# Generating optical supercontinuum and frequency comb in tenuous plasmas

Kenan Qu  Nathaniel J. Fisch  Department of Astrophysical Sciences, Princeton
University, Princeton, New Jersey 08544, USA

###### Abstract

There are several mechanisms by which the frequency spectrum of a laser
broadens when it propagates at near-relativistic-intensity in tenuous plasma.
Focusing on one dimensional effects, we identify two strong optical
nonlinearities, namely, four-wave mixing (FWM) and forward Raman scattering
(FRS), for creating octave-wide spectra. FWM dominates the interaction when
the laser pulse is short and intense; Its combination with phase modulation
produces a symmetrically broadened supercontinuum. FRS dominates when the
laser pulse is long and relatively weak; It broadens the laser spectrum mainly
towards lower frequencies and produces a frequency comb. The creation of the
supercontinuum and frequency combs only frequency modulates, but does not
compress, the laser pulse.

##  I Introduction

Wide-band laser spectra [1, 2, 3] with high intensities are capable of
overcoming plasma instabilities, such as Raman/Brillouin scattering [4, 5, 6,
7, 8, 9, 10] and filamentation [5, 6], and enable efficient laser power
delivery [11, 12, 13]. Numerical simulations shows that they can propagate
over an extended distance through plasmas without incurring significant pulse
distortion [14, 15, 16] or absorption [17, 11, 18]. Hence, they are useful in
numerous laser-plasma applications, such as laser-plasma accelerator [19],
inertial confinement fusion [20, 13], and EUV radiation [21]. But traditional
methods using nonlinear optical crystals cannot efficiently produce wide-band
spectra at high power [22, 23].

Actually, plasma can be used to broaden the laser spectrum because of its
strong optical nonlinear susceptibility and high thermal resistance. It has
been theoretically proposed [24, 25, 26, 27, 28, 29, 30] that a dual-color
continuous laser wave can broaden its spectrum in plasma through cascades of
forward Raman scattering (FRS) [26, 27]. The plasma electron waves couple
laser components that are detuned by the plasma frequency. Hence, a pair of
continuous lasers can be converted into a wide-band spectrum with discrete
spikes, i.e., an optical frequency comb [31, 32, 33, 34].

In this paper, we revisit the cascaded broadening of the laser spectrum in
plasmas and point out a new regime dominated by four-wave mixing (FWM) [35,
36, 37, 38]. The FWM-dominated process produces a wide-band continuous
spectrum without discreteness, i.e., an optical supercontinuum [39, 40]. We
compare the FWM and FRS processes to show that the cascade generates a
supercontinuum when the laser pulse is short relative to a few plasma wave
periods and has near relativistic intensity, and otherwise it generates a
frequency comb. Short pulse duration and high intensity causes strong phase
modulation and chirping. It flattens each frequency sideband and creates a
continuous spectrum. Given sufficient plasma length, either the supercontinuum
or frequency comb can reach almost a full electromagnetic spectrum spanning
multiple octaves.

To create the wide-band spectrum, a pair of co-propagating laser pulses with
detuning of the plasma frequency is sent into plasma. They beat to create a
plasma Langmuir wave [24, 25] if FRS dominates, or an electron mass
perturbation (a virtual phonon) if FWM dominates. Both the plasma wave and the
virtual phonon scatter the laser and broadens the spectrum at multiples of the
plasma frequency. From a quantum point of view, a photon can either split into
a lower-frequency photon and a phonon/virtual phonon, or convert itself into a
higher frequency photon by combining with a phonon/virtual phonon. Since the
frequency downconversion process takes place at a higher interaction rate, the
phonon/virtual phonon number grows and the spectrum expands. The process can
terminate due to wavevector mismatch near plasma frequency before exiting the
plasma medium.

The plasma wave excited through FRS has a phase velocity near the speed of
light but no group velocity. Zero group velocity means that the plasma wave
propagates backward in the laser frame, causing energy transportation from the
laser front to the tail. The growing plasma wave amplitude leads to a wider
spectrum in the laser pulse tail than the front. Due to the energy consumption
by the plasma wave, the laser spectrum shows overall frequency downshift.

FWM is a parametric process which does not excite a plasma wave and thus
conserves the total electromagnetic energy. The virtual phonon amplitude does
not grow or decay and is solely determined by the instantaneous laser waves.
Hence, the virtual phonon has the maximum amplitude at the laser peak where
the widest spectrum broadening takes place. With conserved laser energy and
photon number, the spectrum is broadened symmetrically besides the input laser
frequency. The short pulse duration and high intensity combine to produce a
strong chirp to stretch each frequency sideband and form a supercontinuum.

This paper focuses on the 1D laser pulse evolution to compare the FWM and FRS
processes. It is organized as follows: In Sec. II, we model the laser
propagation problem and explain the laser nonlinearities. In Sec III, we
identify two different interaction regimes dominated by FRS and FWM,
respectively, and find the growth of the plasma wave and virtual phonon. In
Sec. IV, we find the scaling laws of the frequency bandwidth in each regime.
In Sec. V, we analyze the temporal envelope evolution of the laser pulse. In
Sec. VI, we demonstrate using PIC simulations the generation of a
supercontinuum and a frequency comb. In Sec. VII, we present our conclusions.

##  II model

We analyze the laser pulse evolution from the coupled laser plasma equations
in the 1D form [6, 19]

$\displaystyle(\partial_{tt}-c^{2}\partial_{zz})a=\frac{\omega_{p}^{2}}{\gamma}\frac{n}{\bar{n}}a\cong\omega_{p}^{2}(1+\tilde{n}-\frac{a^{2}}{2})a,$
(1)
$\displaystyle(\partial_{tt}+2\nu\omega_{p}+\omega_{p}^{2})\tilde{n}=\frac{c^{2}}{2}\partial_{zz}a^{2},$
(2)

where $a$ is the dimensionless vector potential normalized to the laser
intensity $I$ as $I={1.37\times
10^{18}}(a/\lambda[\mu\mathrm{m}])^{2}[\mathrm{Wcm}^{-2}]$,
${\gamma=\sqrt{1+a^{2}}}$ is the Lorentz factor of the electrons,
$\tilde{n}=(n-\bar{n})/\bar{n}$ is the normalized perturbed electron density,
and $n$ and $\bar{n}$ are the local and average electron densities,
respectively. $c$ is the speed of light in vacuum and $\omega_{p}$ is plasma
frequency at density $\bar{n}$. The expansion in Eq. (1) is valid for
$\tilde{n}\ll 1$ and $a\ll 1$. We include a heuristic damping factor $\nu$ in
Eq. (2) to describe plasma wave damping. Our discussion focuses on laser pulse
duration longer than a plasma wavelength to avoid wake field excitation.

The nonlinear term proportional to $a^{3}$ accounts for the change of mass
when the electrons are driven to near the relativistic velocity in the strong
laser field. Its coefficient, $-\omega_{p}^{2}/2$, results from the first-
order expansion of $\omega_{p}^{2}/\gamma=\omega_{p}^{2}/\sqrt{1+a^{2}}$. The
change of plasma frequency can, in turn, modulate the laser field. This
instantaneous mutual coupling between the laser field and electrostatic
potential is investigated in a more general formalism in Refs. [37, 38]. It is
found that the plasma-to-laser back-action yields a correction term
$(\omega_{p}^{2}/2)c^{2}\partial_{zz}(\partial_{t}^{2}+\omega_{p}^{2})^{-1}$
in additional to the nonlinear coefficient, $-\omega_{p}^{2}/2$. The
correction term is, nevertheless, very small in tenuous plasmas, and we
neglect it in the rest of this article. We will present particle-in-cell (PIC)
simulations in Sec. VI to justify the use of Eqs. (1)-(2).

The plasma wave is most strongly excited at its eigen frequency $\omega_{p}$,
which scatters the laser wave into discrete frequencies. We therefore expand
the laser field and plasma density perturbation as

$\displaystyle a$ $\displaystyle=\sum_{j}a_{j}e^{-i\omega_{j}t+ik_{j}z}+c.c.,$
(3) $\displaystyle\tilde{n}$ $\displaystyle=\delta
ne^{-i\omega_{p}t+ik_{p}z}+c.c.$ (4)

where $\omega_{j}=\omega_{0}+j\omega_{p}>0$, $k_{j}=k_{0}+jk_{p}>0$, and
$\omega_{0}(k_{0})$ is the frequency (wavenumber) of the pump laser. In the
large frequency limit ${\omega_{j}\gg\omega_{p}}$, the dispersion relation
$\omega_{j}^{2}=c^{2}k_{j}^{2}+\omega_{p}^{2}$ is approximated as
$\omega_{j}\approx ck_{j}$, so $\omega_{p}\approx ck_{p}$. Matching the fast
oscillation components in Eqs. (1)-(2) yields

$\displaystyle(\partial_{t}+v_{g,j}\partial_{z})a_{j}=-\frac{\omega_{p}^{2}}{2i\omega_{j}}(\delta
n^{*}a_{j+1}+\delta na_{j-1})$
$\displaystyle\quad+\frac{3\omega_{p}^{2}}{4i\omega_{j}}\left[\sum_{k}|a_{k}|^{2}a_{j}+\sum_{m,n}^{m\neq
n}a_{m}^{*}a_{n}a_{m-n+j}+\mathrm{TSFG}\right],$ (5)
$\displaystyle(\partial_{t}+i\nu)\delta
n=-\frac{\omega_{p}}{2i}\sum_{j}a_{j+1}a_{j}^{*},$ (6)

where ${v_{g,j}=c^{2}k_{j}/\omega_{j}}$ is the group velocity of frequency
component $\omega_{j}$. The first term on right hand side of Eq. (5) describes
the anti-Stokes and Stokes Raman scattering of the plasma wave. A photon is
created either by annihilation of an anti-Stokes photon and creation of a
phonon or by annihilation of both a Stokes photon and a phonon, as illustrated
in Fig. 1(a). Inside the square brackets are the terms for phase modulation
(including self-phase modulation for $k=j$, and cross-phase modulation for
$k\neq j$), FWM, and third-order sum-frequency generation (TSFG),
respectively. The phase modulation nonlinearity only induces a phase shift
without causing energy dissipation or redistribution. The FWM nonlinearity
describes annihilation of two photons to create two new photons at different
frequencies, as illustrated in Fig. 1(b). The FWM nonlinearity can create
photons at new frequencies, similar to the FRS process, without exciting
plasma waves. Each FWM stage is similar to the process discussed in Ref. [38]
in absence of nonlinear resonance broadening effects. The TSFG nonlinearity
describes combination of three photons and create a photon at the three-
photon-sum frequency. The four photons in TSFG have drastically different
frequencies making it difficult to satisfy the phase matching condition.
Hence, we neglect the TSFG term in the following discussion. Equation (6)
describes the plasma wave amplitude driven by the beat of two adjacent
electromagnetic sidebands. A phonon is created when a photon is converted into
a Stokes photon.

Figure 1:  Diagrams of (a) FRS and (b) FWM processes.

For spatial-temporal analysis, we next introduce the coordinate system that is
comoving with the speed of light: $\zeta=t-z/c$ and $\tau=z/c$. Then Eqs.
(5)-(6) are transformed into

$\displaystyle\bigg{[}\partial_{\tau}+\bigg{(}\frac{c}{v_{g,j}}-1\bigg{)}\partial_{\zeta}\bigg{]}a_{j}=-\frac{\omega_{p}^{2}}{2ick_{j}}(\delta
n^{*}a_{j+1}+\delta na_{j-1})$
$\displaystyle\qquad+\frac{3\omega_{p}^{2}}{4ick_{j}}\sum_{l}\chi_{l}a_{j-l},$
(7) $\displaystyle(\partial_{\zeta}+i\nu)\delta
n=-\frac{\omega_{p}}{2i}\sum_{j}a_{j+1}a_{j}^{*},$ (8)
$\displaystyle\chi_{l}=\sum_{j}a_{j+l}a_{j}^{*}.$ (9)

Here, we introduce the virtual phonon parameter $\chi_{l}$ to describe the
phase modulation nonlinearity for $l=0$ and the FWM nonlinearity for $l\neq
0$. It obeys the relation $\chi_{l}^{*}=\chi_{-l}$. Note that the virtual
phonon is not a wave, so it does not propagate or decay.

Although the phase modulation process does not generate new discrete frequency
sidebands, it flattens each sideband by inducing a chirp. For short laser
pulses, the phase modulation, which is proportional to $\chi_{0}(\tau,\zeta)$,
varies rapidly within the pulse duration. It broadens the spectrum by an
amount of $[\omega_{p}^{2}\tau/(2\omega_{j})](\partial_{\zeta}\chi_{0}$). We
will show in Sec. IIIC that $\chi_{0}$ remains quasi-constant during the
interaction. Each sideband thus expands linearly with $\tau$, and eventually
merges with adjacent sidebands when broadens to $\omega_{p}$. For an input
pulse with duration $T$ and a peak value $\chi_{0M}$, The spectrum loses
discreteness and becomes a supercontinuum when

$\omega_{p}\tau\gtrsim\omega_{j}\big{/}(T\chi_{0M}).$ (10)

Figure 2:  (a) Tempo-spectral diagram of a short ($T=2\pi/\omega_{p}$) laser
pulse after propagating in plasma for $250$ plasma wavelengths. The initial
pulse includes components $a_{49}=a_{50}=0.4e^{-(\zeta/T)^{2}}$. (b) Temporal
spectral diagram of the output of a long ($T=40\pi/\omega_{p}$) laser pulse
after propagating in plasma for $800$ plasma wavelengths. The initial pulse
includes components $a_{49}=a_{50}=0.15e^{-(\zeta/T)^{2}}$. Note the
difference of the vertical axis scales. The top panels show the spectra
histograms for the whole pulse. The side panels show the temporal envelopes of
the initial pulse (red curves) and the final pulse (gray shades). Their full-
size envelope plots are shown in Fig. 5 and Fig. 6, respectively.

##  III Two regimes of operation: FWM and FRS

Each new frequency component is created when an existing photon combines with
a phonon or a virtual phonon. Starting with a pump photon $a_{o}$ and a probe
photon $a_{o-1}$ at adjacent frequencies, FWM directly creates the anti-Stokes
sideband $a_{o+1}$ and the Stokes sideband $a_{o-2}$ via the instantaneous
virtual phonon. The pump and probe waves also excite a plasma wave $\delta n$
whose amplitude increases along $\zeta$. The plasma wave then combines with
the pump and probe photon to contribute to $a_{o+1}$ and $a_{o-2}$. But the
mediating plasma wave $\delta n$ introduces a $\pi/2$ phase, hence the two
paths of creating the new sidebands do not coherently add to each other.

FRS and FWM scatter the laser in distinctive manners. First, the plasma wave
amplitude grows in the direction $\zeta$, but the virtual phonon amplitude is
determined solely by the instantaneous laser field. Consequently, FWM causes
the most significant spectrum broadening near the intensity peak of the pulse
and FRS broadening happens mostly in the pulse tail. This also leads to the
second distinction that FWM is prominent only with short pulse duration
relative to plasma wave periods, and otherwise FRS dominates. The short
duration and high laser intensity makes a low threshold to develop into a
supercontinuum. Third, FRS can only change the frequency of a photon by a
single plasma frequency $\pm\omega_{p}$, but FWM can cause changes in multiple
plasma frequencies $\pm j\omega_{p}$ even for non-integer $j$’s. Fourth, FWM
creates both a low-frequency photon and a high-frequency photon
simultaneously, but FRS creates Stokes or anti-Stokes photons independently.
Since the interaction rate is higher for frequency downshift, FRS overall
creates more low frequency photons.

To illustrate the interaction properties of FWM and FRS, we numerically solve
Eqs. (7)-(9) and show the results in Fig. 2. We first consider a short
Gaussian laser pulse with rms duration $T=2\pi/\omega_{p}$. The input pulse
contains two frequency components at $a_{49}=a_{50}=0.4e^{-(\zeta/T)^{2}}$.
The double-hump pulse envelope is the result of beating. Their interaction is
dominated by FWM in the $250$ plasma-wavelength-long plasma. The tempo-
spectral diagram in Fig. 2(a) shows that the maximum spectral broadening
happens near the peak of the pulse. The output spectrum shown in the top bar
plot has a relatively flat distribution with sharp decent in both ends. The
side panel shows that the pulse temporal envelope on the remains its structure
despite some slight distortion.

For comparison, we demonstrate a FRS-dominated spectral broadening interaction
by increasing the laser pulse duration to $T=40\pi/\omega_{p}$. The pulse
amplitude is correspondingly reduced to $a_{49}=a_{50}=0.15e^{-(\zeta/T)^{2}}$
and the total propagation distance is $800$ plasma wavelengths. Plasma wave
decay $\nu$ is zero. The output tempo-spectral diagram in Fig. 2(b) shows
increasingly broader spectrum towards the tail of the laser pulse. It forms a
grid-like structure with alternating dark and bright spots in both $\zeta$ and
$\omega$ directions. It indicates that, for a certain frequency components,
the pulse is transformed into a series of pulse trains. The time-integrated
spectrum, as illustrated in the top bar plot, shows that most of the photons
are shifted to lower frequencies. The spectrum has a constant decreasing trend
in towards both lower and higher frequency limits. The temporal envelope
remains its modulation structure despite higher peak amplitudes in the pulse
tail.

###  III.1 Conservation of energy and photon number

The downshift of photon frequencies in the FRS frequency comb indicates loss
of laser energy, which seems to be conserved in the FWM-dominated frequency
comb, as shown in Fig. 2. For more rigorous analysis, we quantitatively
investigate Eqs. (7)-(9). First, we remind ourselves that the photon energy
density in plasma can be expressed as
$\mathcal{E}_{\mathrm{EM}}=\\{\partial_{\omega}[\omega\epsilon(\omega)]\epsilon_{0}E^{2}+B^{2}/\mu_{0}\\}/2=\epsilon_{0}E^{2}\propto\omega^{2}a^{2}$
according to the Landau-Lifshits formula [41]. The photon number density hence
is $\mathcal{E}_{\mathrm{EM}}/\omega\propto\omega a^{2}$. Note that these two
expressions have a different form in vacuum. The conservation of total photon
number can be obtained exactly from Eq. (7), i.e.,
$\partial_{\tau}\left[\int\sum\omega_{j}|a_{j}|^{2}\mathrm{d}\zeta\right]=0$,
where $\omega_{j}|a_{j}|^{2}$ represents the photon number density of the
component with frequency $\omega_{j}$ in plasma.

For local photon number density and local laser energy density, we obtain from
Eqs. (7)-(9) that

$\displaystyle\partial_{\tau}\sum_{j}\omega_{j}|a_{j}|^{2}=\partial_{\zeta}\sum_{j}(1-\frac{c}{v_{g,j}})\omega_{j}|a_{j}|^{2},$
(11)
$\displaystyle\partial_{\tau}\sum_{j}\omega_{j}^{2}|a_{j}|^{2}=\partial_{\zeta}\sum_{j}(1-\frac{c}{v_{g,j}})\omega_{j}^{2}|a_{j}|^{2}-\omega_{p}^{2}\partial_{\zeta}|\delta
n|^{2}.$ (12)

We see that the photon number density changes only due to group velocity
dispersion (GVD), but the laser energy density is affected by both GVD and
FRS. Equation (12) has a form of
$\partial_{\tau}\mathcal{E}_{\mathrm{EM}}=(1/c)\partial_{\zeta}\mathcal{S}$,
which describes the convection of local laser energy
$\mathcal{E}_{\mathrm{EM}}=\partial_{\tau}\sum_{j}\omega_{j}|a_{j}|^{2}$ out
of the local region by GVD and by conversion into plasma waves. By integrating
over the laser pulse duration from $\zeta=0$ to $\zeta_{f}$, we find the laser
energy dissipation rate
$\partial_{\tau}\int_{0}^{\zeta_{f}}\mathcal{E}_{\mathrm{EM}}\mathrm{d}\zeta=\omega_{p}^{2}|\delta
n(\zeta_{f})|^{2}$ is exactly the plasma wave energy density at the pulse
tail. Although the plasma wave transports energy towards the tail of the laser
pulse, FRS does not lead to spreading of photon number density, as shown in
Eq. (11).

Remarkably, the parametric phase modulation and FWM nonlinearities do not play
a role in either photon number density redistribution or laser energy density
dissipation. They only contribute to the spectrum broadening by affecting the
optical refractive index of local plasmas: The plasma electrons are driven to
near relativistic speed by the strong laser field and begin to oscillate in an
anharmonic manner. The anharmonicity induces optical nonlinear interaction
among different laser frequency components. This parametric nonlinear process
does not induce any growing plasma density perturbation or electrostatic
fields, and hence does not cause laser energy dissipation. But it changes
plasma dispersion relation and induces phase change to local photons. In
another word, FWM induces virtual phonons and FRS induces real phonons.

###  III.2 FRS and growth of plasma waves

broadening of the laser spectrum is mediated by phonons and virtual phonons of
finite amplitudes. With a multi-color input laser, the virtual phonon
amplitude becomes nonzero instantaneously, but the phonon amplitude grows
gradually. To analyze their growth, we separate the different regimes of
interaction depending on the pulse duration.

For pulses with duration longer than a plasma wave length, FRS dominated
dominates. We neglect the FWM interaction and find after combining Eqs.
(7)-(8) that

$\displaystyle(\partial_{\zeta}-i\nu)\partial_{\tau}\delta n$
$\displaystyle=\partial_{\zeta}\sum_{j}(2-\frac{c}{v_{g,j}}-\frac{c}{v_{g,j+1}})a_{j}^{*}a_{j+1}$
$\displaystyle+\frac{\omega_{p}^{4}}{2}\sum_{j}\left(\frac{1}{\omega_{j-1}}-\frac{1}{\omega_{j+1}}\right)|a_{j}|^{2}\delta
n.$ (13)

The result indicates that FRS causes exponential growth of the plasma wave
amplitude. The growth is caused by asymmetric interaction rates of phonon
absorption and phonon creation. Each laser photon $a_{j}$ can emit a phonon by
converting itself into a Stokes photon or absorbing a phonon by converting
into an anti-Stokes photon. Since the coupling strength is larger for lower
frequency components, a photon is more likely to be down-converted and create
a phonon. The increased phonon number, in turn, enhances the photon-photon
interaction. The plasma wave growth rate in the long pulse limit is
$\Gamma_{\mathrm{FRS,4}}\cong\left[\omega_{p}^{2}/(2\sqrt{2}\gamma)\right]\left(\sum_{j}|a_{j}|^{2}/\omega_{j}^{2}\right)^{1/2}$,
assuming $\nu\to 0$. It can be reduced to
$\omega_{p}^{2}a_{0}/(2\sqrt{2}\gamma\omega_{0})$ in the case of a
monochromatic pump, which agrees with Refs.[42, 8, 9]. The phonon peak
amplitude locates at $\zeta=\tau/2$, i.e., $z=ct/2$. This growth rate cannot
be achieved with finite laser duration. The plasma wave amplitude at distance
$\zeta$ grows as $\exp\big{[}2\Gamma_{\mathrm{FRS,4}}\sqrt{\tau\zeta}\big{]}$,
as can be seen from Fig. 3.

Figure 3:  Plasma wave amplitude at various locations $\zeta$ and pulse
propagation distances $\tau$. The parameters are the same with Fig. 2(b).

Note that the growth rate $\Gamma_{\mathrm{FRS,4}}$ is smaller than the FRS
growth rate
$\Gamma_{\mathrm{FRS,3}}=(\omega_{p}^{3}/\omega_{0})^{\frac{1}{2}}(a_{0}/\gamma)$
which only takes into account coupling with the Stokes wave but not the anti-
Stokes wave. The decrease of growth rate $\Gamma_{\mathrm{FRS,4}}$ can be
attributed to the absorption of phonons when generating anti-Stokes photons.

When a phonon interacts with a photon $a_{j}$, whether the scattering creates
an anti-Stokes photon $a_{j+1}$ or a Stokes photon $a_{j-1}$ depends on the
relative phase of $\delta n$ and $a_{j}$. We obtain the photon number dynamics
from Eq. (7) that

$\displaystyle\quad[\partial_{\tau}+(\frac{c}{v_{g,j}}-1)\partial_{\zeta}](\omega_{j}|a_{j}|^{2})$
$\displaystyle=\mathrm{Re}\big{[}\omega_{p}^{3}(a_{j+1}^{*}\delta
n-a_{j-1}^{*}\delta
n^{*})a_{j}\big{]}-\mathrm{Im}\big{[}3\omega_{p}^{2}\sum_{k}a_{j}a_{j+k}^{*}\chi_{k}\big{]},$
(14)

and $\delta
n=\int_{0}^{\zeta}\chi_{1}e^{-\nu\zeta^{\prime}}\mathrm{d}\zeta^{\prime}$. On
the right hand side, the first term describes the FRS interaction and the
second term describes the FWM interaction. The FRS interaction can be
categorized into resonant terms and non-resonant terms.

The resonant photon number growth due to FRS interaction is proportional to
$(|a_{j+1}|^{2}-|a_{j-1}|^{2})|a_{j}|^{2}$ with the $\zeta$ dependence
neglected. In the small time scale, it leads to an exponential growth of the
photon number in mode $a_{j}$. This amplitude-difference-driven interaction
causes a cascade of photon frequency decrease: Starting with a bi-color laser
input $a_{o}$ and $a_{o-1}$, the resonant FRS interaction initially causes
$|a_{o}|$ to decrease and $|a_{o-1}|$ to increase. A low-frequency mode
$|a_{o-2}|$ is created and grows. As $|a_{o-2}|$ approaches and exceeds
$|a_{o}|$, the mode $|a_{o-1}|$ begins to decrease. Without photon
supplementation from higher frequency modes, $|a_{o}|$ eventually reaches zero
amplitude. Overall, modes are created at the low-frequency limit and are
annihilated in the high-frequency limit, causing a successive frequency
downshift. The trend of downshift can be seen as the bright stream in main
plot of Fig. 2(b). The resonant frequency downshift process asymptotically
results in more low-frequency sidebands with monotonically decreasing
amplitudes, as can be seen in the top panel of Fig. 2(b).

The non-resonant FRS interaction is described by the terms proportional to
$Re[(a_{j+1}^{*}\int_{0}^{\zeta}\chi_{1}e^{-\nu\zeta^{\prime}}\mathrm{d}\zeta^{\prime}-a_{j-1}^{*}\int_{0}^{\zeta}\chi_{1}^{*}e^{-\nu\zeta^{\prime}}\mathrm{d}\zeta^{\prime})a_{j}]$.
Since it is proportional to $a_{j}$, the non-resonant FRS has a lower growth
rate. The non-resonant FRS can create both frequency upshifted photons and
frequency downshifted photons, similar to FWM-type interaction.

Figure 4:  Amplitudes of virtual phonon $|\chi_{k}|$ at the pulse peak
$\zeta/(c\omega_{p})=1$. The parameters are the same with Fig. 2(a).

###  III.3 FWM and evolution of virtual phonons

Laser pulses with a short duration near a plasma wavelength are not sufficient
to generate strong plasma waves and hence FWM dominates. FWM creates new
frequency components by scattering the laser photons via virtual phonons. The
virtual phonons do not have an eigenfrequency and are totally determined by
the beating lasers. Hence, FWM can upshift or downshift a laser photon by
multiple times the plasma frequency.

The FWM growth rate depends on the virtual phonon amplitude. With neglected
FRS interaction, the virtual phonon dynamics is obtained by combining Eqs. (7)
and (9) that

$\displaystyle\partial_{\tau}\sum_{k}|\chi_{k}|^{2}=\sum_{j,k}\chi_{k}(1-\frac{c}{v_{g,j}})\partial_{\zeta}(a_{j}a_{j+k}^{*}+a_{j-k}a_{j}^{*}),$
(15)
$\displaystyle\partial_{\tau}(\chi_{k}\pm\chi_{-k})=\sum_{j}(1-\frac{c}{v_{g,j}})\partial_{\zeta}[a_{j}(a_{j+k}^{*}\pm
a_{j-k}^{*})]$
$\displaystyle\qquad+\frac{3\omega_{p}^{2}}{4i}\sum_{j,k^{\prime}}\left(\frac{1}{\omega_{j}}-\frac{1}{\omega_{j-k-k^{\prime}}}\right)\chi_{k^{\prime}}a_{j-k^{\prime}}a_{j-k}^{*}$
$\displaystyle\qquad\pm\frac{3\omega_{p}^{2}}{4i}\sum_{j,k^{\prime}}\left(\frac{1}{\omega_{j}}-\frac{1}{\omega_{j+k-k^{\prime}}}\right)\chi_{k^{\prime}}a_{j-k^{\prime}}a_{j+k}^{*}.$
(16)

Since $\chi_{-k}=\chi_{k}^{*}$, Eq. (16) for $\pm$ describes the evolution of
the real and imaginary parts of $\chi_{k}$, respectively. The identity (15)
indicates that the total virtual phonon amplitude is conserved when neglecting
GVD. Hence, none of $\chi_{k}$ can grow exponentially. With GVD, growth of
$\chi_{k}$ is driven by other $\chi_{k^{\prime}}$ terms provided that
$a_{j}a^{*}_{j-(k-k^{\prime})}$ is nonzeros. Since the coefficients
$(1/\omega_{j}-1/\omega_{j\pm k-k^{\prime}})\sim(k^{\prime}\pm
k)/(j\omega_{j})$ are small values, the growth of $\chi_{k}$ is lower than the
growth of $a_{j}$. Hence, $\chi_{k}$ can be approximated as quasi-constants.
The quasi-conservation is numerically verified and illustrated in Fig. 4 which
shows only nonzero $\chi_{0,\pm 1}$ for the bi-color input laser.

The photon number growth due to FWM is described by the second term of Eq.
(14). Due to the $\mathrm{Im}$ operation, it does not include any resonant
terms. Thus, the spectrum expands to both lower and higher frequencies
equally. Different from FRS interaction, FWM grows fastest at the pulse
intensity peak. Thus, the tempo-spectral diagram in Fig. 2(a) shows broad
bands only at the peak center.

The dominance of low-order terms $\chi_{0,\pm 1}$ in Fig. 4 means that the
laser spectrum broadens at intervals of $\omega_{p}$. This differs from Ref.
[38] which aims at substantially upshifting the laser frequency by injecting
two highly detuned pulses with frequency differences greater than the plasma
frequency. Similar to Ref. [38], however, both upshift and downshift of the
photon frequency coexist in the FWM process we describe here.

##  IV Scaling of frequency bandwidth growth

With a bi-color laser input, only $\chi_{0}$ and $\chi_{\pm 1}$ are nonzero.
For quasi-steady values of $\chi_{k}$, we can find the analytical solution to
the recursion equation in the limit of small bandwidth
$\mathrm{max}(\omega_{j})-\mathrm{min}(\omega_{j})\ll\omega_{j}$, and hence

$a_{j}(\tau,\zeta)=\sum_{o}a_{o}(0,\zeta)e^{i(j-o)(\frac{\pi}{2}-\psi)}\\\
\times\exp\left(\frac{3i\omega_{p}^{2}\chi_{0}\tau}{4\omega_{j}}\right)J_{j-o}\bigg{(}\frac{3\omega_{p}^{2}|\chi_{1}|}{2\omega_{o}}\tau\bigg{)},$
(17)

where $o$’s denote the indices of the input laser fields, and $\psi$ is the
phase of $\chi_{1}$, i.e., $e^{i\psi}=\chi_{1}/|\chi_{1}|\cong 1$. The
solution shows that each input laser component expands to a broad spectrum
whose amplitude is described by the Bessel function $J_{j-o}$. The
contribution from different $a_{o}$’s differ by a phase of $\pi/2$, so they do
not interfere. The photon number in each mode is then

$\omega_{j}|a_{j}(\tau,\zeta)|^{2}\cong\omega_{j}\sum_{o}|a_{o}(0,\zeta)|^{2}J_{j-o}^{2}\bigg{(}\frac{3\omega_{p}^{2}|\chi_{1}|}{2\omega_{o}}\tau\bigg{)}.$
(18)

Since the first peak of $J_{\pm j}(x)$ locates at approximately $x\sim|j|$,
laser spectral width $\Delta\omega_{\mathrm{FWM}}$ expands with the scaling of

$\Delta\omega_{\mathrm{FWM}}\sim\frac{3\omega_{p}^{2}|\chi_{1}|}{\omega_{o}}\tau=\frac{3\omega_{p}^{2}|a_{o}a_{o-1}^{*}|}{\omega_{o}}\tau.$
(19)

For the parameters used in Fig. 2(a), the output bandwidth reaches $\pm
16\omega_{p}$, which agrees well with the simulation results.

Equation (17) has a similar form with the solution of FRS interaction found by
Karttunen and Salomaa [26, 27] augmented with the self-phase modulation term.
For FRS interaction, Eq. (17) is to be modified by replacing $e^{i\psi}=\delta
n/|\delta n|\cong i$ and $|\chi_{1}|/\to|\delta n|$

$a_{j}(\tau,\zeta)\simeq\sum_{o}a_{o}(0,\zeta)(-1)^{j-o}J_{j-o}\bigg{(}\frac{\omega_{p}^{2}|\delta
n|}{\omega_{o}}\tau\bigg{)},$ (20)

The change of $\psi$ causes a $\pi$ phase difference between the spectra from
different $a_{o}$’s. Using the identity $J_{-j}=(-1)^{j}J_{j}$, we find the
estimation of the photon number

$\omega_{j}|a_{j}(\tau,\zeta)|^{2}\simeq\omega_{j}\left[\sum_{o}(\mp
1)^{o}|a_{o}(0,\zeta)|J_{j-o}\bigg{(}\frac{\omega_{p}^{2}|\delta
n|}{\omega_{o}}\tau\bigg{)}\right]^{2},$ (21)

where $\mp$ are for positive and negative values of $j-o$, respectively. Thus,
the spectral broadening to the high frequency band is suppressed due to
destructive interfere; The spectral expansion to the lower frequency band is
enhanced due to constructive interference, as seen from Fig. 2(b). Since the
spectral broadens only to the lower frequency bands, the scaling is

$\Delta\omega_{\mathrm{FRS}}\sim\frac{\omega_{p}^{2}|\delta
n|}{\omega_{o}}\tau=\frac{\omega_{p}^{3}\tau}{2\omega_{o}}\left|\int_{0}^{\zeta}a_{o}a_{o-1}^{*}e^{-\nu\zeta^{\prime}}\mathrm{d}\zeta^{\prime}\right|.$
(22)

For the parameters used in Fig. 2(a), the output spectrum extends to a lower
frequency by $\pm 15\omega_{p}$, which agrees in the order of magnitude with
the simulation results. It should be born in mind that the solution only
approximately describes the spectral evolution because the plasma amplitude
$\delta n$ changes as we see from Fig. 3.

Comparing Eqs. (19) and (22), we find that the spectral bandwidth broadening
due to FWM or FRS have the same dependence on the parameters including
interaction time $\tau$ and input frequency $\omega_{o}$. They both increase
with higher pump amplitude multiplication
$\chi_{1}(0,\zeta)=a_{o}(0,\zeta)a_{o-1}(0,\zeta)$, but FWM grows
proportionally to $|\chi_{1}(0,\zeta)|$ and FRS depends on its integration
$\big{|}\int_{0}^{\zeta}\chi_{1}(0,\zeta^{\prime})e^{-\nu\zeta^{\prime}}\mathrm{d}\zeta^{\prime}\big{|}$.
Therefore, FWM dominates when the pulse duration is as short as a few plasma
wavelengths and it broadens the laser spectrum most significantly at the pulse
intensity peak; FRS dominates with longer pulse duration and it broadens the
laser spectrum most significantly at the pulse tail.

##  V Pulse envelope modulation

A larger laser spectral bandwidth $\Delta\omega$, in principle, supports
shorter laser pulses provided that all the frequency components constructively
interference with the identical phase. However, Eqs. (17) and (20) show that
the frequency comb components have both opposite phases at different $j$’s.
The phase flipping prevents the pulse from forming sharp peaks.

Figure 5:  Top panel: The input (red curve) and output (blue shade) pulse
envelopes of a short laser pulse $T=2\pi/\omega_{p}$. Bottom panel: contour
plot of the time-frequency synchrosqueezed transform. The parameters are the
same with Fig. 2(a). Figure 6:  Top panel: The input (red curve) and output
(blue shade) pulse envelopes of a short laser pulse $T=40\pi/\omega_{p}$.
Bottom panel: contour plot of the time-frequency synchrosqueezed transform.
The parameters are the same with Fig. 2(b).

Actually, we can find the analytical solution to the pulse temporal envelope
in the limit of large spectral width. Note the generating function of the
Bessel function,
$e^{ix\cos(\phi)}=\sum_{j=-\infty}^{+\infty}i^{j}J_{j}(x)e^{ij\phi}$. For FWM
interaction, the laser field $a$ can be found by combining Eqs. (3) and (17)
and using $\omega_{j}-\omega_{o}=(j-o)\omega_{p}$

$\displaystyle a$
$\displaystyle=2\mathrm{Re}\sum_{o,j}a_{o}(0,\zeta)i^{j-o}e^{\frac{3i\omega_{p}^{2}\chi_{0}\tau}{4\omega_{j}}}J_{j-o}\bigg{(}\frac{3\omega_{p}^{2}|\chi_{1}|}{2\omega_{o}}\tau\bigg{)}e^{-i\omega_{j}\zeta}$
$\displaystyle\approx
2\mathrm{Re}\sum_{o}a_{o}(0,\zeta)e^{\frac{3i\omega_{p}^{2}\chi_{0}\tau}{4\omega_{o}}+i(\frac{3\omega_{p}^{2}|\chi_{1}|}{2\omega_{o}}\tau)\cos(\omega_{p}\zeta)+i\omega_{o}\zeta}$
$\displaystyle=\sum_{o}a_{o}(0,\zeta)\cos\left[\frac{3\omega_{p}^{2}\chi_{0}}{4\omega_{j}}\tau+(\frac{3\omega_{p}^{2}|\chi_{1}|}{2\omega_{o}}\tau)\cos(\omega_{p}\zeta)+\omega_{o}\zeta\right].$
(23)

The result shows that each pump pulse $a_{o}$ is frequency modulated by the
FWM interaction: The $\chi_{0}$ term describes phase modulation and it induces
a chirp proportional to
$\partial_{\zeta}\chi_{0}(\tau,\zeta)\tau=\partial_{\zeta}\left|\sum_{o}a_{o}(0,\zeta)\right|^{2}\tau$;
The $\chi_{1}$ term creates new sidebands and modulates the envelope at the
frequency $\omega_{p}$. The frequency modulation index
$\Delta\omega_{\mathrm{FWM}}/2=\omega_{p}|\chi_{1}|\tau/(\omega_{o})$ grows
linearly with $\tau$. The result also shows that the output laser envelope is
a direct summation of the modulated envelope of each pump pulse $a_{o}$.
Therefore, spectral broadening due to FWM does not automatically reduce the
pulse duration or increase peak amplitude.

However, the solution Eq. (V) does not take into account GVD which can change
the duration and amplitude of a chirped pulse. For FWM-dominated spectral
broadening, the pump pulse duration is shorter than a few plasma wavelengths.
The frequency modulation thus causes a frequency chirp near the pulse center
$\zeta_{M}$. Due to the strong $\zeta$ dependence of
$\chi_{0}(\tau,\zeta)\simeq\sum_{o}|a_{o}(0,\zeta)|^{2}$, the phase modulation
enhances the frequency chirp. If the chirp is positive with high-frequency
component in the tail, GVD causes pulse compression and peak amplitude
enhancement. Such a principle is adopted in Refs. [29, 30] to obtain few-cycle
laser spikes.

Figure 5 zooms in the side panel of Fig. 2(a) and shows the frequency chirp.
The filled body represents the fast oscillating pulse envelope at
$2\pi/(50\omega_{p})$. Note that the double-hump structure of the input pulse
arises from beating of the two frequency components. The pulse envelope
maintains the same structure. The output pulse obviously develops negative
frequency chirp caused by the frequency modulation. Thus, the pulse can be
compressed into two sharp and intense peaks through proper dispersion
management.

For FRS interaction, the laser field can be found similarly by combining Eqs.
(3) and (20)

$\displaystyle a$ $\displaystyle\simeq
2\mathrm{Re}\sum_{o,j}a_{o}(0,\zeta)(-1)^{j-o}J_{j-o}\bigg{(}\frac{\omega_{p}^{2}|\delta
n|}{\omega_{o}}\tau\bigg{)}e^{-i\omega_{j}\zeta}$ $\displaystyle\approx
2\mathrm{Re}\sum_{o}a_{o}(0,\zeta)\exp\left[i\bigg{(}\frac{\omega_{p}^{2}|\delta
n|}{\omega_{o}}\tau\bigg{)}\sin(\omega_{p}\zeta)+\omega_{o}\zeta\right]$
$\displaystyle=\sum_{o}a_{o}(0,\zeta)\cos\left[\bigg{(}\frac{\omega_{p}^{2}|\delta
n|}{\omega_{o}}\tau\bigg{)}\sin(\omega_{p}\zeta)+\omega_{o}\zeta\right].$ (24)

The result shows that FRS causes frequency modulation of the laser pulse with
increasing modulation index, which is similarly to FWM. The spectral
broadening does not automatically reduce the pulse duration or increase peak
amplitude.

In contrast to FWM, the FRS-broadened spectrum does not exhibit a linear chirp
and hence cannot be compressed through GVD. Specificaly, the pulse compression
is hindered by two factors. First, since the laser pulses are longer than
several plasma wavelengths, frequency modulation induces several oscillating
chirping periods of the whole pulse. GVD only compresses the region with
positive chirp but it stretches the region with negative chirp. Second, FRS
causes a dominant frequency downshift, i.e., negative chirping, of the whole
pulse due to resonance interaction as explain in Sec. III.2. Since lower-
frequency components propagates slower in plasma, GVD further stretches the
whole pulse rather than compresses it.

Figure 6 zooms in the side panel of Fig. 2(b) and show the instantaneous
frequency within the pulse. The isolated peak structure with a period of
$2\pi/\omega_{p}$ is the result of beat between two pumps at $49\omega_{p}$
and $50\omega_{p}$. The plot exhibits the most change of pulse envelope in the
peak and tail of the pulse where plasma wave is the strongest. Because chirp
is developed individually within each spike, the pulse train cannot be
compressed into a single pulse to increase its peak intensity.

Figure 7: PIC simulation results of a supercontinuum [(a) and (c)] generated
from a short laser pulse in the FWM-dominated regime, and a frequency comb
[(b) and (d)] generated from a long laser pulse in the FRS-dominated regime.
In (a) and (b), the red curves show the initial pulse envelope, and the blue
shades show the output signal. (c) and (d) show the power spectral density
(PSD) of the corresponding output signal.

##  VI PIC simulations

As proof-of-principle demonstration of generating supercontinuum and frequency
comb, we conduct particle-in-cell (PIC) simulations using the full-
relativistic kinetic code EPOCH [43]. The input laser pulse of each simulation
comprises of two frequency components with wavelengths $1\,\mu$m and
$0.98\,\mu$m, respectively. They both have a Gaussian profile, i.e.,
$a=\sum_{o}a_{o}\exp[-\frac{1}{2}(\frac{\zeta}{T})^{2}+i\omega_{o}t]$, and
their beat causes multiple-peak structure. The uniform plasma has a density of
$4.47\times 10^{17}\,\mathrm{cm}^{-3}$, corresponding to a plasma frequency of
$1/50$ of the $1\,\mu$m laser. Hence, the plasma wavelength is
$\lambda_{p}=2\pi/k_{p}=50\,\mu$m. The output laser pulses after propagating
through $40$-mm-long plasma ($800\lambda_{p}$) are shown in Fig. 7.

Figure 7(a) and (c) show frequency broadening of a short and intense pulse
into a supercontinuum. Each input pulse component has duration
$T=0.15\,\mathrm{ps}$ ($0.9$ plasma period) and peak amplitude $a_{o}=0.4$
(corresponding to $I=2.2\times 10^{17}\,\mathrm{Wcm}^{-2}$). The output pulse
envelope shows some degree of compression in the tail. We take Fourier
transform of the laser electric field at the snapshot to obtain its wavevector
spectrum. The spectrum in Fig. 7(c) shows a supercontinuum with bandwidth of
$\sim 0.8\omega_{o}$.

Figure 7(b) and (d) show frequency broadening of a long and less intense pulse
into a frequency comb. Each input pulse component has duration $T=1.5\,$ps
($9$ plasma periods) and peak amplitude $a_{o}=0.12$ (corresponding to
$I=2\times 10^{16}\,\mathrm{Wcm}^{-2}$). The output envelope shows a small
amount of pulse compression in the tail of the pulse. Fourier transform of its
electric field yields a frequency comb with discrete equidistant spikes
spanning from below $20k_{p}$ to $70k_{p}$. The result agrees well with our
analysis, which justifies the use of Eqs. (1)-(2).

##  VII Conclusion and Discussion

In conclusion, we show that a laser pulse can be expanded into a broadband
spectrum when propagating through tenuous plasmas. The spectrum broadening
arises from a cascade of both Stokes and anti-Stokes scattering due to plasma
waves and electron relativistic effects. We point out that a few-cycle pulse
with near-relativistic intensity can produce a octave-wide supercontinuum
through FWM and phase modulation; and a multi-cycle pulse can produce a
octave-wide frequency comb through FRS. As the frequency comb bandwidth
increases, it continues to lose energy to the plasma wave, causing comb
expansion to lower frequencies. But the supercontinuum conserves laser energy
because virtual phonons do not have growing amplitudes. Hence, the
supercontinuum spectrum broadens symmetrically to both lower and higher
frequencies. A notable mention is that the similar FWM process is investigated
in Ref. [38] to achieve resonant laser frequency upshift by arranging the
frequencies and intensities of two highly detuned laser. Our current article
focuses on a different regime that both FRS and FWM processes only broaden a
monochromatic pump around its central frequency with the aid of a nearly
detuned weak seed pulse.

Compared to using optical crystals for generating optical supercontinuum and
frequency combs, plasmas have high thermal damage tolerance and can work in
the near-relativistic regime. The ultra-intense broadband pulse are
particularly useful for minimizing laser scattering and absorption laser-
plasma application like inertial confinement fusion and laser-plasma
accelerators.

The experimental feasibility of this method is demonstrated through the choice
of accessible parameters in the PIC simulations. The parameters of laser
wavelength, peak intensity and plasma length are similar to those in laser
particle accelerators [19]. But the laser pulses are longer and the plasma
density is much lower for generations of supercontinuum or frequency comb.

Our analysis applies to 1D propagation of lasers with below-relativistic
intensity and pulse duration longer than a plasma wavelength. For ultra-
relativistic laser intensity, the laser-plasma interaction becomes fully
nonlinear [44, 45] and our theory is no longer valid. Pulse duration below
plasma wavelength pulse causes strong wake field [19] and significantly alters
the laser envelope evolution. To avoid pulse distortion in long plasmas, the
laser power needs to be below the critical power for transverse filamentation
[5, 6].

###### Acknowledgements.

This work was supported by NNSA Grant No. DE-NA0002948, and AFOSR Grant No.
FA9550-15-1-0391.

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