# Time-dependent theory of optical electro- and magnetostriction

Mikko Partanen  Photonics Group, Department of Electronics and
Nanoengineering, Aalto University, P.O. Box 13500, 00076 Aalto, Finland  Bruno
Anghinoni  Department of Physics, Universidade Estadual de Maringá, Maringá,
PR 87020-900, Brazil  Nelson G. C. Astrath  Department of Physics,
Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil  Jukka Tulkki
Engineered Nanosystems Group, School of Science, Aalto University, P.O. Box
12200, 00076 Aalto, Finland

(February 28, 2023)

###### Abstract

Electrostriction, the deformation of dielectric materials under the influence
of an electric field, is of continuous interest in optics. The classic
experiment by Hakim and Higham [Proc. Phys. Soc. 80, 190 (1962)] for a
stationary field supports a different formula of the electrostrictive force
density than the recent experiment by Astrath _et al._ [Light Sci. Appl. 11,
103 (2022)] for an optical field. In this work, we study the origin of this
difference by developing a time-dependent covariant theory of optical force
densities in photonic materials. When a light pulse propagates in a bulk
dielectric, the field-induced force density consists of two parts: (i) The
optical wave momentum force density $\mathbf{f}_{\mathrm{owm}}$ carries the
wave momentum of light and drives forward a mass density wave of the covariant
coupled field-material state of light. (ii) The optostrictive force density
$\mathbf{f}_{\mathrm{ost}}$ arises from the atomic density dependence of the
electric and magnetic field energy densities. It represents an optical
Lorentz-force-law-based generalization of the electro- and magnetostrictive
force densities well known for static electromagnetic fields and derived from
the principle of virtual work. Since the work done by
$\mathbf{f}_{\mathrm{ost}}$ is not equal to the change of the field energy
density during the contraction of the material, we have to describe this
difference with optostriction-related dissipation terms to fulfill the energy
conservation. The detailed physical model of the dissipation is left for
further works. The optostrictive force density can be understood in terms of
field-induced pair interactions inside the material. Because of the related
action and reaction effects, this force density cannot contribute to the net
momentum transfer of the optical field. The theory is used to simulate the
propagation of a Gaussian light pulse through a dielectric material. We
calculate the electric and magnetic fields of the Gaussian light pulse from
Maxwell’s equations and simultaneously solve Newton’s equation of motion of
atoms to find how the velocity and displacement fields of atoms develop as a
function of time under the influence of the field-induced force density.

##  I Introduction

Electrostriction is a phenomenon where materials mechanically deform under the
action of an applied electric field in a way that the magnitude of this
deformation is proportional to the square of the electric field Landau1984 .
The magnetic analog is called magnetostriction. The theory of electrostriction
dates back to the early work of Helmholtz in 1881 Helmholtz1881 . Despite the
long history, the knowledge of electrostriction in optical fields has remained
in many ways incomplete due to the challenges in measuring weak optical forces
inside materials with an accuracy that would enable detailed comparison with
the theory. This situation is, however, changing as new technologies are
continuously developed.

An interesting recent discovery by Astrath _et al._ Astrath2022  has shown
that the conventional theory of electrostriction, which explains the results
of the classic Hakim-Higham experiment for static fields Hakim1962 , does not
reproduce the magnitude of the electrostriction effect in water at optical
frequencies. The difference between the conventional theory of electro- and
magnetostriction for static fields and the corresponding theory for time-
dependent optical fields and their relation to recent experiments is the
starting point of the present work. Several experiments of electrostriction at
optical frequencies have been carried out also previously Melloni1998 ;
Fellegara1997 ; Fellegara1998 ; Buckland1997 ; Buckland1997 ; Godbout2000 ;
Townsend1996 ; Dianov1990 ; Hui2015 ; Cai2013 ; Beugnot2012 , but their
accuracy has not enabled detailed analysis of the time-dependence of the
optical electro- and magnetostriction effects. We also point out that most
theoretical and experimental studies of the static electro- and
magnetostriction correspond to the thermodynamical equilibrium limit.
Experimental studies of electrostriction have also revealed interesting
phenomena, such as the giant electrostriction Zhang2022 ; ChenBo2018 ;
Wang2022 ; Yu2022 ; ChenXi2022 ; Zhang2021 ; Wu2020 ; Huang2019 ; Jin2019 ;
Karvounis2018 ; Zhang1998 , negative electrostriction Yamashita2018 ;
Neumayer2019 , and deformations of liquid crystals Guo2020 ; ZhangY2022  and
biological cells Torbati2022 . On the theoretical side, electrostriction has
also been under extensive study Wang2016a ; Jimenez2011 ; Khakpour2021 ;
Lee2005 ; Rakich2010 ; Rakich2011 ; Gojani2014 ; Smith2015 ; Hashimoto2018 ;
Pennec2014 ; Tanner2021 ; Lai1981 ; Torchigin2019 ; Suo2010 ; WangJ2010 ;
Washimi1976 ; Zhao2008 ; Li2014 .

When a light pulse propagates in a bulk dielectric, the field-induced force
density consists of two parts: (i) The optical wave momentum force density
$\mathbf{f}_{\mathrm{owm}}$ carries the wave momentum of light and drives
forward a mass density wave of the covariant coupled field-material state of
light Partanen2022b ; Partanen2017c ; Partanen2021b ; Partanen2019a ;
Partanen2019b . It has been extensively studied regarding its relation to the
Abraham-Minkowski controversy of the momentum of light Leonhardt2006a ;
Pfeifer2007 ; Barnett2010b ; Kemp2011 ; Milonni2010 ; Bliokh2017a ; Burt1973 ;
Mansuripur2010 ; Barnett2010a ; Leonhardt2014 ; Bliokh2017b . (ii) The
optostrictive force density $\mathbf{f}_{\mathrm{ost}}$ arises from the atomic
density dependence of the electric and magnetic field energy densities. The
starting point of the present work is the recently developed mass-polariton
theory of light Partanen2022b ; Partanen2017c ; Partanen2021b ; Partanen2019a
; Partanen2019b , which enables a relativistically consistent study of
$\mathbf{f}_{\mathrm{owm}}$ and the momentum transfer associated with the
propagation of light. We generalize the mass-polariton theory to include
$\mathbf{f}_{\mathrm{ost}}$ in a way that preserves the relativistic
covariance when both force densities are included in the theory.

The force density $\mathbf{f}_{\mathrm{ost}}$ has conventionally been
investigated in the limit of thermodynamics and quasi-static fields, and it
has been extended to time-averaged optical fields Boyd2008 . As briefly
discussed in Ref. Partanen2022b , the full time- and position-dependent
relativistically consistent theory of $\mathbf{f}_{\mathrm{ost}}$, in the
optical regime, has not been presented in detail previously to the best of our
knowledge. The present theory enables a formulation of the optical
electrostriction and optical magnetostriction in terms of time- and space-
dependent force fields without the need to introduce time or harmonic
averages. The unified theory of the time-dependent force density of the
optical field on the material has obvious potential in tailoring acousto-
optical coupling Wissmeyer2018 ; Manohar2016 ; Tam1983 ; Lai1982 ; Patel1981 ;
Davies1993 ; Pozar2018 ; Pozar2013  for both fundamental science and
technological needs.

The Lorentz-force-law-based derivation of the optical force density leads to
an expression of the sum of $\mathbf{f}_{\mathrm{owm}}$ and
$\mathbf{f}_{\mathrm{ost}}$. The origin of $\mathbf{f}_{\mathrm{owm}}$ relies
in the definition of the Poynting vector, describing the optical energy flux,
and the covariance principle of light in dielectric and magnetic media
Partanen2022b ; Partanen2017c ; Partanen2021b ; Partanen2019a ; Partanen2019b
. In contradistinction, $\mathbf{f}_{\mathrm{ost}}$ originates from the atomic
density dependence of the electric and magnetic field energy densities. Thus,
$\mathbf{f}_{\mathrm{ost}}$ can be traced back to pair interactions, and due
to this force and counterforce nature, the resulting momentum densities always
integrate to zero over the volume of the material. Therefore, there is no net
momentum transfer related to $\mathbf{f}_{\mathrm{ost}}$.

We assume that the material is lossless, linear, isotropic, optically
nondispersive, and that the Clausius-Mossotti relation is satisfied. These
conditions exclude complex molecular materials, in which, e.g., giant
electrostriction Zhang2022 ; ChenBo2018 ; Wang2022 ; Yu2022 ; ChenXi2022 ;
Zhang2021 ; Wu2020 ; Huang2019 ; Jin2019 ; Karvounis2018 ; Zhang1998  and
negative electrostriction Yamashita2018 ; Neumayer2019  have been discovered.
Thermophotonic effects are also excluded. Since the investigations of the
present work do not deal with thermodynamical equilibrium, we do not consider
extensions of the Clausius-Mossotti relation, which include wavelength,
temperature, or density-dependent corrections Harvey1998 ; Schiebener1990 .
Even with the restrictions above, the theory has a wide range of applications
in photonics technologies. It is, however, obvious that the theory can be
extended to more complex materials, but these extensions are not discussed in
detail in the present work.

This work is organized as follows. Section II presents the conservation laws
of the material, fields, and interactions, and how the force density appears
in them. Section III summarizes the relations of the macroscopic and local
electric and magnetic fields and flux densities and the relations of the
polarization and magnetization fields. These relations determine the density
dependence of the permittivity and permeability of the material. Section IV
presents in detail the different contributions of the force density which the
material experiences under the influence of an electromagnetic field. Section
V describes simulations of optical electrostriction in silicon. Section VI
presents the stress-energy-momentum (SEM) tensors of different parts of the
field-material system and its interactions. Section VII discusses implications
of the present theory to the previous mass-polariton theory of light. Finally,
conclusions are drawn in Sec. VIII.

##  II Conservation laws

As the foundation of the theory, we use the fundamental conservation laws of
energy and momentum. Together with Maxwell’s equations and the Lorenz force
law, this is a widely used and evidently the most fundamental starting point
for describing optical forces Penfield1967 ; Partanen2021b ; Partanen2022b ;
Anghinoni2022 ; Sheppard2014 ; Kemp2017 . In particular, we focus on writing a
relativistically consistent theory, which can be applied to all inertial
observers independent of their velocities with respect to the material. This
condition imposes strong limitations for possible forms of the SEM tensors and
the related force fields as discussed further in Sec. VI. Accordingly, when
writing the conservation laws below and the general expressions of the SEM
tensors in Sec. VI, we assume a general inertial frame. In the presentation of
the force density in Sec IV, we present the equations for the laboratory
frame, which is a special inertial frame, where the material atoms are at rest
before the force density of the electromagnetic field starts to accelerate
them. In the presentation below, we split the field-material system into two
parts: (1) the material subsystem and (2) the field+interaction subsystem and
write the conservation laws for both of these subsystems satisfying the law of
action and reaction.

###  II.1 Force density and conservation laws of the material

We assume that the material is composed of identical atoms. For a single atom,
the rest mass is denoted by $m_{0}$, the velocity is
$\mathbf{v}_{\mathrm{a}}$, and the momentum is
$\mathbf{p}_{\mathrm{a}}=\gamma_{\mathbf{v}_{\mathrm{a}}}m_{0}\mathbf{v}_{\mathrm{a}}$,
where
$\gamma_{\mathbf{v}_{\mathrm{a}}}=1/\sqrt{1-|\mathbf{v}_{\mathrm{a}}|^{2}/c^{2}}$
is the Lorentz factor, in which $c$ is the speed of light in vacuum. In the
laboratory frame, the atomic velocities caused by the optical field are very
small, and thus the Lorentz factor is close to unity. However, for the
consideration of relativistic invariance properties later in this work, we use
the general form that applies to all inertial observers. The atomic number
density is denoted by $n_{\mathrm{a}}$. The fundamental definition of the
force density $\mathbf{f}=n_{\mathrm{a}}\mathbf{F}$, where $\mathbf{F}$ is the
force on a single atom, is given by Newton’s equation of motion, written in
any inertial frame as Penfield1967 ; Partanen2021b

$n_{\mathrm{a}}\frac{d\mathbf{p}_{\mathrm{a}}}{dt}=\mathbf{f}.$ (1)

First, by using the material derivative $\frac{d}{dt}=\frac{\partial}{\partial
t}+\mathbf{v}_{\mathrm{a}}\cdot\boldsymbol{\nabla}$ to Eq. (1), second,
multiplying the conservation law of the atomic number density,
$\frac{\partial}{\partial
t}n_{\mathrm{a}}+\boldsymbol{\nabla}\cdot(n_{\mathrm{a}}\mathbf{v}_{\mathrm{a}})=0$,
by $\mathbf{p}_{\mathrm{a}}$, and, third, adding the resulting two equations
side by side gives after some vector algebra Penfield1967 ; Partanen2021b

$\frac{\partial\mathbf{G}_{\mathrm{mat}}}{\partial
t}+\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{T}}_{\mathrm{mat}}=\mathbf{f}.$
(2)

Here $\mathbf{G}_{\mathrm{mat}}=n_{\mathrm{a}}\mathbf{p}_{\mathrm{a}}$ is the
momentum density of the material and
$\boldsymbol{\mathcal{T}}_{\mathrm{mat}}=n_{\mathrm{a}}\mathbf{p}_{\mathrm{a}}\otimes\mathbf{v}_{\mathrm{a}}$,
where $\otimes$ denotes the outer product of vectors, is the stress tensor of
the material, which does not account for the stresses associated to the
mechanical or electromagnetically induced pressures. Correspondingly, these
phenomena are included through the force density on the right-hand side of Eq.
(2) as described below. Equation (2) is the conservation law of momentum. The
conservation law of energy of the material reads

$\frac{1}{c^{2}}\frac{\partial W_{\mathrm{mat}}}{\partial
t}+\boldsymbol{\nabla}\cdot\mathbf{G}_{\mathrm{mat}}=\frac{\phi}{c^{2}},$ (3)

where $W_{\mathrm{mat}}=\gamma_{\mathbf{v}_{\mathrm{a}}}m_{0}c^{2}$ is the
energy density of the material and
$\phi=\mathbf{v}_{\mathrm{a}}\cdot\mathbf{f}$ is the power conversion density
of the kinetic energy of the material, in accordance with Eq. (2). In previous
literature, the material subsystem described above is also called the kinetic
subsystem Penfield1967 .

###  II.2 Conservation laws of the field and electromagnetic and mechanical
pressure effects

After defining the energy and momentum densities and the stress tensor of the
material and their conservation laws in Eqs. (2) and (3), we will now define
the energy density $W_{\mathrm{emi}}$, the momentum density
$\mathbf{G}_{\mathrm{emi}}$, and the stress tensor
$\boldsymbol{\mathcal{T}}_{\mathrm{emi}}$ and their conservation laws for _the
rest of the field-material system_. We call this subsystem the
field+interaction subsystem in accordance with previous literature, where it
is the sum of the field and interaction subsystems Penfield1967 . The
conservation laws of the momentum and energy of the field+interaction
subsystem are written as Penfield1967 ; Jackson1999 ; Landau1984

$\frac{\partial\mathbf{G}_{\mathrm{emi}}}{\partial
t}+\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{T}}_{\mathrm{emi}}=-\mathbf{f}.$
(4) $\frac{1}{c^{2}}\frac{\partial W_{\mathrm{emi}}}{\partial
t}+\boldsymbol{\nabla}\cdot\mathbf{G}_{\mathrm{emi}}=-\frac{\phi}{c^{2}},$ (5)

The force density and power-conversion density are generally nonzero, which
means that the field+interaction subsystem is exchanging energy and momentum
with the material subsystem, whose conservation laws are given in Eqs. (2) and
(3). Due to the law of action and reaction, there are opposite signs in the
momentum conservation laws in Eqs. (2) and (4) and the energy conservation
laws in Eqs. (3) and (5).

##  III Microscopic and macroscopic fields and the density dependence of
permittivity and permeability

Before studying the electromagnetic force densities in detail, in this
section, we consider the dependence of the permittivity and permeability of
the material on the atomic density. This dependence originates from the
microscopic expressions of the polarization field $\mathbf{P}$ and the
magnetization field $\mathbf{M}$ and from their relation to the macroscopic
fields Panofsky1962 ; Jackson1999 . The macroscopic electric field
$\mathbf{E}$ and magnetic field $\mathbf{H}$ are related to the electric flux
density $\mathbf{D}$, magnetic flux density $\mathbf{B}$, and the fields
$\mathbf{P}$ and $\mathbf{M}$ by the well-known constitutive relations of a
nondispersive material, given by

$\mathbf{D}=\varepsilon_{0}\varepsilon_{\mathrm{r}}\mathbf{E},\hskip
14.22636pt\mathbf{P}=\varepsilon_{0}(\varepsilon_{\mathrm{r}}-1)\mathbf{E}$
(6) $\mathbf{B}=\mu_{0}\mu_{\mathrm{r}}\mathbf{H},\hskip
14.22636pt\mathbf{M}=(\mu_{\mathrm{r}}-1)\mathbf{H}.$ (7)

Here $\varepsilon_{0}$ is the permittivity of vacuum,
$\varepsilon_{\mathrm{r}}$ is the relative permittivity of the material,
$\mu_{0}$ is the permeability of vacuum, and $\mu_{\mathrm{r}}$ is the
relative permeability of the material. In terms of the relative permittivity
and permeability, the refractive index of the material is given by
$n=\sqrt{\varepsilon_{\mathrm{r}}\mu_{\mathrm{r}}}$.

As conventional, we can write the microscopic expressions of the polarization
and magnetization fields as dipole moment densities, given by Panofsky1962 ;
Jackson1999

$\mathbf{P}=n_{\mathrm{a}}\mathbf{p}=n_{\mathrm{a}}\alpha_{\mathrm{e}}\mathbf{E}_{\mathrm{eff}},$
(8)
$\mathbf{M}=n_{\mathrm{a}}\mathbf{m}=n_{\mathrm{a}}\alpha_{\mathrm{m}}\mathbf{B}_{\mathrm{eff}}.$
(9)

Here $n_{\mathrm{a}}$ is the number density of atoms,
$\mathbf{p}=\alpha_{\mathrm{e}}\mathbf{E}_{\mathrm{eff}}$ is the atomic
electric dipole moment,
$\mathbf{m}=\alpha_{\mathrm{m}}\mathbf{B}_{\mathrm{eff}}$ is the atomic
magnetic dipole moment, $\alpha_{\mathrm{e}}$ and $\alpha_{\mathrm{m}}$ are
the atomic polarizability and magnetizability, which are independent of the
atomic density, and $\mathbf{E}_{\mathrm{eff}}$ and
$\mathbf{B}_{\mathrm{eff}}$ are the effective local electric and magnetic
fields at the site of the atom, given for isotropic cubic materials by
Lorentz1952 ; Panofsky1962 ; Jackson1999 ; Griffiths1998 ; Aspnes1982 ;
Shevchenko2010 ; Shevchenko2011 ; Maki1991

$\mathbf{E}_{\mathrm{eff}}=\mathbf{E}+\frac{\mathbf{P}}{3\varepsilon_{0}}=\mathbf{E}\sum_{l=0}^{\infty}\Big{(}\frac{n_{\mathrm{a}}\alpha_{\mathrm{e}}}{3\varepsilon_{0}}\Big{)}^{l}\\!=\frac{\mathbf{E}}{1-\frac{n_{\mathrm{a}}\alpha_{\mathrm{e}}}{3\varepsilon_{0}}},$
(10)
$\mathbf{B}_{\mathrm{eff}}=\mathbf{B}-\frac{2\mu_{0}\mathbf{M}}{3}=\mathbf{B}\sum_{l=0}^{\infty}\\!\Big{(}\\!-\frac{2\mu_{0}n_{\mathrm{a}}\alpha_{\mathrm{m}}}{3}\Big{)}^{l}\\!\\!=\frac{\mathbf{B}}{1+\frac{2\mu_{0}n_{\mathrm{a}}\alpha_{\mathrm{m}}}{3}}.$
(11)

The calculation of the local fields for other symmetries of the materials,
e.g., for anisotropic materials, is discussed in Ref. Aspnes1982 .

Using the relation between $\mathbf{P}$ and $\mathbf{E}$ in Eq. (6) together
with Eqs. (8) and (10) and the relation between $\mathbf{M}$ and $\mathbf{B}$,
obtained from Eq. (7), together with Eqs. (9) and (11), we obtain the well-
known Clausius-Mossotti relation and its magnetic analog, given by
Panofsky1962 ; Jackson1999 ; Bottcher1993 ; Stratton1941

$\frac{\varepsilon_{\mathrm{r}}-1}{\varepsilon_{\mathrm{r}}+2}=\frac{n_{\mathrm{a}}\alpha_{\mathrm{e}}}{3\varepsilon_{0}},\hskip
14.22636pt\varepsilon_{\mathrm{r}}=1+\frac{n_{\mathrm{a}}\alpha_{\mathrm{e}}/\varepsilon_{0}}{1-\frac{n_{\mathrm{a}}\alpha_{\mathrm{e}}}{3\varepsilon_{0}}}$
(12)
$\frac{\mu_{\mathrm{r}}-1}{\mu_{\mathrm{r}}+2}=\frac{\mu_{0}n_{\mathrm{a}}\alpha_{\mathrm{m}}}{3},\hskip
14.22636pt\mu_{\mathrm{r}}=1+\frac{\mu_{0}n_{\mathrm{a}}\alpha_{\mathrm{m}}}{1-\frac{\mu_{0}n_{\mathrm{a}}\alpha_{\mathrm{m}}}{3}}.$
(13)

For anisotropic materials, for which Eqs. (10) and (11) do not apply, the
relations of the permittivity and permeability in Eqs. (12) and (13) also
become more complicated Aspnes1982 . For dielectric materials with
$\varepsilon_{\mathrm{r}}=n^{2}$, the Clausius-Mossotti relation in Eq. (12)
is also known as the Lorentz-Lorenz relation Kragh2018 ; Bottcher1992 . The
Lorentz-Lorenz relation is generally applicable to a wide range of photonic
materials as such. However, to fine tune this relation to account for more
detailed wavelength, temperature, and density-dependent characteristics of
specific materials, experimental parametrizations have been presented so that
the right hand side of Eq. (12) is replaced by a parametrized sum of the
wavelength, temperature, and pressure-dependent terms Schiebener1990 ;
Harvey1998 .

##  IV Force density

Next, we consider different parts of the force density. The force density
$\mathbf{f}$ and the field+interaction subsystem quantities
$W_{\mathrm{emi}}$, $\mathbf{G}_{\mathrm{emi}}$,
$\boldsymbol{\mathcal{T}}_{\mathrm{emi}}$ can be split into electromagnetic
field and mechanical pressure parts. For the force density, this splitting
reads

$\mathbf{f}=\mathbf{f}_{\mathrm{em}}+\mathbf{f}_{\mathrm{mech}},$ (14)

where $\mathbf{f}_{\mathrm{em}}$ is the total electromagnetic force density
and $\mathbf{f}_{\mathrm{mech}}$ is the mechanical pressure force density.
Here we briefly note that $\mathbf{f}_{\mathrm{mech}}$ is given for materials
with negligible shear strain, e.g., for liquids and gases, by Kittel2005

$\mathbf{f}_{\mathrm{mech}}=-\nabla p_{\mathrm{mech}}.$ (15)

Here $p_{\mathrm{mech}}$ is the mechanical pressure. The mechanical pressure
is given in terms of the atomic position field $\mathbf{r}_{\mathrm{a}}$ of a
homogeneous material as
$p_{\mathrm{mech}}=-K\boldsymbol{\nabla}\cdot\mathbf{r}_{\mathrm{a}}$
Kittel2005 . The general elastic force density, from which Eq. (15) is a
special case, is obtained by using the elasticity tensor of the specific
material as described in Sec. IV.3 below.

###  IV.1 Electromagnetic force density

The total electromagnetic force density $\mathbf{f}_{\mathrm{em}}$, appearing
in Eq. (14), can be split into two parts as

$\mathbf{f}_{\mathrm{em}}=\mathbf{f}_{\mathrm{owm}}+\mathbf{f}_{\mathrm{ost}}.$
(16)

The optical wave momentum force density $\mathbf{f}_{\mathrm{owm}}$ acts, in
the laboratory frame, along the propagation direction of light, and it is
responsible for carrying the wave-momentum of light Bliokh2022 ; Partanen2022b
. This force density acts on the material atoms at interfaces in such a way
that the interface takes the difference of the wave momentum of light in the
material and the momentum of light in vacuum. In the bulk,
$\mathbf{f}_{\mathrm{owm}}$ accelerates and decelerates atoms in such a way
that the atoms carry part of the total wave momentum of light, and the rest
energy of the atomic mass density wave needed for the relativistic covariance
of the theory Partanen2017c ; Partanen2019a ; Partanen2019b ; Partanen2021b .
In contradistinction, the optostrictive force density
$\mathbf{f}_{\mathrm{ost}}$ is an electromagnetic force density acting between
the constituents of the material. Therefore, it has a character of action and
reaction between the material constituents, and its volume integral including
the interface contribution is, thus, at any time equal to zero. Although
$\mathbf{f}_{\mathrm{ost}}$ can give rise to nonzero momentum densities, the
total volume integral of the momentum density caused by
$\mathbf{f}_{\mathrm{ost}}$ is zero, and therefore, it does not carry net
momentum. In a recent experiment of Astrath _et al._ Astrath2022 , acoustic
waves caused by the radial component of $\mathbf{f}_{\mathrm{ost}}$ due to a
light pulse were discovered.

The accurate position- and time-dependent form of $\mathbf{f}_{\mathrm{owm}}$
has been described in detail, e.g., in Ref. Partanen2022b . The
electrostrictive and magnetostrictive force densities in
$\mathbf{f}_{\mathrm{ost}}$ are both experimentally and theoretically widely
known only in the thermodynamical and static field limits. The thermodynamical
theory of electrostriction and magnetostriction cannot be applied to optical
fields. Therefore, in the present work, we focus in finding the exact
position- and time-dependent form of $\mathbf{f}_{\mathrm{ost}}$ in the
optical regime. The total force density in Eq. (14), consisting of the terms
discussed in the present work, explains the available experimental results for
the forces at liquid interfaces Ashkin1973 ; Astrath2014 ; Casner2001 , for
the force on a mirror immersed in a liquid Jones1978 ; Jones1954 , and for the
radial optostrictive force component inside the bulk liquid Astrath2022 .

####  IV.1.1 Optical wave momentum force density

In the rest frame of a nondispersive material, the optical wave momentum force
density $\mathbf{f}_{\mathrm{owm}}$ is given by the Abraham stress-energy-
momentum tensor model Partanen2017c ; Milonni2010 . It can be derived from the
conservation laws by requiring that the coupled field-material state of light
satisfies the covariance principle of the special theory of relativity
Partanen2017c . In the rest frame of the material, $\mathbf{f}_{\mathrm{owm}}$
is given by

$\mathbf{f}_{\mathrm{owm}}=-\frac{1}{2}\varepsilon_{0}|\mathbf{E}|^{2}\nabla\varepsilon_{\mathrm{r}}-\frac{1}{2}\mu_{0}|\mathbf{H}|^{2}\nabla\mu_{\mathrm{r}}+\frac{n^{2}-1}{c^{2}}\frac{\partial}{\partial
t}(\mathbf{E}\times\mathbf{H}).$ (17)

The first two terms of $\mathbf{f}_{\mathrm{owm}}$ represent the interface
force density and the last term is the Abraham volume force density, which has
an effect in the direction of the propagation of light. The interface force
density has been experimentally verified in several works Ashkin1973 ;
Astrath2014 ; Casner2001 ; Jones1978 ; Jones1954 , while the Abraham volume
force density still lacks direct compelling experimental verification at
optical frequencies. The Abraham volume force density, however, has
experimental support in the quasistatic limit Walker1975 . There are works
reporting measurements of the Abraham force at optical frequencies Choi2017 ;
She2008 ; Kundu2017 , but the interpretation of the results of these
measurements is nontrivial Brevik2009 ; Mansuripur2009b ; Partanen2021a ;
Brevik2018a ; Brevik2018b ; Partanen2019e .

####  IV.1.2 Optostrictive force density

Next, we investigate the optostrictive force density
$\mathbf{f}_{\mathrm{ost}}$, which originates from the atomic density
dependence of the energy density of the electric and magnetic fields. It can
be calculated using the principle of virtual work as discussed in Sec. IV.2
below, but here we present first a concise Lorentz-force-law-based derivation
of the optostrictive force density. According to Eq. (16),
$\mathbf{f}_{\mathrm{ost}}=\mathbf{f}_{\mathrm{em}}-\mathbf{f}_{\mathrm{owm}}$.
We know that the force density $\mathbf{f}_{\mathrm{em}}$ can be derived from
the Lorentz force law as discussed in Appendix A, and the expression of
$\mathbf{f}_{\mathrm{owm}}$ is given in Eq. (17). Thus, by using Eqs. (17) and
(61), we obtain

$\mathbf{f}_{\mathrm{ost}}=\mathbf{f}_{\mathrm{oes}}+\mathbf{f}_{\mathrm{oms}}.$
(18)

where the optoelectro- and optomagnetostrictive force densities
$\mathbf{f}_{\mathrm{oes}}$ and $\mathbf{f}_{\mathrm{oms}}$ are given by

$\mathbf{f}_{\mathrm{oes}}=\frac{1}{2}\nabla(\mathbf{P}\cdot\mathbf{E}),$ (19)
$\mathbf{f}_{\mathrm{oms}}=\frac{1}{2}\nabla(\mathbf{M}\cdot\mathbf{B}).$ (20)

It is conventional to define the optoelectro- and optomagnetostrictive
pressures $p_{\mathrm{oes}}$ and $p_{\mathrm{oms}}$ related to the force
densities $\mathbf{f}_{\mathrm{oes}}$ and $\mathbf{f}_{\mathrm{oms}}$ in Eqs.
(19) and (20) by $\mathbf{f}_{\mathrm{oes}}=-\nabla p_{\mathrm{oes}}$ and
$\mathbf{f}_{\mathrm{oms}}=-\nabla p_{\mathrm{oms}}$. Thus, these pressures
can be written as

$p_{\mathrm{oes}}=-\frac{1}{2}\mathbf{P}\cdot\mathbf{E}=-\frac{1}{2}\varepsilon_{0}(\varepsilon_{\mathrm{r}}-1)|\mathbf{E}|^{2},$
(21)
$p_{\mathrm{oms}}=-\frac{1}{2}\mathbf{M}\cdot\mathbf{B}=-\frac{1}{2}\mu_{0}\mu_{\mathrm{r}}(\mu_{\mathrm{r}}-1)|\mathbf{H}|^{2}.$
(22)

As discussed in the next section, the compressive work done by the
optoelectro- and optomagnetostrictive force densities in Eqs. (19) and (20) on
the material is found to be less than the change of the field energy.
Therefore, we will introduce dissipation terms in the calculation of the
optostrictive force density using the principle of virtual work. This is a
fundamental change in the conventional way of calculating the electrostrictive
and magnetostrictive force densities.

###  IV.2 Virtual work approach to the optostrictive force density

Next, we investigate the optostrictive force density based on the _principle
of virtual work_ Penfield1967 ; WangN2018 . According to the principle of
virtual work, the atoms tend to convert the atomic-density-dependent part of
the energy of the field into kinetic and strain energies of the material. The
atomic forces following from this principle of virtual work can always be
understood as force pairs between atoms, and the resulting total force on the
material is a sum of such atomic force pairs. Since the sum of any momentum
impulses resulting from pairs of opposite forces is zero, the force density
$\mathbf{f}_{\mathrm{ost}}$ cannot lead to transfer of net momentum in the
material, in contrast to the optical wave momentum force density
$\mathbf{f}_{\mathrm{owm}}$ in Eq. (17).

The energy density of the electromagnetic field depends on the density of the
material through the permittivity and permeability. Depending on which of the
fields $\mathbf{D}$, $\mathbf{B}$, $\mathbf{E}$, and $\mathbf{H}$ are kept
independent variables during the compression or expansion of the material, the
optostrictive force densities $\mathbf{f}_{\mathrm{oes}}$ and
$\mathbf{f}_{\mathrm{oms}}$ can be calculated from the electric and magnetic
energy densities or from their Legendre transforms Landau1984

$F_{\mathrm{e}}(\mathbf{D})=W_{\mathrm{e}},\hskip
14.22636pt\tilde{F}_{\mathrm{e}}(\mathbf{E})=F_{\mathrm{e}}-\mathbf{E}\cdot\mathbf{D}=-W_{\mathrm{e}},$
(23) $F_{\mathrm{m}}(\mathbf{B})=W_{\mathrm{m}},\hskip
14.22636pt\tilde{F}_{\mathrm{m}}(\mathbf{H})=F_{\mathrm{m}}-\mathbf{H}\cdot\mathbf{B}=-W_{\mathrm{m}},$
(24)

in analogy to how the elastic force density is calculated from the strain
energy density of the material Kittel2005 . Here
$W_{\mathrm{e}}=\frac{1}{2}\varepsilon_{0}\varepsilon_{\mathrm{r}}|\mathbf{E}|^{2}=\frac{1}{2\varepsilon_{0}\varepsilon_{\mathrm{r}}}|\mathbf{D}|^{2}$
is the energy density of the electric field and
$W_{\mathrm{m}}=\frac{1}{2}\mu_{0}\mu_{\mathrm{r}}|\mathbf{H}|^{2}=\frac{1}{2\mu_{0}\mu_{\mathrm{r}}}|\mathbf{B}|^{2}$
is the energy density of the magnetic field. The independent variables in Eqs.
(23) and (24) are indicated by parentheses. Conventionally, in the
thermodynamical derivation of the electro- and magnetostrictive force
densities Landau1984 , Eqs. (23) and (24) correspond to the field-dependent
parts of the free energy densities of the system. In previous literature, the
thermodynamical derivation is considered to be extended to time-dependent
fields as such or by considering the time-averages of free energy densities
over the harmonic cycle Landau1984 ; Boyd2008 ; Brevik2018a . For a discussion
of the electro- and magnetostriction for stationary fields, see Appendix B.

In the case of an optical field, the compression or expansion of the material
associated with optostriction takes place simultaneously with the flow of
energy and momentum into direction of the propagation of light. Therefore, we
can define boundary conditions related to energy and momentum fluxes in the
calculation of the optostrictive force density, which are different from the
boundary conditions used in Eqs. (23) and (24), where we keep certain fields
constant. We do not elaborate these flux-based boundary conditions further in
this work.

In the following, we propose that the changes of the dipole moments
$\mathbf{p}$ and $\mathbf{m}$ by the atomic number density variation $\delta
n_{\mathrm{a}}$, in the time-dependent case, introduce a hitherto unknown
dissipation mechanism to be accounted for. This dissipation can, for example,
correspond to radiation loss and be related to non-conservativity of time-
dependent optical forces Sukhov2017 . We determine its contribution by
requiring that the sum of the work done by the Lorentz-force-law-based
optostrictive force density $\mathbf{f}_{\mathrm{ost}}$ in Eq. (18) and the
dissipation terms is equal to the change of the field energy density.
Consequently, in the calculation of the force densities below, from the full
differentials $(\delta\tilde{F}_{\mathrm{e}})_{\mathbf{E}}=-(\delta
W_{\mathrm{e}})_{\mathbf{E}}=-\frac{1}{2}\varepsilon_{0}\frac{\partial\varepsilon_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}|\mathbf{E}|^{2}\delta n_{\mathrm{a}}$ and $(\delta
F_{\mathrm{m}})_{\mathbf{B}}=(\delta
W_{\mathrm{m}})_{\mathbf{B}}=-\frac{1}{2\mu_{0}\mu_{\mathrm{r}}^{2}}\frac{\partial\mu_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}|\mathbf{B}|^{2}\delta n_{\mathrm{a}}$, for fixed $\mathbf{E}$
and $\mathbf{B}$, we subtract the dissipation-related differentials $(\delta
W_{\mathrm{e,diss}})_{\mathbf{E}}=-\frac{1}{2}n_{\mathrm{a}}\big{(}\frac{\partial\mathbf{p}}{\partial
n_{\mathrm{a}}}\big{)}_{\mathbf{E}}\cdot\mathbf{E}\delta n_{\mathrm{a}}$ and
$(\delta
W_{\mathrm{m,diss}})_{\mathbf{B}}=-\frac{1}{2}n_{\mathrm{a}}\big{(}\frac{\partial\mathbf{m}}{\partial
n_{\mathrm{a}}}\big{)}_{\mathbf{B}}\cdot\mathbf{B}\delta n_{\mathrm{a}}$.
Thus, we obtain the dissipation-reduced differentials as
$(\delta\tilde{F}_{\mathrm{e}}^{\prime})_{\mathbf{E}}=(\delta\tilde{F}_{\mathrm{e}})_{\mathbf{E}}-(\delta
W_{\mathrm{e,diss}})_{\mathbf{E}}$ and $(\delta
F_{\mathrm{m}}^{\prime})_{\mathbf{B}}=(\delta
F_{\mathrm{m}})_{\mathbf{B}}-(\delta W_{\mathrm{m,diss}})_{\mathbf{B}}$.
Therefore, the optoelectro- and optomagnetostrictive force densities are given
by

$\displaystyle\mathbf{f}_{\mathrm{oes}}$
$\displaystyle=\sum_{i=1}^{3}\sum_{j=1}^{3}\partial_{j}\Big{(}\frac{\delta\tilde{F}_{\mathrm{e}}^{\prime}}{\delta(\epsilon_{\mathrm{a}})_{ij}}\Big{)}_{\mathbf{E}}\hat{\mathbf{e}}_{i}$
$\displaystyle=-\nabla\Big{[}n_{\mathrm{a}}\Big{(}\frac{\delta\tilde{F}_{\mathrm{e}}^{\prime}}{\delta
n_{\mathrm{a}}}\Big{)}_{\mathbf{E}}\Big{]}$
$\displaystyle=\frac{1}{2}\nabla(n_{\mathrm{a}}\mathbf{p}\cdot\mathbf{E})$
$\displaystyle=\frac{1}{2}\nabla\Big{(}\mathbf{P}\cdot\mathbf{E}\Big{)},$ (25)
$\displaystyle\mathbf{f}_{\mathrm{oms}}$
$\displaystyle=\sum_{i=1}^{3}\sum_{j=1}^{3}\partial_{j}\Big{(}\frac{\delta
F_{\mathrm{m}}^{\prime}}{\delta(\epsilon_{\mathrm{a}})_{ij}}\Big{)}_{\mathbf{B}}\hat{\mathbf{e}}_{i}$
$\displaystyle=-\nabla\Big{[}n_{\mathrm{a}}\Big{(}\frac{\delta
F_{\mathrm{m}}^{\prime}}{\delta n_{\mathrm{a}}}\Big{)}_{\mathbf{B}}\Big{]}$
$\displaystyle=\frac{1}{2}\nabla(n_{\mathrm{a}}\mathbf{m}\cdot\mathbf{B})$
$\displaystyle=\frac{1}{2}\nabla\Big{(}\mathbf{M}\cdot\mathbf{B}\Big{)}.$ (26)

Here $\hat{\mathbf{e}}_{i}$, $i\in\\{x,y,z\\}$ are the three unit vectors of
the Cartesian coordinate system. The subscripts $\mathbf{E}$ and $\mathbf{B}$
indicate that these fields are taken as constants in the calculation of the
differentials. In the second equalities of Eqs. (25) and (26), we have assumed
that the material is isotropic. In this case, we can convert the derivatives
with respect to the components of the atomic position strain tensor
$\boldsymbol{\epsilon}_{\mathrm{a}}=\textstyle\frac{1}{2}[\boldsymbol{\nabla}\otimes\mathbf{r}_{\mathrm{a}}+(\boldsymbol{\nabla}\otimes\mathbf{r}_{\mathrm{a}})^{T}]$,
where the superscript $T$ denotes the transpose, into derivatives with respect
to the atomic number density as
$\sum_{i=1}^{3}\sum_{j=1}^{3}\partial_{j}\big{(}\frac{\partial
F}{\partial(\epsilon_{\mathrm{a}})_{ij}}\big{)}\hat{\mathbf{e}}_{i}=-\nabla(n_{\mathrm{a}}\frac{\partial
F}{\partial n_{\mathrm{a}}})$ Sun2015 ; Landau1984 .

If the derivation of the force densities is done in analogy to Eqs. (25) and
(26) except that the we use the energy functions without the subtraction of
the dissipation terms, the optoelectro- and optomagnetostrictive pressures
become $p_{\mathrm{es}}=-\frac{1}{2}\mathbf{P}\cdot\mathbf{E}_{\mathrm{eff}}$
and $p_{\mathrm{ms}}=-\frac{1}{2}\mathbf{M}\cdot\mathbf{B}_{\mathrm{eff}}$.
This means that the time-dependent macroscopic fields are replaced by the
corresponding local fields. By their definition, these optoelectro- and
optomagnetostrictive pressures are inherently energy-conserving during the
compression. Using Eqs. (10)–(13), we can relate these pressures to the
pressures in Eqs. (21) and (22) as
$p_{\mathrm{es}}=\frac{\varepsilon_{\mathrm{r}}+2}{3}p_{\mathrm{oes}}$ and
$p_{\mathrm{ms}}=\frac{\mu_{\mathrm{r}}+2}{3\mu_{\mathrm{r}}}p_{\mathrm{oms}}$.
The relation of these pressures to experiments is discussed in Sec. VII.

###  IV.3 Optoelastic strain force density

Next, we present how the mechanical pressure and optostrictive force
densities, $\mathbf{f}_{\mathrm{mech}}$ and $\mathbf{f}_{\mathrm{ost}}$, can
be obtained simultaneously using the elasticity theory formalism of the total
strain. As described below, the force densities of Eqs. (15) and (18) are
obtained as special cases of the general force density expressions of this
section. The total strain energy density is given by Kittel2005

$W_{\mathrm{strain}}=\frac{1}{2}\sum_{i=1}^{3}\sum_{j=1}^{3}\sigma_{ij}\epsilon_{ij}.$
(27)

Here $\sigma_{ij}$ are elements of the elastic stress tensor
$\boldsymbol{\sigma}$ and $\epsilon_{ij}$ are elements of the total strain
tensor $\boldsymbol{\epsilon}$. The total strain tensor
$\boldsymbol{\epsilon}$ is given by Kittel2005

$\boldsymbol{\epsilon}=\boldsymbol{\epsilon}_{\mathrm{a}}-\boldsymbol{\epsilon}_{\mathrm{e}}-\boldsymbol{\epsilon}_{\mathrm{m}}.$
(28)

Here the atomic strain tensor $\boldsymbol{\epsilon}_{\mathrm{a}}$ is given by

$\boldsymbol{\epsilon}_{\mathrm{a}}=\textstyle\frac{1}{2}[\boldsymbol{\nabla}\otimes\mathbf{r}_{\mathrm{a}}+(\boldsymbol{\nabla}\otimes\mathbf{r}_{\mathrm{a}})^{T}].$
(29)

For the electric and magnetic strain tensors, we adopt the most general
description from previous literature Newnham1997 . The electric and magnetic
strain tensors are given by

$\displaystyle\boldsymbol{\epsilon}_{\mathrm{e}}$
$\displaystyle=\mathbf{Q}_{\mathrm{e}}^{(1)}:\mathbf{P}+\mathbf{Q}_{\mathrm{e}}^{(2)}:(\mathbf{P}\otimes\mathbf{P})+\cdots,$
$\displaystyle\boldsymbol{\epsilon}_{\mathrm{m}}$
$\displaystyle=\mathbf{Q}_{\mathrm{m}}^{(1)}:\mathbf{M}+\mathbf{Q}_{\mathrm{m}}^{(2)}:(\mathbf{M}\otimes\mathbf{M})+\cdots.$
(30)

The first terms are associated with the piezoelectricity and piezomagnetism
and the second terms are associated with the electro- and magnetostriction.
Also, higher-order terms can be accounted for. In Eq. (30), the quantities
$\mathbf{Q}_{\mathrm{e}}^{(1)}$ and $\mathbf{Q}_{\mathrm{m}}^{(1)}$ are the
piezoelectric and piezomagnetic tensors and $\mathbf{Q}_{\mathrm{e}}^{(2)}$
and $\mathbf{Q}_{\mathrm{m}}^{(2)}$ are the electro- and magnetostriction
tensors.

Using the elasticity tensor $\mathbf{C}$ of an isotropic material with bulk
modulus $K$ and shear modulus $G$, the elastic stress tensor
$\boldsymbol{\sigma}$ is given by

$\boldsymbol{\sigma}=\mathbf{C}:\boldsymbol{\epsilon}=\textstyle(K-\frac{2}{3}G)\mathrm{Tr}(\boldsymbol{\epsilon})\mathbf{I}+2G\boldsymbol{\epsilon}.$
(31)

Here $\mathrm{Tr}(x)$ denotes the trace of a matrix and $\mathbf{I}$ is a
$3\times 3$ unit matrix. In this case, the total optoelastic strain force
density is given by

$\displaystyle\mathbf{f}_{\mathrm{strain}}$
$\displaystyle=\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}=\sum_{i=1}^{3}\sum_{j=1}^{3}\partial_{j}\Big{(}\frac{\partial
W_{\mathrm{strain}}}{\partial(\epsilon_{\mathrm{a}})_{ij}}\Big{)}\hat{\mathbf{e}}_{i}$
$\displaystyle=\textstyle(K+\frac{4}{3}G)\nabla[\boldsymbol{\nabla}\cdot\mathbf{r}_{\mathrm{a}}]-G\boldsymbol{\nabla}\times[\boldsymbol{\nabla}\times\mathbf{r}_{\mathrm{a}}]$
$\displaystyle\hskip
14.22636pt-\textstyle(K+\frac{4}{3}G)\nabla\mathrm{Tr}(\boldsymbol{\epsilon}_{\mathrm{e}}+\boldsymbol{\epsilon}_{\mathrm{m}})-2G\boldsymbol{\nabla}\cdot(\boldsymbol{\epsilon}_{\mathrm{e}}+\boldsymbol{\epsilon}_{\mathrm{m}}).$
(32)

In our special case of a material with no shear strain, i.e., a liquid or gas,
we can set $G=0$. We assume no piezoelectricity and -magnetism, use the
conventional constitutive relations of the fields in a nondispersive material,
and give for the fourth-rank optical electro- and magnetostriction tensors the
values

$\mathbf{Q}_{\mathrm{e}}^{(2)}=-\frac{\mathbf{I}}{2K\varepsilon_{0}(\varepsilon_{\mathrm{r}}-1)},\hskip
14.22636pt\mathbf{Q}_{\mathrm{m}}^{(2)}=-\frac{\mu_{0}\mu_{\mathrm{r}}\mathbf{I}}{2K(\mu_{\mathrm{r}}-1)},$
(33)

where $\mathbf{I}$ is the identity operator for the matrices
$\mathbf{P}\otimes\mathbf{P}$ and $\mathbf{M}\otimes\mathbf{M}$ in Eq. (30).
Thus, we obtain
$\boldsymbol{\epsilon}_{\mathrm{e}}=-\frac{1}{2K}\mathbf{P}\otimes\mathbf{E}$,
$\boldsymbol{\epsilon}_{\mathrm{m}}=-\frac{1}{2K}\mathbf{M}\otimes\mathbf{B}$,
and consequently Eq. (32) becomes

$\mathbf{f}_{\mathrm{strain}}=\mathbf{f}_{\mathrm{mech}}+\mathbf{f}_{\mathrm{ost}},$
(34)

where $\mathbf{f}_{\mathrm{mech}}$ is equal to the expression given in Eq.
(15) and $\mathbf{f}_{\mathrm{ost}}$ is equal to the expression given in Eq.
(18), with the optoelectro- and optomagnetostrictive force densities given in
Eqs. (19) and (20).

Figure 1:  Simulation of a Gaussian light pulse propagating along the positive
$z$ axis marked by a dashed line ellipse centered at position $z=20$ mm (a–f).
The panels represent (a) the mass density disturbance, (b) atomic
displacement, and (c) atomic velocity resulting from the total force density
$\mathbf{f}=\mathbf{f}_{\mathrm{owm}}+\mathbf{f}_{\mathrm{ost}}$ of a Gaussian
light pulse in silicon. The _optoelectrostrictive_ force density
$\mathbf{f}_{\mathrm{ost}}$ is the _dominating_ force density in these
figures. Panels (d)–(f) show the mass density disturbance, atomic
displacement, and atomic velocity contributions resulting from
$\mathbf{f}_{\mathrm{owm}}$. Simulation of optically generated elastic waves
after the light pulse has passed is presented in panels (g)–(i). These panels
show the time dependence of (g) the mass density disturbance, (h) atomic
displacement, and (i) atomic velocity at $z=0$ mm. The colorbars show the
magnitudes, and the arrows show the directions of the vector quantities.

##  V Simulation of optical electrostriction in silicon

Next, we study the implications of the theory above by simulating the
electrostrictive effect using first a Gaussian light pulse and second a
continuous-wave Gaussian light beam. We have selected silicon as an example of
an electrostrictive photonic material for which we carry out the numerical
simulations. The refractive index of silicon is $n=3.4757$ for the selected
vacuum wavelength of $\lambda_{0}=1550$ nm Li1980 . The density of silicon is
$\rho_{\mathrm{a}}=2329$ kg/m3 Lide2004 , the bulk modulus is $K=97.8$ GPa,
and the shear modulus is $G=79.6$ GPa Hopcroft2010 . These values correspond
to the compressibility of $C=1/(K+\frac{4}{3}G)=4.90\times 10^{-12}$ Pa-1. In
the present work, we do not account for the elastic anisotropy of the silicon
crystal, but model it using the scalar elastic parameters above, in the case
of which the elastic strain force density is given by the first two terms of
Eq. (32) Mavko2003 ; Partanen2017e ; Landau1975 ; Kittel2005 .

In more accurate simulations, the elastic parameters $K$ and $G$ and the
associated elastic force density in Eq. (32) can be replaced by the use of the
complete elasticity matrix and the corresponding elastic force density
Kittel2005 . For silicon, and more generally for solids and liquids, which are
hard to compress, the elastic parameters and derivatives with respect to the
density are very accurately equal for isothermal and isentropic processes.
Thus, we can neglect thermal effects in the studies of electrostrictive
compression in low-loss materials, such as silicon at the wavelength used in
our simulations.

###  V.1 Gaussian light pulse

Next, we perform simulations for a continuous-wave Gaussian light beam. The
electric field of a one-dimensional Gaussian light pulse Griffiths1998 ,
linearly polarized in the direction of the $x$ axis and propagating along the
positive $z$ axis in silicon, is given in cylindrical coordinates
$\mathbf{r}=(r,\phi,z)$ by

$\mathbf{E}(\mathbf{r},t)=E_{0}e^{-r^{2}/w_{0}^{2}}\cos[kz-\omega
t]e^{-(\Delta k)^{2}(z-ct/n)^{2}/2}\hat{\mathbf{e}}_{x}.$ (35)

Here $E_{0}$ is the electric field amplitude, $w_{0}$ is the beam waist radius
at which the intensity drops to $1/e^{2}$ of its axial value,
$k=n\omega/c=2\pi n/\lambda_{0}$ is the wave number, and $\Delta k$ is the
standard deviation of the wave number, for which we use $\Delta k=10^{-5}k$.
For the beam waist radius $w_{0}$, we use the value of $w_{0}=3.489$ mm, and
for the electric field amplitude $E_{0}$, we use the value of
$E_{0}=1.972\times 10^{7}$ V/m. These values correspond to the total
electromagnetic energy of the pulse equal to
$U_{0}=\varepsilon_{0}\varepsilon_{\mathrm{r}}E_{0}^{2}\pi^{3/2}w_{0}^{2}/(4\Delta
k)=5.00$ mJ. The magnetic field corresponding to the electric field above is
determined by Maxwell’s equations.

Figure 1(a) shows the atomic mass density disturbance resulting from the total
force density of a Gaussian light pulse in silicon at the instant of time when
the center of the Gaussian light pulse propagating along the positive $z$ axis
is at $z=20$ mm. The mass density disturbance is spatially averaged over the
harmonic cycle. It is seen that the mass density disturbance has positive
values at the $z$ axis behind the light pulse ($z<20$ mm, $|y|<2$ mm), and
negative values are obtained away from the $z$ axis ($z<20$ mm, 2 mm $<|y|<6$
mm). The corresponding atomic displacement distribution is depicted in Fig.
1(b) and the atomic velocity distribution is shown in Fig. 1(c). The radial
component of the optostrictive force density $\mathbf{f}_{\mathrm{ost}}$
dominates over the optical wave momentum force density
$\mathbf{f}_{\mathrm{owm}}$ and the longitudinal component of
$\mathbf{f}_{\mathrm{ost}}$ in producing the atomic distributions in Figs.
1(a)–1(c). The atomic mass density disturbance, atomic displacement, and
atomic velocity components following from $\mathbf{f}_{\mathrm{owm}}$ are
depicted in Figs. 1(d)–1(f). These distributions are fractions of the total
distributions in Figs. 1(a)–1(c).

Figures 1(g)–1(i) show the time-dependence of the elastic relaxation of the
mass density disturbance, atomic displacement, and atomic velocity by elastic
waves at $z=0$ mm after the light pulse has passed. The distributions at $t=0$
$\mu$s correspond to the quantities of Figs. 1(a)–1(c) behind the light pulse
at $z=0$ mm. The relaxation takes place at the velocity of sound after the
light pulse has passed. The sound waves, caused by the optoelectrostrictive
force density, propagate radially outward from the beam axis ($x=0$ mm). The
displacement of the material by the optostrictive force density takes place
also parallel to the beam axis. The relaxation of this displacement is not
depicted in Fig. 1. The emergence of the mass density disturbance and the
related elastic waves have been experimentally detected for a light pulse in
water Astrath2022 . The detection of optostrictively produced elastic waves in
the present example case of silicon should also be experimentally feasible.

Figure 2:  Unified picture of the momentum components for a normally incident
Gaussian light pulse crossing a vacuum-silicon interface with an
antireflective coating. The center of the pulse is at the position of the
interface at $t=200$ ps. The normal components of the momenta are given in
units of $p_{0}=U_{0}/c$, the total momentum of the pulse in vacuum. The
component $p_{\mathrm{vac}}$ is the momentum of the part of the pulse located
in vacuum, and $p_{\mathrm{em}}$ is the momentum of the electromagnetic field
in silicon. The material momentum driven by $\mathbf{f}_{\mathrm{owm}}$ in the
bulk silicon is denoted by $p_{\mathrm{owm,vol}}$, and the material momentum
generated by $\mathbf{f}_{\mathrm{owm}}$ at the interface is denoted by
$p_{\mathrm{owm,int}}$. Correspondingly, the material momentum driven by
$\mathbf{f}_{\mathrm{oes}}$ in the bulk silicon is denoted by
$p_{\mathrm{oes,vol}}$, and the material momentum generated by
$\mathbf{f}_{\mathrm{owm}}$ at the interface is denoted by
$p_{\mathrm{oes,int}}$. Note that the radial momentum densities integrate out
in this figure, which represents only the total volume integrated values.

Figure 2 presents the momentum components that follow from our unified optical
force theory when a Gaussian light pulse crosses a vacuum-silicon interface
with an antireflective coating. For simplicity, the pulse is assumed to
propagate along the surface normal and the antireflective coating is assumed
to be perfect. The center of the pulse reaches the interface at $t=200$ ps. At
$t=0$ ps, the pulse is entirely in vacuum, its momentum is
$p_{\mathrm{vac}}=p_{0}=U_{0}/c$, and the other momentum components are zero.
When the pulse has fully crossed the interface, the momentum of the
electromagnetic field part of the pulse is $p_{\mathrm{em}}=p_{0}/n$, which is
equal to the _Abraham momentum_ of light. The atomic mass density wave, driven
by the optical wave momentum force density $\mathbf{f}_{\mathrm{owm}}$, has
momentum $p_{\mathrm{owm,vol}}=(n-1/n)p_{0}$, which together with the
electromagnetic momentum component $p_{\mathrm{em}}=p_{0}/n$ gives the total
wave momentum of light. The total wave momentum
$p_{\mathrm{owm,vol}}+p_{\mathrm{em}}=np_{0}$ is equal to the _Minkowski
momentum_ of light. In the bulk silicon, the optoelectrostrictive force
density $\mathbf{f}_{\mathrm{oes}}$ gives rise to the momentum component equal
to $p_{\mathrm{oes,vol}}=-\frac{1}{2}(n-1/n)p_{0}$. The material interface
takes the momentum component $p_{\mathrm{owm,int}}=(1-n)p_{0}$ from
$\mathbf{f}_{\mathrm{owm}}$ and $p_{\mathrm{oes,int}}=\frac{1}{2}(n-1/n)p_{0}$
from $\mathbf{f}_{\mathrm{oes}}$. The sum of all momentum components is equal
to $p_{0}$ _at all times_. Since moreover
$p_{\mathrm{oes,vol}}+p_{\mathrm{oes,int}}=0$ at all times, the
optoelectrostriction does not lead to transfer of volume integrated net
momentum.

In the mass-polariton theory of light Partanen2017c ; Partanen2019a , the
momentum of the atomic mass density wave is given, using the notation of the
present work, by $p_{\mathrm{owm,vol}}$ and the momentum of the mass-polariton
state of light is correspondingly
$p_{\mathrm{MP}}=p_{\mathrm{owm,vol}}+p_{\mathrm{em}}=np_{0}$. This is the
refractive-index-proportional momentum of light in a material, which is
observed in most experiments Jones1954 ; Jones1978 ; Campbell2005 . The
present theory is, thus, in full agreement with the covariant quasiparticle
properties of the mass-polariton theory Partanen2017c ; Partanen2019a ;
Partanen2021b .

Figure 3:  Simulations of optoelectrostriction for a radially Gaussian
continuous-wave light beam. (a) The excess mass density and (b) the atomic
displacements for a light beam propagating along the $z$ axis in a homogeneous
silicon crystal. (c) The excess mass density and (d) the atomic displacements
for the standing wave of a light beam incident from silicon along the negative
$z$ axis and reflected from a perfect mirror at $z=0$ nm. The figures show the
quantities for the length of two wavelengths in the cylindrical central region
of the simulation geometry. The colors show the magnitude and the arrows show
the direction of the vector quantities.

###  V.2 Continuous-wave Gaussian light beam

The electric field of an incident continuous-wave Gaussian light beam
propagating along the negative $z$ axis, polarized along the $x$ axis, and
focused to $x=y=z=0$ is given in cylindrical coordinates by Novotny2006

$\displaystyle\mathbf{E}(\mathbf{r},t)$
$\displaystyle=E_{0}\frac{w_{0}}{w(z)}e^{-r^{2}/w^{2}(z)}\cos\\!\Big{[}kz+\omega
t+\frac{kr^{2}}{2R(z)}-\varphi(z)\Big{]}\hat{\mathbf{e}}_{x}.$ (36)

Here $w(z)=w_{0}\sqrt{1+(z/z_{\mathrm{R}})^{2}}$ is the position-dependent
beam radius, $z_{\mathrm{R}}=\pi w_{0}^{2}n/\lambda_{0}$ is the Rayleigh
range, $R(z)=z[1+(z_{\mathrm{R}}/z)^{2}]$ is the radius of curvature of the
wavefronts, and $\varphi(z)=\arctan(z/z_{\mathrm{R}})$ is the Gouy phase. The
electric field amplitude at the focus is given by $E_{0}=\sqrt{4P_{0}n/(\pi
w_{0}^{2}c\varepsilon_{0}\varepsilon_{\mathrm{r}})}$, where $P_{0}$ is the
average power of the beam. In the simulation of the continuous-wave beam, for
the beam waist radius $w_{0}$, we use the value of $w_{0}=40$ $\mu$m, and for
the average power of the beam, we use $P_{0}=1$ W.

In order to obtain a stationary atomic distribution as a result of the
simulation, in Newton’s equation we add the force density term
$\mathbf{f}_{\mathrm{damping}}=-\Gamma\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}$,
which is associated with the damping of mechanical waves MaartenvanDoorn2020 ,
quantified here by the damping frequency $\Gamma$. In the case of a
continuous-wave field, the damping frequency determines the time scale at
which the excess mass density and atomic displacement distributions studied
below are formed, but it does not affect the values of the final
distributions.

Figure 3(a) presents the excess mass density resulting from the
electrostriction induced by the Gaussian light beam of Eq. (36) in a
homogeneous silicon crystal. The field is only slightly focused, and thus, the
variations in the mass density along the $z$ axis are not visible, while in
the transverse plane, the excess mass density follows the Gaussian form of the
light beam. Apart from very small variations, the result is independent of
whether the square of the electric field used in the simulations is averaged
over the harmonic cycle or not. The atomic displacement field corresponding to
the excess mass density of Fig. 3(a) is presented in Fig. 3(b). This
displacement field is radially directed toward the beam axis, where the
displacement field is zero. Far from the beam axis, the atomic displacement
field asymptotically approaches zero again. This takes place outside the
region depicted in the figure.

Figure 3(c) shows the excess mass density resulting from the electrostriction
induced by the Gaussian light beam of Eq. (36) when the beam is reflected from
a perfect mirror positioned at $z=0$ nm. The excess mass density follows the
standing wave pattern of the square of the electric field as expected. Due to
the reflection and the standing wave pattern, the highest values of the excess
mass density in Fig. 3(c) are four times the highest values the same quantity
for the Gaussian light beam without reflection from the mirror in Fig. 3(a).
The atomic displacement field corresponding to the excess mass density of Fig.
3(c) is depicted in Fig. 3(d). Due to the standing wave pattern, the atomic
displacement field associated with electrostriction has an additional
nonradial contribution. This component is, however, so small that it is not
visible in Fig. 3(d).

##  VI Stress-energy-momentum tensors

The SEM tensor of a physical system or a subsystem compiles the energy and
momentum densities and the stress tensor of the conservation laws, exemplified
by Eqs. (2)–(5), in a single second-rank physical quantity. The contravariant
form of an arbitrary SEM tensor in the Minkowski space-time is defined by
$\mathbf{T}=T^{\alpha\beta}\mathbf{e}_{\alpha}\otimes\mathbf{e}_{\beta}$,
where the Einstein summation convention is used, and $\mathbf{e}_{\alpha}$ and
$\mathbf{e}_{\beta}$ are unit vectors of the four-dimensional space-time,
i.e., $(ct,x,y,z)$. The Greek indices range over the dimensions of the space-
time. The matrix representation of $\mathbf{T}$ is given by Landau1989 ;
Jackson1999 ; Misner1973 ; Partanen2022b

$\mathbf{T}=\left[\begin{array}[]{cc}W&c\mathbf{G}^{T}\\\
c\mathbf{G}&\boldsymbol{\mathcal{T}}\\\
\end{array}\right]=\left[\begin{array}[]{cccc}W&cG^{x}&cG^{y}&cG^{z}\\\
cG^{x}&\mathcal{T}^{xx}&\mathcal{T}^{xy}&\mathcal{T}^{xz}\\\
cG^{y}&\mathcal{T}^{yx}&\mathcal{T}^{yy}&\mathcal{T}^{yz}\\\
cG^{z}&\mathcal{T}^{zx}&\mathcal{T}^{zy}&\mathcal{T}^{zz}\end{array}\right].$
(37)

In some previous literature, asymmetric SEM tensors have been introduced
Penfield1967 , but in our work, all SEM tensors are symmetric and strictly
based on the classical definition in Eq. (37).

We require that the SEM tensors of the total system and its subsystems, e.g.,
the material and field+interaction subsystems, must all transform between
inertial frames in a relativistically covariant way. This imposes strong
limitations on the possible forms of the SEM tensors. Since the SEM tensor is
a derived quantity made of basic physical quantities, such as the atomic
density, atomic velocity, and the electric and magnetic fields, it can be
transformed between inertial frames by transforming the basic physical
quantities in its elements. Simultaneously, the SEM tensor must satisfy the
Lorentz transformation of second-rank tensors Penfield1967 ; Kemp2017 ;
Partanen2019a ; Partanen2021b . Thus, arbitrary combinations of basic physical
quantities forming the SEM tensor are not possible. In addition, one cannot
arbitrarily mix tensor elements between the SEM tensors of two subsystems.
Below, the SEM tensors are presented so that they each satisfy the condition
discussed above. They are presented in terms of four-dimensional quantities,
such as four-vectors, field tensors, and Lorentz scalars, which makes the
fulfillment of the relativistic covariance property transparent.

The total SEM tensor of the system of the electromagnetic field and a perfect
fluid is given by a sum of the SEM tensor $\mathbf{T}_{\mathrm{mat}}$ of the
material subsystem, and the SEM tensor $\mathbf{T}_{\mathrm{emi}}$ of the
field+interaction subsystem as

$\mathbf{T}_{\mathrm{tot}}=\mathbf{T}_{\mathrm{mat}}+\mathbf{T}_{\mathrm{emi}}.$
(38)

By defining the four-fource density as
$\boldsymbol{\mathcal{F}}=(\phi/c,\mathbf{f})$, the conservation laws in Eqs.
(2)–(5) are compactly written as
$\partial_{\beta}(T_{\mathrm{mat}})^{\alpha\beta}=\mathcal{F}^{\alpha}$ and
$\partial_{\beta}(T_{\mathrm{emi}})^{\alpha\beta}=-\mathcal{F}^{\alpha}$. For
the total isolated system, the conservation laws are correspondingly given by
$\partial_{\beta}(T_{\mathrm{tot}})^{\alpha\beta}=0$.

In accordance with the splitting of the force density into parts in previous
sections, the SEM tensor of the field+interaction subsystem can be written as

$\mathbf{T}_{\mathrm{emi}}=\mathbf{T}_{\mathrm{owm}}+\mathbf{T}_{\mathrm{ost}}+\mathbf{T}_{\mathrm{mech}}.$
(39)

Here $\mathbf{T}_{\mathrm{owm}}$ is the optical generalized Abraham SEM tensor
of the field corresponding to $\mathbf{f}_{\mathrm{owm}}$,
$\mathbf{T}_{\mathrm{ost}}$ is the electro- and magnetostriction SEM tensor
corresponding to $\mathbf{f}_{\mathrm{ost}}$, and $\mathbf{T}_{\mathrm{mech}}$
is the SEM tensor of mechanical pressure corresponding to
$\mathbf{f}_{\mathrm{mech}}$.

Using the diagonal Minkowski metric tensor $\boldsymbol{g}$ with sign
convention $g_{00}=1$, $g_{11}=g_{22}=g_{33}=-1$, the subsystem SEM tensors
above are given in contravariant forms applicable to a general inertial frame
as

$\mathbf{T}_{\mathrm{mat}}=\rho_{\mathrm{a}}\mathbf{U}_{\mathrm{a}}\otimes\mathbf{U}_{\mathrm{a}},$
(40) $\displaystyle\mathbf{T}_{\mathrm{owm}}$
$\displaystyle=\frac{1}{2}(\boldsymbol{\mathcal{F}}\boldsymbol{g}\boldsymbol{\mathcal{D}}+\boldsymbol{\mathcal{D}}\boldsymbol{g}\boldsymbol{\mathcal{F}})-\frac{1}{4}\boldsymbol{g}\mathrm{Tr}[\boldsymbol{\mathcal{F}}\boldsymbol{g}\boldsymbol{\mathcal{D}}\boldsymbol{g}]$
$\displaystyle\hskip
11.38092pt-\frac{1}{2c^{2}}[(\boldsymbol{\mathcal{F}}\boldsymbol{g}\boldsymbol{\mathcal{D}}-\boldsymbol{\mathcal{D}}\boldsymbol{g}\boldsymbol{\mathcal{F}})\boldsymbol{g}(\mathbf{U}_{\mathrm{a}}\otimes\mathbf{U}_{\mathrm{a}})$
$\displaystyle\hskip
11.38092pt+(\mathbf{U}_{\mathrm{a}}\otimes\mathbf{U}_{\mathrm{a}})\boldsymbol{g}(\boldsymbol{\mathcal{D}}\boldsymbol{g}\boldsymbol{\mathcal{F}}-\boldsymbol{\mathcal{F}}\boldsymbol{g}\boldsymbol{\mathcal{D}})],$
(41)
$\mathbf{T}_{\mathrm{ost}}=(p_{\mathrm{oes}}+p_{\mathrm{oms}})\Big{(}\frac{\mathbf{U}_{\mathrm{a}}\otimes\mathbf{U}_{\mathrm{a}}}{c^{2}}-\boldsymbol{g}\Big{)},$
(42)
$\mathbf{T}_{\mathrm{mech}}=p_{\mathrm{mech}}\Big{(}\frac{\mathbf{U}_{\mathrm{a}}\otimes\mathbf{U}_{\mathrm{a}}}{c^{2}}-\boldsymbol{g}\Big{)}.$
(43)

Here
$\mathbf{U}_{\mathrm{a}}=\gamma_{\mathbf{v}_{\mathrm{a}}}(c,\mathbf{v}_{\mathrm{a}})$
is the four-velocity of the material, and the contravariant forms of the
electromagnetic field tensor $\boldsymbol{\mathcal{F}}$ and the
electromagnetic displacement tensor $\boldsymbol{\mathcal{D}}$ are given by

$\\!\\!\boldsymbol{\mathcal{F}}=\left[\begin{array}[]{cccc}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\\
E_{x}/c&0&-B_{z}&B_{y}\\\ E_{y}/c&B_{z}&0&-B_{x}\\\
E_{z}/c&-B_{y}&B_{x}&0\end{array}\right],$ (44)
$\boldsymbol{\mathcal{D}}=\left[\begin{array}[]{cccc}0&-D_{x}c&-D_{y}c&-D_{z}c\\\
D_{x}c&0&-H_{z}&H_{y}\\\ D_{y}c&H_{z}&0&-H_{x}\\\
D_{z}c&-H_{y}&H_{x}&0\end{array}\right]\\!.$ (45)

The electro- and magnetostrictive pressures $p_{\mathrm{oes}}$ and
$p_{\mathrm{oms}}$, the mechanical pressure $p_{\mathrm{mech}}$, and the mass
density $\rho_{\mathrm{a}}$ are Lorentz scalars.

The Lorentz invariant form of the electrostrictive pressure is given by
$p_{\mathrm{oes}}=-\frac{1}{2}[\mathbf{P}\cdot\mathbf{E}-\mathbf{v}_{\mathrm{a}}\cdot(\mathbf{P}\times\mathbf{B})]$
and the magnetostrictive pressure is given by
$p_{\mathrm{oms}}=-\frac{1}{2}[\mathbf{M}\cdot\mathbf{B}+\mathbf{v}_{\mathrm{a}}\cdot(\mathbf{M}\times\mathbf{E})/c^{2}]$.
The second terms of these expressions are expected to be related to the
Röntgen and Aharonov-Casher interactions of previous literature since they
have similar forms Horsley2006 . As conventional, the factor $\frac{1}{2}$ is
related to the fact that we are here dealing with induced dipoles and not
permanent ones or a system of free charges. In the special case of the
laboratory frame, where the atomic velocity is negligible as
$\mathbf{v}_{\mathrm{a}}\approx\mathbf{0}$, we have
$p_{\mathrm{oes}}\approx-\frac{1}{2}\mathbf{P}\cdot\mathbf{E}$ and
$p_{\mathrm{oms}}\approx-\frac{1}{2}\mathbf{M}\cdot\mathbf{B}$ in agreement
with Eqs. (21) and (22).

The SEM tensor of the material in Eq. (40) is of the well-known form
Misner1973 ; Dirac1996 . The optical generalized Abraham SEM tensor in Eq.
(41) has appeared in previous literature in Refs. Obukhov2008 ; Makarov2011 ;
Partanen2021b , and it has been generalized for dispersive materials in Ref.
Partanen2022b . The SEM tensors of the optostriction and mechanical pressure
in Eqs. (42) and (43) are of the well-known form of the pressure term
appearing in the SEM tensor of a perfect fluid Misner1973 .

In the special case of the laboratory frame, where the atomic velocity is
negligibly small, the general expressions of the SEM tensors in Eqs. (40)–(43)
reduce to the following simple formulas:

$\mathbf{T}_{\mathrm{mat}}=\left[\begin{array}[]{cc}\rho_{\mathrm{a}}c^{2}&\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}^{T}c\\\
\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}c&\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}\otimes\mathbf{v}_{\mathrm{a}}\end{array}\right],$
(46) $\displaystyle\mathbf{T}_{\mathrm{owm}}$
$\displaystyle\\!\\!=\\!\bigg{[}\begin{array}[]{cc}\\!\\!\frac{1}{2}(\mathbf{E}\\!\cdot\\!\mathbf{D}\\!+\\!\mathbf{H}\\!\cdot\\!\mathbf{B})&\frac{1}{c}(\mathbf{E}\\!\times\\!\mathbf{H})^{T}\\\
\frac{1}{c}\mathbf{E}\\!\times\\!\mathbf{H}&\frac{1}{2}(\mathbf{E}\\!\cdot\\!\mathbf{D}\\!+\\!\mathbf{H}\\!\cdot\\!\mathbf{B})\mathbf{I}-\mathbf{E}\\!\otimes\\!\mathbf{D}-\mathbf{H}\\!\otimes\\!\mathbf{B}\\!\\!\end{array}\bigg{]},$
(49) $\mathbf{T}_{\mathrm{ost}}=\left[\begin{array}[]{cc}0&\mathbf{0}\\\
\mathbf{0}&(p_{\mathrm{oes}}+p_{\mathrm{oms}})\mathbf{I}\end{array}\right],$
(50) $\mathbf{T}_{\mathrm{mech}}=\left[\begin{array}[]{cc}0&\mathbf{0}\\\
\mathbf{0}&p_{\mathrm{mech}}\mathbf{I}\end{array}\right].$ (51)

Here, for the optoelectro- and optomagnetostrictive pressures
$p_{\mathrm{oes}}$ and $p_{\mathrm{oms}}$, one can use the laboratory frame
expressions, given in Eqs. (21) and (22).

##  VII Discussion

An obvious question arising from the present theory of optostriction is how it
influences existing theories of the forces of light Anghinoni2022 . We discuss
in particular the implications to the previous mass-polariton theory
Partanen2017c ; Partanen2019a ; Partanen2019b ; Partanen2021b ; Partanen2022b
; Partanen2018a ; Partanen2018b ; Partanen2022a . As we pointed out in the
analysis of Fig. 2, the optostrictive force density does not lead to net
momentum transfer between the field and the material. Thus, it does not
contribute to the transfer of wave momentum by the optical field or the net
momentum carried by the material. Therefore, the optostrictive force density
does not either lead to net mass transfer of the material, i.e., it does not
contribute to the transferred mass of the mass-polariton state. Accordingly,
the law of constant velocity of the center of energy of an isolated system
remains fulfilled within this generalization of the mass-polariton theory.

The theory can, however, lead to a locally nonzero momentum density of the
material, and actually, in homogeneous materials, the optostrictive force
density for light beams and pulses typically _dominates_ over the optical wave
momentum force density. In particular, the optostrictive force density has a
radial component, which for typical light pulses and beams, is larger than
longitudinal force density component of the Abraham force term, which
contributes to the transfer of wave momentum. Consequently, the optostrictive
force density must be included in the force densities of light beams and
pulses, and it is especially important in the modeling of experimentally
observed displacements of the material caused by optical fields.

Both the optical wave momentum force density $\mathbf{f}_{\mathrm{owm}}$ and
the optostrictive force density $\mathbf{f}_{\mathrm{ost}}$ act in the elastic
wave equation of the material as driving forces of the dynamics. For short
optical pulses, in the regime, where the field is nonzero,
$\mathbf{f}_{\mathrm{owm}}$ and $\mathbf{f}_{\mathrm{ost}}$ dominate over
reactive elastic forces. This means that, in the time and displacement scale
of the optical pulse, the changes in the interatomic distances are so small
that elastic forces have negligible contribution. Thus, the optical pulse
gives a forced displacement and impulse of atoms that can be calculated from
the sum of $\mathbf{f}_{\mathrm{owm}}$ and $\mathbf{f}_{\mathrm{ost}}$. After
the field fades out, these displacements and related momenta form the initial
state of the elastic relaxation, which is seen as sound waves. In this
respect, $\mathbf{f}_{\mathrm{owm}}$ and $\mathbf{f}_{\mathrm{ost}}$ behave
qualitatively in the same way.

From the point of view of the SEM tensors, discussed in Sec. VI, the
relativistic covariance of the theory is preserved since the optostriction is
described by the SEM tensor $\mathbf{T}_{\mathrm{ost}}$ in Eq. (42), whose
relativistic covariance is independent of the covariance of
$\mathbf{T}_{\mathrm{owm}}$ in Eq. (41). The relativistic covariance of the
theory would be preserved even if the dissipation terms introduced in the
calculation of the optostrictive force density were neglected. However, in
this case, the form of the optostrictive pressure in a general inertial frame
would become more complicated.

We can conclude that the optostrictive force density is an essential addition
to the optical wave momentum force density of the mass-polariton theory used
in previous works. It is important in describing the position- and time-
dependent dynamics of the material, but it does not influence the relativistic
covariance of the theory. The theory of optoelectro- and optomagnetostrictive
force densities presented in this work gives a possibility to calculate the
full position and time-dependent dynamics of the material under the influence
of the optical field. Thus, time or harmonic cycle averaging is not needed
within this theory.

From the experimental point of view, it is interesting to compare the
optoelectro- and optomagnetostrictive pressures $p_{\mathrm{oes}}$ and
$p_{\mathrm{oms}}$ to the electrostrictive and magnetostrictive pressures
$p_{\mathrm{es}}$ and $p_{\mathrm{ms}}$, derived originally for stationary
fields Brevik2018a . It is found that these pressures are related by
$p_{\mathrm{es}}=\frac{\varepsilon_{\mathrm{r}}+2}{3}p_{\mathrm{oes}}$ and
$p_{\mathrm{ms}}=\frac{\mu_{\mathrm{r}}+2}{3\mu_{\mathrm{r}}}p_{\mathrm{oms}}$.
While the electrostrictive pressure $p_{\mathrm{es}}$ explains the results of
the classic Hakim-Higham experiment Hakim1962  for stationary fields, the
optoelectrostrictive pressure $p_{\mathrm{oes}}$ explains the results of the
recent measurement by Astrath _et al._ Astrath2022  for an optical field. At
optical frequencies, in the case of water, $p_{\mathrm{oes}}$ is about 20%
smaller than $p_{\mathrm{es}}$, so the relative accuracy of the measurement
does not need to be exceptionally high to observe which one of the formulas
agrees with the results better. However, since the amount of quantitatively
accurate measurements of electrostriction is very limited, it is desired to
verify the reproducibility of the results, to narrow down error margins, and
to carry out similar measurements for other materials.

The condition of a linear, isotropic material and the applicability of the
Clausius-Mossotti relation limit the possibility to use the present theory for
all materials. However, extension of the present theory to cover materials
with different relations between the polarizability of the material and the
macroscopic electric field can be developed following the principles presented
in this work.

Regarding the anisotropy of the electrostrictive tensor of a deformed
material, we note that the anisotropy is negligible for typical field
strengths in common photonic materials. In the case of light pulses, the
optostrictive forces give an impulse to the material atoms. The associated
momenta later result in atomic displacement. These displacements develop at
acoustic velocities and do not have time to become significant during the
short optical transient. Anisotropies resulting from faster processes, such as
the Kerr effect, could in principle take place. However, these effects are
also negligible for common photonic materials. For example, in a recent work
in Ref. Astrath2023 , it was found that the change generated _in the
refractive index_ of water due to the Kerr effect is negligible being of the
order of 10-10. Furthermore, the anisotropy in the electrostrictive tensor
must vanish in the limit of low field strength.

##  VIII Conclusions

In conclusion, we have derived a time-dependent theory of force densities
generated on the material by an optical field. The theory extends the previous
mass-polariton theory of light to include optoelectro- and
optomagnetostrictive force densities, which arise from the atomic density
dependence of the energy density of the electric and magnetic fields. By
introducing additional dissipation terms, nonexistent in the conventional
theory of electrostriction and magnetostriction, we were able to explain the
difference between the existing experimental results for a stationary field
and for an optical field. The dissipation terms, we have introduced, are
necessary to conserve the total energy when the optostrictive force does
contraction work on the material. In the present work, we have determined the
magnitude of this dissipation starting from the Lorentz force model of the
optostrictive force and the requirement of the conservation of energy during
the contraction of the material. Developing a physical model for this
dissipation is a topic of further work. We have shown that the theory is
relativistically covariant, meaning that it can be applied to an arbitrary
inertial observer independent of its velocity with respect to the material.
This is a strong condition limiting the number of possible theories. The
understanding of electrostrictive and magnetostrictive forces at optical
frequencies, developed in the present theory, is expected to revive interest
in experimental studies of optical forces in various photonic materials. We
also expect that the unified electromagnetic force theory for optical fields
will open new approaches to develop our understanding of physics and
engineering of thermal and acousto-optical coupling of light and dielectrics.

###### Acknowledgements.

This work has been funded by the Academy of Finland under Contract No. 318197
and 349971. B.A. and N.G.C.A acknowledge CNPq (409403/2018-0, 304738/2019-0)
and CAPES (Finance Code 001) for financial support.

##  Appendix A Lorentz force density

In this appendix, we present the total electromagnetic force density
$\mathbf{f}_{\mathrm{em}}$ based on the Lorentz force law. This force densty
can be split into the force density $\mathbf{f}_{\mathrm{e}}$ for induced
electric dipoles and $\mathbf{f}_{\mathrm{m}}$ for induced magnetic dipoles.
These parts and their sum are given in the subsections below.

###  A.1 Force density on induced electric dipoles

For electric dipoles generating the polarization field $\mathbf{P}$, the
Lorentz force density is well known to be given by Gordon1973 ; Hinds2009 ;
Barnett2006 ; Stenholm1986 ; Penfield1967 ; Landau1984

$\displaystyle\mathbf{f}_{\mathrm{e}}$
$\displaystyle=(\mathbf{P}\cdot\boldsymbol{\nabla})\mathbf{E}+\frac{\partial\mathbf{P}}{\partial
t}\times\mathbf{B}.$ (52)

Using $\mathbf{P}=\varepsilon_{0}(\varepsilon_{\mathrm{r}}-1)\mathbf{E}$ and
applying the mathematical identity
$(\mathbf{E}\cdot\boldsymbol{\nabla})\mathbf{E}=\nabla(\frac{1}{2}|\mathbf{E}|^{2})-\mathbf{E}\times(\boldsymbol{\nabla}\times\mathbf{E})$
with Faraday’s law
$\boldsymbol{\nabla}\times\mathbf{E}=-\partial\mathbf{B}/\partial t$ and the
product rule of differentiation, we can rewrite the force density in Eq. (52)
as

$\mathbf{f}_{\mathrm{e}}=\frac{1}{2}\nabla(\mathbf{P}\cdot\mathbf{E})-\frac{1}{2}\varepsilon_{0}|\mathbf{E}|^{2}\nabla\varepsilon_{\mathrm{r}}+\frac{\partial}{\partial
t}(\mathbf{P}\times\mathbf{B}).$ (53)

The derivation of the force density $\mathbf{f}_{\mathrm{e}}$ from the Lorentz
forces on the individual charges of the induced electric dipole is presented
briefly below.

The force applied on a single electric charge $\pm q_{\mathrm{e}}$ in an
electromagnetic field at position $\mathbf{r}_{\pm q_{\mathrm{e}}}$ is known
as the Lorentz force Jackson1999 ; Landau1989 , and it is given by

$\mathbf{F}_{\pm q_{\mathrm{e}}}(t)=\pm
q_{\mathrm{e}}\Big{[}\mathbf{E}(\mathbf{r}_{\pm
q_{\mathrm{e}}},t)+\frac{d\mathbf{r}_{\pm
q_{\mathrm{e}}}}{dt}\times\mathbf{B}(\mathbf{r}_{\pm
q_{\mathrm{e}}},t)\Big{]}.$ (54)

Let us take a point $\mathbf{r}_{0}$ as the center of mass of two charges,
which make the electric dipole. Then, we write the macroscopic electric and
magnetic fields, $\mathbf{E}$ and $\mathbf{B}$, around $\mathbf{r}_{0}$ by
using two first terms of their truncated Taylor series as

$\mathbf{E}(\mathbf{r},t)=\mathbf{E}(\mathbf{r}_{0},t)+[(\mathbf{r}-\mathbf{r}_{0})\cdot\boldsymbol{\nabla}]\mathbf{E}(\mathbf{r},t)|_{\mathbf{r}=\mathbf{r}_{0}}.$
(55)
$\mathbf{B}(\mathbf{r},t)=\mathbf{B}(\mathbf{r}_{0},t)+[(\mathbf{r}-\mathbf{r}_{0})\cdot\boldsymbol{\nabla}]\mathbf{B}(\mathbf{r},t)|_{\mathbf{r}=\mathbf{r}_{0}}.$
(56)

Next, we write the net force on the center of mass of an electric dipole made
of two opposite charges, given by
$\mathbf{F}_{\mathrm{e}}=\mathbf{F}_{q_{\mathrm{e}}}+\mathbf{F}_{-q_{\mathrm{e}}}$.
Using the field approximations in Eqs. (55) and (56) with dropping out the
second term of Eq. (56), since the resulting force density terms are
negligible in comparison with the terms included Stenholm1986 , the net force
on an electric dipole then becomes Gordon1973 ; Hinds2009 ; Barnett2006 ;
Stenholm1986 ; Penfield1967 ; Landau1984

$\displaystyle\mathbf{F}_{\mathrm{e}}(t)$
$\displaystyle=q_{\mathrm{e}}\Big{[}\mathbf{E}(\mathbf{r}_{q},t)-\mathbf{E}(\mathbf{r}_{-q},t)$
$\displaystyle\hskip
14.22636pt+\frac{d\mathbf{r}_{q}}{dt}\times\mathbf{B}(\mathbf{r}_{q},t)-\frac{d\mathbf{r}_{-q}}{dt}\times\mathbf{B}(\mathbf{r}_{-q},t)\Big{]}$
$\displaystyle=(\mathbf{p}\cdot\boldsymbol{\nabla})\mathbf{E}(\mathbf{r},t)|_{\mathbf{r}=\mathbf{r}_{0}}+\frac{d\mathbf{p}}{dt}\times\mathbf{B}(\mathbf{r}_{0},t).$
(57)

Here we have defined the electric dipole moment as
$\mathbf{p}=q_{\mathrm{e}}(\mathbf{r}_{q}-\mathbf{r}_{-q})$. By defining the
polarization field $\mathbf{P}$ as the dipole moment density through
$\mathbf{P}=n_{\mathrm{a}}\mathbf{p}$, where $n_{\mathrm{a}}$ is the number
density of electric dipoles, the force density
$\mathbf{f}_{\mathrm{e}}=n_{\mathrm{a}}\mathbf{F}_{\mathrm{e}}$ is then
obtained as given in Eq. (52).

###  A.2 Force density on induced magnetic dipoles

For magnetic dipoles, the derivation of the Lorentz force density is less
straightforward than for electric dipoles and involves the enigma of the
hidden momentum Griffiths2015 ; Correa2020 . The resulting force density is
given by Astrath2022 ; Anghinoni2023

$\mathbf{f}_{\mathrm{m}}=(\mathbf{M}\cdot\boldsymbol{\nabla})\mathbf{B}+\mathbf{M}\times(\boldsymbol{\nabla}\times\mathbf{B})-\frac{1}{c^{2}}\frac{\partial}{\partial
t}(\mathbf{M}\times\mathbf{E}).$ (58)

Using the constitutive relations in Eq. (7), which give
$\mathbf{M}=\frac{\mu_{\mathrm{r}}-1}{\mu_{0}\mu_{\mathrm{r}}}\mathbf{B}$, and
applying the mathematical identity
$(\mathbf{B}\cdot\boldsymbol{\nabla})\mathbf{B}=\nabla(\frac{1}{2}|\mathbf{B}|^{2})-\mathbf{B}\times(\boldsymbol{\nabla}\times\mathbf{B})$
and the product rule of differentiation, we can rewrite the force density in
Eq. (58) as

$\mathbf{f}_{\mathrm{m}}=\frac{1}{2}\nabla(\mathbf{M}\cdot\mathbf{B})-\frac{1}{2}\mu_{0}|\mathbf{H}|^{2}\nabla\mu_{\mathrm{r}}-\frac{1}{c^{2}}\frac{\partial}{\partial
t}(\mathbf{M}\times\mathbf{E}).$ (59)

###  A.3 Total force density on atomic dipoles

The total electromagnetic force density $\mathbf{f}_{\mathrm{em}}$ in Eq. (16)
on a material made of both induced electric and magnetic dipoles can be
written as a sum of the electric and magnetic parts in Eqs. (52) and (58) as
$\mathbf{f}_{\mathrm{em}}=\mathbf{f}_{\mathrm{e}}+\mathbf{f}_{\mathrm{m}}$,
resulting in

$\displaystyle\mathbf{f}_{\mathrm{em}}$
$\displaystyle=(\mathbf{P}\cdot\boldsymbol{\nabla})\mathbf{E}+(\mathbf{M}\cdot\boldsymbol{\nabla})\mathbf{B}+\mathbf{M}\times(\boldsymbol{\nabla}\times\mathbf{B})$
$\displaystyle\hskip 14.22636pt+\frac{\partial\mathbf{P}}{\partial
t}\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial}{\partial
t}(\mathbf{M}\times\mathbf{E}).$ (60)

Alternatively, using Eqs. (53) and (59) for $\mathbf{f}_{\mathrm{e}}$ and
$\mathbf{f}_{\mathrm{m}}$ and applying the constitutive relations in Eqs. (6)
and (7) to combine the time derivative terms of $\mathbf{f}_{\mathrm{e}}$ and
$\mathbf{f}_{\mathrm{m}}$ into a single term
$\frac{n^{2}-1}{c^{2}}\frac{\partial}{\partial
t}(\mathbf{E}\times\mathbf{H})$, where we have used
$\varepsilon_{\mathrm{r}}\mu_{\mathrm{r}}=n^{2}$ and taken this factor out of
the time derivative, we obtain

$\displaystyle\mathbf{f}_{\mathrm{em}}$
$\displaystyle=\frac{1}{2}\nabla(\mathbf{P}\cdot\mathbf{E})+\frac{1}{2}\nabla(\mathbf{M}\cdot\mathbf{B})-\frac{1}{2}\varepsilon_{0}|\mathbf{E}|^{2}\nabla\varepsilon_{\mathrm{r}}$
$\displaystyle\hskip
14.22636pt-\frac{1}{2}\mu_{0}|\mathbf{H}|^{2}\nabla\mu_{\mathrm{r}}+\frac{n^{2}-1}{c^{2}}\frac{\partial}{\partial
t}(\mathbf{E}\times\mathbf{H}).$ (61)

##  Appendix B Electrostriction and magnetostriction for stationary fields

In the calculation of the conventional electro- and magnetostrictive force
densities for stationary fields, one makes use of the conservation of energy
when a small change is made in the density of the material Landau1984 ;
Boyd2008 . The Clausius-Mossotti relation in Eq. (12) and its magnetic analog
in Eq. (13) show that the increase of the density of the material corresponds
to the increase of the relative permittivity and permeability. This can be
concluded from the derivatives of the relative permittivity and permeability
with respect to the atomic density, for which Eqs. (12) and (13) give
$n_{\mathrm{a}}\frac{\partial\varepsilon_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}=\frac{1}{3}(\varepsilon_{\mathrm{r}}-1)(\varepsilon_{\mathrm{r}}+2)$
and $n_{\mathrm{a}}\frac{\partial\mu_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}=\frac{1}{3}(\mu_{\mathrm{r}}-1)(\mu_{\mathrm{r}}+2)$. Thus, if
the fields $\mathbf{D}$ and $\mathbf{B}$ are kept constant, the electric and
magnetic field energy densities
$W_{\mathrm{e}}=\frac{1}{2\varepsilon_{0}\varepsilon_{\mathrm{r}}}|\mathbf{D}|^{2}$
and $W_{\mathrm{m}}=\frac{1}{2\mu_{0}\mu_{\mathrm{r}}}|\mathbf{B}|^{2}$
decrease when the density of the material is increased. Then, following the
principle of virtual work, the field energy density acts a potential energy,
which tends to compress the material. The reduction of the field energy
density must be equal to the increase of the elastic and thermal energy
densities of the material. At equilibrium, the electro- and magnetostrictive
force density is then equal to the thermodynamical force density corresponding
to the compressibility of the material for the given thermodynamical process
Boyd2008 . For an isothermal process taking place at constant temperature $T$,
we then obtain the electrostrictive force density $\mathbf{f}_{\mathrm{es}}$
and magnetostrictive force density $\mathbf{f}_{\mathrm{ms}}$ from the
electric and magnetic free energy densities $F_{\mathrm{e}}=W_{\mathrm{e}}$
and $F_{\mathrm{m}}=W_{\mathrm{m}}$ as

$\displaystyle\mathbf{f}_{\mathrm{es}}$
$\displaystyle=-\nabla\Big{[}n_{\mathrm{a}}\Big{(}\frac{\partial
F_{\mathrm{e}}}{\partial n_{\mathrm{a}}}\Big{)}_{\mathbf{D},T}\Big{]}$
$\displaystyle=\frac{1}{2}\varepsilon_{0}\nabla\Big{[}n_{\mathrm{a}}\Big{(}\frac{\partial\varepsilon_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}\Big{)}_{T}|\mathbf{E}|^{2}\Big{]}$
$\displaystyle=\frac{1}{2}\nabla(\mathbf{P}\cdot\mathbf{E}_{\mathrm{eff}}),$
(62) $\displaystyle\mathbf{f}_{\mathrm{ms}}$
$\displaystyle=-\nabla\Big{[}n_{\mathrm{a}}\Big{(}\frac{\partial
F_{\mathrm{m}}}{\partial n_{\mathrm{a}}}\Big{)}_{\mathbf{B},T}\Big{]}$
$\displaystyle=\frac{1}{2}\mu_{0}\nabla\Big{[}n_{\mathrm{a}}\Big{(}\frac{\partial\mu_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}\Big{)}_{T}|\mathbf{H}|^{2}\Big{]}$
$\displaystyle=\frac{1}{2}\nabla(\mathbf{M}\cdot\mathbf{B}_{\mathrm{eff}}).$
(63)

Equations (62) and (63) define the conventional stationary electrostrictive
and magnetostrictive pressures $p_{\mathrm{es}}$ and $p_{\mathrm{ms}}$ through
$\mathbf{f}_{\mathrm{es}}=-\nabla p_{\mathrm{es}}$ and
$\mathbf{f}_{\mathrm{ms}}=-\nabla p_{\mathrm{ms}}$. Thus,
$p_{\mathrm{es}}=-\frac{1}{2}\varepsilon_{0}n_{\mathrm{a}}\big{(}\frac{\partial\varepsilon_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}\big{)}_{T}|\mathbf{E}|^{2}=-\frac{1}{6}\varepsilon_{0}(\varepsilon_{\mathrm{r}}-1)(\varepsilon_{\mathrm{r}}+2)|\mathbf{E}|^{2}$
and
$p_{\mathrm{ms}}=-\frac{1}{2}\mu_{0}n_{\mathrm{a}}\big{(}\frac{\partial\mu_{\mathrm{r}}}{\partial
n_{\mathrm{a}}}\big{)}_{T}|\mathbf{H}|^{2}=-\frac{1}{6}\mu_{0}(\mu_{\mathrm{r}}-1)(\mu_{\mathrm{r}}+2)|\mathbf{H}|^{2}$
Boyd2008 ; Landau1984 . The stationary electrostrictive pressure has been
quantitatively measured for selected non-polar isotropic liquid dielectrics in
the classic Hakim-Higham experiment Hakim1962 .

Alternatively, if the fields $\mathbf{E}$ and $\mathbf{H}$ are kept constant,
the electric and magnetic field energy densities
$W_{\mathrm{e}}=\frac{1}{2}\varepsilon_{0}\varepsilon_{\mathrm{r}}|\mathbf{E}|^{2}$
and $W_{\mathrm{m}}=\frac{1}{2}\mu_{0}\mu_{\mathrm{r}}|\mathbf{H}|^{2}$
increase when the density of the material is increased. At first sight, this
seems to be in contradiction with the argumentation above. However, in this
case one must do external work to preserve the values of the fields
$\mathbf{E}$ and $\mathbf{H}$ at the same time when the density of the
material is increased. When this external work is accounted for, the free
energy densities $F_{\mathrm{e}}$ and $F_{\mathrm{m}}$ become replaced by
$\tilde{F}_{\mathrm{e}}=-W_{\mathrm{e}}$ and
$\tilde{F}_{\mathrm{m}}=-W_{\mathrm{m}}$, and we obtain
$\mathbf{f}_{\mathrm{es}}=-\nabla\big{[}n_{\mathrm{a}}\big{(}\frac{\partial\tilde{F}_{\mathrm{e}}}{\partial
n_{\mathrm{a}}}\big{)}_{\mathbf{E},T}\big{]}$ and
$\mathbf{f}_{\mathrm{ms}}=-\nabla\big{[}n_{\mathrm{a}}\big{(}\frac{\partial\tilde{F}_{\mathrm{m}}}{\partial
n_{\mathrm{a}}}\big{)}_{\mathbf{H},T}\big{]}$. As a result, the values of the
electrostrictive and magnetostrictive force densities are not changed from
those obtained by using Eqs. (62) and (63). In an example of a dielectric
material placed between capacitor plates, keeping the field $\mathbf{D}$
constant corresponds to having fixed charges on the capacitor plates, while
keeping the field $\mathbf{E}$ constant corresponds to having fixed potential
between the capacitor plates (Sec. 4.7 of Ref. Jackson1999 ). In the latter
case, one must do work to increase the charges on the capacitor plates when
the permittivity of the dielectric is increased.

Note that the stationary electrostrictive and magnetostrictive force densities
in Eqs. (62) and (63) could also be calculated as gradient force densities
from the potential energy density resulting from the Stark [$\Delta
E_{\mathrm{e}}^{(n)}=-\frac{1}{2}(\alpha_{\mathrm{e}})_{ik}^{(n)}(E_{\mathrm{eff}})_{i}(E_{\mathrm{eff}})_{k}$]
and Zeeman [$\Delta
E_{\mathrm{m}}^{(n)}=-\frac{1}{2}(\alpha_{\mathrm{m}})_{ik}^{(n)}(B_{\mathrm{eff}})_{i}(B_{\mathrm{eff}})_{k}$]
shifts Landau1977  of atomic energy levels in an external electromagnetic
field. The equivalence of this approach with the last rows of Eqs. (62) and
(63) can be seen by assuming a single ground-state energy level, assuming that
its polarizability and magnetizability tensors
$(\alpha_{\mathrm{e}})_{ik}^{(n)}$ and $(\alpha_{\mathrm{m}})_{ik}^{(n)}$ are
diagonal as $(\alpha_{\mathrm{e}})_{ik}^{(n)}=\alpha_{\mathrm{e}}\delta_{ik}$
and $(\alpha_{\mathrm{m}})_{ik}^{(n)}=\alpha_{\mathrm{m}}\delta_{ik}$, and
noting that $p_{i}=\alpha_{\mathrm{e}}(E_{\mathrm{eff}})_{i}$ and
$m_{i}=\alpha_{\mathrm{m}}(B_{\mathrm{eff}})_{i}$ are the electric and
magnetic dipole moments of the atom.

## References

  * (1) L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, _Electrodynamics of Continuous Media_ , Pergamon, Oxford (1984).
  * (2) H. Helmholtz, “Ueber die auf das Innere magnetisch oder dielectrisch polarisirter Körper wirkenden Kräfte,” _Ann. Phys._ 249, 385 (1881).
  * (3) N. G. C. Astrath, G. A. S. Flizikowski, B. Anghinoni, L. C. Malacarne, M. L. Baesso, T. Požar, M. Partanen, I. Brevik, D. Razansky, and S. E. Bialkowski, “Unveiling bulk and surface radiation forces in a dielectric liquid,” _Light Sci. Appl._ 11, 103 (2022).
  * (4) S. S. Hakim and J. B. Higham, “An experimental determination of the excess pressure produced in a liquid dielectric by an electric field,” _Proc. Phys. Soc._ 80, 190 (1962).
  * (5) A. Melloni, M. Frasca, A. Garavaglia, A. Tonini, and M. Martinelli, “Direct measurement of electrostriction in optical fibers,” _Opt. Lett._ 23, 691 (1998).
  * (6) A. Fellegara, A. Melloni, and M. Martinelli, “Measurement of the frequency response induced by electrostriction in optical fibers,” _Opt. Lett._ 22, 1615 (1997).
  * (7) A. Fellegara and S. Wabnitz, “Electrostrictive cross-phase modulation of periodic pulse trains in optical fibers,” _Opt. Lett._ 23, 1357 (1998).
  * (8) E. L. Buckland and R. W. Boyd, “Measurement of the frequency response of the electrostrictive nonlinearity in optical fibers,” _Opt. Lett._ 22, 676 (1997).
  * (9) N. Godbout, S. Lacroix, Y. Quiquempois, G. Martinelli, and P. Bernage, “Measurement and calculation of electrostrictive effects in a twin-hole silica glass fiber,” _J. Opt. Soc. Am. B_ 17, 1 (2000).
  * (10) P. D. Townsend, A. J. Poustie, P. J. Hardman, and K. J. Blow, “Measurement of the refractive-index modulation generated by electrostriction-induced acoustic waves in optical fibers,” _Opt. Lett._ 21, 333 (1996).
  * (11) E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” _Opt. Lett._ 15, 314 (1990).
  * (12) R. Hui, C. Laperle, M. Reimer, A. D. Shiner, and M. O’Sullivan, “Characterization of electrostriction nonlinearity in a standard single-mode fiber based on coherent detection and cross-phase modulation,” _J. Light. Technol._ 33, 4547 (2015).
  * (13) J. Cai, Y. Ishikawa, and K. Wada, “Strain induced bandgap and refractive index variation of silicon,” _Opt. Express_ 21, 7162 (2013).
  * (14) J.-C. Beugnot and V. Laude, “Electrostriction and guidance of acoustic phonons in optical fibers,” _Phys. Rev. B_ 86, 224304 (2012).
  * (15) H. Zhang, N. Pryds, D.-S. Park, N. Gauquelin, S. Santucci, D. V. Christensen, D. Jannis, D. Chezganov, D. A. Rata, A. R. Insinga, I. E. Castelli, J. Verbeeck, I. Lubomirsky, P. Muralt, D. Damjanovic, and V. Esposito, “Atomically engineered interfaces yield extraordinary electrostriction,” _Nature_ 609, 695 (2022).
  * (16) B. Chen, T. Li, Q. Dong, E. Mosconi, J. Song, Z. Chen, Y. Deng, Y. Liu, S. Ducharme, A. Gruverman, F. D. Angelis, and J. Huang, “Large electrostrictive response in lead halide perovskites,” _Nat. Mater._ 17, 1020 (2018).
  * (17) L. Wang, H. Qi, S. Deng, L. Cao, H. Liu, S. Hu, and J. Chen, “Design of superior electrostriction in batio3-based lead-free relaxors via the formation of polarization nanoclusters,” _InfoMat_ 5, e12362 (2023).
  * (18) J. Yu and P.-E. Janolin, “Defining “giant” electrostriction,” _J. Appl. Phys._ 131, 170701 (2022).
  * (19) X. Chen, H. Qin, X. Qian, W. Zhu, B. Li, B. Zhang, W. Lu, R. Li, S. Zhang, L. Zhu, F. D. D. Santos, J. Bernholc, and Q. M. Zhang, “Relaxor ferroelectric polymer exhibits ultrahigh electromechanical coupling at low electric field,” _Science_ 375, 1418 (2022).
  * (20) H.-Y. Zhang, Z.-X. Zhang, X.-G. Chen, X.-J. Song, Y. Zhang, and R.-G. Xiong, “Large electrostrictive coefficient in a two-dimensional hybrid perovskite ferroelectric,” _J. Am. Chem. Soc._ 143, 1664 (2021).
  * (21) B. Wu, C. Zhao, Y. Huang, J. Yin, W. Wu, and J. Wu, “Superior electrostrictive effect in relaxor potassium sodium niobate based ferroelectrics,” _ACS Appl. Mater. Interfaces_ 12, 25050 (2020).
  * (22) Y. Huang, C. Zhao, J. Yin, X. Lv, J. Ma, and J. Wu, “Giant electrostrictive effect in lead-free barium titanate-based ceramics via A-site ion-pairs engineering,” _J. Mater. Chem. A_ 7, 17366 (2019).
  * (23) L. Jin, J. Qiao, L. Hou, L. Wang, L. Zhang, X. Lu, H. Du, X. Wei, Y. Yan, and G. Liu, “High electrostrictive effect in La3+-doped Ba(Zr0.2Ti0.8)O3 lead-free ferroelectrics,” _J. Alloys Compd._ 776, 599 (2019).
  * (24) A. Karvounis, B. Gholipour, K. F. MacDonald, and N. I. Zheludev, “Giant electro-optical effect through electrostriction in a nanomechanical metamaterial,” _Adv. Mater._ 31, 1804801 (2019).
  * (25) Q. M. Zhang, V. Bharti, and X. Zhao, “Giant electrostriction and relaxor ferroelectric behavior in electron-irradiated poly(vinylidene fluoride-trifluoroethylene) copolymer,” _Science_ 280, 2101 (1998).
  * (26) S. Yamashita, Y. Nakazawa, S. Yamanaka, M. Okumura, T. Kojima, N. Yoshinari, and T. Konno, “Dielectric jump and negative electrostriction in metallosupramolecular ionic crystals,” _Sci. Rep._ 8, 2606 (2018).
  * (27) S. M. Neumayer, E. A. Eliseev, M. A. Susner, A. Tselev, B. J. Rodriguez, J. A. Brehm, S. T. Pantelides, G. Panchapakesan, S. Jesse, S. V. Kalinin, M. A. McGuire, A. N. Morozovska, P. Maksymovych, and N. Balke, “Giant negative electrostriction and dielectric tunability in a van der Waals layered ferroelectric,” _Phys. Rev. Mater._ 3, 024401 (2019).
  * (28) D.-Y. Guo, C.-W. Chen, C.-C. Li, H.-C. Jau, K.-H. Lin, T.-M. Feng, C.-T. Wang, T. J. Bunning, I. C. Khoo, and T.-H. Lin, “Reconfiguration of three-dimensional liquid-crystalline photonic crystals by electrostriction,” _Nat. Mater._ 19, 94 (2020).
  * (29) Y. Zhang, H. Yoshida, F. Chu, Y.-Q. Guo, Z. Yang, M. Ozaki, and Q.-H. Wang, “Three-dimensional lattice deformation of blue phase liquid crystals under electrostriction,” _Soft Matter_ 18, 3328 (2022).
  * (30) M. Torbati, K. Mozaffari, L. Liu, and P. Sharma, “Coupling of mechanical deformation and electromagnetic fields in biological cells,” _Rev. Mod. Phys._ 94, 025003 (2022).
  * (31) S. Wang, J. Ng, M. Xiao, and C. T. Chan, “Electromagnetic stress at the boundary: Photon pressure or tension?” _Sci. Adv._ 2, e1501485 (2016).
  * (32) J. L. Jimenez, I. Campos, and M. A. Lopez-Marino, “A new perspective of the Abraham-Minkowski controversy,” _Eur. Phys. J. Plus_ 126, 50 (2011).
  * (33) O. Khakpour, B. Yang, G. Chao, L. Honghuan, and L. Li, “Electrostriction-induced third-order nonlinear optical susceptibility in metamaterials,” _Appl. Phys. A_ 127, 376 (2021).
  * (34) H. Y. Lee, Y. Peng, and Y. M. Shkel, “Strain-dielectric response of dielectrics as foundation for electrostriction stresses,” _J. Appl. Phys._ 98, 074104 (2005).
  * (35) P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” _Opt. Express_ 18, 14439 (2010).
  * (36) P. T. Rakich, Z. Wang, and P. Davids, “Scaling of optical forces in dielectric waveguides: Rigorous connection between radiation pressure and dispersion,” _Opt. Lett._ 36, 217 (2011).
  * (37) A. B. Gojani, R. Bejtullahu, and S. Obayashi, “On two optomechanical effects of laser-induced electrostriction in dielectric liquids,” _Jpn. J. Appl. Phys._ 53, 092703 (2014).
  * (38) M. J. A. Smith, B. T. Kuhlmey, C. M. de Sterke, C. Wolff, M. Lapine, and C. G. Poulton, “Electrostriction enhancement in metamaterials,” _Phys. Rev. B_ 91, 214102 (2015).
  * (39) Y. Hashimoto, N. Yamamoto, T. Kato, D. Oshima, and S. Iwata, “Spin-valve giant magneto-resistance film with magnetostrictive FeSiB amorphous layer and its application to strain sensors,” _J. Appl. Phys._ 123, 113903 (2018).
  * (40) Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. E. Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” _Nanophotonics_ 3, 413 (2014).
  * (41) D. S. P. Tanner, E. Bousquet, and P.-E. Janolin, “Optimized methodology for the calculation of electrostriction from first-principles,” _Small_ 17, 2103419 (2021).
  * (42) H. M. Lai, W. M. Suen, and K. Young, “Microscopic derivation of the Helmholtz force density,” _Phys. Rev. Lett._ 47, 177 (1981).
  * (43) V. Torchigin, “Optical electrostriction pressure in liquid, solids and gases,” _Optik_ 189, 90 (2019).
  * (44) Z. Suo, “Theory of dielectric elastomers,” _Acta Mech. Solida Sin._ 23, 549 (2010).
  * (45) J. J. Wang, F. Y. Meng, X. Q. Ma, M. X. Xu, and L. Q. Chen, “Lattice, elastic, polarization, and electrostrictive properties of BaTiO3 from first-principles,” _J. Appl. Phys._ 108, 034107 (2010).
  * (46) H. Washimi and V. I. Karpman, “The ponderomotive force of a high-frequency electromagnetic field in a dispersive medium,” _Sov. Phys. JETP_ 44, 528 (1976).
  * (47) X. Zhao and Z. Suo, “Electrostriction in elastic dielectrics undergoing large deformation,” _J. Appl. Phys._ 104, 123530 (2008).
  * (48) F. Li, L. Jin, Z. Xu, and S. Zhang, “Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity,” _Appl. Phys. Rev._ 1, 011103 (2014).
  * (49) M. Partanen and J. Tulkki, “Time-dependent optical force theory for optomechanics of dispersive 3D photonic materials and devices,” _Opt. Express_ 30, 28577 (2022).
  * (50) M. Partanen, T. Häyrynen, J. Oksanen, and J. Tulkki, “Photon mass drag and the momentum of light in a medium,” _Phys. Rev. A_ 95, 063850 (2017).
  * (51) M. Partanen and J. Tulkki, “Covariant theory of light in a dispersive medium,” _Phys. Rev. A_ 104, 023510 (2021).
  * (52) M. Partanen and J. Tulkki, “Lorentz covariance of the mass-polariton theory of light,” _Phys. Rev. A_ 99, 033852 (2019).
  * (53) M. Partanen and J. Tulkki, “Lagrangian dynamics of the coupled field-medium state of light,” _New J. Phys._ 21, 073062 (2019).
  * (54) U. Leonhardt, “Momentum in an uncertain light,” _Nature_ 444, 823 (2006).
  * (55) R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” _Rev. Mod. Phys._ 79, 1197 (2007).
  * (56) S. M. Barnett, “Resolution of the Abraham-Minkowski dilemma,” _Phys. Rev. Lett._ 104, 070401 (2010).
  * (57) B. A. Kemp, “Resolution of the Abraham-Minkowski debate: Implications for the electromagnetic wave theory of light in matter,” _J. Appl. Phys._ 109, 111101 (2011).
  * (58) P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” _Adv. Opt. Photon._ 2, 519 (2010).
  * (59) K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” _Phys. Rev. Lett._ 119, 073901 (2017).
  * (60) M. G. Burt and R. Peierls, “The momentum of a light wave in a refracting medium,” _Proc. R. Soc. A_ 333, 149 (1973).
  * (61) M. Mansuripur, “Resolution of the Abraham-Minkowski controversy,” _Opt. Commun._ 283, 1997 (2010).
  * (62) S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” _Phil. Trans. R. Soc. A_ 368, 927 (2010).
  * (63) U. Leonhardt, “Abraham and Minkowski momenta in the optically induced motion of fluids,” _Phys. Rev. A_ 90, 033801 (2014).
  * (64) K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum and angular momentum in complex media: from the Abraham-Minkowski debate to unusual properties of surface plasmon-polaritons,” _New J. Phys._ 19, 123014 (2017).
  * (65) R. W. Boyd, _Nonlinear Optics_ , Academic Press, New York (2008).
  * (66) G. Wissmeyer, M. A. Pleitez, A. Rosenthal, and V. Ntziachristos, “Looking at sound: optoacoustics with all-optical ultrasound detection,” _Light Sci. Appl._ 7, 53 (2018).
  * (67) S. Manohar and D. Razansky, “Photoacoustics: a historical review,” _Adv. Opt. Photon._ 8, 586 (2016).
  * (68) A. C. Tam and H. Coufal, “Pulsed opto-acoustics: Theory and applications,” _J. Phys. Colloques_ 44, C6–9 (1983).
  * (69) H. M. Lai and K. Young, “Theory of the pulsed optoacoustic technique,” _J. Acoustical Soc. Am._ 72, 2000 (1982).
  * (70) C. K. N. Patel and A. C. Tam, “Pulsed optoacoustic spectroscopy of condensed matter,” _Rev. Mod. Phys._ 53, 517 (1981).
  * (71) S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” _J. Phys. D_ 26, 329 (1993).
  * (72) T. Požar, J. Laloš, A. Babnik, R. Petkovšek, M. Bethune-Waddell, K. J. Chau, G. V. B. Lukasievicz, and N. G. C. Astrath, “Isolated detection of elastic waves driven by the momentum of light,” _Nat. Commun._ 9, 3340 (2018).
  * (73) T. Požar and J. Možina, “Measurement of elastic waves induced by the reflection of light,” _Phys. Rev. Lett._ 111, 185501 (2013).
  * (74) A. H. Harvey, J. S. Gallagher, and J. M. H. L. Sengers, “Revised formulation for the refractive index of water and steam as a function of wavelength, temperature and density,” _J. Phys. Chem. Ref. Data_ 27, 761 (1998).
  * (75) P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” _J. Phys. Chem. Ref. Data_ 19, 677 (1990).
  * (76) P. Penfield and H. A. Haus, _Electrodynamics of Moving Media_ , MIT Press, Cambridge, MA (1967).
  * (77) B. Anghinoni, G. Flizikowski, L. Malacarne, M. Partanen, S. Bialkowski, and N. Astrath, “On the formulations of the electromagnetic stress–energy tensor,” _Ann. Phys._ 443, 169004 (2022).
  * (78) C. J. Sheppard and B. A. Kemp, “Optical pressure deduced from energy relations within relativistic formulations of electrodynamics,” _Phys. Rev. A_ 89, 013825 (2014).
  * (79) B. A. Kemp and C. J. Sheppard, “Electromagnetic and material contributions to stress, energy, and momentum in metamaterials,” _AEM_ 6, 11 (2017).
  * (80) J. D. Jackson, _Classical Electrodynamics_ , Wiley, New York (1999).
  * (81) W. Panofsky and M. Phillips, _Classical Electricity and Magnetism_ , Addision-Wesley, Reading, MA (1962).
  * (82) H. A. Lorentz, _The Theory of Electrons_ , Dover, New York (1952).
  * (83) D. J. Griffiths, _Introduction to Electrodynamics_ , Prentice-Hall, Upper Saddle River, NJ (1998).
  * (84) D. E. Aspnes, “Local‐field effects and effective‐medium theory: A microscopic perspective,” _Am. J. Phys._ 50, 704 (1982).
  * (85) A. Shevchenko and B. J. Hoenders, “Microscopic derivation of electromagnetic force density in magnetic dielectric media,” _New J. Phys._ 12, 053020 (2010).
  * (86) A. Shevchenko and M. Kaivola, “Electromagnetic force density and energy-momentum tensor in an arbitrary continuous medium,” _J. Phys. B_ 44, 175401 (2011).
  * (87) J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and nonlinear optical measurements of the Lorentz local field,” _Phys. Rev. Lett._ 67, 972 (1991).
  * (88) C. J. F. Böttcher, _Theory of electric polarization, Vol. I_ , Elsevier, New York (1993).
  * (89) J. A. Stratton, _Electromagnetic theory_ , McGraw-Hill, New York (1941).
  * (90) H. Kragh , “The Lorenz-Lorentz formula: Origin and early history,” _Substantia_ 2, 7 (2018).
  * (91) C. J. F. Böttcher and P. Bordewijk, _Theory of electric polarization, Vol. II_ , Elsevier, New York (1992).
  * (92) C. Kittel, _Introduction to Solid State Physics_ , Wiley, Hoboken, NJ (2005).
  * (93) K. Y. Bliokh, Y. P. Bliokh, and F. Nori, “Ponderomotive forces, Stokes drift, and momentum in acoustic and electromagnetic waves,” _Phys. Rev. A_ 106, L021503 (2022).
  * (94) A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” _Phys. Rev. Lett._ 30, 139 (1973).
  * (95) N. G. C. Astrath, L. C. Malacarne, M. L. Baesso, G. V. B. Lukasievicz, and S. E. Bialkowski, “Unravelling the effects of radiation forces in water,” _Nat. Commun._ 5, 4363 (2014).
  * (96) A. Casner and J.-P. Delville, “Giant deformations of a liquid-liquid interface induced by the optical radiation pressure,” _Phys. Rev. Lett._ 87, 054503 (2001).
  * (97) R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” _Proc. R. Soc. Lond. A_ 360, 347 (1978).
  * (98) R. V. Jones and J. C. S. Richards, “The pressure of radiation in a refracting medium,” _Proc. R. Soc. Lond. A_ 221, 480 (1954).
  * (99) G. B. Walker and D. G. Lahoz, “Experimental observation of Abraham force in a dielectric,” _Nature_ 253, 339 (1975).
  * (100) H. Choi, M. Park, D. S. Elliott, and K. Oh, “Optomechanical measurement of the Abraham force in an adiabatic liquid-core optical-fiber waveguide,” _Phys. Rev. A_ 95, 053817 (2017).
  * (101) W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” _Phys. Rev. Lett._ 101, 243601 (2008).
  * (102) A. Kundu, R. Rani, and K. S. Hazra, “Graphene oxide demonstrates experimental confirmation of Abraham pressure on solid surface,” _Sci. Rep._ 7, 42538 (2017).
  * (103) I. Brevik, “Comment on “observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” _Phys. Rev. Lett._ 103, 219301 (2009).
  * (104) M. Mansuripur, “Comment on “observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” _Phys. Rev. Lett._ 103, 019301 (2009).
  * (105) M. Partanen, H. Lee, and K. Oh, “Quantitative in situ measurement of optical force along a strand of cleaved silica optical fiber induced by the light guided therewithin,” _Photonics Res._ 9, 2016 (2021).
  * (106) I. Brevik, “Radiation forces and the Abraham-Minkowski problem,” _Mod. Phys. Lett. A_ 33, 1830006 (2018).
  * (107) I. Brevik, “Analysis of recent interpretations of the Abraham-Minkowski problem,” _Phys. Rev. A_ 98, 043847 (2018).
  * (108) M. Partanen and J. Tulkki, “Comment on “Analysis of recent interpretations of the Abraham-Minkowski problem”,” _Phys. Rev. A_ 100, 017801 (2019).
  * (109) N. Wang, S. Wang, and J. Ng, “Electromagnetic stress tensor for an amorphous metamaterial medium,” _Phys. Rev. A_ 97, 033839 (2018).
  * (110) S. Sukhov and A. Dogariu, “Non-conservative optical forces,” _Rep. Prog. Phys._ 80, 112001 (2017).
  * (111) W. Sun, S. B. Wang, J. Ng, L. Zhou, and C. T. Chan, “Analytic derivation of electrostrictive tensors and their application to optical force density calculations,” _Phys. Rev. B_ 91, 235439 (2015).
  * (112) R. E. Newnham, V. Sundar, R. Yimnirun, J. Su, and Q. M. Zhang, “Electrostriction: Nonlinear electromechanical coupling in solid dielectrics,” _J. Phys. Chem. B_ 101, 10141 (1997).
  * (113) H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” _J. Phys. Chem. Ref. Data_ 9, 561 (1980).
  * (114) D. R. Lide, ed., _CRC Handbook of Chemistry and Physics_ , CRC Press, Boca Raton, FL (2004).
  * (115) M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon?” _J. Microelectromech. Syst._ 19, 229 (2010).
  * (116) G. Mavko, T. Mukerji, and J. Dvorkin, _The Rock Physics Handbook_ , Cambridge University Press, Cambridge (2003).
  * (117) M. Partanen and J. Tulkki, “Mass-polariton theory of light in dispersive media,” _Phys. Rev. A_ 96, 063834 (2017).
  * (118) L. D. Landau and E. M. Lifshitz, _Theory of Elasticity_ , Pergamon, Oxford (1975).
  * (119) G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” _Phys. Rev. Lett._ 94, 170403 (2005).
  * (120) L. Novotny and B. Hecht, _Principles of nano-optics_ , Cambridge University Press, Cambridge (2006).
  * (121) J. M. van Doorn, R. Higler, R. Wegh, R. Fokkink, A. Zaccone, J. Sprakel, and J. van der Gucht, “Propagation and attenuation of mechanical signals in ultrasoft 2D solids,” _Sci. Adv._ 6, eaba6601 (2020).
  * (122) L. D. Landau and E. M. Lifshitz, _The Classical Theory of Fields_ , Pergamon, Oxford (1989).
  * (123) C. W. Misner, K. S. Thorne, and J. A. Wheeler, _Gravitation_ , Freeman, New York (1973).
  * (124) S. A. R. Horsley and M. Babiker, “Power–Zienau–Woolley QED: centre of mass-energy and the Aharonov–Casher effect,” _J. Phys. B_ 39, S565 (2006).
  * (125) P. A. M. Dirac, _General Theory of Relativity_ , Princeton University Press, Princeton, N.J. (1996).
  * (126) Y. Obukhov, “Electromagnetic energy and momentum in moving media,” _Ann. Phys._ 520, 830 (2008).
  * (127) V. P. Makarov and A. A. Rukhadze, “Negative group velocity electromagnetic waves and the energy–momentum tensor,” _Phys.-Uspekhi_ 54, 1285 (2011).
  * (128) M. Partanen and J. Tulkki, “Mass-polariton theory of sharing the total angular momentum of light between the field and matter,” _Phys. Rev. A_ 98, 033813 (2018).
  * (129) M. Partanen and J. Tulkki, “Light-driven mass density wave dynamics in optical fibers,” _Opt. Express_ 26, 22046 (2018).
  * (130) M. Partanen and J. Tulkki, “Negative radiation pressure in metamaterials explained by light-driven atomic mass density rarefication waves,” _Sci. Rep._ 12, 6753 (2022).
  * (131) N. Astrath, B. Anghinoni, G. Flizikowski, V. Zanuto, L. Malacarne, M. Baesso, T. Požar, and D. Razansky, “The role of electrostriction in the generation of acoustic waves by optical forces in water,” _Photoacoustics_ 29, 100445 (2023).
  * (132) J. P. Gordon, “Radiation forces and momenta in dielectric media,” _Phys. Rev. A_ 8, 14 (1973).
  * (133) E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” _Phys. Rev. Lett._ 102, 050403 (2009).
  * (134) S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” _J. Phys. B_ 39, S671 (2006).
  * (135) S. Stenholm, “The semiclassical theory of laser cooling,” _Rev. Mod. Phys._ 58, 699 (1986).
  * (136) D. J. Griffiths and H. Hnizdo, “What’s the use of bound charge?” _arXiv:1506.02590_ (2015).
  * (137) R. Corrêa and P. L. Saldanha, “Hidden momentum in continuous media and the Abraham-Minkowski debate,” _Phys. Rev. A_ 102, 063510 (2020).
  * (138) B. Anghinoni, M. Partanen, and N. Astrath, “The microscopic Ampére formulation for the electromagnetic force density in linear dielectrics,” Submitted (2022).
  * (139) L. D. Landau and E. M. Lifshitz, _Quantum Mechanics: Non-relativistic Theory_ , Pergamon, Oxford (1977).