# Status of the DEAP-3600 experiment

Marcin Kuźniak for the DEAP-3600 collaboration  AstroCeNT, Nicolaus Copernicus
Astronomical Center of the Polish Academy of Sciences, Rektorska 4, 00-614
Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract

DEAP-3600 is a single-phase liquid argon (LAr) dark matter detector, located 2
km underground at SNOLAB in Sudbury, Canada, which started taking data in
2016. The detector is sensitive to nuclear recoils induced by scattering of
dark matter particles, which would cause emission of prompt scintillation
light. DEAP-3600 demonstrated excellent performance, holds the leading WIMP
exclusion among LAr detectors, and published several physics results. The WIMP
sensitivity of the detector is currently limited by backgrounds induced by
alpha activity at the LAr inlet, in a shadowed region of detector. The ongoing
hardware upgrade aims at fixing that limitation and, in consequence, at
reaching the full WIMP sensitivity. This paper summarizes the latest results
from DEAP-3600 and details of the upgrade.

##  1 Introduction

The existence of dark matter is well established through astronomical
observations, by a number of independent data sources and with complementary
techniques. A new, as of yet undetected, type of particle is a candidate for
dark matter, with Weakly Interacting Massive Particle (WIMP) being one of the
favored scenarios. Direct detection of such particles is attempted by looking
for evidence of their interactions with ordinary matter. A number of ultra-low
background detectors used for this purpose are located in deep underground
laboratories, in order to evade backgrounds induced by cosmic rays. For a
detailed discussion of the motivation and status of direct experimental
searches, see the recent review [1].

##  2 The DEAP-3600 detector

DEAP-3600 was built to register scintillation light induced by elastic
scattering of WIMPs on argon nuclei within a 3.3-tonne liquid argon (LAr)
target, contained in a spherical acrylic vessel (AV) [2]. The AV is coated on
the inside with tetraphenyl butadiene (TPB) wavelength shifter (WLS), so that
the LAr scintillation light, peaked at 128 nm i.e. in vacuum ultraviolet
(VUV), is converted to visible for efficient detection with 255
photomultiplier tubes (PMTs) surrounding the AV.

In order to reduce backgrounds from natural radioactivity, mainly neutrons and
$\upalpha$ particles, the detector is: (1) constructed from carefully selected
radiopure materials, (2) shielded from ambient radioactivity by a water tank
instrumented with PMTs, which also serves as a Cherenkov muon veto, and (3)
located underground at SNOLAB, where the muon flux is low.

$\upgamma$-induced electron recoil (ER) events are rejected with pulse-shape
discrimination (PSD), which is exceptionally powerful in LAr because of very
significant difference between lifetimes of the excited LAr triplet ($\sim$1.5
$\upmu$s) and singlet ($\sim$6 ns) states, which are preferentially excited by
ER and nuclear recoils, respectively. This property of LAr, together with its
high scintillation yield, ease of purification and high natural abundance,
makes it an ideal and scalable target for large dark matter detectors.

In its second run the DEAP-3600 collected data from November 2016 until March
2020, when it was emptied of LAr in order to prepare it for a COVID-19 related
period of reduced maintenance, as well as for a planned hardware upgrade. The
physics data collection remains 80% blind in the WIMP search region of
interest (ROI) since January 2018.

##  3 Recent physics highlights

Since the commissioning phase in 2016, DEAP-3600 holds the leading spin-
independent WIMP-nucleon scattering cross section upper limit measured in a
LAr target [3, 4]. The latest null result from 231 live days (exposure of 758
tonne-days) was 3.9$\times$10-45 cm2 (1.5$\times$10-44 cm2) for a 100 GeV/c2
(1 TeV/c2) WIMP mass at 90% C. L., assuming the Standard Halo Model (SHM) and
isospin parity conservation.

Recently, this result has been re-interpreted from the standpoint of Effective
Field Theory (EFT) and astrophysical uncertainties on the dark matter halo
substructure [5]. In addition to studying the effects of stellar streams and
debris flows on the DM velocity spectrum and the resulting WIMP exclusions,
the DEAP-3600 result has been also translated to upper limits on coupling
strength of different non-relativistic EFT (NREFT) contact interaction
operators, isovector couplings, as well as millicharge, magnetic dipole,
electric dipole and anapole WIMP-nucleon interactions, see Fig. 1(left).
Isospin-violating scenarios have also been studied, including the so-called
xenonphobic (XP) coupling scenario, where WIMP-proton and WIMP-neutron
interactions are such that the DEAP-3600 limits are world leading at high WIMP
mass, see Fig. 1(right).

Figure 1: (Left) Exclusion curves on the coupling strength of photon-mediated
interactions: anapole, magnetic dipole, electric dipole and millicharged DM
using the SHM. (Right) Constraints on the O1 interaction from XENON1T and
DEAP-3600, for IV (isovector; solid) and XP (xenonphobic; dashed) scenarios
from non-relativistic EFT (NREFT). For more details see Ref. [5].

The full available dataset, 813 live days, was employed for a blind search of
Planck-scale mass dark matter probing so far unexplored parameter space [6].
Multi-site scatter in the detector was the key signal signature for such
super-heavy particles with high scattering cross section. The null search
result permitted setting new leading limits, see Fig. 2.

Figure 2: Exclusions at 90% C. L. set in the Planck scale mass dark matter
search, compared with results from other experiments, for two theoretical
models: (Left) opaque dark matter candidate, relevant for composite DM models,
(Right) assuming the coherent scattering cross section, scaling with A4. For
more details, see Ref. [6].

##  4 Backgrounds

The DEAP collaboration has pioneered the use of pulse-shape discrimination in
LAr for dark matter detection, essential because of the presence of 0.95 Bq/kg
activity of cosmogenically produced 39Ar in atmospheric Ar. More recently, a
complete model of the LAr scintillation event pulse shape was developed,
taking into account the LAr time constants, WLS re-emission time profile and
PMT-related noise contributions from, primarily, charge resolution, dark noise
rate and afterpulsing [7]. That model, combined with an in-situ calibration of
noise characteristics of each individual PMT and a Bayesian single
photoelectron counting algorithm allowed for an unprecedented PSD performance,
resulting in a fraction of $\upbeta$ events misidentified as NR of 10-9 at the
16 keV threshold [8]. Using demonstrated reduction in 39Ar from underground
argon, this is projected to $<$0.2 events in 3000 tonne-years in a large
single-phase detector (such as ARGO [9]).

The ER background model, including $\upbeta$ and $\upgamma$ from internal and
external radioactivity, informed by ex-situ assay data, was shown to be close
to the observed spectrum. A multi-component fit, which achieved good agreement
with data over nine orders of magnitude in event rate, allowed to measure each
contribution, confirm assumptions informing the neutron rate expectation, and,
as a by-product, provide a precise measurement of the 42Ar specific activity
in the atmosphere [10].

The neutron background is dominated by ($\upalpha$, n) reaction in the
detector materials, mainly the PMT glass. The neutron rate is estimated in-
situ using a delayed coincidence analysis searching for neutron capture gammas
following the NR, resulting in 0.10${}^{+0.10}_{-0.09}$ expected number of
neutron background events in the last WIMP search [4].

$\upalpha$ activity itself, from Rn-borne 210Pb/210Po present in the surface
layer of the AV since the detector construction, is efficiently removed with
position reconstruction, leading to an expectation of $<$0.08 background
events. DEAP-3600 has also demonstrated the lowest so far among noble liquid
detectors rate of suspended Rn events from constant emanation, approx. 0.15
$\upmu$Bq/kg (4 nBq/kg) for 222Rn (220Rn) [4].

The above demonstrates excellent control over conventional types of
backgrounds, achieved with a single-phase detector, which makes it an
attractive technology for scale-up. The remaining special two classes of
$\upalpha$ backgrounds, will be mitigated with the ongoing hardware upgrade.

The main limiting background for the previous DEAP-3600 result was the
$\upalpha$ activity in the LAr inlet on top of the detector, the so-called
neck. While filled with gaseous Ar, it contains enough LAr film on the
surfaces and/or mist for $\upalpha$’s to induce prompt scintillation light,
which is shadowed by the acrylic flowguides, resulting in misreconstructed
events with apparent low energy. A detailed Monte Carlo model of such events
has been developed, showing excellent agreement with the data and allowing for
reliable extrapolations to the ROI. While a number of topological features can
be used in the analysis to remove such events, it is only achieved at a cost
of significantly reduced signal acceptance [4].

Another issue is the evidence of $\upalpha$ activity in the dust particulates
present in the LAr volume. This results in events with $\upalpha$ energy
degraded in the particulate material, and a continuum of events extending to
low energies, dependent on the particulate size distribution and material. A
power law distribution describes the data very well at intermediate energies,
however, a reliable extrapolation to the WIMP ROI will be made once the
systematic uncertainty is constrained by characterization of the dust
extracted from the detector. Similarly to neck events, topological features of
dust events are explored with multivariate analysis, due to the expected
effect of partial shadowing of the scintillation light by dust particulates.

##  5 Hardware upgrade

While analysis tools are being refined for better rejection of $\upalpha$
backgrounds in the dataset already collected, a hardware upgrade of the
detector aimed at eliminating the issue, in addition to the necessary
maintenance on cryogenic systems, is currently in progress. For the neck
backgrounds two independent preventive measures have been proposed.

Firstly, a new set of acrylic flowguides has been machined and sanded in low-
Rn atmosphere, and subsequently coated with a WLS with a distinctly slow re-
emission time constant. In consequence, the VUV photons from $\upalpha$
scintillation directly reaching the inside of the detector through the opening
in the neck, will be dominated by delayed visible photons from the slow WLS,
enabling efficient PSD, and without significant effects on the WIMP signal
acceptance. The collaboration has optimized the composition and fully
characterized performance of such coatings, converging on pyrene-doped
polystyrene, which exhibits the light yield of 36-46% of TPB and the
fluorescence lifetime of 279(14) ns, confirmed at cryogenic conditions and VUV
excitation wavelengths, as well as successfully tested these for mechanical
robustness [11, 12, 13]. Based on Monte Carlo simulations, a leakage fraction
of $<$1.2$\times$10-5 is expected with an optimized PSD discriminant, more
than sufficient to remove such events from the WIMP search ROI.

Secondly, an alternate cooling and LAr delivery system has been designed and
fabricated in order to: (1) allow for extraction and filtration of dust
particulates from the detector and (2) increase the temperature of the neck
surfaces, preventing formation of the layer of liquid or mist in that region,
which additionally prevents occurrence of prompt, WIMP signal-like, LAr
scintillation induced by $\upalpha$’s. A stainless steel vacuum-jacketed pipe
deployed all the way to the bottom of the AV will be initially used for
extraction of LAr. Then, lifted to the top of the AV and connected to an
externally located argon liquefier, it will allow for injection of LAr into
the detector and operation with the previously-used neck cooling coil turned
off, resulting in the temperature of the neck increased above the Ar boiling
point.

Installation and commissioning are planned for Winter and Spring, followed by
filling the detector and data taking, starting from Summer 2022.

##  6 Summary and outlook

The DEAP-3600 detector has demonstrated the potential of single-phase
technology showing excellent performance and control over the key
$\upbeta$/$\upgamma$, neutron and surface $\upalpha$ backgrounds. The
sensitivity of the detector is currently limited by neck and dust particulate
$\upalpha$ backgrounds, which are being addressed in the ongoing detector
upgrade with slow WLS coatings in the neck region and a particulate
extraction/filtration system; the upgraded detector will be able to reach its
design sensitivity. In parallel, multivariate analysis techniques are being
employed to improve the sensitivity of the WIMP search using the dataset
already collected; which will be informed by the characterization of the dust
sample extracted from the detector.

Recent analysis highlights include the NREFT-based reinterpretation of the
last WIMP exclusion, as well as leading sensitivity constraints on the Planck
scale mass dark matter.

The collaboration is working on a number of other physics analyses, including
the precision measurement of specific activity of 39Ar in atmospheric argon
and the first measurement of the 8B solar neutrino absorption.

The detector is expected to resume data taking in Summer 2022 and will proceed
with the physics run, as the currently most sensitive LAr detector. In
addition to probing new physics, DEAP-3600, as a single-phase detector, is an
important test-bed that will inform the scale up to the future ultimate LAr
experiment, capable of probing the WIMP paramter space down to the neutrino
floor [9]. We thank the Natural Sciences and Engineering Research Council of
Canada, the Canadian Foundation for Innovation (CFI), the Ontario Ministry of
Research and Innovation (MRI), and Alberta Advanced Education and Technology
(ASRIP), Queen’s University, the University of Alberta, Carleton University,
the Canada First Research Excellence Fund, the Arthur B. McDonald Canadian
Astroparticle Research Institute, DGAPA-UNAM (PAPIIT No. IN108020) and Consejo
Nacional de Ciencia y Tecnología (CONACyT, Mexico, Grant A1-S-8960), the
European Research Council Project (ERC StG 279980), the UK Science and
Technology Facilities Council (STFC) (ST/K002570/1 and ST/R002908/1), the
Russian Science Foundation (Grant No. 16-12-10369), the Leverhulme Trust
(ECF-20130496), the Spanish Ministry of Science and Innovation
(PID2019-109374GB-I00), the International Research Agenda Programme AstroCeNT
(MAB/2018/7) funded by the Foundation for Polish Science (FNP) from the
European Regional Development Fund, and from the European Union’s Horizon 2020
research and innovation programme (DarkWave, Grant No. 962480). Studentship
support from the Rutherford Appleton Laboratory Particle Physics Division,
STFC and SEPNet PhD is acknowledged. We thank SNOLAB and its staff for support
through underground space, logistical, and technical services. SNOLAB
operations are supported by the CFI and Province of Ontario MRI, with
underground access provided by Vale at the Creighton mine site. We thank Vale
for their continuing support, including the work of shipping the acrylic
vessel underground. We gratefully acknowledge the support of Compute Canada,
Calcul Québec, the Centre for Advanced Computing at Queen’s University, and
the Computation Centre for Particle and Astrophysics (C2PAP) at the Leibniz
Supercomputer Centre (LRZ) for providing the computing resources required to
undertake this work.

## References

## References

  * [1] Billard J et al. 2021 Preprint 2104.07634
  * [2] Amaudruz P-A et al. (DEAP Collaboration) 2019 Astropart. Phys. 108 1
  * [3] Amaudruz P-A et al. (DEAP Collaboration) 2018 Phys. Rev. Lett. 121 071801
  * [4] Ajaj R et al. (DEAP Collaboration) 2019 Phys. Rev. D 100 022004
  * [5] Adhikari P et al. (DEAP Collaboration) 2020 Phys. Rev. D 102 082001
  * [6] Adhikari P et al. (DEAP Collaboration) 2021 Preprint 2108.09405
  * [7] Adhikari P et al. (DEAP Collaboration) 2020 Eur. Phys. J. C 80 303
  * [8] Adhikari P et al. (DEAP Collaboration) 2021 Eur. Phys. J. C 81 823
  * [9] GADM Collaboration. Future Dark Matter Searches with Low-Radioactivity Argon, 2018.
  * [10] Ajaj R et al. (DEAP Collaboration) 2019 Phys. Rev. D 100 072009
  * [11] Clark M et al., presented at CAP 2016, https://indico.cern.ch/event/472838/contributions/1150228
  * [12] Gallacher D et al. 2021 Preprint 2109.06819
  * [13] Benmansour H et al. 2021 Preprint 2110.08103


# Disturbance-free single-pixel imaging camera based on P=0.5 Bernoulli
modulation and complementary detection

Wenlin Gong<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Optoelectronic
Science and Engineering, Soochow University, Suzhou 215006, China

###### Abstract

We present a technique called single-pixel imaging camera based on
complementary detection and optimized Bernoulli modulation (CSPI camera),
which can significantly reduce the influence of the disturbance light to
single-pixel imaging (SPI). The numerical simulation demonstrates that when
the probability of the value “1” is P=0.5 for each coded pattern, CSPI camera
is disturbance-free is still disturbance-free even if the intensity
fluctuation of the disturbance light is much larger than the signal’s
intensity. The reconstruction results of both SPI and differential SPI are
also compared. This technique of CSPI camera can dramatically promote real
application of single-pixel imaging Lidar.

###### pacs:

42.30.Va, 42.30 Wb, 42.50.Ar

##  I Introduction

Single-pixel imaging (SPI), as a computational imaging technique, has
attracted a lot of attention recently Cheng ; Bennink ; Cao ; Angelo ; Shapiro
; Graham-Rowe ; Duarte ; Edgar ; Quero . At present, there are two typical
schematics for SPI Edgar ; Quero . One is ghost imaging (GI), where the target
is illuminated by a series of speckle patterns and the photons reflected from
the target are collected onto a single-pixel detector Cheng ; Bennink ; Cao ;
Angelo ; Shapiro . The other is single-pixel camera (SPC), where the target is
usually imaged onto a spatial modulation device and the modulated signals are
received by a single-pixel detector Graham-Rowe ; Duarte . In comparison with
SPC, more investigations were focused on GI because of its potential
applications in remote sensing Zhao ; Erkmen ; Wang , three-dimensional
imaging Sun ; Gong ; Sun1 , phase imaging Gong1 ; Yu ; Vinu ; Peng , optical
encryption Clemente ; Li , and microwave imaging Wang1 . Recently, some works
have demonstrated that SPC has more advantages than GI in remote sensing Edgar
; Quero ; Mei ; Gong2 ; Sun2 . Firstly, the detection range of GI Lidar is
limited by the damage threshold of the modulation device, whereas there is no
need for SPC to be considered because the reflection signal is usually weak
Mei . Secondly, the quality of SPC is better than GI in the same light
disturbance environment Gong2 . Thirdly, the detection process of SPC usual
satisfies the standard modeling of compressive sensing, whereas the case of GI
is deviated because only the low-frequency information reflected from the
target is detected by the single-pixel detector in long-distance imaging Zhao
; Gong ; Quero ; Sun2 . In addition, the structure’s size of SPC is usually
smaller than that of GI. Therefore, the schematic of SPC may be more suitable
than that of GI in the area of Lidar remote sensing.

Digital micro-mirror device (DMD), as a high-speed light modulator, is widely
used for SPC system Duarte ; Edgar ; Quero . By controlling the micro-mirrors
of DMD, we can obtain a series of random binary coded patterns with different
statistical distribution and the property of coded patterns has a great effect
on the quality of SPC Quero ; Wang2 ; Zhou . For example, Hadamard coded
patterns, as a special case of Bernoulli modulation, can usually obtain better
reconstruction result compared with other random Bernoulli coded patterns
Wang2 ; Zhou . What’s more, because of the modulation property of DMD, the
quality of SPC can be also enhanced by complementary detection Yu1 . However,
different from traditional imaging, lots of measurements are required for the
image reconstruction of SPI and the intensity fluctuation of the
background/disturbance light will lead to the degeneration of SPI in the
sampling process especially for the application of SPI Lidar Gong2 ; Li1 ;
Deng . In order to reduce the influence of the background/disturbance light to
SPI Lidar, there are two approaches up to now. One is to increase the
transmitting energy of Lidar and to decay the intensity of the
background/disturbance light by spectral filtering. However, the enhancement
degree is usually confined in practical application. The other is based on
differential measurement, but it is valid only for the background/disturbance
light with gently intensity fluctuation Yang  and its intensity fluctuation
can not be much stronger than the signal’s intensity (we will verify it in
this manuscript). Furthermore, the intensity fluctuation of the
background/disturbance light is usually random in practice. Therefore, it is
imperative to develop a method of SPC against the background/disturbance light
where its intensity fluctuation is rapidly variable and even is much larger
than the signal’s intensity. In this paper, we propose a single-pixel imaging
camera based on complementary detection and optimized Bernoulli modulation
(called CSPI camera) that is disturbance-free when the background/disturbance
light illuminating on the DMD plane of CSPI camera is spatially uniform and
time-variable. Based on previous SPI research achievement, this work can
dramatically quicken real application of SPI Lidar.

##  II Model and image reconstruction

Figure 1: Proof-of-principle schematic of CSPI camera in light disturbance
environment. The disturbance light directly enters into the imaging system and
uniformly illuminates the DMD, but its intensity is time-variation.

Fig. 1 presents the proof-of-principle schematic of CSPI camera in light
disturbance environment. The light emitted from a pulsed laser uniformly
illuminates the target and the target is imaged onto a DMD by an optical
imaging system with the focal length $f_{1}$. At the same time, the
disturbance light directly enters into the imaging system and illuminates the
DMD. By controlling the mirrors of the DMD, both the disturbance light and the
target’s image are modulated, and then the photons reflected by the DMD are
collected onto two single-pixel detectors $D_{\rm{up}}$ and $D_{\rm{down}}$ by
using another two conventional imaging system with the focal length $f_{3}$,
respectively. According to the property of DMD, the patterns at the plane of
the detectors $D_{\rm{up}}$ and $D_{\rm{down}}$ are complementary. We
emphasize that the patterns modulated by the DMD obey Bernoulli distribution.
In addition, the intensity of the pulsed laser on the target plane is stable,
and the disturbance light on the DMD plane is spatially uniform but its
intensity is different for each measurement. The intensity $Y_{\rm{up}}^{i}$
recorded by the detector $D_{\rm{up}}$ can be represented as Goodman

$\displaystyle
Y_{\rm{up}}^{i}=\int{A^{i}(x)\left({I_{0}T(x)+I_{b}^{i}}\right)}dx+I_{n-up}^{i},{\rm{\
}}\forall_{i}=1\cdots K,$ (1)

where $A^{i}(x)$ denotes the distribution of the pattern modulated by DMD for
the $i$th measurement and $T(x)$ is the intensity reflection function of the
target. In addition, $K$ is the total measurement number and $I_{0}$ is the
intensity of the pulsed laser on the target plane. $I_{b}^{i}$ and
$I_{n-up}^{i}$ are the intensity of the disturbance light and the detection
noise of the detector $D_{\rm{up}}$ for the $i$th measurement, respectively.

Because the detection process is complementary, the intensity
$Y_{\rm{down}}^{i}$ recorded by the detector $D_{\rm{down}}$ can be described
as

$\displaystyle
Y_{\rm{down}}^{i}=\int{\left({1-A^{i}(x)}\right)\left({I_{0}T(x)+I_{b}^{i}}\right)}dx+I_{n-down}^{i},{\rm{\
}}\forall_{i}=1\cdots K,$ (2)

here $I_{n-down}^{i}$ is the detection noise of the detector $D_{\rm{down}}$
for the $i$th measurement.

According to the principle of SPI, the target’s image $O_{\rm{SPI}}$ can be
reconstructed by computing the correlation function between the pattern’s
intensity distributions $A^{i}(x)$ modulated by the DMD and the intensities
$Y^{i}$ recorded by the detector Cheng ; Angelo ; Gong2 ; Quero

$\displaystyle
O_{\rm{SPI}}(x)=\frac{1}{{K}}\sum\limits_{i=1}^{K}\left({A^{i}(x)-\left\langle{A(x)}\right\rangle}\right)Y^{i}.$
(3)

where
${\left\langle{A(x)}\right\rangle}=\frac{1}{{K}}\sum\limits_{s=1}^{K}A^{i}(x)$
represents the ensemble average of $A^{i}(x)$.

For CSPI, the image can be achieved by computing the correlation function
between the pattern’s intensity distributions $A^{i}(x)$ and the intensities
of $Y_{\rm{up}}^{i}-Y_{\rm{down}}^{i}$, namely

$\displaystyle
O_{\rm{CSPI}}(x)=\frac{1}{{K}}\sum\limits_{i=1}^{K}\left({A^{i}(x)-\left\langle{A(x)}\right\rangle}\right)Y_{\rm{CSPI}}^{i}.$
(4)

where we denote $Y_{\rm{CSPI}}^{i}=Y_{\rm{up}}^{i}-Y_{\rm{down}}^{i}$, namely

$\displaystyle
Y_{\rm{CSPI}}^{i}=I_{0}\int{\left({2A^{i}(x)-1}\right)T(x)}dx+I_{b}^{i}\int{\left({2A^{i}(x)-1}\right)}dx+I_{n-up}^{i}-I_{n-down}^{i},{\rm{\
}}\forall_{i}=1\cdots K,$ (5)

From Eq. (5), we find that if the speckle patterns $A^{i}(x)$ obey to
Bernoulli distribution (supposed that the probability of the value “1” is $P$
for each speckle pattern, then the probability of the value “0” is 1-$P$) and
$P$=0.5, then the second item of Eq. (5) is 0 and Eq. (5) can be expressed as

$\displaystyle
Y_{\rm{CSPI}}^{i}=I_{0}\int{\left({2A^{i}(x)-1}\right)T(x)}dx+I_{n-up}^{i}-I_{n-down}^{i},{\rm{\
}}\forall_{i}=1\cdots K,$ (6)

It means that the disturbance light has no influence to the quality of CSPI in
this special case.

In Ref. Yang , the optical background noise can be restrained by instant ghost
imaging to some extent. To compared with the result of CSPI, we introduce this
method into the reconstruction of SPI, which is called differential single-
pixel imaging (DSPI) in this manuscript, the reconstruction process can be
described as

$\displaystyle
O_{\rm{DSPI}}(x)=\frac{1}{{K-1}}\sum\limits_{i=1}^{K-1}\left({A^{i+1}(x)-A^{i}(x)}\right)Y_{\rm{DSPI}}^{i}.$
(7)

where $Y_{\rm{DSPI}}^{i}=Y^{i+1}-Y^{i}$.

In order to evaluate quantitatively the quality of images reconstructed by the
methods described above, the reconstruction fidelity is estimated by
calculating the peak signal-to-noise ratio (PSNR):

$\displaystyle{\rm{PSNR}}=10\times\log_{10}\left[{\frac{{(2^{p}-1)^{2}}}{{{\rm{MSE}}}}}\right].$
(8)

where the bigger the value PSNR is, the better the quality of the recovered
image is. For a 0$\sim$255 gray-scale image, $p$=8 and MSE represents the mean
square error of the reconstruction images $O_{\rm{rec}}$ with respect to the
original object $O$, namely

$\displaystyle{\rm{MSE}}=\frac{1}{{N_{pix}}}\sum\limits_{i=1}^{N_{pix}}{\left[{O_{{\rm{rec}}}(x_{i})-O(x_{i})}\right]}^{2}.$
(9)

where $N_{pix}$ is the total pixel number of the image.

##  III Simulated demonstration

Figure 2: The influence of the disturbance light with sinusoidal modulation to
the detection signals and different SPI methods, when the irradiation SNR
$\varepsilon$ is 10 dB. (a) The target’s detection signal and reconstruction
result in the case of no detection noise and no light disturbance; (b) the
time-variation intensity of the disturbance light with sinusoidal modulation;
(c) the signal recorded by the detector $D_{\rm{up}}$ and corresponding
reconstruction result; (d) the detection signal based on the method of DSPI
and its reconstruction result; (e) the detection signal based on complementary
detection and CSPI reconstruction result.

Figure 3: The effect of the intensity of sinusoidal disturbance light on
different SPI reconstruction methods. (a) $\varepsilon$=-10 dB; (b)
$\varepsilon$=0 dB; (c) $\varepsilon$=10 dB; (d) $\varepsilon$=15 dB; (e)
$\varepsilon$=20 dB; (f) $\varepsilon$=25 dB; (g) the curve of
PSNR-$\varepsilon$.

Figure 4: The influence of the disturbance light with Gaussian modulation to
the detection signals and different SPI reconstruction methods in the case of
irradiation SNR $\varepsilon$=10 dB. The description of (a)-(e) is the same as
Fig. 2.

Figure 5: Simulated results of different SPI reconstruction methods in
different $\varepsilon$ for the disturbance light with Gaussian statistical
distribution. (a) $\varepsilon$=-10 dB; (b) $\varepsilon$=0 dB; (c)
$\varepsilon$=10 dB; (d) $\varepsilon$=15 dB; (e) $\varepsilon$=20 dB; (f)
$\varepsilon$=25 dB; (g) the curve of PSNR-$\varepsilon$.

Figure 6: Simulated results of CSPI in different irradiation SNR $\varepsilon$
when the patterns $A^{i}(x)$ is a random bernoulli distribution and the
probability of the value “1” is P=0.4, 0.45 and 0.5, respectively. (a)
$\varepsilon$=-5 dB; (b) $\varepsilon$=0 dB; (c) $\varepsilon$=5 dB; (d)
$\varepsilon$=10 dB; (e) $\varepsilon$=15 dB; (f) $\varepsilon$=20 dB; (g) the
curve of PSNR-$\varepsilon$.

Figure 7: Simulated demonstration of imaging a scene (“tower”) in different
irradiation SNR $\varepsilon$ when the patterns $A^{i}(x)$ is the same as Fig.
5 and the detection SNR $\delta$ is 30 dB. (a) the scene; (b)
$\varepsilon$=-20 dB; (c) $\varepsilon$=-10 dB; (d) $\varepsilon$=10 dB; (e)
$\varepsilon$=20 dB.

To verify the concept, the parameters of numerical simulation based on the
schematic of Fig. 1 are set as follows: the wavelength of the laser is 532 nm,
the transverse size of the patterns at the DMD plane is set as 27.3 $\mu$m and
the modulated area of the DMD is 64$\times$64 pixels (one pixel is equal to
the pattern’s transverse size). In addition,
$z_{11}$=$z_{12}$=$z_{21}$=$z_{22}$=200 mm, $f_{1}$=$f_{2}$=100 mm. The
imaging target, as illustrated in Fig. 2(a), is a gray-scale object (“siom”,
64$\times$64 pixels). According to Eq. (1) and Eq. (2), the detection SNR
$\delta=\frac{{\left\langle{\int{A^{i}(x)I_{0}T(x)dx}}\right\rangle}}{{std(I_{n-up}^{i})}}$
denotes the signal power to the standard deviation of the noise power ratio,
and the irradiation SNR $\varepsilon=\frac{{I_{0}}}{{std(I_{b}^{i})}}$ denotes
the signal power to the standard deviation of the light disturbance power
ratio. The speckle patterns modulated by the DMD obey Bernoulli distribution.
In order to improve the sampling efficiency, the Bernoulli speckle patterns
are arranged as the type of Hadamard codes (where the position of the value
“-1” is set as 0) and the first pattern (namely the value of all the element
is 1) is removed for the demonstration of Fig. 2-Fig. 5. Therefore, the
measurement number used for this special case is $K$=4095. What’s more, the
detection SNR $\delta$ is set as 20 dB. Fig. 2 has given the detection signal
and the reconstruction results of SPI/DSPI/CSPI when the disturbance light
obeys sinusoidal modulation and the irradiation SNR $\varepsilon$ is 10 dB.
When there is no detection noise and no light disturbance in the detection
process, both the ideal detection signal and the target’s reconstruction image
are shown in Fig. 2(a), which is also corresponding to the target’s original
image. Fig. 2(b) and Fig. 2(c) present the time-variation intensity of the
disturbance light and the signal recorded by the detector $D_{\rm{up}}$. It is
clearly seen that the target’s signal (Fig. 2(a)) is nearly overwhelmed by the
disturbance light and traditional SPI reconstruction can not rebuild the
target’s image (Fig. 2(c)). The detection signals and reconstruction images
based on the methods of DSPI and CSPI are displayed in Fig. 2(d) and Fig.
2(e), respectively. By computing the correlation coefficient $\beta$ between
the signal $Y_{\rm{s}}$ and the signals $Y_{\rm{DSPI}}$/$Y_{\rm{CSPI}}$Mei ,
the value $\beta$ of DSPI is 0.659 whereas it approaches to 1 for CSPI, which
means that the effect of the disturbance light is wiped off by CSPI method.
Similar to the result described in Ref. Yang , DSPI can improve the
reconstruction image to some extent in comparison with traditional SPI for
light disturbance with slowly time-variation. However, the target’s image can
be stably reconstructed by CSPI. Furthermore, the reconstruction results of
SPI/DSPI/CSPI in different irradiation SNR $\varepsilon$ are shown in Fig. 3.
It is obviously observed that DSPI will be disabled in low $\varepsilon$
whereas the method of CSPI is always valid.

In practice, the intensity fluctuation of the disturbance light is usually
random. Similar to Fig. 2, Fig. 4 has presented the detection signals and the
reconstruction results of SPI/DSPI/CSPI when the intensity of the disturbance
light obeys Gaussian distribution and the irradiation SNR $\varepsilon$ is
also 10 dB. Different from the case described in Fig. 2, the signals
$Y_{\rm{DSPI}}$ and $Y_{\rm{s}}$ are hardly correlated, thus the image
reconstruction is disabled for DSPI. However, CSPI can still stably recover
the target’s image. Furthermore, Fig. 5 also illustrates simulated results of
the influence of the irradiation SNR $\varepsilon$ to SPI/DSPI/CSPI in the
condition of the disturbance light with Gaussian statistical distribution. It
is clearly seen that SPI is better than DSPI in large $\varepsilon$ and the
reconstruction result of CSPI is still perfect even if the irradiation SNR
$\varepsilon$ is lower than -10 dB.

The patterns $A^{i}(x)$ used in Fig. 2-Fig. 5 are a special case of Bernoulli
distribution. When the patterns $A^{i}(x)$ is a random Bernoulli distribution
and the probability P of the value “1” is changed, Fig. 6 has given the
simulation results of CSPI in different P and different irradiation SNR
$\varepsilon$. Here the simulation parameters are the same as Fig. 5 except
for the measurement number $K$=10000. We find that the quality of CSPI is the
best in the case of P=0.5 and it will be degraded as the increase of the
deviation from the value P=0.5. In addition, compared with the results of CSPI
described in Fig. 3 and Fig. 5, the reconstruction results in the case of
P=0.5 is worse even if the measurement number $K$ is much larger, which
originates from high efficiency in information acquisition for the Bernoulli
speckle patterns with Hadamard codes.

In order to validate the applicability of CSPI for complex scenes, Fig. 7
gives a simulation demonstration of imaging a tower captured by our camera
(Fig. 7(a)). Using the same simulation parameters as Fig. 5 except for the
detection SNR $\delta$=30 dB, the reconstruction results of CSPI are shown in
Fig. 7(b)-Fig. 7(e) when the irradiation SNR $\varepsilon$ is -20 dB, -10 dB,
10 dB and 20 dB, respectively, which is similar to the results described in
Fig. 5. Therefore, we demonstrate that CSPI is disturbance-free and can be
applied to SPI Lidar in the environment of strong background/disturbance
light.

##  IV Conclusion

In conclusion, we have proposed a technique called CSPI that can remove the
influence of the disturbance light to sing-pixel imaging. We also show that
the technique is always valid even if the intensity of the disturbance light
is much stronger than the signal’s intensity and its intensity fluctuation is
random and rapidly variable. This technique can quicken real application of
SPI Lidar and is useful to the imaging in the wavebands without cameras.

## Funding

Natural Science Research of Jiangsu Higher Education Institutions of China
(21KJA140001), Aeronautical Science Foundation of China (2020Z073012001), and
Startup Funding of Soochow University (NH15900221).

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