# Simultaneous Horizontal and Vertical Oscillation of a Quiescent Filament
observed by CHASE and SDO

Jun Dai Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain
Observatory, CAS, Nanjing, 210023, China  Qingmin Zhang Key Laboratory of Dark
Matter and Space Astronomy, Purple Mountain Observatory, CAS, Nanjing, 210023,
China  Yunnan Key Laboratory of the Solar physics and Space Science, Kunming,
650216, China  Ye Qiu School of Astronomy and Space Science, Nanjing
University, Nanjing 210023, China  Key Laboratory for Modern Astronomy and
Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023,
China  Chuan Li School of Astronomy and Space Science, Nanjing University,
Nanjing 210023, China  Key Laboratory for Modern Astronomy and Astrophysics
(Nanjing University), Ministry of Education, Nanjing 210023, China  Zhentong
Li Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain
Observatory, CAS, Nanjing, 210023, China  Shuting Li Key Laboratory of Dark
Matter and Space Astronomy, Purple Mountain Observatory, CAS, Nanjing, 210023,
China  School of Astronomy and Space Science, University of Science and
Technology of China, Hefei, 230026, China  Yingna Su Key Laboratory of Dark
Matter and Space Astronomy, Purple Mountain Observatory, CAS, Nanjing, 210023,
China  School of Astronomy and Space Science, University of Science and
Technology of China, Hefei, 230026, China  Haisheng Ji Key Laboratory of Dark
Matter and Space Astronomy, Purple Mountain Observatory, CAS, Nanjing, 210023,
China  School of Astronomy and Space Science, University of Science and
Technology of China, Hefei, 230026, China Jun Dai and Qingmin Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we present the imaging and spectroscopic observations of the
simultaneous horizontal and vertical large-amplitude oscillation of a
quiescent filament triggered by an EUV wave on 2022 October 02. Particularly,
the filament oscillation involved winking phenomenon in H$\alpha$ images and
horizontal motions in EUV images. Originally, a filament and its overlying
loops across AR 13110 and 13113 erupted with a highly inclined direction,
resulting in an X1.0 flare and a non-radial CME. The fast lateral expansion of
loops excited an EUV wave and the corresponding Moreton wave propagating
northward. Once the EUV wavefront arrived at the quiescent filament, the
filament began to oscillate coherently along the horizontal direction and the
“winking filament” appeared concurrently in H$\alpha$ images. The horizontal
oscillation involved an initial amplitude of $\sim$10.2 Mm and a velocity
amplitude of $\sim$46.5 km s-1, lasting for $\sim$3 cycles with a period of
$\sim$18.2 minutes and a damping time of $\sim$31.1 minutes. The maximum
Doppler velocities of the oscillating filament are 18 km s-1 (redshift) and
$-$24 km s-1 (blueshift), which was derived from the spectroscopic data
provided by CHASE/HIS. The three-dimensional velocity of the oscillation is
determined to be $\sim$50 km s-1 at an angle of $\sim 50^{\circ}$ to the local
photosphere plane. Based on the wave-filament interaction, the minimum energy
of the EUV wave is estimated to be $2.7\times 10^{20}$ J. Furthermore, this
event provides evidence that Moreton waves should be excited by the highly
inclined eruptions.

Solar activity(1475); Solar filaments(1495); Quiescent solar prominence(1321);
Solar coronal waves(1995); Solar oscillations(1515)

##  1 Introduction

The oscillatory phenomena in filaments are ubiquitous and complex dynamics,
and have received more attention since the first systematic investigation
performed by Hyder (1966). Nowadays, based on the observed velocity amplitude
(Oliver & Ballester, 2002; Ballester, 2006), the oscillatory motions in
filaments are classified into large-amplitude oscillations (LAOs, $\geq$20 km
s-1) and small-amplitude oscillations (SAOs, $\leq$10 km s-1). Furthermore,
the LAOs in filaments can be divided into two groups according to the
direction, that is large-amplitude longitudinal oscillations (LALOs, Jing et
al., 2003, 2006; Vršnak et al., 2007; Li & Zhang, 2012; Zhang et al., 2012;
Luna et al., 2014; Bi et al., 2014; Zhang et al., 2017b, 2020) where the
filament material almost oscillate along its spine, and large-amplitude
transverse oscillations (LATOs, e.g., Isobe et al., 2007; Pintér et al., 2008;
Chen et al., 2008) where the filament considered as an long uniform cylinder
oscillating perpendicular to its axis. According to the recent reviews
(Oliver, 2009; Tripathi et al., 2009; Arregui et al., 2018) and statistical
study (Luna et al., 2018), both LATOs and LALOs are usually damping and
lasting for several cycles, while the amplitudes and periods of LATOs are
generally shorter than those of LALOs. Particularly, transverse and
longitudinal oscillations can occur simultaneously in a filament or prominence
(Pant et al., 2016; Wang et al., 2016; Zhang et al., 2017a; Luna & Moreno-
Insertis, 2021; Dai et al., 2021).

Generally, the LATOs are often triggered by Moreton waves (e.g., Eto et al.,
2002; Liu et al., 2013) or extreme-ultraviolet (EUV) waves (Liu et al., 2012;
Dai et al., 2012; Gosain & Foullon, 2012; Shen et al., 2014a, 2017; Zhang &
Ji, 2018). The restoring force of the LATOs is mainly supplied by the magnetic
tension force of the corresponding filament, and the energy loss or
dissipative process is theoretically considered as the reasonable damping
mechanism (Kleczek & Kuperus, 1969). Specifically, when the LATOs are observed
on the limb, the line-of-sight (LOS) and plane-of-sky (POS) motions correspond
to the horizontal and vertical oscillations, while on the disk, the LOS and
POS motions correspond to the nearly vertical and horizontal oscillations,
respectively (Tripathi et al., 2009). The vertical oscillations are usually
associated with a phenomenon known as “winking filament” which are frequently
observed on the disk in the earlier research (Hyder, 1966; Ramsey & Smith,
1966). The winking filament are characterized with the alternative appearance
and disappearance in H$\alpha$ wings images, due to the velocity component
along the line of sight resulting in the Doppler shift from the H$\alpha$
center to the line wings (Eto et al., 2002; Okamoto et al., 2004). The LOS
velocity could be roughly derived from the simultaneous images in several
wavelength channels close to the H$\alpha$ line-center (e.g., $\pm$0.4 Å or
$\pm$0.8 Å, Morimoto & Kurokawa, 2003) and the value obtained in the previous
studies ranged from 6 km s-1 to 50 km s-1 (Isobe & Tripathi, 2006; Gilbert et
al., 2008; Asai et al., 2012; Shen et al., 2014a).

As the prime triggering mechanism for the LATOs, EUV waves are globally
propagating wavelike phenomena characterized with brightening wavefronts
followed by extending coronal dimming, which were first detected in 195 Å
(Moses et al., 1997; Thompson et al., 1998) by Extreme-ultraviolet Imaging
Telescope (EIT; Delaboudinière et al., 1995) on board the Solar and
Heliospheric Observatory (SOHO; Domingo et al., 1995). Now it is generally
accepted that EUV waves are driven by the lateral expansion of coronal mass
ejections (CMEs) flank rather than the pressure pulse of flares (Biesecker et
al., 2002; Cliver et al., 2005; Chen, 2006; Patsourakos & Vourlidas, 2012).
Theoretically, Moreton waves (Moreton, 1960; Moreton & Ramsey, 1960) are
excited by the downward enhanced pressure when the coronal EUV waves front
sweep over the underlying chromospheric plasma (Vršnak et al., 2002; Vršnak &
Cliver, 2008). More recently, after investigating all the coronal-Moreton wave
events over the past decade, Zheng et al. (2023) uncovered that Moreton waves
are more likely to be excited by the highly inclined eruptions.

Benefiting from the high-cadence data provided by the Atmospheric Imaging
Assembly (AIA; Lemen et al., 2012) on board the Solar Dynamics Observatory
(SDO; Pesnell et al., 2012), an increasing number of EUV waves events
containing both fast and slow components were reported (Chen & Wu, 2011; White
et al., 2012; Dai et al., 2012; Asai et al., 2012; Cheng et al., 2012; Kumar
et al., 2013; Shen et al., 2013; Xue et al., 2013; Zong & Dai, 2015; Liu et
al., 2019). The hybrid wave theory (Chen et al., 2002) where the fast
component was considered as the fast-mode MHD wave or shock wave (Thompson et
al., 1998; Wang, 2000; Wu et al., 2001) and the slow component was supposed as
the apparent wave induced by the stretching of magnetic field lines, is thus
well supported by the thermodynamic MHD simulation (Downs et al., 2012) and
observations (Sun et al., 2022; Chen, 2023). Typically, the fast-component
waves present outer sharp wavefronts and propagate with speed from $\sim$500
km s-1 to more than 1500 km s-1, while the slow-component waves present inner
diffuse wavefronts with speeds usually about one-third of the fast-component
waves, sometimes even propagating slower than the coronal sound speed
(Tripathi & Raouafi, 2007; Thompson & Myers, 2009; Zhukov et al., 2009).
However, the physical nature of EUV waves is still under controversy and more
details is available in recent reviews (Liu & Ofman, 2014; Warmuth, 2015;
Chen, 2016; Shen et al., 2020; Chen, 2023).

On 2022 October 02, an X1.0 flare occurred in the active region (AR) 13110 and
13113, which was accompanied by an EUV wave (Moreton wave) and a non-radial
CME. The heating mechanisms related to the flare have been reported by Song et
al. (2023). In this paper, based on the imaging and spectroscopic
observations, we focus on the simultaneous POS horizontal oscillation and
winking phenomenon of a quiescent filament triggered by the EUV wave and
Moreton wave, which was rarely reported in pervious studies. The observation
and data analysis are described in Section 2. The results are presented in
Section 3. A comparison with previous findings and a brief conclusion are
given in Section 4.

Figure 1: The overview of the eruptions and the X1.0 flare in AR 13110 and
13113. Panel (a): SXR light curves of the X1.0 flare in 1-8 Å (red line) and
0.5-4 Å (blue line); Panel (b): HXR fluxes of the X1.0 flare at various energy
bands (4-300 keV); Panels (c-e) give three EUV images in AIA 304 Å and 171 Å.
Three yellow arrows point to erupting filament and loops, respectively; Panel
(f) give an HMI LOS magnetogram with the magnetic field strengths in the range
of -1200 and 1200 G; Panels (g-h) provide an H$\alpha$ image observed by CHASE
and a UV image in AIA 1600 Å. Panels (i1-i2): Time-distance diagrams of the
slices S1 and S2 in AIA 171 Å. The blue diamonds outline the heights of the
erupting filament and loops, and the blue solid lines represent the fitted
curve using Equation (1).

##  2 Observation and data analysis

###  2.1 Data and Instruments

On 2022 October 02, AR 13110 (N17W52) and 13113 (N16W43) were adjacent and
located near the western limb. The X1.0 flare, the EUV wave (Moreton wave) and
the transverse oscillation of the quiescent filament in this study were
simultaneously recorded by H$\alpha$ Imaging Spectrograph (HIS) on board the
Chinese H$\alpha$ Solar Explorer (CHASE; Li et al., 2022) and Atmospheric
Imaging Assembly (AIA; Lemen et al., 2012) on board the Solar Dynamics
Observatory (SDO; Pesnell et al., 2012). Furthermore, it should be noted that
the observation of CHASE for this event was only from 20:15 UT to 20:41 UT.

As the first space solar mission of China, CHASE was lunched into a Sun-
synchronous orbit on 2021 October 14. HIS takes full-disk spectroscopic
observations at H$\alpha$ (6559.7-6565.9 Å) and Fe I (6567.8-6570.6 Å)
wavebands in raster scanning mode (RSM), with a spectral resolution of 0.024 Å
pixel-1, a spatial resolution of 0$\farcs$52 pixel-1 and a temporal resolution
of 60 s. The detailed calibration procedures from the raw data to high-level
products involved the dark field correction, flat field and slit image
curvature correction, wavelength and intensity calibration, and coordinate
transformation (Qiu et al., 2022). We employed both the images at H$\alpha$
line center (6562.82 Å), H$\alpha$ wings (i.e., $\pm$0.4 Å, $\pm$0.8 Å), and
the derived Dopplergrams to study the motions of the oscillating filament and
the Moreton wave.

AIA provides full-disk images in two ultraviolet (UV; 1600 and 1700 Å)
wavelengths with a cadence of 24 s and in seven EUV (94, 131, 171, 193, 211,
304, and 335 Å) wavelengths with a cadence of 12 s, we mainly employed the EUV
images to study the dynamics of the EUV wave. The photospheric LOS
magnetograms of AR 13110 and 13113 were observed by the Helioseismic and
Magnetic Imager (HMI; Schou et al., 2012) on board SDO with a cadence of 45 s.
The level_1 data from AIA and HMI with a spatial resolution of 1$\farcs$2 were
calibrated using the standard Solar SoftWare (SSW) programs aia_prep.pro and
hmi_prep.pro.

In addition, soft X-ray (SXR) light curves of the flare in 0.5$-$4 Å and 1$-$8
Å were recorded by the GOES-16 spacecraft with a cadence of 2 s, and hard
X-ray (HXR) fluxes at various energy bands (4-300 keV) were detected by the
Gamma-ray Burst Monitor (GBM; Meegan et al., 2009) on board the Fermi
spacecraft. The erupting filament across the two active regions and the
related non-radial CME were also observed by the Extreme-UltraViolet Imager
(EUVI; Wuelser et al., 2004) and the COR2 white-light coronagraph on board the
Ahead Solar Terrestrial Relation Observatory (STEREO-A; Kaiser et al., 2008).
The separation angle between STEREO-A and Earth was $\sim$17∘.

###  2.2 Models and Methods

The cloud model (Beckers, 1964; Mein & Mein, 1988; Chae, 2014; Hong et al.,
2014) is widely used to indirectly derive the 3D velocity of erupting or
oscillating filaments based on H$\alpha$ observations (Morimoto & Kurokawa,
2003; Morimoto et al., 2010; Shen et al., 2014a). To obtain the Doppler
velocity, we first define the line center of the averaged spectral profile in
a quiescent region [600$\arcsec$, 615$\arcsec$]$\times$[525$\arcsec$,
535$\arcsec$] as zero shift to avoid the influence of solar rotation and
instrument movement. Then, we recognize the filament region by larger line
depth and lower intensity. At the filament region, we adopt the single cloud
model to acquire the line-of-sight velocity of filament. For the other region
without filament, the line-of-sight velocity is calculated by the moment
analysis (Li et al., 2019; Yu et al., 2020).

The revised cone model is developed to track the 3D evolution of filament
eruptions or CMEs and investigate the properties of the non-radial filament
eruptions or CMEs, which require observations from at least two viewpoints
(Zhang, 2021, 2022). The revised cone model is based on the assumption that
the shape of the eruptions or CMEs is the ice-cream cone, where the cone apex
is located at the source regions and the cone base is spheric. The revised
cone model is characterized by the matrix transforms between the heliocentric
coordinate system (HCS), local coordinate system (LCS) and cone coordinate
system (CCS) to determine the geometric parameters, such as the inclination
angle $\theta_{1}$, $\phi_{1}$ and angular width $\omega$. To construct the
revised cone model, we select the images simultaneously observed by the
STA/EUVI 195 Å and SDO/AIA 193 Å at 20:22 UT.

##  3 Result

###  3.1 X1.0 flare and Eruptions

In Figure 1, the top panels show the SXR (1-8 Å and 0.5-4 Å) and HXR (4-300
keV) fluxes of the X1.0 flare in AR 13110 and 13113, which peaked at
$\sim$20:25 UT and $\sim$20:22 UT, respectively. It is worthy noting that the
X1.0 flare was associated with eruptions of a filament and its overlying loops
crossing AR 13110 and 13113, which are marked by three yellow arrows in Figure
1(c-d). From $\sim$20:05 UT to $\sim$20:22 UT, the overlying loops and the
filament erupted successively, resulting in two roughly parallel flare ribbons
denoted by yellow marks in Figure 1(g-h).

To investigate the kinematics of the eruptions, we select two long straight
slices S1 and S2 with lengths of 97 Mm and 133 Mm in Figure 1(e) to construct
the time-space diagrams in AIA 171 Å of Figure 1(i1-i2), since the eruptions
were essentially in the same direction along S2 while the lateral expansion of
loops moved along S1. As shown in Figure 1(i1-i2), the signals of the loops
lateral expansion and eruptions pointed by yellow arrows can be clearly
recognized. The height of Loop 1 lateral expansion along S1, as well as the
heights of Loop 2 and filament eruptions along S2 can be obtained by outlining
the edge in the diagrams. The height evolution in the plane-of-sky is
characterized by a slow rise with a constant speed followed by an initial
impulsive acceleration indicated by an apparent exponential increase. Hence,
we fit $h(t)$ using the function composed of a linear term and an exponential
term as proposed by Cheng et al. (2013):

$h(t)=c_{0}e^{(t-t_{0})/{\tau}}+c_{1}(t-t_{0})+c_{2},$ (1)

where $t_{0}$, $\tau$, $c_{0}$, $c_{1}$, and $c_{2}$ are free parameters. The
onset of fast-rise phase is defined by setting the parameters as the point
when the exponential velocity is equal to the linear velocity:

$t_{onset}=\tau\ln(c_{1}\tau/c_{0})+t_{0}.$ (2)

The fitted curve is plotted with the blue solid line in Figure 1(i1-i2).
Precisely, loops started to rise slowly from $\sim$20:05 UT while their fast
rise and lateral expansion started at $\sim$20:18 UT. The filament started to
rise slowly from $\sim$20:10 UT and began to rise fast at $\sim$20:20 UT.

Figure 2:  Kinematics of EUV wave and Moreton wave. Panels (a-b): Running
differences of H$\alpha$ $+$0.4 Å images. Panel (c): An AIA 193 Å image.
Panels (d,e,g) give three AIA 193 Å base-difference images. Panel (f): An HMI
LOS magnetogram. Panel (h1-h2) provide time-distance diagrams along the sector
E in panel (g) using the 193 Å and 211 Å base-difference images. The Moreton
wave, the EUV wave and the oscillating filament are separately marked by
white, yellow and blue arrows. The yellow contours in panels (c,d,f) mark the
coronal hole. The white dotted lines in panel (g) partially display the
longitude of the new solar coordinate system centered on the X1.0 flare,
denoting the same sector as the EUV wave. The propagation of EUV wave through
the oscillating filament is marked by sector F. The speeds of the propagating
EUV wave are denoted by numbers in panels (h1-h2). The blue rectangle in panel
(e) shows the FOV of Figure 4(a)-(c) and Figure 6(a). An animation of the
unannotated SDO observations is available that covers $\sim$45 minutes
starting at 20:10 UT and ending the same day at 20:55 UT, with a time cadence
of 24 seconds. (An animation of this figure is available.) Figure 3:
Sequential CHASE/HIS H$\alpha$ images showing the oscillating filament. The
middle column presents the images of H$\alpha$ line-center (6562.82 Å), while
its left and right columns are images of two H$\alpha$ blue wings ($-$0.8 Å,
$-$0.4 Å) and two H$\alpha$ red wings ($+$0.4 Å, $+$0.8 Å), respectively.
Images in the same row are recorded at the same time. An animation of the
unannotated SDO observations is available that covers $\sim$26 minutes
starting at 20:15 UT and ending the same day at 20:41 UT, with a time cadence
of 1 minute. (An animation of this figure is available.)

###  3.2 Triggering and Kinematics of EUV wave and Moreton wave

Figure 2 presents the propagation of the Moreton wave in running difference
images in H$\alpha$ $+$0.4 Å and the EUV wave in base difference images in 193
Å. Combining the online animation of Figure 2(d-e) and the fitted curve in
Figure 1(i1), it can be seen that the onset of the fast lateral expansion of
loops was cotemporal with the first trace of the EUV wave at $\sim$20:18 UT,
implying that the EUV wave was excited by the fast loops lateral expansion
(Patsourakos & Vourlidas, 2009; Patsourakos et al., 2010). Subsequently, the
EUV wave propagated northward in a fan shape centered on the AR 13110 and its
bright segment is pointed by yellow arrows in Figure 2(d-e), while the
simultaneous Moreton-wave dark front is pointed by white arrows in Figure
2(a-b). What is noteworthy is that the bright segment was reflected at the
edge of the polar coronal hole (Gopalswamy et al., 2009), which is marked by
yellow contours in Figure 2.

In order to analyze the kinematics of the EUV wave, we select a sector (E)
along the propagating direction of the bright segment with a width of 10∘ and
a length of 726 Mm in Figure 2(g), and the corresponding time-space diagrams
of 193 Å and 211 Å base-difference images are plotted in Figure 2(h1-h2). It
can be seen that the bright segment of the EUV wave propagated at a speed of
$\sim$710 km s-1 from $\sim$20:18 UT. The perpendicular reflection at the edge
of the polar coronal hole occurred at $\sim$20:35 UT, then the bight segment
traveled for $\sim$ 200 Mm at a speed of $\sim$240 km s-1, after which the
wavefront could not be distinguished in the EUV images. The deceleration of
the EUV wave after reflection here is consistent with the values in previous
reports (Olmedo et al., 2012; Li et al., 2012; Shen & Liu, 2012).

Figure 4:  Panels (a-b): Two Dopplergrams showing the velocities of the
filament material along the LOS direction during the oscillation. Panel (c):
An EUV image in AIA 171 Å of the filament before the oscillation. Panel (d1):
Time-distance diagrams of the slice AB in AIA 171 Å. The white crosses outline
the filament oscillation between 20:20 UT and 21:20 UT. Panel (d2): Extracted
positions of the oscillating filament along the slice AB. The fitted curve is
overlaid with black solid line. Corresponding parameters are labeled.

###  3.3 Simultaneous Horizontal and Vertical Oscillation

A quiescent filament (N40W60) pointed by blue arrows in Figure 2 was located
north of the two active regions at a distance of $\sim$260 Mm. At $\sim$20:24
UT, the bright segment of the EUV wave swept over the quiescent filament,
after which the winking phenomenon and the obvious horizontal oscillation of
the filament was respectively detected in H$\alpha$ and EUV images, indicating
that the oscillation contained both POS and LOS velocity components at the
same time.

Figure 3 presents the time sequence of H$\alpha$ line-center (middle column),
blue wing (left columns) and red wing (left columns) images to illustrate the
upward and downward motions of the filament along the LOS. The first row shows
the filament features before oscillation, from which we can see in the
H$\alpha$ line-center image that the filament consists of a spine and a barb
on its left side, while the barb and the southern leg of the filament can be
observed in $\pm$0.4 Å images, probably due to the counter-streaming of plasma
(Zirker et al., 1998), along the LOS direction therein. At $\sim$20:25 UT, as
shown in the second row in Figure 3, the filament appeared a decrease in
intensity with increasing wavelength in the red wing images, while
disappearing completely in the blue wing images, indicating that the filament
was first undergoing downward motion after being squeezed by the EUV wave.
After about 10 minutes, the filament underwent upward motion and only appeared
in the blue wing images (see the third row and the online animation of Figure
3). The subsequent winking phenomenon of the filament could not be presented
here due to the limitation of the observation time of CHASE, which only
covered the first cycle of the oscillation. Nevertheless, we obtain the
Doppler velocities of the oscillating filament from the spectroscopic data of
HIS and the maximum velocities are shown in Figure 4(a-b), which were 18 km
s-1 (redshift) and $-$24 km s-1 (blue shift), respectively.

In order to investigate the horizontal motions of the filament, we select a
straight slice (AB) perpendicular to the axis with a length of 36$\arcsec$ in
Figure 4(c), and the corresponding time-space diagrams in 171 Å are plotted in
Figure 4(d1). It can be seen that the filament did not exhibit obvious
oscillations from 20:00 UT to 20:23 UT. At $\sim$20:24 UT, when the EUV waves
arrived, the filament began to deviate from the equilibrium position and moved
coherently eastward. Then, the filament moved backward and oscillated with
decaying amplitude for about three cycles. These phenomena indicate that the
filament oscillation was triggered by the EUV wave, rather than the tornado
phenomenon of the quiescent filament (Su et al., 2012; Panasenco et al.,
2014). To accurately determine the parameters of the filament oscillation, we
fit the white curve in Figure 4(d1) using mpfit.pro and the following
exponentially decaying sine function:

$A(t)=A_{0}\sin\bigg{(}\frac{2\pi
t}{P}+\psi\bigg{)}e^{-\frac{t}{\tau}}+A_{1}t+A_{2},$ (3)

where $A_{0}$ is the initial amplitude, $P$ is the period, $\tau$ is the
damping time, $\psi$ is the initial phase, and $A_{1}t+A_{2}$ represents a
linear term of the equilibrium position of the filament. In Figure 4(d2), the
red crosses represent the extracted positions of the oscillating filament
along the slice AB in 171 Å, and the fitting result using Equation 1 was
overlaid with black line. As seen from the fitted parameters listed in Figure
4(d2), the filament oscillation has an initial amplitude of $\sim$10.2 Mm and
a velocity amplitudes of $\sim$46.5 km s-1. The period and the damping time
are $\sim$18.2 minutes and $\sim$31.1 minutes, and the corresponding quality
factor ($\tau/P$) is $\sim$1.7. Therefore, the quiescent filament underwent a
large amplitude transverse oscillation (LATO).

Combining the initial velocity amplitudes (POS) and the Doppler velocity (LOS)
around 20:25 UT, as well as the location of the quiescent filament (N40W60),
schematic diagrams showing the side view of the wave-filament interaction are
provided in Figure 5, the initial maximum displacement and 3D velocity of the
oscillation is determined to be $\sim$12 Mm and $\sim$50 km s-1, the direction
of which was at an angle of $\sim 50^{\circ}$ to the local photosphere plane.

According to the prominence seismology applied to the winking filaments
(Pintér et al., 2008; Shen et al., 2014b), the radial magnetic field strength
of the quiescent filament can be expressed as the following equation (Hyder,
1966) based on the measured parameters of the LATO:

$B_{r}^{2}=\pi\rho\ r_{0}^{2}[\
4\pi^{2}(\frac{1}{P})^{2}+(\frac{1}{\tau})^{2}],$ (4)

where $B_{r}$ is the radial component of the filament magnetic field,
$\rho=N_{\mathrm{H}}m_{\mathrm{H}}$ is the mass density in the filament
($N_{\mathrm{H}}$ is the mean neutral hydrogen number density and
$m_{\mathrm{H}}$ is the mass of the hydrogen atom), $r_{0}$ is the scale
height of the filament, $P$ is the period, and $\tau$ is the damping time. If
we set the value $r_{0}$ = $3\times 10^{9}$ cm (Hyder, 1966), using the
observed value of $P$ = 1092 s, $\tau$ = 1866 s, and taking the value of
$m_{\mathrm{H}}=1.67\times 10^{-24}$ g, $N_{\mathrm{H}}=3\times 10^{11}$ cm-3
in the quiescent filaments (Heinzel et al., 2008), the radial component of the
filament magnetic field is estimated to be about 6.5 Gauss, which is
consistent with the values in previous studies (e.g., Shen et al., 2014b).

Figure 5: Schematic diagrams on side view showing the scenario of the wave-
filament interaction. The yellow curves represent the EUV wave and the black
arrow indicates the propagation of the EUV wave. The dashed gray and solid red
circles indicate the position of filament before the oscillation and the
position of the maximum displacement after the impact of the EUV wave,
respectively. The black (gray) dots indicate the axis of the filament spine.
The light and dark gray layer on the bottom represent chromosphere and
photosphere, respectively.  Figure 6: Panel (a): An H$\alpha$ image in HIS
6562.82 Å of the filament before the oscillation. The spine of the filament is
marked by a dotted yellow line. Panel (b): Normalized intensity distribution
along the line CD across the oscillating filament, which is fitted with a
single-Gauss function. The full width at half maximum (FWHM) representing the
width or LOS depth of the filament is labeled.

###  3.4 Energetics of Filament Oscillation and EUV wave

Since the transverse oscillation of the quiescent filament was triggered by
the impact of EUV wave, the kinetic energy transferred from the EUV wave to
the filament can be estimated based on their interaction. The mass of a
filament can be estimated roughly using the following expression (Labrosse et
al., 2010):

$M=N_{\mathrm{H}}m_{\mathrm{H}}V,$ (5)

where $V$ represents the volume of the filament plasma. Typical range of
$N_{\mathrm{H}}$ in the quiescent filaments is $3\times 10^{10}-10^{11}$ cm-3
at 6000$-$8000 K (Heinzel et al., 2008; Labrosse et al., 2010). In the current
study, as showing the Figure 6, the apparent width and length of the
oscillating filament are determined to be 4.5 Mm and 43.2 Mm, respectively.
Assuming the geometry occupied by the filament plasma is a uniform cylinder,
and considering that the filament is composed of thin threads with the filling
factor of 0.1 (Lin et al., 2005; Martin et al., 2008; Lin et al., 2009), the
total mass of the oscillating filament is estimated to be $3.5\times
10^{12}-1.2\times 10^{13}$ g. Taking the 3D velocity ($\sim$50 km s-1) of the
oscillation and ensuring that the oscillation can be triggered in case of the
maximum mass of the filament (Gilbert et al., 2008), the minimum energy
transferred from the EUV wave to the filament is $1.5\times 10^{19}$ J, which
is actually the initial kinetic energy of the oscillating filament. As shown
in the Figure 2(g), the propagation of the EUV wave is denoted by the partial
longitude of the new solar coordinate system with an angle of $\sim$165∘,
while the wave-filament interaction is marked by the sector F with an angle of
$\sim$10∘. Assuming that the sectoral propagation of the EUV wave is
isotropic, the area ratio between the filament and the EUV wave dome can be
approximated as the angle ratio between the sector F and the propagating
sector of the EUV wave. Thus, the minimum energy of the EUV wave is estimated
to be $2.7\times 10^{20}$ J. Furthermore, the minimum energy on the
interaction between the EUV wave and coronal hole is estimated to be
$1.8\times 10^{20}$ J, since the velocity of the EUV wave was decelerated to
one-third after the reflection.

Figure 7: Panel (a): An STA/EUVI 195 Å image of the erupting filament and
loops. Panels (b-c): The projections of the reconstructed cone are separately
depicted by magenta and red dots, and overlaid on the simultaneous STA/EUVI
195 Å and AIA 193 Å base-difference images at 20:22 UT. Panels (d1-d2): Two
snapshots of the white-light CME observed by STA/CRO2. The relative positions
of the Sun, Earth and STEREO-A at 20:53 UT on 2022 October 02 are plotted in
panel (d1). Panel (e): Height-time plot of the CME leading edges with red
square symbols. The results of a linear fit is overlaid with a dashed line and
the linear speed is labeled.

###  3.5 Non-radial loops eruption and CME

As shown in Figure 7, the eruptions across two active regions and the
associated CME were obviously deviated northward from the radial direction. To
determine the geometrical parameters of the non-radial eruption, we select the
images simultaneously observed by the STA/EUVI 195 Å and SDO/AIA 193 Å at
20:22 UT to construct the revised cone model, and the projections of the ice-
cream cone are overlaid with magenta and red dots in SDO/AIA and STA/EUVI
field of view, which are displayed in the top panels of Figure 7. It can be
seen that the ice-cream cone fits well with the erupting loops. The
inclination angle $\theta_{1}$, $\phi_{1}$ and angular width $\omega$ obtained
from the cone model are 24∘ (deviated northward), 5∘ (deviated eastward) and
80∘, respectively. Thus, the loops eruption was barely deviated from the plane
of sky (cf. the topology of the revised cone model and the transformations of
coordinate systems from Zhang, 2021). Hence, the linear velocity (1270 km s-1)
of the CME estimated from the STA/COR2 is approximately equal to its true
speed.

##  4 Summary and Discussion

In this paper, we report the simultaneous horizontal and vertical oscillation
of a quiescent filament recorded by CHASE and SDO on 2022 October 02. The
filament oscillation was triggered by an EUV wave accompanied with Moreton
wave, which was excited by a non-radial loops eruption in AR 13110 and 13113.
Particularly, the filament oscillation involved winking phenomenon in
H$\alpha$ images and horizontal motions in EUV images. Specifically, the
quiescent filament underwent the POS horizontal oscillation with an initial
amplitude of $\sim$10.2 Mm and a velocity amplitude of $\sim$46.5 km s-1,
lasting for $\sim$3 cycles with a period of $\sim$18.2 minutes and a damping
time of $\sim$31.1 minutes. The Doppler velocities computed from the CHASE/HIS
spectroscopic data are 18 km s-1 (redshift) and $-$24 km s-1 (blueshift),
respectively. Combining the Doppler velocities and the location of the
quiescent filament, we obtain the value and direction of the 3D velocities,
which was $\sim$50 km s-1 at an angle of $\sim 50^{\circ}$ to the local
photosphere plane. According to the prominence seismology, the radial magnetic
field strength of the quiescent filament is determined to be $\sim$6.5 Gauss.
Furthermore, the minimum energy of the EUV wave is estimated to be $2.7\times
10^{20}$ J based on the wave-filament interaction.

The transverse (horizontal) and vertical oscillations of filaments have been
widely reported and studied since 1930s (Hyder, 1966; Tripathi et al., 2009;
Arregui et al., 2018). When the oscillating filament are located far from the
center of the solar disk, there are projection effects for both horizontal and
vertical oscillations (Kleczek & Kuperus, 1969), which can be removed by
combining POS and LOS velocities. However, the simultaneous horizontal
oscillation and winking phenomenon in a filament was rarely reported in
previous studies. To our knowledge, only Isobe & Tripathi (2006) and Gilbert
et al. (2008) reported the similar filament oscillations which involved both
POS horizontal and LOS vertical motions, while the accurate 3D velocities were
not determined due to the limitations of data. Although there was no winking
filament observed (Isobe & Tripathi, 2006), the Doppler velocities derived
from two H$\alpha$ wings ($\pm$0.8 Å) images was up to 50 km s-1, which was
obviously deviated from the observation. Likewise, based on a few available
wavelengths (H$\alpha$ and He I 10830 Å), the LOS velocity component was
estimated to be 41 km s-1 (Gilbert et al., 2008), which was also controversial
since the Doppler velocity obtained by Jackiewicz & Balasubramaniam (2013) was
only a few km s-1 in their reanalysis for the same event. Using the images in
H$\alpha$ line-center and six H$\alpha$ wings ($\pm$0.5 Å, $\pm$0.8 Å,
$\pm$1.2 Å), Shen et al. (2014a) obtained the accurate Doppler velocity
ranging from 6 km s-1 to 14 km s-1 of a winking filament without horizontal
motions. The above illustrates that the accuracy of Doppler velocities are
determined by the data spectral resolution. Benefiting form the high-
resolution spectral data provided by CHASE/HIS, the 3D velocity of the
filament motion in this study have been derived from the unambiguous
observation of the simultaneous horizontal and vertical oscillations, which
almost completely eliminated the projection effect.

It should be noted that the maximum velocity of the Doppler redshift (18 km
s-1) was slightly smaller than that of the blueshift (24 km s-1) during the
first cycle of the filament oscillation. Excluding the calculation errors and
the effect of the solar rotation, we suppose that since the initial
displacement of the filament was greater than its height, the downward motion
of the filament should be impeded by the chromospheric material and magnetic
field below (see Figure 5), after being squeezed, the chromospheric material
and magnetic field could provide gas and magnetic pressure gradient for the
upward motion of the filament, while the resistance of the corona above can be
negligible.

As we known, the direction of the filament oscillation is closely associated
with the wave-filament interaction (Liu et al., 2013; Shen et al., 2014b),
while the direction of the coronal-Moreton wave is determined by the
inclination of the corresponding eruption or CME. The inclination angles
proposed by Zheng et al. (2023) were estimated by the lowest part of coronal
wave fronts with respect to the solar radial direction. Correspondingly, the
inclination angle in this work can be estimated by the lowest part of cone
model with respect to the solar radial direction. Zheng et al. (2023)
concluded that the bright segments of the EUV wavefronts and the simultaneous
Moreton wave should be excited by the non-radial eruptions with inclination
angles of 63${}^{\circ}-$76∘. In this work, we found that the bright segment
of the EUV wave was contemporal with the dark front of the Moreton wave. With
the addition of angular width ($\omega/2$), the inclination angle of the loops
eruption in this event is determined to be 64∘ northward, which reaches the
value of the highly inclination. Thus, the inclination angle of the loops
eruption here could be sufficient to excite the coronal-Moreton wave, which in
turn triggered the simultaneous horizontal and vertical filament oscillation.

##  5 acknowledgements

We thank the referee for reading the manuscript, and we greatly appreciate the
valuable comments that significantly improved this paper. We thank the CHASE,
SDO and STEREO teams for providing data. The observation data is from CHASE
mission supported by China National Space Administration. SDO is a mission of
NASA’s Living With a Star Program. AIA and HMI data are courtesy of the
NASA/SDO science teams. The authors are supported by the Special Research
Assistant Project of Chinese Academy of Sciences, the Project funded by China
Postdoctoral Science Foundation (2023M733734), the Prominent Postdoctoral
Project of Jiangsu Province, the Youth development program of Purple Mountain
Observatory (2022000123), and Yunnan Key Laboratory of Solar Physics and Space
Science under number YNSPCC202206, the National Key R&D Program of China
2021YFA1600500 (2021YFA1600502), and the Chinese foundations NSFC 12173092,
12250014, 12333009, 12373065.

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††thanks: NOTICE: this is the author’s version of a work that was accepted for
publication in Annals of Physics. Changes resulting from the publishing
process, such as peer review, editing, corrections, structural formatting, and
other quality control mechanisms may not be reflected in this document.
Changes may have been made to this work since it was submitted for
publication. A definitive version was subsequently published in Annals of
Physics, 338, 97 (2013).

# Decomposition of the Total Momentum in a Linear Dielectric into Field and
Matter Components

Michael E. Crenshaw<EMAIL_ADDRESS>US Army Aviation and
Missile Research, Development, and Engineering Center, Redstone Arsenal, AL
35898, USA

###### Abstract

The long-standing resolution of the Abraham–Minkowski electromagnetic momentum
controversy is predicated on a decomposition of the total momentum of a closed
continuum electrodynamic system into separate field and matter components.
Using a microscopic model of a simple linear dielectric, we derive Lagrangian
equations of motion for the electric dipoles and show that the dielectric can
be treated as a collection of stationary simple harmonic oscillators that are
driven by the electric field and produce a polarization field in response. The
macroscopic energy and momentum are defined in terms of the electric,
magnetic, and polarization fields that travel through the dielectric together
as a pulse of electromagnetic radiation. We conclude that both the macroscopic
total energy and the macroscopic total momentum are entirely electromagnetic
in nature for a simple linear dielectric in the absence of significant
reflections.

##  I Introduction

In a dielectric environment, the momentum of the electromagnetic field becomes
a contested issue. The matter was first raised over a century ago when
Minkowski BIMin , in 1908, and Abraham BIAbr , in 1909, produced conflicting
expressions for the energy–momentum tensor of an electromagnetic field in a
linear optical medium. While a number of experiments seemed to validate the
Abraham momentum density

${\bf g}_{A}=\frac{1}{c}\left({\bf E}\times{\bf H}\right)$ (1)

other experiments suggested that the Minkowski momentum density

${\bf g}_{M}=\frac{1}{c}\left({\bf D}\times{\bf B}\right)$ (2)

provides the correct description of the momentum of light in a linear medium
BIBrev . The inability to decide between the two energy–momentum tensors and
their underlying momentum densities became known as the Abraham–Minkowski
controversy. The resolution of the Abraham–Minkowski dilemma was provided by
Penfield and Haus BIPenHau , based on earlier work by Møller BIMol , who
showed that the issue is undecidable because neither the Abraham momentum nor
the Minkowski momentum is the total momentum. Instead, each corresponds to an
open description of the system in which the momentum is allocated differently
between the field and the matter BIBrev ; BIPenHau ; BIMol ; BIKran ; BISal ;
BIPfei .

Following the work of Penfield and Haus BIPenHau , Gordon BIGord , in 1973,
assumed the Abraham form for the field momentum subsystem and derived the
material contribution to the momentum in a dilute vapor by integrating the
Lorentz force on each dipole. Gordon derived the total momentum density

${\bf g}_{G}=\frac{1}{c}\left(n{\bf E}\times{\bf B}\right)$ (3)

for a vapor that is sufficiently dilute that reflections can be neglected.
Three years later, Peierls BIPei  followed a similar procedure to obtain a
total momentum density

${\bf g}_{P}=\frac{n^{2}+1}{2c}\left({\bf E}\times{\bf B}\right)$ (4)

that is the arithmetic mean of the Abraham and Minkowski forms. The Gordon and
Peierls momentum densities are equivalent to lowest order in $\delta n=n-1$.
The literature on the subject is extensive and the reader is referred to
reviews of the prior work by Pfeifer, et al. BIPfei , Barnett and Loudon
BIcankin2 , Milonni and Boyd BIMilB , and Kemp BIKemp  and the resource letter
by Griffiths BIGriffRL .

At this point in time, there is a strong consensus in the scientific
literature that only the total energy and the total momentum have physical
meaning. Then any decomposition of the total momentum into a field component
and a matter component is valid as long as their sum is the total momentum
BIPfei , but the separate momentum components are not physical. Nevertheless,
current practice is to acknowledge that consensus and then proceed to assume
some form of field momentum, usually Abraham, Minkowski, or both, and then add
some momentum for the matter, usually in the form of the movement of massive
particles of the medium that are forced into motion by being either pulled or
pushed by the Lorentz force. The recent experiments of She, Yu, and Feng BISYF
, for example, are analyzed in terms of an inward push force as light exits a
nanometer silica filament. In contrast, earlier experiments by Ashkin and
Dziedzic BIAshD  indicate an outward pull force on the surface as light enters
a liquid. Pfeifer, Nieminen, Heckenberg, and Rubinsztein-Dunlop BIPfei , and
also Ramos, Rubilar, and Obukhov BIRRObuk , start with the Abraham tensor for
the electromagnetic field and add a dust tensor for the material and claim
that the result is the total energy–momentum tensor. By definition, the total
energy–momentum tensor can also be derived by starting with the Minkowski
tensor and adding a different material tensor. Barnett BIBarn  identifies the
Abraham momentum as the kinematic momentum of the block of dielectric, as a
whole, and the Minkowski momentum as the canonical momentum that is associated
with the movement of massive particles embedded in the dielectric. Barnett
goes on to say that the total momentum is both the Abraham momentum with a
corresponding partial momentum and the Minkowski momentum with a different
partial momentum, which is again true by definition of total momentum.
Mansuripur BIMani  treats the kinematic momentum of partially reflecting
matter, pushed by a radiation pressure associated with the Fresnel reflection,
together with the forward traveling light momentum. Kemp BIKemp1  formalizes a
subsystem approach for the field and matter components of momentum.

The most rigorous approach to light propagation in a material is to use a
microscopic model in which the microscopic electromagnetic field interacts
with individual electric dipoles in the vacuum. If the distance between
dipoles in the material is much smaller than the wavelength, then the theory
can be considerably simplified by applying an averaging scheme to a volume
containing many atoms per cubic radiation wavelength. Gordon BIGord , for
example, spatially averages the Lorentz force on a single electric dipole to
create a macroscopic Lorentz force density. Integrating the Lorentz force
density, Gordon obtains a definite formula

${\bf g}_{mat}=\frac{n-1}{c}\left({\bf E}\times{\bf B}\right)$ (5)

for the material component of momentum density in the dielectric. Gordon
assumes the Abraham form for the momentum density of the field, then the field
plus material momentum is the total momentum

${\bf G}_{total}=\int_{\sigma}\frac{n}{c}\left({\bf E}\times{\bf B}\right),$
(6)

where integration of the momentum density is performed over a volume $V$ that
includes all fields present that is extended to all space $\sigma$. Saldanha
BISal  and Bradshaw, Shi, Boyd, and Milonni and Boyd BIBrad , among others,
have revisited Gordon’s derivation of the total momentum as the sum of the
Abraham momentum for the field and a material momentum arising from the
Lorentz force on dipoles and derived the Minkowski momentum, which is not
conserved, for the total momentum. In recent work BICB , global conservation
principles were applied to a closed system consisting of a quasimonochromatic
pulse of electromagnetic radiation normally incident on a stationary simple
dielectric with refractive index $n$ through a gradient-index antireflection
coating. Conservation of the total momentum in a thermodynamically closed
system with complete equations of motion BIKran  is decisive. Then, despite
problems with the original derivation, Eq. (6) is the proven total momentum
for the light pulse plus stationary dielectric closed system BICB ; BISPIE .

In this article, we investigate the interaction of the electromagnetic field
with a simple linear dielectric that is modeled microscopically as a
collection of individual electric dipoles composed of a positively charged
particle and a negatively charged particle. The work of Penfield and Haus
BIPenHau  and of Gordon BIGord  is reproduced in part, but we go on to show
that the dielectric can be treated in the usual and familiar way as a
collection of stationary simple harmonic oscillators driven by the electric
field. Because the contribution to the total momentum occurs through the
resulting polarization field, the total momentum can be regarded as
electromagnetic in character. We begin, in Section II, with an analysis of the
macroscopic picture of the total momentum as being composed of arbitrary field
and matter components of momentum. At the level of continuum electrodynamics,
we take the material momentum to be the difference of the total momentum,
which is conserved, and a field momentum that is to be determined. The
temporal derivative of the material momentum density is the macroscopic force
density that can be expressed in terms of the fields and a macroscopic
material property. Assuming that a decomposition of the total momentum into
field and material components exists, we find that each choice of the field
momentum results in a specific relation between the refractive index and the
macroscopic material property. Then, the decomposition of the total momentum
into field and material components is not arbitrary or degenerate, but unique
and definite. In Section III, we summarize Gordon’s derivation of the
macroscopic Lorentz force from the microscopic Lorentz force. We discuss the
problem that occur when we use the macroscopic Lorentz force, along with
similar results by Peierls BIPei , Saldanha BISal , and Bradshaw, Shi, Boyd,
and Milonni BIBrad , to obtain the field momentum. Finally, in Section IV, we
adopt a microscopic model of the material as being composed of discrete
dipoles formed from individual pairs of positive and negative charges. Writing
the microscopic Lagrangian in a center-of-mass coordinate system, we derive
Lagrange equations of motion using both the center-of-mass coordinates and the
relative coordinates. The microscopic Lorentz force used by Gordon BIGord ,
Penfield and Haus BIPenHau , and other authors can be extracted from the
Lagrange equations for the center-of-mass coordinates. However, macroscopic
movement of massive particles is not allowed in a solid block of dielectric.
Setting the velocity of the center-of-mass of the dipoles to zero in the
Lagrange equations for the relative coordinates, we find the elementary and
expected result that the dielectric can be treated as a collection of
stationary simple harmonic oscillators driven by the electric field BIMVK .
The polarization field that is generated by the oscillators contributes to
both the total energy and the total momentum through the refractive index $n$.
We conclude that, in the absence of reflections, the total energy and the
total momentum are electromagnetic in nature and do not separate into field
and material motion components.

##  II Field and matter components of the total momentum

A physical theory is an abstract mathematical model of some limited aspect of
the real world BIRindler , in this case, light traveling at speed $c/n$ in a
simple linear dielectric. Our model dielectric consists of a rectangular prism
of space in which light travels at speed $c/n$, where $n$ is a real constant.
No such idealization of a dielectric exists in the real world. Instead, the
usual conditions for results of the model to correspond to results in the real
world apply: a simple dielectric is linear, isotropic, homogeneous,
transparent, and dispersion-negligible in a regime in which electrostrictive
effects and magnetostrictive effects can be neglected BIMikura . We assume
that a quasimonochromatic optical pulse impinges on the dielectric from the
vacuum at normal incidence in the plane-wave limit and that the dielectric is
covered with a thin gradient-index coating in order to make reflections and
surface radiation pressure negligible.

The Abraham momentum is a common, but not exclusive, choice for the momentum
of the field in a dielectric BIGord ; BIMilB ; BIMani ; BIPfei . Rather than
make this an ansatz, we start with a more general form of the field momentum
in a dielectric

${\bf G}_{field}=\int_{\sigma}\frac{\zeta(n)}{c}\left({\bf E}\times{\bf
B}\right)dv,$ (7)

with $\zeta(n)$ an unspecified function of $n$, that covers most of the
proposed forms of electromagnetic momentum, (e.g., the Abraham, Minkowski, and
Peirels momentums). We note that $\zeta$ is specifically not a function of the
electric and magnetic fields. In addition, $\zeta$ is not arbitrary, but a
definite function of macroscopic property constants that is to be determined.
The momentum of the material ${\bf G}_{mat}$ is the difference of the total
momentum, Eq. (6) and field momentum, Eq. (7). Then

${\bf G}_{mat}={\bf G}_{total}-{\bf
G}_{field}=\int_{\sigma}\frac{n-\zeta(n)}{c}({\bf E}\times{\bf B})dv.$ (8)

The material momentum density

${\bf g}_{mat}=\frac{n-\zeta(n)}{c}({\bf E}\times{\bf B})$ (9)

leads directly to a macroscopic force density

${\bf f}_{mat}=\frac{\partial{\bf g}_{mat}}{\partial
t}=(n-\zeta(n))\left(\frac{\partial{\bf E}}{\partial(ct)}\times{\bf B}+{\bf
E}\times\frac{\partial{\bf B}}{\partial(ct)}\right).$ (10)

We define a macroscopic material property

$\xi_{e}(n)=n-\zeta(n)$ (11)

that relates the electric and magnetic fields to a macroscopic force density

${\bf f}_{mat}=\xi_{e}\left(\frac{\partial{\bf E}}{\partial(ct)}\times{\bf
B}+{\bf E}\times\frac{\partial{\bf B}}{\partial(ct)}\right)$ (12)

on the spatially distributed particles of matter. Then the decomposition of
the total momentum into field and matter components,

${\bf G}_{field}=\int_{\sigma}\frac{n-\xi_{e}}{c}({\bf E}\times{\bf B})dv$
(13) ${\bf G}_{mat}=\int_{\sigma}\frac{\xi_{e}}{c}({\bf E}\times{\bf B})dv,$
(14)

is uniquely defined by macroscopic material properties $n$ and $\xi_{e}$.

Since the 1960s, the resolution of the Abraham–Minkowski controversy has been
that the total momentum can be arbitrarily separated into field and material
components. What has been proved here is that, if the total momentum can be
separated into a component for the field and a component for the motion of
matter driven by a Lorentz force, then there is a specific form for each of
these components in terms of macroscopic material property constants.

##  III Microscopic Treatment of Material Momentum

The total momentum of a closed system with complete equations of motion is
unique by virtue of its conservation. It has been argued that the
decomposition of the total momentum into field and material is arbitrary
BIBrev ; BIPenHau ; BIMol ; BIPfei ; BIGord ; BIPei ; BIMani ; BIKemp1 ;
BIBarn ; BIcankin2 ; BIMilB ; BIKemp . We now know, as a result of the
previous section, that if such a decomposition into a component for the field
and a component for the motion of matter driven by a Lorentz force exists then
it can be uniquely formulated in terms of macroscopic material property
constants. We simply need to determine what those property constants are.

Gordon BIGord  began with the Lorentz force on a single electric dipole with
dipole moment ${\bf p}=\alpha{\bf e}$

${\bf f}_{dipole}=({\bf p}\cdot\nabla){\bf e}+\frac{d{\bf p}}{d(ct)}\times{\bf
b}.$ (15)

Here, ${\bf e}$ is the microscopic electric field, ${\bf b}$ is the
microscopic magnetic field, and ${\alpha}$ is the linear polarizability.
Assuming that the distance between dipoles in the material is much smaller
than the wavelength, then the theory can be considerably simplified by
applying an averaging scheme to a volume containing many atoms per cubic
radiation wavelength. Gordon BIGord  describes an averaging process that he
uses to derive a macroscopic Lorentz force density

${\bf f}_{L}=N\langle{\bf f}_{dipole}\rangle=\frac{N\alpha}{2}\left({\bf
E}\times\frac{\partial{\bf B}}{\partial(ct)}+\frac{\partial{\bf
E}}{\partial(ct)}\times{\bf B}\right)\,,$ (16)

where $N$ is the number density, ${\bf E}=\langle{\bf e}\rangle$ is the
macroscopic electric field, and ${\bf B}=\langle{\bf b}\rangle$ is the
macroscopic magnetic field. Gordon BIGord  employs the low-density
approximation

$n=\sqrt{1+N\alpha}\approx 1+N\alpha/2$ (17)

such that

${\bf f}_{L}=N\langle{\bf f}_{dipole}\rangle\approx(n-1)\left({\bf
E}\times\frac{\partial{\bf B}}{\partial(ct)}+\frac{\partial{\bf
E}}{\partial(ct)}\times{\bf B}\right)\,.$ (18)

Other authors have derived variations of this basic relation. Peierls BIPei
used the exact relation

$n^{2}=1+N\alpha$ (19)

to obtain

${\bf f}_{L}=N\langle{\bf f}_{dipole}\rangle=\frac{n^{2}-1}{2}\left({\bf
E}\times\frac{\partial{\bf B}}{\partial(ct)}+\frac{\partial{\bf
E}}{\partial(ct)}\times{\bf B}\right)\,.$ (20)

Saldanha BISal  and Bradshaw, et al. BIBrad  find the macroscopic Lorentz
force

${\bf f}_{L}=(n^{2}-1)\left(\nabla\left(\frac{1}{2}E^{2}\right)+{\bf
E}\times\frac{\partial{\bf B}}{\partial(ct)}+\frac{\partial{\bf
E}}{\partial(ct)}\times{\bf B}\right).$ (21)

If the Lorentz force, Eq. (21), is separated into components,

${\bf f}_{L1}=(n^{2}-1)\nabla\left(\frac{1}{2}E^{2}\right)$ (22) ${\bf
f}_{L2}=(n^{2}-1)\left({\bf E}\times\frac{\partial{\bf
B}}{\partial(ct)}+\frac{\partial{\bf E}}{\partial(ct)}\times{\bf B}\right)\,,$
(23)

then ${\bf f}_{L2}$ is in the form of the Abraham force and differs from the
Peierls result for the Lorentz force by a factor of 2.

We now have three different formulas for the macroscopic Lorentz force, Eqs.
(18), (20), and (23), that have been derived from the microscopic Lorentz
force on a dipole. Selecting one of these average Lorentz force densities to
equate with the macroscopic force density ${\bf f}_{mat}$, Eq. (12), will
result in a value for the material parameter $\xi_{e}$ and formulas for the
field and material components of momentum in a dielectric, Eqs. (13) and (14).
Let us consider the consequences of using each of these formulas, in turn.
Comparing the Gordon form of the macroscopic force density, Eq. (18), with Eq.
(12) we obtain $\xi_{e}=n-1$. Substituting this result into Eqs. (13) and (14)
yields

${\bf G}_{field}=\int_{\sigma}\frac{1}{c}({\bf E}\times{\bf B})dv$ (24) ${\bf
G}_{mat}=\int_{\sigma}\frac{n-1}{c}({\bf E}\times{\bf B})dv,.$ (25)

The Gordon form of the force, Eq. (18), provides us with the sensible result
that the field momentum is the Abraham momentum. But the Gordon form is an
approximation. Using Peierls’s exact result, Eq. (20), in Eq. (12), we obtain
$\xi_{e}=(n^{2}-1)/2$. As before, we substitute $\xi_{e}$ into Eqs. (13) and
(14) and find that

${\bf G}_{field}=\int_{\sigma}\frac{1+2n-n^{2}}{2c}({\bf E}\times{\bf B})dv$
(26) ${\bf G}_{mat}=\int_{\sigma}\frac{n^{2}-1}{2c}({\bf E}\times{\bf B})dv.$
(27)

Then the Peierls result, Eq. (20), leads to a field momentum, Eq. (26), that
is negative for values of the refractive index $n>1+\sqrt{2}$. Further, there
is a factor of two discrepancy in the derivations of Gordon BIGord  and
Peierls BIPei  compared to the later work BISal ; BIBrad . Applying the same
procedure to the final form of the Lorentz force, Eq. (23), results in

${\bf G}_{field}=\int_{\sigma}\frac{1+n-n^{2}}{c}({\bf E}\times{\bf B})dv$
(28) ${\bf G}_{mat}=\int_{\sigma}\frac{n^{2}-1}{c}({\bf E}\times{\bf B})dv.$
(29)

Then, the Lorentz force given by Eq. (23) presents us again with the problem
of a negative field momentum for some values of $n$, namely
$n>(1+\sqrt{5})/2$. Therefore, none of the versions of the Lorentz force that
are reprised here from the scientific literature are particularly appealing.
Clearly, this procedure to derive the macroscopic Lorentz force is not
sufficiently rigorous to obtain a result that is consistent with the total
momentum. In the next section, we take a more detailed look at the microscopic
Lorentz force.

##  IV Lagrangian dynamics of the dielectric

The macroscopic force density, or Lorentz force, is only significant where the
amplitudes of fields are changing in time. The effect of the macroscopic
Lorentz force, Eq. (16), has been interpreted, in the context of a typical
pulse shape BIGord . There is an acceleration of the physical atoms/dipoles on
the front side of the pulse where the electromagnetic field is increasing in
strength BIGord . Then the atoms in the middle portion of the pulse travel at
a generally constant velocity until they are decelerated due to the decreasing
field strength on the trailing side of the pulse. No momentum is left in the
atoms after the pulse has passed. The process has been described by saying
that the material momentum travels with the pulse BIGord . In this section, we
derive the dynamics of the atoms/dipoles from the Lagrangian.

The standard Lagrangian for a charged particle with mass $m$ and charge $q$
located at a point ${\bf x}$ interacting with the electromagnetic field is
given by BICohen

$L=\frac{1}{2}m{{\bf\dot{x}}^{2}}+\frac{q}{c}\left({\bf\dot{x}}\cdot{\bf
A}({\bf x})-\phi({\bf x})\right)$
$+\int_{\sigma}\frac{1}{2}\left(\left(\frac{1}{c}\frac{\partial{\bf A}({\bf
x})}{\partial t}+\nabla\phi({\bf x})\right)^{2}-(\nabla\times{\bf A}({\bf
x}))^{2}\right)\,dv\,.$ (30)

Here, ${\bf A}$ is the vector potential, $\phi$ is the scalar potential and
overdots denote partial differentiation with respect to time. We take Eq. (30)
as our given starting point.

The derivation and properties of this equation, including the gauge invariance
properties, are discussed in Sec. II.B of Ref. BICohen . We can construct a
Lagrangian for a dipole interacting with the field from a positive charge of
mass $m_{+}$ located at ${\bf x}_{+}$ and a negative charge with mass $m_{-}$
located at ${\bf x}_{-}$ and connected by a restoring constant $\kappa$

$L=-\frac{1}{2}\kappa({\bf x}_{+}-{\bf
x_{-}})^{2}+\frac{1}{2}m_{+}{{\bf\dot{x}}_{+}^{2}}+\frac{1}{2}m_{-}{{\bf\dot{x}}_{-}^{2}}$
$+\frac{q}{c}\left({{\bf\dot{x}}_{+}}\cdot{\bf A}({{\bf x}_{+}})-\phi({\bf
x}_{+})-{{\bf\dot{x}}_{-}}\cdot{\bf A}({{\bf x}_{-}})+\phi({\bf
x}_{-})\right)$
$+\int_{\sigma}\frac{1}{2}\left[\left(\frac{1}{c}\frac{\partial{\bf A}({\bf
x})}{\partial t}+\nabla\phi({\bf x})\right)^{2}-(\nabla\times{\bf A}({\bf
x}))\right]\,dv\,.$ (31)

Transforming to a center-of-mass coordinate system, the Lagrangian is

$L=\left[-\frac{1}{2}\kappa{\bf
r}^{2}+\frac{1}{2}M{\bf\dot{R}}^{2}+\frac{1}{2}\mu{\bf\dot{r}}^{2}+\frac{q}{c}{\bf\dot{r}}\cdot{\bf
A}({\bf R})\right]$
$+\int_{\sigma}\frac{1}{2}\left[\left(\frac{1}{c}\frac{\partial{\bf A}({\bf
x})}{\partial t}+\nabla\phi({\bf x})\right)^{2}-(\nabla\times{\bf A}({\bf
x}))\right]\,dv\,.$ (32)

where $M=m_{+}+m_{-}$ is the total mass, $\mu=m_{+}m_{-}/M$ is the reduced
mass, ${\bf r}={\bf x}_{+}-{\bf x}_{-}$ is the relative coordinate, ${\bf
R}=(m_{+}{\bf x}_{+}+m_{-}{\bf x}_{-})/M$ is the center-of-mass coordinate. In
writing Eq. (32), we have applied the usual dipole approximation in which the
electric field does not change much over the distance between charges in the
dipole BIWSQO , such that

${\bf A}({{\bf x}_{+}})\cong{\bf A}({{\bf x}_{-}})\cong{\bf A}({\bf R})$ (33)
${\phi}({{\bf x}_{+}})\cong{\phi}({{\bf x}_{-}})\cong{\phi}({\bf R})\,.$ (34)

For each dipole in the material, the Lagrange equations of motion are

$\frac{d}{dt}\left(\frac{\partial{L}}{\partial\dot{\bf
R}_{i}}\right)-\frac{\partial{L}}{\partial{\bf R}_{i}}=0$ (35)
$\frac{d}{dt}\left(\frac{\partial{L}}{\partial\dot{\bf
r}_{i}}\right)-\frac{\partial{L}}{\partial{\bf r}_{i}}=0\,.$ (36)

We will make use of the chain rule

$\frac{d}{dt}=\frac{\partial}{\partial
t}+\sum{\bf\dot{R}}_{i}\frac{\partial}{\partial{\bf
R}_{i}}+\sum{\bf\dot{r}}_{i}\frac{\partial}{\partial{\bf r}_{i}}\,.$ (37)

Applying Eq. (35) to the Lagrangian, Eq. (32), we obtain a microscopic force
equation

${\bf f}_{i}=M{\bf\ddot{R}}_{i}=\frac{q}{c}\frac{\partial}{\partial{\bf
R}_{i}}({\bf\dot{r}}\cdot{\bf A}({\bf R}))$ (38)

for each component. Combining the three vector components results in

${\bf f}=M{\bf\ddot{R}}=\frac{q}{c}\nabla_{R}({\bf\dot{r}}\cdot{\bf A}({\bf
R}))$ (39)

where

$\nabla_{R}=\left(\frac{\partial}{\partial R_{x}},\frac{\partial}{\partial
R_{y}},\frac{\partial}{\partial R_{z}}\right)\,.$ (40)

Applying the vector identity

$\nabla({\bf A}\cdot{\bf B})=({\bf B}\cdot\nabla){\bf A}+({\bf
A}\cdot\nabla){\bf B}+{\bf B}\times(\nabla\times{\bf A})+{\bf
A}\times(\nabla\times{\bf B})$ (41)

in the plane-wave limit, we obtain the microscopic force equation

${\bf
f}_{dipole}=M{\bf\ddot{R}}=\frac{q}{c}{\bf\dot{r}}\times(\nabla_{R}\times{\bf
A}({\bf R}))$ (42)

that is the Lorentz force on a single dipole, that is exerted at the center of
mass. In the plane-wave limit, the Lorentz force is in the direction of
propagation, or the opposite direction, such that $\nabla_{R}=\nabla$ without
loss of generality. With definitions for the magnetic field ${\bf
b}=\nabla\times{\bf A}$, and the polarizability ${\bf p}=q{\bf r}$, we have
the Lorentz force on a dipole

${\bf f}_{dipole}=M{\bf\ddot{R}}=\frac{1}{c}{\bf\dot{p}}\times{\bf b}$ (43)

The Lorentz force on a dipole that is given by Eq. (15) contains an extra term
$({\bf p}\cdot\nabla){\bf e}$ that does not appear in the Lorentz force, Eq.
(43). This term went out in the dipole approximation, which is fortunate
because the dipole force should not have a transverse component in the plane-
wave limit.

In the Gordon BIGord  model, the atoms are free particles that are accelerated
by the Lorentz force at the leading edge of the field and travel at constant
velocity until decelerated by the Lorentz force at the trailing edge of the
pulse. Then, the momentum of the atoms travels with the field and contributes
to the total momentum. However, that model assumes a low-density limit in
which there is no scattering of the field and assumes that the atoms move
without being restrained by association with a lattice and without colliding
with any other particles in the material, i.e. a vacuum. Instead, there is
substantial impediment to the motion of atoms in a dielectric that, after all,
is a material that can be modeled as a continuous medium. Then the canonical
momentum

$M{\bf\dot{R}}=\left(\frac{\partial
L}{\partial{\bf\dot{R}}_{1}},\frac{\partial
L}{\partial{\bf\dot{R}}_{2}},\frac{\partial
L}{\partial{\bf\dot{R}}_{3}}\right)$ (44)

makes no appreciable contribution to the total momentum.

The other Lagrange equation of motion

$\mu{\bf\ddot{r}}+\kappa{\bf
r}=-\frac{q}{c}{\bf\dot{A}}-\frac{q}{c}({\bf\dot{R}}\cdot\nabla_{r}){\bf A}$
(45)

that is derived by applying Eq. (36) to the Lagrangian, Eq. (30), and
collecting the components is more interesting because it is evocative of the
other microscopic model of a dielectric as stationary simple harmonic
oscillators driven by the electric field BIMVK . Setting ${\bf\dot{R}}=0$ in
Eq. (45) we recover the usual and familiar model of a dielectric as a
collection of stationary simple harmonic oscillators that are driven by the
electric field BIMVK . Defining a frequency $\omega_{0}=\sqrt{\kappa/\mu}$,
the solution of

${\bf\ddot{p}}+\omega_{0}^{2}{\bf p}=-\frac{q^{2}}{\mu
c}{\bf\dot{A}}=\frac{q^{2}}{\mu}{\bf e}$ (46)

for a time harmonic field ${\bf
e}=({\bf\tilde{e}}e^{-i(\omega_{p}t-kz)}+c.c.)/2$ is

${\bf\tilde{p}}=\frac{q^{2}/\mu}{\omega_{0}^{2}-\omega_{p}^{2}}{\bf\tilde{e}}$
(47)

Assuming the same relation holds for macroscopic fields, we have a macroscopic
polarization field

${\bf\tilde{P}}=\frac{Nq^{2}/\mu}{\omega_{0}^{2}-\omega_{p}^{2}}{\bf\tilde{E}}=\chi{\bf\tilde{E}}$
(48)

given a number density of dipoles $N$. The fields, electric, magnetic, and
polarization, travel together as a pulse of electromagnetic radiation.

It is well known that the polarization field contributes to the
electromagnetic energy

$U_{e}=\int_{\sigma}\frac{1}{2}\left(({\bf E}+{\bf P}){\bf E}+{\bf
B}^{2}\right)dv,$ (49)

as well as the electric and magnetic fields. In terms of time-harmonic fields,
the electromagnetic energy density is

$\rho_{e}=n^{2}\tilde{E}^{2}.$ (50)

Although a material may possess many forms of energy, we intend total energy
to include only those forms of energy that could impact the dynamics or
electrodynamics of the model system. If the material is initially stationary
in the laboratory frame of reference, then it remains stationary with
negligible kinetic energy due to the gradient-index antireflection coating.
Then the electromagnetic energy is the total energy for a stationary linear
dielectric.

Momentum is associated with the movement of energy. The electromagnetic
momentum will contain a contribution from the movement of the polarization
field, as well as contributions from the movement of the electric and magnetic
fields. Writing the total momentum density, Eq. (3), in terms of time-harmonic
fields, we have

${\bf g}_{total}=\frac{n^{2}\tilde{E}^{2}}{c^{2}}c{\bf\hat{e}}_{\bf
k}=\frac{\rho_{e}}{c^{2}}c{\bf\hat{e}}_{\bf k}\,,$ (51)

where ${\bf\hat{e}}_{\bf k}$ is a unit vector in the direction of propagation.
Then, in a stationary dielectric medium, the equality of the total momentum
density and the electromagnetic energy density means that the total momentum
density, Eq. (3), is equal to the electromagnetic momentum density, Eq. (13),
that is, ${\bf G}_{field}={\bf G}_{total}$. Then we have $\xi_{e}=0$ and
substituting this result into Eq. (14), we find ${\bf G}_{mat}=0$. There is no
appreciable matter component of either energy or momentum for a simple anti-
reflection coated linear dielectric medium that is at rest in the laboratory
frame of reference.

The situation changes somewhat in the absence of a gradient-index
antireflection coating. Momentum balance requires, at the cost of some amount
of rigor in the derivation, that we impute a momentum to the material that is
twice the momentum of the reflected field imparted by the surface force of
reflection. But, that is the only significant material momentum, there is no
momentum of the Gordon type due to motion of the atoms inside the dielectric.

##  V Conclusions

The total momentum is uniquely conserved in a thermodynamically closed system
with complete equations of motion. Once the total momentum is known, with
certainty, from conservation principles, then the momentum relations can be
derived. Using a microscopic model of the dielectric medium, we showed that
the propagating polarization field, not the movement of matter, is a component
of the momentum of a system consisting of a stationary dielectric illuminated
by a pulse of quasimonochromatic radiation through a gradient-index
antireflection coating.

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