# Dispersion of the laser pulse through propagation in underdense plasmas

Shivani Choudhary<EMAIL_ADDRESS>Amol R.
Holkundkar<EMAIL_ADDRESS>Department of Physics,
Birla Institute of Technology and Science - Pilani, Rajasthan, 333031, India

###### Abstract

The propagation of the laser pulses in the underdense plasma is a very crucial
aspect of laser-plasma interaction process. In this work, we explored the two
regimes of laser propagation in plasma, one with $a_{0}<1$ and other with
$a_{0}\gtrsim 10$. For $a_{0}<1$ case, we used a cold relativistic fluid
model, wherein apart from immobile ions no further approximations are made.
The effect of the laser pulse amplitude, pulse duration, and plasma density is
studied using the fluid model and compared with the expected scaling laws and
also with the PIC simulations. The agreement between the fluid model and the
PIC simulations are found to be excellent. Furthermore, for $a_{0}\gtrsim 10$
case, we used the PIC simulations alone. The delicate interplay between the
conversion from the electromagnetic field energy to the longitudinal
electrostatic fields results in the dispersion and so the red-shift of the
pump laser pulse. We also studied the interaction of the dispersed pulse
(after the propagation in underdense plasma) with the sub-wavelength two-layer
composite target. The ions from the thin, low-density second layer are found
to be efficiently accelerated to $\sim 70$ MeV, which is not found to be the
case without dispersion.

##  I Introduction

Since the last couple of decades we are witnessing the rapid technological
advancement in the field of high power lasers, promising number of
applications in both applied and fundamental sciences. The table-top setup for
the ion and electron accelerations to relativistic energies is a result of the
technological breakthrough in the field of high power lasers. The idea of the
laser wakefield acceleration as demonstrated in Ref. Mangles  _et al._ (2004);
Faure  _et al._ (2004); Geddes  _et al._ (2004) really paved the possibilities
to accelerate the electrons to GeV of energies by plasma interaction with the
aforementioned high power lasers. Furthermore, the acceleration of the target
ions to MeV of energies is also proved to be feasible with existing ultra-
intense lasers. Depending on the laser and target parameters, there are
numerous acceleration mechanisms are reported, in line with the experimental
findings. The Target Normal Sheath Acceleration (TNSA) Snavely  _et al._
(2000); Passoni  _et al._ (2010), Radiation Pressure Acceleration (RPA) Qiao
_et al._ (2011); Scullion  _et al._ (2017), Breakout Afterburner (BOA) Yin
_et al._ (2011); Weng  _et al._ (2012), Relativistic Self Induced Transparency
(RSIT) Siminos  _et al._ (2012); Tushentsov  _et al._ (2001); Fernández  _et
al._ (2017); Choudhary and Holkundkar (2016) etc are the name to few.

The high contrast laser pulses are desirable for the studies involving the
interaction with the thin foil targets, however, the prepulse of those high
power lasers is intense enough to ionize the target before the arrival of the
main pulse Kaluza  _et al._ (2004). The ionization of the target and the
formation of the plasma ahead of the main target has very dramatic
consequences which in a sense can completely alter the dynamics of the
interaction. The study of the evolution of the laser pulse as it propagates in
the tenuous plasma has drawn considerable research interest around the globe
both theoretically and experimentally Davis  _et al._ (2005); Sullivan  _et
al._ (1994); Najmudin  _et al._ (2002); Sprangle and Hafizi (2014). The
propagation of the laser pulse in the under-dense plasma ($n_{e}<n_{c}$) has
been studied in the past Najmudin  _et al._ (2002); Chen and Sudan (1993). The
effect of the polarization on the dynamics of the laser-plasma interaction has
been reported in Ref. Singh  _et al._ (2012). The influence of the magnetic
fields on the propagation of the laser in the plasma is discussed in Ref.
Wilson  _et al._ (2017). The generation of the magnetic fields during intense
laser channelling in underdense plasma has been reported in Ref. Smyth  _et
al._ (2016). The propagation of the laser or electromagnetic pulses in plasma
also leads to non-linear phenomenon resulting in the soliton formations
Hadžievski  _et al._ (2002); Yin  _et al._ (2011). The existence of the
solitary waves in the plasma and its effect on the laser pulse itself is
reported in Ref. Sánchez-Arriaga  _et al._ (2011). The wakefield generation is
also one of most important phenomenon as a consequence of the laser pulse
propagation in the under-dense plasma Pukhov  _et al._ (2011). As the plasma
density approaches the critical density $n_{c}$, the wakefield generation is
suppressed and instead laser undergoes nonlinear self-modulation Holkundkar
and Brodin (2018).

In this work, we study the evolution of the laser pulse as it propagates in
the underdense plasma. For the moderate laser intensities ($a_{0}<1$) we
invoke the 1D relativistic cold fluid model, avoiding some common
approximations relevant for underdense plasma. The results for this case are
then compared with the 1D PIC simulations and agreement is found to be
excellent. The evolution of the ultra-intense ($a_{0}>1$) laser pulse is
studied by PIC simulations where the effect of the plasma density, and other
laser parameters are also explored. Furthermore, the dispersed ultra-intense
laser pulse is then used to study the acceleration of the ions via
relativistic self induced transparency (RSIT).

The organization of our paper is as follows. In Sec. II the governing
equations for 1D wave propagation are discussed along with the details of the
PIC simulations. Next, in Sec. III we study the pulse dispersion for $a_{0}<1$
and $a_{0}\gtrsim 10$. The ion acceleration by the RSIT mechanism via the
dispersed pulses is also discussed in the Sec. III-C followed by the
concluding remarks in Sec. IV.

##  II Theory and simulation model

Figure 1: The spatial profile of the electromagnetic (transverse) fields (a,c)
and electrostatic (longitudinal) fields (b,d) is presented at different time
instances using fluid simulation (left panel) and PIC simulation (right
panel). Here, we modeled the interaction of the 800 nm, 3 cycles (FWHM)
Gaussian laser pulse ($a_{0}=0.1$) with plasma having density $0.5n_{c}$.
Figure 2: The temporal evolution of the pulse length (red circle, left axis)
and the peak field amplitude (blue filled circles, right axis) as it
propagates in the underdense plasma is compared by fluid (a) and PIC
simulations (b). The pulse length $L_{fwhm}$ is estimated in units of the
fundamental wavelength of the laser pulse, similarly time is presented in
units of fundamental laser cycle. The laser and plasma conditions are same as
Fig. 1.

The objective of this article is to study the dispersion of the
electromagnetic (EM) waves as it propagates in an under-dense plasma. The
propagation and the dispersion of the EM waves can be understood by the
relativistic cold fluid model. Recently we developed a model to study the
transition from the wakefield generation to the soliton formation Holkundkar
and Brodin (2018); Brodin  _et al._ (2006). However, for the sake of
completeness here also we elaborated the cold fluid model. We have considered
the immobile ions, and apart from this no further approximations are made. For
detailed calculations please refer the Appendix.

The laser amplitude is normalized as $\mathbf{a}=e\mathbf{A_{\perp}}/m_{e}c$,
scalar potential as $\varphi=e\phi/m_{e}c^{2}$, time and space with laser
frequency and wave vector ($\omega t\rightarrow t$ and $kx\rightarrow x$)
respectively, velocity as $\beta=\upsilon/c$, momentum is normalized
$\mathbf{p}=\mathbf{P}/m_{e}c$, charge and mass are normalized by electron
charge and mass, the electron density is normalized by critical density
$n_{c}=\varepsilon_{0}\omega^{2}m_{e}/e^{2}$. By using these normalization one
can easily deduce from Eq. 30-36 of the Appendix the following set of
equations

$\frac{\partial^{2}\mathbf{a}}{\partial
z^{2}}-\frac{\partial^{2}\mathbf{a}}{\partial
t^{2}}=n_{e}\frac{\mathbf{a}}{\gamma}$ (1)
$\frac{d\beta}{dt}=\frac{(1-\beta^{2})}{\gamma}\frac{\partial\varphi}{\partial
z}-\frac{1}{2\gamma^{2}}\Big{(}\frac{\partial\mathbf{a}^{2}}{\partial
z}+\beta\frac{\partial\mathbf{a}^{2}}{\partial t}\Big{)}$ (2) $\frac{\partial
n_{e}}{\partial t}+\frac{\partial}{\partial z}\Big{(}n_{e}\beta\Big{)}=0$ (3)
$\gamma=\sqrt{\frac{1+a^{2}}{1-\beta^{2}}}$ (4)
$\frac{\partial^{2}\varphi}{\partial t\partial z}=-n_{e}\beta$ (5)

The above set of equations are the basis of our analysis of the dispersion of
the EM wave in the under-dense plasma. In order to validate the results of our
fluid model, we used a 1D particle-in-cell (PIC) simulation, the details of
the PIC simulations are as follows: The 1D Particle-In-Cell simulation
(LPIC++) Lichters  _et al._ (1997) is carried out to compare the results of
the cold fluid model. In this code the electric fields are normalized as we
earlier discussed ($a_{0}=eE/m_{e}\omega c$). However space and time are taken
in units of laser wavelength ($\lambda$) and one laser cycle $\tau=\lambda/c$
respectively, mass and charge are normalized with electron mass and charge
respectively. We have used 100 cells per laser wavelength with each cell
having 50 electron and ion macro-particles. The spatial grid size and temporal
time step for the simulation are considered to be 0.01$\lambda$ and 0.01$\tau$
respectively.

##  III Results and Discussions

We have numerically solved the Eqs (1)-(5) in the same sequence to study the
evolution of the laser pulse entering the simulation box from the left side.
The simulation box of length $100$ $\lambda$ is considered, with a constant
unperturbed plasma density $n_{0}$ throughout the simulation domain, the
linearly polarized Gaussian laser pulse of wavelength $800$ nm has a full
width half maximum (FWHM) duration of 3 cycles ($\tau_{fwhm}=3\times 2\pi$ ).
The normalized amplitude $a_{0}$ is varied in the different simulations, and
the boundary conditions on the left side read as:

$\mathbf{a}(0,t)=a_{0}\exp\Big{(}-\frac{4\log(2)t^{2}}{\tau_{fwhm}^{2}}\Big{)}\cos(t)\
\ \mathbf{\hat{x}}$ (6) $n_{e}(0,t)=n_{0}$ (7)
$\beta(0,t)=\varphi^{\prime}(0,t)=0\quad(\varphi^{\prime}\equiv\partial\varphi/\partial
z).$ (8)

It should be noted that the cold fluid relativistic model is only valid for
the cases when the laser pulse amplitude is $a_{0}$ is less than unity
($a_{0}<1$) or for that matter the dispersion is in the linear regime. For
ultra-intense laser pulses $a_{0}>1$ the phenomenon of the wave breaking and
other non-linearities limit the applicability of the fluid approach in
describing the density modulations. The results in this sections are divided
in for the cases when $a_{0}<1$ wherein we compared the results of the fluid
and PIC simulations along with the effects of the laser and plasma parameters
on the dispersion of EM waves. On the contrary the dispersion of the ultra
intense laser pulses ($a_{0}>1$) is studied by only using PIC simulations.

###  III.1 Pulse dispersion for $a_{0}<1$

We consider the propagation of the 800 nm, 3 cycles (FWHM), linearly
polarized, Gaussian laser pulse ($a_{0}=0.1$) in the plasma with an
unperturbed plasma density of $0.5n_{c}$. The spatial profiles of the
transverse EM fields and longitudinal electrostatic fields are illustrated at
different time instances in Fig. 1 both by using fluid simulation (left panel)
as well as PIC simulations (right panel), and apparently the agreement between
the two is found to be good. The dispersive nature of the laser pulse can be
seen by increased pulse length and decreased peak amplitudes as estimated at
different time instances. As it propagates deeper into the plasma the pulse
tend to broaden. Furthermore, it can be seen from Fig. 1(b) that the wakefield
or longitudinal field generation is suppressed for the chosen laser and plasma
parameters and on the contrary a kind of a localized structure is co-
propagating with the laser pulse [see Fig. 1(a)]. The suppression of the
wakefield happens mainly when the length of the laser pulse ($c\tau_{L}$) is
larger than the equivalent length of the plasma oscillations
($\upsilon_{g}\tau_{p}$), here $\upsilon_{g}$ is the group velocity of the
plasma waves and $\tau_{p}$ is the duration of the one plasma cycle. In Ref.
Holkundkar and Brodin (2018), we have presented the detailed analysis of the
transition from the wakefield to the soliton formation.

For the same laser and plasma parameters ($a_{0}=0.1,\tau_{fwhm}=3\
\text{cycles},n_{e}=0.5n_{c}$) the time evolution of the peak field amplitude
and the pulse length is presented in Fig. 2, the agreement between the fluid
and PIC simulations is found to be excellent. It can be observed from Fig. 2,
that the pulse length increases almost linearly with time as the pulse
propagates deeper into the plasma, on the other hand, the peak field
amplitudes decreases. As we discussed earlier, for this laser and plasma
parameters the wakefield generation is suppressed and the modulation in plasma
density actually co-moves with the laser pulse. The total energy content of
the pulse is found to be almost constant during its passage in the plasma, as
energy lost to the wakefield generation is almost negligible.

Figure 3: Temporal evolution of the peak laser amplitude (a) and pulse length
(b) is presented for different plasma densities. The results of the PIC
simulations are also shown with open circles in both (a) and (b). The peak
amplitude is normalized to the peak value of the pulse at $t=10\tau$
($a_{p10}$). The value of these parameters are evaluated at 100$\tau$ are also
presented in (c) for different $n_{e}/n_{c}$. The $a_{0}=0.1$ and
$\tau_{fwhm}=3$ cycles is considered for this case.

Next, we present the effect of the plasma density on the temporal evolution of
the pulse length and the peak field amplitude of the laser pulse, during its
passage to the uniform density plasma. For this we have used the same laser
parameters ($a_{0}=0.1$ and $\tau_{fwhm}=3$ cycles) as in Fig. 1 and 2. The
peak field amplitude is normalized to its value at $t=10\tau$ ($a_{p10}$),
this has been done to iron out a slight discrepancy with PIC simulations
because anyway here we are more interested in the rate change of the peak
amplitude as the pulse propagates through the plasma. The constant plasma
density is varied from $0.2-0.6n_{c}$. It can be observed from Fig. 3(b) that
the pulse length increases linearly with time and the rate at which it
increases varies with the plasma density. We have compared the results of our
fluid simulation with the PIC simulation for the case with $n_{e}=0.6n_{c}$
and the agreement is found to be excellent. In Fig. 3(c) we present the pulse
length and the peak amplitude as evaluated at $t=100\tau$ for different plasma
densities.

Figure 4: Temporal evolution of the peak laser amplitude (a) and pulse length
(b) is presented for different pulse duration. The results of the PIC
simulations are also shown with open circles in both (a) and (b). The peak
amplitude is normalized to the peak value of the pulse at it would be at
$t=10\tau$ ($a_{p10}$). The value of these parameters are evaluated at
100$\tau$ and are presented in (c) for different $\tau_{fwhm}$. The
$a_{0}=0.1$ and $n_{e}=0.5n_{c}$ is considered for this case.

The linear broadening of the laser pulse with time can be understood in terms
of the group velocity of the pulse in the plasma. In the linearized theory the
group velocity (normalized to $c$) can be calculated as:

$\upsilon_{g}=\sqrt{1-\omega_{p}^{2}/\omega^{2}}=\sqrt{1-n_{e}/n_{c}}$ (9)

then the time evolution of pulse length ($L$) can be expected to follow the
relation,

$L(t)=L_{0}+(1-\upsilon_{g})t$ (10)

such, that in case of vacuum ($\upsilon_{g}=1$) the pulse length remains
constant, say $L_{0}$. As we have pointed out that for this cases the
wakefield generation is almost suppressed and the energy content of the laser
pulse is almost constant, this indicates toward the fact that as the pulse
broadens, the respective peak amplitude should drop accordingly. The energy of
the pulse will scale as $\propto a_{0}^{2}L$, which indicates the drop in the
peak amplitude scales as $\propto 1/\sqrt{L(t)}$.

The scaling is found to be in accordance with the results presented in Fig.
3(a). Furthermore, the variation of the pulse length (as evaluated at
$t=100\tau$) with the plasma density is presented in Fig. 3(c). As expected
from Eq. 10, the pulse length and pulse amplitude would scale as:

$L(n_{e})\propto 1-\sqrt{1-n_{e}/n_{c}}\quad;\quad
a_{0}\propto\frac{1}{\sqrt{L(n_{e})}}$ (11)

the fitted Eq. 11 is also illustrated in Fig. 3.

The effect of the pulse duration on the dispersion is presented in Fig. 4.
Here, again we studied the time evolution of the pulse length and the peak
field amplitude during the passage of the pulse through the plasma. We varied
the $\tau_{fwhm}$ for fixed $a_{0}=0.1$ and $n_{e}=0.5n_{c}$. The results are
also compared with the PIC simulations as well and an agreement is found to be
excellent. It can be seen from Fig. 4(c) that the length of the pulse at say
$t=100\tau$ decreases as we increase the laser pulse duration. It can be
understood as follows, we know the shorter the pulse higher would be the
bandwidth, that translates to the fact that a different portion of the pulse
will propagate with the different velocity and as a result the larger
broadening of the pulse. On the other hand for longer pulses the bandwidth is
small, and so the associated dispersion. We can compare the time scales of the
laser pulse duration with the time scales typically involved in the plasma
oscillations $\tau_{fwhm}\propto 1/\omega_{p}\sim 1/\sqrt{n_{e}}$ for a rough
estimates related to the dispersive nature of the plasma.

$\tau_{fwhm}\propto\frac{1}{\sqrt{n_{e}}}$ (12)

however, as we saw earlier the pulse length is related to the plasma density
as given by Eq. 10, so in a sense $n_{e}\propto L^{2}$, this implies

$\tau_{fwhm}\propto\frac{1}{L}\implies L\propto\frac{1}{\tau_{fwhm}}$ (13)

and peak amplitude would be,

$a_{0}\propto\frac{1}{\sqrt{L}}\propto\sqrt{\tau_{fwhm}}$ (14)

Next, we further study the effect of the laser amplitude on the dispersion of
the laser pulse. For this purpose we fixed the pulse duration to
$\tau_{fwhm}=3$ cycles and the plasma density to $0.5n_{c}$ and varied the
peak laser amplitude. The time evolution of the peak field amplitude and
length of the laser pulse is presented in Fig. 5. Again as expected the linear
dispersion law is found to be consistent for the laser and plasma parameters
presented. Though for $a_{0}=0.3$ case, we found a bit of discrepancy with
fluid simulation for pulse length evolution, otherwise the rate change of the
field amplitude is consistent with the findings of the PIC simulations. The
value of the field amplitude and pulse length as evaluated at $100\tau$ is
also illustrated in Fig. 5(c). It is understood that high intensity laser
pulses tend to disperse less as compare to the low amplitude pulses, as a
consequence the pulse length of the intense pulses is smaller than then their
low intensity counterpart after certain time of propagation. As we discussed
earlier, the pulse length can be estimated by Eq. 10, however with the
relativistic corrections the Eq. 10 is modified as,

$L(t)=L_{0}+\Big{(}1-\sqrt{1-\frac{n_{e}}{\gamma^{\prime}n_{c}}}\Big{)}t\quad;\quad\gamma^{\prime}\equiv\sqrt{1+a_{0}^{2}}$
(15)

here, $\gamma^{\prime}$ is relativistic factor (Eq. 4), we ignored the
longitudinal motion of the electrons. The scaling of the pulse length with the
initial laser amplitude is carried out using Eq. 15 and the fitted curve is
also presented in Fig. 5(c).

Figure 5: Temporal evolution of the peak laser amplitude (a) and pulse length
(b) is presented for different laser amplitudes. The results of the PIC
simulations are also shown with open circles in both (a) and (b). The peak
amplitude is normalized to the peak value of the pulse at it would be at
$t=10\tau$ ($a_{p10}$). The value of these parameters are evaluated at
100$\tau$ and are presented in (c) for different $a_{0}$. The $n_{e}=0.5n_{c}$
and $\tau_{fwhm}=3$ cycles is considered for this case.

###  III.2 Pulse dispersion for $a_{0}>1$

In the previous section, we discussed the dispersion of the laser pulses with
$a_{0}<1$. We developed an analytical framework based on the cold relativistic
fluid model and benchmarked the results with the 1D PIC simulations. However,
for high intense laser pulses, the cold fluid model is no longer valid, as for
intense laser fields the nonlinear phenomenon like wave-breaking would
prevail, which indeed is outside the purview of the fluid approach. In order
to study the dispersion of the intense laser pulses ($a_{0}>1$) we would be
using the PIC simulations alone.

Figure 6: The temporal snapshots of the laser field as evaluated at 30, 60 and
90$\lambda$ is illustrated for the case when 3 cycle laser with $a_{0}=20$ is
propagating in the plasma with density 0.01$n_{c}$ (a), 0.02$n_{c}$(b) and
0.03$n_{c}$(c). The Fourier spectrum of the laser pulse as evaluated at
$90\lambda$ is also compared for different plasma densities (d).

We consider the propagation of 3 cycles ($\tau_{fwhm}$), linearly polarized,
Gaussian pulse with the peak field amplitudes as $a_{0}=10,15$ and 20 in the
plasma with uniform density $\lesssim 0.03n_{c}$. The reason to consider the
lower plasma density (as compared to the previous section) for $a_{0}>1$ is to
mitigate the formation of the overdense plasma ($n_{e}>n_{c}$) caused by the
ponderomotive force exerted by intense laser pulses $a_{0}\gtrsim 10$. The
overdense plasma then prohibits the further propagation of the laser pulses,
till it becomes sufficiently underdense (by space-charge effect) to allow the
passage of the laser pulse. For this kind of scenario we might have the
reflections of the laser pulse from the different part of the plasma, where it
turns overdense. In order to avoid any reflections by the formation of the
overdense plasma, in this section we would be considering the plasma densities
$\lesssim 0.03n_{c}$.

Figure 7: The temporal snapshots of the laser field as evaluated at 30, 60 and
90$\lambda$ is illustrated for the 3 cycle laser propagating in the plasma
with density 0.03$n_{c}$. The laser amplitude $a_{0}=10$ (a), 15(b) and 20(c)
are considered. The Fourier spectrum of the laser pulse as evaluated at
$90\lambda$ is also compared for different laser amplitudes (d).

We compared the time evolution of the laser ($a_{0}=20$) electric field for
three different plasma densities in Fig. 6. The field profiles are evaluated
after the laser propagated the distances $30\lambda$, $60\lambda$ and
$90\lambda$ in the plasma. It can be observed from this figure that the peak
of the envelope moves roughly with the same velocity for different plasma
densities, this indicates that the group velocity of the laser is more or less
unaffected for the considered laser and plasma parameters, or maybe it would
require longer simulation to see any prominent effect on propagation. We have
also presented the Fourier spectrum of the laser pulse in Fig. 6(d). It can be
seen that for higher densities the broadening of the frequency spectrum is
larger because of the stronger plasma wave generation. The spectrum is found
to be red shifted in direct correlation with the plasma density Pathak  _et
al._ (2018). The red shifting and broadening of the spectrum generally
accounts for the stronger plasma wave generation, because of the energy
transformation from the laser to the plasma. This fact can also be observed
from the Fig. 6(d), wherein the red shift for higher density plasma is larger
as compared to the lower density plasma. In order to elucidate the effect of
the laser pulse amplitude on the dispersion of the laser pulse, in Fig. 7 we
have varied the laser pulse amplitude while keeping the plasma density fixed
at $0.03n_{c}$. The broadening of the spectrum is seen to be prominent for the
$a_{0}=10$ as compare to $a_{0}=20$, because the rate at which the energy is
depleted for lower laser amplitudes would be larger as compared to higher
laser amplitudes.

The time evolution of the electromagnetic and electrostatic field energies are
presented in Fig.8. Here, again we have considered the propagation of the 3
cycle, linearly polarized laser pulse with $a_{0}=10,20$ in the plasma having
densities $0.01,0.03n_{c}$. As time progress the decrease in the
electromagnetic field energy and increase in the electrostatic field energy is
observed which indicates toward the stronger plasma wave generation at the
cost of the electromagnetic energy. As expected it is further observed that
the depletion rate of the electromagnetic field energy is larger for the laser
with peak amplitude $a_{0}=10$ as compare to $a_{0}=20$. This is so because
the dispersion of the high intensity laser pulses would be relatively slower
than the laser pulses with lower intensity. The direct correlation of the
plasma density can also be seen on the depletion rate of the electromagnetic
field energy, and so the growth in the longitudinal field energy.

Figure 8: Temporal evolution of the laser pulse energy (a) and the
longitudinal field energy is presented for 3 cycle Gaussian pulse with
$a_{0}=10,\ 20$ when it propagated in the plasma with density $0.01,\
0.03n_{c}$. The laser pulse energy in (a) is normalized to maximum value at
$t=10\tau$, as we are interested in the depletion rate of the laser pulse
energy for different laser and plasma parameters.

###  III.3 Ion acceleration by intense dispersed pulses

We have recently demonstrated the use of the negatively chirped laser pulses
to accelerate the ions to a few hundreds of the MeV by using a double layer
(Hydrogen plasma) target Choudhary and Holkundkar (2018). The primary layer
having density $6n_{c}$ is found to be transparent for the negatively chirped
laser pulse with $a_{0}=20$, creating a persistent electrostatic field which
actually accelerates the ions from the secondary layer ($0.1n_{c}$). Next, we
deploy the similar geometry of two layer target just after the low density
plasma. The propagation of the laser pulse in underdense plasma actually
causes the dispersion of the pulse, as a result the pulse would be chirped
when it incidents on the two-layer target. In Fig. 9 we present the energy
spectrum of the ions from the secondary layer. We considered the 3 cycle
Gaussian laser pulse with $a_{0}=20$ propagates in the 100$\lambda$ long
underdense plasma having density $0.03n_{c}$. The dispersed pulse then
incidents on the two-layer target, first layer is $0.75\lambda$ thick with
density $6n_{c}$ and adjacent secondary layer is $0.2\lambda$ thick with
density $0.1n_{c}$.

Figure 9: The energy spectrum of the ions from the secondary layer is compared
when the laser directly interacts with the target i.e. $n_{0}=0$ (a) and when
initially its allowed to propagate in the 100$\lambda$ pre-plasma
($n_{0}=0.03n_{c}$) prior to interaction with composite target (b). The
geometry of the setup is also illustrated as an inlet. Here, $n_{0},n_{1}$ and
$n_{2}$ are the plasma density of the pre-plasma, first layer and second layer
respectively. Please refer the text for the physical parameters of first and
second layer.

The energy spectrum of the ions from the secondary layer is compared for the
cases when there is no underdense plasma and when the laser propagated through
the plasma [see, Fig. 9]. It can be seen that the ions from the secondary
layer are very efficiently accelerated to almost mono-energetically when the
dispersed pulse interacts with the two-layer composite target geometry. The
reason being the dispersion, wherein the frequency of the pulse undergone the
modulation in space and time, in other words pulse is somewhat chirped. As the
high frequency component of the pulse interacts with the primary layer, it
transmits through the layer by the relativistic self induced transparency, or
in other words, the critical density for the transmission gets modified for
the chirped laser pulses. The transmitted pulse drags the electrons from the
primary (as well as secondary) layer with them, creating very persistent
longitudinal electrostatic field Choudhary and Holkundkar (2018). The
electrostatic field then pulls the ions from the secondary layer, forming the
mono-energetic ion bunch. However, in the absence of the pre-plasma the
primary target is opaque to the incident unchirped pulse, resulting in the
reflection. If the laser pulse suffers the reflection at the primary layer
then acceleration is mostly caused by the radiation pressure mechanism,
resulting in lower energy yield for the same laser intensity [see, Fig. 9(a)].
The optimization of the degree of the pulse chirping (dispersion) by varying
the pre-plasma length and/or density for most efficient acceleration of the
ions from the secondary layer is beyond the scope of the current manuscript.

##  IV Concluding Remarks

We have studied the dispersion of the laser pulse as it propagates in the
underdense plasma. For the moderate laser intensities ($a_{0}<1$) we invoked a
1D relativistic cold fluid model to evaluate the spatial and temporal
evolution of the laser as it propagates in the plasma with density $\lesssim
0.6n_{c}$. Apart from the immobile ions, no further approximations are made.
The effect of the laser pulse amplitude, pulse duration and the plasma density
is explored using the fluid model and the results are compared with the 1D PIC
simulations along with the expected scaling laws. The agreement between fluid
model and the PIC simulations are found to be excellent. Furthermore, in order
to study the interaction of highly intense laser pulses $a_{0}\gtrsim 10$, we
only relied on the PIC simulations as the nonlinear nature of the interaction
process is beyond the validity of the cold relativistic fluid model. For these
cases we restricted to the plasma density $\lesssim 0.03n_{c}$, or the strong
ponderomotive force of laser pulses tend to make plasma over-dense
($n_{e}>n_{c}$) restricting the further propagation of the laser pulse. The
conversion from the electromagnetic field energy to the electrostatic fields
in the form of plasma waves results in the dispersion and so the red shift of
the pump laser pulses. The dispersed pulse then allowed to be incident on the
sub-wavelength two layer composite target. The ions from the thin, low density
secondary layer are found to be mono-energetically accelerated to $\sim 70$
MeV, which was not the case without the dispersion.

## Appendix-I

Maxwell’s equation using Coulomb gauge can be written as [J D Jackson,
_Classical Electrodynamics_],

$\nabla^{2}\phi=-\rho/\varepsilon_{0}$ (16)
$\nabla^{2}\mathbf{A}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}=-\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\nabla\Big{(}\frac{\partial\phi}{\partial
t}\Big{)}$ (17)

We are considering the 1D case so that the variation of $\phi$ and
$\mathbf{A}$ along $x$ and $y$ are not considered. The electron current
density can be written as $\mathbf{J}=-en_{e}\mathbf{v}$. Furthermore,
$\mathbf{v}=\mathbf{v}_{||}+\mathbf{v}_{\perp}$ and
$\mathbf{v_{z}}\equiv\mathbf{v}_{||}$. By using these approximations above
equations can be written as,

$\frac{\partial^{2}\phi}{\partial z^{2}}=-\rho/\varepsilon_{0}$ (18)
$\frac{\partial^{2}\mathbf{A}}{\partial
z^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}=\mu_{0}en_{e}(\mathbf{v}_{z}+\mathbf{v}_{\perp})+\frac{1}{c^{2}}\frac{\partial}{\partial
z}\Big{(}\frac{\partial\phi}{\partial t}\Big{)}$ (19)

The last equation can be split into two parts, one for perpendicular and
another for parallel directions respectively. It should be noted that here
parallel and perpendicular directions are taken with respect to the direction
of laser pulse propagation. A Laser is polarized in a plane perpendicular to
$z$ axis and hence it will contribute toward the perpendicular component of
the particle velocities ($\mathbf{v}_{\perp}$). On the other hand the
electrostatic potential created would be responsible for the motion of the
particle along the $z$ direction. By equating the perpendicular and parallel
component from RHS and LHS one obtains,

$\frac{\partial^{2}\mathbf{A}}{\partial
z^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}=\mu_{0}en_{e}\mathbf{v}_{\perp}$ (20)
$\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial t\partial
z}=-\mu_{0}en_{e}\upsilon_{z}$ (21)

here we have used $\mathbf{v_{z}}=\upsilon_{z}\mathbf{e_{z}}$

A Laser is considered to be propagating along the $z$ direction and hence
denoted by,

$\mathbf{A}=A_{x}\mathbf{e_{x}}+A_{y}\mathbf{e_{y}}$ (22)

here $\mathbf{e_{x}}$ and $\mathbf{e_{y}}$ are unit vectors along $x$ and $y$
directions respectively. The electric and magnetic fields are denoted by,
$\mathbf{E}=-\nabla\phi-\partial\mathbf{A}/\partial t$ , and
$\mathbf{B}=\nabla\times\mathbf{A}$. The electric field can further be written
as, $\mathbf{E}_{||}=-\nabla\phi$ (Wakefield) and
$\mathbf{E}_{\perp}=-\partial\mathbf{A}/\partial t$ (Laser electric field).

Now consider the Lorentz force equation,

$\frac{d\mathbf{P}}{dt}=-e\Big{[}-\nabla\phi-\frac{\partial\mathbf{A}}{\partial
t}+\mathbf{v\times(\nabla\times\mathbf{A})}\Big{]}$ (23)

It should be noted that $\mathbf{A}$ only varies along $z$ direction and only
contains the perpendicular components and hence
$\partial_{x}=\partial_{y}=A_{z}=0$. The above equation can be written as,

$\mathbf{v\times(\nabla\times
A)}=-\upsilon_{z}\frac{\partial\mathbf{A_{\perp}}}{\partial
z}+\Big{(}\mathbf{v_{\perp}}\cdot\frac{\partial\mathbf{A_{\perp}}}{\partial
z}\Big{)}\mathbf{e_{z}}$ (24)

and hence, the perpendicular and parallel component of the Lorentz force
equation respectively can be written as

$\frac{d\mathbf{P_{\perp}}}{dt}=e\Big{[}\frac{\partial}{\partial
t}+\upsilon_{z}\frac{\partial}{\partial
z}\Big{]}\mathbf{A_{\perp}}=\frac{d}{dt}(e\mathbf{A_{\perp}})$ (25)
$\frac{d\mathbf{P_{z}}}{dt}=e\frac{\partial\phi}{\partial
z}\mathbf{e_{z}}-e\Big{(}\mathbf{v_{\perp}}\cdot\frac{\partial\mathbf{A_{\perp}}}{\partial
z}\Big{)}\mathbf{e_{z}}$ (26)

which translates to the fact that,

$\mathbf{P_{\perp}}=e\mathbf{A_{\perp}}\implies\mathbf{v_{\perp}}=e\mathbf{A_{\perp}}/\gamma
m_{e}$ (27)

Substituting the value of $\mathbf{v_{\perp}}$ from (25) to (26) one obtains
(we omit the $\mathbf{e_{z}}$ for the sake of convenience, as all the
quantities are along $z$ direction only),

$\frac{dP_{z}}{dt}=e\frac{\partial\phi}{\partial z}-\frac{e^{2}}{2\gamma
m_{e}}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial z}$ (28)

The last term in (28) is the Ponderomotive force which is responsible for the
displacement of the electrons from the laser focus.

We need to solve for the $\upsilon_{z}$ so in view of this (28) can further be
written as,

$\frac{dP_{z}}{dt}=\frac{d}{dt}(\gamma
m_{e}\upsilon_{z})=e\frac{\partial\phi}{\partial z}-\frac{e^{2}}{2\gamma
m_{e}}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial z}$ (29)

$\gamma\frac{d\upsilon_{z}}{dt}+\upsilon_{z}\frac{d\gamma}{dt}=\frac{e}{m_{e}}\frac{\partial\phi}{\partial
z}-\frac{e^{2}}{2\gamma
m_{e}^{2}}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial z}$ (30)

$\frac{d\upsilon_{z}}{dt}=\frac{e}{\gamma m_{e}}\frac{\partial\phi}{\partial
z}-\frac{e^{2}}{2\gamma^{2}m_{e}^{2}}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial
z}-\frac{\upsilon_{z}}{\gamma}\frac{d\gamma}{dt}$ (31)

The rate of change of total energy ($\mathcal{E}=\gamma mc^{2}$ ) of the
charge particle can be written as,

$\frac{d\mathcal{E}}{dt}=q\mathbf{v}\cdot\mathbf{E}$ (32)

which in our case ($q=-e$, $m=m_{e}$ and
$\mathbf{E}=-\nabla\phi-\partial\mathbf{A}/\partial t$) can be written as,

$\frac{d\gamma}{dt}=\frac{-e}{m_{e}c^{2}}(\mathbf{v_{\perp}}+\mathbf{v_{z}})\cdot(-\nabla\phi-\partial\mathbf{A}/\partial
t)$ (33)

$\frac{d\gamma}{dt}=\frac{e}{m_{e}c^{2}}\Big{(}\upsilon_{z}\frac{\partial\phi}{\partial
z}+\mathbf{v_{\perp}}\cdot\frac{\partial\mathbf{A}_{\perp}}{\partial
t}\Big{)}$ (34)

Using again the value of $\mathbf{v_{\perp}}$ from (25) in above equation
simplifies to,

$\frac{d\gamma}{dt}=\frac{e}{m_{e}c^{2}}\Big{(}\upsilon_{z}\frac{\partial\phi}{\partial
z}+\frac{e}{2\gamma m_{e}}\frac{\partial\mathbf{A}^{2}_{\perp}}{\partial
t}\Big{)}$ (35)

Substituting (35) in (31) one obtains,

$\frac{d\upsilon_{z}}{dt}=\frac{e}{\gamma
m_{e}}\Big{(}1-\frac{\upsilon_{z}^{2}}{c^{2}}\Big{)}\frac{\partial\phi}{\partial
z}-\frac{e^{2}}{2\gamma^{2}m_{e}^{2}}\Big{(}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial
z}+\frac{\upsilon_{z}}{c^{2}}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial
t}\Big{)}$ (36)

Local charge separation results in electrostatic fields which can be taken
into account by Poisson’s equation,

$\nabla^{2}\phi=-\frac{e}{\varepsilon_{0}}(Zn_{i}-n_{e})$ (37)

here $Z$ and $n_{i}$ are ion charge and density respectively, however, $n_{e}$
is electron density. In 1D case $\nabla^{2}$ is replaced by
$\partial^{2}/\partial z^{2}$.

$\frac{\partial^{2}\phi}{\partial
z^{2}}=-\frac{e}{\varepsilon_{0}}(Zn_{i}-n_{e})$ (38)

The charge is conserved by continuity equation,

$\frac{\partial n_{e}}{\partial t}+\nabla\cdot(n_{e}\mathbf{v})=0$ (39)

which again for 1D case can be written as,

$\frac{\partial n_{e}}{\partial t}+\frac{\partial}{\partial
z}(n_{e}\upsilon_{z})=0$ (40)

Now we deduce the expression for $\gamma$ in terms of vector potential, By
definition,

$\gamma=\frac{1}{\sqrt{1-(\upsilon_{\perp}^{2}+\upsilon_{z}^{2})/c^{2}}}$ (41)
$\upsilon_{\perp}^{2}+\upsilon_{z}^{2}=c^{2}(1-1/\gamma^{2})$ (42)

Using the fact that $\mathbf{v}_{\perp}=e\mathbf{A}_{\perp}/\gamma m_{e}$ we
obtain,

$\frac{e^{2}A_{\perp}^{2}}{\gamma^{2}m_{e}^{2}}+\upsilon_{z}^{2}=c^{2}\Big{(}1-\frac{1}{\gamma^{2}}\Big{)}$
(43)

Solving for $\gamma$ one obtains,

$\gamma=\sqrt{\frac{1+(eA_{\perp}/m_{e}c)^{2}}{1-\upsilon_{z}^{2}/c^{2}}}$
(44)

So finally the complete set of equations can be summarized as follows,

$\frac{\partial^{2}\mathbf{A}_{\perp}}{\partial
z^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}_{\perp}}{\partial
t^{2}}=\mu_{0}en_{e}\mathbf{v}_{\perp}$ (45)
$\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial t\partial
z}=-\mu_{0}en_{e}\upsilon_{z}$ (46)

$\mathbf{P_{\perp}}=e\mathbf{A_{\perp}}\implies\mathbf{v_{\perp}}=e\mathbf{A_{\perp}}/\gamma
m_{e}$ (47)

$\frac{d\upsilon_{z}}{dt}=\frac{e}{\gamma
m_{e}}\Big{(}1-\frac{\upsilon_{z}^{2}}{c^{2}}\Big{)}\frac{\partial\phi}{\partial
z}-\frac{e^{2}}{2\gamma^{2}m_{e}^{2}}\Big{(}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial
z}+\frac{\upsilon_{z}}{c^{2}}\frac{\partial\mathbf{A}_{\perp}^{2}}{\partial
t}\Big{)}$ (48)

$\frac{\partial^{2}\phi}{\partial
z^{2}}=-\frac{e}{\varepsilon_{0}}(Zn_{i}-n_{e})$ (49)

$\frac{\partial n_{e}}{\partial t}+\frac{\partial}{\partial
z}(n_{e}\upsilon_{z})=0$ (50)

$\gamma=\sqrt{\frac{1+(eA_{\perp}/m_{e}c)^{2}}{1-\upsilon_{z}^{2}/c^{2}}}$
(51)

These are the complete set of equations in closed form which need to be solved
numerically with appropriate boundary conditions.

## Acknowledgments

Authors would like to acknowledge the Department of Physics, Birla Institute
of Technology and Science, Pilani, Rajasthan, India for the computational
support. AH acknowledges the computational resources funded by the DST-SERB
project EMR/2016/002675.

## References

  * Mangles  _et al._ (2004) S. P. D. Mangles, C. D. Murphy, Z. Najmudin, A. G. R. Thomas, J. L. Collier, A. E. Dangor, E. J. Divall, P. S. Foster, J. G. Gallacher, C. J. Hooker, D. A. Jaroszynski, A. J. Langley, W. B. Mori, P. A. Norreys, F. S. Tsung, R. Viskup, B. R. Walton, and K. Krushelnick, Nature 431, 535 (2004).
  * 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).
  * Geddes  _et al._ (2004) C. G. R. Geddes, C. Toth, J. van Tilborg, E. Esarey, C. B. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, and W. P. Leemans, Nature 431, 538 (2004).
  * Snavely  _et al._ (2000) R. A. Snavely, M. H. Key, S. P. Hatchett, T. E. Cowan, M. Roth, T. W. Phillips, M. A. Stoyer, E. A. Henry, T. C. Sangster, M. S. Singh, S. C. Wilks, A. MacKinnon, A. Offenberger, D. M. Pennington, K. Yasuike, A. B. Langdon, B. F. Lasinski, J. Johnson, M. D. Perry, and E. M. Campbell, Phys. Rev. Lett. 85, 2945 (2000).
  * Passoni  _et al._ (2010) M. Passoni, L. Bertagna, and A. Zani, New J. of Physics 12, 045012 (2010).
  * Qiao  _et al._ (2011) B. Qiao, M. Zepf, P. Gibbon, M. Borghesi, B. Dromey, S. Kar, J. Schreiber, and M. Geissler, Phys. Plasmas, 18, 043102 (2011).
  * Scullion  _et al._ (2017) C. Scullion, D. Doria, L. Romagnani, A. Sgattoni, K. Naughton, D. R. Symes, P. McKenna, A. Macchi, M. Zepf, S. Kar, and M. Borghesi, Phys. Rev. Lett. 119, 054801 (2017).
  * Yin  _et al._ (2011) L. Yin, B. J. Albright, D. Jung, R. C. Shah, S. Palaniyappan, K. J. Bowers, A. Henig, J. C. Fernández, and B. M. Hegelich, Phys. Plasmas 18, 063103 (2011).
  * Weng  _et al._ (2012) S. M. Weng, M. Murakami, P. Mulser, and Z. M. Sheng, New J. Phys. 14, 063026 (2012).
  * Siminos  _et al._ (2012) E. Siminos, M. Grech, S. Skupin, T. Schlegel, and V. T. Tikhonchuk, Phys. Rev. E 86, 056404 (2012).
  * Tushentsov  _et al._ (2001) M. Tushentsov, A. Kim, F. Cattani, D. Anderson, and M. Lisak, Phys. Rev. Lett. 87, 275002 (2001).
  * Fernández  _et al._ (2017) J. C. Fernández, D. C. Gautier, C. Huang, S. Palaniyappan, B. J. Albright, W. Bang, G. Dyer, A. Favalli, J. F. Hunter, J. Mendez, M. Roth, M. Swinhoe, P. A. Bradley, O. Deppert, M. Espy, K. Falk, N. Guler, C. Hamilton, B. M. Hegelich, D. Henzlova, K. D. Ianakiev, M. Iliev, R. P. Johnson, A. Kleinschmidt, A. S. Losko, E. McCary, M. Mocko, R. O. Nelson, R. Roycroft, M. A. S. Cordoba, V. A. Schanz, G. Schaumann, D. W. Schmidt, A. Sefkow, T. Shimada, T. N. Taddeucci, A. Tebartz, S. C. Vogel, E. Vold, G. A. Wurden, and L. Yin, Phys. Plasmas 24, 056702 (2017).
  * Choudhary and Holkundkar (2016) S. Choudhary and A. R. Holkundkar, Eur. Phys. J. D 70, 234 (2016).
  * Kaluza  _et al._ (2004) M. Kaluza, J. Schreiber, M. I. K. Santala, G. D. Tsakiris, K. Eidmann, J. Meyer-ter Vehn, and K. J. Witte, Phys. Rev. Lett. 93, 045003 (2004).
  * Davis  _et al._ (2005) J. Davis, G. M. Petrov, and A. L. Velikovich, Phys. Plasmas 12, 123102 (2005).
  * Sullivan  _et al._ (1994) A. Sullivan, H. Hamster, S. P. Gordon, R. W. Falcone, and H. Nathel, Opt. Lett. 19, 1544 (1994).
  * Najmudin  _et al._ (2002) Z. Najmudin, M. Tatarakis, K. Krushelnick, E. L. Clark, V. Malka, J. Faure, and A. E. Dangor, IEEE Trans. Plasma Sci. 30, 44 (2002).
  * Sprangle and Hafizi (2014) P. Sprangle and B. Hafizi, Phys. Plasmas 21, 055402 (2014).
  * Chen and Sudan (1993) X. L. Chen and R. N. Sudan, Phys. Fluids B 5, 1336 (1993).
  * Singh  _et al._ (2012) D. K. Singh, J. R. Davies, G. Sarri, F. Fiuza, and L. O. Silva, Phys. Plasmas 19, 073111 (2012).
  * Wilson  _et al._ (2017) T. C. Wilson, F. Y. Li, M. Weikum, and Z. M. Sheng, Plasma Phys. Control. Fusion 59, 065002 (2017).
  * Smyth  _et al._ (2016) A. G. Smyth, G. Sarri, M. Vranic, Y. Amano, D. Doria, E. Guillaume, H. Habara, R. Heathcote, G. Hicks, Z. Najmudin, H. Nakamura, P. A. Norreys, S. Kar, L. O. Silva, K. A. Tanaka, J. Vieira, and M. Borghesi, Phys. Plasmas 23, 063121 (2016).
  * Hadžievski  _et al._ (2002) L. Hadžievski, M. S. Jovanović, M. M. Škorić, and K. Mima, Phys. Plasmas 9, 2569 (2002).
  * Sánchez-Arriaga  _et al._ (2011) G. Sánchez-Arriaga, E. Siminos, and E. Lefebvre, Plasma Phys. Control. Fusion 53, 045011 (2011).
  * Pukhov  _et al._ (2011) A. Pukhov, N. Kumar, T. Tückmantel, A. Upadhyay, K. Lotov, P. Muggli, V. Khudik, C. Siemon, and G. Shvets, Phys. Rev. Lett. 107, 145003 (2011).
  * Holkundkar and Brodin (2018) A. R. Holkundkar and G. Brodin, Phys. Rev. E 97, 043204 (2018).
  * Brodin  _et al._ (2006) G. Brodin, M. Marklund, L. Stenflo, and P. K. Shukla, New J. Phys. 8, 16 (2006).
  * Lichters  _et al._ (1997) R. Lichters, R. E. W. Pfund, and J. Meyer-Ter-Vehn, MPQ Report 225 (1997).
  * Pathak  _et al._ (2018) N. Pathak, A. Zhidkov, T. Hosokai, and R. Kodama, Phys. Plasmas 25, 013119 (2018).
  * Choudhary and Holkundkar (2018) S. Choudhary and A. R. Holkundkar, Phys. Plasmas 25, 103111 (2018).