# Simultaneous polarization transformation and amplification of multi-petawatt
laser pulses in magnetized plasmas

Xiaolong Zheng  Key Laboratory for Laser Plasmas (MoE), School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Collaborative Innovation Center of IFSA, Shanghai Jiao Tong University,
Shanghai 200240, China  Suming Weng<EMAIL_ADDRESS>Key Laboratory for
Laser Plasmas (MoE), School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  Collaborative Innovation Center of IFSA,
Shanghai Jiao Tong University, Shanghai 200240, China  Zhe Zhang  National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing 100190, China  Hanghang Ma  Key Laboratory for Laser
Plasmas (MoE), School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Collaborative Innovation Center of IFSA, Shanghai Jiao
Tong University, Shanghai 200240, China  Min Chen  Key Laboratory for Laser
Plasmas (MoE), School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Collaborative Innovation Center of IFSA, Shanghai Jiao
Tong University, Shanghai 200240, China  Paul McKenna  SUPA, Department of
Physics, University of Strathclyde, Glasgow G4 0NG, UK  Zhengming Sheng
<EMAIL_ADDRESS>Key Laboratory for Laser Plasmas (MoE), School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Collaborative Innovation Center of IFSA, Shanghai Jiao Tong University,
Shanghai 200240, China  SUPA, Department of Physics, University of
Strathclyde, Glasgow G4 0NG, UK  Tsung-Dao Lee Institute, Shanghai 200240,
China

###### Abstract

With increasing laser peak power, the generation and manipulation of high-
power laser pulses becomes a growing challenge for conventional solid-state
optics due to their limited damage threshold. As a result, plasma-based
optical components which can sustain extremely high fields are attracting
increasing interest. Here, we propose a type of plasma waveplate based on
magneto-optical birefringence under a transverse magnetic field, which can
work under extremely high laser power. Importantly, this waveplate can
simultaneously alter the polarization state and boost the peak laser power. It
is demonstrated numerically that an initially linearly polarized laser pulse
with 5 petawatt peak power can be converted into a circularly polarized pulse
with a peak power higher than 10 petawatts by such a waveplate with a
centimeter-scale diameter. The energy conversion efficiency of the
polarization transformation is about $98\%$. The necessary waveplate thickness
is shown to scale inversely with plasma electron density $n_{e}$ and the
square of magnetic field $B_{0}$, and it is about 1 cm for $n_{e}=3\times
10^{20}$ cm-3 and $B_{0}=100$ T. The proposed plasma waveplate and other
plasma-based optical components can play a critical role for the effective
utilization of multi-petawatt laser systems.

##  I Introduction

Higher laser peak power has been continuously pursued since the laser was
invented. Thanks to development of laser technologies, especially chirped
pulse amplification Strickland , peak laser power at the petawatt (PW) level
is now routinely achieved and a number of multi-PW lasers are built or under
construction Danson . When such an ultra-high-power laser pulse is tightly
focused, a laser peak intensity exceeding $10^{21}$ W/cm2 can be achieved,
which not only brings about many prospective applications but also becomes a
unique tool to create extreme conditions for fundamental research MourouRMP ;
Gibbon . With increasing laser peak power, however, conventional solid-state
optical components must be enlarged to avoid laser-induced damage Stuart . In
general, multi-PW laser systems require metre-scale optical components Cotel ,
which are economically costly and technically challenging.

Plasma-based optical components offer an attractive solution to this issue,
since all material will be at least partially ionized by an intense laser
pulse and the resultant plasma can be used to manipulate the laser light. The
design and application of high-power lasers has the potential to be
revolutionized by plasma-based optics Suckewer , which may pave the way for
the study of laser-matter interactions at unprecedented intensities. So far,
planar plasma mirrors are extensively used to the contrast of intense lasers
Doumy ; Thaury1 , while ellipsoidal plasma mirrors or flying plasma mirrors
are used to focus or compress laser pulses toward extreme intensities
Nakatsutsumi ; Bulanov . Plasma-based optics are also extensively studied for
other prospective applications, including plasma amplifiers Shvets ; Andreev ;
Trines ; Chiaramello ; Turnbull2018 , plasma gratings or photonic crystals
Sheng ; WuPoP ; WuAPL ; YuJOSAB ; Lehmann2016 , plasma optical modulators YuNC
, plasma apertures or shutters Gonzalez1 ; Gonzalez2 ; Wei , plasma holograms
Leblanc , and cross beam energy transfer Michel ; Moody . In particular,
plasma gratings can act as polarizers or waveplates Michel2014 ; Turnbull2016
; Lehmann2018  to control the polarization state of intense laser pulses.
Recently, magnetized plasmas have been proposed for the conversion of a
linearly polarized (LP) laser pulse into two circularly polarized (CP) pulses
by use of the Faraday effect under a strong longitudinal magnetic field Weng .
The drawback of this scheme is that it reduces the peak power of the laser
pulse by a factor of two.

In this work, we explore the birefringence effect of strong transverse
magnetic fields induced in plasmas and propose to apply this effect to design
plasma waveplates. In a plasma under a transverse magnetic field, the phase
velocity of a light wave depends on the orientation of its electric field.
Therefore, the magnetized plasma can be used as a waveplate like a
birefringent crystal. In particular, this magnetized plasma waveplate has two
peculiar advantages in manipulating ultra-high-power laser pulses. Firstly, it
is not susceptible to optical damage, as usually found for solid state optics.
Secondly, it can simultaneously alter the polarization state of an intense
laser pulse and effectively boost the peak pulse power via self-compression of
the pulse.

##  II Theoretical analysis

We first consider the propagation of a LP light wave in a plasma that is
subjected to an external transverse magnetic field $\textbf{B}_{0}$. In
general, this LP wave is a combination of two sub-waves: an ordinary wave with
the electric field $\textbf{E}\parallel\textbf{B}_{0}$ and an extraordinary
wave with $\textbf{E}\perp\textbf{B}_{0}$, hereinafter referred to as O-wave
and X-wave, respectively. The O- and X-waves have distinct dispersion
relations as given respectively in the following FFChen

$\displaystyle\frac{c^{2}k^{2}}{\omega^{2}}$ $\displaystyle=$
$\displaystyle\frac{c^{2}}{v_{\phi,O}^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}},$
(1) $\displaystyle\frac{c^{2}k^{2}}{\omega^{2}}$ $\displaystyle=$
$\displaystyle\frac{c^{2}}{v_{\phi,X}^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}\frac{\omega^{2}-\omega_{p}^{2}}{\omega^{2}-\omega_{p}^{2}-\omega_{c}^{2}},$
(2)

where $\omega$ and $k$ are the wave angular frequency and wavenumber,
$\omega_{p}=(n_{e}e^{2}/\epsilon_{0}m_{e})^{1/2}$ and
$\omega_{c}=eB_{0}/m_{e}$ are respectively the plasma frequency and the
electron cyclotron frequency, and $n_{e}$ is the electron number density. The
above equations define the distinct phase velocities $v_{\phi,O}$ and
$v_{\phi,X}$ for the O- and X-waves, respectively. Subsequently, a phase
difference $\Delta\phi$ will be induced between the O- and X-waves, which
makes the incident LP wave elliptically polarized. Therefore, the magnetized
plasma can be considered as a wave retarder or waveplate. A notable case is
that an incident wave LP at $45^{\circ}$ with respect to $\textbf{B}_{0}$ can
be converted into a CP wave as shown in Fig. 1 when the phase difference

$\Delta\phi=2\pi\frac{d}{\lambda}(\frac{c}{v_{\phi,O}}-\frac{c}{v_{\phi,X}})=\frac{\pi}{2},$
(3)

where $d$ is the propagation distance in the plasma and $\lambda$ is the
wavelength (see Visualization 1 for the time evolution of the electric field
vector in this polarization transformation). In the case $\omega_{p}\ll\omega$
and $\omega_{c}\ll\omega$, the phase difference can be simplified to
$\Delta\phi\simeq\pi(d/\lambda)(n_{e}/n_{c})(eB_{0}/m_{e}\omega)^{2}$, where
$n_{c}=\epsilon_{0}m_{e}\omega^{2}/e^{2}$ is the critical plasma density.

Figure 1: Sketch of the magnetized plasma quarter-wave plate. A LP light wave
is incident into a plasma under a transverse magnetic field $\textbf{B}_{0}$.
The incident wave can be equally divided into an ordinary sub-wave with
$\textbf{E}\parallel\textbf{B}_{0}$ and an extraordinary sub-wave with
$\textbf{E}\perp\textbf{B}_{0}$ if its polarization plane is oriented at
$45^{\circ}$ with respect to $\textbf{B}_{0}$. A phase difference $\Delta\phi$
emerges between these two sub-waves because of their different phase
velocities, and a CP wave can result when $\Delta\phi=\pi/2$.

This magneto-birefringence effect in plasmas is analogous to the Voigt effect
in gases found in 1902 or the Cotton-Mouton effect in liquids discovered in
1907 Voigt . However, this effect is usually applied only for the polarimetric
diagnosis of magnetic fields Segre . To the best of our knowledge, it is the
first time to propose a concept of magnetized plasma waveplate based on this
effect for high power laser pulses. Above all, magnetized plasmas as a
restoring medium have unique strengths in manipulating intense laser pulses.
Importantly, an intense laser pulse can be further amplified by nonlinear
effects in a plasma Shvets ; Andreev ; Trines ; Chiaramello ; Turnbull2018 .
In particular, a weakly relativistic laser pulse can be intensified via the
process of self-compression in a plasma Shorokhov . In a plasma, the envelope
of an intense laser pulse can be approximately described by the nonlinear
Schrödinger equation (NLSE) Ren

$2i\frac{\omega^{2}}{\omega_{p}^{2}}\frac{v_{g}^{3}}{c^{3}}\frac{\partial
a}{\partial\tau}+\frac{\omega^{2}}{\omega_{p}^{2}}\frac{v_{g}^{2}}{c^{2}}\nabla_{\bot}^{2}a+\frac{\partial^{2}a}{\partial\psi^{2}}+\frac{v_{g}^{2}}{c^{2}}(1-\frac{1}{\gamma})=0,$
(4)

where $\tau=z\omega/c$, $\psi=(t-z/v_{g})\omega$, $v_{g}=kc^{2}/\omega$, the
normalized electromagnetic vector potential
$\textbf{a}=e\textbf{A}/m_{e}c^{2}$ , and the electron Lorentz factor
$\gamma\simeq\sqrt{1+a^{2}/2}$ in a LP laser pulse. Here the effect of the
external magnetic field can be neglected as long as $\omega_{c}\ll\omega$. The
transverse self-focusing becomes obvious at a distance
$z=Z_{R}(P/P_{c}-1)^{-1/2}$ Gibbon , where $Z_{R}=\pi r_{0}^{2}/\lambda$ is
the Rayleigh length, $r_{0}$ is the pulse waist, $P$ is the pulse power, and
$P_{c}\simeq 17.5n_{c}/n_{e}$ GW is the critical power for relativistic self-
focusing. Since $z\propto r_{0}^{2}$, the self-focusing is usually negligible
within the Rayleigh length when the laser pulse is not tightly focused.
Consequently, Eq. (4) is descended to the one-dimensional (1D) NLSE Shorokhov
; Ren

$2i\frac{\omega^{2}}{\omega_{p}^{2}}\frac{v_{g}^{3}}{c^{3}}\frac{\partial
a}{\partial\tau}+\frac{\partial^{2}a}{\partial\psi^{2}}+\frac{v_{g}^{2}}{c^{2}}(1-\frac{1}{\gamma})=0,$
(5)

which describes the pure longitudinal self-compression.

##  III Simulation verification

To test the performance of magnetized plasma waveplates, we perform a series
of particle-in-cell (PIC) simulations using the Osiris code Fonseca . In each
simulation, a laser pulse with a wavelength $\lambda=1$ $\mu\texttt{m}$ is
incident along the $z$-axis into a magnetized plasma that is located at $0\leq
z\leq d_{0}$. The moving window is employed with a simulation box moving along
the $z$ axis at the speed of light in vacuum. In 1D simulations, the spatial
and temporal resolutions are $\Delta z=\lambda/40$ and $\Delta t\simeq\Delta
z/c$, and each cell has 8 macroparticles. In the 3D simulation, the spatial
resolutions are $\Delta z=\lambda/16$ and $\Delta x=\Delta y=50\lambda$, the
temporal resolution is $\Delta t\simeq\Delta z/c$, and each cell has four
macroparticles. The initial pulse is assumed to be LP at $+45^{\circ}$ with
respect to the $x$-axis, and the external magnetic field $\textbf{B}_{0}$ is
along the $x$-axis. The polarization state of the laser pulse is described by
the Stokes parameters Saleh : $I=|E_{x}|^{2}+|E_{y}|^{2}$,
$Q=|E_{x}|^{2}-|E_{y}|^{2}$, $U=2\texttt{Re}{\\{E_{x}^{*}E_{y}}\\}$ and
$V=2\texttt{Im}{\\{E_{x}^{*}E_{y}}\\}$, where $E_{x}$ ($E_{y}$) is the complex
amplitude of the electric field along the $x$ ($y$) axis, and $E_{x}^{*}$ is
the complex conjugate of $E_{x}$. The parameter $I$ denotes the intensity
regardless of polarization, $Q$ the LP along the $x$ ($+$) or $y$ ($-$) axes,
$U$ the LP at $+45^{\circ}$ ($+$) or $-45^{\circ}$ ($-$) from the $x$ axis,
$V$ the right-handed ($+$) or left-handed ($-$) CP. For comparison, all
parameters are normalized to the initial peak intensity $I_{0}$.

Figure 2: Stokes parameters of a laser pulse (a) before and (b) after a
magnetized plasma quarter-wave plate with $d_{0}\simeq 3.9$ cm,
$n_{e}/n_{c}=0.3$, and $B_{0}=50$ T. To be a quarter-wave plate, the required
plasma thickness $d_{0}$ as a function of (c) plasma density $n_{e}$ and (d)
magnetic field $B_{0}$, where the PIC simulation results (triangles) are in
good agreement with the predictions by Eq. (3) (solid curves). In the
simulation, the laser pulse initially has a FWHM duration of 300 fs and a
relatively weak intensity ($a_{0}=0.001$).

Figure 2 displays 1D PIC-simulation results for a weak laser pulse with an
amplitude $a_{0}=0.001$ ($I_{0}\simeq 10^{12}$ W/cm2). Figures 2 (a) and (b)
show the Stokes parameters of the pulse before and after it passes through a
magnetized plasma with $d_{0}\simeq 3.9$ cm, $n_{e}/n_{c}=0.3$ and
$B_{0}\simeq 50$ T ($eB_{0}/m_{e}\omega=0.005$). According to Eq. (3), this
magnetized plasma will induce a phase difference $\Delta\phi\simeq\pi/2$ and
thus act as a quarter-wave plate. Therefore, the laser polarization is
converted from LP at $+45^{\circ}$ $(U/I\simeq+1)$ to left-handed CP
$(V/I\simeq-1)$. Further, Figs. 2 (c) and (d) show that the required plasma
thickness $d_{0}$ scales inversely with the plasma density $n_{e}$ and the
square of the magnetic field $B_{0}$, as predicted by Eq. (3). In the
simulations with weak laser pulses, the degrees of circular polarization
$|V|_{\max}/I_{\max}$ of the output pulses always exceed $99\%$. However, the
output peak intensities are, to some extent, smaller than the input values due
to the dispersive broadening. For example, it is about $0.67I_{0}$ in the case
shown in Fig. 2(b).

Figure 3: (a) Stokes parameters for a weakly relativistic laser pulse with
$a_{0}=0.015$ after it passes through the magnetized plasma, where the input
intensity profile $I_{\texttt{in}}$ is also drawn for comparison; (b) the
output peak intensity $I_{0}^{\prime}$ as a function of the initial amplitude
$a_{0}$. (c) The maximum peak intensity $I_{\max}$ can be achieved during the
laser propagation in the plasma and (d) the propagation distance $z_{\max}$
where this maximum is achieved as functions of $a_{0}$; where the PIC
simulation results (triangles) are in good overall agreement with the
predictions by NLSE Eq. (5) (squares). Except $a_{0}$, all other parameters
are the same as those in Fig. 2 (b).

When increasing the laser pulse intensity such that it is weakly relativistic,
we find that the dispersive broadening can be easily compensated by the pulse
self-compression. As illustrated in Fig. 3(a), a laser pulse with
$a_{0}=0.015$ is intensified by a factor of $2.4$ in another 1D PIC-simulation
using the same magnetized plasma as used for the results presented in Fig.
2(b). Meanwhile, the output pulse maintains a high degree of circular
polarization ($\sim 95\%$). Fig. 3(b) shows that the output peak intensity
increases with the increasing $a_{0}$ when $a_{0}\leq 0.015$. This is because
the self-compression becomes more and more significant with increasing laser
intensity Shorokhov . Figure 3(c) shows that the maximum achievable
amplification factor $I_{\max}/I_{0}$ of the pulse increases with increasing
$a_{0}$. For cases $a_{0}>0.015$, however, the self-compression becomes so
fast that the maximum amplification factor will be achieved for a decreasing
propagation distance $z_{\max}$ with the increasing $a_{0}$ as shown in Fig.
3(d). Therefore, the maximum amplification of the pulse will be achieved in
the interior of the plasma. Consequently, in the case $a_{0}>0.015$ the final
output peak intensity from the magnetized plasma in Fig. 3(b) will be lower
than the maximum achievable value shown in Fig. 3(c). The small difference
between the maximum achievable peak intensities from the numerical analysis
and PIC simulations in Fig. 3(c) may be due to the simplifications in the
deduction of Eq. (5)Ren . Further, it is worth mentioning that the magnetic
field used in this work has no obvious influence on the pulse compression
since $\omega_{c}\ll\omega_{p}$ Wilson2017 ; Wilson2019 . Therefore, we ignore
the effect of magnetic field in the theoretical model for the pulse self-
compression, i.e., Eq. (5).

Figure 4: Isosurfaces of the electromagnetic field energy density
$E^{2}+B^{2}=\mathcal{E}_{0}/10$ (a) before ($t=-50$), (b) during ($t=950$),
and (c) after ($t=1350$) the laser pulse interaction with a magnetized plasma,
where $\mathcal{E}_{0}$ is the maximum of the initial electromagnetic energy
density; (d) the pulse power as a function of the variable $\psi$ for the
input and output pulse, respectively; (e) Stokes parameters on the $z$-axis of
the laser pulse after it passes through the magnetized plasma, where the input
intensity profile $I_{\texttt{in}}$ is also drawn for comparison.

The above results indicate that the higher is the laser intensity, the faster
the self-compression occurs. So it seems that the magnetized plasma quarter
waveplate should have an extremely thin thickness for laser pulses at
relatively high intensities. Correspondingly, the required magnetic field
should be extremely strong according to Eq. (3). However, it is important to
note that an intense laser pulse can experience a periodic compression and
decompression, in which the peaks of the pulse amplification appear
periodically Shorokhov . Therefore, the plasma waveplate thickness can be set
as the propagation distance where the second or subsequent amplification peak
is achieved. Correspondingly, the magnetic field strength of plasma quarter
waveplate for intense laser pulses might be greatly reduced to a more
accessible level.

The capability of the magnetized plasma waveplate for laser pulse
intensification has also been demonstrated via a 3D PIC simulation as shown in
Fig. 4, where a shorter laser pulse duration 80 fs and a stronger magnetic
field $B_{0}=300$ T are invoked to reduce the computation cost. For
$B_{0}=300$ T, the required thickness of the plasma with $n_{e}/n_{c}=0.3$
will be greatly reduced to $d_{0}\simeq 1083\lambda$ as predicted by Eq. (3).
Accordingly, the initial laser amplitude increases up to $a_{0}=0.1$ in order
to speed up the pulse compression and achieve an obvious intensification at
$d=d_{0}$. The strong self-compression of the laser pulse is clearly visible
in Figs. 4(a), (b) and (c) (see Visualization 2 for the entire self-
compression process), where a large pulse waist $r_{0}=4800\lambda$ is set to
gain an initial peak power $P\simeq 5$ PW. The pulse compression results in an
increasing power. As shown in Fig. 4(d), the peak power is boosted from 5 PW
up to 11.5 PW by the magnetized plasma waveplate. Correspondingly, the peak
intensity is enhanced by about a factor of 2 as shown in Fig. 4(e), which also
illustrates that the output pulse has a high degree of circular polarization
($\sim 94\%$). The energy conversion efficiency of the polarization
transformation is as high as $98\%$. The resultant ultra-high-power CP pulse,
focused properly, could be extensively applied to laser-driven ion
acceleration and ultra-bright X-ray radiation Scullion ; Yu ; Bin ; Shen .

##  IV Discussion

Although circularly polarized laser pulses with peak powers as high as 10
petawatts have already been assumed broadly in recent theoretical studies
Scullion ; Yu ; Bin ; Shen , the generation of such pulses is still one of the
outstanding problems in the field of high-power laser science. Nowadays, the
generation of circularly-polarized laser pulses still relies on the use of
conventional crystal quarter-wave plates. For a 10 petawatt laser pulse,
however, the diameter of the crystal waveplate must be of the order of metre-
scale to avoid laser-induced damage. It is presently not possible to
manufacture such a large-diameter crystal waveplate of the required thickness
and optical quality. The magnetized plasma waveplate introduced above provides
an alternative approach to manipulate the polarization state of a laser pulse.
In the encounter with an ultra-high-power laser pulse, such a magnetized
plasma waveplate has two advantages over the conventional waveplate made from
a birefringent crystal. Firstly, the magnetized plasma waveplate can be more
compact since it can sustain an intense laser pulse. As illustrated in the
above 3D simulation, a plasma waveplate on the centimeter scale (a waist of
4.8 mm) enables the transmission of a 10 PW laser pulse. Secondly, the
magnetized plasma waveplate can compress an intense laser pulse and boost its
peak power. This is of great benefit to ultra-high-power laser systems since
it may partially liberate the role of the grating compressor in directly
achieving the 10 PW level laser pulses.

Figure 5:  Stokes parameters of a laser pulse after it passes through a plasma
under (a) a longitudinal magnetic field or (b) a transverse magnetic field.
Except for the magnetic field direction, all other parameters used in (a) and
(b) are the same. The magnetic field strength is $B_{0}=50$ T. The initial
laser pulse is LP with $a_{0}=0.025$ and a duration of 100 fs. The plasma has
a uniform density $n_{e}=0.3n_{c}$, and a thickness $d\simeq 3.9$ cm. With
$d\simeq 3.9$ cm, the magnetized plasma in (b) plays a role as a quarter-wave
plate according to Eq. (3). Therefore, the output laser pulse is CP with
$|V/I|\simeq 1$ as shown in (b). With $d\simeq 3.9$ cm, the magnetized plasma
in (a) should be long enough to separate the left-handed and right-handed CP
sub-pulses according to Eq. (4) in Ref. Weng . However, no pulse splitting is
observed in (a) since $B_{0}=50$ T is below the threshold for the pulse
splitting.

As magnetized plasmas have been proposed for the polarization transformation
of high-power laser pulses previouslyWeng , we would like to emphasize that
the polarization transformation scheme proposed in the current paper is based
on completely new underlying physics and has a few prominent advantages over
the schemes already proposed. The scheme proposed here utilizes the
birefringence in a plasma under a transverse magnetic field, while the scheme
in Ref. Weng  is based on the chirality (i.e. circular dichroism) in a plasma
under a longitudinal magnetic field. The new underlying physics of the current
scheme gives it two advantages over the scheme in Ref. Weng . Firstly, an
initial LP laser pulse will be split into a left-handed CP and a right-handed
CP laser pulses by using the scheme in Ref. Weng . Therefore, the peak power
of the laser pulse will be reduced by a factor of two in general. In contrast,
neither the pulse splitting nor the significant reduction in the peak power
occurs in the current scheme since it transforms an initial LP laser pulse
into another single CP pulse directly. The slight reduction in the peak power
due to the dispersive broadening could be compensated by the relativistic
self-compression of the pulse. Secondly, the electron cyclotron frequency
($\omega_{c}=eB/m_{e}$) in the magnetic field should be larger than the
frequency spread ($\Delta\omega$) of the laser pulse in order to guarantee
that the pulse splitting is faster than the dispersive broadening by using the
scheme in Ref. Weng . That is to say, the magnetic field is required to be
stronger than a threshold for the pulse splitting

$B\geq B_{\min}=\frac{m_{e}\Delta\omega}{e}.$ (6)

This usually demands an ultra-strong magnetic field for an ultrashort laser
pulse. In contrast, there is no magnetic field strength constraint in the
current scheme since it demands no pulse splitting in principle. As
illustrated in Fig. 5(a), using the scheme in Ref. Weng , an initial 100 fs LP
laser pulse cannot be split into two CP laser pulse no matter how far it
propagates in a plasma under a longitudinal magnetic field $B=50$ T. Since the
frequency spread $\Delta\omega\simeq 0.015\omega$ is larger than the electron
cyclotron frequency $\omega_{c}=eB/m\simeq 0.005\omega$. In this case, the
dispersive broadening dominates over the pulse splittingWeng . In contrast,
Fig .5(b) demonstrates that this initial LP laser pulse can be transformed
into a CP laser pulse by using the current scheme that is based on the
birefringence under a transverse magnetic field with the same strength.

Figure 6: Stokes Parameters of a short laser pulse after it passes through a
magnetized plasma with $d_{0}=0.474$ cm, $n_{e}=0.1n_{c}$ and $B=300$ T, where
the input intensity profile $I_{\texttt{in}}$ is also drawn for comparison.
The initial LP laser pulse has $a_{0}=0.14$ and $t_{p}=30$ fs.

For an ultrashort laser pulse with a broad spectrum, however, magnetized
plasma quarter-wave plates also suffer from a similar problem of conventional
solid-state quarter-wave plates. That is, the resultant phase difference
$\Delta\phi$ between O-wave and X-wave is exactly equal to $\pi/2$ only for
the central wavelength. Therefore, it is difficult to convert an initial
ultrashort LP laser pulse into a CP pulse with a high level of circular
polarization. However, magnetized plasma quarter waveplates may partially
alleviate this problem because it can compress a laser pulse as well as alter
its polarization state. Therefore, an ultrashort CP laser pulse can be
obtained from an initially relatively long LP laser pulse. Although the final
pulse duration is ultrashort in this case, the spectrum that determined by the
initial duration is not so broad. So the level of circular polarization for
the final CP pulse could be relatively high. As illustrated in Fig. 6, an
initial LP laser pulse with $t_{p}=30$ fs can be converted to a CP laser pulse
whose FWHM duration is only about 10 fs and circular polarization level
$|V/I|$ is as high as 92%..

It is worth pointing out that this proposed magnetized plasma waveplate also
has some limitations on the laser and plasma parameters, although plasmas are
usually considered to be free from laser-induced damage. As suggested
previously Weng , it is important to set the laser intensity $a_{0}\sim 0.1$
and the plasma density $n_{e}\ll n_{c}$. The conditions $a_{0}\sim 0.1$ and
$n_{e}\ll n_{c}$ guarantee the collisionless damping small. We find that the
energy loss due to the collisionless damping is maintained at a level $\sim
2\%$ as long as $a_{0}\leq 0.1$ and $n_{e}\leq 0.3n_{c}$ in the simulations.
Taking into account the collisional damping, there will be some additional
limitations on the plasma density and/or magnetic field strength. The losses
due to the inverse bremsstrahlung, which are not included in our PIC
simulations, can be estimated as $K_{ib}=1-\exp{(-\kappa_{ib}d_{0})}$
EliezerBook , where $d_{0}$ is the thickness of magnetized plasma waveplate
and $\kappa_{ib}\simeq\nu_{ei}(n_{e}/n_{c})^{2}(1-n_{e}/n_{c})^{-1/2}/c$ is
the spatial damping rate. For $a_{0}\sim 0.1$, the effective electron-ion
collision frequency is given by $\nu_{ei}\simeq
Z_{i}e^{4}n_{e}\ln\Lambda/(4\pi\epsilon_{0}^{2}m_{e}^{2}v_{eff}^{3})$ Gibbon ;
WengPRE , with the effective electron velocity
$v_{eff}=(v_{te}^{2}+v_{os}^{2})^{1/2}\simeq a_{0}c$, the ionization state
$Z_{i}$ , the Coulomb logarithm $\ln\Lambda$, the electron thermal velocity
$v_{te}$ and oscillatory velocity in the laser field $v_{os}\simeq a_{0}c$.
Therefore, we can estimate that the thickness of magnetized plasma waveplate
should be thinner than 0.3 cm in order to maintain the collisional losses
$<10\%$ in the case $a_{0}=0.1$ and $n_{e}=0.3n_{c}$. According to Eq. (3),
this demands a magnetic field stronger than 200 T, which might be generated
only by laser-driven capacitor coils Fujioka ; Law ; Tikhonchuk . Fortunately,
the collisional damping rate $\kappa_{ib}$ decreases quickly with a decreasing
plasma density. Consequently, the limitation upon the magnetic field strength
can be greatly relaxed with a lower plasma density. For instance, we find that
a magnetic field $>70$ T (correspondingly $d_{0}<9$ cm) is enough to maintain
the collisional losses $<10\%$ in the case $a_{0}=0.1$ and $n_{e}=0.1n_{c}$.

For a high power laser pulse, its waist will be very large if it is not
focused tightly. For instance, the waist $r_{0}\simeq 4800\lambda$ in the case
$a_{0}=0.1$ for a 5 PW laser pulse. Correspondingly, the Rayleigh length
$z_{R}=\pi r_{0}^{2}/\lambda\simeq 7.24\times 10^{7}\lambda$. This extremely
long Rayleigh length guarantees that multi-dimensional effects such as
relativistic self-focusing and transverse filamentation instabilities do not
occur strongly before the polarization transformation of the laser pulse is
completed. We have performed a few 2D test simulations with $r_{0}\simeq
4800\lambda$, and no obvious transverse instabilities is observed within a
propagation distance up to a few centimeters. There is also no obvious Raman
scattering instability in the performed 1D and 2D test simulations with
$n_{e}<n_{c}/4$. This may be because the employed laser pulses are relatively
short (on the order of 100 fs) and weakly relativistic ($a_{0}\leq 0.1$).

It is worth mentioning that it is extremely high demand for computational
resources to carry out 3D PIC simulations with high transverse resolutions
$\delta x,\delta y<\lambda$ due to the huge focal spot size of the laser pulse
($r_{0}=4800\lambda$). To make sure there is no problem to use such large
resolutions in the transverse direction, we have also done a few 2D test
simulations with the transverse resolution ranging from $\lambda/8$ to
$50\lambda$, and we find no obvious difference between the simulation results
as long as the transverse resolution is much smaller than the laser pulse
waist $r_{0}$.

##  V Conclusion

In conclusion, we have shown that a plasma slab under a transverse magnetic
field can act as a waveplate due to magnetic-field induced birefringence. This
magnetized plasma waveplate can not only sustain intensities several orders of
magnitude higher than a crystal waveplate but also boost the peak power of the
laser pulse. Therefore, this scheme could be developed as a powerful means to
manipulate the polarization state of ultra-high-power laser pulses. Such a
magnetized plasma waveplate should make it feasible to achieve 10 PW
circularly polarized laser pulses that are technologically challenging by
other means. The magnetized plasma waveplate can be considered as a new
addition to the plasma-base class of optical components, which have the
potential to revolutionize the design and application of ultra-high-power
lasers.

## Funding

National Natural Science Foundation of China (Grant Nos. 11675108, 11655002,
11721091, 11535001, and 11405108), National Basic Research Program of China
(Grant No. 2013CBA01504), National 1000 Youth Talent Project of China, Science
Challenge Project (No.TZ2018005) and EPSRC (Grant No. EP/R006202/1).

## Acknowledgments

Simulations have been carried out on the Pi supercomputer at Shanghai Jiao
Tong University. Data associated with research published in this paper can be
accessed at https://doi.org/10.15129/8d4b5e09- 5680-4b3d-8501-0cba442892a3.

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