# Light–absorbed orbital angular momentum in the linear response regime

Philippe Scheid<EMAIL_ADDRESS>Université de Lorraine,
LPCT, CNRS, UMR 7019, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France
Université de Lorraine, IJL, CNRS, UMR 7198, BP 70239, 54000 Nancy Cedex,
France  Stéphane Mangin  Université de Lorraine, IJL, CNRS, UMR 7198, BP
70239, 54000 Nancy Cedex, France  Sébastien Lebègue  Université de Lorraine,
LPCT, CNRS, UMR 7019, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

###### Abstract

In exploring the light-induced dynamics within the linear response regime,
this study investigates the induced orbital angular momentum on a wide variety
of electronic structures. We derive a general expression for the torque
induced by light on different electronic systems based on their characteristic
dielectric tensor. We demonstrate that this phenomenon diverges from the
inverse Faraday effect as it produces an orbital magnetization persistent
post-illumination. Indeed, our results reveal that, while isotropic non-
dissipative materials do not absorb orbital angular momentum from circularly
polarized light, any symmetry-breaking arrangement of matter, be it spatial or
temporal, introduces novel channels for the absorption of orbital angular
momentum, or magnetization. Most notably, in dissipative materials, circularly
polarized light imparts a torque corresponding to a change in orbital angular
momentum of $\hbar$ per absorbed photon. The potential of these mechanisms to
drive helicity-dependent magnetic phenomena paves the way for a deeper
understanding of light-matter interactions. Notably, the application of pump-
probe techniques in tandem with our findings allows experimentalists to
quantitatively assess the amount of orbital angular momentum transferred to
electrons in matter, thus hopefully enhancing our ability to steer ultrafast
light-induced magnetization dynamics.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

With the study of phenomena such as the birefringence, the photoelectric
effect and, more recently, a plethora of ultrafast light-induced dynamics, the
interaction between light and matter has remained a very fertile ground for
physics throughout the past centuries. Most notably, the explanation of the
photoelectric effect by EinsteinEinstein (1905) not only shed light on the
mechanisms of absorption of energy carried by light, but also on the quantum
nature of light itself.
Conversely, mechanisms by which the angular momentum of light and matter are
interacting with each other have remained a challenging topic. Michael Faraday
pioneered this field in 1845 with the discovery that linearly polarized light
rotates while propagating through a piece of heavy glass subjected to an
external magnetic field. More than one century later, PitaevskiiPitaevskii
(1961) theoretically showed that circularly polarized light induces a magnetic
moment proportional to its intensity in non-dissipative media, i.e. that light
can also change the magnetic state of matter. He naturally coined this
phenomena “inverse Faraday effect” (IFE). Only a few years later, experimental
confirmation of a static magnetization solely induced by the light was
givenVan Der Ziel  _et al._ (1965). With the advance of femtosecond ultra-
intense light sources and the discovery of the all-optical helicity-dependent
switching effectStanciu  _et al._ (2007); Mangin  _et al._ (2014); Lambert
_et al._ (2014); Scheid  _et al._ (2022), where the polarization of the light
can be used to steer the magnetic state of metallic magnets in a deterministic
way, the interest for the IFE has even been further invigorated. However, due
to the continuum of states available at the Fermi energy and above it, these
compounds are highly absorptive and the traditional IFE could no longer be
invoked. To this end, Battiato et al.Battiato  _et al._ (2012, 2014) developed
a generalized framework to compute it in any type of material.
Before Pitaevskii, and differently from an induced magnetic moment as in the
IFE, BethBeth (1936) experimentally showed that, while light travels in a
transparent birefringent compound, and its polarization evolves from linear to
circular, a torque is exerted on the material. He assigned this applied torque
to the fact that, for light to be able to change its elipticity, it has to
gain or lose angular momentum, which is thus lost or gained by the material
such that the angular momentum remains conserved. More recently, the spin
counterpart of this angular-momentum absorption has been theoretically
exhibited in dissipative magnetic materialsScheid  _et al._ (2019, 2021), and
is enabled by the presence of spin-orbit coupling. Here, building on the
initial work of BethBeth (1936) and Scheid et al.Scheid  _et al._ (2019,
2022), we derive the general expression of the torque induced by light on the
electronic system, directly as a function of a measurable quantity: the
dielectric tensor. Doing so, we readily express the density of absorbed
orbital angular momentum in a wide variety of materials, and, using
experimentally measured dielectric tensors, we provide exact values for the
latter.
Notably, we show that the absorbed orbital angular momentum cancels in
isotropic non-dissipative materials, even for circularly polarized light, thus
contrasting with the IFE. Additionally, we find that a linearly polarized
electric field induces a second harmonic generation of orbital angular
momentum in non-absorptive but magnetic materials. As this effects arises
directly from the same broken symmetries as the Faraday effect, it thus
appears to be the actual counterpart of the latter. In contrast, we show that
in dissipative materials, the absorption of circularly polarized light induces
a torque on the electronic system, which direction is opposite for both
polarization of the light and that corresponds to a change of orbital angular
momentum of $\hbar$ per absorbed photon. Lastly, we examine non-isotropic
materials, which present additional channels for the absorption of orbital
angular momentum, even when they are not dissipative, i.e. do not absorb
energy.

##  II Theory

###  II.1 Light-induced torque and angular momentum as a function of linear
response tensors

We are interested in the calculation of the torque,
$\braket{\bm{\tau}}_{\text{ind}}(t)$, induced by the interaction of the light
with the electronic system. We begin by showing that only the light–induced
variation of the expectation value of the position operator, $\hat{\bm{r}}$,
acting on any arbitrary electronic subspace, and which we write
$\delta\braket{\hat{\bm{r}}}_{\text{ind}}(t)=\braket{\hat{\bm{r}}}(t)-\braket{\hat{\bm{r}}}_{0}$,
is needed to assess both of these quantities. Here the brackets stand for the
expectation value, which are performed on the light-induced time-evolving
states, except when written as $\braket{}_{0}$ in which case it is performed
on the ground state. $\braket{\hat{\bm{r}}}_{0}$ therefore is the ground state
position of the electronic center of mass, static in the frame of the
laboratory. Note that we ignore microscopic variations, i.e. we consider a
macroscopic average, which here is also performed on a homogeneous medium such
that we can neglect any spatial dependence of the average.
In this framework, the torque operator writes as:

$\begin{split}\hat{\bm{\Gamma}}&=\sum_{i}\left(\hat{\bm{r}}_{i}-\braket{\hat{\bm{r}}_{i}}_{0}\right)\times-e\bm{E}(t)\\\
\end{split}$ (1)

where $i$ is indexing the electronic subspaces, $-e$ is the charge of an
electron and $\bm{E}(t)$ is the electric field of the light, such that the
light-induced torque finally writes as:

$\braket{\bm{\Gamma}}_{\text{ind}}(t)=\delta\braket{\hat{\bm{r}}_{\text{tot}}}_{\text{ind}}(t)\times-e\bm{E}(t)$
(2)

where $\hat{\bm{r}}_{\text{tot}}=\sum_{i}\hat{\bm{r}}_{i}$, $i$ is indexing
the electronic subspaces,
Eq. 2 can be further simplified on the account that, due to the symmetry of
the wavefunction:

$\begin{split}\bra{\Psi(t)}\hat{\bm{r}}_{i}\ket{\Psi(t)}&=\int\Psi^{*}(\hat{\bm{r}}_{1},\dots,\hat{\bm{r}}_{i},\dots,\hat{\bm{r}}_{N})\hat{\bm{r}}_{i}\Psi(\hat{\bm{r}}_{1},\dots,\hat{\bm{r}}_{i},\dots,\hat{\bm{r}}_{N})d\hat{\bm{r}}_{1}\dots
d\hat{\bm{r}}_{N}\\\
&=\int\Psi^{*}(\hat{\bm{r}}_{i},\hat{\bm{r}}_{1},\dots\hat{\bm{r}}_{N})\hat{\bm{r}}_{i}\Psi(\hat{\bm{r}}_{i},\hat{\bm{r}}_{1},\dots\hat{\bm{r}}_{N})d\hat{\bm{r}}_{1}\dots
d\hat{\bm{r}}_{N}\\\
&=\int\Psi^{*}(\hat{\bm{r}}_{1},\dots,\hat{\bm{r}}_{N})\hat{\bm{r}}_{1}\Psi(\hat{\bm{r}}_{1},\dots,\hat{\bm{r}}_{N})d\hat{\bm{r}}_{1}\dots
d\hat{\bm{r}}_{N}\end{split}$ (3)

such that we have
$\bra{\Psi(t)}\hat{\bm{r}}_{i}\ket{\Psi(t)}=\bra{\Psi(t)}\hat{\bm{r}}_{1}\ket{\Psi(t)}$,
i.e. the expectation value of an operator acting on any subspace is the same,
which simply translates the indistinguishability of identical particles.
Consequently,
$\braket{\hat{\bm{r}}_{\text{tot}}}(t)=N_{e}\braket{\hat{\bm{r}}}(t)$ where
$N_{e}$ is the total number of electrons. Eq. 2 can therefore equally be
written in terms of the light-induced torque density,
$\braket{\hat{\bm{\tau}}}_{\text{ind}}(t)=\frac{\braket{\bm{\Gamma}}_{\text{ind}}(t)}{V}$,
where $V$ is the volume occupied by the material, and the electronic density,
$\rho_{e}=\frac{N_{e}}{V}$, such as:

$\braket{\bm{\tau}}_{\text{ind}}(t)=\rho_{e}\delta\braket{\hat{\bm{r}}}_{\text{ind}}(t)\times-e\bm{E}(t)$
(4)

Hence, when considering the light-induced polarization density one gets:

$\begin{split}\braket{\hat{\bm{P}}}_{\text{ind}}(t)&=-e\rho_{e}\delta\braket{\hat{\bm{r}}}_{\text{ind}}(t)\\\
&=\varepsilon_{0}\int_{-\infty}^{\infty}dt^{\prime}\chi_{\bm{p}}(t-t^{\prime})\bm{E}(t^{\prime}),\end{split}$
(5)

where the passage from the first to the second line is true within the
framework of linear response. From Eq. 5, the light-induced variation of the
electronic center of mass finally expresses as:

$\begin{split}\delta\braket{\hat{\bm{r}}}_{\text{ind}}(t)&=\frac{\varepsilon_{0}}{-e\rho_{e}}\int_{-\infty}^{\infty}dt^{\prime}\chi_{\bm{p}}(t-t^{\prime})\bm{E}(t^{\prime})\\\
&=\frac{\varepsilon_{0}}{-e\rho_{e}}\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\tilde{\chi}_{\bm{p}}(\omega)\tilde{\bm{E}}(\omega)e^{-i\omega
t}.\end{split}$ (6)

Here $\tilde{\chi}_{\bm{p}}(\omega)$ and $\tilde{\bm{E}}(\omega)$ respectively
are the Fourier transform of the polarizability tensor and of the electric
field of the light. Indeed, we now interest ourselves in the spectrum of
$\braket{\hat{\bm{\tau}}}_{\text{ind}}$ which, given by its obvious dependence
on the polarizability and dielectric tensors are expected to vary with the
frequency of the incident electromagnetic field, as well as their polarization
and direction with respect to the crystallographic axis.
To investigate such a dependence, we now consider the incident light to be a
monochromatic planewave, which electric field can thus be written as:

$\bm{E}(t)=E_{0}\Re(e^{i\Omega t}\bm{u}),$ (7)

with $\bm{u}$ a Jones vector from which all the polarizations and propagation
directions can be expressed and $E_{0}$ the amplitude of the electric field.
It’s Fourier transform expresses as:

$\tilde{\bm{E}}(\omega)=\pi
E_{0}\left(\delta(\Omega+\omega)\bm{u}+\delta(\Omega-\omega)\bm{u}^{*}\right)$
(8)

Using Eq. 4 and 8, we can now compute the light-induced torque
$\braket{\hat{\bm{\tau}}}_{\text{ind}}(t)$ exerted by arbitrarily polarized
incident planewaves. Indeed, expressing Eq. 4 as a function of
$\tilde{\chi}_{\bm{p}}(\omega)$ and $\tilde{\bm{E}}(\omega)$ yields:

$\braket{\hat{\bm{\tau}}}_{\text{ind}}(t)=\frac{\varepsilon_{0}}{(2\pi)^{2}}\int_{-\infty}^{+\infty}d\omega\int_{-\infty}^{+\infty}d\omega^{\prime}\left(\tilde{\chi}_{\bm{p}}(\omega)\cdot\tilde{\bm{E}}(\omega)\right)\times\tilde{\bm{E}}(\omega^{\prime})e^{-i(\omega+\omega^{\prime})t}$
(9)

which, upon injection of Eq. 8 gives:

$\begin{split}\braket{\hat{\bm{\tau}}}_{\text{ind}}(t)&=\frac{1}{2}\varepsilon_{0}E^{2}_{0}\big{[}\mathfrak{Re}\left[e^{-i2\Omega
t}\left(\tilde{\chi}_{\bm{p}}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}^{*}\right]\\\
&+\mathfrak{Re}\left[\left(\tilde{\chi}_{\bm{p}}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}\right]\big{]}.\end{split}$
(10)

Both of the latter quantities can equally be written in terms of the
transversal dielectric response by considering the relation:

$\varepsilon(\Omega)=\mathbb{1}+\chi_{\bm{p}}(\Omega),$ (11)

such that we finally obtain:

$\begin{split}\braket{\hat{\bm{\tau}}}_{\text{ind}}(t)&=\frac{1}{2}\varepsilon_{0}E^{2}_{0}\big{[}\mathfrak{Re}\left[e^{-i2\Omega
t}\left(\tilde{\varepsilon}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}^{*}\right]\\\
&+\mathfrak{Re}\left[\left(\tilde{\varepsilon}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}\right]\big{]},\end{split}$
(12)

or, equivalently in term of the intensity, I:

$\begin{split}\braket{\hat{\bm{\tau}}}_{\text{ind}}(t)&=\frac{1}{c}I\big{[}\mathfrak{Re}\left[e^{-i2\Omega
t}\left(\tilde{\varepsilon}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}^{*}\right]\\\
&+\mathfrak{Re}\left[\left(\tilde{\varepsilon}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}\right]\big{]},\end{split}$
(13)

with $c$ the speed of light.

Using Eq. 13, one can thus compute the total density of orbital angular
momentum absorbed by any compound, as long as the intensity and $\varepsilon$
are known as a function of time.

###  II.2 Case of a planewave propagating along the $z$ axis in different
types of media

We now proceed to the evaluation of the amplitude of the light-induced torque
by using, as much as possible, experimentally measured values of the
dielectric function. In their ground state, this quantity has been measured
for a wide range of materials, such that in what follows $\varepsilon$ is
considered time-independent. By considering this, instead of assessing the
induced torque per unit of intensity, we can compute the induced orbital
angular momentum per unit of fluence. Through the gyroscopic ratio, $\gamma$,
we can thus compute the absorbed magnetization, $\bm{m}_{\text{abs}}$, per
unit of fluence such as:

$\begin{split}\frac{d\bm{m}_{\text{abs}}}{dF}&=\frac{\gamma}{c}\big{[}\mathfrak{Re}\left[e^{-i2\Omega
t}\left(\tilde{\varepsilon}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}^{*}\right]\\\
&+\mathfrak{Re}\left[\left(\tilde{\varepsilon}(\Omega)\cdot\bm{u}^{*}\right)\times\bm{u}\right]\big{]}.\end{split}$
(14)

Here we note the key difference between this induced torque, resulting in an
induced magnetization per unit of fluence, and the induced magnetization per
unit of intensity, as in the IFE.

In what follows, we express all the possible polarizations of the light
propagating along the $z$ axis using Jones vectors writing as:

$\bm{u}=\begin{bmatrix}\cos{\theta}\\\ \sin{\theta}e^{i\phi}\\\
0\end{bmatrix}.$ (15)

Using Eq. 15, arbitrarily defined linear polarizations located in the $xy$
plane are defined by $\phi=0$ and any value of $\theta$. On the other hand,
the two circular polarizations, $\sigma^{\pm}$ are obtained for
$\phi=\pm\frac{\pi}{2}$ and for $\theta=\frac{\pi}{4}$.

####  II.2.1 Isotropic media

In isotropic media, the dielectric tensor writes as:

$\tilde{\varepsilon}=\left(\begin{matrix}\varepsilon^{\prime}_{xx}+i\varepsilon^{\prime\prime}_{xx}&0&0\\\
0&\epsilon^{\prime}_{xx}+i\varepsilon^{\prime\prime}_{xx}&0\\\
0&0&\varepsilon^{\prime}_{xx}+i\varepsilon^{\prime\prime}_{xx}\end{matrix}\right)$
(16)

Just as the energy absorption of light by dissipative media (see Annex VI.3),
the rate of absorption of orbital angular momentum, or magnetization, is
directly proportional to the imaginary part of $\tilde{\varepsilon}$:

$\frac{d\bm{m}_{\text{abs}}}{dF}=-2\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xx}\sin{\left(\phi\right)}\sin{\left(\theta\right)}\cos{\left(\theta\right)}\bm{u}_{z}.$
(17)

Indeed, Eq. 17 shows that a torque is applied by non-linearly polarized light,
i.e. when $\phi\neq n\pi$ and $\theta\neq n\frac{\pi}{2}$. It takes its
maximum amplitude for circularly polarized light and its direction reverses
with the helicity such as:

$\frac{d{\bm{m}_{\text{abs},\sigma^{\pm}}}}{dF}=\mp\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xx}\bm{u}_{z}.$
(18)

Interestingly, by dividing Eq. 18 by the amount of absorbed photons per unit
of fluence as expressed in Eq. 50, we obtain:

$\frac{d\bm{m}_{\text{abs},\sigma^{\pm}}}{dn}=\mp\gamma\hbar\bm{u}_{z},$ (19)

thus showing that, upon absorption of a circularly polarized photon, an
angular momentum of $\hbar$ is indeed transferred to the electronic system.

Also, we note that in this case, the second harmonic generation does not
contribute to the induced torque, and that, as $\bm{m}_{\text{abs}}$ does not
depend on $\varepsilon^{\prime}_{xx}$, orbital angular momentum cannot be
absorbed in the absence of dissipation. These facts are well illustrated in
Fig. 1, where we used experimental measures of the dielectric function is
various isotropic compounds. Indeed, metals are absorbing angular momentum at
all photon energy, even though the phenomenon is particularly strong at low
energy where the contribution of intraband transition (Drude like), is
dominant. We note a singularity in Cu, which presents a peak of absorption at
$\approx 1$eV. On the other hand, in semi-conductor, the absorption of angular
momentum takes place only at energies greater than the gap, where absorption
is possible.

Figure 1: Top: absorbed magnetic moment in $\mu_{B}$ per atom and per 10mJ
cm-2 (fluence) in Ag, Au, Pt and Cu, calculated using the experimentally
measured dielectric functions from Ref. Werner  _et al._ (2009). Bottom:
absorbed magnetic moment in $\mu_{B}$ per conventional unit cell and per mJ
cm-2 (fluence) in Si, Ge, GaP and GaAs, calculated using the experimentally
measured dielectric functions from Ref. Aspnes and Studna (1983).

####  II.2.2 Uniaxial media

In uniaxial media, the dielectric tensor writes as:

$\tilde{\varepsilon}(\Omega)=\left(\begin{matrix}\tilde{\varepsilon}^{\prime}_{xx}+i\tilde{\varepsilon}^{\prime\prime}_{xx}&0&0\\\
0&\tilde{\varepsilon}^{\prime}_{yy}+i\tilde{\varepsilon}^{\prime\prime}_{yy}&0\\\
0&0&\tilde{\varepsilon}^{\prime}_{yy}+i\tilde{\varepsilon}^{\prime\prime}_{yy}\end{matrix}\right),$
(20)

thus leading to the following absorbed density of angular momentum per unit of
fluence:

$\begin{split}\frac{d{\bm{m}_{\text{abs}}}}{dF}=-\frac{\gamma}{c}\left(\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}\right)\sin{\left(\theta\right)}\cos{\left(\theta\right)}\Big{[}\cos{\left(\phi\right)}\\\
+\cos{\left(\phi\right)}\cos{\left(2\Omega
t\right)}-\sin{\left(\phi\right)}\sin{\left(2\Omega
t\right)}\Big{]}\bm{u}_{z}\\\
-\frac{\gamma}{c}\left(\varepsilon^{\prime\prime}_{xx}+\varepsilon^{\prime\prime}_{yy}\right)\sin{\left(\theta\right)}\cos{\left(\theta\right)}\sin{\left(\phi\right)}\bm{u}_{z}\\\
+\frac{\gamma}{c}\left(\varepsilon^{\prime\prime}_{xx}-\varepsilon^{\prime\prime}_{yy}\right)\sin{\left(\theta\right)}\cos{\left(\theta\right)}\Big{[}\sin{\left(\phi\right)}\cos{\left(2\Omega
t\right)}\\\ +\cos{\left(\phi\right)}\sin{\left(2\Omega
t\right)}\Big{]}\bm{u}_{z}\end{split}$ (21)

As shown by the first two lines of Eq. 21, and contrary to the isotropic case,
a component proportional to
$\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}$, which are real parts of
the dielectric tensor is now present. Consequently, in such media, orbital
angular momentum can be absorbed from light, even in absence of energy
absorption. Contrary to the isotropic case, this phenomenon is proportional to
$\cos{\left(\phi\right)}$, i.e. its amplitude is maximal for linearly
polarized light ($\phi=n\pi$) sitting in the directions
$\theta=\pm\frac{\pi}{4}$. Moreover, the sign of this effect reverses with
$\theta$. In such a polarization a second harmonic is also generated such that
the total torque writes as:

$\frac{d\bm{m}_{\text{abs},\pi^{\theta=\pm\frac{\pi}{4}}}^{\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}}}{dF}=\mp\frac{\gamma}{2c}\left(\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}\right)\left(1+\cos{\left(2\Omega
t\right)}\right)\bm{u}_{z}$ (22)

This phenomenon cancels for $\theta=n\frac{\pi}{2}$, i.e. whenever the
polarization of the light is linear and along $x$ or $y$, corresponding to the
eigenstates of the dielectric tensor.

For circular polarizations, the real part of the dielectric tensor is only
responsible for a second harmonic generation, which phase factor depends on
the helicity:

$\frac{d\bm{m}_{\text{abs},\sigma^{\pm}}^{\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}}}{dF}=\pm\frac{\gamma}{2c}\left(\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}\right)\sin\left(2\Omega
t\right)\bm{u}_{z}$ (23)

On top of this non-dissipative contribution, the presence of dissipation
brings a static torque which has the same characteristics as in the isotropic
absorptive media, i.e. cancels for linearly polarized light, has its maximal
amplitude for circular polarizations, and reverses its sign with the latter,
such as:

$\frac{d\bm{m}_{\text{abs},\sigma^{\pm}}^{\varepsilon^{\prime\prime}_{xx}+\varepsilon^{\prime\prime}_{yy}}}{dF}=\mp\frac{\gamma}{2c}\left(\varepsilon^{\prime\prime}_{xx}+\varepsilon^{\prime\prime}_{yy}\right)\bm{u}_{z}$
(24)

Moreover, and as shown by the last two lines of Eq. 21, the complex part of
the dielectric tensor also induces a second harmonic proportional to
$\varepsilon^{\prime\prime}_{xx}-\varepsilon^{\prime\prime}_{yy}$. As the
second harmonic proportional to
$\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}$, the latter cancels for
linearly polarized light within the $x$ and $y$ directions and takes its
maximum amplitude for circularly polarized light, expressing as:

$\frac{d\bm{m}_{\text{abs},\sigma^{\pm}}^{\varepsilon^{\prime\prime}_{xx}-\varepsilon^{\prime\prime}_{yy}}}{dF}=\pm\frac{\gamma}{2c}\left(\varepsilon^{\prime\prime}_{xx}-\varepsilon^{\prime\prime}_{yy}\right)\sin(2\Omega
t)\bm{u}_{z},$ (25)

as well as for linearly polarized light when $\theta=\pm\frac{\pi}{4}$, such
as:

$\frac{d\bm{m}_{\text{abs},\pi^{\theta=\pm\frac{\pi}{4}}}^{\varepsilon^{\prime\prime}_{xx}-\varepsilon^{\prime\prime}_{yy}}}{dF}=\mp\frac{\gamma}{2c}\left(\varepsilon^{\prime\prime}_{xx}-\varepsilon^{\prime\prime}_{yy}\right)\cos(2\Omega
t)\bm{u}_{z}$ (26)

Figure 2: Amplitude of the contributions to the absorbed orbital magnetic
moment per unit cell of rutile TiO2 using experimental dielectric functions
measured by Jellison et al.Jellison  _et al._ (2003), and per unit cell of
monolayer WSe2 using experimental dielectric functions measured by Chu et
al.Chu and Zhang (2020).

Using experimental measurements of the dielectric function in rutile
TiO2Jellison  _et al._ (2003), which space group is P42/mnm and holds 12 atoms
within its conventional cell, and monolayer WSe2Chu and Zhang (2020) which
space group is P63/mmc and holds 6 atoms within its unit cell, we compute the
amplitude of the different aforementioned phenomena, as seen in Fig. 2. As
both of these compounds present a gap, the dissipative part, proportional to
$\varepsilon^{\prime\prime}_{xx}+\varepsilon^{\prime\prime}_{yy}$ is not
contributing up to $\approx 3.5$eV in TiO2, and $\approx 1.5$eV in WSe2. On
the other hand, even below these energies, the non-dissipative contributions,
proportional to $\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}$ are
allowing for the absorption of orbital angular momentum from the light within
the aforementioned conditions of Eq. 22 and Eq. 23. The sign of this
component, which can lead to the absorption of orbital angular momentum even
using a linearly polarized light, depends on the energy of the light in both
compounds. In the two compounds, when above the gap energy, the largest
contribution comes from the regular, static part of the torque, proportional
to $\varepsilon^{\prime\prime}_{xx}+\varepsilon^{\prime\prime}_{yy}$
represented by Eq. 24, and which is maximal for circular polarizations. We
notice however that, even above the absorption threshold, the magnetization
induced by the static part of the torque, proportional to
$\varepsilon^{\prime}_{xx}-\varepsilon^{\prime}_{yy}$, cannot be neglected in
front of its dissipative counterpart. Finally, the same can be said regarding
the second harmonics generations, which are proportional to
$\varepsilon^{\prime\prime}_{xx}+\varepsilon^{\prime\prime}_{yy}$, and thus
only exist above the absorption threshold.

####  II.2.3 Monoclinic media

In monoclinic media, the dielectric tensor writes as:

$\tilde{\varepsilon}(\Omega)=\left(\begin{matrix}\tilde{\varepsilon}^{\prime}_{xx}+i\tilde{\varepsilon}^{\prime\prime}_{xx}&\tilde{\varepsilon}^{\prime}_{xy}+i\tilde{\varepsilon}^{\prime\prime}_{xy}&0\\\
\tilde{\varepsilon}^{\prime}_{xy}+i\tilde{\varepsilon}^{\prime\prime}_{xy}&\tilde{\varepsilon}^{\prime}_{xx}+i\tilde{\varepsilon}^{\prime\prime}_{xx}&0\\\
0&0&\tilde{\varepsilon}^{\prime}_{xx}+i\tilde{\varepsilon}^{\prime\prime}_{xx}\end{matrix}\right)$
(27)

Such that the induced torque is:

$\begin{split}\frac{d\bm{m}_{\text{abs}}}{dF}(t)=-2\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xx}\sin{\left(\phi\right)}\sin{\left(\theta\right)}\cos{\left(\theta\right)}\bm{u}_{z}\\\
-\frac{\gamma}{c}\varepsilon^{\prime}_{xy}\big{[}\cos{\left(2\theta\right)}+\left(\sin^{2}\theta\cos
2\phi-\cos^{2}\theta\right)\cos 2\Omega t\\\
-2\sin^{2}\theta\sin\phi\cos\phi\sin 2\Omega t\big{]}\bm{u}_{z}\\\
-\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\big{[}\left(\sin^{2}\theta\cos
2\phi-\cos^{2}\theta\right)\sin 2\Omega t\\\
-2\sin^{2}\theta\sin\phi\cos\phi\cos 2\Omega t\big{]}\bm{u}_{z}.\end{split}$
(28)

The first line of Eq. 28 is the component of the torque induced by the
dissipative part of the diagonal elements, and is similar to the
aforementioned isotropic case, given by Eq. 17. In addition, two components
proportional to the non-diagonal elements, $\tilde{\varepsilon}^{\prime}_{xy}$
and $\tilde{\varepsilon}^{\prime\prime}_{xy}$, due to the monoclinic nature of
the compound, are appearing. Remarkably, only the former brings a static
contribution to the torque, and therefore leads to the absorption of angular
momentum. It does not depend on the ellipticity of the light, i.e. on $\phi$,
and cancels when $\theta=\frac{\pi}{4}+n\frac{\pi}{2}$, which when linearly
polarized, correspond to the eigenstate polarization of the dielectric tensor.
On the other hand, its amplitude is maximal along the directions $\theta=0$,
and $\theta=\pi$, i.e. in the case of a linearly polarized light along the $x$
and $y$ direction, and writes as:

$\frac{d\bm{m}_{\text{abs},\pi=\left\\{0,\pi\right\\}}^{\text{stat.}}}{dF}=\mp\frac{\gamma}{c}\varepsilon^{\prime}_{xy}\bm{u}_{z}.$
(29)

Still for a linearly polarized light, the second harmonic generation
proportional to $\tilde{\varepsilon}^{\prime}_{xy}$ has the same properties as
the aforementioned static component, thus expressing as:

$\frac{d\bm{m}_{\text{abs},\pi=\left\\{0,\pi\right\\}}^{2^{\text{nd}}\text{h.}}}{dF}(t)=\mp\frac{\gamma}{c}\varepsilon^{\prime}_{xy}\cos\left(2\Omega
t\right)\bm{u}_{z}$ (30)

Thus, when the polarization is eigenvalue of the dielectric tensor, the whole
torque cancels. On the other hand, when circularly polarized, the second
harmonic generation writes as:

$\frac{d\bm{m}_{\text{abs},\sigma^{\pm}}^{2^{\text{nd}}\text{h.}}}{dF}(t)=-\frac{\gamma}{c}\varepsilon^{\prime}_{xy}\cos\left(2\Omega
t\right)\bm{u}_{z},$ (31)

i.e. is the same for both helicities of the light.

Finally, the second harmonics proportional to
$\tilde{\varepsilon}^{\prime\prime}_{xy}$ present exactly the same properties
as the ones proportional to its non-dissipative counterpart, with the addition
of a de-phasing of $-\frac{\pi}{2}$, i.e.:

$\frac{d\bm{m}_{\text{abs},\pi=\left\\{0,\pi\right\\}}^{2^{\text{nd}}\text{h.}}}{dF}(t)=\mp\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\sin\left(2\Omega
t\right)\bm{u}_{z},$ (32)

and

$\frac{d\bm{m}_{\text{abs},\sigma^{\pm}}^{2^{\text{nd}}\text{h.}}}{dF}(t)=-\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\sin\left(2\Omega
t\right)\bm{u}_{z}.$ (33)

Figure 3: Amplitude of the contributions to the absorbed orbital magnetic
moment per unit cell of Ga2O3 using experimental dielectric functions measured
by Sturm et al.Sturm  _et al._ (2015).

To illustrate this case we use the work of Sturm et al.Sturm  _et al._ (2015),
who experimentally measured the dielectric tensor of the monoclinic ($\beta$)
phase of Ga2O3. It possesses 20 atoms per unit cell, which volume represents
211.47 A3. Using the same orientation as Sturm et al., its dielectric tensor
writes as:

$\tilde{\varepsilon}(\Omega)=\left(\begin{matrix}\tilde{\varepsilon}_{xx}&0&\tilde{\varepsilon}_{xz}\\\
0&\tilde{\varepsilon}_{yy}&0\\\
\tilde{\varepsilon}_{xz}&0&\tilde{\varepsilon}_{zz}\end{matrix}\right)$ (34)

Consequently, in addition of the non-diagonal term,
$\tilde{\varepsilon}_{xz}$, and depending on the orientation of the
perturbation, i.e. of the plane in which the electric field is evolving
through time ($xy$, $yz$ or $zx$), the phenomena represented by Eq. 22, 23,
24, 25, and 26 will triggered at three different amplitudes, all of which are
represented in the first 9 graphs of Fig. 3. As, in general, the values taken
by the dielectric function of Ga2O3 are small as compared to the other
compounds studied supra, the amplitude found for the different aforementioned
phenomena are equally smaller. However, and as usual, the absorption of
angular momentum proportional to the dissipative diagonal terms (here, their
sums), occurring in the case of a circularly polarized light are the largest.
Moreover, in this case, their value noticeably depends on the plane containing
the circularly polarized electric field. The latter dependence is only
exacerbated in the case of phenomena proportional to the difference of the
diagonal elements. Finally, and most directly related to the case of
monoclinic media, the phenomena proportional to $\varepsilon^{\prime}_{xz}$
and $\varepsilon^{\prime\prime}_{xz}$ have the same order of magnitude, and
therefore cannot be neglected with respect to the ones proportional to
differences of diagonal elements.

####  II.2.4 Isotropic magnetic material

Lastly, we treat the case of an isotropic magnetic material. Its dielectric
tensor writes as:

$\tilde{\varepsilon}(\Omega)=\left(\begin{matrix}\tilde{\varepsilon}^{\prime}_{xx}+i\varepsilon^{\prime\prime}_{xx}&i(\tilde{\varepsilon}^{\prime}_{xy}+i\tilde{\varepsilon}^{\prime\prime}_{xy})&0\\\
-i(\tilde{\varepsilon}^{\prime}_{xy}+i\tilde{\varepsilon}^{\prime\prime}_{xy})&\tilde{\varepsilon}^{\prime}_{xx}+i\tilde{\varepsilon}^{\prime\prime}_{xx}&0\\\
0&0&\tilde{\varepsilon}^{\prime}_{xx}+i\tilde{\varepsilon}^{\prime\prime}_{xx}\end{matrix}\right)$
(35)

$\begin{split}\frac{d\bm{m}_{\text{abs}}}{dF}(t)&=-2\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xx}\sin{\left(\phi\right)}\sin{\left(\theta\right)}\cos{\left(\theta\right)}\bm{u}_{z}\\\
&+\frac{\gamma}{c}\varepsilon^{\prime}_{xy}\big{[}\left(\cos^{2}\left(\theta\right)+\cos\left(2\phi\right)\sin^{2}\left(\theta\right)\right)\sin\left(2\Omega
t\right)\\\
&+2\sin\left(\phi\right)\cos\left(\phi\right)\sin^{2}\left(\theta\right)\cos\left(2\Omega
t\right)\big{]}\bm{u}_{z}\\\
&+\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\bm{u}_{z}\\\
&+\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\big{[}-\left(\cos^{2}(\theta)+\cos(2\phi)\sin^{2}(\theta)\right)\cos\left(2\Omega
t\right)\\\
&+2\sin^{2}\left(\theta\right)\cos\left(\phi\right)\sin\left(\phi\right)\sin\left(2\Omega
t\right)\big{]}\bm{u}_{z}\end{split}$ (36)

Here, and as in the previous cases, in a non-dissipative medium and when the
light is polarized in an eigenvalue of the dielectric tensor, which here are
the two helicities of the light, no torque is induced and we have:

$\frac{d\bm{m}_{\text{abs}}^{\text{non-dis.}}}{dF}=\bm{0}$ (37)

On the other hand, when the light is not in such a state, and as shown by the
term proportional to $\varepsilon^{\prime}_{xy}$, second harmonics will be
generated. When linearly polarized, the latter writes as:

$\frac{d\bm{m}_{\text{abs}}^{\varepsilon^{\prime}_{xy}}}{dF}(t)=\frac{\gamma}{c}\varepsilon^{\prime}_{xy}\sin(2\Omega
t)\bm{u}_{z}.$ (38)

Here we note that it is precisely the configuration in which the Faraday
effect takes place, i.e. a linearly polarized light, which axis of
polarization rotates as it propagates along the direction of the
magnetization. According to the Onsager reciprocal relations, it would
therefore appear that the IFE is the generation of a time-dependent magnetic
moment by such a linearly polarized light, as shown by Eq. 38, rather than the
generation of a magnetization density by circularly polarized light as
originally pinned by Pitaevskii Pitaevskii (1961). Indeed, this is readily
hinted by the fact that $\varepsilon^{\prime}_{xy}$, which is precisely the
component of the dielectric tensor that dictates the magnitude of rotation of
the light in the Faraday effect occurring in non-lossy materials, is also the
component to which the second harmonic generation described by Eq. 38 is
proportional to.
The presence of dissipation brings many additional contributions. The first
one, represented by the first line of Eq. 36 is the component of the torque
induced by the dissipative part of the diagonal elements, and is similar to
the aforementioned isotropic case, given by Eq. 17. The second one, also
static, writing as:

$\frac{d\bm{m}_{\text{abs}}^{\varepsilon^{\prime\prime}_{xy},\text{stat.}}}{dF}=\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\bm{u}_{z},$
(39)

is the same for any polarization of the light and is due to both the presence
of a magnetic moment and of dissipation. It shows that, in such materials,
even linearly polarized light can induce the absorption of a magnetic moment
by the matter. Also proportional to $\varepsilon^{\prime\prime}_{xy}$, and as
shown by the last two lines of Eq. 36, the dissipative, out of diagonal part
of the response also participates to the generation of second harmonics which,
in the case of a linear polarization writes as:

$\frac{d\bm{m}_{\text{abs}}^{\varepsilon^{\prime\prime}_{xy},\text{sec.
h.}}}{dF}(t)=\frac{\gamma}{c}\varepsilon^{\prime\prime}_{xy}\cos(2\Omega
t)\bm{u}_{z}.$ (40)

As expected because of rotational symmetry around the $z$ axis, and contrary
to the case of a monoclinic material, the induced torque does not depend on
$\theta$.

Fig. 4 presents the amplitude of all the aforementioned phenomena in Ni, Co
and Fe. While $\varepsilon^{\prime\prime}_{xx}$ is extracted from
experimentsWerner  _et al._ (2009), the out-of-diagonal components are
obtained from ab initio density functional linear response calculations (see
annexes). In all the compounds, and by 4 to 5 orders of magnitude, the
component of the torque induced by the dissipative part of the diagonal
elements dominates. Therefore, and contrary to the case of monoclinic
compounds, the out-of-diagonal elements induced by the presence of a magnetic
moment are negligible. This conclusion holds true as much in the optical range
than at the M-edge.

Figure 4: Amplitude of the contributions to the absorbed orbital magnetic
moment per unit cell in the elemental ferromagnets Ni, Co and Fe using
experimental dielectric functions measured by Werner et al.Werner  _et al._
(2009) et al.Sturm  _et al._ (2015).

##  III Discussions

###  III.1 Symmetry breaking and light-induced torque

As evidenced throughout Sec. II.2 the light-induced torque in permitted by any
symmetry-breaking arrangement of the matter, be it spatial and thus due to a
deviation from the isotropic configuration, or related to the breaking of
time-reversal symmetry, manifesting through the presence of dissipation and/or
magnetism. Indeed, in such type of media, the polarization of light can
deviate from the eigenstates of the dielectric tensor such that a static and a
second harmonic torque generation can emerge.
As a consequence, and differently from the light-induced angular momentum as
within the IFE defined by Pitaevskii, no torque is applied by light in
isotropic, transparent and non-magnetic materials. On the other hand, the
largest contribution to the light-induced torque generally directly arises in
lossy materials from the absorption of circularly polarized light.
Interestingly, as shown in Sec. II.2.1, and directly supporting the fact that
the spin-angular momentum of light is absorbed by the electronic system, in
isotropic materials the angular momentum absorbed as a consequence the
integrated torque produced per absorbed circularly-polarized photon is equal
to $\hbar$.
On the other hand, angular momentum can also be absorbed without the presence
of dissipation in uniaxial and monoclinic media. In other words, a static
contribution in the induced torque can exist without a static contribution in
the light-induced work. This is notably remanent of the seminal work of
BethBeth (1936) showing that, when light propagates through a birefringent
material and its polarization evolves, it induces a torque that could be
experimentally measured. Conversely, in non-dissipative magnetic materials,
time-reversal symmetry breaking does not induce a static torque, but rather
generates a dynamic, time-dependent torque that oscillates sinusoidally at the
second harmonic of the incident light frequency.

###  III.2 Differences between the inverse Faraday effect and the light-
induced torque

While discovered more than half a century ago, the inverse Faraday effect as
described by PitaevskiiPitaevskii (1961) has gained much interest in the past
decades. Indeed, as the latter originally described the induction of a
magnetization in non-magnetic transparent media proportional to the intensity
of the light, and which direction depends on its helicity, it stood as a prime
candidate in the explanation of a plethora of light-induced helicity-dependent
effects such as the all-optical helicity-dependent switchingMangin  _et al._
(2014); Lambert  _et al._ (2014), the helicity-dependent demagnetizationTsema
_et al._ (2016); Quessab  _et al._ (2019) and the helicity-dependent domain
wall motionQuessab  _et al._ (2018). While the effect discussed in this work,
namely the absorption of orbital angular momentum, also results in the
induction of a magnetization by the light, it differs from the IFE. This is
most obviously evidenced by the fact that, contrary to the latter, the
absorption of angular momentum cancels in isotropic non-dissipative media.
Furthermore, its magnitude is not proportional to the intensity of light, as
is the IFE, but rather to its fluence. This difference, and as pinpointed in
Scheid et al.Scheid  _et al._ (2022) for the spin counterpartScheid  _et al._
(2019) of the currently discussed effect, originates from the fact that,
rather than being a light-induced moment as is the IFE, it is the consequence
of a light-induced torque. Consequently, and just as the absorption of energy
from light, the angular momentum absorbed as a result of the integrated light-
induced torque stays, even after the light is gone. As such, this effect, as
well as its spin counterpartScheid  _et al._ (2019) have a lasting impact on
the magnetization, that perdure long after the light is gone. Given their
irreversible nature, the light-induced torques are therefore prime candidates
in the explanation of the aforementioned helicity-dependent effects. In the
light of this work one could thus re-qualify theoretical results mis-
attributed to the IFEHurst  _et al._ (2018); Sinha-Roy  _et al._ (2020) as
being the result of the light-induced torque.

###  III.3 Towards a new definition of the inverse Faraday effect?

The Faraday effect, also known as Faraday rotation, is a magneto-optical
phenomenon in which the plane of polarization of linearly polarized light
rotates as it passes through a transparent material under the influence of a
magnetic field, or magnetization, parallel to the direction of light
propagation. On the other hand, the inverse Faraday effect as defined by
PitaevskiiPitaevskii (1961) was expressed in transparent materials that were
non-magnetic, or not submitted to any external magnetic field. In such a
setup, circularly polarized light would induce a magnetization which direction
depends on its helicity and which magnitude is proportional to its intensity.
While this phenomenon has been measured a while agoVan Der Ziel  _et al._
(1965), it does not appear to directly correspond to the reciprocal effect of
the Faraday rotation. Indeed, such an effect would have to take place in a
similar setup as the Faraday rotation, i.e. within a transparent material
under the influence of a magnetic field, or magnetization, in which a linearly
polarized light propagates. On the other hand, and as mentioned in Sec.
II.2.4, linearly polarized light generates a second harmonics magnetization
density when traveling through such a media, thus being the direct “inverse”
effect of the Faraday rotation. This is further evidenced by the fact that
both effects are proportional to $\varepsilon_{xy}$. According to Onsager
reciprocal relations, this effect would therefore be the inverse of the
Faraday rotation, i.e. the IFE.

##  IV Conclusion

Within the framework of linear response theory and the dipole approximation,
we derived the general expression of the torque induced by the spin part of
light on the electronic system. We then showed that this light-induced torque
can readily be expressed in terms of a measurable quantity: the dielectric
tensor. When considering time-independent dielectric properties, this torque
can equivalently be expressed as an induced magnetization per unit of fluence,
thus pinpointing the difference between this phenomenon and the inverse
Faraday effect as defined by PitaevskiiPitaevskii (1961), being a light-
induced magnetization per unit of intensity. This key difference means that,
contrary to the inverse Faraday effect, the light-induced torque generates an
orbital magnetization that remains, even after the light is gone. We showed
that, in dissipative media, the absorption of circularly polarized light
indeed changes the electronic angular momentum by $\hbar$ per photon.
Moreover, and in agreement with the work of BethBeth (1936), orbital angular
momentum can be absorbed even in transparent materials, as long as the spatial
symmetry of the material is not isotropic. When occurring in magnetic media,
it is yet not clear how this absorbed orbital angular momentum –and thus
magnetization– interacts with the existing one. Rather than being considered
to act as a so-called effective ”opto-magnetic field” as usually assumed in
the case of the IFEHertel (2006); Berritta  _et al._ (2016), such an induced
magnetization should more naturally be considered to interact with the
existing one through an exchange. While assessing the magnitude of the latter
is a challenging task, such a phenomenon could thus drive the various light-
induced helicity-dependent phenomena such as the all-optical helicity-
dependent switchingStanciu  _et al._ (2007); Mangin  _et al._ (2014); Lambert
_et al._ (2014), the helicity-dependent domain-wall motionQuessab  _et al._
(2018), and the helicity-dependent demagnetizationTsema  _et al._ (2016);
Quessab  _et al._ (2019), even long after the light is gone. Finally,
mutualizing pump-probe techniques to assess the variations of the dielectric
response through time with results from this work would allow experimentalists
to explicitly compute the amount of orbital angular momentum absorbed from the
pump.

##  V Acknowledgments

Philippe Scheid is grateful for the many fruitful discussions with B. Herzog,
C. Chatelain, A. Bergman and I. P. Miranda.
This work is supported by the French National Research Agency ANR for
ANR-20-CE09-0013 UFO, ANR-15-IDEX-04-LUE (“Lorraine Université d’Excellence”)
and through the France 2030 government grants EMCOM (ANR-22-PEEL-0009).

##  VI Annexes

###  VI.1 Fourier transform

We define the Fourier transform of a time-dependent function, $f$ as:

$\tilde{f}(\omega)=\int_{-\infty}^{\infty}f(t)e^{i\omega t}dt,$ (41)

where $\omega$ is the pulsation of the light, and such that the inverse
transform is:

$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{f}(\omega)e^{-i\omega
t}d\omega.$ (42)

###  VI.2 Work done by an electric field

The density of work per unit of time exerted by the electric field on the
electronic system writes as:

$P_{\text{abs}}(t)=\rho_{e}\frac{d}{dt}\delta\braket{\hat{\bm{r}}}_{\text{ind}}(t)\cdot-e\bm{E}(t)$
(43)

$P_{\text{abs}}(t)=\frac{\varepsilon_{0}}{(2\pi)^{2}}\int_{-\infty}^{+\infty}d\omega\int_{-\infty}^{+\infty}d\omega^{\prime}\left(-i\omega\tilde{\chi}_{\bm{p}}(\omega)\cdot\tilde{\bm{E}}(\omega)\right)\cdot\tilde{\bm{E}}(\omega^{\prime})e^{-i(\omega+\omega^{\prime})t}$
(44)

$\begin{split}P_{\text{abs}}(t)&=\frac{1}{2}\Omega\varepsilon_{0}E^{2}_{0}\big{[}\mathfrak{Im}\left[e^{-i2\Omega
t}\left(\tilde{\chi}_{\bm{p}}(\Omega)\cdot\bm{u}^{*}\right)\cdot\bm{u}^{*}\right]\\\
&+\mathfrak{Im}\left[\left(\tilde{\chi}_{\bm{p}}(\Omega)\cdot\bm{u}^{*}\right)\cdot\bm{u}\right]\big{]}.\end{split}$
(45)

Following the same proceeding as in Sec. II.2, Eq. 45 can be rewritten as an
absorbed energy, $E_{\text{abs}}$, per unit of fluence such as:

$\begin{split}\frac{dE_{\text{abs}}}{dF}(t)&=\frac{\Omega}{c}\big{[}\mathfrak{Im}\left[e^{-i2\Omega
t}\left(\tilde{\chi}_{\bm{p}}(\Omega)\cdot\bm{u}^{*}\right)\cdot\bm{u}^{*}\right]\\\
&+\mathfrak{Im}\left[\left(\tilde{\chi}_{\bm{p}}(\Omega)\cdot\bm{u}^{*}\right)\cdot\bm{u}\right]\big{]}.\end{split}$
(46)

###  VI.3 Number of absorbed photons in an isotropic, dissipative medium

In the case of an isotropic medium, the dielectric tensor writes as in Eq. 16
and the density of absorbed energy per unit of fluence writes as:

$\begin{split}\frac{dE_{\text{abs}}}{dF}=\frac{\Omega}{c}\varepsilon^{\prime\prime}_{xx}\\\
+\frac{\Omega}{c}\varepsilon^{\prime\prime}_{xx}\left(\cos^{2}\left(\theta\right)+\sin^{2}\left(\theta\right)\cos\left(2\phi\right)\right)\cos\left(2\Omega
t\right)\\\
-\frac{\Omega}{c}\varepsilon^{\prime\prime}_{xx}\sin^{2}\left(\theta\right)\sin\left(2\phi\right)\sin\left(2\Omega
t\right)\\\
+\frac{\Omega}{c}\left(1-\varepsilon^{\prime}_{xx}\right)\sin^{2}\left(\theta\right)\sin\left(2\phi\right)\cos\left(2\Omega
t\right)\\\
-\frac{\Omega}{c}\left(1-\varepsilon^{\prime}_{xx}\right)\sin^{2}\left(\theta\right)\sin^{2}\left(\phi\right)\sin\left(2\Omega
t\right)\\\ \end{split}$ (47)

For any polarization of the light, the absorptive part thus writes as:

$\frac{dE_{\text{abs}}^{\text{stat.}}}{dF}=\frac{\Omega}{c}\varepsilon^{\prime\prime}_{xx}$
(48)

On the other hand, the second harmonic generation cancels for circular
polarizations, while it writes as:

$\frac{dE_{\text{abs}}^{\text{2\textsuperscript{nd}h.}}}{dF}=\frac{\Omega}{c}\left(\varepsilon^{\prime\prime}_{xx}\cos\left(2\Omega
t\right)+\left(1-\varepsilon^{\prime}_{xx}\sin\left(2\Omega
t\right)\right)\right)$ (49)

for linearly polarized light.

In the case of a circularly polarized light, the number of absorbed photons
per unit of fluence therefore writes as:

$\frac{dn}{dF}=\frac{1}{c\hbar}\varepsilon^{\prime\prime}_{xx}$ (50)

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