# Merging and band transition of bound states in the continuum in leaky-mode
photonic lattices

Sun-Goo Lee<EMAIL_ADDRESS>Department of Data Information and Physics,
Kongju National University, Gongju, 32588, Republic of Korea  Institute of
Application and Fusion for Light, Kongju National University, Cheonan, 31080,
Republic of Korea  Seong-Han Kim  Advanced Photonics Research Institute,
Gwangju Institute of Science and Technology, Gwangju 61005, Republic of Korea
Wook-Jae Lee  Department of Data Information and Physics, Kongju National
University, Gongju, 32588, Republic of Korea  Institute of Application and
Fusion for Light, Kongju National University, Cheonan, 31080, Republic of
Korea

###### Abstract

Bound states in the continuum (BICs) theoretically have the ability to confine
electromagnetic waves in limited regions with infinite radiative quality ($Q$)
factors. However, in practical experiments, resonances can only exhibit finite
$Q$ factors due to unwanted scattering losses caused by fabrication
imperfections. Recently, it has been shown that ultrahigh-$Q$ guided-mode
resonances (GMRs), which are robust to fabrication imperfections, can be
realized by merging multiple BICs in momentum space. In this study, we
analytically and numerically investigate the merging and band transition of
accidental BICs in planar photonic lattices. Accidental BICs can merge at the
edges of the second stop band, either with or without a symmetry-protected
BIC. We show that as the thickness of the photonic lattice gradually
increases, the merged state of BICs transitions from the upper to the lower
band edge. Using coupled-mode analysis, we present the analytical merging
thickness at which multiple accidental BICs merge at the second-order $\Gamma$
point. Our coupled-mode analysis could be beneficial for achieving
ultrahigh-$Q$ GMRs in various photonic lattices composed of materials with
different dielectric constants.

††preprint: APS/123-QED

##  I Introduction

In photonics, bound states in the continuum (BICs) refer to the special
eigensolutions of electromagnetic wave equations in non-Hermitian physical
systems [1, 2, 3]. Unusually localized BICs can exhibit theoretically infinite
radiative $Q$ factors, even though they can couple with outgoing radiative
waves that can carry electromagnetic energy [4]. Due to their remarkable
ability to increase light-matter interactions in confined regions, extensive
studies on BICs have been conducted for basic research and practical
applications in recent years [5, 6, 7, 8, 9, 10, 11]. In theoretical or
numerical studies, BICs with infinite $Q$ factors can be found in various open
photonic systems, including photonic crystals [12, 13], metasurfaces [14, 14,
16], and plasmonic structures [17, 18, 19]. In experimental implementations,
however, unwanted scattering losses attributed to fabrication imperfections,
such as surface roughness, imperfect boundaries of individual scattering
elements, and structural disorder, give rise to the considerable reduction of
radiative $Q$ factors. In practical photonic structures, the theoretically
perfect BICs appear as quasi-BICs with finite $Q$ factors due to out-of-plane
coupling with continuous radiating waves.

Subwavelength photonic crystal slabs can exhibit abundant BICs if the lattices
possess time-reversal symmetry, up-down mirror symmetry, and proper rotational
symmetry [20, 21]. Recently, topologically-protected BICs in photonic crystal
slabs have attracted particular interest because they are stable [22, 23, 24,
25]. Furthermore, current nanofabrication technology is sufficient to
implement BICs in photnoic crystal slabs [26, 27, 28]. Due to their
topological nature, accidental BICs can be moved along the band diagram by
varying the structural parameters while maintaining the symmetry of the
systems [29, 30]. It has also been demonstrated that multiple accidental BICs
can be merged at the Brillouin zone center, which is also referred to as the
lattice $\Gamma$ point in a photonic band structure, by adjusting structural
parameters [31, 32]. In the merged states of multiple BICs, undesired out-of-
plane scattering losses can be significantly suppressed by increasing the
radiative $Q$ factors of nearby eigenstates over a finite range of
wavevectors. Merging BICs is important for practical applications because it
provides a powerful mechanism to achieve robust ultrahigh-$Q$ resonances, that
enhance light-matter interactions.

Figure 1:  (a) Schematics of a 1D photonic lattice and homogeneous dielectric
waveguide. Leaky modes in the photonic lattice lose their electromagnetic
energy over time, while nonleaky guided modes in the homogeneous waveguide are
protected by total internal reflection. (b) The dispersion curve for the
fundamental $\mathrm{TE}_{0}$ mode of a homogeneous dielectric waveguide. (c)
Conceptual illustration of photonic band structures and transmission curve
through a 1D photonic lattice. Leaky modes of the photonic lattice in the
lower (upper) dispersion curve at the Bloch wavevector $k_{z}=|\delta|$ around
the second band gap ($j=2$) originate from the propagating modes of the
homogeneous waveguide at the wavevector $\beta=K-|\delta|$
($\beta=K+|\delta|$). At the second stop band, one band edge mode generate a
GMR via coupling with normally incident plane wave, whereas the other edge
mode becomes a symmetry-protected BIC.

The concept of merging multiple BICs has been introduced recently and has been
investigated in only a few studies so far. In this paper, we present
analytical and numerical results on the merging and band transition of
accidental BICs in one-dimensional (1D) and two-dimensional (2D) photonic
lattice slabs. Guided modes become BICs when out-of-plane radiation completely
disappears. We present an analytical expression, an overlap integral in the
slab region, associated with the out-of-plane radiation of the guide modes,
and show that accidental BICs can appear at generic $k$ points in reciprocal
space where the value of the overlap integral approaches zero. As the
thickness of photonic lattice slabs increases, accidental BICs gradually move
downward along the upper dispersion curve and eventually meet at the lattice
$\Gamma$ point. As the thickness is further increased, the merged state of
BICs transitions from the upper to lower band edges at the lattice $\Gamma$
point. The BICs move down and away from each other with further increase in
thickness. Merging thicknesses, at which multiple accidental BICs merge at the
second-order $\Gamma$ point, are calculated through the coupled-mode analysis.
Our coupled-mode analysis could be applied to achieve ultrahigh-$Q$ resonances
in diverse 1D and 2D photonic lattices made of materials with different
dielectric constants.

Figure 2:  Merging and band transition of accidental BICs in a 1D leaky-mode
photonic lattice. (a) FEM-simulated dispersion curves near the second stop
band for four different values of $t$. Insets with blue and red colors show
the spatial distribution of the electric field ($E_{y}$) for the band edge
modes at the $y=0$ plane. The vertical dotted lines indicate the mirror planes
in the computational cells. (b) Simulated radiative $Q$ factors of the upper
and lower dispersion curves. The merged state of BICs transitions from the
upper to lower dispersion curve as $t$ increases, while one symmetry-protected
BIC with $+1$ charge always appears at the upper band edge. The structural
parameters used in the FEM simulations were $\epsilon_{0}=4$,
$\Delta\epsilon=0.5$, $\epsilon_{s}=1$, and $\rho=0.35$.

##  II Merging and band transition of BICs in 1D photonic lattices

Figure 1(a) compares a conventional homogeneous slab waveguide and a
representative 1D photonic lattice for studying the merging and band
transition of BICs in this study. The waveguide and photonic lattice are
surrounded by a surrounding medium with a dielectric constant $\epsilon_{s}$.
The waveguide consists of a homogeneous material with a dielectric constant
$\epsilon_{w}$, and the 1D photonic lattice is composed of high
($\epsilon_{h}$) and low ($\epsilon_{l}$) dielectric constant materials. The
period is $\Lambda$, the width of the high dielectric constant part is
$\rho\Lambda$, and the thickness of the photonic lattice is $t$. In the
homogeneous waveguide with $\epsilon_{w}>\epsilon_{s}$, guided modes posses
purely real eigenfrequencies $\Omega=\Omega_{\mathrm{Re}}$ and propagate along
the waveguide without out-of-plane radiation because they are perfectly
protected by the total internal reflection (TIR). As illustrated in Fig. 1(b),
dispersion curves for waveguide modes should be located in yellow region below
the light line. The 1D lattice can also support transverse electric
($\mathrm{TE}$) guided modes because its average dielectric constant
$\epsilon_{0}=\epsilon_{l}+\rho(\epsilon_{h}-\epsilon_{l})$ is larger than
$\epsilon_{s}$. With $\Delta\epsilon=\epsilon_{h}-\epsilon_{l}>0$, as
illustrated in Fig. 1(c), all Bloch guided modes can be plotted in the
irreducible Brillouin zone and photonic band gaps open at the Bragg condition
$k_{z}=jK/2$, where $k_{z}$ is the Bloch wavevector, $K=2\pi/\Lambda$ is the
magnitude of the grating vector, and $j$ is an integer. Near the second stop
band in the white region, guided modes posses complex eigenfrequencies
$\Omega=\Omega_{\mathrm{Re}}+i~{}\Omega_{\mathrm{Im}}$ and exhibit interesting
leaky-wave effects such as BICs and guided-mode resonances (GMRs), also known
as guided resonances, via zero-order diffraction [33, 34]. Guided modes in the
yellow region are not associated with the leaky-wave effects because they do
not couple with external waves in the radiation continuum, and Bloch modes in
the grey region are less practical because they generate unwanted higher-order
diffracted waves as well as the desired zero-order diffraction. In general,
numerous leaky guided modes coexist in the photonic lattices with a slab
geometry and the various modes possess their own dispersion curves, photonic
band gaps, and high-$Q$ BICs. In this study, we focus our attention to the
BICs in the vicinity of the second stop band ($j=2$) open by fundamental
$\mathrm{TE}_{0}$ mode, as this simplest case brings out the key properties of
the accidental BICs. In fact, most of important studies on the BICs are
associated the leaky Bloch modes in the vicinities of the second stop bands.
To elucidate the fundamental properties of BICs, we employ a semianalytical
dispersion model and rigorous finite-element method (FEM) simulations.

Merging and band transition of BICs are possible because they are
topologically protected [29, 30]. Figure 2(a) illustrates the evolution of the
dispersion curves of guided modes in the vicinity of the second stop band
under variation of the thickness $t$. No noticeable changes in the dispersion
curves are observed with variations in thickness. The simulated spatial
electric field ($E_{y}$) distributions in the insets show that regardless of
$t$, upper band edge modes with asymmetric field distributions become nonleaky
symmetry-protected BICs. Conversely, lower band edge modes with symmetric
field distributions can be either leaky or nonleaky depending on the thickness
values. The merging and band transition of accidental BICs can be clearly seen
in Fig. 2(b), which depicts $Q$ factors as a function of $k_{z}$. When
$t=1.13~{}\Lambda$, two accidental BICs with $-1$ charge and one symmetry-
protected BIC with $+1$ charge appear, resulting in three isolated peaks in
the $Q$ factor curve of the upper band. Due to the $180^{\circ}$ rotational
symmetry of the 1D lattice, two accidental BICs appear in pairs at wavevectors
$k_{z}=\pm 0.01995~{}K$ simultaneously. As the value of $t$ increases from
$1.13~{}\Lambda$, two accidental BICs gradually move downward along the upper
dispersion curve and approach to $\Gamma$ point where a symmetry-protected BIC
is located. When $t=1.14489~{}\Lambda$, two accidental BICs and one symmetry-
protected BIC merge at the lattice $\Gamma$ point, resulting in a single
enhanced peak in the $Q$ factor curve. Compared to the isolated symmetry-
protected or accidental BICs, the merged state at $t=1.14489~{}\Lambda$
exhibits significantly improved radiative $Q$ factors over a wide range of
wavevectors. As the value of $t$ further increases to $1.16282~{}\Lambda$,
simultaneous peaks in the $Q$ factor curves are observed at both the upper and
lower band edges. The peak in the upper band is attributed to the symmetry-
protected BIC, evident from the asymmetric field distributions shown in the
inset of Fig. 2(a). We propose that the interband transition of the merged
state of BICs, induced by the variation of $t$, leads to the enhanced peak at
the lower band edge. Upon further increasing $t$ beyond $1.16282~{}\Lambda$,
the merged state at the $\Gamma$ point splits into two isolated peaks with
$-1$ charge.

##  III Coupled-mode anaylsis

In the planar waveguide structures shown in Fig. 1(a), the dispersion
relations of guided modes, including eigenfrequencies and radiative $Q$
factors, can be obtained by solving the 1D wave equation given by [35]

$\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
z^{2}}\right)E_{y}(x,z)+\epsilon(x,z)k_{0}^{2}E_{y}(x,z)=0,$ (1)

where $k_{0}$ denotes the wave number in free space. For the homogeneous
waveguide plotted in Fig. 1(b), Eq. (1) can be solved analytically and spatial
electric field distribution of the $\mathrm{TE}_{0}$ mode satisfies $E_{x}=0$,
$E_{z}=0$, and

$E_{y}(x,z)=\varphi(x)e^{i\beta z},$ (2)

where $\varphi(x)$ characterizes the transverse profile of the
$\mathrm{TE}_{0}$ mode and $\beta$ denotes the propagation constant along the
$z$ direction [36]. The $\mathrm{TE}_{0}$ mode can propagate without radiative
loss along the waveguide due to TIR.

For the photonic lattices in Fig. 1(a), Eq. (1) can be solved numerically by
expanding the electric field $E_{y}(x,z)$ as a Bloch form and the periodic
dielectric function $\epsilon(x,z)$ in a Fourier series [37]. Using the
current state of computational software and hardware, it is possible to obtain
exact solutions of Eq. (1) for certain system parameters. However, relying
solely on direct numerical simulations may not be sufficient to fully
comprehend the fundamental properties of BICs, even though numerical
calculations can provide accurate dispersion relations of leaky guided modes.
To gain a deeper understanding of the merging and band transition of BICs, we
employ the semianalytical dispersion model introduced by Kazarinov and Henry
(KH) [38]. In this model, the spatial dielectric function and electric field
are approximated as:

$\displaystyle\epsilon(x,z)$
$\displaystyle=\gamma_{0}+\sum_{n=1,2}\left(\gamma_{n}e^{inKz}+\gamma_{-n}e^{-inKz}\right),$
(3) $\displaystyle E_{y}(x,z)$
$\displaystyle=\left(Ae^{iKz}+Be^{-iKz}\right)\varphi(x)+\Delta E(x,z).$ (4)

In Eq. (3), the Fourier coefficient $\gamma_{n}$ has a zero value outside the
periodic layer, and $\gamma_{0}=\epsilon_{0}$. In Eq. (4),
$A(z)\sim\exp(ik_{z}z)$ and $B(z)\sim\exp(-ik_{z}z)$ are the slowly varying
envelopes of the two counter-propagating waves, and $\Delta E(x,z)$ represents
the diffracted wave radiating away from the periodic structure. For symmetric
lattices with $\epsilon(x,z)=\epsilon^{\ast}(x,-z)$ and
$\gamma_{-n}=\gamma_{n}$, near the second stop band, the eigenfrequencies of
guided modes can be written as

$\Omega(k_{z})=\Omega_{0}-\frac{ih_{1}\pm\sqrt{k_{z}^{2}+(h_{2}+ih_{1})^{2}}}{Kh_{0}},$
(5)

where $\Omega_{0}$ is the Bragg frequency under vanishing index modulation,
and the coupling coefficients $h_{n=0,1,2}$ are given by

$\displaystyle h_{0}$ $\displaystyle=$
$\displaystyle\frac{k_{0}}{K}\int_{-\infty}^{\infty}{\gamma_{0}}(x)\varphi(x)\varphi^{*}(x)dx,$
(6) $\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle
i\frac{k_{0}^{4}\gamma_{1}^{2}}{2K}\int_{-t}^{0}\int_{-t}^{0}G(x,x^{\prime})\varphi(x^{\prime})\varphi^{*}(x)dx^{\prime}dx,$
(7) $\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle\frac{k_{0}^{2}\gamma_{2}}{2K}\int_{-t}^{0}\varphi(x)\varphi^{*}(x)dx,$
(8)

where $G(x,x^{\prime})$ denotes the Green’s function for diffracted fields [8,
33]. Equation (5) indicates that the leaky stop band with two band edges
$\Omega^{b}=\Omega_{0}+h_{2}/(Kh_{0})$ and
$\Omega^{a}=\Omega_{0}-(h_{2}+i2h_{1})/(Kh_{0})$ opens at $k_{z}=0$. For
symmetric lattices, the coupling coefficient $h_{1}$ is generally a complex
value, while $h_{0}$ and $h_{2}$ are real values, irrespective of the lattice
parameters. With the time dependence of $\exp(-i\Omega t)$, Bloch modes near
the second stop band tend to lose their electromagnetic energy over time
because they have complex frequencies. However, one of the band edge modes
with a purely real eigenfrequency $\Omega^{b}$ can become a nonleaky symmetry-
protected BIC at the lattice’s $\Gamma$ point.

By investigating the dispersion relations in Eq. (5) with the associated
coupling coefficients $h_{0}$, $h_{1}$, and $h_{2}$, one can notice that
$\mathrm{Im}[\Omega]$ becomes zero when $\mathrm{Re}[h_{1}]$ approaches zero.
With $\mathrm{Re}[h_{1}]=0$, dispersion relations in Eq. (5) can be rewritten
as

$\Omega(k_{z})=\Omega_{0}+\frac{\mathrm{Im}[h_{1}]\pm\sqrt{k_{z}^{2}+(h_{2}-\mathrm{Im}[h_{1}])^{2}}}{Kh_{0}}.$
(9)

Equation (9) shows that accidental BICs with purely real eigenfrequencies can
be observed at any generic $k$ point, including the lattice $\Gamma$ point.
Due to their topological nature, these accidental BICs do not destroy but
rather move along dispersion curves as the lattice parameters change
continuously. In 1D photonic lattices possessing $180^{\circ}$ rotational
symmetry, two accidental BICs should appear in pairs at
$k_{z}=\pm|k_{\text{BIC}}|$ simultaneously. It is important to note that the
formation of accidental BICs is independent of the existence of symmetry-
protected BICs. Thus, at the asymmetric edge, two accidental BICs and one
symmetry-protected BIC can merge as illustrated in Fig. 2 when
$\mathrm{Re}[h_{1}]$ approaches 0 due to a variation in lattice parameters. On
the other hand, at the symmetric edge, two accidental BICs can merge with the
appropriate lattice parameters. In previous studies [8, 34], it was shown that
the position of a symmetry-protected BIC changes from the upper to lower band
edge through the band flip, also known as the topological phase transition.
This transition can be achieved by varying the value of the parameter $\rho$.
Before the band flip, as shown in Fig. 2 with $\rho=0.35$, accidental BICs can
merge with a symmetry-protected BIC at the upper band edge. Conversely, after
the band flip with appropriate value of $\rho$, the merged state with a
symmetry-protected BIC appears at the lower band edge.

We now present an analytical expression that is associated with the formation
and band transition of accidental BICs. By examining the value of $h_{1}$ with
the appropriate Green’s function [8], we can demonstrate that
$\mathrm{Re}(h_{1})$ approaches zero when the overlap integral given by

$\Sigma=\int_{-t}^{0}e^{i\mu x}\varphi(x)dx,$ (10)

approaches zero; see Supplemental Material [39] for details. In Eq. (10),
$\mu=\sqrt{\epsilon_{0}k_{0}^{2}-\delta^{2}}$, and the transverse profile of
the $\mathrm{TE}_{0}$ mode is given by $\varphi(x)=ce^{+i\alpha
x}+c^{*}e^{-i\alpha x}$, where
$\alpha=\sqrt{\epsilon_{0}k_{0}^{2}-\beta^{2}}$, and the complex coefficient
$c$ is set to satisfy $\int_{-\infty}^{\infty}|\varphi(x)|^{2}dx=1$. As shown
in Figs. 1(b) and 1(c), waveguide modes with a positive (negative) value of
$\delta$ at $\beta=K+\delta$ correspond to the leaky modes in the upper
(lower) dispersion curve of photonic lattices at the Bloch wavevector
$k_{z}=|\delta|$. Merging and band transition of accidental BICs illustrated
in Fig. 2 can be explained by investigating analytical BIC points
$\delta_{\text{BIC}}$, where the overlap integral $\Sigma$ becomes zero, as a
function of the thickness $t$. Figure 3 illustrates the calculated
$\delta_{\text{BIC}}$ with $\Sigma=0$. In the calculations, the value of $t$
varies from $\Lambda$ to $1.3~{}\Lambda$, and $\beta$ changes in the small
discrete step of $10^{-5}~{}K$. $\delta_{\text{BIC}}(t)$, represented by the
blue solid line, is defined as the first value at which the value of
$|\Sigma|$ becomes smaller than $10^{-5}$ for a given $t$. Figure 3 indicates
that as the thickness gradually increases from $\Lambda$,
$\delta_{\text{BIC}}$ monotonically moves downward and reaches $\delta=0$
($\beta=K$), which corresponds to the second-order $\Gamma$ point, when the
thickness is $t_{0}=1.15464~{}\Lambda$. As the thickness increases further,
$\delta_{\text{BIC}}$ continues to move downward and gets away from the
$\Gamma$ point.

Figure 3:  The locations of the accidental BICs, $\delta_{\text{BIC}}$,
represented by a blue solid line as a function of $t$, are obtained from the
1D coupled-mode analysis. The overlap integral $\Sigma$ in Eq. (10) becomes
zero at $\delta_{\text{BIC}}$. FEM-simulated BIC points, $k_{\text{BIC}}$,
denoted by magenta circles, are also plotted for comparison. In the
calculation of $\delta_{\text{BIC}}$, $\epsilon_{0}$ and $\epsilon_{s}$ are
fixed to 4 and 1, respectively, so that the analytically obtained
$\delta_{\text{BIC}}$ explains the evolution of the accidental BICs simulated
by FEM in Fig. 1. The parameter $t$ was used as the vertical axis, such that
BIC points move downward as $t$ increases. Since the eigenfrequencies of
guided modes get lower as $t$ increases, the FEM-simulated magenta circles
mimic the dispersion curves represented in Fig. 2(a).

The analytical BICs in Fig. 3 and the dispersion curve in Fig. 1(b) indicate
that the eigenfrequencies of accidental BICs decrease continuously with
increasing thickness. However, the effects of dielectric constant modulation,
such as band folding and photonic band gap, cannot be observed in the
analytical BICs represented by the blue line in Fig.3, as the overlap integral
$\Sigma$ in Eq.(10) is determined by the transverse mode profile $\varphi(x)$
and the Green’s function for the unmodulated homogeneous waveguide slab of
thickness $t$. To verify whether accidental BICs indeed occur as $\Sigma$
approaches zero, we conducted additional investigations of the BIC points
$k_{\text{BIC}}$ using FEM simulations. The results are represented as magenta
circles in Fig. 3. Due to the mirror symmetry of the photonic lattice, FEM-
simulated accidental BICs with infinite $Q$ factors appear in pairs at
$\pm|k_{\text{BIC}}|$. When the thickness ($t<t_{0}$) is far from
$t_{0}=1.15464~{}\Lambda$, the values of $\delta_{\text{BIC}}$ and
$k_{\text{BIC}}$ agree well. However, as $t$ approaches $t_{0}$, the FEM-
simulated values of $k_{\text{BIC}}$ slightly deviate from the analytic values
of $\delta_{\text{BIC}}$. The deviation $|k_{\text{BIC}}-\delta_{\text{BIC}}|$
increases until $t$ approaches $t_{u}=1.14489~{}\Lambda$, where
$k_{\text{BIC}}$ reaches the $\Gamma$ point. As $t$ further increases beyond
$t_{u}$, the BIC point is not found for a while, but when
$t_{l}=1.16282~{}\Lambda$, the BIC point appears again at the $\Gamma$ point.
The deviation $|k_{\text{BIC}}-\delta_{\text{BIC}}|$ decreases with additional
increases in thickness ($t>t_{l}$), and $k_{\text{BIC}}$ agrees well with
$\delta_{\text{BIC}}$ if $t$ is far away from $t_{l}$. In the vicinity of the
second-order $\Gamma$ point, photonic band gaps open by the coupling between
two counterpropagating waves [8]. Since the coupling strength is maximum at
$k_{z}=0$ and decreases rapidly as the Bloch wavevector moves away from the
$\Gamma$ point, the deviation $|k_{\text{BIC}}-\delta_{\text{BIC}}|$ could be
noticeable only around the $\Gamma$ point. In the FEM simulations, there exist
a thickness range, $t_{u}<t<t_{l}$, where accidental BIC can not be found due
to the presence of a photonic band gap. At $t=t_{u}$ and $t=t_{l}$, two
accidental BICs meet at the upper and lower band edges, respectively,
resulting in the formation of merged BIC states. A symmetry-protected BIC can
be included in the merged state at the upper or lower band depending on the
value of lattice parameter $\rho$, not $t$.

Figure 4:  Analytical and FEM-simulated normalized merging thicknesses as a
function of $\epsilon_{0}$ for (a) $\epsilon_{s}=1$ and (b)
$\epsilon_{s}=2.25$. Analytical results $t_{0}/\Lambda$ are beneficial to find
the merging thicknesses $t_{u}/\Lambda$ and $t_{l}/\Lambda$. In the FEM
simulations, structural parameters $\rho=0.35$ and $\Delta\epsilon=0.5$ are
kept to constant.

##  IV Ultrahigh-$Q$ resonances robust to fabrication imperfections

Our coupled-mode analysis reveals that the value of $\Sigma$ can be tuned to
zero at the second-order $\Gamma$ point of photonic lattices by adjusting the
lattice parameters. This tuning can cause accidental BICs to merge, resulting
in interesting ultrahigh-$Q$ resonances. The overlap integral $\Sigma$ in Eq.
(10) mathematically depends on the lattice parameters $t$, $\epsilon_{0}$, and
$\epsilon_{s}$. However, thickness variation is essential to achieve merged
states of BICs. While positions of accidental BICs can be substantially moved
along dispersion curves with thickness variation, BICs only move slightly with
variations of $\epsilon_{0}$ and $\epsilon_{s}$. In conventional experimental
configurations at wavelengths around $\lambda=1550$ nm, the value of
$\epsilon_{s}$ typically ranges from 1 to 2.25, and $\epsilon_{0}$ varies from
3 to 12. For photonic lattices with these typical values of $\epsilon_{0}$ and
$\epsilon_{s}$, it is meaningful to investigate the normalized merging
thickness $t_{0}/\Lambda$, where $\Sigma=0$. Red lines in Figs. 4(a) and 4(b)
display the calculated $t_{0}/\Lambda$ as a function of $\epsilon_{0}$ for
$\epsilon_{s}=1$ and $\epsilon_{s}=2.25$, respectively. FEM-simulated merging
thicknesses $t_{u}/\Lambda$ and $t_{l}/\Lambda$, where accidental BICs merge
at the upper and lower band edges, respectively, are also displayed for
comparison. Figure 4 demonstrates that $t_{0}/\Lambda$ obtained from Eq. (10)
could provide useful reference for achieving merged states of BICs for
ultrahigh-$Q$ resonances. In Fig. 4, the values of $t_{0}/\Lambda$ change from
$1.14588$ to $1.16981$ when $\epsilon_{s}=1$, and from $1.08116$ to $1.16066$
when $\epsilon_{s}=2.25$. As the value of $\epsilon_{0}$, representing the
averaged dielectric constant in the waveguide region, decreases, the
calculated and simulated merging thicknesses decrease. Figure 4 also shows
that a higher dielectric constant $\epsilon_{s}$ in the cladding region
results in a decrease in merging thicknesses. Changes in $\epsilon_{0}$ and
$\epsilon_{s}$ have a limited impact on the analytical and FEM-simulated
merging thicknesses. The dependency of merging thicknesses on the dielectric
constants $\epsilon_{0}$ and $\epsilon_{s}$ may be attributed to the strong
confinement of guided modes in the waveguide and the scaling property of
Maxwell’s equations [40].

##  V Merging and band transition of BICs in 2D photonic lattices

We also investigate the merging and band transition of BICs in 2D photonic
crystals with a thin-film geometry. As illustrated in Fig. 5(a), we use a
simple 2D periodic slab composed of square arrays of square-shaped high
dielectric constant ($\epsilon_{h}$) materials in the background medium with a
low dielectric constant ($\epsilon_{l}$). Here, the averaged dielectric
constant is given by $\epsilon_{0}=\epsilon_{l}+\rho^{2}\Delta\epsilon$. FEM-
simulated dispersion curves plotted in Fig. 5(b) shows that there are four
bands, A, B, C, and D, in the vicinity of the second-order $\Gamma$ point.
Additionally, spatial electric field ($|\mathbf{E}|$) distributions of four
band edge modes and radiative $Q$ factors displayed in Figs. 5(c) and 5(d),
respectively, show that the band edge modes in A and B are nonleaky symmetry-
protected BICs. Conversely, band edge mode in C and D are degenerate and
generally radiative out of the photonic crystal slab. Merging and band
transition of accidental BICs can be clearly seen in Fig. 5(e), which depicts
the evolution of radiative $Q$ factors as the slab thickness varies. The
radiative $Q$ factors are simulated along the $\Gamma\mathrm{X}$ and
$\Gamma\mathrm{M}$ directions in the Brillouin zone. In this study, we
investigate the merging and band transition of BICs in the highest band A and
the lowest band D, for convenience. When $t=1.161~{}\Lambda$, two accidental
BICs with $\pm 1$ charge and one symmetry-protected BIC with $-1$ charge
appear, resulting in three distinct peaks in the $Q$ factor curve in band A.
Accidental BICs with $+1$ and $-1$ charge approach the symmetry-protected BICs
along the $\Gamma\mathrm{X}$ and $\Gamma\mathrm{M}$ directions, respectively.
Due to the $C_{4}$ rotational symmetry of the 2D lattice, four sets of
accidental BICs with $+1$ charge and four sets with $-1$ charge appear
simultaneously. As the value of $t$ increases from $1.161~{}\Lambda$, eight
accidental BICs gradually move downward along the band A and approach to
$\Gamma$ point where a symmetry-protected BIC is located. When
$t=1.176~{}\Lambda$, eight accidental BICs and one symmetry-protected BIC
merge at the lattice $\Gamma$ point, resulting in a noticeably enhanced peak
in the $Q$ factor curve. When $t=1.18072~{}\Lambda$, two peaks are observed in
the $Q$ factor curves at the edges of band A and band D. The peak in band A is
attributed to the symmetry-protected BIC, while the enhanced peak in band D
arises from the merged state of accidental BICs. As $t$ is increased beyond
$1.18072~{}\Lambda$, the merged state at the $\Gamma$ point splits into
isolated peaks with $+1$ and $-1$ charges, and these peaks move downward along
band D.

Figure 5:  Merging and band transition of accidental BICs in a 2D photonic
crystal slab. (a) Schematic of a 2D photonic crystal slab. (b) FEM-simulated
dispersion curves near the second-order $\Gamma$ point. (c) Simulated spatial
electric field ($|\mathbf{E}|$) distributions of four band edge modes at $z=0$
plane. White arrows represent in-plane electric field vectors (d),(e)
Simulated radiative $Q$ factors. The structural parameters used in the
simulations were $\epsilon_{0}=9$, $\epsilon_{s}=1$, $\Delta\epsilon=1$, and
$\rho=0.4$. (b)–(d) were simulated with $t=0.5~{}\Lambda$.

In the FEM simulations with 2D photonic crystal slabs, the merging of BICs was
achieved when $t=1.176~{}\Lambda$ and $t=1.18072~{}\Lambda$, which are close
to the analytical merging thickness of $t_{0}/\Lambda=1.167$ from Fig. 4(a).
Furthermore, Jin _et al._ have recently demonstrated experimentally that on-
chip photonic resonances can be made robust against fabrication imperfections
by combining nine BICs in 2D photonic crystal slabs [31]. The merged state of
BICs was achieved at a normalized thickness of $h/a=1.13$, which is also close
to $t_{0}/\Lambda=1.167$. Hence, it is reasonable to conclude that the
analytical merging thickness obtained from Eq. (10) could also be beneficial
for achieving merged states of BICs in practical 2D photonic crystal slabs.
Compared to the merged states in a 1D photonic lattice plotted in Fig. 2(b),
the merged states in a 2D lattice exhibit significantly improved radiative $Q$
factors across a wide range of wavevectors. This improvement is due to the
convergence of a larger number of BICs at the center of the Brillouin zone in
2D configurations. Jin _et al._ showed that there was an improvement in the
scaling property from $Q\propto 1/k_{z}^{-2}$ to $Q\propto 1/k_{z}^{-6}$ by
merging eight accidental BICs and one symmetry-protected BIC in 2D photonic
crystal slabs.

##  VI Conclusion

In conclusion, we analytically and numerically investigated the formation,
merging, and band transition of accidental BICs in 1D and 2D photonic crystal
slabs. Using a simple coupled-mode analysis, we derived an analytical
expression for the overlap integral associated with the out-of-plane radiation
of guided modes. Accidental BICs can emerge at generic $k$ points, including
the lattice $\Gamma$ point, where the value of the overlap integral approaches
zero. Due to the $180^{\circ}$ ($90^{\circ}$) rotational symmetry of 1D (2D)
lattices, two (eight) accidental BICs appear in the momentum space
simultaneously. As the thickness of the slab increases, the accidental BICs
move downward along the dispersion curve, resulting in the merged state at the
edges of the second stop band. As the thickness increases further, the merged
state of BICs transitions from the upper to the lower band edge. With a
further increase in thickness, the merged BICs at the lower band edge split
into multiple isolated ones and move away from each other. We presented the
theoretical merging thickness $t_{0}/\Lambda$, which is the ratio between the
thickness and period of photonic crystal slabs, where multiple BICs merge at
the second-order $\Gamma$ point. The merging of multiple BICs are important
for practical applications because it provides a powerful mechanism to achieve
robust ultrahigh-$Q$ resonances capable of enhancing light-matter
interactions. Our coupled-mode analysis and FEM-simulated results are helpful
for achieving topologically enabled ultrahigh-$Q$ resonances that are robust
to fabrication imperfections in diverse 1D and 2D photonic lattices made of
various dielectric materials.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements
This research was supported by the grant from the National Research Foundation
of Korea, funded by the Ministry of Science and ICT (No. 2022R1A2C1011091).

Conflict of Interest
The authors declare no conflicts of interest.

Data Availability Statement
Data underlying the results in this paper may be obtained from the authors
upon reasonable request.

Keywords
bound states in the continuum, guided-mode resonances, topologically enabled
ultrahigh-$Q$

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This work has been submitted at the IEEE International Conference on
Acoustics, Speech and Signal Processing 2023. Copyright may be transferred
without notice, after which this version may no longer be accessible.

# Fully complex-valued deep learning model for visual perception

###### Abstract

Deep learning models operating in the complex domain are used due to their
rich representation capacity. However, most of these models are either
restricted to the first quadrant of the complex plane or project the complex-
valued data into the real domain, causing a loss of information. This paper
proposes that operating entirely in the complex domain increases the overall
performance of complex-valued models. A novel, fully complex-valued learning
scheme is proposed to train a Fully Complex-valued Convolutional Neural
Network (FC-CNN) using a newly proposed complex-valued loss function and
training strategy. Benchmarked on CIFAR-10, SVHN, and CIFAR-100, FC-CNN has a
4-10% gain compared to its real-valued counterpart, maintaining the model
complexity. With fewer parameters, it achieves comparable performance to
state-of-the-art complex-valued models on CIFAR-10 and SVHN. For the CIFAR-100
dataset, it achieves state-of-the-art performance with 25% fewer parameters.
FC-CNN shows better training efficiency and much faster convergence than all
the other models.

Index Terms—  Deep learning, Image classification, complex-valued neural
network

##  1 Introduction

Deep learning is used for various tasks such as speech processing [1][2],
robotics, signal processing, image processing, and sonar [3], which contain
complex numbers, but the majority of the models rarely use complex numbers.
Complex-valued data can exist in: (1) natural form, involving applications
using Synthetic Aperture Radar (SAR) [4], Magnetic Resonance Imaging (MRI),
and telecommunication, (2) complex-valued representation of real-valued data,
such as Fourier transform [5] and spectrum-based methods [6], and (3) physics
application such as optical physics [7], where both amplitude and phase
information are essential. Complex-valued models (CV-models) have certain
advantages over their real-valued counterparts, such as richer representation
ability [8], easier optimization [9], efficient multi-task learning [10],
superior MRI reconstruction [11], and robust memory retrieval mechanism [12].
CV models can be used for standard computer vision tasks, such as image
classification, because of their rich representation capacity in the complex
domain. DCN [13] extends real-valued deep learning models to the complex
domain by proposing complex-valued batch norm, weight initialization, and
CReLU activation function. It operates only in the first quadrant of the
complex plane. SurReal [14] tackles complex numbers from a manifold deep
learning perspective but does not exploit the complex-valued representation.
CDS models [15] introduce equivariant and invariant counterparts of common
neural network layers to address complex scaling.
All the models mentioned above do not operate entirely in the complex domain
and project data to the real domain to make predictions and compare with
ground truth labels. This causes a loss of phase information, hence, the
complex domain’s representation capacity is not used optimally. [7] shows that
more information is stored in the phase component compared to the amplitude
component in the parameter subspace, proving the heterogeneous nature of
neural networks. Operating completely in the complex domain to train the model
may improve the overall generalization ability and performance of CV models
for visual perception.
In this paper, a Fully Complex-valued Convolutional Neural Network (FC-CNN) is
proposed for image classification with the same number of parameters as its
real-valued counterpart. A novel complex-valued learning scheme is proposed to
train FC-CNN entirely in the complex domain, taking phase information into
account when updating network parameters. The key contributions can be
summarized as follows: (1) A novel fully complex-valued learning scheme is
proposed using orthogonal decision boundary theory [16] to directly convert
shallow real-valued networks to fully complex-valued networks, which improves
the performance while maintaining the same model complexity. (2) A new
regularized complex-valued loss function and a two-step training strategy for
FC-CNN are proposed to address overfitting, leading to faster learning and
better convergence. (3) FC-CNN outperforms its real-valued counterpart on RGB
and other encodings. It achieves state-of-the-art performance on CIFAR-10,
CIFAR-100, and SVHN datasets, with less parameters.

##  2 Fully complex-valued convolutional neural network

In this section, the Fully Complex-valued Convolutional Neural Network (FC-
CNN) is explained in detail. Complex-valued prerequisites are briefly
discussed first, followed by network architecture and the complex-valued
learning scheme.

###  2.1 Complex-valued Prerequisites

Complex-valued convolution operation Complex-valued convolution operator is
defined for complex-valued weight $W$ and complex-valued input $I$ as follows,

$\displaystyle\textbf{W}\ast\textbf{I}$
$\displaystyle=(W_{R}+\emph{i}W_{I})\ast(I_{R}+\emph{i}I_{I})$ (1)

where $W_{R}$ ,$W_{I}$ denote the real and the imaginary parts of the complex-
valued weight, and $I_{R}$ ,$I_{I}$ denote the real and imaginary parts of the
complex-valued input.
Complex-valued activation function Complex differentiable functions are called
analytic or holomorphic if the real and imaginary parts of the complex
function follow Cauchy-Riemann differential equations. A sufficient condition
for complex-valued activation functions to be compatible with complex
backpropagation [9] is that they should be differentiable with respect to each
complex-valued parameter and its complex conjugate [17]. Although there are
numerous complex-valued activation functions, mainly CReLU [13] and complex
cardioid [18] activation functions are considered as both are extensions of
the ReLU function in the complex plane. CReLU can be defined as applying the
ReLU function on real and imaginary parts separately, as shown below,

$\displaystyle CRelu(z)$ $\displaystyle=ReLU(R(z))+i\cdot ReLU(Img(z)),$ (2)

where z is a complex number, R(z) and Img(z) represent the real and imaginary
components of the complex number respectively. While using CReLU, the complex
plain is restricted to only the first quadrant. The complex cardioid is
defined as,

$\displaystyle f(z)$ $\displaystyle=\frac{1}{2}(1+cos(\angle z))\cdot z$ (3)

where z is a complex number, $\angle$z denotes the phase of the complex
number. The cardioid function operates in the whole complex-plain while
preserving phase and scaling the magnitude.

###  2.2 Fully complex-valued learning scheme

Fig. 1: Fully complex-valued learning scheme with two stage training strategy.

Network architecture FC-CNN model has 3 convolutional layers (stride 2) and 1
fully connected layer, as shown in Table 1. The group setting for the second
and third convolutional layers is 2 and 4, respectively. Depthwise-separable
convolution is used as a learnable pooling layer to reduce the feature map
size, with a group setting of 64. The real and imaginary components of the
complex-valued weights are initialized using a uniform distribution. Cardioid
activation function is used to operate in the whole complex plane and not be
restricted to any particular quadrant. The output of the complex-valued fully
connected layer is a complex-valued entity [19]. The last layer is modified
according to the number of classes for a given dataset.
Complex-valued one hot encoding Real-valued labels are one hot encoded to
complex-valued labels to compare with the complex-valued predictions made by
the fully complex-valued network. Let
$\\{(x_{1},c_{1}),(x_{2},c_{2})....(x_{t},c_{t})...(x_{n},c_{n})\\}$ be the
input data, where $x_{t}$ represents the t-th input image and $c_{t}$
represents its label[20][21]. One-hot encoding $y_{k}^{t}$
=$\\{y_{1}^{t},y_{2}^{t},..y_{k}^{t}...y_{n}^{t}\\}$ for its corresponding
label $c_{t}$, is given by :

$y_{k}^{t}=\left\\{\begin{array}[]{ll}1+i,&\text{if}\quad c_{t}=k\\\
-1-i,&\quad\text{otherwise.}\end{array}\right.$

Layer Type Input shape Kernel Stride Output shape CV - Conv1 [3,32,32] 3x3 2
[32,16,16] Cardioid [32,16,16] - - [32,16,16] CV - Conv2 [32,16,16] 3x3 2
[64,8,8] Cardioid [64,8,8] - - [64,8,8] CV - Conv3 [64,8,8] 3x3 2 [64,4,4]
Cardioid [64,4,4] - - [64,4,4] CV- Depthwise Conv [64,4,4] 4x4 2 [128,1,1]
Flatten Layer [128,1,1] - - [128] CV - Linear layer [128] - - [10]

Table 1: FC-CNN model architecture for 10 output classes.

Method # Param CIFAR-10 SVHN CIFAR-100 RGB LAB Sliding RGB LAB Sliding RGB LAB
Sliding Real-valued CNN 22,634/ 34,244 67.27 62.52 65.0 87.3 86.58 87.11 37.81
33.49 36.48 DCN 23,914/ 47,044 65.64 63.72 66.94 86.80 85.77 86.56 35.88 35.25
34.85 Sur-Real 35,274/- 50.68 53.02 54.61 80.51 53.48 80.79 23.57 25.97 26.66
CDS Type - I 24,241/ 47281 69.18 67.89 70.28 89.40 89.27 89.81 40.2 37.94
39.27 CDS Type - E 23,697/ 46737 69.98 67.7 69.5 88.75 74.38 90.50 40.11 38.91
40.97 FC-CNN 22,634/ 34,244 69.82 69.87 69.99 89.65 89.13 89.12 41.71 42.26
41.46

Table 2: Comparison of test accuracy between real-valued CNN, state of the art
complex-valued and fully complex-valued model (FC-CNN) on CIFAR-10, CIFAR-100
and SVHN.

Orthogonal decision boundary theory A fully complex-valued network consists of
two hypersurfaces as its decision boundary - real and imaginary [16]. These
hypersurfaces are orthogonal to each other if the activation function in the
network satisfies Cauchy–Riemann equations (lemma 6.1, [20]). Real and
imaginary hypersurfaces of a fully complex network with complex cardioid are
orthogonal to each other as complex cardioid satisfies Cauchy–Riemann
equations [18]. Complex-valued one hot encoding of the ground truth labels are
projected in the output feature space. Either the real or imaginary
hypersurface can be used to classify the on and off values of the labels.
Using orthogonal decision boundary theory, the real components of the complex-
valued output can be directly compared with the real components of the
complex-valued one-hot encoded labels to compute the accuracy.
Regularized complex-valued hinge loss A new loss function is proposed to
compute the error between complex-valued predictions and complex-valued
labels. This helps the phase information to be taken into account when
updating the network parameters. The complex-valued hinge loss e [21] is given
by ,

$e=\left\\{\begin{array}[]{ll}0&\text{if}\quad Re(y_{l}^{t})\cdot
Re(\hat{y}_{l}^{t})>1\\\
(y_{l}^{t})-(\hat{y}_{l}^{t})&\quad\quad\text{otherwise.}\par\end{array}\right.$

where $(y_{l}^{t})$ is the complex-valued one hot encoding and
$(\hat{y}_{l}^{t})$ is the complex-valued predictions. FC-CNN has an
overfitting issue, which is why the loss function is modified to have a
regularizing effect for the correctly predicted samples. Two terms, error
threshold $e_{thr}$ and max threshold $e_{M}$ are defined, where error
threshold $e_{thr}$ is updated after every epoch and is given by,

$\displaystyle e_{thr}$ $\displaystyle=e^{-(0.05)\cdot(epoch)}$ (4)

Maximum threshold $e_{M}$ represents the maximum value of error from all the
samples in the batch, and is defined as,

$\displaystyle e_{M}$
$\displaystyle=\text{max}(\text{abs}(e(y_{l}^{t},\hat{y}_{l}^{t})))$ (5)

where abs represents the magnitude of complex numbers and max operator returns
the maximum value from all the samples in the batch. The following condition
is used to update e,

$e=\left\\{\begin{array}[]{ll}0&\text{if}\quad e_{M}<e_{thr}\\\
e&\text{if}\quad e_{M}>e_{thr}\par\end{array}\right.$

For misclassified samples, the complex-valued hinge loss e remains the same.
Real-valued loss function E is defined as the product of complex-valued hinge
loss e and its complex conjugate[19],and is given by

$\displaystyle E$ $\displaystyle=\frac{1}{2n}(e^{H})(e)$ (6)

where n is the number of data samples.
Training strategy The fully complex-valued network (FC-CNN) is trained end-to-
end using the proposed complex loss function and complex-valued labels, as
shown in Fig.1(a) training step 1. Once the whole network is trained, the
output of the final complex-valued convolutional layer of the last epoch is
flattened and stored. The model is then finetuned in the second training step
(Fig.1(b)), where only the complex-valued linear layer is re-trained. During
this step, the stored flattened vector is fed to the complex-valued linear
layer and trained for the same number of epochs with the same loss function.
During inference time, the model consists of complex-valued convolutional
layers and the finetuned complex-valued linear layer.

Fig. 2: Performance curves of test accuracy of real-valued CNN, CV-models and FC-CNN on CIFAR-100 (RGB) dataset. Method |  | CIFAR-100
---
(RGB)
| FLOPs
---
(MACS)
| Real -valued CNN
---
37.81 | 1M
| DCN (With CReLU)
---
35.88 | 1.03M
| DCN (With Cardioid)
---
34.29 | 999.65K
| FC-CNN (W/o Finetuning)
---
39.06 | 986.85K
| FC-CNN (With Finetuning)
---
41.71 | 986.85K
Table 3: Comparison of test accuracy and FLOPs (forward pass) of real-valued,
DCN and fully complex-valued (FC-CNN) models. FLOPs are reported during
testing.

##  3 Experimental results

The experiments are conducted on three benchmark datasets, i.e., CIFAR-10
[22], CIFAR-100, and SVHN [23]. CIFAR-10 and CIFAR-100 contain 10 and 100
classes, with 50000 training images and 10000 testing images, respectively.
SVHN contains ten classes, with 73,257 training images and 26,032 test images.
For a fair comparison, real-valued CNN and DCN both have the same network
architecture as FC-CNN and are trained using cross-entropy loss. Relu
activation function is used for the real model, and CReLU activation is used
for the DCN model. Sur-real [14], CDS Type - I, and E models [15] have the
same network architecture as mentioned in [15]. Note that as Sur-real’s code
is not publicly available, the results mentioned in Table 2 were borrowed from
the results cited in [15].111The results for CDS-I and E are reproduced from
the github code : https://github.com/sutkarsh/cds. The results for FC-CNN are
reproduced using the same code and same seed. All the models are trained with
AdamW optimizer using momentum (0.99,0.999), with a learning rate of 0.001,
batch size 256, and weight decay of 0.1.
Test accuracy of real-valued and complex-valued models are shown in Table 2
for real-valued RGB images and their corresponding complex-valued sliding [15]
and LAB encoding [15] for all three datasets. The number of parameters for all
the models drastically increases when changing the output categories from 10
to 100. The real-valued CNN and DCN models have comparable performance to each
other, but for RGB data, real-valued CNN outperforms the DCN model. Comparing
these networks to fully complex-valued network(FC-CNN), a significant
improvement in the performance can be seen. For CIFAR-10 and SVHN datasets,
FC-CNN achieves similar performance to the best-performing CDS model, with
fewer parameters. For the CIFAR-100 dataset, FC-CNN outperforms its real-
valued counterpart on RGB and all the encodings with the same number of
parameters. It achieves state-of-the-art performance compared to other CV
models with considerably less number of parameters. Fig.2 shows the test
accuracy of real-valued, complex-valued, and FC-CNN models on CIFAR-100 (RGB).
FC-CNN outperforms all the other models with much more efficient learning. It
converges within 15k iterations, while all the other models take considerably
more training time. Compared to its real-valued counterpart, with the same
number of parameters, the overall test accuracy and convergence are much
better. Thus, it generalizes better on larger datasets than real-valued and CV
models.
Table 3 shows the test accuracy and model complexity of real-valued, DCN, and
FC-CNN networks on the CIFAR-100 dataset. FLOPs for the real-valued and FC-CNN
networks remain almost the same. FC-CNN performs better than both networks
without the finetuning step. Using either CReLU or cardioid for the DCN model
does not improve its performance compared to the real-valued network. Hence,
we infer that operating in all quadrants of the complex plane is not enough,
and the model must also train entirely in the complex domain and not be
projected back to the real domain. The empirical results of FC-CNN justify
this intuition. The two-stage training strategy, i.e., finetuning the CV-
linear layer, shows a considerable gain in performance.

##  4 Conclusion

This paper proposes a Fully Complex-valued network for image classification of
real-valued images and their complex-valued encodings. It is an extension of a
shallow real-valued CNN to the complex domain with an equal number of
parameters. A novel, fully complex-valued learning scheme is proposed for the
FC-CNN to operate entirely in the complex domain and operate in all four
quadrants of the complex domain, using a novel regularized loss function and
training strategy. Experimental results on CIFAR-10, SVHN, and CIFAR-100 show
that FC-CNN has a gain in accuracy of 4-10% compared to its real-valued
counterpart on RGB, Sliding, and LAB encodings. Compared to the shallow DCN
model, accuracy increases by 4-6% with 5% lesser computational complexity.
Compared to other complex-valued models, it achieves comparable performance to
the state-of-the-art network on CIFAR-10, SVHN. For CIFAR-100, it achieves
state-of-the-art performance with almost 25% fewer parameters.

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