# Transport properties of hybrid single-bilayer graphene interfaces in
magnetic field

Nadia Benlakhouy<EMAIL_ADDRESS>Laboratory of Theoretical Physics,
Faculty of Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El Jadida,
Morocco  Ahmed Jellal<EMAIL_ADDRESS>Laboratory of Theoretical Physics,
Faculty of Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El Jadida,
Morocco  Canadian Quantum Research Center, 204-3002 32 Ave Vernon, BC V1T 2L7,
Canada  Michael Schreiber  Institut für Physik, Technische Universität,
D-09107 Chemnitz, Germany

###### Abstract

The electronic properties of a hybrid system made of single-bilayer graphene
structures subjected to a perpendicular magnetic field are studied for the
zigzag boundaries of the junction, zigzag-1 (ZZ1) and zigzag-2 (ZZ2). These
later examples exhibit different behaviors that have been investigated using
the continuum Dirac model. Our results reveal that the conductance depends on
the width of bilayer graphene for ZZ1 and shows maxima for ZZ2 as a function
of the magnetic field, in contrast to ZZ1. It is found that interfaces have
significant impacts on the transmission probability, with the confinement of
the ZZ1 boundary being more substantial than that of ZZ2.

###### pacs:

73.22.Pr, 72.80.Vp, 73.63.-b
Keywords: Graphene junctions, magnetic field, energy spectrum, transmission,
conductance, Klein tunneling.

##  I Introduction

Graphene is a two-dimensional, single-layer sheet of carbon atoms arranged in
a hexagonal lattice structure [1]. It is the thinnest, strongest, and most
conductive material known to science, with exceptional mechanical, electrical,
and thermal properties. Graphene has sparked widespread advancements in
various disciplines due to its potential applications [1, 2, 3, 4, 5]. Bilayer
graphene (BLG) is a material composed of two layers of graphene sheets stacked
on top of each other. The two layers are separated by a small interlayer
distance $\gamma_{1}$, and can exhibit different electronic properties
depending on the stacking order. When the two graphene layers are aligned in
the same direction, with their atoms directly on top of each other, it is
called AA-BLG. This results in perfect lattice symmetry, which gives rise to a
special electronic band structure that depends on the interlayer spacing [6,
7, 8]. Another interesting stacking is AB-BLG. In the AB-BLG configuration,
the atoms in the two layers are not aligned with each other, leading to a
slight variation in the electronic properties of the bilayer compared to
single-layer graphene (SLG). This results in the opening of a band gap in the
electronic structure, which can be tuned by applying an external electric
field [9]. The development of high-quality samples and theoretical and
experimental research into its distinctive electronic properties make AB-BLG
appealing for a variety of applications [10, 11, 12, 13, 14, 15, 16, 17, 18].

Recent studies have shown that junctions between regions of different numbers
of graphene layers, such as the SLG/BLG interfaces, can result in interesting
properties. For instance, in [19] the transmission probability through SLG/BLG
junction was estimated in the absence of a magnetic field. Theoretical
investigations on the transport characteristics of BLG with locally decoupled
graphene sheets have also been performed [12]. Moreover, a BLG flake
sandwiched between two single zigzag or armchair nanoribbons was studied, and
it was shown that oscillations in the conductance were seen at energies
greater than the interlayer coupling [20]. Recently, it was found that the
interface of these hybrid systems exhibits an unconventional Landau
quantization [21, 22]. Another study was devoted to the effects of an electric
bias and a perpendicular magnetic field on the electron energy spectrum in
SLG/BLG and BLG/SLG structures [23]. It is generally known that graphene has
two main types of edges: zigzag and armchair edges. Regarding the electrical
structure of finite-sized systems, it has been established that graphene with
zigzag edges exhibits localized states close to Fermi energy, but those with
armchair edges do not [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38]. As a consequence, the existence of an edge state results in notable
variations in the transport characteristics.

Various works on the edge states of the hybrid interface attracted our
interest. Inspired by the findings in Refs. [23], and [19], we investigate the
transport properties of SLG and AB-BLG junctions that can be created from the
block shown in Fig. 1. Our configuration consists of SLG, connected into an
AB-BLG subjected to a magnetic field. As seen in Fig. 1, this structure’s ZZ
junctions cannot have the same edge interface on both sides, therefore, they
always have a pair of distinct ZZ boundaries, denoted ZZ1 and ZZ2. When
considering this distinction, our conductance appears in the ZZ1 border to be
independent of energy and exhibits antiresonances that tend to be $0$ for
large values of the magnetic field, because of the coexistence of two
propagating channels. Our findings show that the conductance depends on the
width of the BLG for the highest energies, and the resonances become
significant and dependent on the magnetic field. Turning now to the ZZ2
boundary, the conductance as a function of the magnetic field shows maxima, in
contrast to the ZZ1 boundary, and by increasing the width of BLG and the Fermi
energy, the conductance shows oscillation in the ZZ2 feature. The transmission
probability is substantially influenced by boundaries and the ZZ1 boundary’s
confinement is more significant than the ZZ2 boundary’s.

The present paper is structured as follows. In Sec. II, we consider an SLG/AB-
BLG/SLG structure and use the full-band continuum model to establish the
energy spectrum. We introduce formulation to describe two kinds of zigzag
boundaries of the junction, zigzag-1 (ZZ1) and zigzag-2 (ZZ2) in Sec. III.
Sec. IV is devoted to the numerical analysis of our findings and comparison
with literature. In Sec. V, we summarize our main conclusions. In Appendix, we
mainly formulate the details of the transfer matrix method for the SLG and BLG
interfaces.

##  II THEORY AND MODEL

AB-BLG is a type of bilayer graphene in which the two layers are stacked in an
AB arrangement. In this stacking configuration, the carbon atoms in one layer
sit directly above the centers of the hexagons in the other layer, resulting
in a characteristic Bernal stacking pattern [39]. It contains $A_{1}$ and
$B_{1}$ atoms on layer $1$ and $A_{2}$ and $B_{2}$ on layer $2$, which are
connected by interlayer coupling $\gamma_{1}$.

Figure 1: (Color online) Schematic presentation of hybrid SLG/BLG interfaces
of types ZZ1 and ZZ2, consisting of SLG connected to an AB-BLG subjected to a
magnetic field $B$. The BLG width is $d$ (yellow region). Black thick lines
represent the bottom layer with $A_{1}$ (red) and $B_{1}$ (blue) sites,
whereas blue light lines represent the top layer with $A_{2}$ (white) and
$B_{2}$ (green) sites.

To achieve our goal, we consider the geometry depicted in Fig. 1. Then,
without taking into account the minor contributions of the other interlayer
couplings, the effective Hamiltonian in the vicinity of the $K$ valley is
given by

$\mathcal{H}_{\text{BLG}}=\begin{pmatrix}0&v_{F}\pi^{{\dagger}}&0&0\\\
v_{F}\pi&0&\gamma_{1}&0\\\ 0&\gamma_{1}&0&v_{F}\pi^{{\dagger}}\\\
0&0&v_{F}\pi&0\\\ \end{pmatrix}$ (1)

where $\pi=p_{x}+ip_{y}$, $v_{F}=10^{6}$ m/s is the Fermi velocity for
electrons in each graphene layer, and $\gamma_{1}=0.4$ eV is the nearest-
neighbor interlayer hopping term. The eigenstates of
$\mathcal{H}_{\text{BLG}}$ are four-component spinors
$\Psi(x,y)=\left[\psi_{A_{1}},\psi_{B_{1}},\psi_{A_{2}},\psi_{B_{2}}\right]^{\dagger}$.
Here $\dagger$ denotes the transpose of the row vector. In the presence of a
perpendicular magnetic field $\mathbf{B}=Be_{z}$, one substitutes the
canonical momentum $\mathbf{p}$ by the gauge-invariant kinetic momentum
$\mathbf{p}+e\mathbf{A}$ in Eq. 1. $\mathbf{A}=(0,Bx)$ is the vector potential
chosen in the Landau gauge.

We may solve the eigenvalue problem by separating the variables and writing
the eigenspinors as a plane wave in the $y$-direction because of the
conservation of $p_{y}$. As a result, we write
$\Psi(x,y)=e^{ik_{y}y}\psi(x,k_{y})$ and the envelope functions
$\psi(x,k_{y})\equiv\psi(X)$ depend only on a single combination of the
variables, $X=\frac{x}{\ell_{B}}+k_{y}\ell_{B}$, with $\ell_{B}=\sqrt{1/(eB)}$
being the magnetic length. They satisfy the eigenvalue equation

$\left(\begin{array}[]{cccc}0&\sqrt{2}\epsilon_{0}\hat{a}&0&0\\\
\sqrt{2}\epsilon_{0}\hat{a}^{\dagger}&0&\gamma_{1}&0\\\
0&\gamma_{1}&0&\sqrt{2}\epsilon_{0}\hat{a}\\\
0&0&\sqrt{2}\epsilon_{0}\hat{a}^{\dagger}&0\end{array}\right)\Psi=E\Psi,$ (2)

where the annihilation $\hat{a}=\frac{1}{\sqrt{2}}\left(X+\partial_{X}\right)$
and creation
$\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(-\partial_{X}+X\right)$ operators
are fulfilling the commutation relation
$[\hat{a},\hat{a}^{\dagger}]=\mathbb{1}$, with $\epsilon_{0}=v_{F}/\ell_{B}$.
Now from Eq. 2, we obtain four coupled equations as

$\displaystyle-i\sqrt{2}\epsilon_{0}\hat{a}\psi_{B_{1}}(X)$
$\displaystyle=E\psi_{A_{1}}(X),$ (3) $\displaystyle
i\sqrt{2}\epsilon_{0}\hat{a}^{\dagger}\psi_{A_{1}}(X)+\gamma_{1}\psi_{A_{2}}(X)$
$\displaystyle=E\psi_{B_{1}}(X),$ (4)
$\displaystyle-i\sqrt{2}\epsilon_{0}\hat{a}\psi_{B_{2}}(X)+\gamma_{1}\psi_{B_{1}}(X)$
$\displaystyle=E\psi_{A_{2}}(X),$ (5) $\displaystyle
i\sqrt{2}\epsilon_{0}\hat{a}^{\dagger}\psi_{A_{2}}(X)$
$\displaystyle=E\psi_{B_{2}}(X).$ (6)

By eliminating $\psi_{A_{1}}(X),\psi_{A_{2}}(X)$, and $\psi_{B_{2}}(X)$, we
get the fourth order differential equation

$\begin{array}[]{r}{\left[2\epsilon_{0}^{2}\hat{a}^{\dagger}\hat{a}-E^{2}\right]\left[2\epsilon_{0}^{2}\hat{a}\hat{a}^{\dagger}-E^{2}\right]\psi_{B_{1}}}(X)=\gamma_{1}E^{2}\psi_{B_{1}}(X),\end{array}$
(7)

or equivalently

$\left(\partial_{X}^{2}-X^{2}-1-2\lambda_{+}\right)\left(\partial_{X}^{2}-X^{2}-1-2\lambda_{-}\right)\psi_{B_{1}}(X)=0,$
(8)

where $\lambda_{\pm}$ defines the energy bands

$\lambda_{\pm}=-\frac{1}{2}+\frac{E^{2}}{2\epsilon_{0}^{2}}\pm\frac{\sqrt{\epsilon_{0}^{4}+\gamma_{1}^{2}E^{2}}}{2\epsilon_{0}^{2}}.$
(9)

Therefore, the general solution of Eq. 8 can be written in terms of Weber’s
parabolic cylinder function $D_{\lambda}(Z)$, with $Z=\sqrt{2}X$. Thus, we
have

$\psi_{B_{1}}(Z)=\psi_{B_{1}}^{+}(Z)+\psi_{B_{1}}^{-}(Z),$ (10)

with

$\displaystyle\psi_{B_{1}}^{+}(Z)=c_{+}D_{\lambda_{+}}(Z)+c_{-}D_{\lambda_{+}}(-Z),$
(11)
$\displaystyle\psi_{B_{1}}^{-}(Z)=d_{+}D_{\lambda_{-}}(Z)+d_{-}D_{\lambda_{-}}(-Z),$

with the constants $c_{\pm}$ and $d_{\pm}$. The rest of the components can
derived using the coupled equations.
$\psi_{A_{1}}(Z)=\psi_{A_{1}}^{+}(Z)+\psi_{A_{1}}^{-}(Z)$ can be obtained by
substituting Eq. (10) into Eq. 3

$\displaystyle\psi_{A_{1}}^{+}(Z)=c_{+}\nu\lambda_{+}D_{\lambda_{+}-1}(Z)+c_{-}\nu^{*}\lambda_{+}D_{\lambda_{+}-1}(-Z),$
(12)
$\displaystyle\psi_{A_{1}}^{-}(Z)=d_{+}\nu\lambda_{-}D_{\lambda_{-}-1}(Z)+d_{-}\nu^{*}\lambda_{-}D_{\lambda_{-}-1}(-Z),$

where $\nu=-\frac{i\sqrt{2}\epsilon_{0}}{E}$. Combining $\psi_{A_{1}}$ and
$\psi_{B_{1}}$ in Eq. 4 gives
$\psi_{A_{2}}(Z)=\psi_{A_{2}}^{+}(Z)+\psi_{A_{2}}^{-}(Z)$ with

$\displaystyle\psi_{A_{2}}^{+}(Z)=c_{+}\zeta^{+}D_{\lambda_{+}}(Z)+c_{-}\zeta^{+}D_{\lambda_{+}}(-Z),$
(13)
$\displaystyle\psi_{A_{2}}^{-}(Z)=d_{+}\zeta^{-}D_{\lambda_{-}}(Z)+d_{-}\zeta^{-}D_{\lambda_{-}}(-Z),$

and
$\zeta^{\pm}=\frac{E}{\gamma_{1}}-\frac{2\epsilon_{0}^{2}\lambda_{\pm}}{\gamma_{1}E}$.
Finally from Eq. 6, we obtain
$\psi_{B_{2}}(Z)=\psi_{B_{2}}^{+}(Z)+\psi_{B_{2}}^{-}(Z),$ with

$\displaystyle\psi_{B_{2}}^{+}(Z)=c_{+}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(Z)+c_{-}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(-Z),$
(14)
$\displaystyle\psi_{B_{2}}^{-}(Z)=d_{+}\nu\zeta^{-}D_{\lambda_{-}+1}(Z)+d_{-}\nu\zeta^{-}\lambda_{-}D_{\lambda_{-}+1}(-Z).$

On the other hand, the eigenspinors of SLG are given by solving the time-
independent Schrödinger equation for the Hamiltonian

$\mathcal{H}_{\text{SLG}}=v_{F}\vec{p}\cdot\vec{\sigma},$ (15)

to end up with the wave function

$\Phi(x,y)=e^{ik_{y}y}\begin{pmatrix}\phi_{A}(x)\\\ \phi_{B}(x)\end{pmatrix},$
(16)

where the two $x$-dependent components are given by

$\displaystyle\phi_{A}(x)=a\alpha^{-}e^{ik_{x}x}-b\alpha^{+}e^{-ik_{x}x},$
(17) $\displaystyle\phi_{B}(x)=ae^{ik_{x}x}+be^{-ik_{x}x},$ (18)

and we have

$k_{x}=\sqrt{E^{2}-k_{y}^{2}},\quad\alpha^{\pm}=\frac{k_{x}\pm ik_{y}}{E}$
(19)

with two constants $a$ and $b$.

##  III SLG and BLG junction

Our presumption is that charge carriers always move from left to right. We
consider a system that combines SLG and BLG. In this system, the leads on the
left and right are SLGs, while in between they are connected to an AB-BLG
subjected to a magnetic field. In the following, we take into consideration
two different zigzag boundary types: zigzag-1 (ZZ1) and zigzag-2 (ZZ2).

The front-most line of the bilayer edge is created by $B_{1}$ and $A_{2}$
sites as can be seen in Fig. 1. Note that ZZ1 is aligned to the honeycomb
lattice’s zigzag direction. Then we use the continuity of wavefunctions to
obtain at $x=0$

$\displaystyle\phi_{\text{A}}(x=0)=\psi_{\text{A}_{1}}(Z_{1}),$ (20)
$\displaystyle\phi_{\text{B}}(x=0)=\psi_{\text{B}_{1}}(Z_{1}),$ (21)
$\displaystyle\psi_{\text{B}_{2}}(Z_{1})=0,$ (22)

and at $x=d$

$\displaystyle\psi_{\text{A}_{1}}(Z_{2})=\phi_{\text{A}}(x=d),$ (23)
$\displaystyle\psi_{\text{B}_{1}}(Z_{2})=\phi_{\text{B}}(x=d),$ (24)
$\displaystyle\psi_{\text{B}_{2}}(Z_{2})=0,$ (25)

where we have set $Z_{1}=\sqrt{2}k_{y}\ell_{B}$ and
$Z_{2}=\sqrt{2}(\frac{d}{\ell_{B}}+k_{y}\ell_{B})$.

As far as ZZ2 is concerned, $B_{2}$ sites form the front-most line of the
bilayer region. Also, the continuity leads to set of equations [19, 40]

$\displaystyle\phi_{\text{A}}(x=0)=\psi_{\text{A}_{1}}(Z_{1}),$ (26)
$\displaystyle\phi_{\text{B}}(x=0)=\psi_{\text{B}_{1}}(Z_{1}),$ (27)
$\displaystyle\psi_{\text{A}_{2}}(Z_{1})=0,$ (28)
$\displaystyle\psi_{\text{A}_{1}}(Z_{2})=\phi_{\text{A}}(x=d),$ (29)
$\displaystyle\psi_{\text{B}_{1}}(Z_{2})=\phi_{\text{B}}(x=d),$ (30)
$\displaystyle\psi_{\text{A}_{2}}(Z_{2})=0.$ (31)

The above matching equations are worked out in the Appendix to to establish
the transmission probability $T(E)$ for each boundary. Then a transfer matrix
approach was used to get the two transmission coefficients given in Eqs. (65)
and (75). They can serve to derive the conductance based on the Landauer-
Buttiker formula

$G(E)=G_{0}T(E),$ (32)

with the unit $G_{0}=2e^{2}/h$.

Figure 2: (Color online) Conductance as a function of the magnetic field for
ZZ1 at $k_{y}=0$ for various width of BLG: $d=5$ nm (blue line), $d=10$ nm
(black line), and $d=15$ nm (magenta line), and for various values of the
Fermi energy, (a) $E=\gamma_{1}$, (b) $E=2\gamma_{1}$, (c) $E=2.5\gamma_{1}$,
(d) $E=3\gamma_{1}$.

Figure 3: (Color online) Conductance as a function of the width of BLG $d$ for
ZZ1 at $k_{y}=0$ for various Fermi energy values: $E=0.5\gamma_{1}$ nm (blue
line), $E=2\gamma_{1}$ nm (black line), and $E=3\gamma_{1}$ nm (magenta line),
and for various values the magnetic field $B$, (a) $B=0.1$ T, (b) $B=0.2$ T,
(c) $B=0.3$ T, (d) $B=0.4$ T.

Figure 4: (Color online) Density plot of the transmission probability as a
function of bilayer region length $d$, and Fermi energy $E$ for ZZ1, (a)
$B=0.1$ T, (b) $B=0.2$ T, (c) $B=0.3$ T, (d) $B=0.4$ T.

Figure 5: (Color online) Conductance as a function of the magnetic field for
ZZ2 at $k_{y}=0$ for various width of BLG $d$ values: $d=5$ nm (blue line),
$d=10$ nm ( black line), and $d=15$ nm (magenta line), and for various values
of the Fermi energy: (a) $E=\gamma_{1}$, (b) $E=2\gamma_{1}$, (c)
$E=2.5\gamma_{1}$, (d) $E=3\gamma_{1}$.

Figure 6: (Color online) Conductance as a function of the width of BLG $d$ for
ZZ2 at $k_{y}=0$ for various Fermi energy values: $E=0.5\gamma_{1}$ nm (blue
line), $E=2\gamma_{1}$ nm ( black line), and $E=3\gamma_{1}$ nm (magenta
line), and for various values the magnetic field $B$: (a) $B=0.1$ T, (b)
$B=0.2$ T, (c) $B=0.3$ T, (d) $B=0.4$ T.

Figure 7: (Color online) Density plot of the transmission probability as a
function of bilayer region length $d$, and Fermi energy $E$ for ZZ2, with (a)
$B=0.1$ T, (b) $B=0.2$ T, (c) $B=0.3$ T, (d) $B=0.4$ T.

##  IV RESULTS AND DISCUSSION

In this section, we analyze our key findings numerically and discuss them. In
Fig. 2, we show the conductance as a function of the magnetic field at normal
incidence $(k_{y}=0)$ for different widths of the BLG varying the values of
the Fermi energy $E$. We notice that whatever the value of the energy, the
conductance $G(E)$ seems independent of energy. This is a manifestation of
Klein tunneling due to pseudospin conservation [41, 42]. In Fig. 2 (a), for
$E=\gamma_{1}$, the result shows zero conductance and there are no
antiresonances because there is only one propagating channel at the BLG.
However, it is not the case for the $E>\gamma_{1}$ as depicted in Fig. 2 (b),
(c), and (d), where $G(E)$ presents antiresonances that appear with zero
conductance for large values of the magnetic field because of the coexistence
of two propagating channels.

Fig. 3 shows the conductance as a function of the BLG width for the ZZ1
boundary for four different values of the magnetic field. We notice that for
$E=0.5\gamma_{1}$, for which there is only one propagating channel in the BLG,
the conductance shows an exponential decay of the resonance for sufficiently
long widths and all magnetic field values, as shown in Figs. 3 (a) and (b).
However, for the highest energies, the resonances become important and
dependent on the magnetic field. We observe that the shapes and the numbers of
the resonances change from $B=0.1$ T to $B=0.4$ T. For sufficiently long
widths, i.e., $d>4$ nm the conductance $G$ tends to $0$. This result shows
agreement with our previous results [43]. The presence of the magnetic field
in the bilayer graphene in this SLG-BLG-SLG affects the conductance and
removes the periods of the antiresonances as seen in [20].

We exhibit in Fig. 4 the density plot of the transmission probability as a
function of the Fermi energy $E$ and the width of the BLG at normal incidence
$k_{y}=0$ to examine the impact of the magnetic field. As displayed in the
plots there are two separate regions of energy, set by the interlayer coupling
$E<\gamma_{1}$, there are no antiresonances due to the existence of just one
propagation channel at the BLG, in contrast to $E>\gamma_{1}$, where the
coexistence of two propagating eigenchannels in the BLG lets the zero
antiresonances appear. The behavior is comparable to that investigated by
González et al [20], with a clear distinction between the spatial periods, due
to the presence of the magnetic field, and also we observe that transmission
is completely suppressed for large widths of the BLG ($d>2.5$) and the Klein
tunneling gets not incandescent anymore. Hence, we conclude that in the
presence of the magnetic field, Klein tunneling is hampered, and instead
Febry-Pérot resonances [44] appear for $E>\gamma_{1}$.

We will turn our attention now to the zigzag boundary ZZ2, for which we plot
the conductance as a function of the magnetic field in Fig. 5 for three
different widths of the BLG, taking into account four different values of the
Fermi energy. In Fig. 5 (a) the conductance shows maxima for $B<0.10$ T, in
contrast to the ZZ1 boundary, by increasing the width of BLG and the Fermi
energy, the conductance shows oscillations in the ZZ2 case. These results show
that the transmission probability depends more strongly on boundaries, and
confinement is more important in the ZZ1 boundary than the ZZ2 boundary, which
is in agreement with [19] results.

In Fig. 6, we plot the conductance as a function of the width $d$ using the
same parameters as in Fig. 3. We observe fewer resonances than in the ZZ1 case
for $B\leqslant 0.3$ T. By increasing the BLG width, we see that our
conductance also vanishes for the ZZ2 boundary.

Fig. 7 displays the density plots for the ZZ2 case. There are two important
differences with respect to the ZZ1 case, see Fig. 4. First, for the energy
region $E<\gamma_{1}$ the transmission is more important, with an obvious
difference in the contrast of the spatial resonances. We note too that the
transmission with respect to $d$ is more similar to the $ZZ1$ case, but both
results indicate strong confinement at the ZZ1 boundary.

##  V Conclusion

We examined the effect of a perpendicular magnetic field in BLG surrounded by
SLG regions. We looked at both types of zigzag boundaries. Starting with ZZ1,
we showed that the conductance as a function of the magnetic field at normal
incidence $(k_{y}=0)$ seems independent of energy. Due to pseudospin
conservation, this is an instance of Klein tunneling. For $E=\gamma_{1}$, the
result shows zero conductance and there are no antiresonances because there is
only one propagating channel at the BLG, but for $E>\gamma_{1}$, $G(E)$
presents antiresonances that appear with zero conductance for large values of
the magnetic field because of the coexistence of two propagating channels.

As a function of the BLG width, we have found that for $E=0.5\gamma_{1}$, for
which there is only one propagating channel in the BLG, the conductance shows
an exponential decay of the resonance for a sufficiently long width and all
magnetic field values. For the highest energies, the resonances become
important and dependent on the magnetic field. Our results also show that the
shapes and numbers of the resonances change from $B=0.1$ T to $B=0.4$ T. For
sufficiently long widths, i.e., $d>4$ nm, the conductance $G$ tends to $0$.
For the zigzag boundary ZZ2, we found distinct behaviors compared with the ZZ1
boundary. We observed maxima for small values of the magnetic field in the
conductance plot, in contrast to the ZZ1 boundary, and by increasing the width
of BLG and the Fermi energy, the conductance shows oscillatory behavior in the
ZZ2 feature. Our results showed that the transmission probability depends more
strongly on boundaries, and confinement is more important in the ZZ1 boundary
than in the ZZ2 boundary. We have also analyzed the conductance as a function
of the width and length $d$. We have observed fewer resonances than in the ZZ1
case for $B\leqslant 0.3$ T. Increasing the BLG width, our conductance $G(E)$
vanishes also for the ZZ2 boundary.

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##  Appendix A Transfer matrix for SLG and BLG junction

In this Appendix, we briefly review the main steps of our analytical
calculations. In order to determine the transmission probability, we impose
the appropriate boundary conditions in the context of the transfer matrix
approach. More explicitly, for boundary ZZ1 Eqs. 20 and 23, we obtain six
equations with six unknowns, at $x=0$

$\displaystyle\alpha^{-}-r\alpha^{+}=c_{+}\nu\lambda_{+}D_{\lambda_{+}-1}(Z_{1})+c_{-}\nu^{*}\lambda_{+}D_{\lambda_{+}-1}(-Z_{1})+d_{+}\nu\lambda_{-}D_{\lambda_{-}-1}(Z_{1})+d_{-}\nu^{*}\lambda_{-}D_{\lambda_{-}-1}(-Z_{1}),$
(33) $\displaystyle
1+r=c_{+}D_{\lambda_{+}}(Z_{1})+c_{-}D_{\lambda_{+}}(-Z_{1})+d_{+}D_{\lambda_{-}}(Z_{1})+d_{-}D_{\lambda_{-}}(-Z_{1}),$
(34) $\displaystyle
0=c_{+}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(Z_{1})+c_{-}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(-Z_{1})+d_{+}\nu\zeta^{-}D_{\lambda_{-}+1}(Z_{1})+d_{-}\nu\zeta^{-}\lambda_{-}D_{\lambda_{-}+1}(-Z_{1}),$
(35)

and for $x=d$

$\displaystyle
c_{+}\nu\lambda_{+}D_{\lambda_{+}-1}(Z_{2})+c_{-}\nu^{*}\lambda_{+}D_{\lambda_{+}-1}(-Z_{2})+d_{+}\nu\lambda_{-}D_{\lambda_{-}-1}(Z_{2})+d_{-}\nu^{*}\lambda_{-}D_{\lambda_{-}-1}(-Z_{2})=t\alpha^{-}e^{ik_{x}d},$
(36) $\displaystyle
c_{+}D_{\lambda_{+}}(Z_{2})+c_{-}D_{\lambda_{+}}(-Z_{2})+d_{+}D_{\lambda_{-}}(Z_{2})+d_{-}D_{\lambda_{-}}(-Z_{2})=te^{ik_{x}d},$
(37) $\displaystyle
c_{+}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(Z_{2})+c_{-}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(-Z_{2})+d_{+}\nu\zeta^{-}D_{\lambda_{-}+1}(Z_{2})+d_{-}\nu\zeta^{-}\lambda_{-}D_{\lambda_{-}+1}(-Z_{2})=0.$
(38)

We note that the eigenspinors for SLG at $(x=0)$ consist of the incident and
reflected plane waves, then the Eqs. in 17 rewritten as

$\displaystyle\phi_{A}(x=0)=\alpha^{-}-r\alpha^{+},$ (39)
$\displaystyle\phi_{B}(x=0)=1+r,$ (40)

consists of the transmitted waves at $(x=d)$

$\displaystyle\phi_{A}(x=d)=t\alpha^{-}e^{ik_{x}d},$ (41)
$\displaystyle\phi_{B}(x=d)=te^{ik_{x}d},$ (42)

with $r$ and $t$ denote the reflection and transmission coefficients. Eqs. 33
and 34 as well as Eqs. 36 and 37 can be reformulated using the transfer matrix
approach [45]

$\displaystyle\mathcal{G}^{\text{SLG}}\mathcal{P}_{x=0}\left[\begin{array}[]{l}1\\\
r\end{array}\right]=\mathcal{G}_{+,Z_{1}}^{\text{BLG}}\left[\begin{array}[]{l}c_{+}\\\
c_{-}\end{array}\right]+\mathcal{G}_{-,Z_{1}}^{\text{BLG}}\left[\begin{array}[]{l}d_{+}\\\
d_{-}\end{array}\right],$ (49)
$\displaystyle\mathcal{G}_{+,Z_{2}}^{\text{BLG}}\left[\begin{array}[]{l}c_{+}\\\
c_{-}\end{array}\right]+\mathcal{G}_{-,Z_{2}}^{\text{BLG}}\left[\begin{array}[]{l}d_{+}\\\
d_{-}\end{array}\right]=\mathcal{G}^{\text{SLG}}\mathcal{P}_{x=d}\left[\begin{array}[]{l}t\\\
0\end{array}\right],$ (56)

with

$\displaystyle\mathcal{G}^{\text{SLG}}=\begin{pmatrix}\alpha^{-}&-\alpha^{+}\\\
1&1\\\ \end{pmatrix},$ (57)
$\displaystyle\mathcal{P}_{x}=\begin{pmatrix}e^{ikx}&0\\\ 0&e^{-ikx}\\\
\end{pmatrix},$ (58)
$\displaystyle\mathcal{G}_{\pm,Z_{1/2}}^{\text{BLG}}=\begin{pmatrix}\nu\lambda_{\pm}D_{\lambda_{\pm}-1}(Z_{1/2})&\nu^{*}\lambda_{\pm}D_{\lambda_{\pm}-1}(-Z_{1/2})\\\
D_{\lambda_{\pm}-1}(Z_{1/2})&D_{\lambda_{\pm}-1}(-Z_{1/2})\\\ \end{pmatrix}.$
(59)

Combining Eqs. 35 and 38, its matrix counterpart can be expressed as

$\left[\begin{array}[]{l}d_{+}\\\
d_{-}\end{array}\right]=\Re_{\text{ZZ1}}\left[\begin{array}[]{l}c_{+}\\\
c_{-}\end{array}\right],$ (60)

with

$\displaystyle\Re_{\text{ZZ1}}=$
$\displaystyle-\left[\begin{pmatrix}\nu\zeta^{-}D_{\lambda_{-}+1}(Z_{1})&\nu\zeta^{-}D_{\lambda_{-}+1}(-Z_{1})\\\
\nu\zeta^{-}D_{\lambda_{-}+1}(Z_{2})&\nu\zeta^{-}D_{\lambda_{-}+1}(Z_{2})\\\
\end{pmatrix}\right]^{-1}$ (61)
$\displaystyle\times\left[\begin{pmatrix}\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(Z_{1})&\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(-Z_{1})\\\
\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(Z_{2})&\nu^{*}\zeta^{+}D_{\lambda_{+}+1}(-Z_{2})\\\
\end{pmatrix}\right].$

Combining Eqs. 49, 56, and 60, the transfer matrix of the our structure can be
obtained as

$\left[\begin{array}[]{c}1\\\
r\end{array}\right]=\mathcal{M}_{\text{ZZ1}}\left[\begin{array}[]{c}t\\\
0\end{array}\right],$ (62)

with

$\displaystyle\mathcal{M}_{\text{ZZ1}}=$
$\displaystyle\left(\mathcal{G}^{\text{SLG}}\mathcal{P}_{x=0}\right)^{-1}\left[\mathcal{G}_{+,Z_{1}}^{\text{BLG}}+\mathcal{G}_{-,Z_{1}}^{\text{BLG}}\Re_{\text{ZZ1}}\right]$
(63)
$\displaystyle\times\left[\mathcal{G}_{+,Z_{2}}^{\text{BLG}}+\mathcal{G}_{-,Z_{2}}^{\text{BLG}}\Re_{\text{ZZ1}}\right]^{-1}\mathcal{G}^{\text{SLG}}\mathcal{P}_{x=d}.$
(64)

From Eq. 62, the transmission coefficient can be derived as

$t_{\text{ZZ1}}=\frac{1}{\mathcal{M}_{\text{ZZ1}}}.$ (65)

In the same way, by requiring the continuity using Eqs. 26 and 29, we get

$\displaystyle\alpha^{-}-r\alpha^{+}=c_{+}\nu\lambda_{+}D_{\lambda_{+}-1}(Z_{1})+c_{-}\nu^{*}\lambda_{+}D_{\lambda_{+}-1}(-Z_{1})+d_{+}\nu\lambda_{-}D_{\lambda_{-}-1}(Z_{1})+d_{-}\nu^{*}\lambda_{-}D_{\lambda_{-}-1}(-Z_{1}),$
(66) $\displaystyle
1+r=c_{+}D_{\lambda_{+}}(Z_{1})+c_{-}D_{\lambda_{+}}(-Z_{1})+d_{+}D_{\lambda_{-}}(Z_{1})+d_{-}D_{\lambda_{-}}(-Z_{1}),$
(67) $\displaystyle
0=c_{+}\zeta^{+}D_{\lambda_{+}}(Z_{1})+c_{-}\zeta^{+}D_{\lambda_{+}}(-Z_{1})+d_{+}\zeta^{-}D_{\lambda_{-}}(Z_{1})+d_{-}\zeta^{-}D_{\lambda_{-}}(-Z_{1}),$
(68)

and for $x=d$

$\displaystyle
c_{+}\nu\lambda_{+}D_{\lambda_{+}-1}(Z_{2})+c_{-}\nu^{*}\lambda_{+}D_{\lambda_{+}-1}(-Z_{2})+d_{+}\nu\lambda_{-}D_{\lambda_{-}-1}(Z_{2})+d_{-}\nu^{*}\lambda_{-}D_{\lambda_{-}-1}(-Z_{2})=t\alpha^{-}e^{ik_{x}d},$
(69) $\displaystyle
c_{+}D_{\lambda_{+}}(Z_{2})+c_{-}D_{\lambda_{+}}(-Z_{2})+d_{+}D_{\lambda_{-}}(Z_{2})+d_{-}D_{\lambda_{-}}(-Z_{2})=te^{ik_{x}d},$
(70) $\displaystyle
c_{+}\zeta^{+}D_{\lambda_{+}}(Z_{2})+c_{-}\zeta^{+}D_{\lambda_{+}}(-Z_{2})+d_{+}\zeta^{-}D_{\lambda_{-}}(Z_{2})+d_{-}\zeta^{-}D_{\lambda_{-}}(-Z_{2})=0.$
(71)

All these equations can be written in compact form by introducing the transfer
matrix

$\displaystyle\mathcal{M}_{\text{ZZ2}}=$
$\displaystyle\left(\mathcal{G}^{\text{SLG}}\mathcal{P}_{x=0}\right)^{-1}\left[\mathcal{G}_{+,Z_{1}}^{\text{BLG}}+\mathcal{G}_{-,Z_{1}}^{\text{BLG}}\Re_{\text{ZZ2}}\right]$
$\displaystyle\times\left[\mathcal{G}_{+,Z_{2}}^{\text{BLG}}+\mathcal{G}_{-,Z_{2}}^{\text{BLG}}\Re_{\text{ZZ2}}\right]^{-1}\mathcal{G}^{\text{SLG}}\mathcal{P}_{x=d},$
(73)

where

$\displaystyle\Re_{\text{ZZ2}}=$
$\displaystyle-\left[\begin{pmatrix}\zeta^{-}D_{\lambda_{-}}(Z_{1})&\zeta^{-}D_{\lambda_{-}}(-Z_{1})\\\
\zeta^{-}D_{\lambda_{-}}(Z_{2})&\zeta^{-}D_{\lambda_{-}}(-Z_{2})\\\
\end{pmatrix}\right]^{-1}$ (74)
$\displaystyle\times\left[\begin{pmatrix}\zeta^{+}D_{\lambda_{+}}(Z_{1})&\zeta^{+}D_{\lambda_{+}}(-Z_{1})\\\
\zeta^{+}D_{\lambda_{+}}(Z_{2})&\zeta^{+}D_{\lambda_{+}}(-Z_{2})\\\
\end{pmatrix}\right].$

As a result, we can express the transmission coefficient $t_{\text{ZZ2}}$ as

$t_{\text{ZZ2}}=\frac{1}{\mathcal{M}_{\text{ZZ2}}}.$ (75)


# BanglaBait: Semi-Supervised Adversarial Approach for Clickbait Detection on
Bangla Clickbait Dataset

Md. Motahar Mahtab
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>
&Monirul Haque
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>Mehedi Hasan
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>
&Farig Sadeque
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>
###### Abstract

Intentionally luring readers to click on a particular content by exploiting
their curiosity defines a title as clickbait. Although several studies focused
on detecting clickbait titles in English articles, low-resource language like
Bangla has not been given adequate attention. To tackle clickbait titles in
Bangla, we have constructed the first Bangla clickbait detection dataset
containing 15,056 labeled news articles and 65,406 unlabelled news articles
extracted from clickbait-dense news sites. Each article has been labeled by
three expert linguists and includes an article’s title, body, and other
metadata. By incorporating labeled and unlabelled data, we finetune a pre-
trained Bangla transformer model in an adversarial fashion using Semi-
Supervised Generative Adversarial Networks (SS-GANs). The proposed model acts
as a good baseline for this dataset, outperforming traditional neural network
models (LSTM, GRU, CNN) and linguistic feature-based models. We expect that
this dataset and the detailed analysis and comparison of these clickbait
detection models will provide a fundamental basis for future research into
detecting clickbait titles in Bengali articles. We have released the
corresponding code and dataset
111https://github.com/mdmotaharmahtab/BanglaBait.

##  1 Introduction

Due to the widespread usage of the internet, the news industry has
progressively evolved into an online news industry leading to the explosion of
clickbait titles in recent years. As the concept of clickbait can be hazy to
grasp, the classification of clickbait is a highly subjective endeavor. Biyani
et al. (2016) suggests that clickbait titles can be roughly categorized into
eight types. Table 1 displays this different clickbait categories222’wrong’
category in Biyani et al. (2016) was replaced by ’question’ category - reason
described in details in section 3.1 and their corresponding Bangla articles.

Category |  Reason |  Headline example & Translation
---|---|---
Questions |  Titles pose a query that compels the reader to click to get the answer. |  (Salman’s actress Rombha mysteriously disappeared, what is she up to?).
Inflammatory |  Titles evoke strong emotion. |  (Lost his cool, kicked, then busted out the stamps before abusing the umpire!)
Curiosity Gap/Teasing |  Titles leave the reader in the dark, which tempts them to click. |  (Explore how to preserve meat without a refrigerator!)
Ambiguous |  Imprecise or unclear titles that pique interest. |  (Not even Mir spared Nusrat)
Exaggerate |  Titles overstating what is written on the landing page. |  ! (Three foods when combined with blackberries, could kill you!)
Graphic |  Salacious, unsettling, or implausible subject matter. |  (After he finds her cheating, the mother cuts her son into bits and stuffs him into a bag before tossing it into the water.)
Formatting |  Excessive use of punctuation or other symbols. |  (Faria ’out’ Porimoni ’in’!)
Bait & Switch |  Overpromising titles with under-delivering content; requires additional clicks. |  (One body two souls! One is an English teacher, whereas another is of Mathematics.)
Table 1: Clickbait news titles and their categories.

There are an estimated 11.4 million internet users in
Bangladesh333https://www.cia.gov/the-world-factbook/countries/bangladesh/ who
receive their daily news mostly from online news sites. However, no research
has been conducted on tackling the increasing number of clickbait titles on
these sites and other news websites. For English articles, Potthast et al.
(2018a) built the first large-scale annotated clickbait corpus (Webis
Clickbait Corpus 2017) containing 338,517 articles. In the Bangla language,
the lack of an annotated clickbait-rich dataset is hindering the progress of
Bangla Clickbait Detection. We construct the first Bangla Clickbait Corpus,
which contains an article’s title, content, and other metadata collected from
various clickbait-rich websites upon which future researchers can build an
effective Bangla clickbait detection model. The effectiveness of Semi-
Supervised Generative Adversarial networks (SS-GANs; Salimans et al., 2016)
have been shown for text classification tasks in Croce et al. (2020). From our
experiments, it is evident that fine-tuning a Bangla ELECTRA model in this
setup improves clickbait detection performance outperforming all other model
types.

The main contributions of this paper can be summarized as follows:

  * •

We create an annotated dataset of 15,056 articles and an unannotated dataset
of 65,406 Bangla articles rich with clickbait titles. The dataset contains the
title, body, domain, article category, publication date, and English
translation of title and content. We have released both of these datasets and
the corresponding code to reproduce our experiments.

  * •

We develop the first Bangla Clickbait Detection model for Bangla textual data
by thoroughly experimenting with different statistical machine learning
algorithms, deep neural networks using state-of-the-art embeddings, and
Transformer networks (Vaswani et al., 2017) to discover the best approach for
detecting clickbait. Section 7 analyzes the quantitative comparisons among all
these different models.

  * •

We train a Bangla Transformer model in a Semi-Supervised Generative
Adversarial setup and show that it improves upon existing models trained in a
supervised manner.

##  2 Related Work

The origin of clickbait is rooted in tabloids which have been in journalism
since the 1980’s (Bird, 2008). Generally, clickbait detection features can be
obtained from 3 different origins: clickbait teaser phrase or post text, the
attached article that the post text wants the user to click, and metadata for
both (Potthast et al., 2018a). Apart from the post text, which is used by most
to identify clickbait, the works of Potthast et al. (2016) and Biyani et al.
(2016) also considered the linked article, metadata and used handcrafted
features, TF-IDF similarity between headline and article content and Gradient
Boosted Decision Trees (GBDT). Potthast et al. (2018a) suggested that
clickbait detection should be a regression problem instead of a binary
classification challenge, as the latter provides a way to measure how much
clickbait is in the teaser message. They initiated the Webis clickbait
challenge 2017, which boosted research activity in clickbait detection giving
rise to highly effective and flexible deep learning techniques. For clickbait
challenge 2017, Zhou (2017) first used self-attentive RNN (Elman, 1990) to
select the important words in the title and created a BiGRU (Cho et al., 2014)
network to encode the contextual information. Thomas (2017), on the other
hand, incorporated article content into an LSTM model (Hochreiter and
Schmidhuber, 1997) for the clickbait challenge. Rony et al. (2017) used
continuous skip-gram model (Mikolov et al., 2013) to generate the word
embedding of clickbait titles. However, Indurthi et al. (2020) first
investigated the application of transformer regression models in clickbait
detection and achieved the first position in the clickbait challenge. Besides,
Hossain et al. (2020) created the first Bengali newspaper dataset for Bengali
fake news detection containing an annotated dataset of $\approx 50K$ Bangla
news. To the best of our knowledge, the first attempt to detect clickbait in
Bangla was made by Munna and Hossen (2021). They created a dataset on video-
sharing platforms containing Bangla and English video links and used numerical
features to detect clickbait links. However, no research has been conducted to
tackle clickbaits in written news mediums using the textual features of the
article. We present the first clickbait detection dataset in Bangla and also
provide a comprehensive comparison of various models to detect them.

##  3 First Bangla Dataset for Detecting Bangla Clickbait News Articles

###  3.1 Data Collection

We first compile a list of websites that publish Bangla news articles.
Although Potthast et al. (2018b, 2016) used metrics like the number of
retweets to select the most influential websites, such metric providing
services like Alexa ranking444https://www.alexa.com/ is unavailable for most
prominent Bangla Websites. Instead, we first create a preliminary list of
Bangla news article sites from where we choose a website for scraping if the
homepage seems to contain more clickbait than non-clickbait titles after a
cursory glance by the annotators. We also select some famous Bengali online
news publishers such as Kaler Kantha555https://www.kalerkantho.com/,
SomoyTV666https://www.somoynews.tv/, and RTV news777https://www.rtvonline.com/
for scraping to facilitate future investigation into clickbait practices in
popular Bangla news mediums. Before scraping, we check whether the publishers
we select have terms and conditions against scraping or using their content
for educational or research purposes to avoid copyright infringement.
Utilizing the Python Selenium module, we have scraped data from the first week
of February 2019 to the last week of February 2022.

Although Hossain et al. (2020) published the first dataset of Bangla Fake
news, we find it necessary to create a separate dataset for clickbait in
Bangla as a news title can be a clickbait without necessarily being fake news
(Dong et al., 2019) 888More details can be found in section A.3. To enrich our
dataset size, one thousand titles labeled ’clickbait’ from Bangla Fake News
Dataset (Hossain et al., 2020) are added to our own dataset after their labels
are revised again by annotators.

Information | Value |
---|---|---
Crawling Period | Feb 2019 - Feb 2022 |
Total Clickbait | 5239 |
Total Non-clickbait | 9817 |
Total Unlabelled | 65406 |
Title Analysis | Clickbait | Non-clickbait
Average number of characters | 52.845 | 49.097
Average number of words | 8.983 | 7.8356
Average word length | 4.99 | 5.4
Average Punctuation | 1.003 | 0.805

Table 2: Summary statistics of our dataset.

Column | Value
---|---
Domain | https://www.rtvonline.com/
Date | 2021-05-25
Title |
Content |
Label | 1 (Clickbait)
Translated Title | It’s only Rs. 13!
Translated Content |  That’s the truth, though it sounds surprising. Only 13 rupees can be bought at home…
Category | Science & Technology

Table 3: Sample Data

###  3.2 Annotation Process

The dataset is annotated by three annotators with an MA in Bangla Linguistics.
At first, they study the annotations of popular English clickbait datasets
(Potthast et al., 2018b; Agrawal, 2016; Potthast et al., 2016). Investigating
English titles help the annotators understand how titles induce curiosity in
practice, which they can then use to annotate Bangla titles. As questions
naturally entice interest, a new clickbait category named ’question’ is added
to the clickbait categories in Table 1. No publisher or source of the article
is available to the annotators to avoid any induced publisher-based biases as
reported by Potthast et al. (2018a) to be the case for several clickbait
datasets (Rony et al., 2017; Ganguly, 2016; Agrawal, 2016). A majority vote
among the annotators decides the final annotation. The annotators reach an
inter-annotator agreement Fleiss kappa (Fleiss et al., 1971) of 0.62, which is
substantial Landis and Koch (1977) and enough for a good speculative
conclusion regarding annotator agreement (Artstein and Poesio, 2008).

The annotators mark clickbait news as a numeric value of 1 and non-clickbait
news as a numeric value of 0. Our labeled and unlabelled datasets contain
eight categories - Economy, Education, Entertainment, Politics, International,
Sports, National, and Science & Technology of clickbait and non-clickbait
titles. After removing all duplicates from labeled and unlabeled datasets, our
dataset contains 15,056 unique news articles with 9,817 non-clickbait and
5,239 clickbait articles, and 65,406 unique unlabelled articles. The labeled
and unlabeled datasets do not have any overlapping content or titles. The test
set is further curated by removing titles that have similar titles in the
training set through Levenshtein distance (Levenshtein, 1965). Table 3 shows
that clickbait titles have a slightly higher average number of words and
punctuation than non-clickbait titles. The most frequent fifteen words in
clickbait titles are -

It contains words like viral, video, and intense which usually induce readers
to click. Each data instance contains the title and content of the article,
publishing date, domain, news category, translated title, and translated
content as shown in Table 3.

##  4 Human Baseline

Five human annotators who are undergraduate and regular newspaper readers are
given 200 news article titles from the test set to annotate. They achieve an
inter-annotator agreement of Fleiss’ kappa (Fleiss et al., 1971) score of
0.374, which is fair according to Landis and Koch (1977). Compared to our
dataset annotators’ score, this score is much lower. Our annotation process
includes investigating English titles first to better form a coherent
perception of clickbait titles. By majority voting among the five annotators,
we select the final labels and achieve an F1 score of 76.82% and an accuracy
of 77.01% on the clickbait class, which serves as the human baseline for
Bangla clickbait detection shown in Table 4.

##  5 Approach

###  5.1 GAN-BanglaBERT

Figure 1: GAN-BanglaBERT architecture. Generator $\mathcal{G}$ generates fake
data given random noise, and Discriminator $\mathcal{D}$ takes both real and
this fake data and outputs four labels: 0 for non-clickbait, 1 for clickbait,
2 for real and 3 for fake data.

In Generative Adversarial Network (Goodfellow et al., 2014), a generator
$\mathcal{G}$ is trained to generate a data distribution similar to the real
data to ’fool’ the discriminator $\mathcal{D}$ and $\mathcal{D}$ is trained to
differentiate between the two in an adversarial fashion. Semi-Supervised GANs
(Salimans et al., 2016, SS-GANs;) train the discriminator $\mathcal{D}$ to
predict the classification labels along with the additional task of predicting
whether the data is real or fake. This training technique helps the model
improve its inner representations by utilizing the unlabelled and generated
data (Croce et al., 2020). Following researchers of Croce et al. (2020), we
finetune a BanglaBERT (Bhattacharjee et al., 2021), a state-of-the-art ELECTRA
(Clark et al., 2020) model pre-trained on 35 GB of Bangla textual data from
the web and call it ’GAN-BanglaBERT’ throughout the paper. Figure 1 shows the
overall architecture of the GAN-BanglaBERT model. Generator $\mathcal{G}$ and
discriminator $\mathcal{D}$ both are a 2-layered deep neural network(DNN). A
100-dimensional noise vector is drawn from a standard normal distribution
$N\left(\mu=0,\sigma^{2}=1\right)$ following the initialization practice in
GANs (Goodfellow et al., 2014). Generator $\mathcal{G}$ produces
$h_{\text{fake }}\in\mathbb{R}^{d}$ vector from this noise vector where $d$ is
the last layer size of the pre-trained Transformer network. Discriminator
$\mathcal{D}$ takes in input the concatenation of both real and fake data’s
representation $[h_{\text{real }};h_{\text{fake}}]$. Detailed training loss
calculation is provided in Croce et al. (2020), which remains unchanged in our
implementation. The average of the last hidden layer outputs of BanglaBERT is
the transformer encoding $h_{\text{real }}$ for a real title.

###  5.2 Comparison Methods

We compare the GAN-BanglaBERT model to the following models.

  * -

Statistical Models: For statistical methods, we employ a Logistic and Random
Forest classifier on a combination of various features like TF-IDF (term
frequency–inverse document frequency) of the word and character n-grams
(n-gram range=3-5), Bangla pre-trained word embeddings, punctuation frequency,
and normalized Pars-of-Speech frequency according to Hossain et al. (2020).

  * -

Zhou (2017): employ a BiGRU (Cho et al., 2014) network with a self-attentive
network (Yang et al., 2016) on top of the BiGRU representations and achieve
the first position at Clickbait Challenge 2017 (Potthast et al., 2018a) with
an F1 score of 0.683.

  * -

Agrawal (2016): employ a multi-channel CNN model with one convolutional layer
similar to the model demonstrated by Kim (2014). Pre-trained word embeddings
are passed to multiple filters, and their concatenated representation is sent
to a Max Pooling layer for the final representation.

  * -

Lee et al. (2021): We translate all our article titles using a Bangla-to-
English translator model Bangla-NMT (Hasan et al., 2020a) which outperformed
Google Translate on SUPara-benchmark test set (Hasan et al., 2019). The
translated titles are passed into a state-of-the-art misinformation detection
model UnifiedM2 (Lee et al., 2021) trained on fake, clickbait, rumor, and
news-bias datasets in English. We investigate if translating the titles and
using a state-of-the-art model trained in English suffices for clickbait
detection or whether language-specific training is necessary.

##  6 Experimental Setup

###  6.1 Pre Processing

Normalizer module by Hasan et al. (2020b) and Bangla unicode normalizer by
(Alam et al., 2021) are used for Unicode and nukta normalization, removing
HTML tags, URL links, etc. High punctuation usage is a common trait of
clickbait titles. We preserve all syntactically correct punctuation in our
titles and remove punctuation that appeared in the middle of words causing
words to break and create out-of-vocabulary words for models.

###  6.2 Experimental Settings

For all models, we use the article’s title as input as the title mainly
creates the curiosity gap that is the principal characteristic of a clickbait
title (Potthast et al., 2016). We use Bangla Fasttext (Bojanowski et al.,
2017) and Bangla Word2Vec embedding pre-trained on Bangla Wikipedia Dump
Dataset with coverage of 65.16% and 60.91% respectively, on the total
vocabulary size of article titles as embedding inputs. We extract the Parts of
Speech (POS) tags using BNLP toolkit (Sarker, 2021). We derive a Bangla
punctuation list from Alam et al. (2021). We experiment with both BiGRU and
BiLSTM models for (Zhou, 2017) model and show the better performing one in
section 7. The above models are trained for 40 epochs with Adam optimizer
(Kingma and Ba, 2017) and learning rate = 2e-5, which is changed dynamically
according to 1cycle learning rate scheduler (Smith and Topin, 2018). The GAN-
BanglaBERT and BanglaBERT models are trained for 20 epochs with AdamW
optimizer (Loshchilov and Hutter, 2019), and the learning rate is slowly
increased from zero to 1e-5 within a warmup period. For GAN-BanglaBERT, the
learning rate for the generator and discriminator model is kept the same. For
all models, we pad or truncate titles to lengths of 64. The labeled dataset is
split into 70:10:20 fashion for training, validation, and test splits using
stratified sampling. All models are trained with batch size=64, and the best
model based on the validation result is used to evaluate the final test set.
Each experiment is repeated five times, and the average result on the held-out
test set is used for the final result of all the models.

##  7 Results and Analysis

Model | F1 Score | Precision | Recall | Accuracy
---|---|---|---|---
Zhou et al. (2016) (Fasttext) | 39.37 | 39.88 | 38.87 | 57.87
Agrawal (2016) (Fasttext) | 35.15 | 40.05 | 31.32 | 59.33
Logistic Regression (character 3,4,5 gram) | 66.28 | 75.36 | 59.15 | 78.82
Random Forest (character 3 gram) | 67.01 | 61.06 | 74.25 | 74.27
Lee et al. (2021) | 11.02 | 39 | 6.4 | 63.53
BanglaBERT | 71.72 | 80.42 | 64.71 | 82.04
GAN BanglaBERT | 75.13 | 75.45 | 74.81 | 82.57
Human Baseline | 76.81 | 77.6 | 76.04 | 77.01
Table 4: Performance comparison of GAN-BanglaBERT and all other models on the
test set. F1 score, precision, and recall are for the clickbait class. For
Zhou (2017) model, a better performing BiLSTM-attn model result is shown. GAN-
BanglaBERT outperforms all other models and the performance difference is
statistically significant ($p<0.01$) according to McNemar’s test (Dietterich,
1998)

Table 4 illustrates the performance of all models on our test set. For each
type of model, only the best-performing feature’s result is shown. GAN-
BanglaBERT outperforms all other models regarding F1 score, precision, metric,
and recall. It achieves a 75.13% F1 score on the clickbait class, which is
3.41% greater than the supervised BanglaBERT model. The performance is close
to the human upper bound of 76.8% F1 score. The human baseline score shows
that separating clickbait and non-clickbait titles is a difficult task even
for humans, and clickbait may not be perceptible to all humans (Potthast et
al., 2018b).

Figure 2 shows the ROC curve (receiver operating characteristic curve) for all
models where the GAN-BanglaBERT model achieves the highest AUC (area under ROC
curve) score of 0.8925, which is higher than the BanglaBERT. The high AUC
score of GAN-BanglaBERT suggests that it can distinguish between clickbait and
non-clickbait titles more accurately than other models.

Figure 2: ROC curve for all models where GAN-BanglaBERT achieves the highest
Area Under ROC Curve (AUC) score.

To investigate whether training in a semi-supervised approach improves
BanglaBERT’s inner representations as stated by (Croce et al., 2020), we plot
the average of the last layer hidden representations of GAN-BanglaBERT and
BanglaBERT using a t-SNE projection (van der Maaten and Hinton, 2008) in
Figure 3(b). GAN-BanglaBERT better separates the clickbait class from the non-
clickbait than the BanglaBERT model, proving that training a BERT model in a
semi-supervised adversarial manner can improve the learned representations of
the model and thus improve performance.

(a) BanglaBERT t-SNE

(b) GAN-BanglaBERT t-SNE

Figure 3: Visualization of last layer hidden representations using t-SNE for BanglaBERT (3(a)) and for GAN-BanglaBERT (3(b)). 0 represents Non-Clickbait and 1 represents Clickbait in both figures. Category | Attention Weighted Words | Important Words
---|---|---
Ambiguous | BanglaBERT |  | ,  (forsake, not)
GAN-BanglaBERT |
Title |
Translation | They did not even forsake my mother
Table 5: Comparison between GAN-BanglaBERT and BanglaBERT on ambiguous type
clickbait title prediction. Each word is highlighted according to the
attention weight given by the model.

For creating the unlabelled dataset, we choose clickbait-dense websites from
the web to ensure a higher abundance of clickbait titles. To investigate
whether this helps performance, we create another unlabelled dataset of the
same size from Daily Prothom Alo archive999https://github.com/zabir-
nabil/bangla-news-rnn, which has a substantially lower clickbait ratio. Our
model achieves 72.38% F1 score on this second unlabelled set compared to
75.13% F1 score on the original unlabelled set, proving that a higher
clickbait ratio in the unlabelled set improves performance on the Clickbait
class.

Table 5 shows a prominent clickbait category - ’ambiguous’ where GAN-
BanglaBERT performs better than other models. ’They did not even forsake my
mother! - Bhabna’ is a quotation that implies something ostentatious happened
with the mother, although expressed very vaguely.  (not, forsake) words create
this ambiguity which GAN-BanglaBERT correctly gives more attention to, but
BanglaBERT fails to do so. The high AUC score and better separation in
encoding shown in Figure 3 enables GAN-BanglaBERT to perform better in these
harder-to-detect cases.

Table 4 shows that Lee et al. (2021) model on translated titles performs very
poorly compared to other models. Machine translation produces more synthetic
text, which diminishes the lexical and syntactical style and richness of the
source language (Vanmassenhove et al., 2021). For example,

is translated to ’Monkeys selling fruit in the market at the expense of the
real man, viral video.’ Although this translation is factually correct, it
loses the source language’s exaggerated tone, leading to misclassification.

Logistic regression and Random Forest model on character TF-IDF features
heavily outperform neural network models like BiLSTM with attention network
and CNN (Zhou, 2017; Agrawal, 2016). These models can effectively identify
certain keywords that are very significant in classifying clickbait titles.
For instance, a top character feature returned by logistic regression is
(told), which is a common keyword found in many clickbait titles, e.g.,  (What
Dighi said about the great news). The poor performance of neural network
models can be attributed to Bangla pre-trained Fasttext and Word2Vec
embeddings, which are trained on the Bangla Wikipedia dump and are
significantly smaller in size than English. Training these embeddings on
training data and then initializing the neural models with these embeddings
may improve performance.

All models perform poorly on Bait & Switch type titles as mentioned in Table 1
where titles where the main content under-delivers the title’s statements. As
these types of clickbait require reading the content to predict correctly, all
models underperform as they are trained on only the article’s title.
Effectively combining content features with titles to classify these types of
clickbait titles is a future research endeavor for us.

##  8 Conclusion

We present the first clickbait detection dataset containing 15,056 labeled new
articles and 65,406 unlabelled articles containing article title, content and
metadata to enable researchers to use this dataset to build state-of-the-art
clickbait detection models. By conducting a comprehensive study on various
architectures, we provide a strong baseline for detecting clickbait in Bangla
articles. We show that training a pre-trained Transformer model in a semi-
supervised approach by incorporating unlabeled data improves performance and
inner representation. As simple statistical models perform strongly on
clickbait titles, we aim to investigate how these features can be combined
with word embeddings to pass into neural networks. We also plan to investigate
how features from article content can be utilized to detect clickbait. We wish
to publicly release the dataset and code to further progress into Bangla
clickbait detection.

## Acknowledgments

We thank our data annotators and the volunteers from the Department of CSE,
BRAC University, who participated in the human baseline experiment. We also
thank the CSE Dept. of BRAC University, Bangladesh, for their continued
support and direction. We show our sincerest gratitude to the anonymous
reviewers and the pre-submission mentors for their valuable suggestions which
helped improve the research work.

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##  Appendix A Appendix

###  A.1 Data sources

We choose a site for scraping if the homepage seems to contain more clickbait
than non-clickbait titles after a cursory glance by the annotators. We also
select some famous Bengali online news publishers such as Kaler Kantha4,
SomoyTV5, and RTV news6 for scraping to facilitate future investigation into
clickbait practices in popular Bangla news mediums. Table 6 contains the data
sources for our dataset’s labeled and unlabelled portions.

Labelled
---
Domain | News Count
clickbait | non-clickbait
twentyfourbd | 1727 | 1062
topdhaka | 1004 | 920
rtvonline | 1003 | 86
BanFakeNews | 623 | 415
kureghornews | 634 | 375
newzcitizen | 633 | 351
nbtimes24 | 750 | 228
citynewszet | 503 | 451
authoritynewz | 561 | 385
thebasenewz | 537 | 268
newzauthority | 308 | 281
newsholder21 | 361 | 117
techzoom | 369 | 60
channeldhaka | 291 | 38
kalerkantho | 285 | 37
somoynews | 194 | 38
beanibazarview24 | 109 | 52

Unlabelled | Domain | News Count
---|---|---
mtnews24 | 16194
dakpeon24 | 1567
newsfastcreator | 8836
propernewsbd | 1830
thecityvpn | 14455
usbanglanews | 16099
glamourbd | 6197
jagonews24 | 228

Table 6: Data sources of Bangla Clickbait Datset

###  A.2 Detailed Results

For statistical models, we experimented with Random Forest and Logistic
Regression networks. We passed various types of lexical, syntactical, and
embedding features to these networks to investigate which performs best. For
neural network models, we employ architectures from two previous research
works Zhou et al., 2016; Agrawal, 2016. For Transformer networks, we train
commonly available Bangla pre-trained transformer models in both classic and
semi-supervised GAN manner. Table 7 contains the results of these experiments.

Statistical Classifiers
---
Traditional Linguistic Features |  Logistic Regression |  Random Forest
Unigram (U) |  57.39 |  56.11
Bigram (B) |  29.7 |  53.34
U+B+T |  57.68 |  55.94
C-3 gram |  64.81 |  67.01
C-4 gram |  65.59 |  62.48
C-5 gram |  65.13 |  58.58
C3+C4+C5 |  66.28 |  65.36
All Lexical(L = U+B+T+C3-C5) |  64.29 |  65.6
Parts of Speech(POS) |  33.14 |  40.37
L+POS |  62.23 |  65.97
Embedding Word2Vec (E W) |  53.11 |  51.35
Embedding Fasttext (E F) |  50.19 |  49.4
L+POS+E W |  64.2 |  65.04
L+POS+E F |  64.43 |  65.05
Punctuation (P) |  5.88 |  52.06
L+POS+E W+P |  63.34 |  64.7
L+POS+E N+P |  64.34 |  64.91
All features |  64.73 |  63.26

Transformer Networks |  Classic |  SS-GAN
---|---|---
BERT base multilingual cased |  62.37 |  70.21
Bangla BERT Base |  68.13 |  68.54
Indic-BN-BERT |  72.21 |  73.36
Indic-BN-RoBERTa |  67.76 |  70.52
DistilBERT base multilingual cased |  69.61 |  70.38
Indic-BN-DistilBERT |  71.32 |  72.35
Bangla-Electra |  66.79 |  67.77
Indic-BN-XLM-RoBERTa |  71.82 |  70.75
CSENLP-BanglaBert |  71.72 |  75.13
CSENLP-BanglaBert_Large |  71.66 |  72.07

Neural Networks
---
CNN (Agrawal, 2016) |  35.15
Bi-LSTM (Zhou et al., 2016) |  39.37

Table 7: Detailed result of all experiments conducted on BanglaBait dataset

###  A.3 Difference between Clickbait and Fake news

Although Hossain et al. (2020) published the first dataset of Bangla Fake
news, we don’t focus on the misinformation, fabricated or fake content within
the articles, or their authenticity to detect clickbait in this dataset. The
following two examples explain the difference between fake and clickbait
titles in detail-
Example 1: Buying land on the Moon is the current craze. Explore how you can
do that too!

Example 2: ’Hawa’ got nominated for the Oscars

Example 1 presents an accurate title (verified by renowned news publishers
such as the Kalerkantho and the Somoynews) in a clickbait-style by using
hyperbolic words like ’craze’ and alluring phrases like ’Explore how you can
do that too.’ It proves a clickbait article does not have to be fake to be
clickbait. Example 2, on the other hand, is fake news verified from the
official Facebook page of the movie ’Hawa’, however, the title style is not
exactly luring readers to click, proving that an article can be fake without
being clickbait. In short, clickbait headlines do not necessarily have to be
fake news; they may contain genuine information but in an exaggerated fashion
(Dong et al., 2019). Biyani et al. (2016) includes factually wrong articles in
the ’wrong’ category of clickbait articles.


# Analysis of Fourier ptychographic microscopy with half of the captured
images

Ao Zhou  Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of
Sciences, Shanghai 201800, China  University of Chinese Academy of Sciences,
Beijing 100049, China.  Ni Chen  Shanghai Institute of Optics and Fine
Mechanics, Chinese Academy of Sciences, Shanghai 201800, China  Corresponding
authors<EMAIL_ADDRESS>Haichao Wang  Shanghai Institute of Optics and
Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
University of Chinese Academy of Sciences, Beijing 100049, China.  Guohai Situ
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences,
Shanghai 201800, China

###### Abstract

Fourier ptychography microscopy (FPM) is a new computational imaging technique
that can provide gigapixel images with both high resolution and a wide field
of view (FOV). However, time consuming of the data-acquisition process is a
critical issue. In this paper, we make an analysis on the FPM imaging system
with half number of the captured images. Based on the image analysis of the
conventional FPM system, we then compare the reconstructed images with
different number of captured data. Simulation and experiment results show that
the reconstructed image with half number captured data do not show obvious
resolution degradation compared to that with all the captured data, except a
contrast reduction. In particular in the case when the object is close to
phase-only/amplitude-only, the quality of the reconstructed image with half of
the captured data is nearly as good as the one reconstructed with full data.

##  1 Introduction

It’s well known that in the conventional microscope, the low numerical
aperture (NA) of the objective lens produces a wide FOV image, but induces low
image resolution. FPM is a newly developed computational optical imaging
technique, which breaks the diffraction limit of the objective lens by using
an angularly varying light emitting diode (LED) illumination and a phase
retrieval algorithm [1, 2, 3]. In the FPM system, a programmable LED array is
usually used as the light source. After capturing a stack of low resolution
images under different illumination angles, an iterative phase retrieval
process [4, 5] is used to reconstruct the object’s complex field with enhanced
resolution without sacrificing the wide FOV. Because the FPM breaks the
limitation of the space-bandwidth product (SBP) of the optical system, and
achieves gigapixel imaging, it has a great potential in a variety of
applications, such as biomedical imaging [6, 7, 8, 9, 10, 11], characterizing
unknown optical aberrations of lenses [12, 13].

However, FPM’s high SBP imaging capability is time consuming, limiting its
application to only static objects. This limitation comes from the requirement
of a large amount of low resolution images. In addition, the low illumination
intensity of the LED array induces long exposure time during image
acquisition. To improve the capture efficiency, a variety of techniques have
been proposed. In principle, these techniques can be devided into two
categories. The first one is to improve the FPM illumination. For example,
Kuang _et al._ have been proposed to use a laser instead of the LED array to
enhance the intensity of the illumination, so as to reduce the exposure time
during the capture procedure [14]. Other approaches in this category include
the lighting up of several LEDs simultaneously [10, 15, 16, 17], and non-
uniform sampling of the object’s spectrum [18, 19, 20, 6, 21] to reduce the
required number of images . These techniques usually change the illumination
structure and need to find an optimal strategy, which sacrifices the
simplicity of the original FPM. The other one is to improve the reconstruction
algorithm [22, 23, 24]. For example, the Wirtinger flow optimization algorithm
has been proposed to save around 80% of the exposure time [22]. But this
algorithm increases the calculation cost than the original FPM [1].

In this paper, we propose an alternative approach by analyzing the imaging
process. Based on the analysis, we perform simulations and experiments to
demonstrate our method. In section 2, we make a theoretical analysis on the
FPM imaging process. In section 3 and 4, simulations and experiments are
conducted to prove the analysis.

##  2 Analysis of the FPM imaging system

We suppose that, in a FPM system, a complex object with the transmittance
function of $O(x,y)=A(x,y)\exp[j\phi(x,y)]$ is located at the front focal
plane of the microscopic objective, where $A(x,y)$ and $\phi(x,y)$ are the
amplitude and phase of the object. A plane wave parallel to the optical axis
illuminates the object with a wavelength of $\lambda$. Without considering the
aperture size, the Fourier spectrum $H(u,v)$ located at the back focal plane
of the microscopic objective is written as [25, 26]

$\displaystyle
H(u,v)=\iint\limits_{-\infty}^{\infty}O(x,y)\exp\left[-j\frac{2\pi}{\lambda
f}(xu+yv)\right]\mathrm{d}x\mathrm{d}y,$ (1)

where $f$ is the focal length of the objective, and $(u,v)$ is the spatial
frequency coordinate. In the case that the illumination angle is
$(\theta_{x},\theta_{y})$, the Fourier spectrum becomes

$\displaystyle H_{\theta_{x},\theta_{y}}(u,v)$ $\displaystyle=$
$\displaystyle\iint\limits_{-\infty}^{\infty}O(x,y)\times\exp\left[j2\pi\left(\frac{\sin\theta_{x}}{\lambda}x+\frac{\sin\theta_{y}}{\lambda}y\right)\right]$
$\displaystyle\times\exp\left[-j\frac{2\pi}{\lambda
f}(xu+yv)\right]\mathrm{d}x\mathrm{d}y$ $\displaystyle=$ $\displaystyle
H(u-u_{0},v-v_{0}),$ (2)

which indicates the Fourier spectrum is shifted to $(u_{0},v_{0})$ due to the
illumination angle. It has been proved that $u_{0}=f\sin\theta_{x}$ and
$v_{0}=f\sin\theta_{x}$ [27]. Because of the limited NA of the system, only a
part of the Fourier spectrum transmits the tube lens. Let the radius of the
transmitted sub spectrum $H_{\theta_{x},\theta_{y}}^{sub}(u,v)$ be $r$, it
then can be written as

$\displaystyle H_{\theta_{x},\theta_{y}}^{sub}(u,v)$ $\displaystyle=$
$\displaystyle
H(u-u_{0},v-v_{0})\times\text{circ}\left(\frac{\sqrt{u^{2}+v^{2}}}{r}\right),$
(3)

where $\text{circ}\left({\sqrt{u^{2}+v^{2}}}/{r}\right)$ is the circle
function. Suppose the tube lens has a same focal length as the objective lens,
the captured image under specific illumination angle thus can be written as

$\displaystyle I_{\theta_{x},\theta_{y}}(x,y)$ $\displaystyle=$
$\displaystyle\left|\mathcal{F}^{-1}\left\\{H(u-u_{0},v-v_{0})\times\text{circ}\left(\frac{\sqrt{u^{2}+v^{2}}}{r}\right)\right\\}\right|^{2}$
$\displaystyle=$
$\displaystyle\Bigg{|}O\left(x,y\right)\times\exp\left[j{2\pi}\left(\frac{\sin\theta_{x}}{\lambda}x+\frac{\sin\theta_{y}}{\lambda}y\right)\right]$
$\displaystyle\otimes J\left(x,y\right)\Bigg{|}^{2},$ (4)

where $\mathcal{F}^{-1}$ is the inverse Fourier transform,
$J(x,y)=\frac{\lambda fr}{\sqrt{x^{2}+y^{2}}}J_{1}\left(\frac{2\pi
r\sqrt{x^{2}+y^{2}}}{\lambda f}\right)$ is the inverse Fourier transform of
the circle function, and $J_{1}\left(x,y\right)$ is the first order Bessel
function.

In the case that the object is amplitude-only, i.e., $O(x,y)=A(x,y)$, Eq. 4
can be written as

$\displaystyle I_{\theta_{x},\theta_{y}}(x,y)$ $\displaystyle=$
$\displaystyle\Bigg{|}A(x,y)\times\exp\left[j{2\pi}\left(\frac{\sin\theta_{x}}{\lambda}x+\frac{\sin\theta_{y}}{\lambda}y\right)\right]$
$\displaystyle\otimes J\left(x,y\right)\Bigg{|}^{2}.$ (5)

From Eq. 5, it can be seen that the captured intensity images
$I_{\theta_{x},\theta_{y}}(x,y)$ equals $I_{-\theta_{x},-\theta_{y}}(x,y)$
strictly.

In the case that the object is phase-only, i.e., $O(x,y)=\exp[j\phi(x,y)]$,
Eq. 4 can be written as

$\displaystyle I_{\theta_{x},\theta_{y}}(x,y)$ $\displaystyle=$
$\displaystyle\Bigg{|}\exp\left\\{j\left[\phi(x,y)+{2\pi}\left(\frac{\sin\theta_{x}}{\lambda}x+\frac{\sin\theta_{y}}{\lambda}y\right)\right]\right\\}$
$\displaystyle\otimes J\left(x,y\right)\Bigg{|}^{2}.$ (6)

When the phase-only object is thin enough so that
$\left|{2\pi}\left(x\sin\theta_{x}/\lambda+y\sin\theta_{y}/\lambda\right)\right|\gg\phi(x,y)$,
Eq. 6 can be written as

$\displaystyle I_{\theta_{x},\theta_{y}}(x,y)$ $\displaystyle\approx$
$\displaystyle\left|\exp\left[j{2\pi}\left(\frac{\sin\theta_{x}}{\lambda}x+\frac{\sin\theta_{y}}{\lambda}y\right)\right]\otimes
J\left(x,y\right)\right|^{2}.$ (7)

From Eq. 7, we can find that the assumption is better satisfied when the
$\theta_{x}$ and $\theta_{y}$ are larger, which means that the images captured
at larger illumination angles are more close to the ground-truth than those
that are captured at smaller illumination angles.

In the case that the object is complex, Eq. 4 can be written as

$\displaystyle I_{\theta_{x},\theta_{y}}(x,y)$ $\displaystyle=$
$\displaystyle\Bigg{|}A(x,y)\exp[j\phi(x,y)]$
$\displaystyle\exp\left[j{2\pi}\left(\frac{\sin\theta_{x}}{\lambda}x+\frac{\sin\theta_{y}}{\lambda}y\right)\right]\otimes
J\left(x,y\right)\Bigg{|}^{2}.$ (8)

Compared to phase-only object, influenced by the amplitude $A(x,y)$, the
difference between $I_{\theta_{x},\theta_{y}}(x,y)$ and
$I_{-\theta_{x},-\theta_{y}}(x,y)$ is complex, which is hard to compare.

From the theoretical analysis of the FPM imaging system, we get a preliminary
conclusion that the images captured with circular symmetrical illumination
angles have little intensity difference when the object is amplitude-only or
phase-only. Considering the fact that the FPM uses only intensity images to
reconstruct the wide FOV and high resolution images, we speculate that the
reconstructed images with a half number of low resolution images can reach a
comparable resolution to that with the whole images in the FPM.

##  3 Simulation verification

We performed two simulations to verify the above analysis. In these
simulations, we used a 4$\times$ 0.1 NA microscopic objective and a $15\times
15$ LED array. The wavelength of the LED array was $630\text{\,}\mathrm{nm}$,
and the spacing between two adjacent LEDs was $4\text{\,}\mathrm{mm}$. The
distance between the LED array and the sample was $110\text{\,}\mathrm{mm}$.
The camera sensor has $128\times 128$ pixels, the pitch of which is
$6.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Figure 1:  Amplitude and phase image of the object used in the simulation.

(a)

(b)

(c)

Figure 2:  Simulation 1: Comparison of the images captured under symmetrical
illumination angles using (a) amplitude-only, (b) phase-only and (c) complex
objects.

In the first simulation, we compared the images captured by a microscope under
symmetrical illumination angles. Figure 1 shows the amplitude and phase
profiles of the object used in this simulation. The original object has
$512\times 512$ pixels. The pixel number of the low resolution pictures
captured by the camera is $128\times 128$. To make the object thin enough, the
range of phase value was set to $[0,0.5\pi]$. The object was set to be at the
focal plane of the microscope.

To compare the captured images, we plot the gray values along the horizontal
center lines of the captured images and calculate the root-mean-square error
(RMSE) between them. The images were compared in three cases, i.e., the object
is amplitude-only, phase-only and complex. Figure 2 shows the results.
$\theta_{1}$ and $\theta_{2}$ are the illumination angles. $\theta_{1}$ is
$(\theta_{x},\theta_{y})=(4.2^{\circ},4.2^{\circ})$ and $\theta_{2}$ is
$(\theta_{x},\theta_{y})=(8.3^{\circ},8.3^{\circ})$. Figure 2(a) shows the
results when the object is amplitude-only. The RMSE of the images are all 0 in
$\theta_{1}$ and $\theta_{2}$ respectively, we can observe that the curves are
exactly identical. Figure 2(b) shows the results when the object is phase-
only. The RMSE of the images are 5.71 and 3.63 in $\theta_{1}$ and
$\theta_{2}$ respectively, we can observe that the difference between the
curves become smaller when the illumination angle $\theta_{1}$ is increasing.
Figure 2(c) shows the results of the complex object. The RMSE of the images
are 28.01 and 5.65 in $\theta_{1}$ and $\theta_{2}$ respectively. There is
obvious difference between the curves in different illumination angles. The
simulation results are coincided with the analysis in section 2. Considering
both the numerical analysis and the simulations, we come to a conclusion that
the captured images under symmetrical illumination angles are absolutely the
same when the sample is amplitude-only, are almost the same when the sample is
thin phase-only, and are different when the object is complex.

(a) Amplitude-only object reconstructions.

(b) Phase-only object reconstructions.

(c) Complex object reconstruction.

Figure 3:  Simulation 2: Comparison of the FPM reconstructions using a/an (a)
phase-only, (b) amplitude-only, and (c) complex object. Compared with the
images in the left column, the RMSE of the images in the center column are
6.25, 17.23, 12.82, 28.84 respectively.

The second simulation was performed to verify our speculation. The amplitude
and phase profiles of the object used in the FPM simulation are the same as in
the first simulation. The results are plotted in Fig. 3. In Fig. 3(a-c), the
left and center columns show the reconstructed amplitude and phase with whole
$(15\times 15)$ and half $(15\times 8)$ camera images respectively. The right
column shows the intensity comparison between the left and center columns
along the middle horizontal intersection. Figure 3(a) and (b) show that the
shape and distribution of the amplitude and phase profiles are similar. We
also performed a simulation when the object is complex. From the results shown
in Fig. 3(c), we can observe obvious difference between using whole and half
captured images. The reason is that the captured images are totally different
when the object is illuminated by symmetrical angles. Besides, during the
iteration process of phase retrieval algorithm in FPM, there are some
crosstalk between the amplitude and phase reconstructions. From the simulation
results, we observe that the reconstructions using half captured images do not
show obvious resolution degradation in the cases of amplitude-only and phase-
only objects.

##  4 Experimental verification

We built an FPM system by replacing the light source of an Olympus IX 73
inverted microscope with a programmable LED array ($32\times 32$ LEDs,
$4\text{\,}\mathrm{mm}$ spacing) controlled by an Arduino. The LEDs have a
central wavelength of $629\text{\,}\mathrm{nm}$ and a bandwidth of
$20\text{\,}\mathrm{nm}$.

Two experiments were performed to verify our speculation. One used a USAF-1951
resolution target as the object, which can be regarded as an amplitude-only
object. The other one used a microscopic object that has phase information.
Both samples were imaged with a 4$\times$ 0.1 NA microscopic objective and a
complementary metal oxide semiconductor (CMOS) camera (PCO. edge 4.2). The
pixel size of the CMOS is $6.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

(a)

(b)

(c)

(d)

(e)

Figure 4:  (a) is the captured raw image of the USAF resolution target, (b) is
image within the red rectangle in (a). (c)(d) are the reconstructed amplitudes
with the whole and half of the captured images. Compared with (c), the RMSE of
(d) is 10.90. (e) is the amplitude profiles along the red horizontal lines in
(c) and (d).

In the first experiment, the central $15\times 15$ LEDs of the array were on
sequentially so that 225 low resolution images were captured by the sCMOS
camera. The exposure time of the camera was $600\text{\,}\mathrm{ms}$ per
acquisition. The distance between the LED array and the sample was
$108\text{\,}\mathrm{mm}$. The expected synthesized NA of the imaging system
is 0.45. The experimental results are shown in Fig. 4. Figure 4(b) is one of
the segment of the original captured images. Figure 4(c) shows the
reconstructed result using 225 images. Figure 4(d) shows the reconstructed
result using 120 images. Figure 4(e) shows the intensity profile along the red
vertical red lines in Figs. 4(c) and 4(d). From the two curves, we can observe
that their shape are almost the same, but the gray values of the
reconstruction with all the acquired images are higher than that using half of
them. This means that the image resolution of Figs. 4(c) and 4(d) has no
significant difference except the image contrast.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5: (a) is the captured raw image of a plant root, (b) is the image
within the red rectangle in (a). (c)(d) are the reconstructed amplitudes with
the whole and half of the captured images. Compared with (c),The RMSE of (d)
is 12.91. (f)(g) are the reconstructed phases with the whole and half of the
captured images. Compared with (f), the RMSE of (g) is 10.67. (e) is the
amplitude profiles along the red horizontal lines in (c) and (d). (h) is the
phase profiles along the red horizontal lines in (f) and (g).

In the second experiment, the central $17\times 17$ LEDs of the array were on
sequentially so that 289 low resolution images were captured. The distance
between the LED array and the sample was $113.5\text{\,}\mathrm{mm}$. The
expected synthesized NA of the imaging system is 0.48. The other experimental
specifications were the same as in the first experiment. The experimental
results are shown in Fig. 5. Figure 5(b) is one of the segment of the original
captured images. Figure 5(c) and 5(f) show the reconstructed amplitudes and
phases using 289 images. Figure 5(d) and 5(g) show the reconstructed amplitude
and phase using 153 images. Figure5(e) shows the intensity profile along the
red horizon lines in Fig. 5(c) and 5(d). Figure 5(h) shows the intensity
profile along the red horizon lines in Figs. 5(f) and 5(g). We can also
observe that the reconstructions don’t show obvious resolution degradation
both in the reconstructed amplitudes and phases, only the image contrast
reduction can be observed. Because the real object’s amplitude and phase exist
similarity in structure, they perform like a phase-only object. Therefor,
there is no significant crosstalk between the amplitude and phase.

##  5 Conclusion

Under the assumption of thin biological samples of FPM, both the theoretical
analysis and simulations of a microscopic imaging system have shown that
images captured with circular symmetrical illumination angles have little
intensity difference when the sample is amplitude-only or phase-only,
therefore, the FPM reconstructions show insignificant difference. In the FPM
experiments, both amplitude-only and complex objects were used to perform the
verification. The reconstructions didn’t show obvious resolution degradation
between using full and half images, only the image contrast reduction can be
observed. It also shows the reconstructions are degraded very less when the
phase and amplitude of the object have similar distribution. Since many
biological samples satisfy this condition, half number of captured images can
be used in these cases. Owing to half number images reduction, time costs of
both image capture and computational processing were also reduced in half.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(61705241, 61327902 and 61377005), Natural Science Foundation of Shanghai
(17ZR1433800), and Chinese Academy of Sciences (QYZDB-SSW-JSC002).

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