the previous case (O-mode configuration), here the polarization of the higher harmonics, both second and third, is the same as that of the incident laser pulse.

Figure 4. The generation of higher harmonics is depicted here for the case where the polarization of incident laser has been chosen to be in the X-mode configuration, i.e. $\boldsymbol {\tilde {E}}_{l} \perp \boldsymbol {B}_0$. Here, we have considered $B_0 = 2.5$. (a) The EM part of the magnetic field along $\hat z$, $B_z$, at a particular instant of time $t = 1000$. (b) The FFT of $E_y$ and $B_z$ at the location $x = 500$ (vacuum). It is clearly seen that in addition to the original reflected laser field ($\omega \approx 0.4$), higher harmonics ($\omega \approx 0.8, 1.2$) are also present in the reflected radiation. (c) The existence of these higher harmonics inside the bulk plasma, where the FFTs have been performed at the location $x = 2500$.

3.3. Mechanism of harmonic generation in a magnetized plasma

Let us now understand the mechanism of the harmonic generation. The fundamental mechanism of high-order harmonic generation at the vacuum–magnetized plasma interface has been previously studied by Mu et al. (Reference Mu, Li, Sheng and Zhang2016). In the present work, we briefly discuss the same extending it for both O- and X-mode configurations of the incident laser pulse. Additionally, we have also developed a simple approximate mathematical model for a qualitative description of the observed characteristics in both O- and X-mode configurations, which is provided in Appendix A.

When a laser pulse with O-mode configuration, i.e. ${\boldsymbol {E}}_l \parallel {\boldsymbol {B}}_0$, is incident on the vacuum–plasma interface, plasma particles will experience a force ($\propto \tilde {v}_{z}\tilde {B}_{l}\exp ({\textrm {i}2\omega _l t)}$) due to the Lorentz force (${\boldsymbol {v}} \times {\boldsymbol {B}}_l$) along the $\hat x$ direction. Here, ${\boldsymbol {v}}$ is the particle quiver velocity initiated due to the laser electric field $\boldsymbol {E_l}$ in $\hat z$. As a result, plasma electrons wiggle, forming an oscillating current at the surface of the plasma in the $\pm \hat x$ direction with a frequency twice the incident laser frequency (equation (A10a,b)). This can also be understood by recalling that in the intense EM fields of the incident laser, electron motion would initially follow the well-known ‘figure-of-eight’ path, which contains a transverse component with a frequency $\omega _l$ and a longitudinal component with frequency $2\omega _l$ (Teubner & Gibbon Reference Teubner and Gibbon2009). The electrons being magnetized in the presence of the external magnetic field $B_0\hat {z}$, revolve in the $x$–$y$ plane. Thus, the oscillatory motion of electrons along $\hat x$ generated by the process discussed above is coupled with the motion along $\hat {y}$, having the same frequency. Consequently, an oscillating current with a frequency twice the laser frequency is produced in the $\hat y$ direction, as shown in figure 5(a1,a2). This acts as an oscillating current sheet antenna and radiates EM waves with fields $\tilde {E}_y$ and $\tilde {B}_z$ propagating along both $\pm \hat x$ directions. These are the second harmonics that have been captured in our simulations in both vacuum and plasma regimes. It is easy to understand that the oscillating current sheets in the $\hat y$ direction might have all the even higher harmonics depending upon the strength of the nonlinearity involved in the electron dynamics. In our simulations, we have captured up to the sixth harmonic and can be easily seen in the FFT of $J_{ey}$ in figure 5(a2).

Figure 5. The time evolution of (a1) $y$ component and (b1) $z$ component of electron currents $J_{ey}$ and $J_{ez}$ at the vacuum–plasma interface ($x = 1000$), respectively. The FFTs of (a2) $J_{ey}$ and (b2) $J_{ez}$. Here, the red solid lines and blue dotted lines represent the cases with $B_0 = 2.5$ and $B_0 = 0$, respectively.

In addition, the electron motion along $\hat x$ will couple to the laser magnetic field $\tilde {B}_l$ (which is along $\hat y$ for O-mode configuration). As a result, it will create an oscillating current sheet along $\hat z$ with a frequency three times the laser frequency and also at higher odd harmonic values, as shown in figure 5(b1,b2). Thus, another radiation will be produced propagating in the $\pm x$ direction, but, in this case, the EM fields associated with this radiation are $\tilde {E}_z$ and $\tilde {B}_y$. It is to be noticed that ideally, all the odd higher harmonics might be present in $J_{ez}$. In our simulations, we have identified up to the fifth harmonic, as shown in figure 5(b2). It is to be noticed that the external magnetic field is necessary to generate the $\hat y$ component of surface current $J_{ey}$, oscillating with the even harmonic frequencies, producing the even higher-harmonic EM radiations. This is shown by red solid and blue dotted lines in figure 5(a1,a2) and is also consistent with the approximate mathematical expression of $J_{ey}$ in (A10a,b). On the other hand, the $\hat z$ component of surface current, i.e. $J_{ez}$, oscillating with odd harmonic frequencies is produced only due to the nonlinear electron dynamics in the laser fields and does not require any external magnetic field (Bulanov et al. Reference Bulanov, Naumova and Pegoraro1994; Lichters et al. Reference Lichters, Meyer-ter Vehn and Pukhov1996; Teubner & Gibbon Reference Teubner and Gibbon2009), as shown in figure 5(b1,b2). This is also apparent from the expression of $J_{ez}$ in (A13).

For the laser polarization in the X-mode configuration, the magnetic field $\tilde {B}_l$ of the laser is parallel to ${B_0}$ (along $\hat z$). For this case, using similar arguments, both second and third harmonics (in fact, all the high harmonics) will be generated for which the EM fields are $\tilde {E}_y$ and $\tilde {B}_z$. Thus, the generated harmonic radiation also has the X-mode configuration. This is exactly what we have observed in our simulations, as shown in figure 4.

As mentioned in § 2, in most of our simulation studies we have considered the electron temperature to be very low ($T_e = 0.05$ eV). However, we have also done a comparative study by considering different values of electron temperature to check if the fundamental mechanism of harmonic generation has any dependence on the plasma temperature. The time FFTs of surface currents $J_{ez}$ and $J_{ey}$ are shown for different values of electron temperature ($T_e = 0.05$–500 eV) in figures 6(a) and 6(b), respectively. It is seen that surface currents oscillating with high harmonic frequencies have been generated in all cases. There is no significant change observed in the generation of high-harmonic radiations for different values of electron temperature.

Figure 6. The FFTs in time for (a) $\hat {z}$ component of surface current $J_{ez}$ and (b) $\hat {y}$ component of surface current $J_{ey}$ at the vacuum–plasma interface ($x=1000 c/\omega _{pe}$) for different electron temperatures ($T_e=0.05$, 5, 50 and 500 eV).

3.4. Effect of external magnetic field on harmonic generation

We have shown in the previous section that the generation of harmonics depends on the surface current oscillations, and odd harmonics can be generated even when there is no external magnetic field. However, it has been shown that an external magnetic field is necessary to generate higher even harmonics. In this section, we show that the amplitude of harmonics (both even and odd) actually has a strong dependence on applied external magnetic fields.

For this analysis, we have done a comparative study by simulating the same geometry with $\omega _l=0.4\omega _{pe}$ for different values of the external magnetic field. In each run, we have obtained the amplitude (peak value) of FFT from time-series data of $J_{ey}$ and $J_{ez}$ corresponding to the second- and third-harmonic frequency, respectively. Next, we have calculated the absolute value of second- and third-harmonic current density from the approximated model given in Appendix A, i.e. (A10a,b) and (A13), for different magnetic field values. We have plotted these two quantities as a function of the external magnetic field $B_0$ as shown in figure 7. One can observe that the trends are similar for quantities obtained from simulation and the theoretical model. This plot provides a qualitative understanding of the mechanism presented in this paper. This observation also establishes that harmonic generation in plasma is boosted in the presence of an external magnetic field. There is an optimum value of the magnetic field for which better efficiency of harmonic generation can be found, and this value is where $\omega _{ce} \rightarrow 2\omega _{l}$.

Figure 7. (a) The peak value of the FFT spectrum of $J_{ey}$ corresponding to the second-harmonic frequency and theoretical value of $|{J_{ey}}|$ obtained from (A10a,b). (b) The peak value of the FFT spectrum of $J_{ez}$ corresponding to the third-harmonic frequency and theoretical value of $|{J_{ez}}|$ obtained from (A13).

3.5. Characterization of harmonics

We now analyse in further detail and characterize the high-harmonic radiations. As has been shown previously, these higher harmonics can be observed for the laser polarization in both O-mode (${\boldsymbol {E}}_l \parallel {\boldsymbol {B}}_0$) and X-mode (${\boldsymbol {E}}_l \perp {\boldsymbol {B}}_0$) configurations. We consider here the observations corresponding to the O-mode configuration. As discussed in previous sections, the higher-harmonic radiations are EM. Thus, in vacuum, these radiations will travel with the speed of light, i.e. the frequency and wavenumber (in normalized units) will have the same values. On the other hand, when they propagate through the plasma medium, they will have to follow the appropriate dispersion relation of that medium.

Thus, the group velocity and the phase velocity of these radiations will have different values depending upon the dispersion relation. These properties are clearly shown in figure 8. The spatial FFTs of the transverse fields ($E_z$, $B_y$) for vacuum and bulk plasma are shown in figures 8(a) and 8(b), respectively. It is clearly seen from figure 8(a) that, as expected, the reflected laser pulse ($\omega _l = 0.4$) and the third-harmonic radiation ($\omega \approx 1.2$) are associated with the wavenumbers $k_x \approx 0.4$ and $k_x \approx 1.2$, respectively, as they travel in vacuum. On the other hand, the spectrum of the spatial FFTs of $E_z$ and $B_y$ in the bulk plasma region shows a distinct peak at a particular value of $k_x \approx 0.67$. It is interesting to realize that the third-harmonic radiation is associated with the transverse electric field ($E_z$) parallel to the external magnetic field $B_0$ and travelling perpendicular to $B_0$. Thus, it matches the condition of plasma O-mode. We have evaluated the theoretical dispersion curve for the O-mode for our chosen values of system parameters, as shown in figure 8(e). It is seen that the value of wavenumber corresponding to the frequency $\omega = 1.2$ is approximately equal to $0.66$. Thus, it matches well with the properties of third-harmonic radiation observed in our simulation inside the bulk plasma region.

Figure 8. Fourier transforms in space of the EM fields after the laser beam is reflected from the vacuum–plasma interface. The FFT of ($E_z$, $B_y$) along $\hat x$ at time $t = 600$ and $1600$ for (a) vacuum and (b) bulk plasma. (c,d) The same for the fields ($E_y$, $B_z$). Dispersion curves (Boyd, Boyd & Sanderson Reference Boyd, Boyd and Sanderson2003) of (e) O-mode and (f) X-mode for the chosen values of the system parameters of this study.

The FFTs of transverse fields $E_y$ and $B_z$ in space associated with the second-harmonic radiation are depicted in figures 8(c) and 8(d) for vacuum and bulk plasma, respectively. The FFT spectrum reveals that the second-harmonic radiation ($\omega \approx 0.8$) propagates with a finite wavenumber in a vacuum, $k_x \approx 0.8$, which is expected as it has to travel with the speed of light. On the other hand, inside the plasma, it is associated with a different wavenumber $k_x \approx 0.76$. The theoretical model analysis again affirms that this second-harmonic radiation matches the condition for plasma X-mode and propagates through the passband lying between left-hand cutoff ($\omega _L$) and upper hybrid frequency ($\omega _{UH}$) of the X-mode dispersion curve. This is demonstrated in figure 8(f).

The mode analysis of harmonic radiations has a direct significance in the sense that we can now have control over the excitation of these radiations inside the plasma medium. For a given set of plasma parameters, if we change the value of $B_0$, the dispersion curves of the plasma modes (figure 8) will be modified accordingly. Thus, the value of $B_0$ determines whether harmonic radiation with a particular frequency generated at the vacuum–plasma interface will be able to transmit inside the plasma or not. Alternatively, if the value of $\omega _l$ is changed, the frequency of the higher harmonics will also be modified accordingly. Thus, there might be some situations where these harmonics will not be allowed to pass through the plasma medium for a particular value of $B_0$. For instance, we have considered a particular case where the frequency of the incident laser pulse is chosen to be $0.2 \omega _{pe}$ and all other system parameters have been kept the same as for the previous case. It has been observed that the second-harmonic radiations initiated at the vacuum–plasma interface are travelling in both vacuum and plasma. This is clearly demonstrated in figure 9(a). It is to be noticed that the frequency of the second-harmonic radiation, which happens to be $0.4\omega _{pe}$ in this case (figure 9b1,b2), is still higher than the left-hand cutoff $\omega _L = 0.376$, and thus lies within the passband region between $\omega _L$ and $\omega _{UH}$. Hence, the second-harmonic radiation is allowed to pass through the plasma. On the other hand, the third-harmonic radiation, which has the O-mode characteristics, will have a frequency $0.6\omega _{pe}$. Thus, in this particular case, the third-harmonic radiation lies below the cutoff ($\omega = \omega _{pe}$) of the O-mode dispersion curve and is forbidden to propagate inside the plasma. This is clearly illustrated in figure 10. In figures 10(a) and 10(b), we show the $y$ component of the transverse magnetic field $\tilde {B}_y$ at a particular instant of time $t = 1000$ for two different incident laser frequencies $\omega _l = 0.2\omega _{pe}$ and $0.4\omega _{pe}$, respectively. It is seen that for $\omega _l = 0.2\omega _{pe}$ (10a), EM field $\tilde {B}_y$ of the incident laser pulse has been reflected from the vacuum–plasma interface and no signal of $\tilde {B}_y$ exists inside the plasma. Whereas, for $\omega _l = 0.4\omega _{pe}$ (10b), a part of $\tilde {B}_y$ is also present inside the plasma and which is associated with the third-harmonic radiation, as also demonstrated in § 3.1.

Figure 9. (a) The $z$ component of the magnetic field $B_z$ (EM) with respect to $x$ at time $t = 1000$. The green dotted line at the location $x = 1000$ represents the vacuum–plasma interface. The FFTs of ($E_y$, $B_z$) at the locations (b1) $x = 500$ and (b2) $x = 2000$ in the frequency domain clearly demonstrate that the second harmonic is present in both reflected and transmitted radiations, respectively.

Figure 10. The $y$ component of magnetic field $B_y$ (EM) with respect to $x$ at a particular instant of time $t = 1000$ for incident laser frequency (a) $\omega _l = 0.2$ and (b) $\omega _l = 0.4$. In both cases, the polarization of the incident laser has been chosen to be in O-mode configuration, i.e. $\boldsymbol {\tilde {E}}_{l} \parallel \boldsymbol {B}_0$ (along $\hat z$), and the value of $a_0$ is considered to be 0.5. Here, the green dotted line at $x = 1000$ represents the vacuum–plasma interface.

It is straightforward that in the presence of an external magnetic field $B_0$, a plasma wave with electric field ${\boldsymbol {E}} \perp {\boldsymbol {B}}_0$ and propagation vector ${\boldsymbol {k}} \perp {\boldsymbol {B}}_0$ (X-mode configuration) always tends to be elliptically polarized instead of plane-polarized (Chen Reference Chen1984). That is, when an EM wave propagates through the plasma, an electric field component parallel to the propagation direction will also be present. The wave, therefore, has both EM and electrostatic features. In our study, the observed second-harmonic radiation is associated with an electric field perpendicular to $B_0$ and propagates along $\hat x$, as shown in previous sections. Thus, an electric field component along $\hat x$, $\tilde {E}_x$, is also expected to be present and will be travelling along with the second-harmonic EM radiation. This is clearly depicted in figure 11(a). The $E_x$ field profile is seen to be associated with the second-harmonic radiation. The Fourier spectra of this particular profile in both the $\omega$ domain and $k$-space also show the same characteristic properties as the EM second-harmonic radiation, as is shown in figure 11(b1,b2). It is to be noticed that no $\hat x$ component of electric field ($E_x$) is observed inside the bulk plasma for the case without an external magnetic field, as is shown by blue dotted lines in figure 11. This is consistent with the results discussed in §§ 3.1 and 3.3. In this case, $E_x$ is present only at the vacuum–plasma interface where the disturbances are made due to the ponderomotive pressure of the incident laser pulse. It cannot propagate inside the bulk plasma without an external magnetic field. It should be noted that the odd harmonics that are generated and present inside the bulk plasma for $B_0 = 0$ (figure 2) can also induce longitudinal electric field $E_x$. However, this will be negligible for $B_0 = 0$ as the longitudinal electric field will, in this case, get generated by the coupling of dynamics of electrons in the electric and magnetic field of the harmonic EM wave radiation (i.e. $\boldsymbol {E}_{nh}\times \boldsymbol {B}_{nh}$, where $n$ represents the order of harmonic). Since the intensity of the harmonic radiation is typically much weaker, this effect will be very small compared to $E_x$ that gets generated by the coupling of the electric field of the harmonic radiation with the external magnetic field. This is clearly illustrated in figure 11.

Figure 11. (a) The longitudinal electric field $E_x$ with respect to $x$ at a particular instant of time $t = 1000$. The FFTs of $E_x$ in (b1) the frequency domain and (b2) $k$-space. Here, the red solid lines and blue dotted lines represent the cases with $B_0 = 2.5$ and $B_0 = 0$, respectively.

We also observe from figure 11(a) that a large-scale disturbance (red solid line) is present in $E_x$ near the plasma–vacuum interface for $B_0 = 2.5$. Such a disturbance has also been observed in the transverse fields $E_y$ and $B_z$, as shown in figure 2(b). This fluctuation is EM but becomes elliptically polarized in the presence of an external magnetic field $B_0$, as discussed earlier. It is also observed that this disturbance travels with a much slower velocity than the higher-harmonic radiation present in the system. Such a disturbance has also been observed in both electron and ion density profiles, as can be seen in figure 12. The normalized density profiles of electrons and ions at a particular instant of time $t = 1000$ are shown by the solid red line and blue dotted line, respectively, in figure 12. A spatial electron density profile travels along with the generated second-harmonic pulsed structure wave and is thus associated with it. As the second-harmonic frequency is much higher than the ion response time scale, this structure appears only in the electron density profile, not in that of ions. On the other hand, near the plasma surface, the fluctuations are present in both electron and ion density profiles, and they are almost identical. These fluctuations are associated with low-frequency magnetosonic perturbations. They are initiated due to the ponderomotive force associated with the finite pulse width of the laser. However, for the choice of infinitely massive ions, these magnetosonic perturbations do not appear. Such a disturbance has also been reported in a recent study by Vashistha et al. (Reference Vashistha, Mandal and Das2020a) for the X-mode configuration. We confirm that this is also present in the O-mode configuration.

Figure 12. Electron and ion density fluctuations are shown by red solid and blue dotted lines, respectively. Here, the external magnetic field $B_0$ is considered to be $2.5$.