# Spatiotemporal control of high-intensity laser pulses with a plasma lens

D. Li<EMAIL_ADDRESS>Gérard Mourou Center for Ultrafast Optical Science,
University of Michigan, Ann Arbor, Michigan 48109, USA  K.G. Miller
University of Rochester, Laboratory for Laser Energetics, Rochester, New York
14623-1299, USA  J.R. Pierce  Department of Physics and Astronomy, University
of California, Los Angeles, California 90095, USA  W.B. Mori  Department of
Physics and Astronomy, University of California, Los Angeles, California
90095, USA  A.G.R. Thomas  Gérard Mourou Center for Ultrafast Optical Science,
University of Michigan, Ann Arbor, Michigan 48109, USA  J.P. Palastro
<EMAIL_ADDRESS>University of Rochester, Laboratory for Laser
Energetics, Rochester, New York 14623-1299, USA

###### Abstract

Spatiotemporal control encompasses a variety of techniques for producing laser
pulses with dynamic intensity peaks that move independently of the group
velocity. This controlled motion of the intensity peak offers a new approach
to optimizing laser-based applications and enhancing signatures of fundamental
phenomena. Here, we demonstrate spatiotemporal control with a plasma optic. A
chirped laser pulse focused by a plasma lens exhibits a moving focal point, or
“flying focus,” that can travel at an arbitrary, predetermined velocity.
Unlike currently used conventional or adaptive optics, a plasma lens can be
located close to the interaction region and can operate at an orders of
magnitude higher, near-relativistic intensity.

##  I Introduction

Spatiotemporal pulse shaping uses advanced optical techniques to construct
laser pulses with dynamic and controllable properties Longhi (2003); Kondakci
and Abouraddy (2017); Sainte-Marie  _et al._ (2017); Froula  _et al._ (2018);
Kondakci and Abouraddy (2019); Li and Kawanaka (2020a, b); Palastro  _et al._
(2020); Jolly  _et al._ (2020); Caizergues  _et al._ (2020); Simpson  _et al._
(2022); Besieris and Saari (2022); Yessenov  _et al._ (2022); Pierce  _et al._
(2023); Ramsey  _et al._ (2023); Liang  _et al._ (2023); Ambat  _et al._
(2023); Pigeon  _et al._ (2023). The laser pulses created with these
techniques can exhibit an intensity peak that travels faster than the speed of
light Sainte-Marie  _et al._ (2017); Froula  _et al._ (2018), accelerates
Liang  _et al._ (2023); Ambat  _et al._ (2023), or even oscillates Sainte-
Marie  _et al._ (2017); Ambat  _et al._ (2023), all while maintaining a near-
constant profile over distances far greater than a Rayleigh range. The
flexibility to control the motion of the peak intensity has provided new
opportunities to optimize laser-based applications and enhance signatures of
fundamental phenomena Turnbull  _et al._ (2018); Palastro  _et al._ (2018,
2020); Caizergues  _et al._ (2020); Howard  _et al._ (2019); Di Piazza (2021);
Ramsey  _et al._ (2022); Formanek  _et al._ (2022); Kabacinski  _et al._
(2023); Simpson  _et al._ (2023). Nevertheless, many of these opportunities
require intensities that would damage the conventional and adaptive optics
used to structure the laser pulse.

Plasma optics, having been ionized, can withstand orders of magnitude higher
intensities than conventional or adaptive optics Durfee and Milchberg (1993);
Malkin  _et al._ (1999); Ping  _et al._ (2000); Hubbard  _et al._ (2002);
Doumy  _et al._ (2004); Lévy  _et al._ (2007); Katzir  _et al._ (2009);
Nakatsutsumi  _et al._ (2010); Palastro  _et al._ (2015); Turnbull  _et al._
(2016); Lehmann and Spatschek (2016); Leblanc  _et al._ (2017); Qu  _et al._
(2017); Vieux  _et al._ (2017); Kirkwood  _et al._ (2018); Peng  _et al._
(2021); Edwards  _et al._ (2022); Edwards and Michel (2022). The refractive
index in a plasma depends on the electron density and the frequency of the
laser pulse. As a result, spatial variation, temporal evolution, or
nonlinearity in the electron density can be used to reflect Doumy  _et al._
(2004); Lévy  _et al._ (2007), refract Hubbard  _et al._ (2002); Katzir  _et
al._ (2009); Palastro  _et al._ (2015), diffract Leblanc  _et al._ (2017);
Kirkwood  _et al._ (2018); Edwards and Michel (2022), disperse, frequency
convert Wilks  _et al._ (1989); Peng  _et al._ (2021); Sandberg and Thomas
(2023), or amplify laser pulses Malkin  _et al._ (1999); Ping  _et al._
(2000); Vieux  _et al._ (2017). In fact, several experiments already make
routine use of plasma optics based on these processes: plasma gratings tune
the symmetry of implosions at the National Ignition Facility Michel  _et al._
(2009); Glenzer  _et al._ (2010); plasma waveguides extend the interaction
lengths in laser wakefield accelerators Geddes  _et al._ (2004); Miao  _et
al._ (2022); and plasma mirrors enhance the intensity contrast in ultrashort
pulse lasers Doumy  _et al._ (2004); Lévy  _et al._ (2007).

Figure 1:  A chirped laser pulse focused by a plasma lens results in a moving,
or “flying,” focus with an intensity peak that moves independently of the
group velocity. The plasma lens focuses different frequencies to different
longitudinal locations, while the chirp controls the arrival time of the
frequencies at these locations. In this case, the velocity of the moving focus
is superluminal ($v_{f}>c$).

Here, we demonstrate that a plasma optic can be used for the spatiotemporal
control of high-intensity laser pulses. A preformed plasma channel functions
as a thick, chromatic lens that focuses different frequencies in a laser pulse
to different locations along the propagation axis. The chirp of the laser
pulse determines the arrival time of the frequencies at these locations. The
time-dependent focusing of the pulse produces a moving focal point with an
arbitrary velocity that can be tuned by adjusting the chirp (Fig. 1). This
configuration is a plasma-based version of the original, chromatic flying
focus Froula  _et al._ (2018), which has been proposed for a range of
experiments, including Raman amplification Turnbull  _et al._ (2018), photon
acceleration Howard  _et al._ (2019); Franke  _et al._ (2021), nonlinear
Thomson scattering Ramsey  _et al._ (2022), and vacuum birefringence Formanek
_et al._ (2023). However, unlike the diffractive optic used in the original
flying focus, a plasma lens can be placed close to the interaction region and
can operate at an orders of magnitude higher, near-relativistic intensity.

The remainder of this article begins with a model for the propagation of a
laser pulse focused (or guided) by a plasma lens (Section II). The model is
general enough to describe the focusing of arbitrary, space-time structured
laser pulses, with or without orbital angular momentum. The model predicts
that the chromatic aberration of the plasma lens and the chirp of a laser
pulse can be used to produce an intensity peak with a specified, constant
velocity (Section III). The plasma lens is designed to produce an extended
focal region, while avoiding resonance absorption and mitigating parametric
instabilities (Section IV). Particle-in-cell simulations based on this design
validate the model for moderate intensities and determine the intensities and
durations at which the plasma lens ceases to operate as expected (Section V).
The article concludes with a summary of the results and a discussion of future
prospects (Section VI).

##  II Plasma Lens

Consider a linearly polarized laser pulse propagating in the positive
$\hat{\mathbf{z}}$ direction. The pulse is normally incident on a preformed
plasma channel with an entrance and exit located at $z=0$ and $z=L_{p}$,
respectively. The transverse electric field of the pulse can be expressed as a
superposition of its frequency components:

$E(\mathbf{x},\xi)=\frac{1}{4\pi}\int
e^{-i\omega\xi}\tilde{E}(\mathbf{x},\omega)d\omega+\mathrm{c.c.},$ (1)

where $\xi=t-z/c$ is the moving-frame coordinate. At the entrance to the
channel, the transverse profile of each component is given by

$\displaystyle\tilde{E}(\mathbf{x}_{\perp},z=0,\omega)$
$\displaystyle=\sum_{q,\ell}\alpha_{q,\ell}A_{I}\Bigl{(}\frac{\sqrt{2}r}{w_{I}}\Bigr{)}^{|\ell|}L^{|\ell|}_{q}\Bigl{(}\frac{2r^{2}}{w^{2}_{I}}\Bigr{)}$
(2) $\displaystyle\text{exp}\left[-\frac{r^{2}}{w_{I}^{2}}+\frac{i\omega
r^{2}}{2cR_{I}}+i\ell\theta+i\phi_{I}\right],$

where $r=(x^{2}+y^{2})^{1/2}$, $\theta$ is the azimuth, $L_{q}^{|\ell|}$ is a
generalized Laguerre polynomial with radial and orbital angular momentum mode
numbers $q$ and $\ell$, and $\alpha_{q,\ell}$ quantifies the projection of
each mode onto the initial profile. The incident amplitude $A_{I}$, phase
$\phi_{I}$, spot size $w_{I}$, and radius of curvature $R_{I}$ may all depend
on frequency.

Each frequency component of the pulse will refract from the plasma, advance in
phase, and diffract by a different amount. This frequency-dependent evolution
is described by the paraxial wave equation

$\left(2i\omega\frac{\partial}{\partial
z}+c\nabla^{2}_{\perp}\right)\tilde{E}(\mathbf{x},\omega)=\frac{1}{c}\omega_{p}^{2}(\mathbf{x})\tilde{E}(\mathbf{x},\omega),$
(3)

where $\omega_{p}^{2}(\mathbf{x})=e^{2}n(\mathbf{x})/m_{e}\varepsilon_{0}$ is
the square of the plasma frequency and $n(\mathbf{x})$ is the electron
density. Here, the plasma response is assumed to be linear, such that
$n(\mathbf{x})$ is independent of $E$ (see Section V for particle-in-cell
simulations that include nonlinear effects). The plasma channel is modeled
using a parabolic density profile

$n(\mathbf{x})=n_{0}+\frac{1}{2}n_{2}r^{2}$ (4)

for $0\leq z\leq L_{p}$, and $n(\mathbf{x})=0$ otherwise. With this profile,
the refractive index in the plasma,
$\mu(\mathbf{x},\omega)=[1-\omega_{p}^{2}(\mathbf{x})/\omega^{2}]^{1/2}$ has
an on-axis maximum, which bends the “rays” of the pulse towards the optical
axis ($r=0$) like a lens.

The transverse profile of each frequency component at any $z>0$ can be found
by solving the paraxial wave equation with the initial condition in Eq. (2).
The solution is

$\displaystyle\tilde{E}(\mathbf{x},\omega)$
$\displaystyle=\sum_{q,\ell}\alpha_{q,\ell}A(z)\Bigl{[}\frac{\sqrt{2}r}{w(z)}\Bigr{]}^{|\ell|}L^{|\ell|}_{q}\Bigl{[}\frac{2r^{2}}{w^{2}(z)}\Bigr{]}$
(5) $\displaystyle\text{exp}\left[-\frac{r^{2}}{w(z)^{2}}+\frac{i\omega
r^{2}}{2cR(z)}+i\ell\theta+i\phi^{q,\ell}(z)\right].$

Within the plasma channel ($0\leq z\leq L_{p}$), the frequency-dependent
amplitude, phase, spot size, and radius of curvature are given by

$\displaystyle A(z)$ $\displaystyle=\frac{w_{I}}{w(z)}A_{I}$ (6)
$\displaystyle\phi^{q,\ell}(z)$
$\displaystyle=\phi_{I}-\frac{\omega_{p0}^{2}}{2c\omega}z-(2q+\ell+1)\text{atan}\left[\frac{w_{I}^{2}Z_{m}}{w_{m}^{2}R_{I}}+\Big{(}\frac{w_{m}^{2}}{w_{I}^{2}}+\frac{w_{I}^{2}Z_{m}^{2}}{w_{m}^{2}R_{I}^{2}}\Big{)}\tan\Big{(}\frac{z}{Z_{m}}\Big{)}\right]$
(7) $\displaystyle w(z)$
$\displaystyle=w_{I}\left[1+\frac{Z_{m}}{R_{I}}\sin\Big{(}\frac{2z}{Z_{m}}\Big{)}+\Big{(}\frac{Z_{m}^{2}}{R_{I}^{2}}+\frac{w_{m}^{4}}{w_{I}^{4}}-1\Big{)}\sin^{2}\Big{(}\frac{z}{Z_{m}}\Big{)}\right]^{1/2}$
(8) $\displaystyle R(z)$
$\displaystyle=R_{I}\frac{w^{2}(z)}{w_{I}^{2}}\left[\cos\Big{(}\frac{2z}{Z_{m}}\Big{)}+\frac{R_{I}}{2Z_{m}}\Big{(}\frac{Z_{m}^{2}}{R_{I}^{2}}+\frac{w_{m}^{4}}{w_{I}^{4}}-1\Big{)}\sin\Big{(}\frac{2z}{Z_{m}}\Big{)}\right]^{-1},$
(9)

where $w_{m}=(8c^{2}/\omega_{p2}^{2})^{1/4}$ is the ‘matched’ spot size of the
plasma channel, $Z_{m}=\omega w_{m}^{2}/2c$ is the Rayleigh range associated
with the matched spot size, $\omega_{p0}^{2}=e^{2}n_{0}/m_{e}\varepsilon_{0}$,
and $\omega_{p2}^{2}=e^{2}n_{2}/m_{e}\varepsilon_{0}$.

After the plasma channel ($z>L_{p}$), the frequency-dependent amplitude,
phase, spot size, and radius of curvature can be expressed in terms of their
values at $z=L_{p}$. Denoting a quantity at the exit of the plasma channel
with the subscript $l$ [e.g., $A_{l}\equiv A(L_{p})$], one finds

$\displaystyle A(z)$ $\displaystyle=\frac{w_{l}}{w(z)}A_{l}$ (10)
$\displaystyle\phi^{q,\ell}(z)$
$\displaystyle=\phi^{q,\ell}_{l}-(2q+\ell+1)\text{atan}\Big{(}\frac{z-L_{p}}{Z_{l}}\Big{)}$
(11) $\displaystyle w(z)$
$\displaystyle=w_{l}\left[\Big{(}\frac{z-L_{p}+R_{l}}{R_{l}}\Big{)}^{2}+\Big{(}\frac{z-L_{p}}{Z_{l}}\Big{)}^{2}\right]^{1/2}$
(12) $\displaystyle R(z)$
$\displaystyle=R_{l}\frac{w^{2}(z)}{w_{l}^{2}}\left[1+\Big{(}1+\frac{R_{l}^{2}}{Z_{l}^{2}}\Big{)}\Big{(}\frac{z-L_{p}}{R_{l}}\Big{)}\right]^{-1},$
(13)

where $Z_{l}=\omega w_{l}^{2}/2c$. Equations (10)–(13) describe the focusing
(or defocusing) of an arbitrary Laguerre-Gaussian mode by a lens with a focal
length

$f_{l}=-\frac{R_{l}}{1+R_{l}^{2}/Z_{l}^{2}}$ (14)

to a minimum spot size

$w_{f}=\frac{2cf_{l}}{\omega
w_{l}}\bigg{(}\frac{2}{1+\sqrt{1-4f_{l}^{2}/Z_{l}^{2}}}\bigg{)}$ (15)

in the plane $z=L_{p}+f_{l}$. Thus, a plasma channel can be used as a lens.

For nearly all parameters of interest, $|R_{I}|\gg Z_{I}\gg Z_{m}$ and
$Z_{l}\gg|R_{l}|$ (see Appendix A). Under these conditions, the focal length
and focused spot size of the plasma lens reduce to the relatively simple
expressions

$\displaystyle f_{l}(\omega)$ $\displaystyle\approx\frac{\omega
w_{m}^{2}}{2c}\cot\Big{(}\frac{2cL_{p}}{\omega w_{m}^{2}}\Big{)}$ (16)
$\displaystyle w_{f}(\omega)$
$\displaystyle\approx\frac{w_{m}^{2}}{w_{I}}\left|\csc\Big{(}\frac{2cL_{p}}{\omega
w_{m}^{2}}\Big{)}\right|,$ (17)

for $j\pi\leq L_{p}/Z_{m}\leq(j+\tfrac{1}{2})\pi$ and $j$ an integer, which
ensures $f_{l}>0$. If in addition $L_{p}\ll Z_{m}$, the laser pulse does not
appreciably diffract or refract within the plasma lens, $w_{l}\approx w_{I}$,
and Eqs. (16) and (17) further reduce to $f_{l}(\omega)\approx
Z_{m}^{2}/L_{p}$ and $w_{f}(\omega)\approx 2cf_{l}/\omega w_{l}$. This is the
“thin” lens limit. Otherwise, the plasma lens is considered “thick.”
Regardless of the thickness, the focal length of the plasma lens depends on
frequency, i.e., the lens is chromatic.

##  III Plasma Lens Flying Focus

The plasma lens focuses each frequency component of the laser pulse to a
different longitudinal location $z_{f}(\omega)=L_{p}+f_{l}(\omega)$. This
produces an extended focal range with a length determined by the minimum and
maximum frequencies:

$L_{f}=f_{l}(\omega_{\text{max}})-f_{l}(\omega_{\text{min}}).$ (18)

Each frequency arrives at its focal location at a different time
$t_{f}(\omega)$. The focal time is the sum of two contributions: the relative
timing of each frequency within the laser pulse at the exit of the plasma
lens, i.e., $\partial_{\omega}\phi^{q,\ell}_{l}(\omega)$, and the time it
takes each frequency to travel a distance $f_{l}(\omega)$. In total,

$t_{f}(\omega)=\frac{1}{c}f_{l}(\omega)+\partial_{\omega}\phi^{q,\ell}_{l}(\omega).$
(19)

The frequency-dependent focal location and time results in a moving focal
point with a velocity

$v_{f}(\omega)=\frac{dz_{f}}{d\omega}\Big{(}\frac{dt_{f}}{d\omega}\Big{)}^{-1}=c\left[1+c\Big{(}\frac{\partial\omega}{\partial
f_{l}}\Big{)}\Big{(}\frac{\partial^{2}\phi^{q,\ell}_{l}}{\partial\omega^{2}}\Big{)}\right]^{-1}.$
(20)

The focal velocity can be tuned by adjusting the properties of the plasma
lens, the mode numbers, or the initial spectral phase of the laser pulse
$\phi_{I}(\omega)$.

Up to this point, the analysis has considered a laser pulse with an arbitrary
frequency spectrum. For the remainder, it is convenient to consider a laser
pulse characterized by a “central” frequency $\omega_{0}$ and spectral width
$\tau^{-1}\ll\omega_{0}$. When the conditions $|R_{I}|\gg Z_{I}\gg Z_{m}$ and
$Z_{l}\gg|R_{l}|$ are satisfied (see Appendix A), the spectral phase at the
exit of the plasma lens is independent of $q$ and $\ell$:

$\phi^{q,\ell}_{l}(\omega)\approx\phi_{I}(\omega)-\frac{\omega_{p0}^{2}}{2c\omega}L_{p}\equiv\phi_{l}(\omega),$
(21)

where constant phase terms have been dropped. Because the focal velocity
depends on $\partial_{\omega}^{2}\phi_{l}$, an initial second-order spectral
phase

$\phi_{I}(\omega)=\frac{1}{2}\phi_{2}(\omega-\omega_{0})^{2}$ (22)

is the simplest phase that provides control over the velocity. This is
equivalent to chirping the laser pulse. More specifically, one can write
$\phi_{2}=\hat{\eta}\tau^{2}/2$, where the chirp parameter $\hat{\eta}$
quantifies the temporal elongation of the pulse.

Figure 2:  Density contours of the plasma lens (grey scale) and intensity
contours (color scale) of a laser pulse traversing the plasma lens. For this
design, the plasma lens is “thick,” and the spot size of the laser pulse
decreases appreciably from the entrance to the exit of the lens (left to
right). The incident laser pulse has a normalized amplitude $a_{I}=0.5$ and
chirp parameter $\eta=-5$. All other parameters can be found in Table I.

The plasma contribution to the spectral phase [Eq. (21)] can be compensated by
the initial spectral phase $\phi_{I}$. To second order in $\omega-\omega_{0}$,

$\phi_{l}(\omega)\approx\frac{1}{4}\left(\hat{\eta}-\frac{2\omega_{p0}^{2}L_{p}}{c\omega_{0}^{3}\tau^{2}}\right)\tau^{2}(\omega-\omega_{0})^{2},$
(23)

where constant and linear phase terms have been dropped (the latter would only
result in an overall temporal delay). Thus, setting the chirp parameter
$\hat{\eta}=\eta+2\omega_{p0}^{2}L_{p}/c\omega_{0}^{3}\tau^{2}$ compensates
the second-order phase acquired in the plasma lens and results in a spectral
phase determined solely by $\eta$. Higher-order phase could also be
compensated by introducing higher-order terms in $\phi_{I}$. However, for the
cases considered here, the second-order phase is already small:
$2\omega_{p0}^{2}L_{p}/c\tau^{2}\omega_{0}^{3}\ll 1$.

With the compensated spectral phase [Eq. (23)], the chromatic focusing of the
chirped pulse results in the focal velocity

$\frac{v_{f}(\omega)}{c}=\left[1+\frac{\eta(\frac{c\tau}{w_{m}})^{2}\frac{Z_{m}}{L_{p}}\sin^{2}(\frac{L_{p}}{Z_{m}})}{1+\frac{Z_{m}}{2L_{p}}\sin(\frac{2L_{p}}{Z_{m}})}\right]^{-1}.$
(24)

The focal velocity depends on frequency through the matched Rayleigh range
$Z_{m}=\omega w_{m}^{2}/2c$. To first approximation, this dependence can be
ignored, and $\omega$ can be replaced by $\omega_{0}$ because
$\omega_{0}\tau\ll 1$. As a result, the focal velocity is nearly constant
throughout the focal range $L_{f}$.

For large values of the chirp parameter ($\eta\gg 1$), the stationary phase
approximation can be used in Eq. (1) to find the electric field of the laser
pulse in the time domain:

$E(\mathbf{x},\xi)\approx\frac{\text{sgn}(\eta)}{(4\pi|\eta|\tau^{2})^{1/2}}\tilde{E}(\mathbf{x},\omega_{s})e^{-i(\omega_{s}\xi-\frac{\pi}{4})}+\mathrm{c.c.},$
(25)

where $\omega_{s}=\omega_{0}+2\xi/\eta\tau^{2}$. The time-domain envelope of
the laser pulse $\tilde{E}(\mathbf{x},\omega_{s})$ is given by Eq. (2) with
$\omega$ replaced by $\omega_{s}$. Using Eqs. (6), (10), and the replacement
$\omega\rightarrow\omega_{s}$ in Eq. (12) yields the temporal profile of the
moving focus within the focal range:

$I_{f}\propto\frac{w_{I}^{2}A_{I}^{2}}{w^{2}(z)}\approx\frac{w_{I}^{2}A_{I}^{2}}{w_{f}^{2}}\left[1+\Big{(}\frac{t-z/v_{f}-\zeta_{0}}{\tau_{f}}\Big{)}^{2}\right]^{-1},$
(26)

where

$\tau_{f}=\left|\frac{c-v_{f}}{cv_{f}}\right|\frac{\omega_{0}w_{f}^{2}}{2c}$
(27)

is the duration of the intensity peak and
$\zeta_{0}=(v_{f}-c)[L_{p}+f_{l}(\omega_{0})]/cv_{f}$ is the focal time with
respect to the coordinate $\zeta=t-z/v_{f}$ for $\omega=\omega_{0}$. The
maximum value of the moving intensity peak will be approximately constant
within the focal range if the spectral amplitude $A_{I}(\omega)$ is constant
for $\omega_{\mathrm{min}}\leq\omega\leq\omega_{\mathrm{max}}$.

##  IV Design Considerations

A distinguishing property of a flying focus is that the peak intensity can
maintained over a distance greater than the Rayleigh range of the focused
spot. To take advantage of this property and to ensure that the focal
trajectory can be clearly identified, the plasma lens should be designed so
that

$\displaystyle L_{f}>\frac{\omega_{0}w_{f}^{2}}{2c}.$ (28)

In addition, the parameters for the plasma lens should be chosen to avoid
resonance absorption and mitigate parametric instabilities. This can be
accomplished by keeping the maximum density experienced by the pulse
$n_{\text{max}}\sim n_{2}w_{I}^{2}$ well below the critical density
$n_{cr}=m_{e}\varepsilon_{0}\omega_{0}^{2}/e^{2}$. Equation (28) can be re-
expressed in terms of these densities and the length of the plasma lens as
follows:

$\displaystyle\left(\frac{\Delta\omega
L_{p}}{2c}\right)\left(\frac{n_{\text{max}}}{n_{cr}}\right)>1,$ (29)

where $\Delta\omega\equiv\omega_{\text{max}}-\omega_{\text{min}}$ is the total
bandwidth of the laser pulse and
$L_{f}\approx\Delta\omega\partial_{\omega}f_{l}$ has been used. Preventing
instabilities like two-plasmon decay and absolute stimulated Raman scattering
requires $n_{\text{max}}/n_{cr}\lesssim 1/4$. Thus, for a fixed bandwidth
$\Delta\omega$, the length of the focal region relative to the Rayleigh range
is determined solely by $L_{p}$. Note, however, that (1) $L_{p}$ can only be
increased up to $\pi Z_{m}/2$ before the plasma lens becomes a defocusing lens
[see Eq. (16)], and (2) a longer plasma lens may exacerbate instabilities like
stimulated Raman forward scattering.

With these considerations in mind, a plasma lens was designed to produce
flying foci over a range $L_{f}=\omega_{0}w_{f}^{2}/c$. The parameters,
displayed in Table I, were motivated by commercially available Ti:sapphire
laser systems and experimentally demonstrated plasma channels Durfee and
Milchberg (1993); Durfee  _et al._ (1995). The rightmost column of the table
provides the parameters in normalized units to facilitate scaling the results
presented here to other laser wavelengths or plasma densities. For this
design, $L_{p}\approx Z_{m}$. As a result, the plasma lens is “thick,” and the
spot size evolves significantly within the lens (Fig. 2).

Table 1:  Laser pulse and plasma lens parameters used in the simulations. In the rightmost column, space, time, and density are normalized by $c/\omega_{0}$, $1/\omega_{0}$, and $n_{cr}=m_{e}\varepsilon_{0}\omega_{0}^{2}/e^{2}$. The vacuum wavelength $\lambda_{0}=2\pi c/\omega_{0}$. The chirp parameter $\eta$ was varied to change the focal velocity. Pulse parameters | Value | Normalized
---|---|---
$\lambda_{0}$ | 800 nm | 2$\pi$
$\omega_{0}$ |  $2.4\times 10^{15}$ rad/s | 1
$\Delta\omega$ (FWHM) |  $1.4\times 10^{14}$ rad/s | 0.061
$\tau$ | 21 fs | 51
$R_{I}$ | $\infty$ | $\infty$
$w_{I}$ |  $60$ $\mu$m | 470
Plasma lens parameters | Value | Normalized
$n_{0}$ | $1\times 10^{18}\text{cm}^{-3}$ | 5.7$\times 10^{-4}$
$w_{m}$ |  $12.5$ $\mu$m | 98
$L_{p}$ |  $0.61$ mm | 4.8$\times 10^{3}$
$f_{l}(\omega_{0})$ |  $0.4$ mm | 3.1$\times 10^{3}$
$w_{f}(\omega_{0})$ | 3.1 $\mu$m | $24$
$L_{f}$ |  $77$ $\mu$m | 604
$f/\\#$ | 6.1 | 6.1
Figure 3:  Spatiotemporal profiles of the moving focus within the focal range
for different focal velocities $v_{f}$ (or chirps $\eta$) at $r=0$. The
intensity peak has a near-constant velocity and a maximum intensity that is
consistent with the theory. In each case, the incident laser pulse had a
normalized amplitude $a_{I}=0.5$. The distance from the nominal focus
$f_{l}(\omega_{0})$ is normalized to the Rayleigh range of the focused spot at
the central frequency $Z_{f}\equiv cw_{f}^{2}/2\omega_{0}$.

##  V Simulation Results

The model presented in the previous sections illustrates the salient phenomena
underlying a plasma-lens-based flying focus. Nevertheless, the model neglects
non-paraxial and nonlinear propagation effects, such as stimulated Raman
scattering, self-focusing, ponderomotively driven density modifications, and
the increase in the effective electron mass due to relativistic motion. This
section presents the results of quasi-3D particle-in-cell (PIC) simulations
that include these effects (see Appendix B for details).

For incident intensities up to $I_{I}=5.4\times 10^{17}\;\text{W/cm}^{2}$, the
simulations demonstrate that the plasma-lens-based flying focus works as
designed, validating the model. For incident intensities $I_{I}\gtrsim 2\times
10^{18}\;\text{W/cm}^{2}$, the simulations show that nonlinear effects disrupt
the focusing of the plasma lens and formation of a flying focus. Notably,
these intensities correspond to normalized amplitudes $a_{I}\equiv
eE_{I}/m_{e}c\omega_{0}$ equal to 0.5 and 1.0, respectively, where
$E_{I}=\text{max}[E(\mathbf{x}_{\perp},z=0,\xi)]$, which straddle the
transition from non-relativistic to relativistic electron motion.

The incident laser pulse was initialized in the frequency domain as in Eq. (2)
with a transverse Gaussian profile ($q=\ell=0$). The initial spectral
amplitude had the super-Gaussian profile

$\displaystyle
A_{I}(\omega)=\exp\left\\{-[\tfrac{1}{2}\tau(\omega-\omega_{0})]^{4}\right\\},$
(30)

with a corresponding full width at half maximum
$\Delta\omega=(4/\tau)[\text{ln}(2)/2]^{1/4}$. The relatively flat spectral
amplitude was chosen to ensure a near-constant peak intensity within the focal
region. The initial spectral phase was specified as in Eq. (22) with
$\phi_{2}=\eta\tau^{2}/2$. The on-axis plasma density $n_{0}$ was low enough
that the plasma contributed a negligible second-order phase. All other
parameters can be found in Table I.

Figure 3 demonstrates that a plasma lens can produce a moving intensity peak
with a predesigned velocity $v_{f}$. The panels show the on-axis ($r=0$)
spatiotemporal profile of the intensity peak as it traverses the focal range
for four different chirp values. In each case, the moving focus has a near-
constant velocity as predicted by Eq. (24) (white dashed lines). Note that
with respect to the moving-frame coordinate $\xi=t-z/c$, the intensity peak
follows the trajectory $z=cv_{f}\xi/(c-v_{f})$.

Figure 4 compares the focal velocities calculated from the theory and
simulations for all simulated chirp values. For focal velocities ranging from
negative to positive superluminal, the two are in excellent agreement. Slight
discrepancies can be observed for small values of the chirp parameter
($|\eta|\leq 2$). At these chirps, the effective duration of the moving focus
$\tau_{f}$ [Eq.(27)] becomes larger than the duration of the laser pulse. In
addition, the temporal profile of the laser pulse begins to approach the
transform-limited profile, which no longer resembles the spectral amplitude
(i.e., the stationary phase approximation breaks down). The combination of
these effects makes it difficult to discern the intensity peak of the moving
focus from the inherent intensity peak of the temporal profile.

In Fig. 3, the incident laser pulses had an amplitude $a_{I}=0.5$,
corresponding to an intensity of $I_{I}=5.4\times 10^{17}\;\text{W/cm}^{2}$.
For the larger chirp values, the peak intensity at focus is consistent with
Eqs. (6) and (10): $I_{f}=(w_{I}/w_{f})^{2}I_{I}=2\times
10^{20}\,\text{W/cm}^{2}$. The slightly lower intensity when $\eta=2$ results
from modifications to the temporal profile not captured within the stationary
phase approximation as discussed above (i.e., the temporal profile is
approaching its transform limit).

Figure 4:  Comparison of the focal velocities predicted by the “thick” lens
theory [Eq. (24)] and the PIC simulations as a function of the chirp parameter
$\eta$. The theory and simulations are in excellent agreement when the
effective duration $\tau_{f}$ is larger than the stretched pulse duration
(outside the red shaded area).

As the incident amplitude of the laser pulse is pushed beyond $a_{I}=0.5$,
nonlinear propagation within the plasma lens and the nonlinear plasma response
begin to disrupt the formation of a flying focus. Figure 5 shows the on-axis
intensity profile at $z=L_{p}+f_{l}(\omega_{0})$ normalized to the maximum of
the incident intensity for $a_{I}=$ 0.1, 0.5, and 1.0, and three stretched
pulse durations (chirp values). For the shorter pulse durations ($\eta=-4$ and
$-8$), the $a_{I}=0.1$ and $a_{I}=0.5$ profiles are nearly identical,
indicating that nonlinear effects have not modified the plasma lens focusing.
For the longest pulse duration ($\eta=-12$), the profile of the $a_{I}=0.1$
pulse remains unmodified, while the back half of the $a_{I}=0.5$ pulse has
become distorted. When $a_{I}=1$, this distortion occurs earlier within the
pulse and is already apparent at the intermediate duration $\eta=-8$. At the
longest pulse duration ($\eta=-12$), the entire back half of the pulse has
deteriorated.

Mitigating these nonlinear modifications to the laser pulse requires either
lowering the incident amplitude or shortening the stretched pulse duration. To
shorten the duration for a fixed focal spot size (f-number) and focal velocity
[Eqs. (17) and (24)], one can use the scalings $\tau\rightarrow\chi\tau$,
$\Delta\omega\rightarrow\chi^{-1}\Delta\omega$ , $w_{m}\rightarrow\chi w_{m}$,
$L_{p}\rightarrow\chi^{2}L_{p}$, $w_{I}\rightarrow\chi^{2}w_{I}$, and
$n_{\text{max}}\rightarrow n_{\text{max}}$. With these scalings, the right-
hand side of Eq. (29) $\rightarrow\chi(\Delta\omega
L_{p}n_{\text{max}})/(2cn_{cr})$. This scaling shows that decreasing the
duration ($\chi<1$) makes it more difficult to satisfy Eq. (29). As a result,
lowering the amplitude may be the preferable option for mitigating nonlinear
modifications to the pulse.

Figure 5:  Temporal profiles of the on-axis intensity at the nominal focal
point $z=L_{p}+f_{l}(\omega_{0})$ for three different focal velocities (or
chirps $\eta)$ and incident amplitudes $a_{I}$. In all cases, the intensity is
normalized to the incident intensity $I_{I}$. Pulses with longer durations and
larger amplitudes are more susceptible to modifications due to nonlinear
propagation and plasma evolution.

##  VI Conclusions and Prospects

Plasma optics allow for spatiotemporal control at orders of magnitude higher
intensities than conventional optics. In the specific case considered here, a
chirped laser pulse focused by a plasma lens exhibits a dynamic or “flying”
focus that moves independently of the group or phase velocities. By adjusting
the chirp, the velocity of the moving focus can be varied from sub to
superluminal in either the forward or backward directions. The plasma lens can
be created using the same experimental techniques used to produce plasma
channels and operates as designed up to near-relavistic intensities ($I\approx
6\times 10^{17}\,\mathrm{W/cm^{2}}$).

The parameters of the plasma lens considered were motivated by experimentally
demonstrated plasma channels Durfee and Milchberg (1993); Durfee  _et al._
(1995). With these parameters, the incident intensity $I_{I}=5.4\times
10^{17}\,\mathrm{W/cm^{2}}$ resulted in a focused intensity of
$I_{f}=2.0\times 10^{20}\,\mathrm{W/cm^{2}}$. The focused intensity can be
increased or decreased by scaling the parameters of the plasma lens. For a
fixed incident intensity and focused spot size (or f-number), the scalings
$w_{I}\rightarrow\chi w_{I}$, $w_{m}\rightarrow\chi^{1/2}w_{m}$,
$L_{p}\rightarrow\chi L_{p}$, and $f_{l}\rightarrow\chi f_{l}$ result in a
focused intensity $I_{f}\rightarrow\chi^{2}I_{f}$. As an example, achieving a
plasma-lens-based flying focus with a world-record intensity of $I_{f}=1\times
10^{23}\,\mathrm{W/cm^{2}}$ Yoon  _et al._ (2021) would require a plasma lens
with a maximum radius of $r_{\mathrm{max}}\sim w_{I}\sim 1.2\,\mathrm{mm}$ and
a length of $L_{p}\sim 1.2$ cm.

As an alternative to plasma channels, a plasma lens can also be created by
using the interference of two laser beams to structure the plasma density
through ionization or ponderomotive forces Edwards  _et al._ (2022). These
holographic plasma lenses operate like a diffractive lens and are inherently
chromatic. A chirped laser pulse focused by such a plasma lens would also
produce a flying focus. Future work will consider this alternative and explore
other plasma configurations that can controllably modify the space-time
structure of laser pulses. For instance, a plasma-based version of the
ultrafast flying focus Palastro  _et al._ (2020) or the flying focus X Simpson
_et al._ (2022) could allow for arbitrary focal velocities and an ultrashort-
duration intensity peak that travels distances much greater than a Rayleigh
range.

###### Acknowledgements.

The authors would like to thank J. Vieira, T.T. Simpson, D. Ramsey, and D.H.
Froula for productive discussions. The work of DL and AGRT is supported by the
National Science Foundation under Award Number NSF-2108075 and the Office of
Fusion Energy Sciences under Award Number DE-SC0022109. The work of JRP and
WBM is supported by the LLE subcontract SUB00000211/GR531765, the National
Science Foundation under Award Number NSF-2108970, and the Office of Fusion
Energy Sciences under Award Number DE-NA0004131. The work of KGM and JPP is
supported by the Office of Fusion Energy Sciences under Award Number DE-
SC00215057, the Department of Energy National Nuclear Security Administration
under Award Number DE-NA0004144, the University of Rochester, and the New York
State Energy Research and Development Authority.

##  Appendix A Operating Regime

This appendix justifies the conditions $|R_{I}|\gg Z_{I}\gg Z_{m}$ and
$Z_{l}\gg|R_{l}|$, which were used to simplify the equations derived in
Sections II and III. In any situation of interest, a plasma lens would be
located far from an initial optical assembly. Suppose then that the laser
pulse incident on the plasma lens was originally collimated and subsequently
focused by a conventional lens with a focal length $f_{I}\approx|R_{I}|$
located at $z=-z_{0}<0$. At the entrance to the plasma lens, the spot size
would be

$w_{I}\approx
w_{0}\left[\Big{(}\frac{\Delta}{f_{I}}\Big{)}^{2}+\Big{(}\frac{w_{D}}{w_{0}}\Big{)}^{2}\right]^{1/2},$
(31)

where $w_{0}$ is the spot size incident on the conventional lens,
$\Delta\equiv|f_{I}-z_{0}|\ll f_{I}$ is the distance between the conventional
focus and the entrance to the plasma lens, and $w_{D}=2cf_{I}/\omega w_{0}$ is
the diffraction-limited spot size. If the entrance to the plasma lens is
located within the confocal region of the conventional lens, i.e.,
$\Delta\leq\omega w_{D}^{2}/2c$, then $w_{I}\approx w_{D}$. As a result,
$|R_{I}|/Z_{I}\approx f_{I}/Z_{I}\approx w_{0}/w_{D}$. For the lenses of
interest $w_{0}\gg w_{D}$, such that $|R_{I}|\gg Z_{I}$. If the plasma lens is
outside of the confocal region, i.e., $\Delta>\omega w_{D}^{2}/2c$, then
$w_{I}\approx w_{0}\Delta/f_{I}$. In this case, $|R_{I}|/Z_{I}\approx
f_{I}/Z_{I}\approx(w_{D}/w_{0})(f_{I}/\Delta)^{2}$, which easily satisfies the
first condition when the plasma lens is far from the conventional lens:
$\Delta\ll(w_{D}/w_{0})^{1/2}f_{I}$.

The condition $Z_{I}\gg Z_{m}$ is equivalent to $w_{I}^{2}/w_{m}^{2}\gg 1$.
When $|R_{I}|/Z_{I}\gg 1$, the spot size of the laser pulse can oscillate
between $w_{I}$ and $w_{m}^{2}/w_{I}$, depending on the length of the plasma
lens [Eq. (8)]. A ratio $w_{I}/w_{m}>1$ ensures that the laser pulse exits the
plasma lens with a smaller spot than it entered with. Perhaps more
importantly, the radius of curvature can oscillate in the plasma lens [Eq.
(9)]. These oscillations can result in unintended defocusing of the laser
pulse. When $R_{I}<0$ (e.g., due to preliminary focusing by a conventional
lens), $w_{I}>w_{m}$ ensures that the laser pulse accumulates a stronger
focusing phase over the initial length of the plasma lens. This initial length
$L_{i}$ and the ultimate strength of the focusing phase reach their maximum in
the limit $w_{I}^{2}/w_{m}^{2}\rightarrow\infty$. More specifically,
$L_{i}\rightarrow\pi Z_{m}/2$ and $R\rightarrow 0$ from below. Thus, operating
in a regime where $w_{I}^{2}/w_{m}^{2}\gg 1$ provides a larger range of
available focusing powers and a larger margin on the length of the plasma
before one needs to worry about a sign reversal in the curvature phase.
Finally, for any focusing plasma lens of use, the focused spot size $w_{f}$
will be much smaller than the spot size at the exit of the lens $w_{l}$, such
that $Z_{l}/|R_{l}|=\omega w_{l}^{2}/2c|R_{l}|\approx w_{l}/w_{f}\gg 1$.

##  Appendix B Simulation Details

All of the simulations presented in this work were performed using osiris with
the moving-window and quasi-3D capabilities Fonseca  _et al._ (2002, 2013);
Davidson  _et al._ (2015). The transverse domain consisted of two azimuthal
modes and 600 cells over $240\;\mu$m in the radial direction. The longitudinal
domain was scaled with the chirp to contain the stretched duration of the
laser pulse. The minimum and maximum number of longitudinal cells were 3200
and 9600, corresponding to lengths of 80 and 240 $\mu$m. The longitudinal
resolution was fixed at $\Delta\xi\sim$25 nm to maintain 30 grid points per
the shortest wavelength in the pulse.

The electromagnetic fields and particle motion were evolved with the dual
solver, which ensures accurate dispersion of the waves and eliminates time-
staggering errors in the Lorentz force Li  _et al._ (2021). To accomplish
this, the dual solver employed finite-difference operators with 16
coefficients. A 0.083 fs time step was set to satisfy the Courant condition,
and the simulation duration was $\sim$6 ps. Open boundary conditions were
applied for both the fields and particles. In the region occupied by the
plasma lens, the ions were fixed, and 32 particles per cell were used for the
electrons. The plasma lens had $10\,\mu$m density ramps at its entrance and
exit.

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