# Electromagnetic wave transparency of X mode in strongly magnetized plasma

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  Amita Das
Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New
Delhi 110016, India

###### Abstract

An Electromagnetic (EM) pulse falling on a plasma medium from vacuum can
either reflect or propagate inside the plasma depending on whether it is
overdense or underdense. In a magnetised plasma, however, there are usually
several pass and stop bands for the EM wave depending on the orientation of
the magnetic field with respect to the propagation direction. The EM wave
while propagating in a plasma can excite electrostatic disturbances in the
plasma Vashistha et al. (2020, 2021). In this work Particle - In - Cell
simulations have been carried out to illustrate the complete transparency of
the EM wave propagation inside a strongly magnetised plasma. The external
magnetic field is chosen to be perpendicular to both the wave propagation
direction and the electric field of the EM wave, which is the X mode
configuration. Despite the presence of charged electron and ion species the
plasma medium behaves like a vacuum. The observation is understood with the
help of particle drifts. It is shown that though the two particle species move
under the influence of EM fields their motion does not lead to any charge or
current source to alter the dispersion relation of the EM wave propagating in
the medium. Furthermore, it is also shown that the stop band for EM wave in
this regime shrinks to a zero width as both the resonance and cut-off points
approach each other. Thus transparency to the EM radiation in such a strongly
magnetised case appears to be a norm. This may have important implications in
astrophysical scenarios. For instance, the plasma surrounding objects like
pulsars and magnetars is often threaded with strong magnetic fields.

##  I Introduction

In astrophysical plasmas, a wide range of magnetic fields exist e.g. in galaxy
clusters, shock formation in gamma ray bursts, magnetospheres of neutron stars
Medvedev (2006); Vikhlinin et al. (2001); Vogt and Enßlin (2003); Taylor et
al. (2002). Several instabilities also lead to the generation of intense
magnetic field in the astrophysical plasmas e.g. magnetic field generation via
kelvin-Helmholtz instability, counter-streaming electron flows etc Medvedev et
al. (2006); Alves et al. (2012, 2014). Thus plasmas associated with
astrophysical objects are often threaded with magnetic fields. These magnetic
fields can be very strong in some cases. For instance, near pulsars and
magnetars they could be of the order of Giga Teslas Mazur and Heyl (2011);
Parent et al. (2011); Camilo et al. (2000). It is thus possible that
electromagnetic wave would encounter a strongly magnetized plasma in such
astrophysical scenarios.

Such scenarios involve understanding the interaction of EM waves with matter
which has been covered with tremendous interest for many decades now Kaw and
Dawson (1969); Dawson and Oberman (1963); Kaw (2017). The principle mechanisms
in plasmas depend on EM wave frequency and plasma permittivity. Plasma
permittivity can be altered by suitably choosing the plasma density and
applied magnetic field. The free charges and their associated currents in the
plasma medium act as sources and influence the plasma dielectric constant
making it different from vacuum. Propagation of EM wave through strongly
magnetized plasma sources, therefore, needs to be understood.

The issue of EM wave achieving complete transparency is important and has been
considered earlier in many contexts. An attempt to achieve transparency using
strong fields generated by intense $fs$ laser pulse has been studied by
Teychenné et al. (1998); Giulietti et al. (1997); Siminos et al. (2012);
Kulagin et al. (2007); Palaniyappan et al. (2012). Total transmission was
observed when $30fs$ laser pulse of intensity $3\times 10^{18}Wcm^{-2}$ passes
through $0.1\mu m$ plastic foil targets (Giulietti et al. (1997)). This
mechanism is operative when the target width is much smaller than the laser
wavelength Teychenné et al. (1998). However, for thick targets, relativistic
laser would lead to excitation of coherent structures and/or instabilities
leading to turbulence in the systemMandal et al. (2020); Chatterjee et al.
(2017).

External magnetic fields have been applied in several contexts to achieve
transparency. The dressing of resonance states for RHCP( Right Hand Circularly
Polarized) waves with the combination of axial and wiggler magnetic field is
attempted to acquire a window of transparency in the opaque magnetized plasma
for the EM wave Gad et al. (2012). Another technique to seek transparency is
by employing pump electromagnetic wave to transmit the probe wave (Kawamori et
al. (2012); Shvets and Wurtele (2002); Shvets et al. (2005); Hur et al.
(2003)). This method is analogous to a quantum mechanical phenomena known as
EIT(Electromagnetically Induced Transparency). In this phenomena, an
electromagnetic wave is made to propagate in an normally opaque medium in
presence of powerful secondary EM wave. This is possible due to the
destructive interference between several energy levels connecting the ground
and excited states of the atom. This method is heavily used in non-linear
optics to manipulate the energy levels of atomic states or slow down the
wavesBoller et al. (1991); Fleischhauer et al. (2000); Harris (1993). In
plasma, the use of pump and probe EM wave is used to make plasma transparent
to RHCP wave. But this study is limited to propagation of RHCP waves along the
magnetic field lines and generating an additional wiggler magnetic field is a
complication from application viewpoint. More importantly these methods focus
on R-wave mode where propagation is along the magnetic field, while L-wave and
X-mode propagation geometries are yet to be explored.

We have carried out Particle - In - Cell simulations to study the propagation
of EM wave in a strongly magnetized plasma for which both the electron and ion
species are strongly magnetized. An interesting observation of complete
transparency of the plasma medium is observed for the propagation of EM wave
in the X mode configuration. The width of the stop band of the X mode reduces
to zero and a completely transparent propagation of EM wave is observed. For
X-mode, dispersion curve is given in fig.1, there are two stop bands
($\omega_{LH}-\omega_{L}$ $\&$ $\omega_{UH}-\omega_{R}$, where $LH,UH,L,R$
stands for Lower Hybrid, Upper Hybrid, Left Hand Cutoff, Right Hand Cutoff
respectively). Whenever laser/EM wave frequency lying on these bands is
incident on plasma, it generates a shielding electric field in response to
avoid penetration of EM wave inside it. The propagation of EM wave in plasma
thus depends on the intensity and/or frequency of the incident EM wave. We
show that when the strong external magnetic field dominates the motion of both
charged species, _i.e._ $\omega_{ce}>\omega_{ci}>\omega_{l}$, or it strongly
magnetizes electrons but the perturbations are at faster time scales than that
of ions $\omega_{ce}>\omega_{l}>\omega_{LHC}$, the electromagnetic wave
propagates undisturbed inside plasma. The inequalities gives an insight to
plasma in terms of anisotropy and modification in its collective behaviour
that gets affected by introduction of magnetic field. Such caveats are vital
for absorption phenomena as well.

A strong magnetic field in which the two charged species remain closely tied
to the magnetic field at the EM wave frequency, ensures that they do not
provide the plasma a chance to respond to the incoming laser/ EM wave. We show
that as the strength of magnetic field increases, the magnitude of self
generated fields in bulk plasma due to interaction of laser with plasma
decreases and the EM wave propagation speed inside the plasma approaches the
speed of light $c$. The plasma medium thus acts as a transparent to the
incoming EM wave. These observations would have important implications in
astrophysics. For instance, the strength of magnetic field in neutron stars
and magnetars goes upto Giga-Tesla where this study is expected to be
applicable.

The paper has been arranged as follows: Section II describes our PIC
simulation, Section III contains the analysis and the discussion under
different sub-sections and Sec.IV provides concluding remarks.

##  II  Simulation Details

Table 1: Details of various simulation runs used in this study based on different regions of X-mode shown in fig. 1 |  $B_{0}$ criteria I ($\omega_{ce}>\omega_{l}>\omega_{ci}$) |  $B_{0}$ criteria II ($\omega_{ce}>\omega_{ci}>\omega_{l}$) |  $\omega_{l}$ (in $\omega_{pe}$) |  $a_{0}$ |  $E_{0}$
---|---|---|---|---|---
Region I (0-$\omega_{LH}$) |  3, 8 |  20 |  0.05 |  0.6 |  0.03
Region II ($\omega_{LH}-\omega_{L}$) |  3, 8 |  20, 40 |  0.2 |  0.15 |  0.03
Region III ($\omega_{L}-\omega_{UH}$) |  (0.25,0.5  $\omega_{ce}<\omega_{l}$),3, 8 |  40 |  0.5 |  0.06 |  0.03

We have carried out series of one dimensional (along $\hat{x}$) PIC
simulations in X-mode configurations using OSIRIS-4.0 Hemker (2000); Fonseca
et al. (2002, 2008). For X-mode configuration, uniform external magnetic
field($B_{0}(m_{e}\omega_{pe}c/e)$) has been applied in $\hat{z}$ direction. A
uniform plasma density comprising of electrons and ions has been considered.
Ion mass is taken to be 100 times mass of electrons ($m_{i}=100m_{e}$) for
faster computation. Plasma boundary extends from $x=850c/\omega_{pe}$ to
$x=2000c/\omega_{pe}$ whereas total length of simulation box is
$3000c/\omega_{pe}$. Boundary condition for particles as well as fields are
absorbing. A p-polarized, plane $CO_{2}$ laser pulse is incident normally at
plasma from the left boundary. Laser is propagating along $\hat{x}$ with its
spatial profile centered at x= $450c/\omega_{pe}$ and ranging from x= $0$ to
$800c/\omega_{pe}$. We also want to clarify that this work focuses on proof of
concept so the mechanism presented in this paper depends on the magnetisation
of the charge species with respect to the incoming EM pulse frequency. We have
carried out a parametric study on magnetic fields such that broadly they
follow either criteria I ($\omega_{ce}>\omega_{l}>\omega_{ci}$) or criteria II
($\omega_{ce}>\omega_{ci}>\omega_{l}$). This parametric study has been done
with laser pulse of intensity lying in non-relativistic regime such that
amplitude of laser electric field($E_{0}=0.03(m_{e}\omega_{pe}c/e)$) is
constant for all runs. This has been done to avoid other relativistic
mechanism to play a role. A schematic of simulation geometry has been shown in
Fig.2. Different laser frequency maintaining criteria I and criteria II has
also been chosen according to regional frequency which is explained more
elaborately in next section. A tabular form of simulation parameters is given
in table I.

##  III Results

###  III.1 Theory and Analytical assessment

It is well known that when EM wave is propagating perpendicular to external
magnetic field, plasma supports two kinds of waves, O-mode (ordinary wave) and
extraordinary mode (X-mode). O-mode is independent of applied magnetic field
(ordinary wave). The general dispersion relation for perpendicular propagation
in cold plasma ($\vec{k}\perp\vec{B}$) is given by the matrix,

$\begin{bmatrix}S&-iD&0\\\ iD&S-n^{2}&0\\\
0&0&P-n^{2}\end{bmatrix}\begin{bmatrix}E_{x}\\\ E_{y}\\\ E_{z}\end{bmatrix}=0$
(1)

where, $S=\frac{1}{2}\left(R+L\right)$, $D=\frac{1}{2}\left(R-L\right)$,
$P=1-\frac{\omega^{2}_{p}}{\omega^{2}}$

$R=1-\frac{\omega_{pe}^{2}+\omega_{pi}^{2}}{(\omega+\omega_{ci})(\omega-\omega_{ce})}$
(2)

$L=1-\frac{\omega_{pe}^{2}+\omega_{pi}^{2}}{(\omega-\omega_{ci})(\omega+\omega_{ce})}$
(3)

The X-mode has R and L mode having cut-offs at $\omega_{R}$ and $\omega_{L}$
respectively. $\omega_{R}$ and $\omega_{L}$ are given as follows:

$\omega_{R,L}=[\omega_{pe}^{2}+\omega_{pi}^{2}+(\omega_{ci}+\omega_{ce})^{2}/4]^{1/2}\mp(\omega_{ci}-\omega_{ce})/2$
(4)

Dispersion curve of X-mode is shown in fig.1. We have classified dipersion
curve into three regions depending on the dominant role played by the species.
Region I is dominated by dynamics of ions and Region III for electrons. Region
II is stop band as it lies between $\omega_{LH}$ (resonance point) and
$\omega_{L}$ (cut-off point).

Dispersion relation for X-mode is obtained

$n^{2}=\frac{RL}{S}$ (5)

Principal resonance occur when $S\rightarrow 0$

$\omega^{4}-\left(\omega_{pe}^{2}+\omega_{pi}^{2}+\omega_{ce}^{2}+\omega_{ci}^{2}\right)\omega^{2}+\omega_{ci}^{2}\omega_{ce}^{2}+\omega_{pe}^{2}\omega_{ci}^{2}+\omega_{pi}^{2}\omega_{ce}^{2}=0$
(6)

This is a bi-quadratic equation, it’s lower end solution is plotted as
function of applied magnetic field in fig.4. As can be seen from the figure,
when $B_{0}<10$(at $B_{0}=10,\omega_{ci}=\omega_{pi}$) it falls in criteria I
and solution of eq. 6 matches perfectly with reduced expression of
$\omega_{LH}$. At higher magnetic fields, $\omega_{LH}$ saturates at
$\omega_{pi}$ while solution of eq. 6 approaches left hand cut off (
$\omega_{L}$) asymptotically which concludes that at this higher magnetic
field the resonance point and cut off approach each other thus effectively
reduce the width of the stop band. This was checked by simulation as well for
frequency parameter lying in region II(i.e. stop band). Under criteria I,
laser reflected back. On the other hand, under criteria II laser pulse was
able to propagate through the plasma. This was possible due to effective
reduction of stop band and resonance point lying well above EM frequency. So,
effectively this case does not lie in region II but in region I.

In Fig.3, we have plotted the dispersion curve for X-mode in two criteria. As
one can observe from the left subplot that in criteria I, all the regions are
well separated while in the criteria II stop band has shrunk. Moreover, the
dispersion follows $\omega=k$(as $c=1$). Therefore, it rules out any other
mode excitation when $\omega_{ce}>\omega_{ci}>\omega_{l}$.

To summarize our propagation characteristics according to their region of
dispersion curve in fig. 1. is given in table II.

Table 2: Details of propagation characteristics observed in various simualtion runs for different regions of dispersion curve |  criteria I |  criteria II
---|---|---
Region I |  LH |  Transparent
Region II |  Stop band |  Transparent
Region III |  Transparent |  N.A

Detail quantitative analysis to calculate absorption, reflection, transmission
coefficients has also been done which is presented here in tabulated form
(Table III).

Table 3: A comparison of various coefficients with different external magnetic field |  $B_{0}$ |  R |  T |  A
---|---|---|---|---
Region I |  3 |  0.67 |  8.9$\times 10^{-9}$ |  0.32
8 |  0.0663 |  0.872 |  0.0246
20 |  0.0035 |  0.9931 |  0.02
Region II |  3 |  0.99 |  7.95$\times 10^{-9}$ |  1.9$\times 10^{-5}$
8 |  0.998 |  $8\times 10^{-7}$ |  0.0036
20 |  0.0089 |  0.9823 |  0.0047
40 |  5.6$\times 10^{-4}$ |  0.9989 |  0.0018
Region III |  3 |  0.013 |  0.9742 |  0.003
8 |  0.006 |  0.9988 |  0.0019
40 |  0.0035 |  0.9931 |  0.02

One comment should be made here about another solution in the upper end of
frequency scale, it was found that at high magnetic field region $\omega_{ce}$
dominates all modes and cut off points and hence they merge very well. Plot of
exact solution of eq.6 as function of $B_{0}$ applied is given in fig.5. As
one can observe here gap between $\omega_{R}$ and $\omega_{UH}$ is very thin
and at high magnetic fields they also merge indicating that stop band at upper
frequency also vanishes with application of strong magnetic field.

###  III.2 $\omega$ and $\vec{k}$ Analysis

In any dispersive medium, as the refractive index of media change spatially,
the frequency of the EM wave remains same while its wavelength suffers a
change. In this section we calculate the modified $k$ and the phase velocity
of incident laser pulse. We observe that by varying ambient magnetic field,
phase velocity of laser pulse also changes (approaches velocity of light in
vacuum,$c$) while decreasing the perturbations in the plasma.

Fig.6 shows a comparison of all three cases lying in Region I. Initially (at
t=0), electric field due to laser is present in the system. In case(A)
($B_{0}=0$), the laser interacts with plasma and gets reflected back from the
plasma surface. However, for case(B) ($B_{0}=3$, satisfying the condition
$\omega_{ce}>\omega_{l}>\omega_{ci}$), there are certain modes generated in
plasma and as a result we observe a finite magnitude of $E_{x}$ in the system.
$E_{x}$ that get generated in plasma have higher magnitude than $E_{y}$. On
the other hand, in case(C) ($B_{0}=20$, satisfying the condition
$\omega_{ce}>\omega_{ci}>\omega_{l}$), the plasma seems to be completely
undisturbed by the laser as pulse freely propagates inside it without creating
any perturbations in the medium and goes into vacuum space in the right side.
The transparency induced in plasma on applying external magnetic field is the
key observation of this work. Plasma density plots show that in case(A),
plasma density at the interface is modified, on the contrary, for case(B),
density perturbations are present in the bulk plasma as well. Ion density
fluctuates more than electrons which propagates in longitudinal direction as
can be seen at later times in fig.7. For case(C), there being density
perturbations that can be seen at $t=500$ is due to laser field _i.e._
electrons and ions fluctuates with same amplitude, justifying our observation
that plasma remains undisturbed via interaction with laser in this case[fig.
7].

Table 4: Velocity of EM wave (normalised to c) in plasma with changing magnetic field $B_{0}$ |  Velocity from method I |  Modified $K_{x}$ |  Velocity from method II
---|---|---|---
8 |  0.78 |  0.2518 |  0.79
15 |  0.87 |  0.228 |  0.88
20 |  0.92 |  0.2137 |  0.94
40 |  0.96 |  0.2030 |  0.98

Fig. 8 shows the FFT of $B_{z}$ of laser with respect to time for four
different value of magnetic field, where transparency has been induced. It can
be seen that the frequency of laser($\omega_{l}=0.2$) does not change while
propagating inside plasma (we show FFT of $B_{z}$ with time at two different
values of x in the bulk plasma and obtain the same peak.). However, $k_{x}$ of
EM wave gets modified on propagation inside plasma (figure 9). The shift from
the initial value of $k_{x}$ decreases on increasing applied magnetic field.
We calculate the modified velocity of EM wave in plasma by peak frequency of
the wave from the FFT and modified $k_{x}$ value (method II in table 4) and
found that velocity of the wave inside plasma approaches to $c$ on increasing
applied magnetic field ( Table 4). In table 4, we calculate velocity by two
methods. In Method I, we choose a point on the waveform and calculate the time
taken by that point to cover a particular distance and method II includes
calculation of velocity by modification in $k_{x}$. So we conclude that strong
magnetization can stop pulse modification while pulse waveform is propagating
through plasma media.

###  III.3 Reversible and Irreversible exchange of energy

In this study, we observed that depending on region and criteria, the laser
energy exchange is either reversible or irreversible. As one can notice from
table3, in region I criteria I there is significant absorption and this region
is well explored in refVashistha et al. (2020). From these studies we know
that in region I criteria I energy is dominantly coupled to ions and this
coupling process is irreversible. While in region I criteria II, due to
tansparency, energy transfer is observe to be reversible. As when laser is
present in the plasma, electrons and ions oscillate due to oscillating
electric field and when the field passes through, they come to rest.
In region II criteria I, laser reflects back due to formation of shielding
fields so there’s no exchange of energy altogether. On the other hand region
II criteria II is effectively Region I criteria II so there’s similar exchange
of energy which is reversible.
In region III criteria I, we observe $97\%$ transparency and reversible
exchange of energy. This is quite different from other two regions, the reason
behind this is simple. As the time scales of region III are same as electrons,
with $B_{0}=3$ electrons are strongly magnetized. So that’s why laser is not
able to couple its energy into electron effectively. To couple laser energy
into electron irreversibly one has to weakly magnetise the electron and that
can be achieved by ensuring another inequality _i.e._
$\omega_{ce}<\omega_{l}<\omega_{L}$. When we simulated with this condition by
taking $B_{0}=0.25$, we observe $10.2\%$ absorption into electrons and about
$2.2\%$ energy to ions irreversibly while $88\%$ of laser pulse was reflected
back.
We concluded that when the species are tightly bound to external magnetic
field, they are not able to take energy from EM pulse irreversibly. That’s why
in region I and criteria II when both the species were tightly magnetized to
external magnetic field they were unable to couple their motion to laser pulse
and that’s how pulse was transparent in this medium. In region III where it is
in propagating region when electrons were tightly bounded we observe
transparency for similar reason. One can argue that ions are not magnetized in
this condition but this region’s time scales are fast so only electron motion
is important here.

###  III.4 Drift Velocity Comparison in Different Regions

In this section, we will provide proof on how charge species are magnetized or
unmagnetized in different regimes and their analytic estimates are compared to
their simulation values. Under the effect of oscillating electric field and
external magnetic field, the longitudinal drift can be written by eq.7

$\vec{V}_{\vec{E}\times\vec{B}}(t)=\frac{\omega_{cs}^{2}}{\omega_{cs}^{2}-\omega_{l}^{2}}\frac{\vec{E}(t)\times\vec{B}}{B^{2}}$
(7)

Here, the suffix $s=e,i$ represents the electron and ion species respectively.
Like in Criteria II, both ions and electrons are strongly magnetised and the
dynamics are governed by the Lorentz force i.e. eq.7. And as longitudinal
velocity is independent of mass of the species for this criteria, there’s no
possibility of charge separation such that there are no shielding fields to
restrict the EM pulse propagation. A comparative analysis has been done
between analytical drift given in eq.7 and numerical longitudinal drift
experienced by both ion and electron in subplot(B) of fig. 10 and it can be
observed that there’s no velocity difference between species which results in
no net charge density separation and hence EM wave is able to propagate
unhindered. In Criteria I, electrons follow eq.7 while ion motion is governed
by electric fields as they are unmagnetized. This is shown in subplot (A) and
(C) of fig.10. We observe that they match well. When
$\omega_{ce}<\omega_{l}$(named as Criteria 0), both species are unmagnetized
and they follow the longitudinal electric field (see subplot (D) of fig.10).

##  IV conclusion

A detail PIC simulation has been carried out by us to show complete
transparency of EM wave radiation through a plasma in the presence of strong
ambient field. The strength of the magnetic field has to be strong enough to
elicit magnetised response from both electron and ion species at the EM wave
frequency.

The effect is understood by realising that at such strong magnetic field the
the hybrid modes essentially vanish and in that regime resonance point and
cut-off points approach each other resulting in reduction and eventually
disappearance of the width of the stop band. Thus, the EM wave freely
propagates inside the plasma medium for any choice of frequency in this
particular regime. It has also been shown that the particle drifts are
identical for both electron and ion species at strong magnetic field. Thus,
plasma provides for no charge and current sources in the EM wave propagation.
The electrostatic perturbations, ( which lead to irreversible transfer of EM
wave energy to plasma) normally occur due to differential drift of the two
species. In this case the drifts being identical there is no conversion to
electrostatic perturbations of the EM wave.

We feel that these observations will have important significance in the
context of astrophysical plasma near pulsar and magnetars where the magnetic
field is quite strong and would elicit magnetised plasma response for typical
EM frequencies of interest.

## Acknowledgements

The authors would like to acknowledge the OSIRIS Consortium, consisting of
UCLA ans IST(Lisbon, Portugal) for providing access to the OSIRIS4.0 framework
which is the work supported by NSF ACI-1339893. AD would like to acknowledge
her J. C. Bose fellowship grant JCB/2017/000055 and the CRG/2018/000624 grant
of DST for the work. The simulations for the work described in this paper were
performed on Uday, an IPR Linux cluster.

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Figure 1: Dispersion of X-mode when $\vec{k}\perp\vec{B_{0}}$, classified into
three regions. The numbers on the right hand side correspond to resonance and
cutoff frequencies for $m_{i}=100m_{e}$ in normalized units.  Figure 2:
Schematic of simulation geometry. Plasma slab is placed in vacuum, direction
of external Magnetic field($B_{0}$) is along the same direction as that of the
magnetic field of laser($B_{l}$) _i.e._ $\hat{z}$. Electric field of laser is
along $\hat{y}$ direction.  Figure 3: Dispersion curve for two
criteria(criteria I $(\omega_{ce}>\omega_{l}>\omega_{ci})$ and criteria II
$\omega_{ce}>\omega_{ci}>\omega_{l})$) are shown in subplot (a) and (c)
respectively). In subplot (b) and (d), we expand the dispersion curves shown
in (a) and (c) respectively. From these curves, we can observe that while
criteria I has two stop bands and three pass bands, criteria II dispersion has
$\omega=ck$ all along.  Figure 4: Comparison of exact solution for $\omega$
with cut off frequency and analytical expression of LH as function of $B_{0}$.
As one can observe that at lower magnetic field value there’s gap between
values of $\omega_{L}$ and $\omega_{LH}$ but at high value of magnetic field
where exact solution of $\omega$ will play a role it matches with $\omega_{L}$
signifying the vanishing of stop band. Gray dash-line indicates the
$\omega_{pi}$ position in the plot. Figure 5: Comparison of one exact solution
for $\omega$ with cut off frequency and analytical expression of UH as
function of $B_{0}$. As one can observe that at lower magnetic field value
there’s gap between values of $\omega_{R}$ and $\omega_{UH}$ but at high value
of magnetic field where exact solution of $\omega$ will play a role it matches
with $\omega_{R}$ indicative to the vanishing of stop band. Figure 6: Spatial
variation of $E_{x}$ and $E_{y}$ for the three cases at different times.
Initially, only $E_{y}$(due to laser field) is present in the system. In case
(A), laser gets reflected from the plasma boundary without being able to
interact with it. On the other hand, laser is able to interact with plasma in
Case(B) and we observe generation of $E_{x}$ in the system. In case (C),
medium becomes transparent to laser and laser just passes through the plasma
medium unhindered. Figure 7: Spatial variation of number density of electrons
and ions for the three cases at different times showing that plasma density
gets perturbed at the interface for case(A) whereas for case(B), we observe
density fluctuations in the bulk plasma as well which retain themselves even
after laser has moved out of the simulation box($t=1000$).For case(C), on the
other hand, we observe some density perturbations in the bulk plasma at the
time of interaction with laser($t=500$) but these fluctuations are not
retained by plasma after the interaction with laser is over. Figure 8: FFT of
$B_{z}$ with time at two different locations in bulk plasma, showing that the
peak frequency of electromagnetic wave does not change and it propagates
unhindered in the plasma medium for different magnetic
fields($B_{0}=8,15,20,40$). Figure 9: FFT of $B_{z}$ with x at different times
showing that as laser interacts with plasma, $k_{x}$ shifts to a larger value
than the initial $k_{x}$. This shift reduces as we increase the magnitude of
applied magnetic field(shown in different subplots). This shift in $k_{x}$ is
also reflected in velocity of EM wave approaching c in the medium (Table 4)
and the transmission coefficient approaching unity (Table 3).  Figure 10:
Comparison of electron and ion drift theoretically(eq. 7) and simulation
results. In criteria II both species are magnetised and their drift velocity
in longitudinal direction is also same such that there’s no charge separation
which can inhibit the propagation of EM wave (B). In the case when
$\omega_{l}>\omega_{ce}$, there’s generation of electrostatic perturbation
which helps in coupling laser energy into plasma irreversibly shown in
subplot(D). For criteria I in region I and III, ions are un-magnetised while
electrons are tightly bound to magnetic field(in subplot(A) and (C)).