# Quantitative Analysis of Shock Wave Dynamics in a Fluid of Light

T. Bienaimé  Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL
Research University, Collège de France, Paris 75005, France  M. Isoard
Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France  Physikalisches
Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3,
D-79104 Freiburg, Germany  Q. Fontaine  Laboratoire Kastler Brossel, Sorbonne
Université, CNRS, ENS-PSL Research University, Collège de France, Paris 75005,
France  A. Bramati  Laboratoire Kastler Brossel, Sorbonne Université, CNRS,
ENS-PSL Research University, Collège de France, Paris 75005, France  A. M.
Kamchatnov  Moscow Institute of Physics and Technology, Institutsky lane 9,
Dolgoprudny, Moscow region, 141700, Russia  Institute of Spectroscopy, Russian
Academy of Sciences, Troitsk, Moscow, 108840, Russia  Q. Glorieux  Laboratoire
Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University,
Collège de France, Paris 75005, France  N. Pavloff  Université Paris-Saclay,
CNRS, LPTMS, 91405, Orsay, France

###### Abstract

We report on the formation of a dispersive shock wave in a nonlinear optical
medium. We monitor the evolution of the shock by tuning the incoming beam
power. The experimental observations for the position and intensity of the
solitonic edge of the shock, as well as the location of the nonlinear
oscillations are well described by recent developments of Whitham modulation
theory. Our work constitutes a detailed and accurate benchmark for this
approach. It opens exciting possibilities to engineer specific configurations
of optical shock wave for studying wave-mean flow interaction.

In many different fields such as acoustics [1], plasma physics [2],
hydrodynamics [3, 4, 5], nonlinear optics [6], ultracold quantum gases [7, 8,
9, 10], the short time propagation of slowly varying nonlinear pulses can be
described discarding the effects of dispersion and dissipation. The prototype
of such an approach is given by the system of equations governing compressible
gas dynamics [11]. This type of treatment typically predicts that, due to
nonlinearity, an initially smooth pulse steepens during its time evolution,
eventually reaching a point of gradient catastrophe. This is the wave-breaking
phenomenon, which results in the formation of a shock wave [12, 13]. If, after
wave breaking, dispersive effects dominate over viscosity, the shock
eventually acquires a stationary nonlinear oscillating structure for which the
width increases with diminishing dissipation [14]. In the case of weak
dissipation the time for reaching a stationary regime can be quite long.
Gurevich and Pitaevskii [15] made a major contribution to the field when they
first realized the interest of studying the evolution of the associated
dynamical structure, now called a dispersive shock wave (DSW). Besides, they
understood that a DSW can be described as a modulated nonlinear traveling wave
and studied in the framework of the Whitham theory of modulations [16].

In the present work we study the propagation of an optical beam in a nonlinear
defocusing medium. Wave-breaking and (spatial or temporal) dispersive shocks
have already been observed in such a setting [17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30]. However, all previous theoretical descriptions of
experimental optical shocks either remained only qualitative or resorted to
numerical simulations for reaching accurate descriptions. Indeed, a realistic
quantitative characterization of the experimental situation requires to take
into account a number of nontrivial effects which sum up to a quite difficult
task. For instance, saturation effects, such as occurring in semiconductor
doped glasses [31] and in photorefractive media [32], can only be taken into
account by using a nonintegrable nonlinear equation, even for a medium with a
local nonlinearity. Besides, both “Riemann invariants” typically vary during
the prebreaking period and this complicates the description of the
nondispersive stage of the pulse spreading, even in a quasi unidimensional
(1D) geometry. Moreover, for realistic initial intensity pulse profiles, the
post-breaking evolution corresponds, at best, to a so-called “quasisimple”
dispersive shock [33], the characterization of which requires an elaborate
extension of the Gurevich-Pitaevskii scheme. Finally, the nonintegrability of
the wave equation significantly complicates the post-breaking description of
the nonlinear oscillations within the shock. Despite these difficulties, it
has recently become possible to combine several theoretical advances [34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45] to obtain a comprehensive treatment of
the nonlinear pulse spreading and the subsequent formation of a dispersive
shock in a realistic setting [46, 47]. In this Letter, we provide a
nonambiguous experimental evidence for the accuracy of this theory with a
precise description of the main features of the shock. This universal and
quantitative benchmark is a major advance for manipulation and engineering of
optical shockwaves.

Figure 1: Sketch of the experimental setup. To create the initial state we
overlap the background and the hump beams on a beam splitter with their
relative phase precisely adjusted such that they interfere constructively.
This state then propagates inside the nonlinear medium consisting of a hot
85Rb vapor cell of length $L$. The insets represent cuts of the relevant
intensity profiles in the plane perpendicular to the direction $z$ of
propagation. The output intensity is recorded on a camera by direct imaging
through two lenses in $4f$ configuration.

We study the propagation of a laser field in a $L=7.5$ cm-long cell filled
with an isotopically pure 85Rb vapor (99% purity) warmed up to a controlled
temperature of $120\,^{\circ}\text{C}$ to adjust the atomic vapor density. We
use a Ti:sapphire laser detuned by $-3.9\,\text{GHz}$ with respect to the
$F=3\rightarrow F^{\prime}$ transition of the D2-line of 85Rb at
$\lambda_{0}=2\pi/k_{0}=780\,\text{nm}$. For such a large detuning, the
natural Lorentzian shape of the line dominates and the Doppler broadening
$k_{0}v\simeq 240\,\text{MHz}$ can be safely neglected. In these experimental
conditions, the system is self defocusing (repulsive photon-photon
interaction) and the transmission through the cell is $60\,\%$. We find that
this medium is well described by local photon-photon interactions, but
contrary to previous works [48, 49, 50, 51, 52] we find it important to take
into account the saturation of the nonlinearity to quantitatively describe the
dynamics of the shock waves.

The input intensity profile is a cross-beam configuration of two vertically
polarized laser beams, both propagating along the axis of the cell (denoted as
O$z$), with their respective phase precisely adjusted such that the two beams
interfere constructively, see Fig. 1. One of the beams (which is denoted the
_hump_) is extended along the $y$ direction and significantly more intense
than the other one (the _background_) which is extended along the $x$
direction. At the entrance of the cell ($z=0$) both beams have an elliptic
Gaussian profile. The background beam has a power $P_{0}$ and waists
$w_{x,0}>w_{y,0}$, whereas the hump has power $P_{1}$ and waists
$w_{x,1}<w_{y,1}$ ; see Supplemental Material [53]. During the initial
nondiffractive stage of evolution, nonlinearity acts as an effective pressure
which favors spreading of the hump in the $x$ direction along which is
initially tighter collimated. Conversely, the low intensity background
experiences almost no spreading and behaves as a pedestal which triggers wave
breaking of the hump during its spreading. Each beam has a maximum entrance
intensity ${\cal I}_{\alpha}=2P_{\alpha}/(\pi w_{x,\alpha}w_{y,\alpha})$
($\alpha=0$ or 1), and we explore the DSW dynamics for a fixed ratio ${\cal
I}_{1}/{\cal I}_{0}$. We work in the deep nonlinear regime, with ${\cal
I}_{1}=20\,{\cal I}_{0}$. This large value corresponds to a wave breaking
distance typically shorter than the cell length, and makes it possible, by
changing the total power $P_{\text{tot}}$ of the beams, to observe several
stages of evolution of the DSW. The total power can be increased up to
$700\,\text{mW}$ and is limited by the laser maximum output power.

Figure 2: (a) Left: experimental profile $\widetilde{I}_{\text{out}}$ taken
for $P_{\text{tot}}=680\,\text{mW}$. Right: Two-dimensional numerical
simulations at the same total entrance power. (b) $x$ and $y$ profiles along
the cuts represented by the two white lines on the two-dimensional profiles
(a). The solid blue line represents the experimental data, the dashed green
line the two-dimensional numerical simulation. On the $x$ profile, the orange
line is a one-dimensional numerical simulation, from Eqs. (2) and (3). (c)
$\widetilde{I}_{\rm out}(x,0)$ for various total beam powers. The color code
is the same as in (b). The vertical pink and gray bars on the right part of
each intensity profile indicate the positions of the solitonic edge of the DSW
and of the first maxima of oscillations within the DSW. The thickness of each
bar represents the experimental uncertainty.

We image the total field intensity $I_{\text{out}}(x,y)$ at the output of the
cell on a camera. In order to minimize the effect of absorption and increase
the visibility of the DSW, we determine the _normalized_ output intensity

$\widetilde{I}_{\text{out}}(x,y)\equiv\frac{I_{\text{out}}(x,y)-I_{\text{out}}^{0}(x,y)}{{\cal
I}_{\text{out}}^{0}},$ (1)

where $I_{\text{out}}^{0}(x,y)$ is the intensity profile at the cell output
when _only_ the background beam propagates through the medium (the hump beam
is blocked). ${\cal I}_{\text{out}}^{0}$ is the maximal value of
$I_{\text{out}}^{0}(x,y)$. $\widetilde{I}_{\text{out}}(x,y)$ is represented in
Fig. 2(a). Our theoretical description relies on only two parameters which
characterize the photon-photon interaction, namely, the Kerr coefficient,
$n_{2}$, and the saturation intensity $I_{\text{sat}}$ [cf. Eq. (2)]. Their
values $n_{2}=1.5\times 10^{-4}$ mm2/W, and $I_{\text{sat}}=0.6$ W/mm2 have
been determined by comparing the experimental results with large-scale 2D
numerical simulations [53]. The excellent agreement reached in Fig. 2
indicates that two effects – saturable nonlinearity and linear absorption –
are the relevant physical ingredients for a theoretical description of our
experiment.

In the regime $w_{1,x}\ll w_{0,x}$ and ${\cal I}_{1}\gg{\cal I}_{0}$ we
consider, the normalized output density $\widetilde{I}_{\text{out}}$ becomes
independent on the precise shape of the background beam. As a result,
$\widetilde{I}_{\text{out}}(x,0)$ can be described by using a simplified 1D
theoretical description, where a hump propagates over a background of uniform
intensity ${\cal I}_{0}$. Within the cell, the complex field amplitude at
$y=0$, denoted as ${\cal A}(x,0,z)\equiv a(x,z)$, then obeys a 1D nonlinear
Schrödinger equation where the position $z$ along the axis of the beam plays
the role of an effective “time” [54]. The equation, once included the
nonlinearity saturation and the linear absorption [32], reads

$i\,\partial_{z}a=-\frac{1}{2n_{0}k_{0}}\partial^{2}_{x}a+\frac{k_{0}\,n_{2}\,|a|^{2}}{1+|a|^{2}/I_{\text{sat}}}\,a-\frac{i}{\Lambda_{\rm
abs}}\,a,$ (2)

where $n_{0}\simeq 1$ is the linear index of refraction and $\Lambda_{\rm
abs}=30$ cm, which corresponds to a 60 % transmission for a cell of length
$L=7.5$ cm. The value of the effective amplitude at the entrance of the cell
is taken as

$a(x,0)=\sqrt{{\cal I}_{0}}+\sqrt{{\cal
I}_{1}}\exp\left(-\frac{x^{2}}{w^{2}_{x,1}}\right),$ (3)

In order to evaluate the accuracy of the mapping to the 1D model of Eq. (2),
we compare in the upper panel of Fig. 2(b) the corresponding value of
$|a(x,L)|^{2}\exp(2L/\Lambda_{\rm abs})/{\cal I}_{0}-1$ with the experimental
$\widetilde{I}_{\rm out}(x,0)$ and with the result of 2D simulations. The
excellent agreement is confirmed in Fig. 2(c) for the whole range of beam
powers $P_{\rm tot}$.

The mapping to a 1D problem enables us to compare our measurements with recent
analytical predictions. In particular, if one neglects the linear absorption
within the cell, for the initial intensity profile (3), wave breaking occurs
at a propagation distance [45]

$z_{\rm\scriptscriptstyle
WB}=4\,\sqrt{\frac{n_{0}I^{*}}{n_{2}}}\,\frac{(1+I^{*}/I_{\rm
sat})^{2}}{3+I^{*}/I_{\rm sat}}\cdot\frac{1}{{\rm
max}\left|\frac{dI(x,0)}{dx}\right|},$ (4)

where $I(x,0)=|a(x,0)|^{2}$ is the entrance intensity and $I^{*}$ is the value
$I(x^{*},0)$ at point $x^{*}$ where $|dI(x,0)/dx|$ reaches its maximum. For
low entrance power, no DSW is observed because $z_{\rm\scriptscriptstyle WB}$
is larger than the cell length. Wave breaking first occurs within the cell for
a total power $P_{\rm\scriptscriptstyle WB}$ such that
$z_{\rm\scriptscriptstyle WB}=L$. For our experimental parameters we obtain
$P_{\rm\scriptscriptstyle WB}=48$ mW. Numerical tests show that taking
absorption into account does not modify notably this value.

For a total power larger than $P_{\rm\scriptscriptstyle WB}$, the DSW is
formed and develops within the cell. The physical phenomenon at the origin of
the DSW is the following: large intensity perturbations propagate faster than
small ones, so there exist values of $x$ reached at the same “time” by
different intensities. When this occurs first, the density gradient is
infinite. This corresponds to the onset of a cusp catastrophe [55, 56], the
nonlinear diffractive dressing of which is a dispersive shock wave. This takes
the form of a modulated oscillating pattern consisting asymptotically (i.e. at
large $z$, or equivalently large $P_{\rm tot}$) in a train of solitons which,
away from the center of the beam, gradually evolves into a linear
perturbation. The position of its “solitonic edge” on the $y=0$ axis at the
cell output ($z=L$) is denoted as $x_{s}$. It is located in Figs. 2(c) by a
vertical red bar whose thickness represents the uncertainty on the estimation
of $x_{s}$ from the experimental $\widetilde{I}_{\rm out}(x,0)$. This
uncertainty limits the experimental determination of $x_{s}$ to powers larger
than 120 mW. The following maxima of oscillations, represented by vertical
gray lines, are more precisely determined experimentally. The technique
devised in Ref. [47] makes it possible to theoretically determine $x_{s}$ and
the corresponding intensity $\widetilde{I}_{\rm out}(x_{s},0)$. As illustrated
in Fig. 3 the results of this analytical approach (green solid lines) compare
well with the experimental data, although it does not take absorption into
account. Importantly, omission of the nonlinearity saturation leads to
incorrect results (brown solid line).

Figure 3: Characterization of the solitonic edge of the DSW as a function of
the beam’s power. The upper panel represents the position $x_{s}$ of the
shock, and the lower one the corresponding intensity $\widetilde{I}_{\rm
out}(x_{s},0)$. In each panel, the red points with error bars are experimental
results, from Fig. 2(c) and, in the upper one, the orange stars are the
results of 1D numerical simulations of Eq. (2). The green solid line is the
theoretical result, from Ref. [47]. The brown solid line is the theoretical
result in the absence of saturation.

One may study the DSW in an even more detailed way by locating the position of
the maxima of the nonlinear oscillations. While the theoretical results for
$x_{s}$ essentially rely on an approach due to El [34, 40, 43] which is valid
for any type of nonlinearity, the precise intensity profile within the DSW can
be computed only for exactly integrable systems, i.e., by neglecting
saturation effects. The position-dependent oscillation period ${\cal L}(x,z)$
was computed in this framework in Ref. [46] for a parabolic initial intensity
distribution. Fitting the center of the intensity profile (3) by an inverted
parabola, the positions $x_{1}$, $x_{2}$, and $x_{3}$ of the first maxima of
oscillation of the DSW at the output of the cell are determined by

$x_{1}-x_{s}={\cal L}\left(\frac{x_{s}+x_{1}}{2},L\right),$ (5)

and by similar formulas obtained by replacing $x_{1}$ by $x_{2}$ (then
$x_{3}$) and $x_{s}$ by $x_{1}$ (then $x_{2}$). The results are compared with
the experimental ones in the upper half ($x>0$) of Fig. 4.

Figure 4: Color plot of the experimental intensity profiles
$\widetilde{I}_{\text{out}}(x,0)$ as a function of $P_{\text{tot}}$. The
purple dot-dashed line represents the edge $x_{s}$ of the DSW extracted from
Whitham theory (upper part of the figure: $x>0$) and from 1D numerical
simulations (lower part, $x<0$). In each half of the figure ($x\lessgtr 0$)
the white dashed lines are the corresponding analytic predictions (5) for the
maxima of oscillation.

The small offset in the position of the theoretical maxima with respect to the
experimental ones observed in the figure is due to an initial small overshoot
in the theoretical position of $x_{s}$ (cf. the green solid line in Fig. 3)
which is itself due to the absence of absorption in the model. Indeed, the 1D
numerical simulations – which do take absorption into account – are in
slightly better agreement with the experimental results for $x_{s}$ (cf. the
orange stars in Fig. 3). Using the numerical $x_{s}$ in Eq. (5) instead of the
analytical one yields, for the maxima of oscillations, an excellent agreement
with experiment, cf. the lower half of Fig. 4. Such a good agreement despite
the fact that Eq. (5) does not take saturation into account is not surprising:
the rapid decrease of intensity away from the solitonic edge (cf. Fig. 2)
significantly reduces the importance of saturation within the DSW.

It thus appears possible to give a detailed description of precise
experimental recordings of the intensity pattern of an optical shock wave, not
only thanks to numerical simulations, but on the basis of Whitham’s modulation
theory. This is an important validation of recent advances in this approach,
which is no longer restricted to integrable systems or idealized initial
configurations. We are reaching a point where these progresses make it
possible not only to study DSWs per se, but also as tools for prospecting new
physical phenomena, such as the type of wave-mean flow interaction recently
identified in Ref. [57]: our platform is ideally designed to investigate
scattering of elementary excitations by a DSW, a study which is also relevant
to the domain of analogue gravity. Indeed, as discussed in the Supplemental
Material [53], a dispersive shock can be considered as an exotic model of
acoustic white hole, and the good experimental control and theoretical
understanding of this structure demonstrated in the present work opens the
prospect of a detailed investigation of the corresponding induced background
fluctuations.

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