#  Harmonic generation in the interaction of laser with a magnetized overdense
plasma

Srimanta Maity<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India  Devshree
Mandal  Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India
Homi Bhabha National Institute, Mumbai, 400094, India  Ayushi Vashistha
Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India  Homi
Bhabha National Institute, Mumbai, 400094, India  Laxman Prasad Goswami
Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New
Delhi 110016, India  Amita Das  Department of Physics, Indian Institute of
Technology Delhi, Hauz Khas, New Delhi 110016, India

###### Abstract

The mechanism of harmonic generation in both O and X-mode configurations for a
magnetized plasma has been explored here in detail with the help of Particle-
In-Cell (PIC) simulations. A detailed characterization of both the reflected
and transmitted electromagnetic radiation propagating in the bulk of the
plasma has been carried out for this purpose. The efficiency of harmonic
generation is shown to increase with the incident laser intensity. Dependency
of harmonic efficiency has also been found on magnetic field strength. This
work demonstrates that there is an optimum value of the magnetic field at
which the efficiency of harmonic generation maximizes. The observations are in
agreement with theoretical analysis. For O-mode configuration, this is
compelling as the harmonic generation provides for a mechanism by which laser
energy can propagate inside an overdense plasma region.

##  I Introduction

Laser-plasma interaction has been an important field attracting research
interest for several decades Liu et al. (2019); Kaw (2017). The research has
been focused on many exciting areas of physics, e.g., long scale magnetic
field generation Bret (2009); Das et al. (2020), nonlinear electromagnetic
structure formation Verma et al. (2016); Mandal et al. (2020); Yadav et al.
(2021), wave breaking Modena et al. (1995); Bera et al. (2021), particle
acceleration Tajima and Dawson (1979); Modena et al. (1995); Faure et al.
(2004), x-ray source Rousse et al. (2004); Corde et al. (2013) and gamma-ray
source Cipiccia et al. (2011), etc. Traditionally, laser-plasma interaction
study has essentially been focused and limited to the regime of unmagnetized
plasma response. This has been so as the required external magnetic field to
elicit a magnetized response from plasma medium at the laser frequency is
quite high and not possible to achieve in the laboratory. Lately, there have
been technological developments in this direction, and magnetic fields of the
order of kilo Tesla has been achieved Nakamura et al. (2018).

The regime of laser interacting with a magnetized plasma has thus caught
attention lately. Studies investigating new possibilities of laser energy
absorption, penetration, and characteristics mode propagation inside the
plasma, have been carried out in detail Kumar et al. (2019); Vashistha et al.
(2020a, b). These studies have used the X-mode geometry for which the laser
electric field is normal to the applied external magnetic field. The O-mode
configuration, however, has not been studied so far. It is generally believed
that such a configuration will exhibit a similar response as that of an
unmagnetized plasma and would have nothing new to offer. With the help of
Particle-In-Cell (PIC) simulation, we have shown, that the O-mode geometry has
surprises to offer. While the laser energy cannot penetrate an unmagnetized
overdense plasma, in the case of O-mode configuration, this happens with the
help of harmonic generation (HG) at the laser-plasma interface. It is shown
that a part of laser energy gets converted into higher harmonics in the
presence of an externally applied magnetic field and can propagate inside the
plasma if the plasma is underdense at this high frequency.

The physics of harmonic generation of electromagnetic (EM) radiation is itself
an important area of investigation Margenau and Hartman (1948); Sodha and Kaw
(1970). It has been studied in the context of laser-plasma interaction for
several decades now Burnett et al. (1977); Carman et al. (1981); Teubner and
Gibbon (2009); Ganeev (2012). The high-harmonic observations in laboratory
plasmas have opened up a wide range of applications. It is considered as one
of the most efficient techniques known to obtain electromagnetic waves of
higher frequency in a controlled manner Tsakiris et al. (2006); Dromey et al.
(2006). HG has been used as a distant probe for the detection of turbulence in
a toroidal magnetically confined plasma Ajendouz et al. (2007). Polarization
measurement of the harmonic has also been used to detect the poloidal magnetic
field profile in Tokamak devices Cano et al. (1975). Recently, the second
harmonic radiation generated in the interaction of laser beams with an
underdense plasma has been used to experimentally verify some of the
fundamental properties of photons, including the conservation of total angular
momentum Huang et al. (2020).

In the review article by Teubner and Gibbon (2009), a nonlinear fluid model
has been discussed to predict the harmonic generation in the reflected
radiation from an overdense plasma surface in the absence of an external
magnetic field. An equivalent model was also formulated by Bulanov et al.
(1994) and Lichters et al. (1996) predicting the existence of so-called
selection rules for the polarization of reflected harmonics from an overdense
plasma surface. As already stated, we show here, with the help of PIC
simulations, that a magnetized plasma provides another mechanism of harmonic
generation. High-order harmonic generation in the interaction of laser with a
magnetized plasma has been reported in several previous studies. The second
harmonic generation in a uniform magnetized plasma has been studied by Jha et
al. (2007). Second and third-harmonic generation in the interaction of laser
fields with a magnetized plasma having a density below the critical density
has been reported by Ghorbanalilu (2012). Second harmonics generation in the
reflected and transmitted radiation by an obliquely p-polarized laser pulse
propagating through a homogeneous, underdense, and transversely magnetized
plasma is studied by Ghorbanalilu and Heidari (2017). The second harmonic
generation of a relativistic chirped laser pulse propagating through
homogeneous magnetized plasma is studied by Kant and Thakur (2016). These
studies are mostly analytical, involving approximations. In all these studies,
plasma was assumed to be homogeneous and underdense so that a laser could
propagate through the plasma. In the present study, we have performed PIC
simulations considering a finite laser pulse (O-mode configuration) incident
on an overdense magnetized plasma surface where the original laser pulse
cannot penetrate inside the bulk plasma. It only interacts with the plasma
species at the vacuum-plasma interface.

In the PIC study reported by Mu et al. (2016), they observed second harmonic
generation in the reflected radiation from a solid, dense plasma surface in
the presence of an external magnetic field. They explored the harmonic
generation efficiency with the variation of external magnetic field and pre-
plasma scale length. In the present study, we have observed the presence of
higher harmonics in the reflected as well as transmitted radiations. Here, we
have mainly concentrated on the characterization of the harmonic radiations
transmitted inside the bulk plasma, which was unexplored in the previous
study. The harmonics get generated at the vacuum plasma interface and
propagate inside the plasma as well as in the vacuum region. The conditions
for channeling the harmonic radiations generated at the vacuum plasma
interface inside the bulk plasma have been identified and analyzed. It occurs
in both O and X-mode configurations. We have provided a comparison and
contrasted both configurations for a harmonic generation. Simulation
observations for the case with and without externally applied magnetic fields
have been compared and discussed. The conversion efficiency of HG for a wide
range of external magnetic fields has been analyzed. We consider the laser to
be incident normal to the plasma surface for various laser and plasma
parameters and have observed conversion efficiency as high as $1.77\%$ for the
second harmonic. The conversion efficiency for these higher harmonics both in
the transmitted (inside the plasma) and reflected (in vacuum) radiations has
been shown to depend on the strength of the laser and external magnetic field.
Furthermore, we have also characterized the following properties of harmonic
radiation in detail: (i) the dispersion properties of the observed higher
harmonics, (ii) polarization of the higher harmonic radiation, and (iii)
forbidden frequencies for given plasma and EM wave parameters.

The paper has been organized as follows. In section 2, we describe the
simulation setup. Section 3 contains the observations. The various subsections
therein describe the generation and characterization of harmonics. In section
4, we provide a summary and conclusion. We have provided an approximated
analytical calculation for the mechanism of HG in the appendix.

##  II Simulation Details

We have carried out one-dimensional Particle-In-Cell Simulations using OSIRIS
4.0 framework Fonseca et al. (2002, 2008); Hemker (2015) for our study. Our
simulation geometry has been shown in Fig. 1. It has a longitudinal extent of
$3000d_{e}$ with plasma boundary starting from $x=1000d_{e}$. Here, $d_{e}$ is
the electron skin depth $c/\omega_{pe}$ and $c$ is the speed of light in
vacuum. We have chosen $60000$ grid points, which corresponds to
$dx=0.05d_{e}$. The number of particles per cell has been chosen to be $8$.
Time has been normalized by $t_{N}=\omega_{pe}^{-1}$, where $\omega_{pe}$ is
the plasma frequency corresponding to the density $n_{0}$. The length is
normalized by $x_{N}=c/\omega_{pe}=d_{e}$, and fields by
$B_{N}=E_{N}=m_{e}c\omega_{pe}/e$, where $m_{e}$ and $e$ represent the rest
mass of an electron and the magnitude of an electronic charge, respectively.
The external magnetic ($B_{0}=2.5$ in normalized units) field has been applied
along the $\hat{z}$ direction.

Table 1 presents laser and plasma parameters in normalized units. A possible
set of values in the standard unit has also been provided. We have treated
both electron and ion dynamics in our simulations with a reduced mass of ions
$m_{i}=100m_{e}$. For some cases, we have also treated the ions to be
infinitely massive and acting merely as a stationary background. The
inferences on HG were not altered for the two cases. A laser pulse with
intensity $\approx 3\times 10^{15}$ $W/cm^{2}$ (for normalized vector
potential, $a_{0}=eE_{l}/m\omega_{l}c=0.5$) and frequency
$\omega_{l}=0.4\omega_{pe}$ is incident normally from the left side of plasma.
The electric field of the laser $E_{l}$ is chosen to be along $\hat{z}$ for
the O-mode configuration and along $\hat{y}$ for the X-mode configuration. In
Fig. 1 the O-mode configuration has been depicted in a schematic
representation. The longitudinal profile of laser pulse is a polynomial
function with rise and fall time of $100\omega_{pe}^{-1}$ which translates to
$200fs$ and it starts from $x=950d_{e}$. The value of the external magnetic
field is chosen to elicit magnetized response of electrons while ions remain
unmagnetized at the laser frequency i.e. $\omega_{ce}>\omega_{l}>\omega_{ci}$,
where $\omega_{ci},\omega_{ce}$ are cyclotron frequencies of ion and electron
respectively while $\omega_{l}$ is laser frequency. Absorbing boundary
conditions are used for fields and particles. For an unmagnetized plasma
(i.e., in the absence of an external magnetic field), the laser cannot
penetrate inside the plasma. In the X-mode configuration, even in the
overdense case, if the laser frequency lies in the passband of magnetized
plasma, it propagates inside and generates lower hybrid and magnetosonic
excitation, as has been illustrated in some of the earlier works Kumar et al.
(2019); Vashistha et al. (2020a, b). However, for O-mode configuration, the
laser cannot propagate inside plasma if it is overdense. In this case, the
laser light only penetrates the plasma up to skin depth order. We observe that
this is sufficient for the generation of harmonics in the O-mode
configuration. For the higher frequency of harmonics that get generated, the
plasma becomes underdense. In such a situation, the generated harmonic
radiation is free to propagate inside the plasma.

In our PIC simulation study, we have not included Coulomb collisions. The
quiver velocity of electrons would be high at the laser intensities of $\sim
10^{15}$ $W/cm^{2}$ considered in our study. We, therefore, feel that the
collisions will not have any significant role to play Dendy (1995). In our
study, the initial temperature of electrons are assumed to be very low
($T_{e}=0.05$ $eV$). Such an assumption is valid because the temperature
remains small compared to the typical oscillation energy of electrons in
intense laser fields Gibbon (2005). Thus, the plasma temperature will not
affect the HG mechanism presented here. We have done a comparative study of
temperature effect on the generation of harmonics too (in section III.3) which
illustrates the above point.

Table 1: Simulation parameters: In normalized units and possible values in standard units. Parameters |  Normalized Value |  A possible value in standard unit
---|---|---
Laser Parameters
Frequency ($\omega_{l}$) |  $0.4\omega_{pe}$ |  $1.78\times 10^{14}$rad/s
Wavelength |  $15.7c/\omega_{pe}$ |  $10.6\mu m$
Intensity |  $a_{0}=0.5$ |  $3.04\times 10^{15}W/cm^{2}$
Plasma Parameters
Number density($n_{0}$) |  $1$ |  $6.2\times 10^{19}$ $cm^{-3}$
Electron Plasma frequency ($\omega_{pe}$) |  $1$ |  $4.44\times 10^{14}$rad/s
Electron skin depth ($c/\omega_{pe}$) |  $1$ |  $0.68\mu m$
External Fields
Magnetic Field ($B_{0}$) |  $2.5$ |  $\approx$ 6.3 kT

##  III Observations and Discussion

It is well known that in the X-mode configuration, the EM radiation of the
laser penetrates the plasma in the respective permitted passbands. In this
case, bulk plasma can interact with the incident radiation although
$\omega_{l}<\omega_{pe}$. In the O-mode, however, the dispersion relation
being identical to the unmagnetized case, there is no propagation when the EM
wave frequency is smaller than the plasma frequency. The laser-plasma
interaction, in this case, is thus confined only within the electron skin
depth layer. The plasma within the skin depth responds to the Lorentz force
acted upon by the electric and magnetic field of the EM wave and the applied
external magnetic field. The incident EM radiation has a finite spatial pulse
profile in the longitudinal direction. The finite spatial profile of the laser
pulse provides for an additional ponderomotive force to the plasma medium. We
have chosen to work in the frequency domain ( shown in the table 1) for which
the condition $\omega_{ci}<\omega_{l}<\omega_{ce}$ is satisfied. Here,
$\omega_{l}$ defines the laser frequency, and $\omega_{ce}$ and $\omega_{ci}$
represent the electron and ion cyclotron frequency, respectively. Thus,
electrons would have a magnetized response to offer in the time period
corresponding to a laser cycle, whereas ions would be unmagnetized. We now
present various features of our observations in the following subsections.

###  III.1 HG in O-mode configuration
$(\vv{\bm{E}}_{l}\parallel\vv{\bm{B}}_{0})$

Figure 1: A summary of the observations of this study has been shown in this
schematic. We have performed 1D PIC simulation (along $\hat{x}$) with a laser
being incident on the plasma surface at $x=1000$. The external magnetic field
$B_{0}$ has been applied along $z-$direction. The polarization of the incident
laser has been chosen in O-mode configuration in this schematic, i.e., the
electric field of the incident laser pulse is oscillating along the direction
of the external magnetic field $B_{0}$. As the laser interacts with the plasma
surface, it generates higher harmonics with different polarization in the
reflected and transmitted radiations, as shown in the schematic. Magnetosonic
disturbance has also been observed in these interactions. Figure 2: Transverse
time varying magnetic fields $B_{y}$ and $B_{z}$ with respect to $x$ have been
shown at a particular instant of time $t=1000$ (when laser already gets
reflected back from the system) in subplots (a) and (b), respectively. In
subplot (a1), $B_{y}$, which exists inside the plasma, has been shown on a
different scale. Here, the black dotted line at $x=1000$ represents the plasma
surface. It is to be noted that the electromagnetic fields $\vec{B_{l}}$ and
$\vec{E_{l}}$ of the incident laser pulse were along $\hat{y}$ and $\hat{z}$
directions, respectively. In this figure, red lines represent for $B_{0}=2.5$
with $E_{l}\parallel B_{0}$ and blue dotted lines are for $B_{0}=0$. Figure 3:
Fourier transform of electromagnetic fields with time after laser reflected
from the plasma surface. In subplot (a) and (b), the FFT of $E_{z}$ and
$B_{y}$ with time in vacuum ($x=500$) and the bulk plasma ($x=2000$) have been
shown, respectively. The same has been shown for the fields ($E_{y}$, $B_{z}$)
in subplots (c) and (d), respectively. In the subplots (a)-(d), the external
magnetic field ($B_{0}$) is considered to be $2.5$. These FFTs clearly
indicate that the higher harmonics have been generated, and they are present
in both vacuum and the bulk plasma. Subplots (e) and (f) represent the time
FFT of $E_{z}$ and $B_{y}$ without any external magnetic field ($B_{0}=0$) in
vacuum and plasma, respectively.

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 were
considered to be in O-mode configuration, i.e.,
$\vv{\bm{E}}_{l}\parallel\vv{\bm{B}}_{0}$ (in $\hat{z}$ direction). Here,
$\bm{E}_{l}$ is the laser electric field, and $\bm{B}_{0}$ is the externally
applied magnetic field. The transverse $\hat{y}$ and $\hat{z}$ components of
the magnetic field, $B_{y}$ and $B_{z}$ have been shown by red solid line for
$B_{0}=2.5$ at a particular instant of time $t=1000$ in subplots (a) and (b)
of Fig. 2, respectively. It is to be noted that at time $t=0$, laser pulse
with the electromagnetic fields $B_{y}$ and $E_{z}$ was set to propagate along
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 subplot (a), is associated with the reflected part of the incident laser
propagating along $-\hat{x}$ direction. A small fraction of $B_{y}$ is also
present inside the bulk plasma, as depicted in the zoomed scale in subplot
(a1). We will identify this structure as the third harmonic radiation (and the
higher odd harmonics) in the consecutive section. In the subplot (b) of Fig.
2, it is seen that the $\hat{z}-$component of the oscillating magnetic field
$B_{z}$ which was not present before the laser hits 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 the subplot (b). One is the large-scale disturbance near the plasma
surface, which will be identified as the magnetosonic perturbation. And
another 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 the plasma
particles. The simulation observations for $B_{0}=0$ have also been depicted
in Fig. 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
subplot (b) of Fig. 2.

The time FFT (Fast Fourier Transform) of these reflected and transmitted
radiations has been shown in Fig. 3. The FFTs of $E_{z}$ and $B_{y}$ at the
location $x=500$ (vacuum) show two distinct peaks at the frequency
$\omega\approx 0.4\omega_{pe}$ and $\omega\approx 1.2\omega_{pe}$ (subplot (a)
of Fig. 3). It is important to recall that at $t=0$, the incident laser pulse
was located in 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 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 has been
shown in subplot (b) of Fig. 3. Thus, it is now clear that the small
disturbance in $B_{y}$ present inside the plasma, as shown in subplot (a1) of
Fig. 2 is essentially associated with the third harmonic radiation. The FFT of
$E_{y}$ and $B_{z}$ in time at the locations $x=500$ and $2000$ have been
shown in subplots (c) and (d) of Fig. 3, respectively. The FFT at the location
$x=2000$ (plasma), in subplot (d), has been evaluated within a time window
$t=1000$ to $3000$. This is to eliminate the slowly moving magnetosonic
disturbance as shown in the subplot (b) of Fig. 2. 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. (2016). However, in
our study, we have observed the harmonic radiations in both reflected and
transmitted radiations, as has been shown in Fig. 3. In the case without an
external magnetic field, the third harmonic (odd harmonics) shows up with the
polarization ($E_{z}$, $B_{y}$) in both reflected and transmitted radiations,
as shown in subplots (e) and (f) of Fig. 3. It is to be noticed that, unlike a
magnetized case, no harmonics with the polarization ($E_{y}$, $B_{z}$) is
generated in the absence of external magnetic field, as has been shown in
subplot (b) of Fig. 2. 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. (1996). 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 in our manuscript is the same as
reported by Lichters et al. (1996). 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
electromagnetic nature of harmonic radiations. It is also important to observe
that the polarization of third harmonic radiation is the same as the incident
laser pulse. In contrast, the polarization of the second harmonic radiation is
different from that of the incident laser pulse. In subsequent subsection
III.3, we will discuss the reason behind this.

The conversion efficiencies of second and third harmonics have been provided
in table 2 for different values of $a_{0}(=eE_{l}/m\omega_{l}c)$ and laser
frequency $\omega_{l}$. The conversion efficiency has been calculated by
taking the ratio of spatially integrated electromagnetic field (emf) 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 harmonics’ efficiencies. 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 has
been clearly shown in table 2.

Table 2: Conversion efficiencies of harmonics in O-mode configuration of incident laser pulse for $B_{0}=2.5$. $a_{0}$ |  $\omega_{l}$ |  $a_{0}\omega_{l}$ |  $\eta_{2^{nd}}(ref)(\%)$ |  $\eta_{2^{nd}}(trans)(\%)$ |  $\eta_{3^{rd}}(trans)(\%)$
---|---|---|---|---|---
0.5 |  0.7 |  0.35 |  0.346 |  0.209 |  0.0342
0.5 |  0.6 |  0.30 |  0.264 |  0.173 |  0.024
0.5 |  0.5 |  0.25 |  0.193 |  0.128 |  0.016
0.5 |  0.4 |  0.20 |  0.130 |  0.085 |  0.0094
0.4 |  0.5 |  0.20 |  0.125 |  0.0833 |  0.009

###  III.2 HG in X-mode configuration $(\vv{\bm{E}}_{l}\perp\vv{\bm{B}}_{0})$

The higher harmonics can also be observed for the case when the polarization
of the incident laser pulse is chosen to be in X-mode configuration, i.e.,
$\vv{\bm{E}}_{l}\perp\vv{\bm{B}}_{0}$. This has been clearly illustrated in
Fig. 4. Laser pulse with frequency $0.4\omega_{pe}$ was set up initially
($t=0$) to propagate in $\hat{x}$ 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 in
between 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 has been clearly illustrated in subplot (a) of
Fig. 4 where we have shown 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 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 subplot (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 FFT in time for
this signal of the transverse fields $E_{y}$ and $B_{z}$ in subplots (b) and
(c) of Fig. 4 corroborate this. The FFTs of $E_{y}$ and $B_{z}$ inside the
plasma ($x=2500$), as have been shown in subplot (c), 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 both the subplots correspond to the
second and third harmonic, 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 the incident laser
pulse.

Figure 4: The generation of higher harmonics has been depicted here for the
case where the polarization of incident laser has been chosen to be in X-mode
configuration, i.e., $\mathbf{\widetilde{E}}_{l}\perp\mathbf{B}_{0}$. Here, we
have considered $B_{0}=2.5$. The electromagnetic part of the magnetic field
along $\hat{z}$, $B_{z}$ has been shown in subplot (a) at a particular instant
of time $t=1000$. In the subplot (b), the FFT of $E_{y}$ and $B_{z}$ at the
location $x=500$ (vacuum) has been sown. 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. The
existence of these higher harmonics inside the bulk plasma has been depicted
in subplot(c), where the FFTs have been performed at the location $x=2500$.

###  III.3 Mechanism of HG in a magnetized plasma

Figure 5: Figure shows 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. In subplots (a2) and (b2), the FFTs of
$J_{ey}$ and $J_{ez}$ have been shown, respectively. Here, the red solid lines
and blue dotted lines represent the case with $B_{0}=2.5$ and $B_{0}=0$,
respectively.

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. (2016).
In the present work, we have briefly discussed 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
configuration, and provided in the appendix.

When a laser pulse with O-mode configuration, i.e.,
$\vv{\bm{E}}_{l}\parallel\vv{\bm{B}}_{0}$ is incident on the vacuum-plasma
interface, plasma particles will experience a force
($\propto\widetilde{v}_{z}\widetilde{B}_{l}\exp({i2\omega_{l}t)}$) due to the
Lorentz force ($\vv{\bm{v}}\times\vv{\bm{B}}_{l}$) along $\hat{x}$ direction.
Here, $\vv{\bm{v}}$ is the particle’s quiver velocity initiated due to the
laser electric field $\bm{E_{l}}$ in $\hat{z}$. As a result, plasma electrons
wiggle, forming an oscillating current at the surface of the plasma in
$\pm\hat{x}$ direction with a frequency twice the incident laser frequency
(Eq. 7 ). This can also be understood by recalling that in the intense EM
fields of incident laser, electron motion would initially follow the well
known “figure-of-8” path, which contains a transverse component with a
frequency $\omega_{l}$ and a longitudinal part with $2\omega_{l}$ frequency
Teubner and Gibbon (2009). 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 its 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 have been shown in
subplots (a1) and (a2) of Fig. 5. This acts as an oscillating current sheet
antenna and radiates electromagnetic wave with the fields $\widetilde{E}_{y}$
and $\widetilde{B}_{z}$ 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 non-linearity involved in the
electron dynamics. In our simulations, we have captured up to sixth harmonic
and can be easily seen in the FFT $J_{ey}$ in subplots (a2) of Fig. 5.

In addition, the electron motion along $\hat{x}$ will couple to the laser
magnetic field $\widetilde{B}_{l}$ (which is along $\hat{y}$ for O-mode
configuration). As a result, it will create an oscillating current sheet along
$\hat{z}$ with the frequency thrice of laser frequency and also at higher odd
harmonic values, as have been shown in subplots (b1) and (b2) of Fig. 5. Thus,
another radiation will be produced propagating in $\pm x$ direction, but in
this case, the electromagnetic fields associated with this radiation are
$\widetilde{E}_{z}$ and $\widetilde{B}_{y}$. It is to be noticed that ideally,
all the odd higher harmonics might be present in the $J_{ez}$. In our
simulations, we have identified up to fifth harmonic and shown in subplot (b2)
of Fig .5. 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 has been shown by red solid and blue dotted lines in the
subplot (a1) and (a2) of Fig. 5 and is also consistent with the approximate
mathematical expression of $J_{ey}$ in Eq. 7. 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 Lichters et al.
(1996); Teubner and Gibbon (2009); Bulanov et al. (1994), as has been shown in
the subplots (b1) and (b2) of Fig. 5. This is also apparent from the
expression of $J_{ez}$ in Eq. 10.

For the laser polarization in the X-mode configuration, the magnetic field
$\widetilde{B}_{l}$ 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 electromagnetic
fields are $\widetilde{E}_{y}$ and $\widetilde{B}_{z}$. Thus, the generated
harmonic radiation also has the X-mode configuration. This is exactly what we
have observed in our simulations, as have been shown in Fig. 4.

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

Figure 6: FFT 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 ($x=1000c/\omega_{pe}$) for different electron temperature
($T_{e}=0.05eV,5eV,50eV$ and $500eV$).

###  III.4 Effect of external magnetic field on HG

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 the Eq. 7. (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 the Eq. 10.

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 will 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 _i.e._ Eq. 7 and 10 for different
magnetic field values. We have plotted these two quantities as a function of
the external magnetic field $B_{0}$ and shown in Fig. 7. One can observe that
trend is similar for quantities obtained from simulation and 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
$\omega_{ce}\rightarrow 2\omega_{l}$.

###  III.5 Characterization of harmonics

Figure 8: Fourier transforms in space of the electromagnetic fields after the
laser is reflected from the vacuum-plasma interface. In subplots (a) and (b),
the FFT of ($E_{z}$, $B_{y}$) along $\hat{x}$ at time $t=600$ and $1600$ have
been shown for vacuum and bulk plasma, respectively. On the other hand, the
same has been shown for the fields ($E_{y}$, $B_{z}$) in subplots (c) and (d),
respectively. Dispersion curves (Boyd et al. (2003)) of ordinary ($O$) mode
and extraordinary ($X$) mode for the chosen values of the system parameters of
this study have been shown in subplots (e) and (f), respectively. Figure 9:
The z-component of the magnetic field $B_{z}$ (electromagnetic) with respect
to $x$ at time $t=1000$ has been shown in subplot (a). The green dotted line
at the location $x=1000$ represents the vacuum-plasma interface. The FFTs of
($E_{y}$, $B_{z}$) at the locations $x=500$ and $x=2000$ in the frequency
domain, as have been shown in subplots (b1) and (b2), clearly demonstrate that
second harmonic is present in both reflected and transmitted radiation,
respectively. Figure 10:  The y-component of magnetic field $B_{y}$
(electromagnetic) with respect to $x$ at a particular instant of time $t=1000$
has been shown for incident laser frequency (a) $\omega_{l}=0.2$ and (b)
$\omega_{l}=0.4$. In both the cases, the polarization of the incident laser
has been chosen to be in O-mode configuration, i.e.,
$\mathbf{\widetilde{E}}_{l}\parallel\mathbf{B}_{0}$ (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. Figure 11: The longitudinal
electric field $E_{x}$ with respect to $x$ has been shown in the subplot (a)
at a particular instant of time $t=1000$. In the subplots (b1) and (b2), the
FFTs of $E_{x}$ in frequency and $k-$space have been shown, respectively.
Here, the red solid lines and blue dotted lines represent the case with
$B_{0}=2.5$ and $B_{0}=0$, respectively. Figure 12: The electrons and ions
density fluctuations have been shown by red solid and blue dotted lines,
respectively. Here, the external magnetic field $B_{0}$ is considered to be
$2.5$.

We now analyze 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
($\vv{\bm{E}}_{l}\parallel\vv{\bm{B}}_{0}$) and X-mode
($\vv{\bm{E}}_{l}\perp\vv{\bm{B}}_{0}$) configurations. We consider here the
observations corresponding to the O-mode configuration. As discussed in the
previous sections, the higher harmonic radiations are electromagnetic. Thus,
in the 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 the plasma medium.

Thus, the group velocity and the phase velocity of these radiations will have
different values depending upon the dispersion relation. These properties have
been clearly shown in Fig. 8. The spatial FFTs of the transverse fields
($E_{z}$, $B_{y}$) for vacuum and bulk plasma have been shown in subplots (a)
and (b) of Fig. 8, respectively. It is clearly seen from subplot (a) that, as
expected, the reflected laser pulse ($\omega_{l}=0.4$) and the third harmonic
($\omega\approx 1.2$) radiation 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 traveling perpendicular to $B_{0}$.
Thus, it matches the condition of plasma ordinary ($O$) mode. We have
evaluated the theoretical dispersion curve for the $O$ mode for our chosen
values of system parameters and have shown in the subplot (e) of Fig. 8. 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.

The FFT of transverse fields $E_{y}$ and $B_{z}$ in space associated with the
second harmonic radiation have been depicted in subplots (c) and (d) of Fig. 8
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 extraordinary ($X$) mode and propagates through the
passband lying in between left-hand cutoff ($\omega_{L}$) and upper hybrid
frequency ($\omega_{UH}$) of the $X$ mode dispersion curve. This has been
demonstrated in the subplot (f) of Fig. 8.

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 (Fig. 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
the previous case. It has been observed that the second harmonic radiations
initiated at the vacuum-plasma interface are traveling in both vacuum and
plasma. This has been clearly demonstrated in the subplot (a) of Fig. 9. It is
to be noticed that the frequency of the second harmonic radiation, which
happens to be $0.4\omega_{pe}$ in this case (subplots (b1) $\&$ (b2)), is
still higher than the left-hand cutoff $\omega_{L}=0.376$ and thus, lies
within the passband region in 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 the frequency $0.6\omega_{pe}$. Thus, in this
particular case, the third harmonic radiation is lying below the cutoff
($\omega=\omega_{pe}$) of the O-mode dispersion curve and is forbidden to
propagate inside the plasma. This has been clearly illustrated in Fig. 10. In
subplot (a) and (b) of Fig. 10, we have shown the y-component of the
transverse magnetic field $\widetilde{B}_{y}$ 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 $\omega_{l}=0.2\omega_{pe}$ (subplot (a)), electromagnetic field
$\widetilde{B}_{y}$ of the incident laser pulse has been reflected from the
vacuum-plasma interface and no signal of $\widetilde{B}_{y}$ exists inside the
plasma. Whereas, for $\omega_{l}=0.4\omega_{pe}$ (subplot (b)), a part of
$\widetilde{B}_{y}$ is also present inside the plasma and which is associated
with the third harmonic radiation, as also has been demonstrated in the
section III.1.

It is straightforward that in the presence of an external magnetic field
$B_{0}$, a plasma wave with the electric field
$\vv{\bm{E}}\perp\vv{\bm{B}}_{0}$ and the propagation vector
$\vv{\bm{k}}\perp\vv{\bm{B}}_{0}$ (X-mode configuration) always tend to be
elliptically polarized instead of plane-polarized Chen et al. (1984). 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 electromagnetic 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 $\hat{x}$,
$\widetilde{E}_{x}$ is also expected to be present and will be traveling along
with the second harmonic EM radiation. This has been clearly depicted in the
subplot (a) of Fig. 11. In this subplot, the $E_{x}$ field profile is seen to
be associated with the second harmonic radiation. The Fourier spectra of this
particular profile in both $\omega$ and $k$-space also show the same
characteristic properties as the electromagnetic second harmonic radiation, as
has been shown in subplots (b1) and (b2) of Fig. 11. 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 external magnetic field, as has been shown by blue
dotted lines in Fig. 11. This is consistent with the results discussed in
section III.1 and section III.3. In this case, $E_{x}$ is present only at the
vacuum-plasma interface where the disturbances were made due to the
ponderomotive pressure of the incident laser pulse. It can not propagate
inside the bulk plasma without an external magnetic field. It should be noted
that the odd harmonics getting generated and present inside the bulk plasma
for $B_{0}=0$ (Fig. 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., $\vec{E}_{nh}\times\vec{B}_{nh}$, 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 has been clearly illustrated in Fig. 11.

We also observe from the subplot (a) of Fig. 11 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 has been shown in subplot (b) of Fig. 2. This
fluctuation is electromagnetic 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 Fig. 12. The normalized density profiles of electrons and ions at a
particular instant of time $t=1000$ have been shown by the solid red line and
blue dotted line in Fig. 12, respectively. 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 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 were 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. (2020b) for X-mode configuration. We confirm that this is also present in
the O-mode configuration.

##  IV Summary

We have shown that the interaction of laser with a magnetized plasma has
interesting and rich facets. The O-mode geometry, which is typically
considered to exhibit unmagnetized plasma response, has certain distinctive
features. Specifically, we have demonstrated the generation of harmonics in a
magnetized plasma even in the O-mode configuration for overdense plasma
regime. The dynamics of a laser pulse interacting with plasma have been
followed up by one-dimensional PIC simulations using OSIRIS-4.0. A laser pulse
coming from a vacuum is chosen to fall on an overdense plasma medium
($\omega_{l}<\omega_{pe}$) in the presence of an externally applied magnetic
field. The dynamical mechanisms leading to higher harmonic generation in both
O-mode ($\vv{\bm{E}}_{l}\parallel\vv{\bm{B}}_{0}$) and X-mode configurations
($\vv{\bm{E}}_{l}\perp\vv{\bm{B}}_{0}$) have been demonstrated and analyzed.
It has been shown that when the incident laser has the polarization in the
O-mode configuration, the higher even harmonics will be produced in the X-mode
configuration, and odd harmonics will be in the O-mode configuration. Whereas
both even and odd harmonics will be in the X-mode configuration for the
incident laser pulse in the X-mode configuration. A comparison of simulation
results with and without an external magnetic field has been provided and
discussed. The required conditions for the propagation of the harmonic
radiation inside the plasma have been identified. Our study reveals that the
conversion efficiency for the harmonic generation increases with laser
intensity. We have also shown that the conversion efficiency has a strong
dependence on the externally applied magnetic field. There is an optimum value
of the magnetic field for which the amplitude of harmonics is maximum. Our
study also demonstrates a conversion efficiency of about $1.77\%$ for second
harmonic for a laser intensity of $a_{0}=0.5$ and $B_{0}=0.6$, which is higher
than the previously reported value in the analytical studies done in the
context of magnetized plasma (Jha et al. (2007); Ghorbanalilu (2012)). We feel
that this can be improved further by appropriate tailoring of the plasma and
magnetic field profiles.

##  V Acknowledgements

The authors would like to acknowledge the OSIRIS Consortium, consisting of
UCLA and IST (Lisbon, Portugal) for providing access to the OSIRIS 4.0
framework which is the work supported by NSF ACI-1339893. This research work
has been supported by the J. C. Bose fellowship grant of AD (JCB/2017/000055)
and the CRG/2018/000624 grant of DST. The authors thank IIT Delhi HPC facility
for computational resources.

##  Appendix A Surface current oscillation in magnetized configuration

We demonstrate the possible generation of harmonics by the surface electron
currents, which would get excited by the laser field falling on an overdense
plasma target in both O and X - mode configuration. The spatial variation
along $x$ is ignored for simplicity.

###  A.1 Surface current oscillation in O-mode configuration

In the O-mode configuration of incident laser wave, we have the external
magnetic fields and the electromagnetic (EM) fields of an incident wave as
follows:

$\displaystyle\vv{B_{0}}=B_{0}\hat{z}\hskip 14.22636pt;\hskip
14.22636pt\vv{\widetilde{E}_{l}}=E_{lz}exp(-i\omega_{l}t)\hat{z}\hskip
14.22636pt;\hskip
14.22636pt\vv{\widetilde{B}_{l}}=\frac{E_{lz}}{c}exp(-i\omega_{l}t)\hat{y}$
(1)

The electron velocity is expanded in terms of various orders in the laser
field amplitude as

$\vv{\widetilde{v}}=\vv{\widetilde{v}}^{(1)}+\vv{\widetilde{v}}^{(2)}+\vv{\widetilde{v}}^{(3)}+...$

From the Lorentz force equation we obtain for electrons

$\frac{\partial{\vv{\widetilde{v}}}}{\partial
t}=-\frac{e}{m}\left[\vv{\widetilde{E}_{l}}+\vv{\widetilde{v}}\times(\vv{B_{0}}+\vv{\widetilde{B}_{l}})\right]$
(2)

Balancing the first order linear term in Eq. 2 we obtain for the $\hat{z}$
component

$\displaystyle\frac{\partial\widetilde{v}_{z}^{(1)}}{\partial
t}=-\frac{e}{m}\widetilde{E}_{l}\hskip 14.22636pt;\hskip
14.22636pt\widetilde{v}_{z}^{(1)}=-\frac{e}{m}\frac{i}{\omega_{l}}\widetilde{E}_{l}$

$\widetilde{v}_{z}^{(1)}=-\frac{ie}{m\omega_{l}}E_{lz}exp(-i\omega_{l}t)$ (3)

The x and y components represents the gyromotion and depends on the thermal
velocity of electrons and are independent of laser field. For a cold plasma
this can be ignored compared to the quiver velocity of the electron in the
laser field. Thus, for the O-mode configuration
($\vv{\widetilde{E_{1}}}\perp\vv{B_{0}}$), $\widetilde{v}_{x}^{(1)}$ and
$\widetilde{v}_{y}^{(1)}$ can be neglected.

The second order term from Eq. 2

$\frac{\partial\widetilde{v}_{z}^{(2)}}{\partial
t}=-\frac{e}{m}(\widetilde{v}_{x}^{(1)}\widetilde{B}_{l})\approx 0\hskip
14.22636pt$ (4)

$\frac{\partial\widetilde{v}_{x}^{(2)}}{\partial
t}=\frac{e}{m}\left[\widetilde{v}_{z}^{(1)}\widetilde{B}_{l}-\widetilde{v}_{y}^{(2)}B_{0}\right]\hskip
14.22636pt;\hskip 14.22636pt\frac{\partial\widetilde{v}_{y}^{(2)}}{\partial
t}=\frac{e}{m}(\widetilde{v}_{x}^{(2)}B_{0})$ (5)

From the coupled expressions in Eq. 5, we will get

$\displaystyle\frac{\partial^{2}\widetilde{v}_{x}^{(2)}}{\partial
t^{2}}+\omega_{c}^{2}\widetilde{v}_{x}^{(2)}=\frac{e}{m}\left[\frac{\partial(\widetilde{v}_{z}^{(1)}\widetilde{B}_{l})}{\partial
t}\right]\hskip 14.22636pt;\hskip
14.22636pt\frac{\partial^{2}\widetilde{v}_{y}^{(2)}}{\partial
t^{2}}+\omega_{c}^{2}\widetilde{v}_{y}^{(2)}=\frac{e}{m}\omega_{c}\frac{(-ie)}{(m\omega_{l})}\widetilde{E}_{l}\widetilde{B}_{l}$
(6)

Here, $\omega_{c}=\frac{eB_{0}}{m}$. Separating the time dependence we have

$\displaystyle\widetilde{v}_{x}^{(2)}\approx
v_{x}^{(2)}exp(-i2\omega_{l}t)\hskip 14.22636pt;\hskip
14.22636pt\widetilde{v}_{y}^{(2)}\approx v_{y}^{(2)}exp(-i2\omega_{l}t)$

Solving the Eq. 6, obtain the expressions of the $x$ and $y$ component of
current as:

$\widetilde{J}_{ey}^{(2)}=in_{e}e\left(\frac{e}{m}\right)^{2}\frac{\omega_{c}}{\omega_{l}}\frac{E_{lz}^{2}}{c\left(\omega_{c}^{2}-4\omega_{l}^{2}\right)}exp(-i2\omega_{l}t)\hskip
5.69046pt;\hskip
5.69046pt\widetilde{J}_{ex}^{(2)}=2n_{e}e\left(\frac{e}{m}\right)^{2}\frac{E_{lz}^{2}}{c\left(\omega_{c}^{2}-4\omega_{l}^{2}\right)}exp(-i2\omega_{l}t)$
(7)

Eq. 7 represent the expression of currents in the $x-y$ plane oscillating with
a frequency twice of incident wave (second harmonic). This oscillating current
acts as an antenna and radiates EM radiation at the second harmonic frequency.

Similarly, various components of the third order velocity terms in Eq. 2 obey

$\frac{\partial\widetilde{v}_{x}^{(3)}}{\partial
t}=-\omega_{c}\widetilde{v}_{y}^{(3)}\hskip 14.22636pt;\hskip
14.22636pt\frac{\partial\widetilde{v}_{y}^{(3)}}{\partial
t}=\omega_{c}\widetilde{v}_{x}^{(3)}$ (8)

Eq. 8 represents the cyclotron motion in the plane ($x-y$) perpendicular to
$B_{0}$ with an amplitude associated with the third order correction of
perpendicular velocity which can only have thermal contribution. Thus, as
mentioned previously, we can neglect this for our case.

The $z$ component of Eq. 2 for third order term will give

$\frac{\partial{\widetilde{v}_{z}^{(3)}}}{\partial
t}=-\frac{e}{m}\left[\widetilde{v}_{x}^{(2)}\widetilde{B}_{l}\right]$ (9)

Using the expression of $\widetilde{v}_{x}^{(2)}$ from Eq. 7 it can be shown
that,

$\widetilde{J}_{ez}^{(3)}=-i2n_{e}e\left(\frac{e}{m}\right)^{3}\frac{E_{lz}^{3}}{3\omega_{l}c^{2}\left(\omega_{c}^{2}-4\omega_{l}^{2}\right)}exp(-i3\omega_{l}t)$
(10)

Eq. 10 represents the $z$ component of current $\widetilde{J}_{z}$ oscillating
with $3\omega_{l}$ frequency(third harmonic).

It is to be noted that for O-mode configuration of the incident wave the
second harmonic radiation has the polarization in x-y plane
($\widetilde{J}_{x}$ and $\widetilde{J}_{y}$) and the polarization of third
harmonic radiation is along $\hat{z}$ $(\widetilde{J}_{z}$). If we calculate
for all the higher order terms, it can be shown that for O-mode configuration,
all the even harmonics will have the polarization in the x-y plane and all the
odd harmonics will have polarization along $\hat{z}$. This has also been
observed in our simulation, as has been shown in Fig. 5.

###  A.2 Surface current oscillation in X-mode configuration

In the X-mode configuration of incident laser wave, we have the directions of
external magnetic fields and the EM fields of incident wave as follows:

$\displaystyle\vv{B_{0}}=B_{0}\hat{z}\hskip 14.22636pt;\hskip
14.22636pt\vv{\widetilde{E}_{l}}=E_{ly}exp(-i\omega_{l}t)\hat{y}\hskip
14.22636pt;\hskip
14.22636pt\vv{\widetilde{B}_{l}}=\frac{E_{ly}}{c}exp(-i\omega_{l}t)\hat{z}$
(11)

We will calculate the expression of higher order terms of currents using Eq. 2
and 11.

Solving for the first order term of Eq. 2,

$\displaystyle\frac{\partial\widetilde{v}_{z}^{(1)}}{\partial t}=0\hskip
14.22636pt;\hskip 14.22636pt\frac{\partial\widetilde{v}_{y}^{(1)}}{\partial
t}=-\frac{e}{m}\left[\widetilde{E}_{l}-\widetilde{v}_{x}^{(1)}B_{0}\right]\hskip
14.22636pt;\hskip
14.22636pt\widetilde{v}_{x}^{(1)}=-\frac{e}{m}\left(\widetilde{v}_{y}^{(1)}B_{0}\right)$
(12)

The solutions of Eq. 12 can be written as

$\widetilde{v}_{x}^{(1)}=\frac{e\omega_{c}}{m(\omega_{c}^{2}-\omega_{l}^{2})}\widetilde{E}_{l}\hskip
14.22636pt;\hskip
14.22636pt\widetilde{v}_{y}^{(1)}=\frac{ie\omega_{l}}{m(\omega_{c}^{2}-\omega_{l}^{2})}\widetilde{E}_{l}\hskip
14.22636pt;\hskip 14.22636pt\widetilde{v}_{z}^{(1)}=0$ (13)

The second order terms of Eq. 2 can be expressed as

$\frac{\partial\widetilde{v}_{x}^{(2)}}{\partial
t}=-\frac{e}{m}\left[\widetilde{v}_{y}^{(1)}\widetilde{B}_{l}+\widetilde{v}_{y}^{(2)}{B}_{0}\right]\hskip
14.22636pt;\hskip 14.22636pt\frac{\partial\widetilde{v}_{y}^{(2)}}{\partial
t}=\frac{e}{m}\left[\widetilde{v}_{x}^{(1)}\widetilde{B}_{l}+\widetilde{v}_{x}^{(2)}{B}_{0}\right]\hskip
14.22636pt;\hskip 14.22636pt\frac{\partial\widetilde{v}_{z}^{(2)}}{\partial
t}=0$ (14)

Eq. 14 can be expressed as the second order differential equations as follows:

$\displaystyle\frac{\partial^{2}\widetilde{v}_{x}^{(2)}}{\partial
t^{2}}+\omega_{c}^{2}\widetilde{v}_{x}^{(2)}=-\frac{e}{m}\left[\frac{\partial\left(\widetilde{v}_{y}^{(1)}\widetilde{B}_{l}\right)}{\partial
t}+\omega_{c}\widetilde{v}_{x}^{(1)}\widetilde{B}_{l}\right]$ (15)

and,

$\displaystyle\frac{\partial^{2}\widetilde{v}_{y}^{(2)}}{\partial
t^{2}}+\omega_{c}^{2}\widetilde{v}_{y}^{(2)}=\frac{e}{m}\left[\frac{\partial\left(\widetilde{v}_{x}^{(1)}\widetilde{B}_{l}\right)}{\partial
t}-\omega_{c}\widetilde{v}_{y}^{(1)}\widetilde{B}_{l}\right]$ (16)

We can solve the Eq. 15 and 16 using the expressions in Eq. 13. The solutions
can be expressed as follows:

$\displaystyle\widetilde{v}_{x}^{(2)}=F_{1}exp(-i2\omega_{l}t)\hskip
14.22636pt;\hskip 14.22636pt\widetilde{v}_{y}^{(2)}=F_{2}exp(-i2\omega_{l}t)$
(17)

Here,

$\displaystyle
F_{1}=-\frac{(e/m)^{2}E_{ly}^{2}\left(\omega_{c}^{2}+2\omega_{l}^{2}\right)}{c\left(\omega_{c}^{2}-4\omega_{l}^{2}\right)\left(\omega_{c}^{2}-\omega_{l}^{2}\right)}\hskip
14.22636pt;\hskip
14.22636ptF_{2}=-i\frac{3(e/m)^{2}E_{ly}^{2}\omega_{c}\omega_{l}}{c\left(\omega_{c}^{2}-4\omega_{l}^{2}\right)\left(\omega_{c}^{2}-\omega_{l}^{2}\right)}$
(18)

From Eq. 17 we can obtain the $x$ and $y$ components of current oscillating
with the frequency $2\omega_{l}$ (second harmonic) and is given by,

$\displaystyle\widetilde{J}_{ex}^{(2)}=-n_{e}eF_{1}exp(-i2\omega_{l}t)\hskip
14.22636pt;\hskip
14.22636pt\widetilde{J}_{ey}^{(2)}=-n_{e}eF_{2}exp(-i2\omega_{l}t)$ (19)

Similarly, the third order terms of Eq. 2 can be expressed as

$\frac{\partial\widetilde{v}_{x}^{(3)}}{\partial
t}=-\frac{e}{m}\left[\widetilde{v}_{y}^{(2)}\widetilde{B}_{l}+\widetilde{v}_{y}^{(3)}{B}_{0}\right]\hskip
14.22636pt;\hskip 14.22636pt\frac{\partial\widetilde{v}_{y}^{(3)}}{\partial
t}=\frac{e}{m}\left[\widetilde{v}_{x}^{(2)}\widetilde{B}_{l}+\widetilde{v}_{x}^{(3)}{B}_{0}\right]\hskip
14.22636pt;\hskip 14.22636pt\frac{\partial\widetilde{v}_{z}^{(3)}}{\partial
t}=0$ (20)

From Eq. 20, we can write

$\displaystyle\frac{\partial^{2}\widetilde{v}_{x}^{(3)}}{\partial
t^{2}}+\omega_{c}^{2}\widetilde{v}_{x}^{(3)}=-\frac{e}{m}\left[\frac{\partial\left(\widetilde{v}_{y}^{(2)}\widetilde{B}_{l}\right)}{\partial
t}+\omega_{c}\widetilde{v}_{x}^{(2)}\widetilde{B}_{l}\right]$ (21)

and,

$\displaystyle\frac{\partial^{2}\widetilde{v}_{y}^{(3)}}{\partial
t^{2}}+\omega_{c}^{2}\widetilde{v}_{y}^{(3)}=\frac{e}{m}\left[\frac{\partial\left(\widetilde{v}_{x}^{(2)}\widetilde{B}_{l}\right)}{\partial
t}-\omega_{c}\widetilde{v}_{y}^{(2)}\widetilde{B}_{l}\right]$ (22)

Solving Eq. 21 and Eq. 22 by using the expression of $\widetilde{v}_{x}^{(2)}$
and $\widetilde{v}_{y}^{(2)}$ given in Eq. 17, we can get

$\displaystyle\widetilde{J}_{ex}^{(3)}=-n_{e}eF_{3}exp(-i3\omega_{l}t)\hskip
14.22636pt;\hskip
14.22636pt\widetilde{J}_{ey}^{(3)}=-n_{e}eF_{4}exp(-i3\omega_{l}t)$ (23)

Here, the expressions of $F_{3}$ and $F_{4}$ can be shown as

$\displaystyle
F_{3}=-\frac{(e/m)E_{ly}}{c\left(\omega_{c}^{2}-9\omega_{l}^{2}\right)}\left[F_{1}\omega_{c}-3iF_{2}\omega_{l}\right]\hskip
2.84544pt;\hskip
2.84544ptF_{4}=-\frac{(e/m)E_{ly}}{c\left(\omega_{c}^{2}-9\omega_{l}^{2}\right)}\left[F_{2}\omega_{c}+3iF_{1}\omega_{l}\right]$
(24)

Eq. 23 represents the $x$ and $y$ component of currents oscillating with the
frequency $3\omega_{l}$ (third harmonic). It is to be noted that, unlike
O-mode configuration, in this case we can get all the higher harmonics to be
in the x-y plane. This has also been observed in our simulations.

## References

  * Liu et al. (2019) C. Liu, V. Tripathi, and B. Eliasson, _High-power laser-plasma interaction_ (Cambridge University Press, 2019).
  * Kaw (2017) P. Kaw, Reviews of Modern Plasma Physics 1, 1 (2017).
  * Bret (2009) A. Bret, The Astrophysical Journal 699, 990 (2009).
  * Das et al. (2020) A. Das, A. Kumar, C. Shukla, R. K. Bera, D. Verma, D. Mandal, A. Vashishta, B. Patel, Y. Hayashi, K. Tanaka, et al., Physical Review Research 2, 033405 (2020).
  * Verma et al. (2016) D. Verma, R. K. Bera, A. Das, and P. Kaw, Physics of Plasmas 23, 123102 (2016).
  * Mandal et al. (2020) D. Mandal, A. Vashistha, and A. Das, Journal of Plasma Physics 86 (2020).
  * Yadav et al. (2021) S. K. Yadav, R. K. Bera, D. Verma, P. Kaw, and A. Das, Contributions to Plasma Physics 61, e202000101 (2021).
  * Modena et al. (1995) A. Modena, Z. Najmudin, A. Dangor, C. Clayton, K. Marsh, C. Joshi, V. Malka, C. Darrow, C. Danson, D. Neely, et al., nature 377, 606 (1995).
  * Bera et al. (2021) R. K. Bera, A. Mukherjee, S. Sengupta, and A. Das, Physics of Plasmas 28, 092102 (2021).
  * Tajima and Dawson (1979) T. Tajima and J. M. Dawson, Physical Review Letters 43, 267 (1979).
  * Faure et al. (2004) J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J.-P. Rousseau, F. Burgy, and V. Malka, Nature 431, 541 (2004).
  * Rousse et al. (2004) A. Rousse, K. T. Phuoc, R. Shah, A. Pukhov, E. Lefebvre, V. Malka, S. Kiselev, F. Burgy, J.-P. Rousseau, D. Umstadter, et al., Physical review letters 93, 135005 (2004).
  * Corde et al. (2013) S. Corde, K. T. Phuoc, G. Lambert, R. Fitour, V. Malka, A. Rousse, A. Beck, and E. Lefebvre, Reviews of Modern Physics 85, 1 (2013).
  * Cipiccia et al. (2011) S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, et al., Nature Physics 7, 867 (2011).
  * Nakamura et al. (2018) D. Nakamura, A. Ikeda, H. Sawabe, Y. Matsuda, and S. Takeyama, Review of Scientific Instruments 89, 095106 (2018).
  * Kumar et al. (2019) A. Kumar, C. Shukla, D. Verma, A. Das, and P. Kaw, Plasma Physics and Controlled Fusion 61, 065009 (2019).
  * Vashistha et al. (2020a) A. Vashistha, D. Mandal, A. Kumar, C. Shukla, and A. Das, New Journal of Physics (2020a).
  * Vashistha et al. (2020b) A. Vashistha, D. Mandal, and A. Das, Nuclear Fusion (2020b).
  * Margenau and Hartman (1948) H. Margenau and L. Hartman, Physical Review 73, 309 (1948).
  * Sodha and Kaw (1970) M. S. Sodha and P. K. Kaw, in _Advances in Electronics and Electron Physics_ (Elsevier, 1970), vol. 27, pp. 187–293.
  * Burnett et al. (1977) N. Burnett, H. Baldis, M. Richardson, and G. Enright, Applied Physics Letters 31, 172 (1977).
  * Carman et al. (1981) R. Carman, C. Rhodes, and R. Benjamin, Physical Review A 24, 2649 (1981).
  * Teubner and Gibbon (2009) U. Teubner and P. Gibbon, Rev. Mod. Phys. 81, 445 (2009), URL https://link.aps.org/doi/10.1103/RevModPhys.81.445.
  * Ganeev (2012) R. Ganeev, _High-order Harmonic Generation In Laser Plasma Plumes_ (World Scientific Publishing Company, 2012), ISBN 9781848169821, URL https://books.google.co.in/books?id=rs26CgAAQBAJ.
  * Tsakiris et al. (2006) G. D. Tsakiris, K. Eidmann, J. Meyer-ter Vehn, and F. Krausz, New Journal of Physics 8, 19 (2006).
  * Dromey et al. (2006) B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, et al., Nature physics 2, 456 (2006).
  * Ajendouz et al. (2007) A. Ajendouz, T. Pierre, M. Boussouis, and K. Quotb, EPL (Europhysics Letters) 81, 15001 (2007).
  * Cano et al. (1975) R. Cano, I. Fidone, and J. Hosea, The Physics of Fluids 18, 1183 (1975).
  * Huang et al. (2020) C.-K. Huang, C. Zhang, Z. Nie, K. A. Marsh, C. E. Clayton, and C. Joshi, Communications Physics 3, 1 (2020).
  * Bulanov et al. (1994) S. V. Bulanov, N. Naumova, and F. Pegoraro, Physics of Plasmas 1, 745 (1994).
  * Lichters et al. (1996) R. Lichters, J. Meyer-ter Vehn, and A. Pukhov, Physics of Plasmas 3, 3425 (1996).
  * Jha et al. (2007) P. Jha, R. K. Mishra, G. Raj, and A. K. Upadhyay, Physics of plasmas 14, 053107 (2007).
  * Ghorbanalilu (2012) M. Ghorbanalilu, Laser and Particle Beams 30, 291 (2012).
  * Ghorbanalilu and Heidari (2017) M. Ghorbanalilu and Z. Heidari, Laser and Particle Beams 35, 551 (2017).
  * Kant and Thakur (2016) N. Kant and V. Thakur, Optik 127, 4167 (2016).
  * Mu et al. (2016) J. Mu, F.-Y. Li, Z.-M. Sheng, and J. Zhang, Laser and Particle Beams 34, 545 (2016).
  * Fonseca et al. (2002) R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk, W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T. Katsouleas, et al., _OSIRIS: A Three-Dimensional, Fully Relativistic Particle in Cell Code for Modeling Plasma Based Accelerators_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2002), pp. 342–351, ISBN 978-3-540-47789-1, URL http://dx.doi.org/10.1007/3-540-47789-6_36.
  * Fonseca et al. (2008) R. A. Fonseca, S. F. Martins, L. O. Silva, J. W. Tonge, F. S. Tsung, and W. B. Mori, Plasma Physics and Controlled Fusion 50, 124034 (2008), URL http://stacks.iop.org/0741-3335/50/i=12/a=124034.
  * Hemker (2015) R. G. Hemker, arXiv preprint arXiv:1503.00276 (2015).
  * Dendy (1995) R. O. Dendy, _Plasma physics: an introductory course_ (Cambridge University Press, 1995).
  * Gibbon (2005) P. Gibbon, _Short pulse laser interactions with matter: an introduction_ (World Scientific, 2005).
  * Boyd et al. (2003) T. Boyd, T. Boyd, and J. Sanderson, _The physics of plasmas_ (Cambridge University Press, 2003).
  * Chen et al. (1984) F. F. Chen et al., _Introduction to plasma physics and controlled fusion_ , vol. 1 (Springer, 1984).