# Tunable all-optical logic gates based on nonreciprocal topologically
protected edge modes

Jie Xu1,2,7, Panpan He3,4, Delong Feng1,2, Yamei Luo1,2, Siqiang Fan5, Kangle
Yong1,2,8, Kosmas L. Tsakmakidis6,9 1School of Medical Information and
Engineering, Southwest Medical University, Luzhou 646000, China  2Medical
Engineering & Medical Informatics Integration and Transformational Medicine of
Luzhou Key Laboratory, Luzhou 646000, China  3Luzhou Key Laboratory of
Intelligent Control and Application of Electronic Devices, Luzhou Vocational &
Technical College, Luzhou 646000, China  4School of Electrical and Electronic
Engineering, Luzhou Vocational & Technical College, Luzhou 646000, China
5Chongqing Key Laboratory of Photo-Electric Functional Materials, Chongqing
401331, China  6 Section of Condensed Matter Physics Department of Physics
National and Kapodistrian University of Athens Panepistimioupolis, Athens
GR-157 84, Greece  7<EMAIL_ADDRESS>8<EMAIL_ADDRESS>9
<EMAIL_ADDRESS>
###### Abstract

All-optical logic gates have been studied intensively for their potential to
enable broadband, low-loss, and high-speed communication. However, poor
tunability has remained a key challenge in this field. In this paper, we
propose a Y-shaped structure composed of Yttrium Iron Garnet (YIG) layers that
can serve as tunable all-optical logic gates, including, but not limited to,
OR, AND, and NOT gates, by applying external magnetic fields to magnetize the
YIG layers. Our findings demonstrate that these logic gates are based on
topologically protected one-way edge modes, ensuring exceptional robustness
against imperfections and nonlocal effects while maintaining extremely high
precision. Furthermore, the operating band of the logic gates is shown to be
tunable. In addition, we introduce a straightforward and practical method for
controlling and switching the logic gates between ”work”, ”skip”, and ”stop”
modes. These findings have important implications for the design of high-
performance and precise all-optical integrated circuits.

##  I Introduction

Since the invention of the transistor in 1947, human society has experienced
an unprecedented boom in electronic communications based on electrical signals
to meet the needs of everyday life and scientific research riordan:Invention ;
Das:2021 . However, with the development of integrated circuits, transistors
are becoming increasingly miniaturized, resulting in increased energy waste.
Additionally, electronic communication still suffers from defects such as high
error rates and cross-talk markov:Limits . On the other hand, optical
communication has advantages such as high-speed signal processing, error-free
transmission Mueller:2010 , parallel computation Pierangeli:Large , and low
loss Tong:2003 , making it a potential candidate for the next-generation
communication technology. In recent decades, the concept of integrated optical
circuits has been introduced, greatly developed, and studied.

All-optical logic gates (LGs) are an important component of integrated optical
circuits and have received considerable attention in recent years, with
interesting results in this field. Researchers have constructed various types
of all-optical LGs, such as photonic crystal and Mach-Zehnder interferometer
structures, using nonlinear processes almeida:all ; wang:thermal ;
jandieri:realization  and/or interferometry he:topological ; fu:all ;
gao:logic ; wang:coherent , and have implemented all basic logic operations.
However, most LGs suffer from low contrast ratios (CRs), typically less than
30 dB. This is understandable because reflections are unavoidable in
conventional optical LGs, and imperfections in their manufacturing affect the
accuracy of the gates to some extent, particularly in nonlinearity-based LGs
fushimi:all ; chao:novel ; Chai:Ultrafast . In many studies on sub-wavelength
all-optical LGs, researchers often neglect the impact of nonlocal effects on
logical operations. While this is generally true in near-wavelength cases,
non-local effects should be considered when the device’s scale is
subwavelength or even deep-subwavelength. In fact, the impact of nonlocal
effects on nonreciprocal/one-way surface magnetoplasmons (SMPs) has been
widely discussed in the past several years Hassani:Do ; buddhiraju:Absence ;
Tsakmakidis:Topological . SMPs are edge modes sustained in magneto-optical
(MO) heterostructures, and many interesting and meaningful results, such as
slow light Xu:Br ; hu:slowing ; hu:surface , overcoming the time-bandwidth
limit Tsakmakidis:Br , and rainbow trapping Xu:Sl ; Shen:Tr , have been
discovered. Recently, we proposed a method to implement (sub-wavelength) all-
optical logic operations using one-way SMP modes Xu:Alloptical . This type of
one-way electromagnetic (EM) mode has been proven to be topologically
protected Wang:Ob ; Wang:To  in the microwave regime by several research
groups, and no significant impact of non-local effects has been observed.
Therefore, in this paper, we focus on such nonlocality-immune SMPs to study
tunable LGs. Additionally, since guided wave modes have only one transmission
direction, the problem of preparation process defects is well overcome, and
unidirectional modes are immune to backscattering. More importantly, all-
optical LGs based on unidirectional EM modes theoretically have an infinite
contrast ratio, which means unparalleled accuracy.

Note that in Ref.Xu:Alloptical , the designed all-optical LGs relied on
Yttrium Iron Garnet (YIG) with remanence. Consequently, although
unidirectional SMPs-based all-optical LGs were implemented using MO
heterostructures, their lack of tunability hindered their application in
future integrated optical circuits. In this paper, we propose a Y-shaped
structure composed of three YIG layers under different bias magnetic fields
and theoretically analyze the dispersion relation in the three arms, which are
all YIG-YIG heterostructures. We observe interesting phenomena, such as
reverse propagation direction, and close and/or reopen one-way regions. More
importantly, we discover highly tunable characteristics of the Y-shaped
structure and the LGs, which are confirmed by full-wave simulation. Our
proposed (subwavelength) tunable LGs have the potential to be applied in the
design of high-performance and programmable integrated optical circuits.

##  II Physical model and topologically protected SMPs

The Y-shaped configuration is a commonly used physical model in the field of
all-optical LGs, which has been extensively studied in recent decades
song:electrochemical ; gao:lithographically ; pan:optical ; assunccao:phase ;
fu:all . In Fig. 1(a), we propose a Y-shaped YIG-based model that enables
tunable all-optical logic operations. The model comprises three straight arms,
each containing two layers of YIG. Unlike our previous work Xu:Alloptical ,
where YIG with remanence was used, all the YIG layers in this study are
subjected to an external magnetic field ($\mathrm{H}_{\mathrm{0}}$) to further
enhance the tunability of the LGs. It should be noted that metals can always
be considered as perfect electric conductor (PEC) walls in the microwave
regime Pozar:Mi . For simplicity, as shown in Fig. 1(b), the arm with YIG
layers having the same magnetization is referred to as ’EYYE-s’, where ”E”
represents the PEC boundary, ”Y” represents YIG, and ”s” symbolizes the same
magnetization direction. Similarly, the structure with YIG layers having
opposite magnetization directions is labeled ’EYYE-r’. To achieve basic logic
operations based on one-way modes, the key is to establish two separate one-
way channels that allow efficient transfer of the EM wave/signal. The question
then arises as to how to design suitable arms and how to efficiently tune the
structure according to our needs.

Figure 1:  (a) The schematic of the Y-shaped structure of all-optical logic
operations. (b) Two types of arms are shown, i.e. the ’EYYE-s’ and the
’EYYE-r’. (c) Pre-designed two one-way channels in our proposed structure.
Note that, in this paper, we use $\omega_{\mathrm{0}}^{\mathrm{a}}$,
$\omega_{\mathrm{0}}^{\mathrm{b}}$, and $\omega_{\mathrm{0}}^{\mathrm{c}}$ to
clarify the procession angular frequencies ($\omega_{\mathrm{0}}$) for green-
colored YIG, yellow-colored YIG and blue-colored YIG layers, respectively.

To achieve this, one must first study the dispersion relation of the SMPs in
those arms. The ’EYYE-r’ contains two layers of YIG with two different
relative permeability and for the lower ($\bar{\mu}_{\mathrm{a}}$) and upper
($\bar{\mu}_{\mathrm{b}}$) YIG, we have

$\bar{\mu}_{\mathrm{a}}=\left[\begin{array}[]{ccc}\mu_{1}^{\mathrm{a}}&-i\mu_{2}^{\mathrm{a}}&0\\\
i\mu_{2}^{\mathrm{a}}&\mu_{1}^{\mathrm{a}}&0\\\
0&0&1\end{array}\right],\quad\bar{\mu}_{\mathrm{b}}=\left[\begin{array}[]{ccc}\mu_{1}^{\mathrm{b}}&i\mu_{2}^{\mathrm{b}}&0\\\
-i\mu_{2}^{\mathrm{b}}&\mu_{1}^{\mathrm{b}}&0\\\ 0&0&1\end{array}\right]$ (1)

where
$\mu_{1}=1+\frac{\omega_{\mathrm{m}}\left(\omega_{\mathrm{0}}-iv\omega\right)}{\left(\omega_{\mathrm{0}}-iv\omega\right)^{2}-\omega^{2}}$
and
$\mu_{2}=\frac{\omega_{\mathrm{m}}\omega}{\left(\omega_{\mathrm{0}}-iv\omega\right)^{2}-\omega^{2}}$.
$\omega$, $\omega_{\mathrm{m}}$, $\nu$ and
$\omega_{\mathrm{0}}=\mu_{\mathrm{0}}\gamma\mathrm{H}_{\mathrm{0}}$ refer
respectively to the angular frequency, the characteristic circular frequency,
the damping factor, and the procession angular frequencyPozar:Mi . Please note
that the superscripts ’a’ and ’b’ represent the lower and upper layers,
respectively. In this paper, we assume that the magnetic-field direction in
the lower layer is permanently oriented in the -z direction. By applying
Maxwell’s equations and three boundary conditions in the ’EYYE-r’ arm, one can
easily calculate the dispersion relation of the SMPs sustained on the YIG-YIG
interface. The dispersion relation takes the following form

$\mu_{v}^{\mathrm{a}}\left[\frac{\mu_{2}^{\mathrm{b}}}{\mu_{1}^{\mathrm{b}}}k+\frac{\alpha_{\mathrm{b}}}{\tanh\left(\alpha_{\mathrm{b}}d\right)}\right]+\mu_{v}^{\mathrm{b}}\left[\frac{\mu_{2}^{\mathrm{a}}}{\mu_{1}^{\mathrm{a}}}k+\frac{\alpha_{\mathrm{a}}}{\tanh\left(\alpha_{\mathrm{a}}d\right)}\right]=0$
(2)

where
$\alpha_{\mathrm{a}}=\sqrt{k^{2}-\varepsilon_{\mathrm{m}}\mu_{v}^{\mathrm{a}}k_{\mathrm{0}}^{2}}$,
$\alpha_{\mathrm{b}}=\sqrt{k^{2}-\varepsilon_{\mathrm{m}}\mu_{v}^{\mathrm{b}}k_{\mathrm{0}}^{2}}$,
and $\mu_{v}=\mu_{1}-\mu_{1}^{2}/\mu_{2}$. Equation (2) reveals that the SMPs
in the ’EYYE-r’ arm exhibit different propagation properties for opposite
wavenumbers, i.e., $k_{1}=-k_{2}$, which is a well-known nonreciprocity
effect. More importantly, adjusting the external magnetic field can create a
special one-way region where the waves propagate in only one specific
direction. The asymptotic frequencies (AFs) of the SMPs in the ’EYYE-r’ arm
can be derived and calculated from Eq. (2). We found four AFs, which can be
described by the following equations:

$\omega_{\mathrm{sp}}^{(+)}=\begin{aligned}
&\begin{cases}&\omega_{\mathrm{sp}}^{(+1)}=\omega_{\mathrm{0}}^{\mathrm{a}}+\omega_{\mathrm{m}}\\\
&\omega_{\mathrm{sp}}^{(+2)}=\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{m}}\end{cases}\\\
\end{aligned}$ (3) $\omega_{\mathrm{sp}}^{(-)}=\begin{aligned}
&\begin{cases}&\omega_{\mathrm{sp}}^{(-1)}=\frac{(\omega_{\mathrm{0}}^{\mathrm{a}}+\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{m}})+\sqrt{(\omega_{\mathrm{0}}^{\mathrm{a}}+\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{m}})^{2}-2(2\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{m}}+\omega_{\mathrm{0}}^{\mathrm{b}}\omega_{\mathrm{m}})}}{2}\\\
&\omega_{\mathrm{sp}}^{(-2)}=\frac{(\omega_{\mathrm{0}}^{\mathrm{a}}+\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{m}})-\sqrt{(\omega_{\mathrm{0}}^{\mathrm{a}}+\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{m}})^{2}-2(2\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{0}}^{\mathrm{b}}+\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{m}}+\omega_{\mathrm{0}}^{\mathrm{b}}\omega_{\mathrm{m}})}}{2}\end{cases}\\\
\end{aligned}$ (4)

$\omega_{\mathrm{sp}}^{+}$ and $\omega_{\mathrm{sp}}^{-}$ indicate the AF as
$k\to+\infty$ and $k\to-\infty$, respectively. In fact, the value of
$\omega_{\mathrm{sp}}^{+}$ corresponds to the zero point of
$\mu_{v}^{\mathrm{a}}$ or $\mu_{v}^{\mathrm{b}}$. Similarly, the dispersion
relation of the SMPs in the ’EYYE-s’ arm can be directly obtained from Eq. (2)
by replacing $\mu_{2}^{\mathrm{b}}$, $\mu_{1}^{\mathrm{b}}$,
$\mu_{v}^{\mathrm{b}}$, and $\alpha_{\mathrm{b}}$ with
$-\mu_{2}^{\mathrm{c}}$, $\mu_{1}^{\mathrm{c}}$, $\mu_{v}^{\mathrm{c}}$, and
$\alpha_{\mathrm{c}}$, respectively. In this case, the permeability
($\bar{\mu_{\mathrm{c}}}$) of the upper YIG has the same form as
$\bar{\mu_{\mathrm{a}}}$, and the corresponding dispersion equation can be
written as follows:

$\mu_{v}^{\mathrm{a}}\left[\frac{-\mu_{2}^{\mathrm{c}}}{\mu_{1}^{\mathrm{c}}}k+\frac{\alpha_{\mathrm{c}}}{\tanh\left(\alpha_{\mathrm{c}}d\right)}\right]+\mu_{v}^{\mathrm{c}}\left[\frac{\mu_{2}^{\mathrm{a}}}{\mu_{1}^{\mathrm{a}}}k+\frac{\alpha_{\mathrm{a}}}{\tanh\left(\alpha_{\mathrm{a}}d\right)}\right]=0$
(5)

There are also four potential AFs in the ’EYYE-s’ arm, which have the
following form:

$\omega_{\mathrm{sp}}^{(+)}=\begin{aligned}
&\begin{cases}&\omega_{\mathrm{sp}}^{(+1)}=\omega_{\mathrm{0}}^{\mathrm{a}}+\omega_{\mathrm{m}}\\\
&\omega_{\mathrm{sp}}^{(+2)}=\frac{(\omega_{\mathrm{0}}^{\mathrm{c}}-\omega_{\mathrm{0}}^{\mathrm{a}})+\sqrt{(\omega_{\mathrm{0}}^{\mathrm{c}}-\omega_{\mathrm{0}}^{\mathrm{a}})^{2}+2(2\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{0}}^{\mathrm{c}}+\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{m}}+\omega_{\mathrm{0}}^{\mathrm{c}}\omega_{\mathrm{m}})}}{2}\end{cases}\\\
\end{aligned}$ (6) $\omega_{\mathrm{sp}}^{(-)}=\begin{aligned}
&\begin{cases}&\omega_{\mathrm{sp}}^{(-1)}=\omega_{\mathrm{0}}^{\mathrm{c}}+\omega_{\mathrm{m}}\\\
&\omega_{\mathrm{sp}}^{(-2)}=\frac{(\omega_{\mathrm{0}}^{\mathrm{a}}-\omega_{\mathrm{0}}^{\mathrm{c}})+\sqrt{(\omega_{\mathrm{0}}^{\mathrm{c}}-\omega_{\mathrm{0}}^{\mathrm{a}})^{2}+2(2\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{0}}^{\mathrm{c}}+\omega_{\mathrm{0}}^{\mathrm{a}}\omega_{\mathrm{m}}+\omega_{\mathrm{0}}^{\mathrm{c}}\omega_{\mathrm{m}})}}{2}\end{cases}\\\
\end{aligned}$ (7)

Figure 2:  (a-c) The dispersion diagrams of three arms are shown, in which the
lower YIG has $\omega_{\mathrm{0}}$ values of $0.6\omega_{\mathrm{m}}$,
$0.4\omega_{\mathrm{m}}$, and $0.6\omega_{\mathrm{m}}$, respectively, while
the upper YIG has $\omega_{\mathrm{0}}$ values of $0.3\omega_{\mathrm{m}}$,
$0.3\omega_{\mathrm{m}}$, and $0.4\omega_{\mathrm{m}}$”, respectively. Note
that ’r’ and ’s’ indicate the ’EYYE-r’ arm and ’EYYE-s’ arm, respectively, and
the magnetization orientation of the lower YIG is permanently -z. The cyan
lines represent the edge of the bulk zones, while the black arrows indicate
the location of $\omega=\omega_{\mathrm{s}}$. Stars show the corresponding
asymptotic frequencies in each case. (d) The simulated electric field
distributions of the three cases are shown for $f=0.8f_{\mathrm{m}}$. The
other parameters are (a-c) $d=0.02\lambda_{\mathrm{m}}$, $\nu=0$ and (d)
$\nu=0.001\omega_{\mathrm{m}}$.

Based on Eqs. (4) and (5), we plot the dispersion curves for the SMPs in both
the ’EYYE-r’ and ’EYYE-s’ arms as $d=0.02\lambda_{\mathrm{m}}$
($\lambda_{\mathrm{m}}=2\pi c/\omega_{\mathrm{m}}=c/f_{\mathrm{m}}$) and
$\nu=0$ (lossless condition). Three different values of $\omega_{\mathrm{0}}$
($\mathrm{H}_{\mathrm{0}}$) are applied in the three arms, and for
convenience, we introduce a simple notation -
’$\left[\alpha,\beta,\theta\right]$’ - in which $\alpha$ and $\beta$ represent
the absolute values of the normalized $\omega_{\mathrm{0}}$
($\bar{\omega_{\mathrm{0}}}=\omega_{\mathrm{0}}/\omega_{\mathrm{m}}$) for the
lower and upper YIG, while $\theta$ could be either ’r’ referring to the
’EYYE-r’ arm or ’s’ referring to the ’EYYE-s’ arm. For example, [0.6, 0.3, r]
in Fig. 2(a) implies that the dispersion curve is calculated in the ’EYYE-r’
arm, where $\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{0}}^{\mathrm{b}}=0.3\omega_{\mathrm{m}}$. In Fig. 2(a), the
green and blue stars represent $\omega_{\mathrm{sp}}^{(+)}$ and
$\omega_{\mathrm{sp}}^{(-)}$, respectively. The red and black lines indicate
the dispersion curves of SMPs on the YIG-YIG interface, and due to the deep-
subwavelength thickness of the YIG layers, the bulk zones are significantly
compressed Xu:Sl ; Zhou:realization . Therefore, it is believed that almost
all the SMPs on the red and black lines are one-way EM modes except for the
SMPs located near the resonant frequencies of YIG
($\omega_{\mathrm{s}}=\sqrt{\omega_{\mathrm{0}}^{2}+\omega_{\mathrm{0}}\omega_{\mathrm{m}}}$),
which are marked by black arrows. As depicted in the inset of Fig. 2(a), the
case of [0.6, 0.3, r] can be treated as one of the input arms (arm ’A’) of the
Y-shaped heterostructure. We also calculate the dispersion curves for the
other arms in Figs. 2(b) and 2(c). As a result, similar to the first case,
there are two one-way regions in both cases. However, in the case of [0.6,
0.4, s], the EM waves within the lower one-way region have negative group
velocities ($v_{\mathrm{g}}<0$).

Based on Eqs. (3), (4), (6), and (7) as mentioned earlier, the one-way regions
are defined by the AFs (green and blue stars in Fig. 2). For the three cases
discussed above, the regions are: (a) [0.428$f_{\mathrm{m}}$,
1.3$f_{\mathrm{m}}$] and [1.472$f_{\mathrm{m}}$, 1.6$f_{\mathrm{m}}$], (b)
[0.3475$f_{\mathrm{m}}$, 1.3$f_{\mathrm{m}}$] and [1.3525$f_{\mathrm{m}}$,
1.4$f_{\mathrm{m}}$], and (c) [0.766$f_{\mathrm{m}}$, 0.966$f_{\mathrm{m}}$]
and [1.4$f_{\mathrm{m}}$, 1.6$f_{\mathrm{m}}$]. Therefore, to design two one-
way channels, the frequencies used must be located within the
[0.766$f_{\mathrm{m}}$, 0.966$f_{\mathrm{m}}$] region (the red line region in
Fig. 2(c)). In addition, the loss effect and the robustness of the one-way
propagation of SMPs are examined using full-wave simulations, as illustrated
in Fig. 2(d). In this case, we consider $\nu=0.001\omega_{\mathrm{m}}$ and
$f=0.8f_{\mathrm{m}}$, and air holes with a radius of $r=0.5$ mm ($\sim
0.008\lambda_{\mathrm{m}}$) are placed on the YIG-YIG interface. The
simulation results show good agreement with the theoretical analysis, and the
imperfections have a negligible impact on the one-way SMPs. It is also worth
noting that recent studies have questioned the robustness of one-way SMPs,
with the nonlocal effect being a major focus of these worksHassani:Do ;
buddhiraju:Absence . Here, we emphasize that the one-way SMPs studied in this
paper are theoretically topologically protected, which has been theoretically
demonstratedsilveirinha:chern ; Wang:Re  and experimentally
provedholmes:Experimental ; Wang:Ob  by many groups. This nonlocality-immune
property is particularly evident in cases where the waveguide is relatively
thick or the wavenumber (k) is relatively smallTsakmakidis:Topological . Our
proposed logic gates in this paper are believed to be largely unaffected by
nonlocal effects, given the tunability of the SMPs, which will be discussed in
the next subsection.

##  III Tunable all-optical logic gates

In our theoretical analysis, we have shown that a Y-shaped structure
consisting of magnetized YIG layers can support two independent one-way
channels, making it suitable as a logical gateXu:Alloptical . More
importantly, benefiting from the tunability of the topologically protected
one-way SMPs, the proposed LGs should be easily tunable by changing the bias
magnetic fields. In the following sections, we demonstrate the tunability of
our proposed logical gates in detail. Firstly, we study the impact of
$\mathrm{H}_{\mathrm{0}}$ on AFs, which always define the one-way regions. As
displayed in Fig. 3(a), four AFs in the ’EYYE-r’ arm ’A’ (’C’), are plotted as
a function of $\omega_{\mathrm{0}}^{\mathrm{a}}$
($\omega_{\mathrm{0}}^{\mathrm{c}}$) and $\omega_{\mathrm{0}}^{\mathrm{b}}$.
Figure 3(b) depicts the similar relationship between AFs and
$\omega_{\mathrm{a}}$ and $\omega_{\mathrm{c}}$. To differentiate between the
four distinct AFs in Eqs. (3, 4) (’EYYE-r’) and (6, 7) (’EYYE-s’), we use the
names $\omega_{\mathrm{sp}}^{(+1)}$, $\omega_{\mathrm{sp}}^{(+2)}$,
$\omega_{\mathrm{sp}}^{(-1)}$, and $\omega_{\mathrm{sp}}^{(-2)}$. Notably, as
$\omega_{\mathrm{0}}$ ($\mathrm{H}_{\mathrm{0}}$) changes, the values of AFs
and their numerical relationships may also change. This can lead to a reversal
of the group velocity and the transmission direction of EM signals in LGs.
Therefore, any changes in the AFs can affect the functionality of the LGs.

Figure 3: (a,b) The asymptotic frequencies (AFs) are plotted as a function of
$\omega_{\mathrm{0}}^{\mathrm{a}}$ (or $\omega_{\mathrm{0}}^{\mathrm{c}}$) and
$\omega_{\mathrm{0}}^{\mathrm{b}}$ for (a) the ’EYYE-r’ arm and (b) the
’EYYE-s’ arm. (c-e) AFs are plotted as a function of
$\omega_{\mathrm{0}}^{\mathrm{b}}$ when (c,d)
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$ and (e)
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$. (f) Four
constructed equations ($y_{1}$, $y_{2}$, $y_{3}$, and $y_{4}$) as shown in Eq.
(8) are plotted as functions of $\omega_{\mathrm{0}}^{\mathrm{b}}$ and
$\omega_{\mathrm{0}}^{\mathrm{c}}$ when
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$.

To illustrate the changes in AFs and one-way regions, we set the lower YIG
$\omega_{\mathrm{0}}$ to $0.6\omega_{\mathrm{m}}$ and assume
$0<\omega_{\mathrm{0}}<\omega_{\mathrm{m}}$ for the upper YIG in both ’EYYE-r’
(Fig. 3(c)) and ’EYYE-s’ (Fig. 3(d)) arms. As $\omega_{\mathrm{0}}$
($\omega_{\mathrm{0}}^{\mathrm{b}}$) of the upper YIG varies from 0 to
$0.6\omega_{\mathrm{m}}$, the lower one-way region gradually widens, while the
upper one-way region becomes smaller and eventually closes at
$\omega_{\mathrm{0}}^{\mathrm{b}}=0.6\omega_{\mathrm{m}}$. The black dashed
line represents the [0.6, 0.3, r] case discussed earlier in Fig. 2(a), in
which two clear one-way regions are present (excluding the local area near
$\omega=\omega_{\mathrm{s}}$). As $\omega_{\mathrm{0}}^{\mathrm{b}}$ increases
further, for the ’EYYE-r’ arm, the first one-way region is compressed
slightly, while a new one-way region bounded by $\omega_{\mathrm{sp}}^{(-1)}$
and $\omega_{\mathrm{sp}}^{(+2)}$ emerges with a forward propagation direction
($v_{\mathrm{g}}>0$).The inset of Fig. 3(c) displays a zoomed-in dispersion
curve for the case of
$\omega_{\mathrm{0}}^{\mathrm{b}}=0.8\omega_{\mathrm{m}}$ (blue line). In
contrast, the ’EYYE-s’ arm behaves differently. As shown in Fig. 3(d), when
$\omega_{\mathrm{0}}^{\mathrm{c}}$ (in the upper YIG) is increased, the
propagation direction of SMPs in the lower one-way region changes from
backward ($v_{\mathrm{g}}<0$) to forward ($v_{\mathrm{g}}>0$), and the one-way
region closes and reopens. Similar phenomena of reversing propagation
direction and close-reopen one-way regions are observed in the higher regime
as well. The black and blue dashed lines in Fig. 3(d) indicate cases where
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.4\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.8\omega_{\mathrm{m}}$, respectively, with
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$ in both cases. The
insets in Fig. 3(d) show the reversed one-way regions and the dispersion
curves of SMPs.

The question of whether the group velocity will reverse in one-way systems can
be answered by determining if the system’s symmetry or chirality is broken. As
demonstrated in Figure 3(c), within the lower one-way region of the ’EYYE-r’
arm, the SMPs can propagate only in the forward direction, regardless of which
layer has a higher $\mathrm{H}_{\mathrm{0}}$ ($\omega_{\mathrm{0}}$). We
consider two conditions, [0.6, 0.4, r] and [0.4, 0.6, r], where the
propagation directions are the same. This is because the second case can be
treated as the entire system of the first case revolving 180 degrees around
the propagation direction, and thus the system’s symmetry/chirality is
conserved. In contrast, [0.6, 0.4, s] and [0.4, 0.6, s] have opposite
propagation directions because they cannot be obtained by simply rotating each
other, and thus the system’s symmetry/chirality is broken when changing
$\omega_{\mathrm{0}}$ accordingly.

To achieve a relatively broad one-way band, it is necessary that
$\omega_{\mathrm{0}}^{c}$ in arm ’B’ is small enough, as shown in Figure 3(d).
Thus, we select $\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$
(marked by the red dashed line in Figure 3(d)), and Figure 3(e) depicts the
corresponding AFs and one-way regions as functions of
$\omega_{\mathrm{0}}^{\mathrm{b}}$. For this case,
$\omega_{\mathrm{sp}}^{(-2)}\simeq 0.048\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{sp}}^{(+2)}=\omega_{\mathrm{m}}+\omega_{\mathrm{b}}$. With
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$ fixed, the only
remaining unknown parameter in the Y-shaped structure is
$\omega_{\mathrm{0}}^{\mathrm{b}}$. Ideally, we aim for the whole one-way
region with $v_{\mathrm{g}}<0$ in arm ’B’ to be the working band of the LGs.
Based on our calculations, we can achieve this goal if
$0<\omega_{\mathrm{0}}^{\mathrm{b}}<0.31\omega_{\mathrm{m}}$. However, in most
cases, to ensure that the entire one-way region of arm ’B’ is the working band
of LGs, we need to ensure that $\omega_{\mathrm{sp}}^{(-2)}$ (the blue line in
Figure 3(d)) and $\omega_{\mathrm{sp}}^{(+2)}$ (the green line in Figure 3(d))
are both inside the one-way regions with $v_{\mathrm{g}}>0$ in arms ’A’ and
’C’. To accomplish this, we construct the following equations:

$\displaystyle\begin{cases}&y_{1}=({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(-2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_A}}^{(-2)}})({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(-2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_A}}^{(+2)}})\\\
&y_{2}=({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(-2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_C}}^{(-2)}})({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(-2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_C}}^{(+2)}})\\\
&y_{3}=({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(+2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_A}}^{(-2)}})({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(-2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_A}}^{(+2)}})\\\
&y_{4}=({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(+2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_C}}^{(-2)}})({\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_B}}^{(-2)}}-{\color[rgb]{0,0,0}\omega_{\mathrm{sp\\_C}}^{(+2)}})\\\
\end{cases}$ (8)

where ’A/B/C’ represent arm ’A’/’B’/’C’. In this context, it is worth noting
that arms ’A’ and ’C’ belong to the ’EYYE-r’ type, while arm ’B’ belongs to
the ’EYYE-s’ type. The AFs are represented by $\omega_{\mathrm{sp}}^{(+)}$ and
$\omega_{\mathrm{sp}}^{(-)}$, which are given by Eqs. (3), (4), (6), and (7).
Equation (8) determines whether $\omega_{\mathrm{sp}}^{(-2)}$ in arm ’B’ lies
within the one-way region of arm ’A’, which occurs for $y_{1}<0$. If
$y_{1}<0$, $y_{2}<0$, $y_{3}<0$, and $y_{4}<0$ at the same time, it means that
the entire one-way region with $v_{\mathrm{g}}<0$ in arm ’B’ lies within the
one-way regions of both arms ’A’ and ’C’. Figure 3(f) represents the functions
of $y_{1}$ ((f-1)), $y_{2}$ ((f-2)), $y_{3}$ ((f-3)), and $y_{4}$ ((f-4))
based on $\omega_{\mathrm{0}}^{\mathrm{b}}$ and
$\omega_{\mathrm{0}}^{\mathrm{c}}$ when
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$. We observe that
$y_{1}$, $y_{2}$, and $y_{4}$ are always negative, while $y_{3}$ can be
positive for relatively large $\omega_{\mathrm{0}}^{\mathrm{b}}$ and small
$\omega_{\mathrm{0}}^{\mathrm{c}}$. Therefore, we set
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$, and keep
$\omega_{\mathrm{0}}^{\mathrm{b}}$ relatively small, such as
$\omega_{\mathrm{0}}^{\mathrm{b}}=0.1\omega_{\mathrm{m}}$ (marked by red balls
in Fig. 3(f)), to ensure that the entire one-way region in arm ’B’ corresponds
to the working band of LGs.

Figures 4(a)-4(c) present dispersion curves for the scenario where
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{0}}^{\mathrm{b}}=\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$.
Similar to Fig. 2, there is a one-way region with $v_{\mathrm{g}}>0$ in arm
’A’ (depicted as a red-line region in Fig. 4(a)) and arm ’C’ (depicted as a
red-line region in Fig. 4(b)). Additionally, there is a one-way region with
$v_{\mathrm{g}}<0$ in arm ’B’ (depicted as a red-line region in Fig. 4(c)).
Moreover, the backward one-way region is much larger than that illustrated in
Fig. 2(c). Consequently, the working band of the LGs in this situation should
be significantly broader. Figs. 4(d) and 4(f) show the coupling effect between
arms when $f=0.8f_{\mathrm{m}}$, which falls within the one-way regions of
interest. Consequently, two one-way channels (’A-C’ and ’B-C’) are
established, while the EM signal cannot propagate from arm ’A’ to arm ’B’. It
should be noted that the first part ([0.1, 0.6, s]) of the ’B-C’ channel
differs from that of the ’A-B’ channel ([0.6, 0.1, s]) due to the geometrical
relationship between the arms. As per symmetry, the SMPs in the [0.1, 0.6, s]
and [0.6, 0.1, s] structures must have opposite propagation directions. In the
simulations, the EM signal can transfer efficiently in the one-way channels,
while the forward transferring signal halts at the interface of arm ’A’ and
arm ’B’.

Figure 4:  (a,b) Dispersion diagrams of three arms with optimized parameters,
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$,
$\omega_{\mathrm{0}}^{\mathrm{b}}=0.1\omega_{\mathrm{m}}$ and
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$. (d,e) The simulated
electric field distribution obtained from coupling simulations containing two
arms, with each arm being either the ’EYYE-r’ type or ’EYYE-s’ type.

##  IV Realization of basic logic gates

Figure 5:  (a) Theory of all-optical logic operation using the positive and/or
negative logic. (b) Numerical simulations in the Y-shaped module as
$f=0.8f_{\mathrm{m}}$, and air holes with $r=0.5$ mm were set on the YIG-YIG
interfaces to verify the robustness of logic operations. (c) The truth tables
of the OR, AND, and NOT operations.

The Y-shaped structure that is designed can function within the one-way region
and perform as basic logic gates, including, but not limited to, OR, AND, and
NOT gates, as presented in Fig. 5. During logical operations, arms ’A’ and ’B’
are considered as two input ports, with arm ’C’ regarded as the output port.
Primarily, the structure operates as a natural OR gate, where any input one-
way EM signal can and must propagate to the output port. If we consider the
presence of the EM signal as logic ’1’ and the absence of the EM signal as
logic ’0,’ i.e., positive logic, then the Y-shaped structure functions as a
broad OR gate that can be adjusted by external magnetic fields. Negative logic
(where the presence of the EM signal is recognized as logic ’0’) is used for
the AND gate and NOT gate. In the AND operation, any input EM signal is
treated as logic ’0,’ resulting in the output EM signal also being logic ’0.’
However, the NOT operation employs negative logic in either the input or
output port, with positive logic used in the remaining port. Figure 5(b)
depicts simulations for the Y-shaped structure with $f=0.8f_{\mathrm{m}}$ when
the EM signal is excited in only one of the input ports. Air holes with
$r=0.5$ mm were set as imperfections on the YIG-YIG interface to show the
robust function of our proposed LGs. As a result, the LGs work fine and
perform extremely high CR which is larger than 200 dB (infinity in theory).
Figure 5(c) shows the corresponding truth tables of the OR, AND, and NOT
operations.

The Y-shaped structure is designed to function within the one-way region and
can serve as basic logic gates, including but not limited to OR, AND, and NOT
gates, as illustrated in Fig. 5. During logical operations, arms ’A’ and ’B’
are the input ports, while arm ’C’ is the output port. The structure operates
as a natural OR gate, where any input one-way EM signal must propagate to the
output port. If we assume that the presence of the EM signal is logic ’1’ and
its absence is logic ’0’ (positive logic), then the Y-shaped structure
functions as a versatile OR gate that can be externally adjusted by magnetic
fields. However, the AND and NOT gates use negative logic, where the presence
of the EM signal is recognized as logic ’0.’ In the AND operation, any input
EM signal is treated as logic ’0,’ resulting in the output EM signal also
being logic ’0.’ On the other hand, the NOT operation employs negative logic
in either the input or output port, with positive logic used in the remaining
port. Figure 5(b) shows simulations of the Y-shaped structure with
$f=0.8f_{\mathrm{m}}$ when the EM signal is excited in only one of the input
ports. Imperfections on the YIG-YIG interface were introduced as air holes
with $r=0.5$ mm to demonstrate the robustness of our proposed LGs. As a
result, the LGs exhibits high CR of over 200 dB (infinity in theory). Figure
5(c) presents the corresponding truth tables for the OR, AND, and NOT
operations.

Figure 6:  (a) Switch theory for tunable LGs. The red arrow refers to the
electromagnetic wave path that allows passage. (b,c) Simulation for tunable
LGs by switching the external magnetic fields for input signals (b) [’1’, ’0’]
and (c) [’0’, ’1’]. Input ’1’ could be alternatively programmed to ’0’ (”stop”
mode) or ’1’ (”skip” or ”work” mode.)

As mentioned earlier in Fig. 3, we can control the LGs based on the Y-shaped
structure with external magnetic fields ($\mathrm{H}_{\mathrm{0}}$). By
changing the direction(s) of $\mathrm{H}_{\mathrm{0}}$, we can reverse the
propagation direction of the one-way SMPs, as illustrated by the lower one-way
regions in Fig. 3(c,d), such as [0.6, 0.4, r] transitioning to [0.6, 0.4, s].
Another notable case is that changing the direction(s) of
$\mathrm{H}_{\mathrm{0}}$ can cause the previous one-way region to close, such
as [0.6, 0.6, r] shifting towards [0.6, 0.6, s]. In this scenario, there are
no one-way regions, and the entire band becomes a band gap, preventing the
propagation of EM signals. Additionally, altering the value of
$\mathrm{H}_{\mathrm{0}}$, either by increasing or decreasing it, can
significantly affect the logic operations. Therefore, the operating band of
our proposed LGs can be easily tunned by changing the external magnetic
fields.

Besides, we suggest an innovative approach to tune LGs, as demonstrated in
Fig. 6. This method achieves three modes of LGs by switching
$\mathrm{H}_{\mathrm{0}}$ or $\omega_{\mathrm{0}}$, namely the ”work,” ”stop,”
and ”skip” modes. As initial magnetic-field parameters, we set
$\omega_{\mathrm{0}}^{\mathrm{a}}=0.6\omega_{\mathrm{m}}$,
$\omega_{\mathrm{0}}^{\mathrm{b}}=-0.1\omega_{\mathrm{m}}$, and
$\omega_{\mathrm{0}}^{\mathrm{c}}=0.1\omega_{\mathrm{m}}$. It is noteworthy
that the ’-’ sign indicates the external magnetic field is in the +z
direction. Upon exchanging $\omega_{\mathrm{0}}^{\mathrm{b}}$
($\mathrm{H}_{\mathrm{0}}^{\mathrm{b}}$) and
$\omega_{\mathrm{0}}^{\mathrm{c}}$ ($\mathrm{H}_{\mathrm{0}}^{\mathrm{c}}$),
it is easy to calculate that SMPs with $f=0.8f_{\mathrm{m}}$ have opposite
propagation directions in the original arms. Similarly, exchanging
$\omega_{\mathrm{0}}^{\mathrm{c}}$ ($\mathrm{H}_{\mathrm{0}}^{\mathrm{c}}$)
and $\omega_{\mathrm{0}}^{\mathrm{a}}$
($\mathrm{H}_{\mathrm{0}}^{\mathrm{a}}$) reverses the propagation direction of
SMPs in arm ’B,’ whereas the direction remains unchanged in other arms.
Moreover, exchanging $\omega_{\mathrm{0}}^{\mathrm{a}}$
($\mathrm{H}_{\mathrm{0}}^{\mathrm{a}}$) and
$\omega_{\mathrm{0}}^{\mathrm{b}}$ ($\mathrm{H}_{\mathrm{0}}^{\mathrm{b}}$)
reverses the propagation direction of SMPs in arm ’A,’ whereas the direction
remains unaltered in other arms. The simulation of the three modes of LG is
illustrated in Figs. 6(b) and 6(c). The original mode is designated the ”work”
mode since it can work as LGs. Two methods can achieve the ”stop” mode with EM
signals being halted, and one method can accomplish the ”skip” mode with EM
signals skipping the present calculation. Therefore, our proposed Y-shaped LGs
offers rich manipulation possibilities and is promising for programmable
optical communication/devices. In contrast, traditional all-optical LGs
typically operate at fixed frequencies and can be challenging to tune.

##  V Conclusion

In summary, we have devised a Y-shaped structure made of YIG layers with
distinct magnetizations. The arms of the structure were categorized into two
types: the ’EYYE-r’ type with opposing magnetization directions and the
’EYYE-s’ type with identical magnetization directions. Our theoretical
analysis of the ’EYYE-r’ and ’EYYE-s’ arms led to the construction of two one-
way channels capable of supporting topologically protected one-way SMPs.
Furthermore, the implementation of basic logic gates, such as OR, AND, and NOT
gates, is achieved through these broadband and topological one-way SMPs,
resulting in highly robust (resistant to backscattering and imperfections) and
precise (theoretically infinite contrast ratio) LGs. In addition, we explored
the tunability of these LGs. By adjusting external magnetic fields, the one-
way region can be easily modulated, either broadened or narrowed, the
propagation directions of the SMPs within the region can be completely
reversed, or the region can be closed. Given the intriguing tunability of the
operating band of the Y-shaped LGs. In addition, we proposed a potential
application for the structure/LGs: by switching external magnetic fields,
three switchable modes (”work”, ”skip”, and ”stop”) can be achieved. Our
proposed LGs, based on magnetized YIG, may pave the way for tunable all-
optical logic operations and hold promise for high-efficiency programmable
optical communication circuits.

## Acknowledgement

This work was supported by the National Natural Science Foundation of Sichuan
Province (No. 2023NSFSC1309), and the open fund of Luzhou Key Laboratory of
Intelligent Control and Application of Electronic Devices (No. ZK202210),
Sichuan Science and Technology Program (No. 2022YFS0616), the Science and
Technology Strategic Cooperation Programs of Luzhou Municipal People’s
Government and Southwest Medical University (No. 2019LZXNYDJ18). J.X., K.Y.,
and Y.L. thanks for the support of the Innovation Laboratory of Advanced
Medical Material & Physical Diagnosis and Treatment Technology. K.L.T. was
supported by the General Secretariat for Research and Technology (GSRT) and
the Hellenic Foundation for Research and Innovation (HFRI) under Grant No.
4509.

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