# Optical Shock-Enhanced Self-Photon Acceleration

P. Franke<EMAIL_ADDRESS>D. Ramsey  T. T. Simpson  D. H. Froula  J.
P. Palastro  University of Rochester, Laboratory for Laser Energetics,
Rochester, New York 14623, USA

###### Abstract

Photon accelerators can spectrally broaden laser pulses with high efficiency
in moving electron density gradients. When driven by a conventional laser
pulse, the group velocity walk-off experienced by the accelerated photons and
deterioration of the gradient from diffraction and refraction limit the extent
of spectral broadening. Here we show that a laser pulse with a shaped space-
time and transverse intensity profile overcomes these limitations by creating
a guiding density profile at a tunable velocity. Self-photon acceleration in
this profile leads to dramatic spectral broadening and intensity steepening,
forming an optical shock that further enhances the rate of spectral
broadening. In this new regime, multi-octave spectra extending from
$400\,\mbox{nm}-60\,\mbox{nm}$ wavelengths, which support near-transform
limited $<400\,\mbox{as}$ pulses, are generated over $<100\,\mu\mbox{m}$ of
interaction length.

††preprint: OSESPA ARXIV

Broadband sources of coherent radiation find utility across diverse scientific
disciplines as experimental drivers and diagnostic tools. State-of-the-art
supercontinuum sources, which primarily achieve spectral broadening through
Kerr-induced self-phase modulation of ultrashort laser pulses in either gas-
filled fibers [1] or self-guided filaments [2], routinely generate multi-
octave spectra in the infrared to ultraviolet wavelength range
$(15\,\mu\mbox{m}-100\,\mbox{nm}$) [3, 4, 5, 6, 7]. Such sources have thus far
been limited to wavelengths $>100\,\mbox{nm}$, due to a lack of dispersion
control and strong absorption in the extreme ultraviolet
($\lambda=10\,\mbox{nm}-100\,\mbox{nm}$). Extending coherent broadband sources
into the extreme ultraviolet would open new wavelength regimes for
spectroscopy and increase the achievable spatial and temporal resolution for
applications including single-shot spectral interferometry [8], transient
spectroscopy [9], and coherence tomography [10].

Figure 1:  (a) A Gaussian beam drives a (gray) radially convex ionization
front at the group velocity of light over approximately a Rayleigh length.
Photons diverge from the optical axis due to diffraction and plasma-
refraction. Frequency up-shifted photons dephase from the ionization front due
to group velocity walk-off, limiting the system to relatively small frequency
shifts. (b) A SFF pulse drives a concave ionization front at a tunable focal
velocity $v_{f}\gtrsim v_{g0}$ over a distance much greater than a Rayleigh
length. Photons are concentrated near the optical axis and stay in phase with
the ionization front, resulting in many photons undergoing a large frequency
shift. Transverse intensity profiles of a typical (c) Gaussian pulse and (d)
SFF pulse used in 2D simulations.

Photon acceleration, _i.e._ , the continuous increase in frequency accrued by
a photon in a temporally-decreasing refractive index, offers a promising
alternative to Kerr-based supercontinuum sources, with the potential to
efficiently generate broadband extreme ultraviolet radiation. A photon
accelerated in a rapidly ionizing medium has frequency $\omega$ that evolves
in time according to $d\omega^{2}/dt=\partial\omega_{p}^{2}/\partial t$, where
$\omega_{p}\propto\sqrt{n_{e}}$ is the plasma frequency and $n_{e}$ is the
electron density [11, 12]. Maximizing the frequency shift requires the photon
to propagate within a sharp electron density gradient over an extended
distance. An intense laser pulse can photoionize a dense gas triggering an
extremely rapid rise in the electron density. In principle, this allows for
large frequency shifts over relatively short interaction lengths. Thus far,
however, photon acceleration from optical to extreme ultraviolet frequencies
has met with limited success [13, 14, 15, 16].

A typical photon accelerator uses a laser pulse focused by conventional optics
to drive an axial electron density gradient that moves at the group velocity
of the pulse ($v_{g}<c$) [Fig 1(a)]. Witness photons injected into the
gradient undergo a frequency up-shift, increasing their group velocity. The
resulting group velocity mismatch causes witness photons to outrun, or dephase
from the gradient, terminating their frequency increase. Diffraction and
plasma-refraction cause the drive pulse to diverge from the optical axis,
reducing the sharpness of the gradient and the accelerator length, further
limiting the maximum frequency shift.

Here we introduce a scheme [Fig. 1 (b)], which largely eliminates the adverse
effects of diffraction, refraction and dephasing by combining spatiotemporal
[17, 18, 19, 20] and transverse intensity profile shaping of the laser pulse
[21, 22, 23]. This structured flying focus (SFF) pulse drives a guiding plasma
density profile that moves at a tunable focal velocity, $v_{f}$. When a
properly designed SFF pulse propagates in a homogeneous partially ionized
plasma, it undergoes extreme self-steepening and spectral broadening,
culminating in the formation and collapse of an optical shock.

In this Letter, we discover that this novel self-shocked photon acceleration
can produce multi-octave spectra extending well into the extreme ultraviolet
$(400\,\mbox{nm}-60\,\mbox{nm})$ over $<100\,\mu\mbox{m}$ of interaction
length. A simple short-pass filter can isolate coherent, high intensity
$(>10^{15}\,\mbox{W cm}^{-2})$, sub-femtosecond pulses
($<400\,\mbox{as},\,9\times$ temporal compression) from the accelerator
output, without the need for post compression. Spatiotemporal shaping produces
a focal region much longer than a Rayleigh length and allows the peak laser
intensity to move through the accelerator at a velocity $v_{g0}<v_{f}<c$,
where $v_{g0}$ is the initial group velocity of the pulse. Under these
conditions, the intensity peak moves forward within the pulse temporal
envelope, such that self-accelerated photons at the back of the pulse catch up
to and remain in phase with the intensity peak formed by the un-shifted
photons as they come into focus. The transverse intensity profile of the pulse
has higher off-axis intensity. The resulting optical field ionization creates
a guiding, radial electron density gradient that dynamically forms just ahead
of the central axial density gradient responsible for the photon acceleration.
Accelerated photons, thus confined near the optical axis, overlap temporally
and spatially with other photons of varying frequency. This local increase in
bandwidth and photon density leads to dramatic self-steepening and elevated
amplitude of the main intensity peak. The resulting sharpened axial gradient
causes faster frequency shifting and more sharpening, which in turn causes
even faster frequency shifting. Eventually, optical wavebreaking terminates
the self-steepening and limits the maximum extent of spectral broadening.

Figure 2: (a) The electron density driven by a SFF pulse after propagating for
288 fs. A guiding channel has formed over 5 $Z_{R}$. The inside pulse
intensity profiles (colorbar) at (b) 96 fs, (c) 192 fs and (d) 288 fs are
shown with the corresponding on-axis electron density (red) and intensity
(black) profiles. The neutral target density was $n_{0}=1.5\times
10^{20}\mbox{cm}^{-3}$, and $v_{f}=1.015\,v_{g0}=0.9817\,c$. The inside pulse
intensity was $1.2\times 10^{16}\,\mbox{W cm}^{-2}$ and the outside pulse
intensity was $4.0\times 10^{16}\,\mbox{W cm}^{-2}$. The color scales were
normalized to the peak intensity at each time.

Self-photon acceleration of SFF pulses propagating in preionized, underdense
nitrogen gas was simulated using a 2D finite-difference time-domain method.
Calculations were limited to the $x-z$ plane, with $\hat{z}$ the pulse
propagation direction. The 3D transverse structure of the SFF was projected in
2D using a sum of orthogonally polarized Hermite-Gaussian modes ($H_{nm}$), as
$H_{00}\,\hat{x}\,+\,H_{10}\,\hat{y}$, where $n$ and $m$ are the $x$ and $y$
indices respectively. $H_{00}$ is a typical Gaussian mode, with peak intensity
on the optical axis; $H_{10}$ has zero intensity on axis and symmetrically
increases to a peak off axis in the $\hat{x}$ direction [Fig. 1 (d)]. The
$\hat{y}$ component (outside pulse) was injected into the simulation with
higher intensity than the $\hat{x}$ component (inside pulse), such that the
optical field ionization rate was high on the optical axis, but increased with
absolute transverse coordinate, $|x|$, before falling to zero for large $|x|$.
The SFF was spatiotemporally shaped according to the desired ionization front
velocity, taking into account the nominal laser group velocity. The flying
focus effect was achieved by sweeping the electric field radius of curvature
and transverse spot size at the injection plane such that adjacent time-slices
of the pulse were focused in succession at adjacent focal positions in the far
field, with the diffraction limited spot size and far-field intensity held
constant. The sweep-rate and initial and final focal lengths were adjusted to
produce a focal spot moving at the desired velocity across the extended focal
region. The target gas (neutral atomic density
$n_{0}=10^{20}-10^{21}\,\mbox{cm}^{-3}$) was preionized to Z=3, such that the
initial electron density of the target was $n_{e0}=3n_{0}$. This partial
preionization serves three essential purposes: 1) it reduces absorption of
high-frequency light, 2) it reduces the nominal laser group velocity to
$v_{g0}<c$, and 3) it provides a significant level of anomalous (negative)
group velocity dispersion.

The outer ring of the SFF pulse forms a co-located plasma channel, dynamically
guiding the inside pulse [Fig. 2]. The final plasma channel extends over the
entire extended focal region of the SFF—nearly five Rayleigh lengths
($Z_{R}=18\,\mu$m) [Fig. 2 (a)]. Co-timing of the inside and outside pulses
ensures that the guiding structure forms just ahead of the focal position of
each time slice within the inside pulse. As a result, the guiding structure
does not interfere with the propagation of the inside pulse before it comes to
focus, and only applies guiding to light that is beginning to defocus from
diffraction and self-generated plasma refraction.

The inside pulse begins with high intensity at the back of the pulse envelope
[Fig. 2 (b)], where notably, the focal spot size was compressed to 60% of the
diffraction limit due to additional focusing by the plasma waveguide. As the
high intensity light at the back of the pulse propagates with a small
diameter, the on-axis intensity and electron density increase and steepen
[Fig. 2 (c)-(d)]. The effective duration of the pulse decreases by a factor of
1.6, from 3.3 fs to 2 fs over only $90\,\mu$m. At higher incident intensities,
the spot size was only compressed to 80-100% of the diffraction limit, but the
duration was compressed 4.7$\times$ to 700 as.

The spectrum of the inside pulse evolves rapidly as it propagates along the
optical axis [Fig. 3 (a)]. Significant broadening about the fundamental
frequency was observed within the first $15\,\mu$m of interaction length. The
spectral width at the $10^{-4}$ level more than doubled within only
$45\,\mu$m, and nearly quadrupled within $90\,\mu$m. At this point, almost all
of the light was shifted to the high-frequency side of the spectrum. Although
some ionization-induced odd-harmonic generation was observed [24], comparisons
with photon-kinetics simulations, which do not include nonlinear effects [25,
12], indicate that it did not play a role in the overall spectral broadening.

Figure 3:  (a) Evolution of the power spectral density (PSD) on the optical
axis with interaction length ($z$) where $\omega_{0}$ is the frequency
corresponding to the peak wavelength of the incident pulse,
$\lambda_{0}=400\,$nm. The black line indicates the position of the short-pass
filter applied to all light passing through $z=90\,\mu\mbox{m}$ to obtain the
pulse shown in (b). The simulation parameters were the same as in Fig. 2,
except that the inside pulse intensity was $4.0\times 10^{16}\,\mbox{W
cm}^{-2}$ and the outside pulse intensity was $4.8\times 10^{16}\,\mbox{W
cm}^{-2}$.  Figure 4:  Spectrograms obtained by wavelet transform of the
inside pulse electric field on the optical axis for a SFF pulse at (a) the
source input plane, (b) 15 $\mu$m, (c) 45 $\mu$m, and (d) 90 $\mu$m for the
same parameters as Fig. 3. For comparison, the spectrograms at higher density
($n_{0}=6.5\times 10^{20}\,\mbox{cm}^{-3}$) are shown for (e) a SFF pulse with
$v_{f}=1.030\,v_{g0}=0.8740\,c$ and (f) a standard Gaussian pulse. The color
scale in each plot is normalized to the peak intensity. On-axis intensity
profiles (black) are normalized to $10^{17}\,\mbox{W cm}^{-2}$.

The total duration of the pulse was compressed considerably during the
interaction, and spectral filtering reveals an even shorter high-frequency
structure [Fig. 3 (b)]. Applying a simple short-pass filter to the output of
the photon accelerator ($z=90\,\mu$m) isolates a spatially coherent, extreme
ultraviolet pulse. The pulse, containing only wavelengths shorter than
$100\,\mbox{nm}$ and 0.16% of the total incident pulse energy has a duration
of 1.5 optical cycles, or $384\,\mbox{as}$ intensity full-width at half-
maximum — 1.2 times the transform limit for the filtered bandwidth. With a
nearly flat quadratic phase in the transverse direction, the pulse has ideal
focusability. Such a pulse could be used to drive or probe an experiment
without any further manipulation.

The formation of an optical shock drives the extreme refractive index
gradients that are responsible for the rapid spectral broadening and temporal
compression of the pulse [Fig. 4]. The intensity of the incident pulse peaks
at the back of the pulse envelope where the first focus occurs [Fig. 4(a)],
and the electron density gradient created by this peak induces a frequency up-
shift of the photons in the initial focus. These photons experience an
increase in group velocity due to the anomalous (negative) group velocity
dispersion in a plasma, _i.e._ , $v_{g}=c(1-\omega_{p}^{2}/\omega^{2})^{1/2}$.
Higher frequencies, initially generated at the back of the pulse, catch up to
the un-shifted downstream photons, causing a local increase in the intensity
and bandwidth, which creates the observed self-steepening and temporal
compression.

After 15$\,\mu$m [Fig. 4 (b)], the bandwidth temporally overlapped with the
intensity peak at the fundamental frequency has increased. The peak intensity
has steepened on the leading edge and moved forward within the pulse envelope.
By 45$\,\mu$m [Fig. 4 (c)], significant local spectral broadening and self-
steepening has doubled the intensity. At 90$\,\mu$m [Fig. 4 (d)], a
temporally-localized, extremely broadband signal has developed. Notably the
intensity increases from near zero to its peak in half an optical cycle,
indicating that a strong optical shock has formed on the leading edge of the
pulse. This time-localized spectral broadening is a unique feature of the
self-shocked photon acceleration regime that facilitates attosecond pulse
generation without any post-accelerator phase compensation.

For an initial density 4.3$\times$ larger than in the previously discussed
simulations, the shock catastrophically collapsed over less than
$60\,\mu\mbox{m}$ [Fig. 4 (e)]. The spectrogram can be considered a time-
velocity phase space representation, and the observed “folding” of the
spectrogram, such that non-adjacent frequencies (velocities) overlap in time
indicates optical wavebreaking [26, 27]. A strongly modulated intensity
profile developed on the leading edge of the pulse, and distinct peaks emerged
in the spectrum due to nonlinear four wave mixing. These two effects terminate
the self-steepening and spectral broadening of the main intensity peak by
extracting energy and causing deleterious plasma formation.

Figure 4 (f) demonstrates the limitations of a standard 15 fs Gaussian pulse
that was focused at $z=10\,\mu$m. After $40\,\mu$m, the leading edge of the
pulse shifted from 400 nm to 370 nm, but the trailing edge completely diverged
from the optical axis due to ionization refraction and diffraction. Refraction
and diffraction reduced the peak intensity by $>4\times$, such that ionization
and photon acceleration ceased.

Self-steepening of Gaussian pulses was observed in simulations, but the effect
was weak compared to the SFF pulse. As photons frequency shifted at the peak
of the Gaussian envelope, their increase in group velocity caused them to
temporally overlap with the lower intensity time-slices ahead, which were
already diffracting out of the focus. This resulted in a small increase in
local bandwidth, but no increase in the intensity, such that self-steepening
was negligible. For the SFF pulse, photons accelerated in the initial focus
were immediately injected into a new high-intensity focus. Each subsequent
time slice that came to focus had the original intensity and bandwidth of the
pulse, in addition to the accelerated photons, leading to a coherent buildup
of intensity and bandwidth. The accelerating gradient was continually pumped
by a higher, sharper intensity peak until it collapsed due to wavebreaking.

Figure 5:  Parameter scaling of focal velocity $v_{f}$ and initial density
$n_{0}$ for a $90\,\mu$m accelerator. $\eta$ is the energy efficiency to
produce all light at frequencies above the frequency indicated on the
horizontal axis.

The exact combination of density and flying focus velocity must be optimized
to deliver the greatest bandwidth increase [Fig. 5]. Clearly, $v_{f}\gtrsim
v_{g0}$ is optimal for all densities. For $v_{f}<v_{g0}$, photons focused to
shorter focal lengths are contained in earlier time slices of the pulse
envelope, and plasma formation prevents the ionization front from propagating
continuously over a long distance [28, 29, 30]. For $v_{f}\gg v_{g0}$, photons
up-shifted at the start of the accelerator do not obtain a large enough group
velocity to catch up to the main intensity peak, so a strong shock cannot
form.

A target with higher initial density has the potential to reach a higher
maximum electron density, in principle supporting sharper density gradients
and faster frequency shifting. However, a correspondingly larger initial
electron density also yields greater group velocity dispersion and a smaller
nominal group velocity, which leads to a shorter wave-breaking distance. At
large initial electron density, for the observed optimum $v_{f}\gtrsim
v_{g0}$, the initial intensity peak forms a sharp density gradient, and
dispersion is strong, quickly accelerating photons to $v_{g}\gg v_{f}$. A
shock forms and collapses over a short distance since even a small increase in
frequency results in a large increase in group velocity, and the initial
$v_{f}\ll c$ is easy to overtake for a photon of higher frequency.

At smaller initial densities, a shallower initial density gradient formed, and
dispersion was weak, so the shock was formed and sustained over a longer
distance. Here a large increase in frequency resulted in a much smaller
increase in group velocity, and since $v_{f}\lesssim c$, the shifted photons
took longer to catch up to, and then dephase from the intensity peak at the
fundamental frequency. The low initial density case leads to larger overall
spectral broadening because the shock can build up a density profile with a
sharper scale length, sustain its propagation over a longer distance and
remain in phase with accelerated photons over a longer distance than in the
high density case. Notably, the spectrum was less sensitive to the exact
$v_{f}$ at higher densities, since initial frequency shifts were larger and
dispersion was more significant.

In summary, a new approach to coherent, broadband radiation generation in the
extreme ultraviolet, enabled by the combination of state-of-the-art pulse
shaping methods, was proposed and demonstrated computationally. Optical shock-
enhanced photon acceleration resulted in coherent spectral broadening to
extreme ultraviolet frequencies over a fraction of the distance required in a
typical Kerr-based supercontinuum source. Self-compression of the incident
pulse was significant, and ideal attosecond pulses could be isolated with a
simple short-pass filter. Such a pulse could be useful for enabling a single-
shot version of the coherence tomography application discussed in Ref. [10]
and providing sub-fs time resolution to observe transient effects. A photon
accelerator based on SFF driven plasma could be optimized further to deliver
higher frequencies, shorter pulses and better efficiency. By tailoring the
initial axial density profile of the target gas and the spatiotemporal shaping
together, a shock could be generated at higher densities, accelerating and
then stabilizing over long-distances at lower densities [31, 32], which could
enable the generation of shorter extreme ultraviolet wavelengths, and possibly
a table top-source of coherent soft x-rays.

###### Acknowledgements.

This material is based upon work supported by the Office of Fusion Energy
Sciences under Award Number DE-SC0019135, the Department of Energy National
Nuclear Security Administration under Award Number DE-NA0003856, the
University of Rochester, and the New York State Energy Research and
Development Authority.This report was prepared as an account of work sponsored
by an agency of the U.S. Government. Neither the U.S. Government nor any
agency thereof, nor any of their employees, makes any warranty, express or
implied, or assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product, or process
disclosed, or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial product, process, or
service by trade name, trademark, manufacturer, or otherwise does not
necessarily constitute or imply its endorsement, recommendation, or favoring
by the U.S. Government or any agency thereof. The views and opinions of
authors expressed herein do not necessarily state or reflect those of the U.S.
Government or any agency thereof. The authors would like to thank D. Turnbull
and J. L. Shaw for fruitful discussions and support.

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