# Fabrication method of a low-loss plasmonic waveguide containing both
“plasmonic- friendly” and “plasmonic- unfriendly” metals

Vadym Zayets  Platform Photonics Research Center, National Institute of
Advanced Industrial Science and Technology, Tsukuba, Japan  Corresponding
author<EMAIL_ADDRESS>Iryna Serdeha  Educational and Scientific
Institute of High Technologies, Taras Shevchenko National University of Kyiv,
Kyiv, Ukraine  Valerii Grygoruk  Educational and Scientific Institute of High
Technologies, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

###### Abstract

Fabrication technology, which allows a substantial decrease of the plasmonic
propagation loss for both “plasmon- friendly” metals like Au, Cu or Al and
“plasmon- unfriendly” metals like Co, Fe or Cr, has been developed and
experimentally demonstrated. Optimization of the optical confinement is used
to reduce the propagation loss below 1 dB per plasmonic device.

††journal: ol

The fabrication of a denser Photonic Integrated Circuit (PIC) allows to
increase both the speed and the functionality of data processing optical
circuits. In the case of an electronic circuit, the size of MOSFET transistors
is very small and billions of transistors can be integrated into one
electrical circuit. Because of the dense integration, the electronic devices
have a complex functionality at a low cost. In comparison to the electronic
components, the size of the optical components is not as small, and a dense
integration of optical components is still challenging. The plasmonic devices
might be the key for a denser integration, because of their small size of
about 10-30 $\mu m$. The uniqueness of a surface plasmon is its topological
nature, meaning that the properties of a surface plasmon are defined by the
optical properties of a surface, but not of the bulk as in the case of a
conventional optical waveguide. As a result, even a small change of optical
properties in close proximity of the interface substantially modifies
plasmon‘s propagation constants. This makes it possible to modulate light
within the propagation distance of only a few wavelengths and, therefore, to
fabricate a plasmonic device [1, 2, 3, 4, 5, 6, 7, 8] of a substantially
smaller size than a device, which is made from conventional optical
waveguides.

Another reason, why plasmonic devices are important for PIC, is that a
plasmonic device is very fast. An operation speed of 170 GHz of a plasmonic
modulator has already been demonstrated [9] and is still very far from the
speed limit of a plasmonic device [10, 9, 11]. Plasmonic devices are very fast
because their length is short and they do not have the problem of phase
mismatch. Additionally, the capacitance of the controlling electrodes is
small, because of the small size of a plasmonic device.

The optical loss is a key issue for any plasmonic device and should be
carefully addressed. A metal absorbs light and, therefore, the propagation
loss is unavoidable for a surface plasmon. However, when the design and
fabrication technology of a plasmonic device are optimized, the loss due to
the light absorption by the metal is negligibly small and does not degrade the
plasmonic device [1, 2, 12]. Since light is absorbed by the metal in a
plasmonic device, the choice of the metal seems to be very important and it
might be suggested that only a few metals can be used in a low-loss plasmonic
structure. This Letter clarifies this misconception and describes the
technological method of fabricating a plasmonic device with a negligibly small
propagation loss, in which a variety of different metals can be used. The
successful experimental demonstration of the proposed method of the loss
reduction in a plasmonic device containing "plasmonic-unfriendly" cobalt is
described. All calculations below are done by the method of rigorous solution
of Maxwell equations in a multilayer structure [13, 2, 14], in which no
approximations or simplifications have been used.

Figure 1:  (a) Propagation loss for light propagating in the bulk of a metal
(b) Propagation loss of a surface plasmon propagating along a $metal/Si$
interface. Note, the loss unit in each graph is different due a substantially
smaller loss of a plasmon in comparison to bulk light propagation.

At present, gold is the first choice as a metal material for the plasmonic
device and most present plasmonic devices are made of gold [3, 15, 16, 10, 9,
11]. The gold has a yellowish color, in fact, due to plasmonic absorption. It
would appear to be the best choice for a plasmonic device. However, gold is
not a good material for fabrication technology. It has a poor adhesion with
other material and a large diffusion coefficient. It is important to
investigate whether other metals can be used for a plasmonic device instead of
gold.

There is a misconception that there are “plasmon- friendly” metals like $Au$,
$Al$, $Cu$, in which the absorption of light is weakest and only which should
be chosen for use in plasmonic devices, and there are “plasmon- unfriendly”
metals like $Co$, $Fe$, $Cr$, in which light absorption is much stronger and
which should not be used in a plasmonic device. Figure 1(a) shows the
calculated propagation loss of light propagating in the bulk of a metal using
optical constants measured in Ref.[17]. In fact, the calculations show that
the absorption loss in the “plasmon- friendly” metals is larger than in the
“plasmon- unfriendly” metals. The smallest absorption is in the “plasmon-
unfriendly” chromium and the largest absorption is in the “plasmon- friendly”
aluminum.

Figure 1 (b) shows the calculated propagation loss of a surface plasmon
propagating along a metal/ Si interface. Indeed, the propagation loss is
moderate (about 1 $dB/\mu m$ at $\lambda$=1550 nm) in the plasmonic structure
containing either $Au$, $Al$ or $Cu$. In the case of a 5-$\mu m$\- long
plasmonic device made of these metals, the insertion loss due to metal
absorption is about 5 dB. It is still large but could be acceptable for some
applications. In contrast, the propagation loss is extremely large ( 20
$dB/\mu m$) in the plasmonic structure containing either $Fe$, $Co$ or $Cr$.
Even in a 1-$\mu m$-long plasmonic device, nearly all of the light is absorbed
and, as a result, such plasmonic structure cannot be used in a PIC.

The data shown in Fig. 1(b) may be incorrectly interpreted to indicate that it
is true that there are “plasmonic-friendly” and “plasmonic-unfriendly” metals,
because the plasmonic loss is substantially smaller in $Au$, $Al$ and $Cu$
than in $Co$, $Fe$ and $Cr$. In order to clarify this issue it is necessary to
understand the reason why the absorption of a plasmon and the light absorption
in the bulk of metal are so different and the reason why the absorption of a
plasmon in $Al$ is smallest (Fig.1(b), even though the light absorption in the
bulk of $Al$ is largest (Fig. 1(a)).

Figure 2:  Surface plasmon at a $metal/Si$ interface. (a) Percentage of
optical field of the plasmon, which is inside of the metal; (b) skin depth in
the metal; (c) 1/e penetration depth in $Si$.

Figure 2(a) shows the relative amount of optical field in metal. In the case
of iron and chromium, the amount of optical field inside a metal is large
reaching 5 % . In contrast, the amount is less than 0.5 % in the case of
aluminum. That is the reason why the plasmonic absorption is substantially
smaller in aluminum in comparison to the iron and chromium despite the bulk
absorption in Al being substantially larger.

It is not only the bulk absorption in a metal, but also the refractive index
of the metal that determines the absorption of a surface plasmon. Figures 2(b)
and 2(c) show the skin depth in the metal and the penetration depth of the
optical field into the dielectric. Both the smaller skin depth in the metal
and the larger penetration depth into Si are reasons why the relative amount
of light penetrating in Al is much smaller than in $Fe$ and $Cr$.

The example of $Al$, in which the plasmonic loss is smallest despite of its
largest bulk absorption, indicates a possibility of engineering of a plasmonic
structure for a smaller loss by optimizing the field distribution of a surface
plasmon and, specifically, by the reduction of the amount of light field
penetrating into the metal.

The amount of light penetrating into the metal can be engineered and reduced
in a more complex plasmonic structure, which contains more dielectric layers.
For example, when a very thin $SiO_{2}$ layer is inserted at the $Si/metal$
interface. Figure 3 shows the calculated propagation loss in this plasmonic
structure vs. $SiO_{2}$ thickness. For both the “plasmon- friendly” and
“plasmon- unfriendly” metals, the propagation loss can be reduced down to 0.01
dB/$\mu m$ at an optimum $SiO_{2}$ thickness. The optimum $SiO_{2}$ thickness
is different for each metal. Such a small loss means that for a 10-$\mu
m$-long plasmonic device the absorption loss is only 0.1 dB, which is
negligible in comparison to the unavoidable coupling loss.

Figure 3:  Propagation loss of a surface plasmon propagating along a
$metal/SiO_{2}/Si$ interface. For each metal there is an optimum $SiO_{2}$
thickness, at which the plasmonic loss becomes small.

The optimum thickness is a little smaller than the cutoff thickness. The
structure, in which the $SiO_{2}$ insertion layer is thicker, has no plasmon
confinement and does not support plasmon propagation along the surface. The
incident light is reflected and absorbed by the metal in such a structure. In
case of a substantially-thicker $SiO_{2}$ layer (/ 500 nm), there is another
cutoff thickness, above which the structure supports plasmon propagation again
[14].

Figure 3 clarifies an important fact that the propagation loss of a surface
plasmon can be reduced to a nearly- equal low number for both the “plasmonic-
friendly” and “plasmonic- unfriendly” metals. The intrinsic absorption of
light in the bulk of the metal is only one of several parameters, which define
the propagation loss of a surface plasmon.

Figure 4:  Surface plasmon at a $metal/SiO_{2}/Si$ interface. (a) Percentage
of optical field of the plasmon, which is inside of the metal; (b) skin depth
in the metal; (c) 1/e penetration depth in Si. $\lambda$ =1550 nm.

Figure 4(a) shows the percentage of light penetrating into the metal as a
function of the $SiO_{2}$ thickness. At the optimum thickness, the penetration
is the smallest. Figures 4(b) and 4(c) show the skin depth in the metal and
the penetration depth in the dielectric. If the skin depth in the metal is
practically independent of the $SiO_{2}$ thickness, the penetration depth in
the dielectric increases substantially near the optimum $SiO_{2}$ thickness.
The increase of the penetration depth into the dielectric is the reason why
the percentage of optical field in metal decreases with an increase of the
$SiO_{2}$ thickness (Fig.4(a)).

This means that there is a trade-off between the volume of a surface plasmon
and the plasmon propagation loss, which cannot be overcome in any metal. The
smaller the loss is, the larger plasmon volume becomes. The larger plasmon
volume is not a problem for many plasmonic applications. For example, for a
data- processing application, the short length of a plasmonic device and the
large magnitude of the electro-optical or the magneto-optical effect are
important. The volume of a surface plasmon is not as important.

Figure 3 shows that, at least theoretically, it is possible to reduce the
plasmonic propagation loss to an ever- small number. However, practically
there are technological limitations, which limit the possible reduction. The
first limitation is the fabrication precision and the roughness of the thin
inserted layer. Near the region of the smallest loss, the dependence of loss
on the $SiO_{2}$ thickness becomes very steep and a tiny deviation of the
$SiO_{2}$ thickness from the optimum thickness causes a substantial
propagation loss. The required fabrication precision is more severe for
$Cr,Fe,Co$ than for $Au,Al,Cu$, because of their steeper slopes in Fig. 3.

Another technological limitation for the loss reduction is the penetration
depth of the optical field into the dielectric. It should roughly not exceed
about 10 $\mu m$. Otherwise; the plasmonic properties may become dependent on
inhomogeneities and imperfections of the substrate. This limits a possible
reduction to about 0.01 dB/$\mu m$ at least in the case of the metals, which
were used in calculations shown in Fig.3.

As an example of the effectiveness of the proposed technology, we have
fabricated a low-propagation-loss plasmonic waveguide made of a “plasmonic-
unfriendly” $Co$ integrated with a $AlGaAs$ waveguide. As an insertion layer,
the 17- nm- thick $MgO$ layer was used. The experimentally-measured plasmon’
propagation loss was reduced down to 0.17 dB/$\mu m$ [2].

Another example of the effectiveness of the proposed technology is the
fabrication of a low-propagation-loss plasmonic waveguide made from a
“plasmonic-unfriendly” $Co$ and the integration of the $Co$\- based plasmonic
waveguide with a $Si$ nanowire waveguide. The cobalt and iron are
ferromagnetic metals, which have large magneto-optical coefficients. The use
of a ferromagnetic metal in a plasmonic structure is essential for the
fabrication of an effective plasmonic isolator [1, 18, 14, 2, 19, 3].

The challenge for the integration of the low-propagation loss plasmonic
waveguides with a $Si$ nanowire waveguides is that the $Si$ nanowire
waveguides are fabricated on a thick $SiO_{2}$ layer having a very small
refractive index. As is shown in Supplement, the reduction of propagation loss
of a surface plasmon can be achieved only in a structure, in which the
refractive index of a thin insertion layer is smaller than that of the
dielectric substrate. It is difficult to find a material whose refractive
index is substantially smaller than that of $SiO_{2}$. The solution to this
problem is the etching out and the replacement of $SiO_{2}$ by a material of a
higher refractive index in the plasmonic section.

Figure 5:  (a) Experimental setup for measuring the propagation loss of a
surface plasmon; (b) Top- view image of the fabricated waveguides. Each
waveguide has a different length of the plasmonic section; (c) Fiber-to-fiber
transmission as a function of wavelength measured for a different length of
the plasmonic section [1].

We have used $TiO_{2}$ as a material of a high refractive index and $SiO_{2}$
as a material of a thin insertion layer. Details of the fabricated structure
and the fabrication technology are described in Supplement. Figure 5(a) shows
the measurement setup for evaluation of the plasmon’ propagation loss. The
fabricated device consists of a $Si$ nanowire waveguide, a part of which is
etched out, and the consequent layers of $TiO_{2}$, $SiO_{2}$ and $Co$ are
deposited into the gap. A set of identical waveguides having a different
length of the plasmonic section were fabricated on the same wafer. Figure 5(b)
shows a top view of the fabricated device. Figure 5(c) shows the measured
fiber-to-fiber transmission as a function of the wavelength for different
lengths of the plasmonic waveguide. From this dependence the propagation loss
of a surface plasmon is evaluated to be 0.7 dB/$\mu m$. Extrapolation of the
propagation loss to the zero length of the plasmonic section gives the
coupling loss between the plasmonic and the Si nanowire waveguides of 4 dB per
a facet [1].

The measured propagation loss is still above the theoretical limit of 0.01
dB/$\mu m$, but is already sufficiently small for a practical application. For
example, the achieved propagation loss in a plasmonic structure containing
"plasmon-unfriendly" cobalt or iron is smaller than that in conventional
gold/dielectric plasmonic structure. (See Fig.1(b)).

The propagation loss can be further reduced when the Atomic Layer Deposition
(ALD) is used instead of sputtering for deposition of the thin insertion
layer. ALD allows for extremely precise control of the film thickness and
uniformity, which are both critically important for the reduction of the
propagation loss of a surface plasmon.

In conclusion, the fabrication technology of a low- loss plasmonic waveguide
is described. The proposed technology allows the use of different metals
having different unique properties in low-loss plasmonic structures,
additionally to traditionally-used gold.

The physical principle of the loss reduction is the redistribution of the
optical field of a surface plasmon across the metal/dielectric interface. A
reduction of the optical confinement and an increase of the volume of surface
plasmon lead to a decrease of the optical field inside the metal and, as a
result, to a decrease of the propagation loss. An optimization of the optical
confinement can be achieved either by an insertion of a thin dielectric
layer[14, 20] or by the modulation of interface roughness[1] or sharpness or
by an optimization of the in-plane confinement of a surface plasmon[20, 1].

The effectiveness of the proposed technology was experimentally demonstrated
for a loss reduction in a plasmonic structure containing “plasmon- unfriendly”
cobalt. The plasmon’s propagation loss was reduced from 11 dB/$\mu m$ down to
0.7 dB/$\mu m$ in a Co-base plasmonic structure.

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##  1 Supplementary Material

Abstract The supplement describes examples of a reduction of the propagation
loss of a surface plasmon in a different plasmonic structure. It demonstrates
that a substantial reduction of the propagation loss of a surface plasmon is a
general rule for a variety of a different material of the metal and the
dielectric. The propagation constants and the field distribution of a surface
plasmon is calculated for a $Co/SiO_{2}/TiO_{2}$ plasmonic structure. Similar
to a $Co/SiO_{2}/Si$ plasmonic structure, which is described in the main part
of the manuscript, the $Co/SiO_{2}/TiO_{2}$ plasmonic structure has an optimum
$SiO_{2}$ thickness, at which the plasmonic propagation loss is reduced below
0.01 dB/$\mu m$. In both cases, the refractive index of a thin inserted layer
of $SiO_{2}$ is smaller than that of the dielectric ($TiO_{2}$ or $Si$). In
contrast, the insertion of a thin layer of a larger refractive at the
metal/dielectric interface does not lead to a substantial reduction of the
plasmonic loss. As an example, a $Co/TiO_{2}/SiO_{2}$ plasmonic structure is
studied, in which the propagation loss of a surface plasmon is calculated to
be substantial independently of the $TiO_{2}$ thickness. It is because the
refractive index of the inserted $TiO_{2}$ thin layer is larger than that of
$SiO_{2}$. Additionally, a $Co/SiO_{2}/TiO_{2}/SiO_{2}/Si$ plasmonic structure
was studied and optimized. This plasmonic structure was used in our
experimental verification of the proposed method of the loss reduction. The
details of the fabrication technology of the integration of a Co-based
plasmonic waveguide and a Si nanowire waveguide were explained.

Figure 6:  Propagation loss of a surface plasmon propagating along a
$metal/SiO_{2}/TiO_{2}$ interface. For each metal there is an optimum
$SiO_{2}$ thickness, at which plasmonic loss becomes negligibly small.
$\lambda$ =1550 nm

###  1.1 Insertion layer of a smaller refractive index effective for reduction
of plasmonic loss

In the case when the refractive index of a thin insertion layer at the
metal/dielectric interface is smaller than the refractive index of the
dielectric, the propagation loss of a surface plasmon can be reduced below
0.01 dB/$\mu m$. The reason for the reduction is that the insertion layer
redistributes the optical field of the surface plasmon near the interface
pushing it out of the metal and deeper into the dielectric.

The $metal/SiO_{2}/TiO_{2}$ plasmonic structure is an example of such a
structure. The refractive index of $SiO_{2}$ ($n_{SiO2}$=1.444 at $\lambda$ =
1550 nm) [nSiO2] is smaller than the refractive index of $TiO_{2}$
($n_{TiO2}$=2.4538 at $\lambda$ = 1550 nm) [nTiO2]. Figure 6 shows the
calculated propagation loss in the $metal/SiO_{2}/TiO_{2}$ plasmonic structure
as a function of the $SiO_{2}$ thickness. Similar to the $metal/SiO_{2}/Si$
plasmonic structure (See Fig.3), there is an optimum $SiO_{2}$ thickness,
which is different for each metal and at which the plasmonic propagation loss
can be reduced below 0.01 dB/$\mu m$. For the same metal, the optimum
$SiO_{2}$ thickness is larger for the $metal/SiO_{2}/TiO_{2}$ plasmonic
structure than for the $metal/SiO_{2}/Si$ plasmonic structure. The slopes of
curves near the optimum $SiO_{2}$ thickness are smaller in the case of the
$metal/SiO_{2}/TiO_{2}$ structure. It means that the requirements for the
roughness of $SiO_{2}$ layer are softer for the $metal/SiO_{2}/TiO_{2}$
structure.

Figure 7:  (a) Surface plasmon at a $metal/SiO_{2}/TiO_{2}$ interface. (b)
Percentage of optical field of the plasmon, which is inside of the metal; (b)
skin depth in the metal; (c) 1/e penetration depth in $TiO_{2}$. $\lambda$
=1550 nm.

Figure 7(a) shows the calculated percentage of light penetrating into the
metal as a function of the $SiO_{2}$ thickness. The dependency is very similar
to that of Fig.4(a). At the optimum $SiO_{2}$ thickness the penetration is the
smallest. As a result, there is a very small amount of optical field inside
the metal and the propagation loss of a surface plasmon is small.

Figures 7(b) and Figure 7(c) show the skin depth in the metal and the
penetration depth in the dielectric. Similar to Fig.4(b), the skin depth in
the metal is practically independent of the $SiO_{2}$ thickness, but the
penetration depth in the dielectric increases substantially near the optimum
$SiO_{2}$ thickness. From compression of Figs. 4(b) and 7(b), the skin depth
changes only slightly between two plasmonic structures and practically can be
considered as independent of the dielectric material.

In both structures, the $Co/SiO_{2}/TiO_{2}$ and the $Co/SiO_{2}/Si$, the
insertion of a thin $SiO_{2}$ layer causes a shift of the optical field out of
the metal into the dielectric. The reason why the optical field is pushed out
from the metal can be understood from Fig. 8. The amplitude of the optical
field in the inserted thin $SiO_{2}$ is an order of magnitude higher than in
$TiO_{2}$. As a result, even though the insertion layer is very thin, it
substantially affects the propagation constants of a surface plasmon. The
effective refractive index $n_{plasmon}$ of a surface plasmon is slightly
higher than the refractive index of $TiO_{2}$. The insertion of a layer of a
smaller refractive index reduces $n_{plasmon}$. When the $SiO_{2}$ thickness
increases, there is more optical field inside $SiO_{2}$ and $n_{plasmon}$
becomes smaller and, therefore, more close to the refractive index of
$TiO_{2}$. As a consequence, the perpendicular- to - interface component of
the wave vector in $TiO_{2}$ becomes smaller and, therefore, the penetration
depth in $TiO_{2}$ increases. At the cut-off thickness, the effective
refractive index of the surface plasmon becomes equal to the refractive index
of $TiO_{2}$ and the surface plasmon loses its optical confinement to the
metal surface.

Figure 8:  Distribution of the optical field across thickness in a
$Co/SiO_{2}/TiO_{2}$ plasmonic structure at a different $SiO_{2}$ thickness
(a) linear scale (b) logarithmic scale; $\lambda$ =1550 nm.

###  1.2 Insertion layer of a larger refractive index ineffective for loss
reduction

A thin insertion layer at a metal/dielectric interface, whose refractive index
is larger than the refractive index of the dielectric, is not effective for
the reduction of propagation loss of a surface plasmon. A $Co/TiO_{2}/SiO_{2}$
plasmonic structure is studied as an example of such an ineffective plasmonic
structure. Figures 9(a) shows the calculated propagation loss as a function of
$TiO_{2}$ thickness. The insertion of the $TiO_{2}$ thin layer does not reduce
the propagation loss. The tendency is opposite to that of Fig. 6 and Fig.4 as
the propagation loss increases when the $TiO_{2}$ thickness increases. The
loss increases, because the percentage of the optical field in the metal
increases (See Fig. 9(b)).

The reason why the insertion of a thin $TiO_{2}$ does not lead to reduction of
the propagation loss of a surface plasmon is the following. The effective
refractive index $n_{plasmon}$ of a surface plasmon in the $Co/SiO_{2}$
plasmonic structure is slightly larger than the refractive index $n_{SiO2}$ of
$SiO_{2}$. The 1/e penetration depth of the evanescent field in the $SiO_{2}$
layer is larger, when the difference between $n_{plasmon}$ and $n_{SiO2}$ is
smaller. The insertion of a thin $TiO_{2}$ layer of a higher refractive index
makes the plasmon’ propagation constant $n_{plasmon}$ larger and, therefore,
the difference between $n_{plasmon}$ and $n_{SiO2}$ becomes even larger. As a
result, the 1/e penetration depth into $SiO_{2}$ decreases, which leads to a
larger amount of optical field inside the metal and, therefore, to a larger
propagation loss of a surface plasmon. Figure 9(c) shows the distribution of
the optical field across the interface for the $Co/TiO_{2}/SiO_{2}$ plasmonic
structure.

Figure 9:  Surface plasmon at a $Co/TiO_{2}/SiO_{2}$ interface. (a)
Propagation loss of a surface plasmon. (b) Percentage of optical field of the
plasmon, which is inside of the metal; (c) Distribution of optical field
across thickness at a different $TiO_{2}$ thickness. $\lambda$ =1550 nm.

###  1.3 Technological considerations for integration of plasmonic and silicon
nanowire waveguides

In order to be competitive, the plasmonic devices must be integrated into the
Photonic Integrated Circuits (PIC) and, therefore, the fabrication technology
of a plasmonic device should be compatible with the existing fabrication
technology of PIC. The standard fabrication technology of PIC uses a Si(220
nm)/$SiO_{2}$(3000 nm)/Si wafer (SOI wafer) and a plasmonic device should be
fabricated on the same SOI wafer. The problem of the SOI wafer for the
application of the proposed reduction method of the plasmonic loss is very
small refractive index of $SiO_{2}$ ($n_{SiO2}$=1.444 at $\lambda$ = 1550 nm)
[nSiO2]. As was discussed above, the refractive index of the insertion layer
should be smaller than the refractive index of the dielectric layer. It is
difficult to find a material whose refractive index is substantially smaller
than that of $SiO_{2}$. The solution of this problem is the etching out and
the replacement of $SiO_{2}$ by a material of a higher refractive index in the
plasmonic section.

The first option for the material of a high refractive index, which we have
tested, was silicon. The refractive index of $Si$ is high ($n_{Si}$=3.477 at
$\lambda$ = 1550 nm) [nSi]. Following etching of $SiO_{2}$ by the Inductively
Coupled Plasma - Reactive Ion Etching (ICP-RIE ), $Si$ and a thin $SiO_{2}$
insertion layer (See Fig.3) were deposited. However, the measured plasmon’
propagation loss was high of about 4 dB/$\mu m$. Presumably, the high loss is
due to a substantial bulk optical absorption of the sputtered $Si$.

The second option for the material of a high refractive index, which we have
tested, is $TiO_{2}$. The refractive index of $TiO_{2}$ is smaller than that
of $Si$ , but still high ($n_{TiO2}$=2.4538 at $\lambda$ = 1550 nm) [nTiO2].
Even though the achieved material quality of the sputtered $TiO_{2}$ is better
that that of $Si$, it is still technologically difficult to obtain the
required material quality of $TiO_{2}$ in a deep-etched micro-sized plasmonic
section. A reduction of the etching depth and the $TiO_{2}$ thickness are
desirable to keep a required interface quality. Figure 10 shows the calculated
propagation loss of a surface plasmon for a different etch depth and,
correspondingly, a different thickness of $TiO_{2}$. The black line shows the
case when the 3-$\mu m$-thick $SiO_{2}$ is fully etched. In this case, the
propagation loss can be below 0.01 db/$\mu m$, when thickness of the top
$SiO_{2}$ is about 25 nm. Even in the case when only a half of the $SiO_{2}$
layer is etched, the propagation loss is reasonably low.

Figure 10:  Propagation loss of a surface plasmon in a
$Co/SiO_{2}/TiO_{2}/SiO_{2}/Si$ structure as a function of thickness of the
top $SiO_{2}$ layer. Each curve corresponds a different thickness of $TiO_{2}$
when the sum of thicknesses of the $TiO_{2}$ and bottom $SiO_{2}$ layers is
kept to be 3 $\mu m$. $\lambda$ =1550 nm

###  1.4 Details of fabricated structure and fabricated technology

The device shown in Fig.5 was fabricated on a SOI wafer. A sputtered 40-nm-
thick $SiO_{2}$ was used as a hard mask. Following an electron-beam (EB)
lithography, a 450-nm-wide Si nanowire waveguide with a 150-nm-wide spot-size
converter was dry etched by an ICP-RIE. Next, a 2 $\mu m$ of $SiO2$ was dry-
etched by an ICP-RIE in the plasmonic section and a $TiO2$ (2.1 $\mu m$) /
$SiO2$ (40 nm) / $Si$ (100 nm) was sputtered and lift-off, so that it remains
only in the plasmonic section. Next, following the EB lithography, a 70-nm-
wide Si bridge was etched in the plasmonic section. The Si bridge is required
for the in-plane confinement of a surface plasmon [1, 12]. Finally, $SiO2$ (10
nm) / $Co$ (200 nm)/$Au$ (500 nm) was sputtered into the plasmonic section.


3 1 May 17, 2022 1

# Dynamical Systems and $\lambda$-Aluthge transforms

Linh Tran  *
Department of Mathematics
Chungnam National University
Daejon 305-764, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this note, we verify that the bounded shadowing property and quasi-
hyperbolicity of bounded linear operators on Hilbert spaces are preserved
under Aluthge transforms.

###### keywords:

Aluthge transform, quasi-hyperbolic, $\lambda$-Aluthge transform, bounded
shadowing property

###### 1991 Mathematics Subject Classification:

Primary: 47A15; Secondary: 47B49, 47B37, 37C20

##  1\. Introduction

Let $\left(\mathbb{H},\left<.\right>\right)$ be a Hilbert space and
$B\left(\mathbb{H}\right)$ be the algebra of bounded linear operators on
$\mathbb{H}$. For each $T\in B\left(\mathbb{H}\right)$, the polar
decomposition of $T$ is uniquely defined by

$\displaystyle T=U\left|T\right|$

in which $\left|T\right|=\left(T^{*}T\right)^{\frac{1}{2}}$ and
$\ker\left(U\right)=\ker\left(\left|T\right|\right)=\ker\left(T\right)$.
Without further mention, the spectrum of $T$ is

$\displaystyle\sigma\left(T\right)=\left\\{\lambda\in\mathbb{C}\mid\nexists\left(T-\lambda
I\right)^{-1}\in B\left(\mathbb{H}\right)\right\\}.$

During the last 30 years, Operator Theorists have paid their attention to
Aluthge transforms because they convert the initial operators into one which
are closer to normal operators. I. B. Jung [11, Theorem 2.1] showed that

$\displaystyle\sigma\left(T\right)=\sigma\left(\Delta\left(T\right)\right)$

where $\sigma\left(.\right)$ are the corresponding spectra. Moreover, Jung-Ko-
Pearcy [10, Corollary 1.16] proved that $T$ has nontrivial invariant subspaces
if and only if so does $\Delta\left(T\right)$.
On the other sides, another reason relates with the Aluthge iterates defined
by

$\displaystyle\begin{cases}\Delta^{(0)}\left(T\right)=T\\\
\Delta^{(n)}\left(T\right)=\Delta\left(\Delta^{(n-1)}\left(T\right)\right),\quad\quad\forall
n\geq 1,n\in\mathbb{Z}^{+}.\end{cases}$

The Aluthge iterates of bounded linear operators are convergent to normal
operators for $\dim\left(\mathbb{H}\right)<\infty$ that has been verified in
[1, Theorem 4.2] but not in general as shown in [11, Corollary 3.2].
We have the following remark.

######  Remark 1.1.

Let $T=U\left|T\right|\in B\left(\mathbb{H}\right)$ be the polar decomposition
of $T$.

  1. (i)

For every $\alpha\in\mathbb{C}$, $\Delta\left(\alpha
T\right)=\left|\alpha\right|\Delta\left(T\right)$;

  2. (ii)

[10] $T$ is quasinormal if and only if $U\left|T\right|=\left|T\right|U$.
Hence, $\Delta\left(T\right)=T$;

  3. (iii)

[15]
$\lim_{n\rightarrow\infty}\left\|\Delta^{\left(n\right)}\left(T\right)\right\|=r\left(T\right)$.

Regarding to the spectral properties, we have the following remark.

######  Remark 1.2.

[10, Theorem 1.3] Let $T\in B\left(\mathbb{H}\right)$ be a bounded linear
operator with the polar decomposition $T=U\left|T\right|$ and
$\Delta\left(T\right)=\left|T\right|^{\frac{1}{2}}U\left|T\right|^{\frac{1}{2}}$
be the Aluthge transform of $T$. Then the following properties hold.

  1. (i)

$\sigma\left(T\right)=\sigma\left(\Delta\left(T\right)\right)$;

  2. (ii)

$\sigma_{ap}\left(T\right)=\sigma_{ap}\left(\Delta\left(T\right)\right)$;

  3. (iii)

$\sigma_{p}\left(T\right)=\sigma_{p}\left(\Delta\left(T\right)\right)$.

where $\sigma_{ap}\left(.\right)$ (resp. $\sigma_{p}\left(.\right)$) are the
corresponding approximate point (resp. point) spectra.

Aluthge transforms and Aluthge iterates of bounded linear operators $T$ on
Hilbert space $\mathbb{H}$ were generalized to $\lambda$-Aluthge transforms
and $\lambda$-Aluthge iterates.

######  Definition 1.3.

Let $T=U\left|T\right|\in B\left(\mathbb{H}\right)$ be the polar decomposition
of $T$. For every $\lambda\in\left(0,1\right)$, the $\lambda$-Aluthge
transform of $T$ is defined by

$\displaystyle\Delta_{\lambda}\left(T\right)=\left|T\right|^{\lambda}U\left|T\right|^{1-\lambda}.$

For each $\lambda\in\left(0,1\right)$, the $\lambda$-Aluthge iterates of $T$
is determined by

$\displaystyle\begin{cases}\Delta_{\lambda}^{(0)}\left(T\right)=T\\\
\Delta_{\lambda}^{(n)}\left(T\right)=\Delta_{\lambda}\left(\Delta_{\lambda}^{(n-1)}\left(T\right)\right),\quad\quad\forall
n\geq 1,n\in\mathbb{Z}^{+}.\end{cases}$

On the other hand, dynamicists recently published various results related to
the behaviors of Dynamical Systems under Aluthge transforms such as in [4, 5,
6]. In fact, I. B. Jung et al. verified that $T$ and its Aluthge transform
$\Delta\left(T\right)$ are similar if $T$ is invertible in [10, Lemma 1.1].
Therefore, they share shadowing property, topological transitivity and chaotic
behaviors. In addition, K. Lee and C. A. Morales introduced bounded shadowing
property in [13].

######  Definition 1.4.

A homeomorphism $g:Y\longrightarrow Y$ has the bounded shadowing property if
for every $\epsilon>0$, there is $\delta>0$ such that every bounded
$\delta$-pseudo orbit can be $\epsilon$-shadowed.

Actually, every shadowing operators has bounded shadowing property but the
converse does not held. In the light of these results, our first result
verifies that bounded shadowing property is preserved under $\lambda$-Aluthge
transform in Theorem 2.2.
In addition, using the spectral properties of $\lambda$-Aluthge transform (for
every $\lambda\in\left(0,1\right)$), we show that quasi-hyperbolicity is
invariant under the action of $\lambda$-Aluthge transforms in Proposition 2.5.

##  2\. Main results

Let us begin this section by recalling that on metric spaces
$\left(X,d_{1}\right)$ and $\left(Y,d_{2}\right)$, a homeomorphism
$L:\left(X,d_{1}\right)\longrightarrow\left(Y,d_{2}\right)$ is called
Lipeomorphism if $L$ satisfies Lipschitz condition, i.e for every $x,y\in X$,
there are constants $A,B>0$ such that

$\displaystyle Ad_{1}\left(x,y\right)\leq
d_{2}\left(L\left(x\right),L\left(y\right)\right)\leq Bd_{1}\left(x,y\right).$

More precisely, $L$ is Lipeomorphism if and and only there is a constant $L>0$
so that

$\displaystyle d_{2}\left(L\left(x\right),L\left(y\right)\right)\leq
Ld_{1}\left(x,y\right).$

Lipeomorphisms draw the attention of mathematicians because they preserve
boundedness and completeness of the domain. Furthermore, Lee-Morales verify
that bounded shadowing property is invariant under certain topological
conjugacies as shown in the following lemma.

######  Lemma 2.1.

[13, Lemma 6] Let $Y$, $Z$ be metric spaces and $H:Z\longrightarrow Y$ be a
Lipeomorphism (i.e Lipschitz homeomorphism with Lipschitz inverse). If
$P:Z\longrightarrow Z$ is a homeomorphism with the bounded shadowing property,
then so does $H\circ P\circ H^{-1}:Y\longrightarrow Y$.

As a consequence, we have our first result.

######  Theorem 2.2.

Let $T=U\left|T\right|\in B\left(\mathbb{H}\right)$ be the polar decomposition
of invertible bounded linear operator $T$. Then $T$ has bounded shadowing
property if and only if $\Delta\left(T\right)$ does.

###### Proof.

Let us assume that $T=U\left|T\right|\in B\left(\mathbb{H}\right)$ be the
polar decomposition of invertible bounded linear operator $T$. Then
$\left|T\right|=\left(T^{*}T\right)^{\frac{1}{2}}$ and
$\left|T\right|^{\frac{1}{2}}$ are invertible bounded linear operators either.
Moreover, since $T$ is bounded, we have

$\displaystyle\left\|T\left(x\right)-T\left(y\right)\right\|=\left\|T\left(x-y\right)\right\|\leq\left\|T\right\|\left\|x-y\right\|,\quad\forall
x,y\in\mathbb{H}.$

Similarly,

$\displaystyle\left\|\left|T\right|^{\frac{1}{2}}\left(x\right)-\left|T\right|^{\frac{1}{2}}\left(y\right)\right\|=\left\|\left|T\right|^{\frac{1}{2}}\left(x-y\right)\right\|\leq\left\|\left|T\right|^{\frac{1}{2}}\right\|\left\|x-y\right\|,\quad\forall
x,y\in B\left(\mathbb{H}\right).$

So, both $T$ and $\left|T\right|^{\frac{1}{2}}$ satisfy Lipschitz condition
and turn out to be Lipeomorphism. Additionally,

$\displaystyle\Delta\left(T\right)=\left|T\right|^{\frac{1}{2}}U\left|T\right|^{\frac{1}{2}}=\left|T\right|^{\frac{1}{2}}\left(U\left|T\right|\right)\left|T\right|^{-\frac{1}{2}};$
$\displaystyle
T=U\left|T\right|=\left|T\right|^{-\frac{1}{2}}\left(\left|T\right|^{\frac{1}{2}}U\left|T\right|^{\frac{1}{2}}\right)\left|T\right|^{\frac{1}{2}}.$

Now, let us suppose that $T$ has bounded shadowing property. Using Lemma 2.1,
we completed the proof and got the first result.

∎

In fact, the spectral picture of bounded linear operator $T\in
B\left(\mathbb{H}\right)$ under Aluthge transformation has been described in
[11]. It can be observed that not only the spectrum $\sigma\left(T\right)$ but
also its components $\sigma_{p}\left(T\right)$ (point spectrum of $T$) and
$\sigma_{ap}\left(T\right)$ (appropriate spectrum of $T$) are preserved. We
now show that the spectral properties of $T$ are preserved under
$\lambda$-Aluthge transforms for all $\lambda\in\left(0,1\right)$.

######  Theorem 2.3.

Let $T=U\left|T\right|\in B\left(\mathbb{H}\right)$ be the polar decomposition
of $T$. For every $\lambda\in\left(0,1\right)$, we suppose that the
$\lambda$-Aluthge transform of $T$ is
$\Delta_{\lambda}\left(T\right)=\left|T\right|^{\lambda}U\left|T\right|^{1-\lambda}$.
Then the following properties held.

  1. (i)

$\sigma\left(T\right)=\sigma\left(\Delta_{\lambda}\left(T\right)\right)$;

  2. (ii)

$\sigma_{ap}\left(T\right)=\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)$.

###### Proof.

The equality
$\sigma\left(T\right)=\sigma\left(\Delta_{\lambda}\left(T\right)\right)$ was
proved before in [8, Lemma 5].
Let us assume that $T=U\left|T\right|\in B\left(\mathbb{H}\right)$ is a
bounded linear operator on Hilbert space $\mathbb{H}$. For arbitrary
$\lambda\in\mathbb{C}$, the $\lambda$-Aluthge transform of $T$ is determinded
by

$\displaystyle\Delta_{\lambda}\left(T\right)=\left|T\right|^{\lambda}U\left|T\right|^{1-\lambda}.$

Firstly,
$\sigma_{ap}\left(T\right)\setminus\left\\{0\right\\}=\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)\setminus\left\\{0\right\\}$
as a result of Theorem 1 in [9].
We now suppose that $0\in\sigma_{ap}\left(T\right)$. By definition, there
exists a sequence of unit vectors
$\left\\{x_{n}\right\\}_{n\rightarrow\infty}$ with $\left\|x_{n}\right\|=1$
(for every $n\in\mathbb{N}$) satisfying

$\lim_{n\rightarrow\infty}\left\|U\left|T\right|x_{n}\right\|=0.$

Consequently, $\left\|\left|T\right|^{1-\lambda}x_{n}\right\|\rightarrow 0$ as
$n\rightarrow\infty$ for every $\lambda\in\left(0,1\right)$. It follows that

$\displaystyle\Delta_{\lambda}\left(T\right)=\left\|\left|T\right|^{\lambda}U\left|T\right|^{1-\lambda}x_{n}\right\|\rightarrow
0\quad\text{as }n\rightarrow\infty.$

That means $0\in\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)$ for
every $\lambda\in\left(0,1\right)$.
Therefore,
$\sigma_{ap}\left(T\right)\subseteq\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)$
for all $\lambda\in\left(0,1\right)$.
On the other sides, we suppose that
$0\in\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)$ for every
$\lambda\in\left(0,1\right)$. By hypothesis, there is a sequence
$\left\\{x_{n}\right\\}_{n\in\mathbb{Z}}$ of unit vectors in $\mathbb{H}$ so
that

$\displaystyle\lim_{n\rightarrow\infty}\left\|\left|T\right|^{\lambda}U\left|T\right|^{1-\lambda}x_{n}\right\|=0.$

There are two subcases.
Firstly, for an arbitrary $\lambda\in\left(0,1\right)$, if
$\left\|\left|T\right|^{1-\lambda}x_{n}\right\|\rightarrow 0$ then obviously
$\left\|Tx_{n}\right\|=\left\|U\left|T\right|^{\lambda}\left|T\right|^{1-\lambda}x_{n}\right\|\rightarrow
0$ as $n\rightarrow\infty$. Secondy, for every $\lambda\in\left(0,1\right)$,
if $\left\|\left|T\right|^{1-\lambda}x_{n}\right\|$ does not converge to $0$
then $\left\|U\left|T\right|^{1-\lambda}\right\|$ does not converge to $0$ as
$n\rightarrow\infty$. However, $\left|T\right|^{1-\lambda}$ maps
$\left\\{U\left|T\right|^{1-\lambda}x_{n}\right\\}$ to a null sequence in norm
so
$\left\|T\left(U\left|T\right|^{1-\lambda}x_{n}\right)\right\|\longrightarrow
0$ when $n\longrightarrow\infty$ as a consequence. Since $U$ is an isometry,
we get $\left\|U\left|T\right|^{1-\lambda}x_{n}\right\|=1$. Hence,
$0\in\sigma_{ap}\left(T\right)$ and
$\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)\subseteq\sigma_{ap}\left(T\right)$.
Totally, the proof has been completed. ∎

As a consequence, the invariance of quasi-hyperbolicity of $T$ under the
action of $\lambda$-Aluthge transforms for all $\lambda\in\left(0,1\right)$ is
excuted.

######  Definition 2.4.

Let $T\in B\left(X\right)$ be an invertible operator on a Banach space $X$.
$T$ is said to be a hyperbolic operator if
$\sigma\left(T\right)\cap\mathbb{T}=\varnothing$ in which $\mathbb{T}$ is the
unit circle of complex plane $\mathbb{C}$.
We shall say that $T\in B\left(\mathbb{H}\right)$ is quasi-hyperbolic if there
exists $n\in\mathbb{N}$ (independent of $x$) such that

$\displaystyle\max\left(\left\|T^{2n}x\right\|,\left\|x\right\|\right)\geq
2\left\|T^{n}x\right\|,\quad\text{for all }x\in\mathbb{H}.$

######  Proposition 2.5.

For a bounded linear operator $T\in B\left(\mathbb{H}\right)$, $T$ is quasi-
hyperbolic if and only if its $\lambda$-Aluthge transform is for every
$\lambda\in\left(0,1\right)$.

###### Proof.

C. J. K. Batty and Y. Tomilov showed in [3] that $T$ is quasi-hyperbolic if
and only if $\sigma_{ap}\left(T\right)$ does not intersect with the unit
circle $\mathbb{T}$ on complex plane $\mathbb{C}$. In other words,

$\displaystyle\sigma_{ap}\left(T\right)\cap\mathbb{T}=\varnothing.$

For every $\lambda\in\left(0,1\right)$, the equality
$\sigma_{ap}\left(T\right)=\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)$
in 1.2 provides
$\sigma_{ap}\left(\Delta_{\lambda}\left(T\right)\right)\cap\mathbb{T}=\varnothing$.
Thus $\Delta_{\lambda}\left(T\right)$ is quasi-hyperbolic for every
$\lambda\in\left(0,1\right)$. ∎

## References

  * [1] J. Antezana, E. R. Pujals and D. Stojanoff, The iterated Aluthge transforms of matrix converge, Adv Math (N Y), 226 (2011), 1591–1620.
  * [2] A. Aluthge, On $p$-hyponormal operators for $0<{p}<1$, Integral Equ. Oper. Theory 13 (1990), 307–315.
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# Active Connectivity Fundamentals for TSCH Networks of Mobile Robots

Charalampos Orfanidis, Paul Pop, Xenofon Fafoutis  Department of Applied
Mathematics and Computer Science, Technical University of Denmark
<EMAIL_ADDRESS>
###### Abstract

Time Slotted Channel Hopping (TSCH) is a medium access protocol defined in the
IEEE 802.15.4 standard which have been proven to be one of the most reliable
options when it comes to industrial applications. TSCH has been designed to be
utilized in static network topologies. Thus, if an application scenario
requires a mobile network topology, TSCH does not perform reliably. In this
paper we introduce active connectivity for mobile application scenarios, such
as mobile robots. This is a feature that enables the option to regulate
physical characteristics such as the speed of a node as it moves, in order to
keep being connected to the TSCH network. We model the active connectivity
approach through a basic example where two nodes are moving towards the same
direction to infer the main principles of the introduced approach. We evaluate
the active connectivity feature through simulations and quantify trade-off
between connectivity and application-layer performance.

###### Index Terms:

TSCH, Mobility, Dependable IoT, Reliability.

##  I Introduction

Time Slotted Channel hopping (TSCH) [1] is a Medium Access Control (MAC)
protocol which became very popular in Industrial Internet of Things (IoT) [2]
since it can provide robustness and reliability towards the network
performance. The TSCH protocol is based on a channel hopping function that
mitigates the impact of interference and multipath fading that diminish the
communication in IoT networks. All the nodes in TSCH networks share a common
time source to enable synchronization among the nodes. Provided that the nodes
are synchronized, a schedule is calculated including a predefined channel
offset and time offset which declares the communication details within the
TSCH network.

TSCH is a flexible protocol which can be tailored towards the requirements of
each application. A very popular customization approach is modifying the TSCH
schedule. Since the traffic demands, network topology and routing might vary a
lot depending on the application, the TSCH schedule is a crucial part which
can be adjusted based on the application demands. Therefore, there is an
extended scientific literature around TSCH scheduling schemes, which includes
three main categories: centralized [3], distributed [4] and autonomous [5].

As TSCH networks emerged, they started being utilized in different kind of
dependable applications with mobile nodes, that each one has various set of
requirements with different challenges every time. For example, wearable
sensors [6] for health, are mobile devices which could benefit from a reliable
mobile TSCH. In [7], the node mobility is distinguished between macro and
micro mobility: macro is the mobility between different network domains and
micro is the mobility within the current network domain. In this paper we
focus on micro mobility and more specifically mobility within the TSCH
network. In our previous work [8] we demonstrated that the current state of
the art in TSCH schedules cannot support mobility. Thus, TSCH performance is
far from reliable when mobile communication is required. There are several
factors to be taken into account in a mobile scenario like the traffic volume,
the mobility model, the power budget and the operating environment.

Mobile robotics is another application that could benefit from reliable TSCH-
based robot-to-robot communication, such as Unmanned Aerial Vehicles (UAV)
[9], autonomous vehicles [10] and many more. Indeed, similar to wireless
sensor networks, mobile robots are cyber-physical systems that depend on
traditional (e.g., [11]) and nontraditional sensors (e.g., [12]) to perceive
the environment and act to maximise their application objectives. This synergy
is recently investigated as part of the Internet of Robotic Things [13, 14].

Unlike wearable sensors, whereby the device has no control on the mobility of
the user, mobile robots routinely adjust their mobility to maximize their
objectives, e.g. planning based on energy consumption estimates [15, 16]. When
reliable communication is vital, a mobile robot can make similar adjustments
for maintaining reliable connectivity based on link quality estimates. In this
spirit, we introduce the concept of Active Connectivity (AC). Namely, a node
has the ability to regulate its physical characteristics (i.e., speed, route,
topology) to remain within coverage of its neighbours and therefore keep being
connected to the TSCH network. While this approach can increase the
reliability and robustness of the TSCH network, it may have negative impact on
other objectives, such as the time the robot needs to cover a certain distance
in order to complete a task and consequently to the overall cost. Thus, we
investigate the tradeoff between the reliable connectivity and application
performance in terms of distance covered. In order to acquire a fundamental
insight into the performance of AC we opted for studying a basic scenario,
whereby two mobile nodes move towards the same direction. For the scope of the
evaluation we used the Cooja simulator [17], a cross layer network simulator
which includes different levels from physical to application layer.

This paper is organized as follows. A brief technical description of TSCH is
mentioned in Section II. Section III presents the state of the art and how it
is related with the current work. Section IV illustrates the AC operating
principles. The experimental scenario, the evaluation details the obtained
results and a discussion are presented in Section V. Finally Section VI
concludes the paper and discusses future work.

##  II TSCH Primer

This Section provides a brief overview with the technical aspects of TSCH
protocol which is part of the IEEE 802.15.4 standard [18] and an amendment in
IEEE 802.15.4e-2015 [19].

TSCH operates at the link layer and two of its main features is the frequency
hopping and the synchronization of the nodes. The nodes of the TSCH network
are following a duty cycle which is dictated by the schedule. They have the
option to sleep for reserving power, transmit or receive a packet. In order to
avoid collisions, it is not allowed to multiple transmissions at the same time
and channel. The schedule is organized a two-dimensional table called the
slotframe. Figure 1 illustrates a slotframe with a topology. The time offset
is represented on the x-axis and the channel offset at the y-axis. Every cell,
which is called timeslot, represents a specific time and channel offset and
most of the times has a duration of $10$ ms which is enough to send an IEEE
802.15.4 packet and receive an ACK. In Figure 1 there are $6$ timeslots and
$5$ channel offsets. The number of the timeslots is defining the length of the
slotframe and has an impact on several tradeoffs affecting the network
performance.

Figure 1: TSCH slotframe on the left and routing topology on the right.

The TSCH coordinator is constructing the TSCH network, it defines the network
ID, slotframe size and the Frequency Hopping Sequence (FHS). The coordinator
is also responsible to initialize the Absolute Slot Number (ASN) to $0$ which
is increased with a new timeslot. To join the TSCH network a node has to
receive an Enhanced Beacon (EB) packet, which is broadcast from the already
joined nodes. EB contains essential information like the time source which is
necessary to synchronize with the other nodes. Equation 1 is computing a
channel for a given cell, CO stands for Channel Offset. There is a priority
mechanism in case there is more than one transmission cell at the same time.

$Channel=FHS(ASN+CO)\;mod\;||FHS||$ (1)

###  II-A Minimal Scheduling Function (MSF)

During the evaluation part we used the MSF schedule to keep this part simple
as well. The MSF is defined in the RFC 9033 [4] as a scheduling mechanism for
TSCH implemented at the top of 6top protocol. MSF include three type of cells:
minimal, autonomous, negotiated. Minimal cells are used for EB packets and
routing purposes; autonomous are used for unicast communication and a
negotiated cell is used when a node has just joined the TSCH network and
intends to let the other nodes about its presence.

##  III Related Work

Utilizing TSCH under mobile scenarios is a research topic which is not covered
adequately by the research literature. This section illustrates the existing
research endeavours but also a few instances presenting how RPL (Routing
Protocol for Low-Power and Lossy Networks) performs in mobile scenarios as
well. RPL is used in several TSCH approaches as the default routing protocol
and information obtained from RPL are used to construct the TSCH schedule at
some case. Therefore, it is interesting to understand how it performs on
mobile scenarios and what may be its impact on TSCH.

Al-Nidawi et al [20], evaluated the Low Latency Deterministic Network (LLDN)
and TSCH protocols under mobility scenarios. The authors identify four main
issues TSCH that affect network performance and quantify their impact on the
energy consumption and the network coverage. The main issues mentioned are the
long channel offset which increases the scanning process, mobile nodes cannot
identify coordinators due to beaconing in multiple frequencies and the absence
of a flexible timeslot scheme which will enable a smoother function when a
node is added or deleted. After gaining an insight into the challenges of TSCH
in mobile scenarios Al-Nidawi et al. [21] propose Mobile TSCH to decrease the
delay that arises during the TSCH joining procedure. In order to achieve this
they encapsulate Enhanced Beacon (EB) packets into ACK packets which are
transmitted on a fixed channel. The proposed approach decreases the Radio Duty
Cycle (RDC) by an average of $30$% and increases the connectivity by $25$%. An
evaluation of TSCH in mobile scenarios which considers both mobile and
stationary nodes presents that TSCH can can achieve adequate connectivity with
the premise that there are adequate amount of nodes either stationary or
mobile to enable a proper coverage [22]. It is introduced though an overhead
due to the increased amount of messages used to maintain synchronization which
has an impact on the energy consumption and the latency of the network. A
novel TSCH schedule called Instant [23], is utilized in indoor localization
and data collection services in application scenarios designed for remote
healthcare in residential environments. Instant includes both stationary and
mobile nodes and the new feature offered in this schedule is that the mobile
nodes are able to know information about reservations of subsequent unicast
cell block of the stationary nodes in advance through a simple probe–ACK
transaction. Therefore, if they want to transmit a large amount of data, they
select a stationary node based on this information to select an available node
and avoid delay and energy overhead. In a previous work we investigated how
popular TSCH schedules perform heterogeneous mobility patterns [8]. More
specifically, we examined through simulations how MSF [4], Orchestra[5] and
Alice[24] schedules perform on mobility patterns representing an autonomous
agricultural vehicles application scenario and a smart warehouse where a
synergy between machinery, autonomous robots/vehicles and workers communicate.
We illustrate that neither of the schedules is providing a reliable
performance because a number of nodes are suffering from coverage issues or
the schedule cannot handle the mobility and most of the times is outdated.

An extended survey on a significant amount of mobility models and their impact
on RPL protocol is presented in [25]. The authors provide a comprehensive
taxonomy, a classification of the mobility models and they conduct a
comparison based on their main specification. Moreover, they evaluate RPL used
in several mobility models using Cooja simulator to quantify metrics such as
power consumption, reliability, latency and control overhead. An interesting
observation is that the RPL’s trickle timer, which is used to control traffic
overhead by dividing the time into intervals, has an impact on the
performance. Barceló et al. propose an extension of the 6TiSCH routing in
[26], considering scenarios where static and mobile nodes co-exist. The
routing between static nodes is carried out by the regular 6TiSCH but the
routing among mobile nodes and static nodes is performed by a novel approach
which is utilizing the end to end reliability estimations with a blacklisting
function based on the position of the node. They evaluate their approach
through simulations and the results show that the reliability between the
mobile and static nodes is increased even when there are occurring high
positioning errors compared to previous routing approaches. Another approach
to improve mobile scenarios which use RPL is presented in [27]. The authors
follow a Software Defined Network (SDN) technique where a central controller
has a holistic view of the network, can predict the handovers and update the
routing tables. In addition, the central controller is not limited in terms of
resources like the regular IoT nodes. Thus, it can execute computational
intensive algorithms such as particle and Kalman filter. The authors evaluate
their approach through Cooja simulations and the results suggest that the
proposed approach can achieve higher reliability compared with the baseline
RPL but also with the current state of the art mRPL.

##  IV Active Connectivity

The AC approach entails several challenges depending on the application
requirements. This section is establishing the foundations of enabling AC in
TSCH networks with mobile nodes. The main principle is that if a link between
two mobile nodes is weak and is about to die then the nodes have the ability
to regulate physical characteristics like their speed, their topology or their
route to keep this link alive. In this paper we focus only on the speed due to
limited space but still this is very general principal that generates a set of
technical concerns that have to be addressed.

The first consideration to be taken is to decide which of the two nodes is
going to regulate its speed. The routing in TSCH networks is organized with a
tree topology most of the times and we decided that it will be convenient that
the child node will be the one which will regulate its speed because it will
probably affect less nodes in the network. In that sense if the child node has
its own children, then these nodes have to regulate their speed as well in the
same rate to remain within the connectivity range. Another consideration was
to select the metric which would indicate that a link is about to die. The IoT
devices are using low cost radios which are prone to noise and interference
[28, 29]. There are metrics like Radio Signal Strength Indicator (RSSI) which
is the captured power level after a packet reception. The RSSI measurement
unit is decibels per milliwatt (dBm). A similar metric is the Link Quality
Indicator (LQI) which describes the quality of the received packet but it is
less depended by the environment activity compared with RSSI. Both of them
have significant spatial or temporal dependencies especially during a mobile
scenario. This makes it challenging to find a balance between a representing
quantity of samples and be coherent with the limited amount of resources
offered by IoT devices. To this end we decided to determine a weak link based
on the Exponentially Weighted Moving Average (EWMA) of RSSI values, which is
described in Equation 2.

$EWMA_{RSSI_{i}}=\alpha*RSSI_{i}+(1-\alpha)*EWMA_{i-1}$ (2)

Figure 2: Representation of potential speed profiles. Example of a moderate
acceleration and deceleration adaptation (a), within the gray intervals the
acceleration $A_{1}$ and deceleration $A_{2}$ represent the speed change,
which is activated based on the Minimum and Maximum RSSI Thresholds. In (b) it
is denoted the steep acceleration moderate deceleration, in (c) the moderate
acceleration steep deceleration and in (d) the steep acceleration steep
deceleration technique.

EWMA is not consuming a lot of resources and at the same time it is able to
represent a series of RSSI values. The selection of the alpha parameter, which
is defining the weight of the current RSSI value and the weight of the
previous RSSI values, is examined in the evaluation part in Section V along
with different RSSI thresholds used to notify the prospect of a weak link. The
rate that the speed is going to be regulated was also one of the
considerations we had to take. Apparently the acceleration or the deceleration
can affect the performance of the AC operation. Especially when a node is at
the very limits of the connectivity range it is important to regulate its
speed faster. We illustrate some potential speed profiles we in Figure 2.
Namely, a moderate approach both for acceleration and deceleration illustrated
in Figure 2a, an approach which includes a steep acceleration and a moderate
deceleration in Figure 2b, another one where the acceleration is moderate and
the deceleration steep in Figure 2c and one where both acceleration and
deceleration are steep in Figure 2d. In this paper we focus only at the speed
profile illustrated in 2a for practical reasons.

The metrics to evaluate the performance of AC have to be considered as well.
Networking metrics like throughput, packet delivery ratio are able to
represent the network performance of the TSCH network but we also use the
downtime which we define as the time a node spends not being connected to the
TSCH network after the first time it gets connected. Moreover, the AC will
have an impact on the distance a node will cover. Therefore, we consider the
moving distance as a metric towards the application layer, because if a node
covers a shorter distance due to AC purposes, it might increase the operating
cost since the node would require more time to operate. For instance, if the
node is an autonomous vehicle, it would require more fuel in that case.

##  V Evaluation

This section describes all the technical details during the evaluation part,
the obtained results and a discussion upon them. In order to evaluate the AC
we used the Cooja simulator [17] and we defined a simplistic topology with two
nodes depicted in Figure 3. The mobility model we use is also very simple,
node A in Figure 3 is the root node and it is moving to the same direction
with node B with a distance of $130$ m. They are moving in a straight line
form until they reach the end of the grid and then they return back. They
repeat this pattern for $30$ minutes of simulation time. Node A is moving with
$1$ m/s and node B with $3$ m/s. The maximum speed a node can go is up to $3$
m/s and the minimum $0$ m/s. We define these regulations for safety reasons (a
mobile node can be a vehicle, a robot, a UAV and cannot move too fast) but
also to consider the covered distance as a meaningful metric to evaluate the
AC impact on the application layer. All the simulations were repeated with
$20$ different random seeds to avoid statistical bias. The configuration
parameters of the simulation are listed in Table I.

Figure 3: The rudimentary mobility pattern Node A and Node B follow to model
the design principles of the Active Connectivity mechanism. $3V_{A}=V_{B}$

We evaluate two versions of AC mechanism. In the first one we assume that a
node knows the position of the rest of the nodes in the network, thus when AC
notifies it about a weak link it knows if it has to accelerate or decelerate.
We do this assumption due to the fact that several application scenarios such
as autonomous agricultural vehicles most probably have GPS and can aggregate
their position through the TSCH network but also in the spirit of abstracting
complexities. We refer to this version as AC. In the other version a node does
not know the positions of the other nodes and the mechanism does a random
guess to decide if it has to accelerate of decelerate when it receives a
notification about a weak link. This decision is evaluated during the
operation and if the EWMA level drops down to threshold after this decision
has been taken, the mechanism changes its decision and does the opposite
action. We refer to this mechanism as AC Random (ACR). In the ACR case the
topology depicted in Figure 3 could change if node B decrease a lot its speed
and node A overtake it and it depends up to the configurations parameters that
node B increase its speed in time to continue being in range with node A. We
also run a version with the AC mechanism deactivated to have it as a benchmark
comparison.

Figure 4: Throughput using AC for different alpha values and RSSI thresholds. The colors representing the alpha values are not visible because the throughput performance is homogeneous close to its maximum value for most of the cases. Figure 5: Throughput using ACR for different alpha values and RSSI thresholds. TABLE I: Configuration parameters used during the simulation Parameter |  Value
---|---
Grid size |  2000 x 2000 m
Transmission range |  450 m
Radio model |  Logistic Loss
Mobility model |  Mobile Ad Hoc Network
Mobility speed |  0 to 3 m/s
Traffic rate |  1 packet/5 sec
Number of nodes |  2
Slotframe length |  3
Slot duration |  10 ms
TSCH schedule |  Minimal Scheduling Function
Transceiver |  CC2420
Frequency spectrum |  2.4 GHz
Simulation duration |  30 min

Figures 4 and 5 illustrate the first set of results which is the achieved
throughput. Thus, Figure 4 presents the throughput for three different alpha
values used in EWMA and six different RSSI threshold values which will
activate the AC mechanism. Notice that the RSSI threshold values are mentioned
in couples as the lower will activate the AC mechanism when the link is weak
and the higher when the link is in a better status and regulate its speed
towards to its initial speed. All the following Figures present the results on
this manner.

AC was able to achieve approximately $33$ bps for most of the cases which is
the maximum it can be achieved in this simulation. However when the alpha
value was $0.25$ there are few cases when the throughput was lower close to
$25$ bps. We speculate that this due to the manner EWMA functions, namely with
a lower alpha value the weight of the last RSSI value is lower in the EWMA
equation. Hence, AC requires more time to respond to weak link case.

Figure 5 shows the throughput for the ACR case. The throughput is lower for
some cases, for instance when the minimum and maximum thresholds were $-90$
and $-85$ dBm. An explanation might be that the high values in the thresholds
notify ACR about a weak link too late and if the random decision is mistaken
there is no time to change the decision before the node goes out of range.
When the alpha is $0.75$, ACR performance drops. The explanation in this case
is that alpha value increases a lot the weight of the last RSSI value making
ACR too aggressive which in combination with the random guesses increases the
mistaken decisions. In general even though ACR performs worse than the AC
version which is expected but if we compare the ACR with the version without
AC features there is a significant improvement.

In Figures 6 and 7 we see the downtime of AC and ACR. The bullet points here
stand for the mean value and the error bar for the standard deviation. The
mean downtime for AC was approximately $10\%$ and the standard deviation
approximately $20$. Apparently, the trends are the same with the throughput
since they are obtained from the same simulations. For the ACR case we see
higher percentages of downtime, the alpha that achieves the lowest downtime,
considering all the threshold values is $0.25$. If we compare the version
without AC features with the worst performance of ACR when thresholds were
$-90$ and $-85$ dBm, the downtime is reduced by $30\%$ approximately.

Figure 6: Downtime using AC for different alpha values and RSSI thresholds.
Figure 7: Downtime using ACR for different alpha values and RSSI thresholds.

The distance covered by node B is represented in Figures 8 and 9 for the case
of AC and ACR respectively. The bars represent the mean distance and the
errorbars the standard deviation. We decided to use this performance metric as
well since it can have on impact on the overall cost. The distance node B
covers without AC features is around $11$ km/h and when AC is activated in
Figure 8 the distance drops to $4$ km/h since it has to reduce its speed to be
in range with node A. This is an overhead that some applications might not be
able to afford. Respectively, some applications might be able to afford a
lower throughput in exchange with more covered distance. This is achieved by
ACR in Figure 9 where the mean covered distance is $7$ km/h when the threshold
values were $-95,-85$. This happened because node B spent more time
disconnected from the TSCH network of course as it is indicated from Figure 7.
The tradeoff described from the results is that the throughput is
disproportionate with the downtime and the covered distance.

Figure 8: Distance using AC for different alpha values and RSSI thresholds.
Figure 9: Distance using ACR for different alpha values and RSSI thresholds.

##  VI Conclusion

We introduced AC approach for TSCH networks including mobile nodes. AC can
regulate the speed of the nodes to increase the reliability by keeping the
nodes associated within the TSCH network by notifying them when they are going
out of coverage. Utilizing Cooja simulator and generating a simplistic
scenario we manage to quantify the performance of two versions of AC. One
where a node knows the position of the other nodes and therefore knows if it
should accelerate or decelerate its speed to within range and another one
where the node does not know and have to guess using a random function. The
results show that the reliability can be increased but there is an impact on
the distance and consequently on the application performance. We describe this
tradeoff and discuss that depending the application requirements it can be
crucial or not. In Future work we plan to evaluate AC with more complex
topologies and real experiments including smart car kits and real IoT nodes.
Moreover, the case of ACR would be investigated if it can be implemented with
Reinforcement Learning methods [30] to replace the random guessing.

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