# Absence of decoherence in the electron-wall system

Zilin Chen  Wayne Cheng-Wei Huang  Herman Batelaan  Physics Department,
University of Nebraska, Lincoln

###### Abstract

Decoherence is associated with a dissipative environment as described by the
Caldeira-Leggett equation. Anglin and Zurek predicted that a resistive surface
could act as such a dissipative environment for a free electron wave passing
close to it. We scrutinize Zurek’s and other promising decoherence theoretical
models by observing electrons passing by an optically excited GaAs surface and
through a gold channel. The high resistivity of the GaAs surface and close
proximity to the gold surface leads to strong decoherence within these
decoherence models. In contradistinction, the observed contrast is high in our
electron diffraction patterns. This implies lower decoherence rates than
suggested by these models, making electron-matter-wave-guides and other
technologies, where quantum coherence of electrons close to materials is
important, a possibility.

††preprint: APS/123-QED

Decoherence has been an important research area in the past few decades. It
provides an explanation for the absence of macroscopic superposition and it
may help to formulate a solution to the quantum measurement problem Zurek
(2003). Quantum decoherence has been investigated in a number of physical
systems. For example, one qubit can be decohered by interacting with an
increasing number of other qubits that serve as a controllable environment
Brydges  _et al._ (2019). Such a bottom-up approach moving to an ever
increasing size of the environment can be complemented by a top-down approach
where a quantum system interacts with a decreasingly complex mesoscopic
environment. For example, a coherent molecular beam was decohered by a thermal
bath. With lowered temperature, decoherence due to the bath was reduced
Hackermüller  _et al._ (2004). In this example, a blackbody photon gas serves
as an environmental bath without local control parameters. The electron-wall
system discussed in this paper may provide a locally-controllable decohering
environment and serves as a testbed for decoherence theories. Moreover, both
dissipation and decoherence can be measured for one system thus pointing to a
test of quantum dissipation theory, i.e., the Caldeira-Leggett equation.

Theories specific for electron-wall interaction have been established based on
different mechanisms. In 1997, Anglin, Paz, and Zurek Anglin  _et al._ (1997)
proposed environment-induced decoherence in the electron-wall system. As a
free electron flies over the surface, dissipation due to Joule heating of the
image charge in the wall occurs, which is linked to decoherence as in the
well-known Caldeira-Leggett theory Zurek (1986). Theories by Scheel and
Buhmann Scheel and Buhmann (2012), Machnikowski Machnikowski (2006), and Howie
Howie (2011) also suggest the presence of environment-induced decoherence (but
not based on dissipation). On the experimental side, observation of free
electron decoherence was first claimed by Peter Sonnentag and Franz Hasselbach
Sonnentag and Hasselbach (2007) in an experiment using an interferometer
placed in close proximity to a silicon wall. The interference pattern contrast
loss was identified to be a result of decoherence since the loss in contrast
scales as a function of the distance between electrons and wall as Zurek’s
decoherence model predicts. However, the interaction with the wall was two
orders of magnitude smaller than that predicted. Beierle et al. Beierle  _et
al._ (2018) and Kerker et al. Kerker  _et al._ (2020) followed up with similar
experiments and confirmed the discrepancy. An alternative explanaton for
contrast loss is dephasing. While decoherence is time irreversible due to
increased entropy, dephasing is reversible and can be caused by spurious
effects such as patch potential. Chen et al. Chen and Batelaan (2020) modeled
a general type of dephasing effect that caused contrast loss in an electron-
wall system, suggesting that decoherence effects in previous experiments can
actually be due to dephasing instead of decoherence. This may explain the
inconsistency between experiments and current decoherence models. Therefore,
an electron-wall system that exhibits unambiguously controllable decoherence
in accordance with the Caldeira-Leggett theory for quantum dissipation is
desirable.

In this paper, we test the experimental bounds of decoherence theories for the
electron-wall system in two scenarios: (a) electrons passing over a GaAs wall
where the resistivity is controlled by laser illumination, and (b) electrons
passing through a gold channel. While Zurek’s theory predicts a stronger
decoherence rate in the GaAs experiment than that in Hasselbach’s experiment,
the observed diffraction pattern remains unaltered after electron interact
with the wall. In the gold channel experiment, the electrons in close
proximity to the surface were also expected to experience strong decoherence
according to the decoherence theories, but again no contrast decrease was
observed. We therefore conclude that no current theories are consistent with
the absence of decoherence in our experiments.

We note that the amount of energy dissipation predicted was also not observed
in our experiments. The absence of both dissipation and decoherence may imply
that with improved resolution, electron-wall experiments may still provide a
test for the Caldeira-Leggett theory.

In our setup, the electron beam is first collimated with two rectangular slits
(Fig. 1(a)). The slits are separated by 24 cm. The width and height of the
first and second slit are $13~{}\mu m\times 19~{}\mu m$ and $2~{}\mu m\times
10~{}\mu m$ respectively. The electron beam size is $3~{}\mu m\times 12~{}\mu
m$ at the nano-grating (whose periodicity is 100 nm), and its transverse
coherence length is $400~{}nm$. The electron beam density is low so that no
electron-electron interaction is involved. Paths that start from different
slits interfere at the screen and produce individual diffraction peaks. The
idea is that a surface brought close to these paths can cause decoherence
between these paths and cause broadening of the diffraction peaks. This effect
occurs at all diffraction peaks which serves as a consistency test. Therefore,
the diffraction pattern is used to measure the decoherence rate in the
experiment.

Figure 1: Experimental setup. (a) The electron beam (yellow line) starts at
the left top and travels to the bottom right. The first collimation slit is
12.7 $\mathrm{\mu m}$ wide and 19 $\mathrm{\mu m}$ tall. The second
collimation slit, placed 25 cm downstream of the first collimation slit, is 2
$\mathrm{\mu m}$ wide and 10 $\mathrm{\mu m}$ tall. The grating periodicity is
100 nm with 50 nm slit widths. The diffracted electron beam passes 12
$\mathrm{\mu m}$ above the surface and intersects the laser beam. Laser
induced charge carriers affect the electron diffraction pattern observed on
the microchannel plate (MCP). The cylindrical lens is used to focus the beam
on the GaAs surface. (b) For the gold gap experiment the GaAs plate is
replaced by a gold gap. A tungsten wire is placed between two gold surfaces at
one end to create the gap. The diameter of the tungsten wire is 15
$\mathrm{\mu m}$. The gold surface is 1.5 mm wide and 15 mm long. (c) A front
view of the electron beam passing through the gold gap.

In the first experiment, the undoped GaAs plate (1 cm $\times$ 1 cm) is placed
$L=3$ mm downstream after the grating with its surface parallel to the
electron beam direction. The beam center is at a height of 12 $\mathrm{\mu m}$
corresponding to the surface. The electrons passing closest to the surface (at
a height of 6 $\mathrm{\mu m}$) experience the strongest interaction and
provide the most useful information for testing decoherence. The surface is
illuminated with focused laser beam. The beam size is $150~{}\mathrm{\mu m}$
perpendicular to the electron beam, and $~{}1~{}\mathrm{cm}$ parallel to the
electron beam. The GaAs surface was illuminated with a 632 nm (super-bandgap)
or a 1064 nm (sub-bandgap) laser for comparison. A charge distribution forms
in the plate due to the laser illumination and its field deflects the electron
beam. The electron deflections are used to model the carrier behavior and
obtain the resistivity which is given in Table. 1. The details of the model
are presented in the appendix. The resistivity is used as the input parameter
in the decoherence models to calculate the decoherence rate and the energy
dissipation rate. The second experiment involves the replacement of the GaAs
surface with a gold gap (Fig. 1(a,b)) and will be described after the first
experiment.

In Anglin and Zurek’s original model Anglin  _et al._ (1997), the Caldeira-
Leggett master equation gives the evolution of the free electron’s density
matrix affected by the quantum Brownian motion. The equation shows that the
resistivity influences the decoherence rate and the energy dissipation rate.
It was stated that the model should work for resistivity in the range of
$10^{-8}$ to $10^{8}~{}\Omega\mathrm{cm}$ consistent with the experimental
values. The decoherence time of the electron flying above a semiconductor wall
is given by Anglin  _et al._ (1997); Beierle  _et al._ (2018),

$\tau_{\mathrm{dec}}^{\text{Zurek }}=\frac{4h^{2}}{\pi
e^{2}k_{\mathrm{B}}T\rho}\frac{z^{3}}{(\Delta x)^{2}},$ (1)

where $h$ is Planck’s constant, $k_{\mathrm{B}}$ is Boltzmann’s constant,
$T=300$ $\mathrm{K}$ is the temperature, $\rho$ is the resistivity of the
sample, $z$ is the electron height above the surface, $\Delta x$ is the path
distance. Additionally, in our experiment, we should add two correction
factors. The first factor is needed as the image charge patches under the
electron path have overlap which reduces their capability to distinguish
”which-way” information by a factor of $(z/\Delta x)^{2}$ Zurek . The second
factor is needed as the GaAs surface location is placed in the transition
region between the near- and far-field of the electron diffraction. This means
that paths from the slits to the detection do not separate entirely. Thus the
overlap of two states reduces the environmental induced decoherence rate as
compared to the rate for two well separated initial states (see Appendix D for
details). The factor is $(\omega/\Delta x)^{2}$ where $\omega$ is the average
width of a single slit beam above the surface.

We give numerical results from Zurek’s model as well as other decoherence
models (See appendix for details) in Table. 1. In the table, Zurek’s model
predicts weak decoherence when GaAs is illuminated, while strong decoherence
is expected without laser illumination. This is not observed in the
diffraction pattern. The energy loss that accompanies the decoherence is
computed to be so large for GaAs without illumination that the electron would
lose most of its kinetic energy. This is not observed either and thus
indicates that the method of calculating energy loss doesn’t describe the
physics for electron-semiconductors interaction.

Figure 2: Simulation and experiment. (a) The simulation (red solid line) is
compared with the expeirmental data (open circles) for the vertical deflection
of the electron beam as a function of the position of the electron beam. The
laser is focussed at a position of approximately x=7.25 mm and has a waist of
0.15 mm. ). The laser-induced charging cause the deflection and changes the
resistivity as a function of position by more than an order of magnitude. (b)
At location (1), (2) and (3) the observed diffraction pattern is consistent
with a weak loss of contrast (the decoherence amount is set to 0.15). A modest
change of resistance accompanied by a change in the decoherence amount to 1
was not observed and supports the conclusion that decoherence does not play a
role for electron-wall interaction in this experiment.

To illustrate that we believe that our device is sensitive to changes in the
number of decoherence events, $Rd=t_{tof}/\tau_{\mathrm{dec}}^{\text{Zurek
}}$, where $t_{tof}$ is the time of flight, we plot the experimental data for
an illuminated GaAs surface together with the simulated diffraction pattern
for $Rd=0.15$ and $Rd=1$ (Fig. 2) . The simulation and experiment agree for
$Rd=0.15$, but for $Rd=1$ the simulated diffraction peaks are clearly wider
than the experimental ones. For dark GaAs and the gold channel the decoherence
amounts are so large that no simulated diffraction peaks (for Zurek’s case)
should be resolvable, while the experimental data show no change when the
surface is brought towards the electron beam.

In the second experiment, we place a wedge-shaped gold gap at $L=5$ mm from
the grating with a gap size varying from 15 $\mathrm{\mu m}$ to 1 $\mathrm{\mu
m}$. This gap is made of two separate pieces of gold-coated glass plates by
clamping a tungsten wire on one end (Fig. 1(b) and 1(c)). The orientation of
the wedge can be changed with respect to the incident direction of the
electron beam (Fig. 3 (a,b)). When the tilt angle exceeds the geometric limit
for unobstructed transmission, electrons reflect multiple times with the gold
surfaces of the gap. The average distance is very close considering multiple
reflections. The predicted loss of energy and corresponding decoherence, both
of a magnitude that is observable, are of interest for the purpose of
confirming the Caldeira-Leggett equation. However, and surprisingly, no loss
of contrast was observed (Fig. 3 (c)).

Table 1: The decoherence amount and energy loss are given for three models. | GaAs (illuminated) | GaAs (dark) | Gold channel | Silicon111Using parameters of Beierle’s experiment. | Gold11footnotemark: 1
---|---|---|---|---|---
resistivity | $5\Omega m$ | $2.5\times 10^{6}\Omega m$ |  $19\times 10^{-8}\Omega m$222For a gold layer of 15 nm thickness. Gilani and Rabchuk (2018) | $1.5\times 10^{-2}\Omega m$ | $2.4\times 10^{-8}\Omega m$
height | $6\mathrm{\mu m}$ | $6\mathrm{\mu m}$ |  $0.85\mathrm{\mu m}$333This is the maximum height as the electron collides with the surface. The gold gapsize is $1.7\mu m$. | $1.7\mathrm{\mu m}$ | $1.7\mathrm{\mu m}$
time of flight |  $4.1\times 10^{-10}$s |  $4.1\times 10^{-10}$s |  $6.2\times 10^{-11}$ s |  $4.1\times 10^{-10}$s |  $4.1\times 10^{-10}$s
| Zurek |
energy loss | 18eV | ($9.0\times 10^{6}$eV) | 48eV | 2.4eV |  $3.8\times 10^{-6}$eV
decoherence amount444The ratio $Rd$ of time of flight over decoherence time. All values are calculated with corrections C1 and C2 (see appendix). | 0.13 | $6.4\times 10^{4}$ | 16 | 0.21 | $3.4\times 10^{-7}$
| Howie |
energy loss |  $0.19$ eV |  $4.7\times 10^{-6}$eV | $<6\times 10^{-4}$ | 0.18eV |  $5.6\times 10^{-4}$eV
decoherence amount | $1.49$ | $1.5$ | $0.25$ | $5.8$ | $0.32$
| Scheel-Markov |
decoherence amount | $0.16$ | $0.16$ | $2.2$ | $1.4(0.12)$ | $0.0082$
Figure 3: Gold gap experiment. By changing the tilt angle of the gap, the
transmission changes gradually. The circles in the figure shows the
transmission rate in different tilt angle. The angle where the electron beam
can straight pass is found with the maximum transmission rate. We present
three diffraction patterns at different tilt angles. Diffraction pattern (a)
corresponds to the beam having multiple reflections and the pattern (b)
corresponds to the beam passing straight through the gap.

In our GaAs experiments, the electron is far away from the GaAs surface:
$\Delta x/z<1$ for our lateral coherence length of approximately 400 nm and
the closest height of about 6 micron. This is taken into account in the
correction factors. If we move the GaAs surface closer to the electron beam to
decrease $\Delta x/z$ then the surface can accumulate charges due to secondary
emissions Chen and Batelaan (2020) whose fields strongly distort the
diffraction pattern. This overwhelms any potential decoherence signature in
the electron diffraction pattern. Adding a second laser either continuously or
chopped at a high rate can strongly reduce surface charges J. Lagowski and
Morawski (1992), which may provide a pathway to a controllable decohering
environment for coherent electron beams.

In principle, there are several ways to identify decoherence. Auto-correlation
imaging may assist in ruling out slow dephasing processes Chen  _et al._
(2018). Holography can be used to directly measure coherence terms in the
density matrix Tonomura (1987). Agreement between experiment and model as a
function of physical parameters may provide insight Barwick  _et al._ (2006),
and the simultaneous observation of energy dissipation and decoherence may
establish the presence of decoherence, if the underlying entanglement follows
the Caldeira-Leggett master equation Zurek (1986). The only model following
that approach, Zurek’s model, surprisingly does not provide an answer close to
the experiment. This leaves the unexplained problem that the classical
dissipation of electromagnetic waves is approximately correct Boyer (1974),
while the dissipation of the electromagnetic field of an electron traveling
near a surface, or at least its feedback on the electron’s motion, is not.

In the gold-gap experiment, the electrons are nominally very close to the
material surface ($\Delta x/z>1$), and still the decoherence rate is low. Our
findings indicate that the search for electron-wall decoherence is currently
at an interesting point. If decoherence rates are close to what can currently
be probed by experiment, then it is likely that improved experimental
techniques will provide a decoherence test-bed, and that improved modeling
will provide insight. On the other hand, the observed rate is already small
enough that it is likely that electron matter-optics can be confined to nearby
material structures possibly leading to new technologies such as an ”optical
fiber” for electron waves. The transmission of an electron diffraction pattern
through the gold gap supports this idea.

In terms of other decoherence models, Howie’s model gave agreement with
Hasselbach’s experiment. In our experiments, some decoherence is predicted in
both cases for GaAs experiment, and stronger decoherence for the earlier
silicon experiment Beierle  _et al._ (2018). This was not observed in the
previous or current experiment. The model developed by Scheel and Buhmann
Scheel and Buhmann (2012) didn’t appear to be promising since no agreement was
achieved until Kerker et. al Kerker  _et al._ (2020) took out some
approximations and reported an agreement with their experiments using a
silicon wall. The same calculation for our experimental parameters, is
consistent for the GaAs experiments, but the gold channel prediction is not in
agreement with our observation.

In conclusion, all theoretical models to date fail to predict correct
decoherence rates for existing experiments. Decoherence is smaller in practice
than predicted. This makes the search for an electron-wall testbed for theory
harder. On the other hand, we found the diffraction pattern to remain
unchanged as it passes through a gold gap with a large tilt angle. With such a
tilt angle, the beam reflects multiple times. This may lead to electron-wave
guide a possibility.

###### Acknowledgements.

We thank Prof. Zurek for suggesting the essential correction factor, Prof.
Harry Ruda for advice on the effect of EL2 states, Prof. Scheel for pointing
out the thermal correction factor, Prof. Stibor for pointing out the Karstens’
thesis, and Prof. Liou for nano fabrication advice. The research was performed
in part in the Nebraska Nanoscale Facility: National Nanotechnology
Coordinated Infrastructure and the Nebraska Center for Materials and
Nanoscience (and/or NERCF), which are supported by the National Science
Foundation under Award ECCS: 2025298, and the Nebraska Research Initiative.
The authors wish to thank Sai Mu and Jörg Malindretos for helpful discussions.
We gratefully acknowledge support by the U.S. National Science Foundation
under Grant No. PHY-1912504.

##  Appendix A Three Theoretical models

In Zurek’s theory, the relation between decoherence time and relaxation time
is given by

$\tau_{\mathrm{dec}}^{\mathrm{Zurek}}=\left(\frac{\lambda_{\mathrm{th}}}{\Delta
x}\right)^{2}\tau_{\mathrm{relax}},$ (2)

where $\Delta x$ is the coherence length for our experiment and
$\lambda_{\mathrm{th}}=\hbar/\sqrt{mk_{b}T}$ is the thermal de Broglie
wavelength of the electron at room temperature (300 K). This equation,
stemming from the Caldeira-Leggett master equation, describes the decoherence
associated with energy loss when the electron passes above a resistive
surface. The definition of relaxation time is $\tau_{\text{relax }}\equiv
v/\left|\frac{dv}{dt}\right|$ according to Zeh Joos  _et al._ (2013). The
energy loss due to Ohmic dissipation used in Zurek’s model comes from Boyer’s
theory Boyer (1974):

$P=\frac{e^{2}\rho v^{2}}{16\pi z^{3}}.$ (3)

By definition,

$P=mv\dot{v}=\frac{mv^{2}}{\tau_{\text{relax
}}}\rightarrow\tau_{\text{relax}}=\frac{mv^{2}}{P},$ (4)

in our experiment, the decoherence amount is calculated by

$R^{Zurek}_{dec}=\int{\frac{dt}{\tau_{dec}^{Zurek}}}=\left(\frac{\Delta
x}{\lambda_{\mathrm{th}}}\right)^{2}\frac{\Delta E}{mv^{2}},$ (5)

where $\Delta E$ is the energy loss during the interaction.

In Anglin and Zurek’s original model Anglin  _et al._ (1997), the Caldeira-
Leggett master equation gives the evolution of the free electron’s density
matrix affected by the quantum Brownian motion. The model shows that the
resistivity sets the decoherence rate and the energy dissipation rate. Anglin
and Zurek state that the model should work for resistivity in the
$10^{-8}\Omega$ cm to $10^{8}\Omega$ cm range. The decoherence time of the
electron flying above the wall (semiconductor) is given by Anglin  _et al._
(1997); Beierle  _et al._ (2018),

$\tau_{\mathrm{dec}}^{\text{Zurek }}=\frac{4h^{2}}{\pi
e^{2}k_{\mathrm{B}}T\rho}\frac{z^{3}}{(\Delta x)^{2}}/(C1\times C2),$ (6)

where $h$ is Planck’s constant, $k_{\mathrm{B}}$ is Boltzmann’s constant,
$T=300$ $\mathrm{K}$ is the temperature, $\rho$ is the resistivity of the
sample, $z$ is the electron height above the surface and $\Delta x$ is the
path distance. The path distance or equivalently the transverse coherence
length $\Delta x$ can be estimated using for the first slit a width of 12.7
$\mu$m and for the second slit a width of 2.0 $\mu$m and the distance between
the slits, $D_{s}=24$ cm. The resulting angular divergence is multiplied with
the electron’s momentum to find the momentum uncertainty spread. From the
uncertainty relation the position uncertainty spread can now be found and this
equals the transverse coherence length, which is the maximum path distance and
for which the decoherence time reaches is largest value. We used the
resitivity of gold for a thin film (15 nm estimated from 5-7 minutes
sputtering time) Gilani and Rabchuk (2018) as the channel substrate was
sputtered with gold as opposed to the resistivity of bulk gold.

In our experiment, we should add two correction factors $C1$ and $C2$ (see
Appendix D for details). The first factor is needed as the image charge
patches under the electron path have overlap which reduces their capability to
distinguish flying electron superposition states by a factor of $(z/\Delta
x)^{2}$ Zurek . The second factor follows the same idea and is another overlap
factor. It is needed as the GaAs surface location is placed in the transition
region between the near- and far-field electron diffraction pattern. This
means that the superposition of paths from the slits to the detection do not
separate entirely. This correction shares the same idea that the overlap of
two entangled states reduce the environmental induced decoherence rate
compared to rate of two well separated initial states. It is estimated to be a
factor of $(\omega/\Delta x)^{2}$ where $\omega$ is the average width of a
single slit beam above the surface (Appendix C).

The model developed by Howie Howie (2011) is established in the electron
microscopy community. In the model, decoherence is related to single
excitation event probability $e^{-P}$ of plasmons in the surface. In other
words, if on average $P$ events occur, the coherence is reduced by a factor
$e^{-P}$. Howie’s model gave agreement with Hasselbach’s experiment. Some
decoherence is predicted in both cases for the GaAs experiment, and stronger
decoherence for the earlier silicon experiment Beierle  _et al._ (2018). The
latter was not observed in the previous or current experiment (column 2 and 4
in Table 1 in the main text).

The general differential form of the single event probability $P$ is

$\frac{d^{2}P\left(y,\omega,q_{x}\right)}{d\omega
dq_{x}}=\frac{e^{2}L}{2\pi^{2}\varepsilon_{0}\hbar
v^{2}}\operatorname{Im}\\{\lambda_{e}\\}\exp\left[-2\nu_{0}z\right],$ (7)

where $\lambda_{e}$ is the retarded loss function,

$\lambda_{e}=-\frac{2}{\nu+\nu_{0}\varepsilon}+\frac{2\beta^{2}}{\nu+\nu_{0}}.$
(8)

The parameters are defined by: $\beta=v/c$,
$\nu_{0}^{2}=\left[q_{x}^{2}+(\omega/v)^{2}-(\omega/c)^{2}\right]$,
$\nu^{2}=\left[q_{x}^{2}+(\omega/v)^{2}-\varepsilon(\omega)(\omega/c)^{2}\right]$.
The dielectric function is given by $\varepsilon(\omega)\approx
1+i\sigma(T)/\varepsilon_{0}\omega$m, where $\sigma$ is the conductivity of
the surface. Previously, the event probability was given in different forms
Howie (2011); Forstmann  _et al._ (1991). First, Eq. (7) is integrated over
the plasmon frequency $\omega$, and

$P^{\text{Howie }}=\left(\frac{e^{2}L\omega_{m}^{2}}{4\pi^{2}\hbar\sigma
v^{2}}\right)\int_{z/4\Delta x}^{\infty}\frac{\exp(-s)}{s}\mathrm{~{}d}s,$ (9)

is obtained, where $L$ is the path length which is the sample length,
$\omega_{m}$ is the cut-off frequency for a semiconductor, and $\sigma=1/\rho$
is the conductivity. A full transition from quantum to classical is predicted
for parameters where $P>>1$. The upper bound of the plasmon frequency can be
estimated from Eq. (7) which falls off quickly so that above the upper bound
it doesn’t contribute to the integral Howie (2011); Kerker  _et al._ (2020);
Beierle  _et al._ (2018). To decohere, the flying electron must get a momentum
recoil larger than its momentum spread and thus ”sufficient to distinguish
between the two interference paths” Howie (2011). Relating this to our the
experiment this can be rephrased as follows. The transverse coherence length
sets the momentum spread of the motional quantum state according to
Heisenberg’s uncertainty principle. A momentum recoil larger than this
momentum spread can thus distinguish the quantum state and cause decoherence.
This implies that one should integrate the wave vector in the range of
$q_{x}>\alpha\Delta x$, where $\alpha$ depends on the specific wavefunction
and is of the order of one:

$\int_{\alpha/\Delta
x}^{\infty}\frac{\exp(-2zk_{x})}{k_{x}}\mathrm{~{}d}k_{x}.$ (10)

The value used in Howie (2011) here is $1/8$ to get agreement with the
experiment by Hasselbach Sonnentag and Hasselbach (2007) (note that there was
a typographical error in Howie (2011); $\alpha$ was mistakenly written as two
Howie . The relative dielectric function,
$\varepsilon=1+i\sigma/\varepsilon_{0}\omega$ given by the Drude model, is
used in these papers. Not previously used for the calculation of decoherence
rates, is that Eq. (7) needs to be multiplied with the usual Boltzmann thermal
correction factor $\operatorname{coth}\left(\frac{\hbar\omega}{2kT}\right)$
Howie ; Howie (2019). Note that even if Howie’s model also predicts energy
loss, it is not associated with the decoherence as an example of the Caldeira-
Leggett equation. A newer version of this theory includes a thermal correction
Howie (2019)

$\begin{array}[]{l}\frac{\displaystyle
d^{2}P\left(y,\omega,q_{x}\right)}{\displaystyle d\omega dq_{x}}=\\\ \\\
\coth(\hbar\omega/2k_{B}T)\frac{e^{2}L}{2\pi^{2}\varepsilon_{0}\hbar
v^{2}}\operatorname{Im}\\{\lambda_{e}\\}\exp\left[-2\nu_{0}z\right]*C2,\end{array}$
(11)

Now the dielectric function $\epsilon(\omega)$ is given in reference Howie
(2019); Karstens (2014)(eq 3.18 in Karstens (2014)). The bounds on the
$\omega$ integral are zero to the cut-off frequency. Howie uses $0.6*10^{12}$.
For large cutoffs, the decoherence rate increases and the experimental test
becomes stronger. The calculated decoherence amount should have affected the
experimental diffraction significantly, but this is not observed.

The model developed by Scheel and Buhmann Scheel and Buhmann (2012) also
considers the image charge of the electron, but instead of using a classical
approach to find the response Anglin  _et al._ (1997); Boyer (1974) a full
quantum electrodynamics treatment is employed.

The interference visibility is given by $e^{-\Gamma[c]}$, where

$\begin{array}[]{lll}\Gamma[c]&=-\frac{q^{2}t}{\epsilon_{0}\hbar(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}k_{x}\mathrm{~{}d}k_{y}\left(1-\cos\left(k_{x}L\right)\right)&\\\
&&\\\
&\times\left(2\bar{n}_{\mathrm{th}}\left(\left|k_{y}v\right|\right)+1\right)\times\frac{\mathrm{e}^{-2k_{\|}z\gamma\left(k_{x},k_{y}\right)}}{2k_{\|}\gamma\left(k_{x},k_{y}\right)}&\\\
&&\\\
&\times\operatorname{Im}\left(r_{\mathrm{p}}\left(k_{\|},\left|k_{y}v\right|\right)\gamma_{v}^{2}+r_{\mathrm{s}}\left(k_{\|},\left|k_{y}v\right|\right)\frac{v^{2}}{c^{2}}\frac{k_{x}^{2}}{k_{\|}^{2}}\right),&\end{array}$
(12)

where the reflection coefficients $r_{\mathrm{s}}$ and $r_{\mathrm{p}}$ depend
on the resistivity, and

$\bar{n}_{\mathrm{th}}(\omega)=\left[e^{\hbar\omega/\left(k_{B}T\right)}-1\right]^{-1}.$
(13)

is the Bose-Einstein distribution. Other parameters are defined as follows:

$\gamma_{v}^{2}(\varphi)=1-\frac{v^{2}}{c^{2}}\cos^{2}\varphi$ (14)

and

$\cos(\varphi)=k_{y}/k_{\parallel},$ (15)

where $k_{\parallel}$ is the projection of the electron’s wavevector in the
x,y plane. These expressions were evaluated using some approximations. The
theory didn’t appear to be promising since no agreement with experiment was
achieved until Kerker et. al Kerker  _et al._ (2020) calculated (A8)
numerically and report an agreement with their experiments using a silicon
wall.

$\begin{array}[]{ll}\epsilon(w)=&1-\frac{\omega_{p}^{2}}{\omega(\omega+\mathrm{i}\gamma)}=\\\
&\\\
&1-\frac{\omega_{p}^{2}}{\omega^{2}+\gamma^{2}}+\mathrm{i}\frac{\omega_{p}^{2}\gamma}{\omega\left(\omega^{2}+\gamma^{2}\right)}=\epsilon_{1}+\mathrm{i}\epsilon_{2}\end{array}$
(16)

Palik (1998), where $r_{s}=\frac{\lambda-k}{\lambda+k}$ and
$r_{p}=\frac{\epsilon(\omega)\lambda-k}{\epsilon(\omega)\lambda+k}$ are the
reflectivities, and the paramaters are defined by
$\lambda=\left(K^{2}-\frac{\omega^{2}}{c^{2}}\right)^{1/2}$,
$k=\left(K^{2}-\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\right)^{1/2}$ and
$K=k_{\parallel}$. The plasma frequency is given by
$\omega_{p}=\sqrt{\frac{Ne^{2}}{\varepsilon_{0}m^{*}}}$, and the damping by
$\gamma=\frac{Ne^{2}\rho}{m^{*}}$. In the last equation the effective mass for
GaAs electrons sub bandgap excitation or our ”dark” case is given by Van
Zeghbroeck (2004)

$m^{*}=m*0.067,$ (17)

while the effective mass for GaAs holes super bandgap excitation or our
”illuminated” case is given by Van Zeghbroeck (2004)

$m^{*}=m*0.34.$ (18)

We repeated the same calculations for our experimental parameters, and find
consistency for the GaAs experiment, but the gold channel prediction is not in
agreement with our observation (column 3 bottom row in Table 1).

##  Appendix B GaAs response modeling

In the GaAs experiment, since the electron beam width is much smaller than the
laser focus width and thus the charge distribution, the lateral motion of the
laser focus results in a measurement of the charge distribution on the GaAs
surface. This charge distribution is used to model the carrier behavior.
Super-bandgap and sub-bandgap illumination induce different charge
distributions, and are recorded individually.

The surface charge due to the surface photovoltage (SPV) effect has been
calculated when the surface is uniformly illuminated Kronik and Shapira
(1999). For non-uniform illumination on the surface lateral electron transport
can occur, which affects the magnitude of the SPV effect. In our experiment,
the sample surface is illuminated in an oval shape which induces SPV gradients
across the surface, leading to a non-uniform charge distribution. To explain
the magnitude and shape of the charge distribution for super-bandgap
illumination, a rate equation model is used. The central elements in the model
are the following. For 632 $\mathrm{nm}$ illumination, the photons can only
penetrate for about $1/\alpha=0.3$ $\mathrm{\mu m}$ Sturge (1962) into the
surface. Electron and hole pairs are generated near the surface and start to
diffuse. Electrons spread into the surroundings much faster than holes because
the electron’s diffusion coefficient typical value is 200 $\mathrm{cm^{2}/s}$
compared to the hole’s diffusion coefficient of 10 $\mathrm{cm^{2}/s}$. Most
generated electrons in the locally illuminated area are trapped by surface
states. Consequently most holes, but only few electrons, escape. In this way,
carriers form a charge distribution with a negative center and positive wings.
The electric field created by the charge distribution prevents the holes from
moving further away, and an equilibrium is reached (Fig. 4(a)). The sub-
bandgap situation is described and modeled after the super-bandgap situation
for comparison.

###  B.1 Super-bandgap model

In the super-bandgap model model, rate equations (called steady state
continuity equations in Liu  _et al._ (1993)) are used to simulate diffusion,
mobility, excitation, recombination and trapping:

$\frac{dn(x)}{dt}=\frac{1}{e}\frac{dJ_{n}(x)}{dx}+G_{n}(x)-R_{n}(x),$ (19)

$\frac{dp(x)}{dt}=-\frac{1}{e}\frac{dJ_{p}(x)}{dx}+G_{p}(x)-R_{p}(x),$ (20)

$\frac{dn_{t}(x)}{dt}=G_{t}(x)-R_{t}(x),$ (21)

where $n$, $p$ and $n_{t}$ are the negative, positive and trapped carrier
densities, $G$ and $R$ are the generation and recombination rate, and $e$ is
the elementary charge magnitude. The current flows $J_{n}$ and $J_{p}$ are,

$J_{n}(x)=e\mu_{n}n(x)\xi(x)+eD_{n}\frac{dn(x)}{dx},$ (22)
$J_{p}(x)=e\mu_{p}p(x)\xi(x)-eD_{p}\frac{dp(x)}{dx},$ (23)

where $\xi$ is electric field, $\mu_{p}=300$ $\mathrm{cm^{2}/Vs}$,
$\mu_{n}=8000$ $\mathrm{cm^{2}/Vs}$ are typical mobilities, and $D_{p}=10$
$\mathrm{cm^{2}/s}$, $D_{n}=200$ $\mathrm{cm^{2}/s}$ are typical diffusion
coefficients for positive and negative carriers. In Eq. (19), the total
thermal recombination rate $R_{n}$ consists of the capture rate into the
valence band, $r_{n\rightarrow p}^{th}$, and the capture rate into the
trapping states $r_{n\rightarrow t}^{th}$,

$\displaystyle R_{n}=r_{n\rightarrow p}^{th}+r_{n\rightarrow t}^{th}$ (24a)
$\displaystyle r_{n\rightarrow p}^{th}=c_{n\rightarrow p}^{th}np$ (24b)
$\displaystyle r_{n\rightarrow t}^{th}=c_{n\rightarrow t}^{th}np_{t},$ (24c)

where the thermal recombination rate is proportional to both the positive
($p$) and negative ($n$) carrier density, and the proportionality constants
(for capture driven by thermal recombination) are given by $c_{n\rightarrow
p}^{th}$ and $c_{n\rightarrow t}^{th}$. Generation term for the conduction
band $G_{n}$ are composed of the thermal generation term $g_{p\rightarrow
n}^{th}$ and the optical generation term $g_{p\rightarrow n}^{opt}$ which
depicts electron excitation from valence band to conduction band; and the
thermal generation term $g_{t\rightarrow n}^{th}$ and optical generation term
$g_{t\rightarrow n}^{opt}$ describes excitation from the trapping states to
the conduction band,

$\displaystyle G_{n}=g_{p\rightarrow n}^{th}+g_{p\rightarrow
n}^{opt}+g_{t\rightarrow n}^{th}+g_{t\rightarrow n}^{opt}$ (25a)
$\displaystyle g_{p\rightarrow n}^{th}=c_{n\rightarrow p}^{th}n_{0}p_{0}$
(25b) $\displaystyle g_{p\rightarrow n}^{opt}=F(1-R)\eta
e^{-(x-x_{0})^{2}/\omega^{2}}$ (25c) $\displaystyle g_{t\rightarrow
n}^{th}=e_{t\rightarrow n}n_{t}$ (25d) $\displaystyle g_{t\rightarrow
n}^{opt}=\sigma_{t\rightarrow n}^{opt}Fn_{t},$ (25e)

where $n_{0}$ and $p_{0}$ are the intrinsic carrier densities (about
$10^{6}\mathrm{cm^{-3}}$) without illumination, $F$ is the photon flux
($\approx 10^{18}\mathrm{cm^{-2}s^{-1}}$), $R=0.3$ is reflectivity Philipp and
Ehrenreich (1963), $\eta=0.7$ is the quantum efficiency M. Milanova and
Smekalin (1999), $x_{0}$ is the beam center location and $\omega$ is the beam
width. The thermal excitation rates and optical excitation rate to the
trapping states are, $e_{t\rightarrow n}$ and $\sigma_{t\rightarrow n}^{opt}$,
respectively. And in the Eq. (20) and Eq. (21) the equalities

$\displaystyle G_{p}=G_{n}$ (26a) $\displaystyle G_{t}=r_{n\rightarrow
t}^{th}$ (26b) $\displaystyle R_{p}=r_{n\rightarrow p}^{th}$ (26c)
$\displaystyle R_{t}=g_{t\rightarrow n}^{th}+g_{t\rightarrow n}^{opt}$ (26d)

hold.

Figure 4: Super-bandgap excitation. (a) A schematic picture of super-bandgap
illumination is shown. Laser light (632 nm) penetrates to a depth of about 0.3
micron at the GaAs sample surface, and excites electrons from the valence band
to the conduction band. After excitation, electrons and holes start to
diffuse, and most electrons are trapped in surface states. (b) The laser beam
is scanned laterally ($x$-axis) and the electron beam deflection (black
circles) is measured. In the center area, the deflection is away from the
surface. The result of a charge carrier model (red solid line) is shown. (c)
The inferred resistivity (blue solid line) shows several orders of magnitude
variation. The electron diffraction patterns exhibit varying degrees of
rotation but the contrast of the diffraction pattern does not change.

In addition to modeling the carrier dynamics, we use a semi-classical electron
trajectory simulation to model the deflection of the beam. The electron
diffraction is described by the usual quantum mechanical distribution, while
the electron motion above the GaAs plate is obtained by numerically solving
Newton’s equation. The electric field used in the equation of motion is
obtained by summing Coulomb’s law over the charge distribution given by the
carrier dynamics model. Details for this method can be found in our previous
work Chen and Batelaan (2020). We found that the beam shift in the vertical
direction is proportional to the total charge integrated over the surface
directly below the beam path. This means that the curvature of the electron
trajectories is negligible for the experimental parameters. It also suggests
that the vertical deflection is a good measurement for the longitudinal
($y$-axis) integrated charge distribution.

No integration over the charge distribution in the $z$-direction is done. This
is motivated by the small laser penetration depth as compared to the height
above the surface. The carrier dynamics are thus simplified to one dimension
($x$). The electron beam deflection was observed while the laser focus was
scanned laterally ($x$). The charge distribution is then obtained from the
combination of the mathematical model and the trajectory simulation (Fig.
4(b)). The experimental data matches the model well. The experimentally higher
side wing on the right can be caused by non uniformities in the laser beam.
The diffraction pattern show no broadening (Fig. 4(c)) and the rotations of
the diffraction orders is due to local field gradients.

The temporal behavior is another clue for the existence of trapping states. To
investigate the temporal response, the deflection was measured while the laser
was chopped (Fig. 5) to reveal charging effects in timescales of $0.1-1$ s. In
GaAs, SPV is usually formed in microseconds J. Lagowski and Morawski (1992);
Liu  _et al._ (1993), but that is without the presence of trapping states. The
charge distribution recovery time is also slow and exceeds one second (Fig.
5). This confirms earlier observations J. Lagowski and Morawski (1992). We now
describe the details of the laser chopping experiment. The electron beam
position is recorded with our camera. When the laser is turned off, the
electron beam starts at a location defined as the zero. Now the chopper is
rotating and the laser is turned on. The electron beam position moves up
further with light pulse when the chopper is in the open position.The beam
position does not fully recover when the chopper is in the closed position at
a chopping frequency of 4-20 Hz. This implies a charging time slower than 1/4
s. When reaching a semi-steady state, the beam oscillates between two
locations whose distance is smaller than the deflection from the zero
location. These two beam locations are the corresponding chop-close (blue
circular dots) and chop-open (red triangular dots) deflection data for each
chopping frequency (Fig. 5). The simulation (solid lines) agrees fairly well
with the experiment.

Figure 5: Temporal dependence of deflection. The data shown was taken at the
center peak location (see Fig. 4) with the laser chopped continuously at a
range of frequencies. The chop-opening time $\Delta t$, corresponds to the
chopping frequency (the laser on and off times are equal). Deflection zero is
defined as the deflection without any laser present.

###  B.2 Sub-bandgap model

In the sub-bandgap model, photons are much less strongly absorbed than for the
super-bandgap situation and can penetrate deeply into the bulk Sturge (1962).
Even if the photon energy is less than the bandgap energy, electrons are still
pumped to the conduction band due to defect-assisted excitation Martin (1981).
Electrons and holes generated at different depths diffuse into the bulk.
Surface states trapping is not the leading effect anymore, and it is not
included in the model for simplicity. A simple sketch illustrates the main
elements of the model (Fig. 6(a)). The detection beam is about 12 microns away
from the surface, while the bulk distribution extends throughout the 300
microns thick GaAs plate. The charges near the surface make the dominant
contribution to the deflection of the free electrons. The deflection shows
large side wings and a relatively small center peak across the sample (Fig.
6(b)) indicating that the surface has a net negative charge and more electrons
are accumulated near the surface. Given that charges are generated throughout
the bulk and a non-zero-sum charge distribution resides at the surface, the
dimension perpendicular to the surface must be included in the model.
Therefore, the continuity equations are extended to two dimensions,

$\frac{dn(x,y)}{dt}=\frac{1}{e}\vec{\nabla}_{x,y}\cdot\vec{J}_{n}(x,y)+G_{n}(x)-R_{n}(x),$
(27)

$\frac{dp(x,y)}{dt}=-\frac{1}{e}\vec{\nabla}_{x,y}\cdot\vec{J}_{p}(x,y)+G_{p}(x)-R_{p}(x),$
(28)

where,

$\vec{J}_{n}(x,y)=e\mu_{n}n(x,y)\vec{\xi}(x,y)+eD_{n}\vec{\nabla}_{x,y}n(x,y),$
(29)
$\vec{J}_{p}(x,y)=e\mu_{p}p(x,y)\vec{\xi}(x,y)-eD_{p}\vec{\nabla}_{x,y}p(x,y),$
(30)

Comparing with the equations for super-bandgap excitations, generation and
recombination only have the terms $r_{n\rightarrow p}^{th}$, $g_{p\rightarrow
n}^{th}$, and $g_{p\rightarrow n}^{opt}$; there are no trapping terms. Because
of the internal scattering and reflection, the laser width deep in the bulk in
the model is bigger than that given by free propagation. The sub-bandgap laser
is composed of two Gaussians beams whose widths are about 150 $\mathrm{\mu m}$
and 750 $\mathrm{\mu m}$, where the wider Gaussian beam is five times weaker.
When calculating the detection beam trajectories, charges near the surface
within 10 $\mathrm{\mu m}$ are used. Positive and negative charges in deep
layers tend to cancel each other’s field close to the surface. If there would
have been uniform charge component from the charge distribution in the deeper
layers, then one could expect an electric field with a long range above the
surface. This can be ruled out experimentally by moving the probing electron
beam to a distance of 100 micron above the surface. At this distance there is
no discernible deflection. The experimental deflection data agrees well with
the simulation (Fig. 6(b)). The experimental diffraction data again shows no
appreciable broadening of the diffraction peaks (Fig. 6(c)).

Figure 6: Sub band-gap excitation. (a) A schematic picture of sub-bandgap
illumination in the bulk is shown. Laser light (1064 nm) can penetrate deeply
into the GaAs bulk. Electrons and holes can diffuse freely in the bulk. The
detection electron beam above the surface is sensitive to charges near the
surface because the bulk thickness is 30 times larger than the distance
between detection beam and the surface. (b) The laser beam is scanned
laterally ($x$-axis) and the electron beam deflection (black circles) is
measured. In the center area, the deflection is towards the surface. The
result of a charge carrier model (red solid line) is shown. (c) The inferred
resistivity (blue solid line) is shown. Several electron diffraction MCP
images exhibit little distortion as compared to super-bandgap excitation
because of smaller induced surface charge (weaker fields). Charge is
distributed throughout the whole sample, while in the super-bandgap situation,
the charges distribute only in limited area.

The assumptions in the simple model can be investigated further to represent
to complex nature of the surface more accurately. For example, the laser can
heat the surface to create a temperature gradient in the material. This might
form an internal electric field referred to as the Peltier-Seebeck effect
Goldsmid (2011). However, in the experiment, the laser intensity is only 2 mW,
and such a low intensity laser cannot create a significant temperature
gradient in a time duration of about one second. The fitting parameters are
consistent with previous experimental data. For example, in the sub-band gap
situation, defect-assisted excitation is assumed in the model. EL2 defects are
one of the well known defects with a high density Germanova and Hardalov
(1987).

Table 2:  Decoherence rate correction factors. C1 and C2 are used with Zurek’s model, while C2 is used for Howie’s and Scheel’s model.  | GaAs | Gold channel | Silicon/Gold555Using parameters of Beierle’s experiment.
---|---|---|---
decoherence correction | $C1,C2=0.0044,0.1$ | $C1,C2=1,0.18$ | $C1,C2=0.055,0.1$

##  Appendix C Correction factors

Figure 7: Correction factors. A symbolic picture of diffraction from a
grating. The image charge patches (red disks) are located underneath the
electron positions (black dots). The electron positions are located on paths
(black lines). Diffaction from single slits is indicated by blue cones to
illustrate the transition from the near- to far-field. As the image charge
patch overlap increases, or the single slit diffraction cones overlap
increases, which-way information is reduced and the decoherence rate slows by
a correction factor (see text).

To calculate the decoherence rate it is assumed in the theoretical models that
the surface states induced by the electron-surface interaction are non-
overlapping. To illustrate this, consider two slits separated by $\Delta x$ on
the left and right (Fig. 7). The surface states, $|s_{L}>$, and $|s_{R}>$
(indicated by red disks), that are induced by the electron paths $|L>$ and
$|R>$, can provide complete which-way information on the electron when they
are non-overlapping. Tracing, for example, the resulting entangled state
$|L,s_{L}>+|R,s_{R}>$ over the surface states completely removes the electron
path coherence. However, when the surface states are partially overlapping,
the which-way information is not completely removed by tracing. The magnitude
of this effect is given by $|<s_{L}|s_{R}>|^{2}$. For an electron at a height
h above the surface, and a Coulombic interaction potential that scales with
$1/r$, the approximate size $s$ of the image charge patch is estimated by $h$
itself. As $w\sim\Delta x$, the correction factor applied to the decoherence
rate is $C1=\sim(\Delta x/h)^{2}$, as suggested by Zurek Zurek . When $\Delta
x$ exceeds $h$ the correction factors equals one. An additional correction
factor needs to be applied. When the electron paths are in the near-field the
paths are completely distinguishable. However, in the transition from near- to
far-field they are not. To illustrate this, instead of considering the paths,
it is useful to consider single slit diffraction from the two slits considered
(Fig. 7, blue shade triangles). This is estimated by propagating a Gaussian
beam to the middle of the surface. The width at that location is given by
$w=w_{0}\sqrt{1+(L\lambda_{dB}/\pi w_{0})^{2}}$. The overlap between the two
diffracted beams is approximated by the second correction factor $C2=((\Delta
x/w)^{2}$ for $\Delta x<w$. Both correction factors reduce the decoherence
rate. The divergence angle of the single slit diffraction pattern is given by
$\theta=\Delta p_{y}/p\sim\Delta y_{s}/\hbar p$, where $\Delta y_{s}$ is the
slit width and $p$ the electron’s momentum . The transition to the far-field
regime occurs when $L\theta\sim\Delta y_{s}/2$.

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# Characterization and Correction of the Scattering Background Produced by
Dust on the Objective Lens of the Lijiang 10-cm Coronagraph

Feiyang Sha  Yu Liu  Xuefei Zhang  Tengfei Song  Yunnan Observatories, Chinese
Academy of Sciences, Kunming, China  School of Physical Science and
Technology, Southwest Jiaotong University, Chengdu, China  School of Astronomy
and Space Science, University of Chinese Academy of Sciences, Beijing, China

###### keywords:

Corona, coronagraph, scatter, stray light, dust

##  1 Introduction

The corona is the outermost layer of the solar atmosphere, containing a thin,
highly ionized plasma that reaches a temperature of a few million degrees. It
has an extremely low brightness and diminishes quickly with height, i.e.
radial distance from the solar center. Therefore, the corona is hidden by the
much brighter solar disk and is generally not visible in normal conditions
(e.g. Habbal et al., 2010; Liang et al., 2021). The corona can be observed if
the disk light is occulted, e.g. during a total solar eclipse or using a Lyot
coronagraph (Lyot, 1933).

Around the beginning of the 20th century, many attempts had been made to
observe the solar corona during noneclipse periods. The most difficult part of
coronal observation was suppressing the relatively strong stray light, which
is typically two orders of magnitude higher than coronal brightness (Aliev,
Guseva, and Tlatov, 2017). It was not until 1930 that Lyot (1933) directly
observed the solar corona during noneclipse periods at the Pic du Midi
Observatory at a latitude of 2780 m. The instrument he used is now known as
the Lyot coronagraph, which employs internal occultation. The basic principle
of this instrument involves placing an occulting disk inside the telescope to
block the solar disk formed by the objective lens and enable the corona light
to pass through. However, it requires strict suppression of stray light. Stray
light can originate from various sources inside the instrument (Lyot, 1939):

  1. i)

Direct sunlight from the solar photosphere.

  2. ii)

The ghost image formed by multiple reflections of the Sun on the two faces of
the objective lens.

  3. iii)

The stray light from direct sunlight striking the telescope tube and the edge
of the objective lens.

  4. iv)

The scattered light produced by surface roughness, scratches, and internal
bubbles of the objective lens under direct sunlight.

  5. v)

The stray light produced by dust particles on the objective lens.

The conjugate-blocking method is used to suppress the first three types of
stray light by positioning a stop at the conjugate image of the source. To
minimize stray light resulting from surface roughness and defects on the
objective lens, high-quality materials and ultrasmooth polishing are utilized
in lens construction. Regular lens cleaning is also crucial to suppress stray
light originating from dust particles on the objective lens.

The stray light outside the instrument primarily originates from atmospheric
scattering, known as sky brightness. Therefore, ideal coronal-observation
sites should be located at high altitudes with low sky brightness.
Additionally, stable and gentle winds, high levels of sunshine and a high
number of sunny days are also necessary.

It is crucial to understand that there are two types of stray light: fixed and
variable. Fixed stray light remains constant, while variable stray light
increases rapidly with time and deterioration of cleanliness level. For
routine ground-based coronagraph observations, the variable stray light is
contributed by sky brightness and dust on the objective lens.

Therefore, coronal data observed at different times can exhibit varying levels
of scattered stray light, leading to different degrees of scattering
background in coronal images. This makes it challenging to analyze subtle
coronal structures and calibrate coronal intensity accurately. In addition,
many methods have been proposed to measure the coronal magnetic field using
coronagraphs (e.g., Yang et al., 2020; Chen et al., 2023a, b), which cannot be
achieved without accurate data calibration.

Therefore, there is an urgent need for better methods to remove the scattering
background. One simple way is to clean the objective frequently or use gas
purging. However, this method has many drawbacks. First, dust accumulation can
still occur and produce variable stray light, because of the atmospheric
particles or low cleanness level, which is beyond our control. Secondly, there
is no standard on whether the objective should be cleaned or not. Hence,
cleaning is quite subjective, leading to over- or undercleaning. Finally,
frequent cleaning of the objective lens can damage the lens itself, causing
extra stray light.

Another possible way is image correction, which requires modeling the
scattered light from dust on the objective lens. Spyak and Wolfe (1992)
measured the scattered light from dust-contaminated mirrors at $\lambda=10.6\
\rm\mu m$ and the results were compared to those predicted by a modified Mie
theory. However, this study applies only to reflective optics and not to the
refractive optics used in most ground-based coronagraphs. Based on Mie theory,
Dittman (2002) simulates the Bidirectional Scattering Distribution Function
(BSDF) of scatter from particulate contaminated optics, where the forward and
backward scatter BSDF are separated, allowing the analyst to consider these
scatter contributions separately for refractive systems. Gallagher et al.
(2016) use the Harvey Shack BSDF model to simulate the scattering distribution
of dust on the objective lens of the Coronal Solar Magnetism Observatory Large
Coronagraph (COSMO-LC). However, these studies are based on theoretical
simulations and few empirical measurements.

Based on the experimental data of Lijiang 10-cm Coronagraph, Sha et al. (2023)
derived an empirical model for the relation between the dust on the objective
lens and the resulting scattering background on the coronal images. However,
the employed strategy does not take into account the spatial distribution of
dust particles and a one-dimensional model that is only related to height is
obtained.

In this study, we propose a new dust-scattering background empirical model
based on the experimental data of Lijiang 10-cm Coronagraph (Ichimoto et al.,
1999; Liu and Zhang, 2018; Liu, 2014).

##  2 Experiment

###  2.1 Experimental Conditions and Instruments

We conducted the experiment at Lijiang Observatory (3200 m, E:
$100^{\circ}01^{\prime}48^{\prime\prime}$, N:
$26^{\circ}41^{\prime}42^{\prime\prime}$), the first observation site in China
capable of coronal observations, with an average sky brightness of less than
20 ppm of the solar-disk intensity (Zhao et al., 2018). The instrument used is
the Lijiang Coronagraph (YOGIS), the only routinely operated coronagraph in
China. It has been in continuous operation since it was established at Lijiang
Observatory in 2013.

The observation setup consists of a 10-cm coronagraph, a tunable Lyot filter,
and a cooled CMOS camera, allowing it to image the coronal green line (Fe XIV,
530.3 nm, Sakurai et al., 1999) within 1.03 to 2.5$\rm\ R_{\odot}$ field of
view. The transmission curve of the Lyot filter can be modulated via two
liquid-crystal variable readers, providing quick wavelength tuning and an
efficient subtraction of sky background (Ichimoto et al., 1999).

In addition, YOGIS has the ability to obtain 2D Doppler velocity maps of the
corona from which Sakurai et al. (2002) found slow sound waves in the corona
and Hori, Ichimoto, and Sakurai (2007) detected an MDH kink oscillation in
coronal loops for the first time. Using this instrument, Zhang et al. (2022a)
found that the coefficients between the green-line coronal data and the 211 Å
data of the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics
Observatory (SDO) (Lemen et al., 2012) always kept high values. Zhang et al.
(2022b) also suggested that the correlation between the coronal green line and
the magnetic field was strong. Li et al. (2023) conducted an analysis of
intensity attenuation of the coronal structures and loops within the inner
corona region. The above-mentioned study was affected by the dust-scattering
background overlapped with the coronal data. If this background is removed,
more accurate results can be obtained.

We seek to determine the relationship between dust on the objective lens and
the resulting scattering background of the coronal image. Therefore, it is
crucial to first obtain their respective individual contributions.

###  2.2 Scattering Background

Figure 1.: The optical layout of YOGIS. The objective lens, with a diameter of
10 cm and a focal length of 1490 mm are not depicted in the diagram. The
occulting disk, on the far left of the figure, is positioned at the primary
focus of the objective lens and it is attached at the center of the field
lens. The Lyot stop is used to suppress the stray light originating from the
telescope tube and the edges of the objective lens, while the Lyot spot
suppresses the ghost image. Two polarizing beamsplitter (PBS1 and PBS2) are
placed before and after the Lyot filter, not only splitting the beam but also
working as the entrance and exit polarizers of the Lyot filter (Ichimoto et
al., 1999). The lens unit, not included in the optical path, is used for
observing the objective lens.

The dust on the objective adds a scattering background to the coronal image.
This scattering background can be derived by removing the pure coronal part of
the image, which can be approximately obtained after wiping out the dust on
the objective.

The optical layout of YOGIS (Sha et al., 2023) is shown in Figure 1. When
imaging the corona, the filter is first adjusted to 530.3 nm with a bandwidth
of 0.1 nm, corresponding to the Fe XIV line center, to obtain a “single”
image. Then, the filter is adjusted to the line wing (530.3 $\pm$ 0.2 nm) to
obtain a “Double” image for removing the sky background. The coronal image is
obtained using the following equation:

$\mbox{Corona}=\frac{\mbox{Single}-\mbox{Double}}{\mbox{Flat}-\mbox{Dark}}\times\overline{\mbox{Flat}-\mbox{Dark}}.$
(1)

To obtain the scattering background, the corona is imaged with dust on the
objective first, denoted as $C_{0}$. Then, the lens is cleaned three times to
remove the dust, with $C_{1},C_{2}$, and $C_{3}$ denoting the images obtained
after each cleaning, respectively. We define three scattering-background
images:

$\mbox{Scatter}_{i}=C_{i}/s_{i}-C_{3}/s_{3}\ \ (i=0,1,2),$ (2)

where $s_{i}$ is the solar-radiation intensity and is used to correct for its
diurnal variation. These coronal images are aligned in advance.

###  2.3 Dust on the Objective Lens

Inside YOGIS, there is a lens unit mounted on a one-dimensional linear stage,
which enables making an image of the objective lens on the camera to evaluate
the scattered light produced by dust (Ichimoto et al., 1999). The optical
layout, when imaging the objective is shown in Figure 2.

Figure 2.: The optical layout of YOGIS when imaging the objective lens. The
lens unit is moved in the optical path compared to Figure 1. It serves to
extend the optical path, bringing the original position of the objective lens
image forward to the location of the camera.

The image of the objective lens, shown in Figure 3, is produced by three
sources. First, the sunlight scattered by the surface roughness of the
objective lens. Secondly, the sunlight scattered by the dust on the objective
lens, forming points of different sizes with approximately circular shapes.
The third source is the scattered light from Earth’s atmosphere, scattered by
dust on the objective lens. Among them, the first source only depends on the
solar-radiation intensity and the objective lens surface-processing
technology, which is fixed with respect to the change in environmental
cleanliness level. In the third source, the scattered light from Earth’s
atmosphere is too weak compared to the direct sunlight of the second source,
and thus can be ignored. Therefore, the second source can be used to describe
the properties of dust.

Figure 3.: Image of the objective in arbitrary units. The large circle in the
image represents the image of the objective lens, resulting from the scattered
light produced by the surface roughness of the objective lens. The numerous
small spots are scattering points formed by dust particles on the objective
lens surface.

In the experiment, the objective lens is imaged after each measurement of the
corona. We denote such lens images as $O_{0},O_{1},O_{2}$, and $O_{3}$.

###  2.4 Experimental Procedure

Two experiments were conducted on November 17, 2021, from 9:00 to 12:30 a.m.
(local time), when the weather was clear and cloudless. The experimental
procedure is the following:

  1. i)

Wait for dust to accumulate on the objective lens, approximately equivalent to
not wiping for 4 – 7 days.

  2. ii)

Image the corona and objective lens and measure the solar intensity, recorded
as $C_{0}$, $O_{0}$, and $s_{0}$.

  3. iii)

Slightly clean the objective lens and repeat step ii to obtain $C_{1}$,
$O_{1}$, and $s_{1}$.

  4. iv)

Slightly clean the objective lens again and repeat step ii to obtain $C_{2}$,
$O_{2}$, and $s_{2}$.

  5. v)

Completely clean the objective lens and repeat step ii to obtain $C_{3}$,
$O_{3}$, and $s_{3}$.

The solar-radiation intensity is obtained by a calibrated commercial weather
station-Davis Vantage Pro2 of the Astronomical Site Monitoring System (ASMS,
Xin et al., 2020) at Lijiang Observatory. It measures the solar radiation-
intensity at a resolution of $1\rm\ W/m^{2}$ and an accuracy of 5%.

##  3 Data Analysis

The data resulting from one of the two experiments is shown in Figure 4.

Figure 4.: A set of experimental data. The first row shows the corona image
(log scale), while the second row displays the objective image. Both images
have been adjusted for solar-radiation intensity. From left to right, the
amount of dust on the objective decreases sequentially, leading to a gradual
reduction in the scattering background in the first-row image and a gradual
decrease in the number of scattering points in the second-row image.

From left to right, the dust on the objective gradually decreases, along with
the number of scattering points in the lens images. Note that, the scattering
background intensity from Figures 4a-c decreases and there is almost no
difference between $C_{2}/s_{2}$ and $C_{3}/s_{3}$. Three scattering
backgrounds can be obtained using Equation 2, as shown in Figure 5.

It is worth mentioning that some coronal structure can still be seen in the
scattering background given in Figure 5c. The background given by Equation 2
contains two parts, one is the true scattering background and the other is the
coronal difference produced by the constantly changing solar corona observed
from the ground. Therefore, the time interval between the different images of
a single experiment should be short enough to reduce the effect of the coronal
variations. The longest time interval in our experiments is about one hour,
which has been shown to have almost no influence on the analysis of the
scattering background (Sha et al., 2023).

Figure 5.: The scattering background of Figures 4a-c obtained using Equation
2. The intensity gradually weakens from left to right.

In the following, two models will be used to describe the data, which depend
on two and three parameters, respectively. The former does not consider the
angular distribution of the scattering points, while the latter does.

###  3.1 Two-Parameter Model

The radial profiles of the scattering background (the median at each radius)
are shown in Figure 6 (left), along with the linear fitting performed on them,
showing strong correlations. Therefore, the slopes ($k$) and intercepts ($b$)
of these lines can be used as the parameters of the scattering background and
they are related to the corresponding dust.

The dust is parameterized by the total intensity of scattering points in
Figures 4e and f. Using Otsu’s segmentation method (Otsu, 1979), a threshold
is calculated for each lens image. Areas with pixel values greater than this
threshold are considered dust-scattering points and the intensity of each
point can be obtained by summing the pixel values of each area.

Assuming a spatially uniform distribution of scattering points, only one
parameter, the total intensity of the scattering points ($I$), can be used to
parameterize the dust. Therefore, the parameters of the scattering background,
$k$ and $b$, are considered functions of $I$. As shown in Figure 6, all five
$k$ and $b$ obtained from the two experiments (one measurement was discarded
due to excessive dust) are linearly fitted to $I$, obtaining:

Figure 6.: Left: The radial distribution of the scattering background of
Figure 5, where $k_{i}$, $b_{i}$, and $cor_{i}$ correspond to the slope,
intercept, and Pearson correlation coefficients of the least squares fitting.
The $y$-axis represents the median intensity of the scattering background
obtained from Equation 2. Right: The relationship between $k,b$ calculated in
the left figure and $I^{\prime}$, where $I^{\prime}=I/s$.

$k(I^{\prime})=-0.43\times 10^{-7}I^{\prime}-0.1\ \ \ \
b(I^{\prime})=0.99\times 10^{-7}I^{\prime}+0.17,$ (3)

where $I^{\prime}=I/s$. Using Equation 3, the scattering background can be
parameterized with the heliocentric distance ($r$) and the total intensity of
the scattering points I, as follows:

$\begin{array}[]{lr}\mbox{Scatter}(r,I)=k(I^{\prime})\times
r/\mathrm{R_{\odot}}+b(I^{\prime})\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=(0.99-0.43\ r/\mathrm{R_{\odot}})\times 10^{-7}\ I/s+0.17-0.1\
r/\mathrm{R_{\odot}}.\end{array}$ (4)

A simulated background image (Scatter2D) can be obtained, and used to correct
the coronal image as follows:

$\mbox{Corona}^{\prime}=\mbox{Corona}-s\times\mbox{Scatter}_{2D}.$ (5)

The above calibration process is summarized by the flowchart in Figure 7. The
inputs to the flowchart are the scattering background and objective image.
After a series of parameterization and data fitting, Equation 4 is obtained.

Figure 7.: The flowchart of the calibration process. The inputs are the images
of the scattering background and the objective. The radial distribution of the
scattering background is linearly fitted to obtain $k$ and $b$, satisfying
$\mbox{Scatter}=k\times r/R_{\odot}+b$. Otsu’s threshold segmentation method
is applied to the objective images to obtain the total intensities of the
scattering points $I$. Then, these parameters are linearly fitted to obtain
$k(I^{\prime})$, $b(I^{\prime})$, and their correlation coefficients
($cor^{k}$ and $cor^{b}$). Finally, the two distributions are substituted back
to the original equation to obtain $\mbox{Scatter}(r,I)$.

###  3.2 Three-Parameter Model

The angle of the scattering background is not included in the two-parameter
model, because we assumed a uniform dust distribution. Therefore, the model
will not be accurate when the scattering points are not uniformly distributed.

To find the relationship between the angular distribution of the scattering
points and the scattering background, both the objective images and the
scattering-background images are divided into $N$ equal circular sectors
(Figure 8).

Figure 8.: The left image illustrates the scattering background, whereas the
right image shows the objective lens. Each figure is divided into $N$ circular
sectors by solid white lines (here $N=8$), and each region is labeled from 1
to $N$.

We randomly select the $m$th and $n$th ($1\leq m,n\leq N$) sectors from the
objective images and the scattering-background images respectively, and use
them as the inputs to the flowchart in Figure 7. Thus, we obtain
$k(I^{\prime})$, $b(I^{\prime})$, and their linear correlation coefficients,
denoted as $cor^{k}_{m,n},cor^{b}_{m,n}$. By iterating through all values of
$m$ and $n$, a total of $2N^{2}$ correlation coefficients can be obtained.

The distance between the two circular sectors is defined as follows:

$\mbox{Distance}=\left\\{\begin{array}[]{lr}n-m&m\leq n\\\
n-m+N&m>n\end{array}\right..$ (6)

We compute the average correlation coefficient versus the distance of all
circular sectors pairs with N=360, i.e. the distance is equal to the
difference of the central position angles of the two circular sectors involved
($\phi$), see Figure 9.

Figure 9.: The distribution of average correlation coefficient between the
total intensity of the scattering points in the circular sectors and the
parameters $k$ (blue), $b$ (orange) of the scattering background versus the
distance of the two sectors involved. $N$ is set to 360.

It can be seen that the largest correlation occurs when the distance between
the two circular sectors is approximately $180^{\circ}$. The weakest
correlation corresponds to a distance of about zero ($\phi\approx 0$). This
antisymmetric correlation is attributed to the disparity in imaging directions
between the objective and the corona. We conclude that the scattering
background in any given direction $\theta$ is contributed to varying degrees
by the intensity of scattering points in all angular directions.

A sinusoidal function is fitted to the curves in Figure 9 to determine an
ideal maximum value at $\phi=191^{\circ}$. We then modify the parameter $I$ in
Equation 4 as follows:

$I_{\theta}(\theta)=I-\sum_{\phi=1}^{360}I_{\theta+\phi}\times\sin(\theta+\phi+79^{\circ}),$
(7)

where $I_{\theta+\phi}$ represents the total intensity of scattering points in
the $(\theta+\phi)$th circular sector. Then, Equation 4 can be rewritten as:

$\mbox{Scatter}(r,I_{\theta}(\theta))=(0.99-0.43\ r/\mathrm{R_{\odot}})\times
10^{-7}I_{\theta}(\theta)/s+0.17-0.1\ r/\mathrm{R_{\odot}}.$ (8)

##  4 Results

###  4.1 Model Validation

Figure 10.: Corrected coronal data. Panels of the first row show the original
data, the same as Figures 4(a)$\sim$(c). Panels in the second and third rows
represent the images corrected using the two- and three-parameter models,
respectively. Note that, for the structures enclosed by the dashed boxes,
panel g exhibits a superior detail structure compared to panel d, indicating a
more effective correction by the three-parameter model. This is more evident
in Figure 11.

The two models described in Section 3 were separately applied to correct the
coronal images, as shown in Figure 10. The pixel values of the coronal images
processed by both models fall within a similar range, suggesting the
effectiveness of the models in removing an important proportion of the
superimposed dust-scattering background. Additionally, the second model
outperforms in capturing finer details, compare Figures 10a, d, and g, whereas
the remaining sets of corrected images exhibit no significant difference. This
indicates that the advantage of the three-parameter model becomes more
prominent under high dust concentrations.

More clearly, we divide the corrected image by the cleanest coronal image
($C_{3}/s_{3}$) , as shown in Figure 11. The images of the first row
represents the error of the raw data, with most of the areas in red indicating
a large amount of scattering background in the raw data. The following two
rows represent the results of the correction using two models, with the
original red area turning yellow or blue, indicating that the correction
effectively removes the scattering background and improves the signal-to-noise
ratio.

When there is a large amount of dust, a large blue area appears in the
corrected image, which is relatively small in the three-parameter model
corrected images. However, the colors in Figures 11f and i are slightly
bluish, indicating an overcorrection. This is due to the fact that $O_{3}$ is
not completely clean in the experiment, which increases the error of the model
when there is very little dust. In future experiments, we will find ways to
improve the cleanliness of $O_{3}$ and improve the accuracy of the model.

Figure 11.: Ratio of the raw and corrected coronal image in Figure 10 by the
quasidust-free coronal image ($C_{3}/s_{3}$). The occulter regions are masked.
The areas within the frames in panel g are yellow
($C_{0}^{\prime\prime}/s_{0}\approx C_{3}/s_{3}$), while in panels (a) and (d)
they are red ($C_{0}/s_{0}\gg C_{3}/s_{3}$) and blue ($C_{0}^{\prime}/s_{0}\ll
C_{3}/s_{3}$), respectively. The regions enclosed by the black dashed line are
used to calculate the mean relative error in Figure 12.

The differences between the corrected and the quasidust-free ($C_{3}$) coronal
images are quantified by the mean relative error with respect to $C_{3}/s_{3}$
. It can be reflected by the pixel values in the low-latitude ($<45^{\circ}$)
region within $1.2\ \rm R_{\odot}$ of Figure 11 (the regions enclosed by the
black dashed line in Figure 11). We show the mean relative error with respect
to $C_{3}/s_{3}$, see Figure 12. The average value after correction is greatly
reduced and close to 1 but slightly less than 1, indicating overcorrection.
When there is a large amount of dust, e.g., $C_{0}$, the three-parameter model
can better reduce the variance of the correction results. When the amount of
dust is small, there is no significant difference in the results between the
two models. It should be noted that this value also includes any error
produced by intrinsic changes in the solar corona and by estimating $s_{i}$.

Figure 12.: The boxplot of the relative error with respect to $C_{3}/s_{3}$.
It is calculated from the pixel values of the areas enclosed by the black
dashed line in Figure 11. The line in the middle of the box represents the
average value, and the top and bottom sides of the box represent the 25th and
75th percentile values. The left panel shows the errors of the raw data and
the blue and red boxes in the right panel represent the correction results of
the two- and three-parameter models, respectively.

As a more quantitative comparison, the angular variation of intensity at
$r=1.08\ \rm R_{\odot}$ was extracted for each image in Figure 10, see Figure
13.

Figure 13.: Angular intensity distribution at $r=1.08\ \rm R_{\odot}$ in the
coronal images of each row in Figure 10. The dashed line represents the corona
obtained when the objective lens is at the cleanest ($C_{3}$), while the red,
green, and blue lines correspond to $C_{0}$, $C_{1}$, and $C_{2}$. The
leftmost panel represents the original data, while the middle and rightmost
panels present the results corrected using the two- and three-parameter
models, respectively. In the latter two, the corrected intensity shows a
significant reduction and better matches the intensity of the solar corona,
e.g., see the red arrows.

After removing the scattering background, the angular distribution becomes
consistent with $C_{3}/s_{3}$, which is the cleanest condition. Additionally,
they are more consistent when applying the three-parameter model correction,
which not only validates the effectiveness of the model but also emphasizes
the necessity of using the three-parameter model.

Future observations hold the potential for enhancing data utilization and
accuracy by periodically capturing images of the objective lens to obtain
$I(\theta)$ and correcting the coronal data using Equation 8.

###  4.2 Radial Distribution of Dust

Given that different angular distributions of scattering points can result in
different scattering backgrounds, it is expected that their different radial
distributions will also have varying impacts. Similar to the approach
described in Section 3.2, in this case, the objective image is no longer
divided into circular sectors, but into concentric rings. The intensity of
scattering points per unit area in different concentric rings is calculated
separately, followed by determining the correlation between these intensities
and the parameters of the scattering background, $k$ and $b$, as illustrated
in Figure 14.

Figure 14.: The distribution of the correlation coefficients between the total
intensity of the scattering points in concentric rings of different radii and
the parameters $k$ and $b$ of the scattering background. Recall that the
objective lens diameter is 10 cm.

It can be seen that the correlation between the scattering background and the
scattering points, within a radius of approximately 2.5 cm from the center of
the objective lens, is very high, approaching a value of 1. Conversely, as the
radius of the ring increases beyond about 2.5 cm, the correlation with the
scattering background progressively diminishes. This suggests that dust
particles in the outer region of the objective lens contribute less to the
scattering background.

##  5 Conclusion

We evaluate the influence of dust particles on coronagraph objective lenses on
coronal imaging and correct the coronal data using empirical models to enhance
its quality. YOGIS was used to capture images of dust-induced scattering
points and the scattering background on the coronal image. The connection
between them was analyzed using two models. The first two-parameter model
describes a radially symmetric dust-scattering background based only on the
radial position and the intensity obtained from the objective-lens image. The
three-parameter model, also takes into account the angular distribution of
scattering points, thereby allowing us to establish a connection between the
scattering background and height, angle, and intensity of the scattering
points (Equation 8).

We conclude that it is advantageous to periodically capture objective images
to obtain $I(\theta)$ and then correct the coronal data using Equation 8. The
quality of the resulting coronal data can be improved to enhance coronal
structures, which is beneficial for the discovery of coronal mass ejections
(CMEs), low-speed out flows (e.g. Xia, Marsch, and Curdt, 2003; Marsch,
Wiegelmann, and Xia, 2004; Tian et al., 2021) , among others. We consider that
this model is an important step in the precise calibration of large ground-
based coronagraphs. Our technique can be particularly useful to support future
routine measurements of the coronal magnetic field using coronagraphs (e.g.
Liu and Lin, 2008; Liu, 2009; Liu, Lin, and Kuhn, 2010).

In addition, we note that the dust on the inner part of the objective
contributes more to the scattering background, which can serve as a valuable
guide for the development of superior scattering-background models in the
future.

The findings of this study highlight the importance of dust on the objective
lens and its impact on the stray light of YOGIS. The proposed method involves
capturing images of the corona and the objective lens, making it a simple and
clear process. This technique can be applied to various internally occulted
coronagraphs and serves as a crucial reference for accurate calibration of
images obtained with other ground-based coronagraphs. This technique may be
also applicable to other optical astronomical instruments. It should be noted
that the actual calibration images presented in this study are only applicable
to YOGIS due to variations in optical systems, filters, cameras, and other
factors that differ among coronagraphs. As a result, the conclusions derived
from this study cannot be directly extrapolated to other coronagraphs.
Nevertheless, conducting experiments using the methodology outlined in this
paper can provide calibrations applicable to other instruments.

The upcoming 20-cm coronagraph of the Chinese Meridian Project Phase II, which
will be stationed at the Lijiang Observatory, is also considering adding an
objective-lens imaging system to perform simultaneous lens imaging during
coronal observations. Furthermore, analyzing the scattered light generated by
dust particles on the objective lens can also be used to diagnose the current
level of stray light in the instrument.

To improve the quantity and accuracy of data in this experiment, we need to
address the limitations in the observing conditions. One way to do this is by
minimizing the duration of each set of experiments to reduce the effect of
solar rotation and the evolution of the corona. We also plan to conduct
experiments at the future coronagraph site in Daocheng, Sichuan province,
which is situated at a higher altitude of 4700 m (Liu et al., 2018). This can
help to gather more high-quality data and enhance the calibration accuracy.

#### Acknowledgements

We sincerely thank Prof. T. Sakurai-san and the NAOJ team for the 10 cm
coronagraph collaboration. This work is supported by the National Natural
Science Foundation of China under grants (12173086, 11873090, 12373063,
11533009), the Light of West China program of CAS and Yunnan Key Laboratory of
the Solar Physics and Space Science [202205AG070009], and the National Key R&D
Program of China No. 2022YFF0503800. We acknowledge the data resources from
the National Space Science Data Center, National Science & Technology
Infrastructure of China (http://www.nssdc.ac.cn).

## Author Contribution

Feiyang Sha, Xuefei Zhang, and Tengfei Song conducted the experiments. Feiyang
Sha, Yu Liu, and Xuefei Zhang wrote the main manuscript text. Feiyang Sha
prepared the figures. All authors reviewed the manuscript. Yu Liu supervised
this work and Xuefei Zhang partly assisted the assembly and guidance of the
instruments in Lijiang Observatory.

## Data Availability

The data that support the findings of this study are available from the
authors, upon reasonable request.

## Conflict of interest

We declare that we have no conflict of interest.

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