# Visualizing coherent molecular rotation in a gaseous medium

Ilia Tutunnikov  AMOS and Department of Chemical and Biological Physics, The
Weizmann Institute of Science, Rehovot 7610001, Israel  Emilien Prost
Laboratoire Interdisciplinaire CARNOT de Bourgogne Franche-Comté, UMR 6303
CNRS-Université de Bourgogne, BP 47870, 21078, Dijon, France  Uri Steinitz
AMOS and Department of Chemical and Biological Physics, The Weizmann Institute
of Science, Rehovot 7610001, Israel  Soreq Nuclear Research Centre, Yavne
8180000, Israel  Pierre Béjot  Laboratoire Interdisciplinaire CARNOT de
Bourgogne Franche-Comté, UMR 6303 CNRS-Université de Bourgogne, BP 47870,
21078, Dijon, France  Edouard Hertz  Laboratoire Interdisciplinaire CARNOT de
Bourgogne Franche-Comté, UMR 6303 CNRS-Université de Bourgogne, BP 47870,
21078, Dijon, France  Franck Billard  Laboratoire Interdisciplinaire CARNOT de
Bourgogne Franche-Comté, UMR 6303 CNRS-Université de Bourgogne, BP 47870,
21078, Dijon, France  Olivier Faucher<EMAIL_ADDRESS>Laboratoire Interdisciplinaire CARNOT de Bourgogne Franche-Comté, UMR 6303
CNRS-Université de Bourgogne, BP 47870, 21078, Dijon, France  Ilya Sh.
Averbukh<EMAIL_ADDRESS>AMOS and Department of Chemical and
Biological Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

Inducing and controlling the ultrafast molecular rotational dynamics using
shaped laser fields is essential in numerous applications. Several approaches
exist that allow following the coherent molecular motion in real-time,
including Coulomb explosion-based techniques and recovering molecular
orientation from the angular distribution of high harmonics. We theoretically
consider a non-intrusive optical scheme for visualizing the rotational
dynamics in an anisotropic molecular gas. The proposed method allows
determining the instantaneous orientation of the principal optical axes of the
gas. The method is based on probing the sample using ultra-short circularly
polarized laser pulses and recording the transmission image through a vortex
wave plate. We consider two example excitations: molecular alignment induced
by an intense linearly polarized laser pulse and unidirectional molecular
rotation induced by a polarization-shaped pulse. The proposed optical method
is promising for visualizing the dynamics of complex symmetric- and
asymmetric-top molecules.

##  I Introduction

Over the years, the scientific field of molecular alignment and orientation
control generated an extensive set of tools, ranging from adiabatic/impulsive
alignment by single linearly polarized laser pulses to orientation by combined
laser fields with intricately tailored polarizations. Molecular alignment and
unidirectional rotation (UDR) have been reviewed [1, 2]. For broader reviews
on molecular control using electromagnetic fields, the reader is referred to
[3, 4]. The problem of control comes hand in hand with the problem of
visualization of the resulting dynamics.

In the early days, mainly Coulomb explosion technique was utilized for
visualizing laser-induced one dimensional molecular alignment. An intense
time-delayed probe pulse ionized the molecules, and the yield of fragments
ejecting along/against the polarization of the probe was detected [5, 6]. This
allowed recovering the degree of alignment, usually quantified by a single
observable $\braket{\cos^{2}\theta}$, where the angle brackets denote the
average value. The angle $\theta$ is the angle between the molecular axis and
the polarization direction of the aligning pulse. Later on, the approach
evolved into a nowadays standard technique—velocity map imaging (VMI),
allowing reconstruction of molecular angular distribution as a function of
time [7]. For a recent example of the state of the art experiment using VMI,
the reader is referred to [8] and the references therein. Another powerful
tool providing access to 3D information is the cold target recoil ion momentum
spectroscopy (COLTRIMS) [9], where the electron and ion momenta are
coincidentally detected. Methods based on imaging of charged fragments were
successfully applied for imaging the dynamics of unidirectionally rotating
molecules and orientation dynamics of asymmetric top molecules [10, 11, 12].
While VMI and COLTRIMS provide a complete characterization of molecular
rotation, this usually comes at a price of strict experimental conditions
(rarefied gases/molecular beam), complex experimental setups, and long
acquisition times.

Another class of methods relies on optical techniques which have many
attractive practical advantages, e.g., a much-extended working range of
pressures and temperatures [13]. So far, the optical detection schemes have
been limited to one-dimensional measurements of the ensemble-averaged
quantities, such as the degree of molecular orientation or alignment,
quantified by $\braket{\cos\theta}$ [14, 15] and $\braket{\cos^{2}\theta}$
[16, 15], respectively. Higher-order moments $\braket{\cos^{n}\theta}$, with
$n>2$, of the molecular angular distribution can be measure using harmonic
generation [17, 18]. Recently, angle-resolved high-order-harmonic spectroscopy
was used for generating molecular “rotational movies” with the help of machine
learning tools [19].

Here, we present a theoretical analysis of the recently demonstrated [20]
purely optical approach allowing detecting the instantaneous orientation of
the principal optical axes of laser-excited molecular gas. The approach relies
on ultrafast birefringence measurement using delayed femtosecond probe pulses.
The paper is organized as follows: Sec. II qualitatively describes the
proposed imaging technique in the case of homogeneously excited molecular
sample. In Section III, we present the numerical results obtained by solving
the paraxial beam propagation equation. We use two example excitations:
impulsive excitation by linearly polarized pulses resulting in a molecular
dynamics that toggle between alignment and antialignment, and excitation by
femtosecond polarization-twisted pulses which cause unidirectional rotation of
the alignment axis. Sections IV and V describe the case of the inhomogeneous
excitation and present the corresponding numerical results, respectively.
Section VI concludes the paper.

##  II Qualitative Description – Planar wave case

We begin with a qualitative description of the proposed optical imaging
approach. As an example, we consider the case of rigid linear molecules in the
gas phase excited by a short _linearly polarized_ in the $XY$ plane pulse
(pump pulse), propagating along the $Z$-axis. Such an excitation results in a
transient molecular alignment along the polarization axis of the pump pulse
[1, 2, 3, 4]. Due to quantum revivals [21, 22], the alignment recurs
periodically with a well-defined period. Each revival event consists of
alignment and antialignment stages. During both stages, the molecular gas
develops anisotropy—the refractive index for light polarized along the pump
polarization axis differs from the refractive index for light polarized along
the orthogonal direction. The two orthogonal optical principal axes in the
$XY$ plane, $X^{\prime}$ and $Y^{\prime}$, have refractive indices
$n_{X^{\prime}}$ and $n_{Y^{\prime}}$, respectively. In the case of linearly
polarized pump pulse, we let $X^{\prime}$ to be the axis of the pump
polarization. Here, the direction of $X^{\prime}$ and $Y^{\prime}$ axes are
fixed, while the values of $n_{X^{\prime}}$ and $n_{Y^{\prime}}$ depend on the
probe delay. We use a circularly polarized probe pulse, whose polarization can
be decomposed onto the $X^{\prime}$ and $Y^{\prime}$ axes.

Since, generally $n_{X^{\prime}}\neq n_{Y^{\prime}}$, the two components of
the probe light accumulate a relative phase as they propagate through the
anisotropic medium. After passing the medium, the probe polarization becomes,
therefore, elliptical. The orientation of the major axis of the ellipse
depends on the orientation of the optical principal axes. For a homogeneous
medium, the major axis of the polarization ellipse is at $\pm\pi/4$ to the
optical axes. The sign depends on the sense of circular polarization and the
stage of the molecular dynamics (alignment/antialignment). One way to extract
the information encoded in the polarization of the probe pulse is to use
polarization axis finder (PAF) [23, 24], which renders a spatial intensity
pattern having the same directionality as the polarization ellipse. In the
present work, the PAF is composed of a vortex plate acting as a radial
polarizer [25, 26, 27, 28, 29] and a linear polarizer. Recently, a similar PAF
was successfully applied to imaging of coherent molecular rotors [20].

FIG. 1: Polar plots of the intensity in Eq. (9). Left - alignment stage,
$\delta<0$. Right - antialignment stage, $\delta>0$.

Qualitatively, given a circularly polarized light at the input plane, the
electric field at an output plane can be determined with the help of Jones
calculus [30]. Here, we adopt the same phase convention as in [30], namely the
phase of a monochromatic plane wave is defined as $kz-\omega t$, where $k$ is
the wave number, $z$ is the position along the $Z$ axis, $\omega$ is the
angular frequency of light, and $t$ is time. At the input plane, the Jones
vector (expressed in the $X^{\prime}Y^{\prime}$ basis) of left-circular (from
the point of view of the receiver) probe light propagating along the $Z$ axis
reads

$\mathbf{u}_{X^{\prime}Y^{\prime}}(z_{i})=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
i\end{array}\right)$ (1)

where $z_{i}$ denotes the position of the input plane. While propagating
through the sample, the $X^{\prime}$ and $Y^{\prime}$ components of the probe
light accumulate a relative phase $\delta$, such that at the output plane
$z_{o}$

$\displaystyle\mathbf{u}_{X^{\prime}Y^{\prime}}(z_{o})=$
$\displaystyle\left(\begin{array}[]{cc}1/\sqrt{2}&0\\\
0&e^{i\delta}/\sqrt{2}\end{array}\right)\left(\begin{array}[]{c}1\\\
i\end{array}\right)$ (6)

where the sign of $\delta$ depends on the stage of molecular dynamics
(alignment/antialignment). During the alignment stage,
$n_{X^{\prime}}>n_{Y^{\prime}}$ making $X^{\prime}$-axis the slow axis,
accordingly, the $Y^{\prime}$ component leads the $X^{\prime}$ component and,
in the chosen phase convention, $\delta<0$. Transforming back to the $XY$
basis is achieved by
$\mathbf{u}_{XY}(z=z_{o})=R_{Z}(\chi)\mathbf{u}_{X^{\prime}Y^{\prime}}(z=z_{o})$,
where $R_{Z}(\chi)$ is the canonical rotation matrix about the $Z$ axis, where
$\chi$ is the angle between $X^{\prime}$, the slow axis of the medium (here,
the pump polarization axis), and $X$ axes. This results in

$\mathbf{u}_{XY}(z=z_{o})=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\cos\chi-
ie^{i\delta}\sin\chi\\\ ie^{i\delta}\cos\chi+\sin\chi\end{array}\right).$ (7)

For $\delta<0$ $(\delta>0)$, the Jones vector in Eq. (7) represents left
(right) elliptical light. The major axis of the ellipse is oriented at angle
$\pi/4+\chi$ $(-\pi/4+\chi)$ relative to the positive $X$ axis (or at angle
$\pi/4$ $(-\pi/4)$ relative to the principal $X^{\prime}$ axis).

For every point (with azimuth $\varphi$) on the PAF plane in the laboratory
frame, the PAF in the $XY$ basis has a specific Jones matrix representation,

$M_{\mathrm{R}}=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\begin{pmatrix}\cos\varphi&\sin\varphi\\\
\sin\varphi&-\cos\varphi\end{pmatrix},$ (8)

where the right matrix represents a $m=1$ vortex plate [25, 26] (the first
element of the PAF). The transformation induced by the vortex plate is
equivalent to the transformation induced by a half-wave plate with a
continuously varying angle of the fast axis, $\varphi/2$. The left matrix in
Eq. (8) represents a linear polarizer oriented along the $X$ axis (the second
element of the PAF). The intensity as a function of the azimuthal angle at the
image plane is given by

$\displaystyle I(\varphi;\delta,\chi)=$
$\displaystyle|M_{\mathrm{R}}\mathbf{u}_{XY}(z=z_{o})|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{2}\sin(\delta)\sin\left[2(\varphi-\chi)\right].$
(9)

The case of $\chi=0$ corresponds to the molecular alignment along the $X$
axis. Since the function in Eq. (9) depends on the difference $\varphi-\chi$,
for $\chi\neq 0$, the images are simply rotated counterclockwise by angle
$\chi$. Figure 1 shows polar plots of the intensity in Eq. (9) for the
alignment $(\delta<0)$ and antialignment $(\delta>0)$ stages, and for two
angles $\chi=0,\pi/6$. For $\delta=\mp\pi/2$, the out-coming probe becomes
linearly polarized, such that the eight-shaped intensity pattern is most
emphasized and it is oriented at an angle of $+\pi/4$ to the slow axis of the
birefringent medium. When $\delta$ approaches zero, the pattern continuously
transforms into a unit circle (isotropic case).

##  III Beam Propagation Simulations Homogeneous case

We consider a gas of rigid linear molecules excited by nonresonant laser pulse
(pump) propagating along the $Z$ axis with linear or shaped polarization
(restricted to the $XY$ plane). Generally, laser excitation results in time-
dependent anisotropy and inhomogeneity of the molecular gas.

The gaseous medium is probed by collinearly propagating time-delayed laser
pulses (probe pulses). We neglect the difference between the group velocities
of the pump and probe pulses (the Rayleigh length is short enough). Also, we
assume that the probe pulse duration is much shorter than the time scale of
molecular rotation, such that for a fixed probe delay, the probe pulse passes
through effectively a time-independent medium. At the first stage, we assume
that the molecular medium is homogeneous which is equivalent to assuming that
the pump intensity is uniform across the $XY$ plane and along the $Z$ axis.
This corresponds to the case when the waist radius and the Rayleigh range of
the probe beam are smaller than those of the pump beam, such that the probe
passes through a relatively homogeneous portion of the molecular sample. The
propagation of the probe electric field $\mathbf{E}$ through the anisotropic
molecular gas is modeled using the wave equation

$\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}(\tau)\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=\mathbf{0},$ (10)

where $c$ is the speed of light in vacuum, and
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}$
is the tensor of relative permittivity depending on the probe delay, $\tau$.
Derivation of the more general version of Eq. (10) is summarized in the
Appendix A. We define the complex amplitude $\mathbf{U}$ by
$\mathbf{E}(x,y,z,t)=\mathbf{U}(x,y,z)\exp\left[i(k_{0}z-\omega t)\right]$,
neglecting the pulse nature of the probe, with $\omega$ and $k_{0}$ being the
carrier frequency and the vacuum wave number of the probe light, respectively.
Here, we use the same phase convention as in [30] (see Sec. II). Applying the
paraxial approximation to Eq. (10), we obtain the simplified equation
describing the propagation of the complex amplitude $\mathbf{U}=(U_{X},U_{Y})$
along the $Z$ axis [see Eq. (42)]

$\frac{\partial\mathbf{U}}{\partial
z}=\frac{i}{2k_{0}}\nabla_{T}^{2}\mathbf{U}+\frac{ik_{0}}{2}(\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}-\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{I}})\mathbf{U}.$
(11)

Here, $\nabla_{T}^{2}$ is the transverse Laplace operator,
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\mathbf{I}}$ is the
identity matrix. Under ordinary conditions, the relative permittivity of a
molecular gas is simply related to the ensemble-averaged molecular
polarizability tensor,
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$

$\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}=\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{I}}+\frac{N}{\varepsilon_{0}}\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}},$
(12)

where $N$ is the number density of the gas, and $\varepsilon_{0}$ is the
vacuum permittivity. All the physical quantities in Eq. (12) are expressed in
SI units. In terms of
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$,
Eq. (11) reads

$\frac{\partial\mathbf{U}}{\partial
z}=\frac{i}{2k_{0}}\nabla_{T}^{2}\mathbf{U}+i\frac{Nk_{0}}{2\varepsilon_{0}}\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}\mathbf{U}.$
(13)

Equation (13) is solved using standard numerical tools. Let us consider two
examples of ensemble dynamics and their visualizations: (A) The transition
from alignment to antialignment, and (B) the rotation of the alignment axis.

###  III.1 Excitation by linearly polarized pump pulse

As the first example, we consider excitation by a short femtosecond linearly
polarized (along $X$ axis) pump pulse. Such an excitation induces molecular
alignment [1, 2]—a transient confinement of molecular axes along the line
defined by the pump polarization. The relatively strong transient alignment
appears immediately after the excitation and disappears shortly after that
because of the molecular angular velocities dispersion. Due to quantum
revivals [21, 22], the transient alignment periodically recurs with a well
defined period (in case of linear molecules) $T_{r}=1/(2Bc)$, where
$B=\hbar/(4\pi Ic$) is the molecular rotational constant, and $I$ is the
moment of inertia (more directly, $T_{r}=2\pi I/\hbar$). There are also
fractional revivals emerging at multiples of
$T_{r}/4,\,T_{r}/2,\,\mathrm{etc.}$ The degree of alignment is quantified by
the expectation value $\braket{\cos^{2}\theta_{X}}$ where $\theta_{X}$ is the
angle between the laboratory $X$ axis and the molecular axis.

FIG. 2: (a) Degree of alignment, quantified by $\braket{\cos^{2}\theta_{X}}$.
Pump peak intensity $I_{0}=20\;\mathrm{TW/cm^{2}}$. The full width at half
maximum of the pulse is $100\,\mathrm{fs}$ and the initial molecular
rotational temperature is $T=300\,\mathrm{K}$. (b) Enlarged portion of panel
(a). The dashed line denotes the degree of alignment in undisturbed gas,
$1/3$, which is considered the threshold between alignment to anti-alignment
(alignment values lower than $1/3$). FIG. 3: Intensity of the probe beam after
the PAF (see text) as a function of the probe delay, $\tau$, during the
fractional revival at $t\approx 3T_{r}/4$. Here, the molecules are excited by
a linearly polarized along $X$ axis pump pulse. The probe pulse at the input
plane is given by Eq. (19). Probe light propagates through $\approx
7\,\mathrm{mm}$ of the molecular gas. The shown images are taken after an
additional $1\,\mathrm{mm}$ propagation through an undisturbed gas.

We consider the alignment of $\mathrm{CO_{2}}$ molecules at initial rotational
temperature $T=300\,\mathrm{K}$. The moment of inertia of $\mathrm{CO_{2}}$
molecule is $I=280207$ a.u., and the molecular polarizabilities (in atomic
units) along and perpendicular to the molecular axis are
$\alpha_{\parallel}=30.1\,\mathrm{a.u.}$ and
$\alpha_{\perp}=14.7\,\mathrm{a.u.}$, respectively. Figure 2(a) shows the
degree of alignment as a function of time (probe delay), following the
excitation at $t=0$.

For quantum mechanical simulations of the expectation values
$\braket{\cos^{2}\theta_{X}}$, the linear molecules were modeled as rigid
polarizable rotors. The interaction energy with nonresonant pump pulse is
given by
$V=-\mathbf{E}_{\mathrm{pump}}\cdot\mathbf{d}_{\mathrm{ind}}/2=-\mathbf{E}_{\mathrm{pump}}\cdot(\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}\mathbf{E}_{\mathrm{pump}})/2,$
where
$\mathbf{d}_{\mathrm{ind}}=\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}\mathbf{E}_{\mathrm{pump}}$
is the induced dipole, and $\mathbf{E}_{\mathrm{pump}}$ is the pump electric
field. Our numerical scheme for simulating the laser driven dynamics of
complex rigid molecules is described in [31]. In the present case, the scheme
is specialized to the case of linear molecules.

The alignment factor shown in Fig. 2 is used to calculate the components of
ensemble averaged polarizability tensor in the laboratory frame [13],
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$

$\displaystyle\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab},XX}$
$\displaystyle(\tau)=\alpha_{\perp}+\Delta\alpha\braket{\cos^{2}\theta_{X}}(\tau),$
(14)
$\displaystyle\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab},YY}$
$\displaystyle(\tau)=\alpha_{\perp}+\frac{\Delta\alpha}{2}\left[1-\braket{\cos^{2}\theta_{X}}(\tau)\right],$
(15)

where $\Delta\alpha=\alpha_{\parallel}-\alpha_{\perp}$. In the considered
case, the pump pulse is polarized along the $X$ axis, therefore the off-
diagonal elements of
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
vanish,

$\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab},XY}=\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab},YX}=0$
(16)

The polarizability tensor components in Eqs. (14) and (15) are used in Eq.
(13) to simulate the probe propagation through the molecular gas. As described
in Sec. II, the probe light is initially circularly polarized. The parameters
we used are a Gaussian beam with a wavelength
$\lambda_{\mathrm{probe}}=400\,\text{nm}$, waist
$\omega_{0,\mathrm{probe}}=20\,\mu\text{m}$, and Rayleigh length
$z_{R,\mathrm{probe}}=3.14\,\mathrm{mm}$.

A typical length (along the $Z$ axis) of the effectively excited molecular
medium is about $7\,\mathrm{mm}$. Using Eq. (13), we propagate the probe beam
starting at the input plane $z=z_{i}=-3.5\,\mathrm{mm}$ up to the output plane
$z=z_{o}=3.5\,\mathrm{mm}$. The initial beam radius is

$\displaystyle w(z_{i})$
$\displaystyle=w_{0,\mathrm{probe}}\sqrt{1+\left(\frac{z_{i}}{z_{R,\mathrm{probe}}}\right)^{2}}=30\,\mathrm{\mu
m}.$ (17)

FIG. 4: Comparison between the images obtained after the PAF. Left - assuming
the vortex plate is attached to the linear polarizer. Right - when the vortex
plate and the linear polarizer are spaced apart by $2\,\mathrm{mm}$. Here, the
probe delay is $\tau=32.02$ (see Fig. 3), and the vortex plate is at a
distance of $1\,\mathrm{mm}$ away from the medium.

The radius of curvature of the Gaussian beam at the input plane is

$\displaystyle R(z_{i})$
$\displaystyle=z_{i}\left[1+\left(\frac{z_{R,\mathrm{probe}}}{z_{i}}\right)^{2}\right]=-6.32\,\mathrm{mm},$
(18)

while the initial complex amplitude is

$\displaystyle U_{X}(x,y,z_{i})$
$\displaystyle=\exp\left(-\frac{x^{2}+y^{2}}{w^{2}(z_{i})}+ik_{0}\frac{x^{2}+y^{2}}{2R(z_{i})}\right)$
(19) $\displaystyle U_{Y}(x,y,z_{i})$ $\displaystyle=iU_{X},$

where $w(z_{i})$ and $R(z_{i})$ are defined in Eqs. (17) and (18),
respectively. The complex amplitude describing the probe beam at $z_{o}$ is
found by numerically solving the equation in Eq. (13) with the initial
condition in Eq. (19). Here, $X$ and $Y$ components of the complex amplitude
vector are decoupled, because the off-diagonal elements of the polarizability
in the lab frame vanish [see Eq. (16)]. Exiting the gas sample, the probe
passes through the PAF (see Sec. II).

Figure 3 shows a series of intensity maps of the probe beam after the PAF at
several delays during the fractional revival at $t=3T_{r}/4$ (see Fig. 2),
which is characterized by a transition from molecular alignment to
antialignment. Here, the number density of the gas is set to $N=0.02504\times
10^{26}\,\mathrm{m}^{-3}$ [see Eq. (13)], corresponding to a pressure of
approximately $0.1\,\mathrm{atm}$. During the alignment stage, $t\approx
31.75-32.12\,\mathrm{ps}$ [see Fig. 2(b)], the intensity pattern is at angle
$+\pi/4$ relative to the $X$ axis (axis of pump polarization, principal
optical axis $X^{\prime}$). During the antialignment stage, $t\approx
31.25-31.75\,\mathrm{ps}$ and $t\approx 32.12-32.60\,\mathrm{ps}$ [see Fig.
2(b)], the intensity pattern is at angle $-\pi/4$ relative to the $X$ axis.
These results are in line with the qualitative discussion in Section II.
Notice, between the revivals, there is a small persistent alignment which is
reflected in weak anisotropy visible in the last panel in Fig. 3.

Let us note that in practice, the vortex plate and linear polarizer (forming
together the PAF) are spaced apart, and the vortex plate applies an angle-
dependent phase mask [see Eq. (8)] to the incident beam. Therefore, while
propagating between the vortex plate to the polarizer, the incident Gaussian
probe beam disperses and evolves into a doughnut-like shape, such that the
experimental images look more like shown in the right panel of Fig. 4 (also,
see the experimental results in [20]). Moreover, in experiments, for practical
reasons, the image plane is located much farther from the excitation volume
(far-field measurement). The images shown here do not represent the actual
scale expected in the experiments. The scale depends on the optical distance
and the lenses system between the gas cell and the imaging device.

###  III.2 Excitation by polarization-twisted pump pulse

As an additional example, we consider the excitation by a polarization shaped
pulse with twisted polarization [32]. A polarization-twisted pulse is modeled
as a pair of two in-phase overlapping orthogonally polarized laser pulses with
a delay $\tau_{p}$ between them. The corresponding electric field reads

$\mathbf{E}_{\mathrm{pump}}(t)=E_{0}[f(t)\mathbf{e}_{X}+f(t-\tau_{p})\mathbf{e}_{Y}]\cos(\omega
t),$ (20)

where $E_{0}$ is the peak amplitude, $\mathbf{e}_{X}$ and $\mathbf{e}_{Y}$ are
the unit vectors along the laboratory $X$ and $Y$ axes. The envelope $f(t)$ of
each constituent linearly polarized pulse is Gaussian. Figure 5 shows an
illustration of the polarization vector twisting from the $X$ axis to the $Y$
axis.

FIG. 5: Illustration of the polarization twisting, see Eq. (20).

In contrast to the excitation by a linearly polarized pump pulse, here, the
$Z$ projection of molecular angular momentum is not conserved as the twisting
polarization induces molecular unidirectional rotation (UDR) about the $Z$
axis [33]. In this case, the description of molecular rotations required two
dynamical degrees of freedom. It is convenient to use the two standard
spherical coordinate system angles: polar angle $\theta$ between the $Z$ axis
and the molecular axis, and azimuthal angle $\phi$ between the projection of
the molecular axis on the $XY$ plane and the positive $X$ axis. In the case of
UDR the polarizability tensor,
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
is no longer diagonal, because the principal optical axes ($X^{\prime}$ and
$Y^{\prime}$) do not generally overlap with $X$ and $Y$ axes, and their
orientation changes with time. The off-diagonal elements of
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$,
reflecting the unidirectional rotation of the molecular axis [34, 32], read

$\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab},XY}=\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab},YX}=\frac{\Delta\alpha}{2}\braket{\sin(2\phi)\sin^{2}(\theta)}.$
(21)

Explicit expressions for all the elements of
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
parameterized by $\theta$ and $\phi$ are given in Appendix B, see Eqs.
(47)-(50). Figure 6(a) shows the time dependence of the off diagonal element
(divided by $\Delta\alpha$) in Eq. (21). Figure 6(b) shows the magnified
portion during the fractional quantum revival at $t\approx 3T_{r}/4$.

FIG. 6: (a) The curve is proportional to the off-diagonal element of the
polarizability tensor, see Eq. (21). Parameters of the polarization twisted
pump pulse: peak intensity $I_{0}=20\;\mathrm{TW/cm^{2}}$, full width at half
maximum of each constituent linearly polarized pulse is $100\,\mathrm{fs}$,
the delay between the pulses is $\tau_{p}=150\,\mathrm{fs}$. Initial molecular
rotational temperature is $T=300\,\mathrm{K}$. (b) Enlarged portion of panel
(a).

Figure 7 shows a series of intensity images of the probe beam after the PAF at
several delays during the fractional revival at $t\approx 3T_{r}/4$. In
addition to the anisotropy (like in the case of excitation by a linearly
polarized pulse) the principal optical axes, $X^{\prime}$ and $Y^{\prime}$,
also rotate in the $XY$ plane. This rotation is the manifestation of molecular
UDR.

FIG. 7: Intensity of the probe beam after the PAF as a function of the probe
delay, $\tau$, during the fractional revival at $t\approx 3T_{r}/4$. Here, the
molecules are excited by a polarization twisted pulse [see Eq. (20)]. The
delay $\tau$ is measured from the center of the first linearly polarized
(along $X$) pulse forming the polarization twisted pulse. The rest is the same
as in Fig. 3.

##  IV Qualitative description – Inhomogeneous Case

So far, we considered the case of uniformly excited molecular medium, and only
the birefringence effect has been taken into account. Generally, however, when
the probe beam waist is comparable with the waist of the pump pulse, the
inhomogeneity needs to be taken into account. The aforementioned inhomogeneity
stems from the fact that the pump intensity decreases away from the optical
axis, resulting in the variation of the molecular alignment across the
transverse plane. In the same manner, on the $Z$ axis, the pump intensity (and
its induced molecular effect) reduces with the distance from the focal plane.

Once again, for simplicity, we consider the case of excitation by a linearly
polarized pump pulse, polarized at an angle $\chi$ to the $X$ axis. The
transverse molecular alignment inhomogeneity acts as an effective lens, as it
manifests in a graded refractive index. Generally, the two orthogonal probe
components, $X^{\prime}$ and $Y^{\prime}$, focus differently. During the
alignment stage, the $X^{\prime}$ components is focused, while the
$Y^{\prime}$ component is defocused and vice versa during the antialignment
stage [35]. Qualitatively, the focusing effect can be described by introducing
the attenuation parameter $0\leq a\leq 1$, such that [compare to Eq. (6)]

$\displaystyle\mathbf{u}_{X^{\prime}Y^{\prime}}(z_{o})$
$\displaystyle=\left(\begin{array}[]{cc}\sqrt{1-a^{2}}&0\\\
0&ae^{i\delta}\end{array}\right)\left(\begin{array}[]{c}1\\\
i\end{array}\right).$ (26)

For $a=1/\sqrt{2}$, Eq. (26) reduces to Eq. (6). Notice, the definition of
parameter $a$ implies that the beam’s energy at a fixed radial distance is
conserved. In reality, however, the _total energy_ is conserved. Therefore,
$a$ should be treated as an effective parameter accounting for the unequal
intensities of the two polarization components of the probe beam at a
particular radial distance from the optical axis and at particular $z_{o}$.
The intensity at the output plane reads [compare with Eq. (9)]

FIG. 8: Polar plots of the intensity in Eq. (27) calculated for alignment
(blue) and antialignment (red) induced by a pump polarized along the $X$ axis.
The angle $\gamma$ is given by Eq. (28).

$\displaystyle I(\varphi;\delta,\chi,a)$
$\displaystyle=\frac{1}{2}-a\sqrt{1-a^{2}}\sin(\delta)\sin[2(\varphi-\chi)]$
$\displaystyle+\left(\frac{1}{2}-a^{2}\right)\cos[2(\varphi-\chi)].$ (27)

FIG. 9: Polar plots for left- (blue) and righ-handed (red) probes [see Eqs.
(27) and (29)]. The angle $\gamma$ for the case of left circular probe is
given by Eq. (28). The other parameters are $\delta=-\pi/4,\,\chi=0$, and
$a=0.2$.

Figure 8 shows polar plots of the intensity in Eq. (27). During the alignment
stage $(\delta<0,\,a<1/\sqrt{2})$, the probe component polarized along the
$X^{\prime}$ axis is focused, while the $Y^{\prime}$ components is defocused.
The eight-shaped intensity pattern is effectively pushed closer to the
$X^{\prime}$ axis. During the antialignment stage $(\delta>0,\,a>1/\sqrt{2})$,
the probe component polarized along the $Y^{\prime}$ axis is focused, while
the $X^{\prime}$ components is defocused. The eight-shaped intensity pattern
is effectively pushed closer to the $Y^{\prime}$ axis.

The angle $\gamma$ (in radians) between the $X^{\prime}$ ($Y^{\prime}$) axis
for the case $\delta<0,\,a<1/\sqrt{2}$ $(\delta>0,\,a>1/\sqrt{2})$ and the
long axis of the eight-shaped intensity pattern (see Fig. 8) can be found from
$\partial_{\varphi}I(\varphi;\delta,\chi,a)|_{\gamma}=0$. Explicitly,

$\gamma=\frac{1}{2}\arctan\left(\frac{2a\sin(\delta)\sqrt{1-a^{2}}}{2a^{2}-1}\right),$
(28)

Generally, angle $\gamma$ also depends on the radial distance from the optical
axis and the position along the optical axis. The reason is that the ratio of
intensities of the two polarization components changes with radial and
longitudinal positions. In the case of negligible focusing, $a\rightarrow
1/\sqrt{2}$, $\gamma\rightarrow\pi/4$ [see Eq. (9) and Fig. 1].

At this point, we would like to draw the reader’s attention to that in the
case of non-negligible focusing, the orientation of the optical principal axes
is ambiguous. Indeed, consider e.g., the blue curve in Fig. 8. What is the
reason behind $\gamma\neq\pm\pi/4$? There are two possible reasons (or a
combination thereof): (i) the principal optical axes are rotated $\chi\neq 0$,
while $\gamma=\pm\pi/4$, (ii) $X^{\prime}$ and $Y^{\prime}$ axes coincide with
$X$ and $Y$ axes ($\chi=0$), but there is a non-negligible focusing,
$\gamma\neq\pm\pi/4$. One way to resolve the ambiguity is to perform an
additional measurement using circular probe light of opposite handedness. It
can be shown that for the right circular probe, the intensity after the PAF
reads [compare with Eq. (27)]

$\displaystyle I(\varphi;\delta,\chi,a)$
$\displaystyle=\frac{1}{2}+a\sqrt{1-a^{2}}\sin(\delta)\sin[2(\varphi-\chi)]$
$\displaystyle+\left(\frac{1}{2}-a^{2}\right)\cos[2(\varphi-\chi)].$ (29)

Notice the plus sign in front of the second terms compared to Eq. (27).
Moreover, angle $\gamma$ [see Eq. (28)] has an opposite sign for the opposite
circular polarizations. Figure 9 compares the intensities obtained for the
probes of opposite handedness in the case of non-negligible focusing.

The pictures obtained from the two measurements together, in principle, enable
to determine the orientations of the optical principal axes (and to assess the
focusing strength). As shown in Fig. 9, the principal axis bisects the angle
between the long axis of the two intensity patterns.

##  V Beam Propagation Simulations – Inhomogeneous case

In the case of non-negligible inhomogeneity, the permittivity of the medium
becomes coordinate-dependent and the wave equation reads [compare with Eq.
(10)]

$\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}(x,y,z;\tau)\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=0.$ (30)

Defining the complex amplitude $\mathbf{U}$, and using the paraxial
approximation, we arrive at the same equation as Eq. (13), but with coordinate
dependent polarizability tensor
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$.
The derivation is summarized in Appendix A. Again, we consider the dynamics of
alignment to antialignment transition, and the rotation of the alignment axis.

###  V.1 Excitation by linearly polarized pump pulse

Considering first the excitation by a linearly polarized along the $X$ axis
pump pulse. Similar to Subsec. III.1, the polarizability
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
is diagonal, and its components are given by Eqs. (14) and (15). However, in
contrast to Subsec. III.1, the degree of alignment now also depends on the
coordinates
$\braket{\cos^{2}\theta_{X}}=\braket{\cos^{2}\theta_{X}}(x,y,z;\tau)$.
Following [20], we assume that the probe light is derived from the pump light
by frequency doubling, and both have Gaussian profiles. Table 1 summarize the
beams parameters used. To simplify matters, we neglect the inhomogeneity along
the propagation direction ($Z$ axis), such that
$\braket{\cos^{2}\theta_{X}}=\braket{\cos^{2}\theta_{X}}(x,y;\tau)$.

Beam | $\lambda\,(\mathrm{nm})$ | $w_{0}\,(\mathrm{\mu m})$ | $z_{R}\,(\mathrm{mm})$
---|---|---|---
Pump | $800$ | $30$ | $3.53$
Probe | $400$ | $20$ | $3.14$
Table 1: Summary of the beams parameters. Here $\lambda$, $w_{0}$, $z_{R}$ are
vacuum wavelength, waist radius and Rayleigh range.

Figure 10 shows several curves of time-dependent alignment factors at various
transverse distances, $r$ from the optical axis for the case of
$\mathrm{CO_{2}}$ molecules. Since the intensity of the pump decreases with
the distance from the optical axis, the overall degree of alignment decreases
as well. Here, the initial rotational temperature is $T=300\,\mathrm{K}$.

FIG. 10: (a) Degree of alignment at various transverse distances $r$ from the
optical axis. The pump peak intensity is $I(r=0)=20\;\mathrm{TW/cm^{2}}$ and
the pump pulse duration at full width at half maximum (FWHM) is
$100\,\mathrm{fs}$. Initial molecular rotational temperature is
$T=300\,\mathrm{K}$. (b) Enlarged portion of panel (a). Black vertical lines
denote the delays shown in Fig. 11.  FIG. 11: The spatial variation of the
degree of alignment at two times denoted by vertical lines in Fig. 10.

Figure 11 shows the spatial dependence of the degree of alignment in the $XY$
plane. The left panel $(t=31.58\,\mathrm{ps})$ corresponds to the
antialignemnt stage, while the right panel $(t=31.97\,\mathrm{ps})$
corresponds to the alignment stage. These two moments are denoted by the
vertical lines in Fig. 10. Considering Eqs. (14) and (15) and the wave
equation in Eq. (13), the $X^{\prime}$ and $Y^{\prime}$ components (here, same
as $X$ and $Y$) of the probe pulse pass through effective lenses with
curvatures $[\braket{\cos^{2}\theta_{X}}(x,y)-1/3]$ and
$[\braket{\cos^{2}\theta_{Y}}(x,y)-1/3]$ [with
$\braket{\cos^{2}\theta_{Y}}(x,y)=-2\braket{\cos^{2}\theta_{X}}(x,y)]$,
respectively (see Fig. 11). Accordingly, the $X^{\prime}$ component is
focused, while the $Y^{\prime}$ component is defocused, and vice versa during
the antialignment stage.

Figure 12 shows a series of intensity images of the probe beam after the PAF
at several delays during the fractional revival at $t=3T_{r}/4$ (see Fig. 10).
Here, the number density of the gas is set to $N=0.1252\times
10^{26}\,\mathrm{m}^{-3}$ [see Eq. (13)], corresponding to pressure of
approximately $0.5\,\mathrm{atm}$. Panel (a) shows the intensities obtained
using a left circular probe [see Eq. (19)], while panel (b) shows the results
obtained using a right circular probe ($U_{Y}=-iU_{X}$). In both cases, due to
the focusing effect, the long axes of the intensity patterns slightly deviate
from the diagonals. As mentioned previously (see Sec. IV), the degree of
deviation of the long axis of the eight-shaped intensity from $\pm
45^{\circ}$, generally, depends on the radial distance $r$ and the PAF/output
plane’s position. Here, the radial dependence is barely noticeable, while the
change with the longitudinal distance, $z$ is visible (not shown). The
deviation is in the opposite directions for the opposite circular
polarizations. This allows determining the orientations of the principal
optical axes unambiguously. It is important to emphasize that due to the
symmetry of excitation, the orientations of the optical principal axes,
$X^{\prime}$ and $Y^{\prime}$, are independent of the distance from the
optical axis.

FIG. 12: Intensity of the probe beam after the PAF as a function of the delay,
$\tau$, during the fractional revival at $t\approx 3T_{r}/4$. Here, the
molecules are excited by a linearly polarized along $X$-axis pump pulse. Probe
light propagates through $\approx 7\,\mathrm{mm}$ of the molecular gas. The
beams’ parameters are defined in Table 1. The shown images are taken after an
additional propagation of $1\,\mathrm{mm}$ through an undisturbed gas. (a)
Left circular probe. (b) Right circular probe.

###  V.2 Excitation by polarization-twisted pump pulse

Next, we consider the excitation by the polarization-twisted pulse [see Eq.
(20) and Fig. 5]. Similar to Subsec. III.2, the polarization-twisted pulse
induces molecular UDR, which recurs during the fractional revivals. Figure 13
shows the off-diagonal element of the polarizability [divided by
$\Delta\alpha$, see Eq. (21)] for various transverse distances, $r$ from the
optical axis. The existence of non-zero time-dependent (probe delay dependent)
off-diagonal elements of the polarizability imply that the orientation of the
optical principal axes changes with time.

FIG. 13: The curves are proportional to the off-diagonal element of the
polarizability tensor at various transverse distances, $r$ from the optical
axis. Parameters of the polarization-twisted pump pulse: peak intensity
$I(r=0)=20\;\mathrm{TW/cm^{2}}$ and the width (FWHM) of each constituent
linearly polarized pulse is $100\,\mathrm{fs}$, the delay between the pulses
is $\tau_{p}=150\,\mathrm{fs}$. Initial molecular rotational temperature is
$T=300\,\mathrm{K}$. (b) Enlarged portion of panel (a).

There is an important difference between the excitation by polarization-
twisted pulse having inhomogeneous intensity profile and the excitation by
polarization-twisted pulse with homogeneous intensity considered in Subsec.
III.2. In the case of twisted pulse the orientation of the optical axes
generally depends on the radial distance from the optical axis. The physical
reason for this is that the efficiency of the induced molecular UDR depends on
the parameters of the polarization-twisted pulses.

Figure 14 shows a series of intensity images like in Fig. 12, but for the case
of excitation by polarization twisted pulse. The white arrows denote the
principal optical axes. As mentioned above, the orientation of $X^{\prime}$
and $Y^{\prime}$ axes depends on the radial distance, however here the effect
remains marginal within the waist of the pump pulse
$(w_{0,\mathrm{pump}}=30\,\mathrm{\mu m})$. The shown principal axes were
found by diagonalizing
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
at $r=20\,\mathrm{\mu m}$.

Thanks to the weak $r$ dependence, the long axes of the intensity patterns
deviate from the chosen principal axes in approximately the opposite sense for
the probes of opposite circular polarization.

FIG. 14: Intensity patterns after the PAF as a function of the probe delay,
$\tau$, during the fractional revival at $t\approx 3T_{r}/4$. Here, the
molecules are excited by a polarization-twisted pulse [see Eq. (20)]. The
delay is measured from the peak of the first linearly polarized pulse. Rest of
the parameters are similar to Fig. 12. (a) Left circular probe. (b) Right
circular probe.

##  VI Conclusions

We theoretically analyzed the optical imaging scheme introduced in [20],
allowing direct visualization of the instantaneous orientation of the optical
principal axes in a gas of laser-excited coherently rotating molecules. Time-
delayed circularly polarized probe pulses are used to investigate the time-
dependent optical birefringence of the molecular medium. A polarization axis
finder (PAF) is used to visualize the polarization of the probe pulse, which
contains the information on the optical principal axes. While in this work, we
considered two examples of molecular excitation by short pulses, the same
approach applies to the visualization of other complex molecular states
created by means of tailored laser pulses, e.g., molecular super rotors [36].

Future work may generalize the proposed scheme to asymmetric-top molecules,
which generally have three distinct polarizability axes. The scheme may be
promising for imaging alignment and unidirectional rotation of complex
molecules (including chiral ones). Visualization of the dynamics of such
molecules in traditional methods, e.g., Coulomb explosion-based methods, is
challenging due to the multitude of possible fragments and ionization
channels. The optical measurement of alignment dynamics may stimulate the
development of imaging schemes suitable for tracing out intricate molecular
_orientation_ dynamics.

###### Acknowledgements.

This work was supported by the Israel Science Foundation (Grant No. 746/15),
the CNRS, the ERDF Operational Programme-Burgundy, the EIPHI Graduate School
(Contract No. ANR-17-EURE-0002), and the Associate (CNRS&Weizmann)
International ImagiNano Laboratory. IA acknowledges support as the Patricia
Elman Bildner Professorial Chair. This research was made possible in part by
the historic generosity of the Harold Perlman Family.

##  Appendix A Wave Equation

Here, we derive the wave equation describing the propagation of a Gaussian
beam in a non-magnetic, inhomogeneous, and time independent molecular gas.
Maxwell’s equations (SI units) in matter which is free of currents and charges
are

$\displaystyle\nabla\times\mathbf{H}$
$\displaystyle=\frac{\partial\mathbf{D}}{\partial t},$ (31)
$\displaystyle\nabla\times\mathbf{E}$
$\displaystyle=-\frac{\partial\mathbf{B}}{\partial t},$ (32)
$\displaystyle\nabla\cdot\mathbf{D}$ $\displaystyle=0,$ (33)
$\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0,$ (34)

where $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic field
vectors, $\mathbf{H}$ and $\mathbf{D}$ are the magnetizing and displacement
field vectors. In addition to the Maxwell’s equation, we use the constitutive
relations

$\mathbf{D}=\varepsilon_{0}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}\mathbf{E}\qquad\mathbf{B}=\mu_{0}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\mu}}_{r}\mathbf{H},$
(35)

where $\varepsilon_{0}$ and
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}$
are the vacuum and relative permittivities, respectively, $\mu_{0}$ and
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\mu}}_{r}$
are the vacuum and relative permeabilities, respectively. We assume a non-
magnetic medium,
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\mu}}_{r}=1$.
Relative permittivity
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}$
is position dependent, but time independent tensor. Applying the curl operator
to the Eq. (32) results in

$\displaystyle\nabla\times\left[\left(\nabla\times\mathbf{E}\right)+\frac{\partial\mathbf{B}}{\partial
t}\right]$ $\displaystyle=\nabla\times\left(\nabla\times\mathbf{E}\right)$
$\displaystyle+\mu_{0}\frac{\partial}{\partial
t}\left(\nabla\times\mathbf{H}\right)$ $\displaystyle=0,$ (36)

where the time derivative and curl operators were interchanged. Substituting
$\nabla\times\mathbf{H}=\partial_{t}\mathbf{D}$ [see Eq. (31)], we get

$\displaystyle\nabla\times\left(\nabla\times\mathbf{E}\right)+\mu_{0}\frac{\partial^{2}\mathbf{D}}{\partial
t^{2}}$ $\displaystyle=\nabla\times\left(\nabla\times\mathbf{E}\right)$
$\displaystyle+\mu_{0}\varepsilon_{0}\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}$ $\displaystyle=0.$ (37)

Using the vector identity
$\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^{2}\mathbf{A}$,
we can simplify Eq. (37)

$\nabla^{2}\mathbf{E}-\mu_{0}\varepsilon_{0}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=\nabla\left(\nabla\cdot\mathbf{E}\right).$ (38)

For our applications, we would like to simplify Eq. (38) by neglecting
$\nabla\left(\nabla\cdot\mathbf{E}\right)$. To estimate the relative size of
$\nabla\left(\nabla\cdot\mathbf{E}\right)$, we assume that
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}=\boldsymbol{\varepsilon}_{r}(x,y,z)$
is a scalar and use Eq. (33),
$\nabla\cdot\mathbf{D}=\varepsilon_{0}\nabla\cdot(\varepsilon_{r}\mathbf{E})=0$.
This corresponds to the case when the electric field is linearly polarized
along one of the principal axes of
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}$.
Using the vector identity
$\nabla\cdot\left(\psi\mathbf{A}\right)=\psi\nabla\cdot\mathbf{A}+(\nabla\psi)\cdot\mathbf{A}$,
we have
$\varepsilon_{0}\nabla\cdot(\varepsilon_{r}\mathbf{E})=\varepsilon_{r}\nabla\cdot\mathbf{E}+(\nabla\varepsilon_{r})\cdot\mathbf{E}=0$.
In other words, in case of a scalar relative permittivity the right hand side
of the equation is
$-\nabla\\{[(\nabla\varepsilon_{r})/\varepsilon_{r}]\cdot\mathbf{E}\\}$.
Compared with the second term on the left hand, the right hand side can be
neglected when the relative change in $\varepsilon_{r}$ over the distance of
one wave-length must be much less than unity [37]. Finally, we obtain the
following wave equation [38]

$\nabla^{2}\mathbf{E}-\mu_{0}\varepsilon_{0}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=0$ (39)

To further simplify Eq. (39), we assume that the electric field propagates
along the $Z$ axis and substitute
$\mathbf{E}(x,y,z,t)=\mathbf{U}(x,y,z)\exp\left[i(k_{0}z-\omega t)\right]$.
The chosen form of $\mathbf{E}$ implies that we neglect the pulse nature of
the light. Here $\mathbf{U}(x,y,z)=(U_{X},U_{Y},0)$ is the complex amplitude.
For consistency with Section II, we use Hecht’s phase convention [30],
$k_{0}z-\omega t$. The result reads

$(\omega^{2}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}-c^{2}k_{0}^{2}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{I}})\mathbf{U}+c^{2}\left(2ik_{0}\frac{\partial\mathbf{U}}{\partial
z}+\nabla^{2}\mathbf{U}\right)=0,$ (40)

where $c=1/\sqrt{\mu_{0}\varepsilon_{0}}$, $\mathbf{I}$ is the identity
matrix, and $\mathbf{e}_{Z}$ is the unit vector along the $Z$ axis. Next, we
make the paraxial approximation and neglect the second derivative with respect
to $z$, such that Eq. 40 becomes

$(\omega^{2}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}-c^{2}k_{0}^{2}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{I}})\mathbf{U}+c^{2}\left(2ik_{0}\frac{\partial\mathbf{U}}{\partial
z}+\nabla_{T}^{2}\mathbf{U}\right)=0,$ (41)

where $\nabla_{T}^{2}$ is the transverse Laplacian operator. Rearrangement
yields

$\frac{\partial\mathbf{U}}{\partial
z}=\frac{i}{2k_{0}}\nabla_{T}^{2}\mathbf{U}+\frac{ik_{0}}{2}\left(\frac{\omega^{2}}{c^{2}k_{0}^{2}}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}-\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{I}}\right)\mathbf{U}.$

Finally, we substitute $\omega^{2}/(c^{2}k_{0}^{2})=1$, such that

$\frac{\partial\mathbf{U}}{\partial
z}=\frac{i}{2k_{0}}\nabla_{T}^{2}\mathbf{U}+\frac{ik_{0}}{2}(\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\varepsilon}}_{r}-\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{I}})\mathbf{U}.$
(42)

##  Appendix B Rotational Dynamics

For free rigid linear molecules in the gas phase
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
is simply a constant, where $\alpha_{\parallel,\perp}$ are the
polarizabilities along and perpendicular to the molecular axis, respectively.
In case of laser excited molecular gas, however
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
is generally position dependent anisotropic tensor. The relation between the
polarizability expressed in the molecule-fixed frame and the polarizability
expressed in the laboratory-fixed frame

$\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}_{\mathrm{lab}}=\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{R}}^{T}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}_{\mathrm{mol}}\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{R}},$
(43)

where

$\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\mathbf{R}}\left(\theta,\phi,\chi\right)=\left(\begin{array}[]{ccc}\mathrm{c}(\theta)\mathrm{c}(\phi)\mathrm{c}(\chi)-\mathrm{s}(\phi)\mathrm{s}(\chi)&\mathrm{c}(\theta)\mathrm{c}(\chi)\mathrm{s}(\phi)+\mathrm{c}(\phi)\mathrm{s}(\chi)&-\mathrm{c}(\chi)\mathrm{s}(\theta)\\\
-\mathrm{c}(\chi)\sin(\phi)-\mathrm{c}(\theta)\mathrm{c}(\phi)\mathrm{s}(\chi)&\mathrm{c}(\phi)\mathrm{c}(\chi)-\mathrm{c}(\theta)\mathrm{s}(\phi)\mathrm{s}(\chi)&\mathrm{s}(\theta)\mathrm{s}(\chi)\\\
\mathrm{c}(\phi)\mathrm{s}(\theta)&\mathrm{s}(\theta)\mathrm{s}(\phi)&\mathrm{c}(\theta)\end{array}\right)$
(44)

is an orthogonal rotation matrix parametrized by Euler angles as defined in
[39]. In the rotating reference frame, it is convenient to choose a basis
including the three principal axes of inertia. In this basis, the
polarizabilty tensor has a simple representation

$\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}_{\mathrm{mol}}=\left(\begin{array}[]{ccc}\alpha_{\perp}&0&0\\\
0&\alpha_{\perp}&0\\\ 0&0&\alpha_{\parallel}\end{array}\right).$ (45)

The explicit expression for
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
in terms of the Euler angles reads

$\braket{\overset{\text{\text{\tiny{\hbox{\bm{\leftrightarrow}}}}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}=\left(\begin{array}[]{ccc}A&D&0\\\
D&B&0\\\ 0&0&C\end{array}\right),$ (46)

where

$\displaystyle A$
$\displaystyle=\frac{1}{4}\left[\alpha_{\parallel}+3\alpha_{\perp}-\Delta\alpha\braket{\cos(2\theta)-2\cos(2\phi)\sin^{2}(\theta)}\right],$
(47) $\displaystyle B$
$\displaystyle=\frac{1}{4}\left[\alpha_{\parallel}+3\alpha_{\perp}-\Delta\alpha\braket{\cos(2\theta)+2\cos(2\phi)\sin^{2}(\theta)}\right],$
(48) $\displaystyle C$
$\displaystyle=\frac{1}{2}\left[\alpha_{\parallel}+\alpha_{\perp}+\Delta\alpha\braket{\cos(2\theta)}\right],$
(49) $\displaystyle D$
$\displaystyle=\frac{\Delta\alpha}{2}\braket{\sin(2\phi)\sin^{2}(\theta)},$
(50)

with $\Delta\alpha=\alpha_{\parallel}-\alpha_{\perp}$.

Pump pulse(s) initiate rotational dynamics, such that, generally, the various
expectation values appearing in Eqs. (47), (48), (49), and (50) depend on the
probe delay. The evaluation of the elements of
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
requires simulating the rotational dynamics of laser driven linear molecules.
For the quantum mechanical simulations, we expressed the trigonometric
functions involved in the matrix elements of
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}_{\mathrm{lab}}$
in terms of Wigner D-functions as follows

$\displaystyle\cos(2\theta)-2\cos(2\phi)\sin^{2}(\theta)$
$\displaystyle=\frac{4D_{00}^{2*}-1}{3}-\sqrt{\frac{8}{3}}\left[D_{20}^{2*}+D_{-20}^{2*}\right],$
(51) $\displaystyle\cos(2\theta)+2\cos(2\phi)\sin^{2}(\theta)$
$\displaystyle=\frac{4D_{00}^{2*}-1}{3}+\sqrt{\frac{8}{3}}\left[D_{20}^{2*}+D_{-20}^{2*}\right],$
(52) $\displaystyle\cos(2\theta)$ $\displaystyle=\frac{4D_{00}^{2*}-1}{3},$
(53) $\displaystyle\frac{1}{2}\sin(2\phi)\sin^{2}(\theta)$
$\displaystyle=i\sqrt{\frac{1}{6}}\left[D_{-20}^{2*}-D_{20}^{2*}\right].$ (54)

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