# Formation of amorphous carbon multi-walled nanotubes from random initial
configurations

###### Abstract

Amorphous carbon nanotubes (a-CNT) with up to four walls and sizes ranging
from 200 to 3200 atoms have been simulated, starting from initial random
configurations and using the Gaussian Approximation Potential [Phys. Rev. B
95, 094203 (2017)]. The important variables (like density, height, and
diameter) required to successfully simulate a-CNTs, were predicted with a
machine learning random forest technique. The models were validated ex post
facto using density functional codes. The a-CNT models ranged from 0.55 nm - 2
nm wide with an average inter-wall spacing of 0.31 nm. The topological defects
in a-CNTs were discussed and new defect configurations were observed. The
electronic density of states and localization in these phases were discussed
and delocalized electrons in the $\pi$ subspace were identified as an
important factor for inter-layer cohesion. Spatial projection of the
electronic conductivity favors axial transport along connecting hexagons,
while non-hexagonal parts of the network either hinder or bifurcate the
electronic transport. A vibrational density of states was calculated and is
potentially an experimentally testable fingerprint of the material and the
appearance of a low-frequency radial breathing mode was discussed. The thermal
conductivity at 300 K was calculated using the Green-Kubo formula.

###### keywords:

amorphous solids, nanotubes, carbon

Chinonso Ugwumadu*, Rajendra Thapa, Yahya Al Majali, Jason Trembly, D. A.
Drabold

C. Ugwumadu, Prof. D. A. Drabold
Department of Physics and Astronomy,
Nanoscale and Quantum Phenomena Institute (NQPI),
Ohio University, Athens, Ohio 45701, USA
E-mail<EMAIL_ADDRESS>

Prof. Y. Al Majali, Prof. J. Trembly
Russ College of Engineering and Technology,
Ohio University, Athens, Ohio 45701, USA

Dr. R. Thapa
Institute for Functional Materials and Devices
Lehigh University,Bethlehem, PA 18015, USA

##  1 Introduction

Owing to their unique structure, mechanical strength, variable electrical
properties, and high thermal conductivity, carbon nanotubes (CNT) have been
the focus of intense research with a range of applications in nano-scale
engineering and electronics [1, 2, 3, 4, 5]. Methods of CNT production
involving the catalytic decomposition of hydrocarbons in the presence of
metal, as a by-product of arc-discharge or hydrocarbon flame production of
fullerenes, laser vaporization, plasma-enhanced and thermal chemical vapor
deposition, are well-established [6, 7, 8, 9, 10, 11]. The initial conception
of CNTs as being perfectly seamless cylindrical structures of graphene layers
without defects (except at the caps), has been shown to be oversimplified from
experimental evidence [12]. Zhou and coworkers performed
intercalation/deintercalation of potassium and rubidium on compressed CNTs and
showed that the samples formed “paper-mache” structures of small groups of
graphite sheets instead of crystalline CNTs (c-CNTs) [13]. Also, using
electron nano-diffraction techniques, Liu and Cowley showed that both helical
and non-helical CNTs exist, and the tube cross-section could be either
circular or polygonal [14]. In recent years, amorphous CNTs (a-CNT, CNTs with
non-hexagonal rings that may not be represented by the Russian doll model [15,
16]) have received increased attention due to their ease of growth [17, 18,
19] and their applications in areas like field-emission display devices,
gaseous adsorbents, energy storage, and catalyst support materials [20, 21,
22]. Evidently, the development of these a-CNT-based devices requires a good
understanding of a-CNT’s properties at the atomistic level. This reason,
coupled with the cost and technical difficulties associated with nano-scale
experimental analysis, emphasizes the need to employ simulations as a tool for
a preliminary and fundamental understanding of the atomistic structure and
properties of a-CNTs.

An underlying motive for this work is discovering synthetic forms of graphite
from available carbonaceous materials like coal. Carbon allotropes have a
remarkable proclivity to form layers, whether as graphene layers in graphite,
spherical shells in ”buckyballs” fullerenes, or cylindrical tubes in CNTs. In
earlier works, we predicted the formation of amorphous graphite, which is
layers of graphene with 5- and 7- member rings in the hexagonal network [23],
and multi-shell fullerenes with topological defects [24]. Theoretical studies
of c-CNT are available [25], but this is not the case for a-CNTs. A notable
exception is the work of Ghosh [26]. In this paper, we explore the atomistic
formation and properties of capped and hollow structures of Single-Wall a-CNT
(a-SWCNT) and Multi-Walled a-CNT (a-MWCNT) obtained from initially random
configurations of carbon atoms using the accurate density-functional-theory
(DFT) trained machine-learning Gaussian Approximation Potential (ML-GAP) [27]
as implemented in the “Large-scale Atomic/Molecular Massively Parallel
Simulator” (LAMMPS) software package [28]. The a-CNT models were validated
using ab-initio molecular dynamics (MD) packages including the plane-wave
basis set DFT code, VASP (Vienna Ab initio Simulation Package) [29] and the
atomic orbital-based package, SIESTA (Spanish Initiative for Electronic
Simulations with Thousands of Atoms) [30]. The electronic structure and
transport, as well as the origin of inter-wall cohesion in a-CNTs, are
discussed in detail. We explored vibrations in a-CNT by computing the density
of states and their corresponding inverse participation ratio, phase quotient,
and bond stretching character. Finally, we provide an estimate for the average
thermal conductivity in a-CNT. We have also included some animations from our
work and described them in sect.S1 in the supplementary material [31]

##  2 Computational Method

The ML-GAP C-C potential, which was trained on an extended set of reference
data that was obtained from simulations based on first principle methods [27,
32, 33, 34, 35], was used to simulate the amorphous carbon nanotubes. The
starting models were arranged as a cylindrical bulk of randomly positioned
carbon atoms at least 1.43 Å apart. The symmetry of CNTs required that the
models were formed by including the periodic boundary condition (PBC) in one
dimension, call it the $z$-axis, while the capped a-CNTS were formed by adding
a narrow space of 3.5 Å in the $z$-axis between the cylinders. Without PBC in
the $x$\- and $y$\- directions, the choice in the values of the input
variables (like the density, diameter, height …) required to successfully
simulate a-CNT becomes non-trivial. To obtain good a-CNT models as shown in
Fig. 1, we trained a Random Forest (RF) classifier [36, 37, 38] on a set of
forming/non-forming models to predict the important input variables (features)
as well as their acceptable value limits. RF is a parallel ensemble learning
technique that combines multiple decision tree algorithms [39] and involves a
voting process that exploits those algorithms to obtain a better predictive
performance. It uses random sampling of the training data set (with
replacement) in building decision trees and for splitting the nodes in the
network. We note here that the height (H) and diameter (d) of the models were
substituted by a unitless parameter (aspect ratio; AR = H/d). At the initial
stage, the parameters used in building the training dataset were sampled
uniformly within reasonable limits (AR: 1-12; system size: 200 - 1500 atoms
(in multiples of 40); vacuum in the $x$-$y$ plane: 1 - 10 Å; cutoff radius:
1.4 - 1.7), which ensured that the initial configuration was constrained to
form a cylindrical shape. The final configurations obtained after MD
simulations were labeled as ”formed” (label = 1) or ”not-formed” (label = 0)
a-CNT. In addition to human validation, we automated the labeling process by
using an ”in-house” Model Filtering and Scoring Algorithm (MoFaSa), which
assigns labels to models from a scoring metric. This metric is based on a
combination of clustering analysis (to form non-connecting walls) and shape
descriptors (like asphericity, acylindricity, and relative shape anisotropy)
on a surface reconstruction of the atomic coordinates of each cluster using
the alpha-shape method [40] on a constructed Delaunay tessellation [41, 42]
that estimate the shape of the model. An illustration of the decisions made by
MoFaSa on different configurations is presented in Fig. 2. With a total of 850
labeled models, the next step involved ranking the parameters on an importance
scale to reduce the number of variables in building an additional training
dataset. The feature importance is an implicit selection process implemented
in random forest implementation that can be obtained using the Gini impurity
criterion given as [43]:

$\mathcal{G}(t)=1-\sum_{i=1}^{m}\mathbf{P}(i|t)^{2}$ (1)

The summation is over all the classes (t) in a training set m. The Gini scores
for ranking the parameters (feature importance) for the K-fold cross-
validation (K =4) set are shown in Table 1 and plotted as a radar chart in
Fig. S1 in the supplementary material [31]. The prediction revealed that the
Aspect Ratio and system size are the most important features. We note that,
while the average energy is obtained at the end of the simulation, it serves
as an important feature during training since well-formed a-CNTs have lower
total energies compared to more defective structures. Also, as our ML model is
applied post-MD to find the feature importance, it is still reasonable to use
the energy as a feature. Next, we trained a decision tree classifier to obtain
a decision boundary that infers a range of suitable AR for the initial
configuration (see a representative tree in Fig S2 [31]. We generated 634 new
training models with a different parameter distribution based on the previous
results. The cutoff radius was fixed at 1.43 Å. The amount of vacuum in the
$x$-$y$ plane and the system size was uniformly sampled between 3 - 6 Å and
200 - 3200 atoms (in increments of 20) respectively, and AR was sampled in a
normal distribution around a tested value of 4.3 as the mean and as a standard
deviation of 1.2. An instance of the decision boundary is shown in Fig. 3. The
orange region in the plot indicates that the optimal AR range is between 0.32
- 0.51 nm. The black and grey scatter points are predictions made by our ML
model on a test set for a range of input variables that formed (value = 1) or
did not form (value = 0) a-CNT. It shows that the decision boundary from our
classifier makes a close to accurate prediction on unseen data. Before
implementing the decision tree algorithm, the features were scaled by
subtracting the mean of the distribution and dividing the result by the
standard deviation. The mean and standard deviation used for AR and system
size are shown in Table 1.

For all a-CNT models in this work, the MD simulation protocol involved
sampling the atomic positions and velocities on the canonical ensemble using a
Nosé–Hoover thermostat at a fixed temperature of 3000 K for 240 ps with a
timestep of 1 fs. Next, the models were allowed to find a more energetically
and structurally favorable configuration by cooling to room temperature at a
rate of 2.7$\times$1013 K/s. We stress that the cooling rate does not alter
bonds formed during/after heating. Finally, the structures were relaxed using
a conjugate gradient (CG) as implemented within LAMMPS with a force tolerance
of 10-6 eV/Å to obtain a representative structure. We also validated the
energies and structure of the final models by independent CG relaxation using
DFT-based Molecular dynamics (MD) packages, VASP and LAMMPS. The energy
difference between the ML-GAP models and DFT-validated models ranged between
0.03 eV/atom $\leq\delta E\leq$ 0.07 eV/atom and DFT relaxation never led to
bond breaking or forming, indicating that, in the configuration space, the
atoms are formed in an energetically stable and realistic local environment.
Model sizes ranged from 200 atoms to 3200 atoms, forming both a-SWCNT and
a-MWCNTs (up to 4 walls). These systems will henceforth be referred to as
a-CNTN, where N is the number of atoms. In the discussions that follows, we
selected 10 independent models with N = 200 (a-SWCNT), 400, 840, and 2000
(a-MWCNT) atoms (40 models in total).

##  3 Results and Discussion

###  3.1 Formation and nano-Structure

We have provided animations showing the time evolution in the formation of
capped and hollow a-CNT in the supplementary material [31]. The a-CNTs were
observed to form an outermost tube first, followed by inner tube(s). This
formation process is similar to multi-shell fullerenes [24]. The final models
have diameters ranging from 0.5 nm to 2 nm for a-CNT with up to 4 walls, and
the average gallery (space between any 2 consecutive layers) in the a-MWCNTs
models was calculated to be 0.31 nm, which is within the range that had been
reported in experiments [44] and close to the inter-layer spacing in graphite.
The structural order of the a-CNT models was analyzed from the radial
distribution function (RDF). In Fig. 5, the peaks obtained for the a-CNT
models were compared with those from crystalline SWCNT (c-CNT; n,m = 10,10)
[45] and amorphous Carbon (a-Carbon) [46]. The RDF peak for the first shell in
all a-CNTs was close to the C-C bond length observed in c-CNT and graphite.
Notably, the RDF of a-CNT200, which is a single-walled a-CNT, has similar
features resembling a-carbon and only the multi-walled a-CNT models reproduced
the third c-CNT peak around 2.84 Å. This is a consequence of the higher ratio
of hexagonal to non-hexagonal rings (6:n; n = 5 or 7) observed in a-CNTs which
have larger system sizes. The size of the topological defects, which are the
non-hexagonal arrangements of carbon atoms incorporated in the hexagonal
network was calculated using King’s criterion for ring statistics [47] (see
inset in Fig. 5) and it showed similar values for all the a-MWCNTs with a
slight decrease observed for a-SWCNTs.

Defects in nanostructures like CNT play a role in their morphology and
chemical properties. Experimentally and otherwise, an accurate and
quantitative description of the defects or a standard for distinguishing them
is still wanting. To this end, we studied the ring defects in the a-CNT
models, Fig. 4 (a-f) shows all the possible defects that exist in a-CNT. Fig 4
(a), (b) and (c) are the Stone-Wales (SW; 5665), 5675, and Stone-Thrower-Wales
(STW; 5775) defects respectively. We refer to these defects (a-c) as primary
defects because they can exist alone in the hexagonal C network. Also, Fig. 4
(d), (e), (f) are the inverse Stone-Thrower-Wales (ISTW; 7557 + 57) (e) 7558 +
57 (f) 7557 + 7558 defects. These are referred to as secondary defects since
they are the union of two distinct defects. The blue ring indicates the two
connecting vertex atoms (see Fig. 4 a-c) or the two shared basal atoms (Fig. 4
d-f) in pentagon-forming rings, from which we classify the defects. The 57,
STW and ISTW defects are well-known defects in carbon nanotubes [48, 49, 50,
51, 52, 53], but to our knowledge, there has been no report on 5675 and 7558
defects, hence this is the first time such defects will be reported for any
layered carbon network. We stress that the heptagon-pentagon and octagon-
pentagon still preserve the connectivity of the sp2 hybridized lattice and a
challenge for future research is in the modification of these defects in
MWCNTs to achieve specific morphologies with attractive electronic, optical,
and thermal properties.

###  3.2 Electronic structure, Transport and $\pi$-electron Delocalization

The electronic density of states (EDoS) for a-CNT was computed within VASP and
the extent of localization of Kohn-Sham states ($\phi$) was calculated as the
participation ratio (EIPR) using the following equation:

$I(\phi_{n})=\frac{\sum_{i}{|a_{n}^{i}|^{4}}}{(\sum_{i}{|a_{n}^{i}|^{2}})^{2}}$
(2)

where a${}_{n}^{i}$ is the contribution to the eigenvector ( $\phi_{n}$) from
the ith atomic orbital. High (low) values of EIPR indicate localized
(extended) states. The Fermi-level ($E_{f}$) in the EDoS and EIPR plots in
Fig. 6 was shifted to zero. The EDoS plots for a-CNT do not show any spectral
gap at $E_{f}$, and a few electronic states appear localized around the Fermi
level. The non-zero gap (NZG) observed for a-CNTs has been reported for
armchair SWCNTs, as well as SWCNTs with $n-m\neq 3\zeta$, where $\zeta$ is a
positive integer [54, 55]. In addition to the NZG similarity in armchair c-CNT
and a-CNT EDoS, we found that replicated periodic images of both structures in
the $z$-direction share a degree of structural similarity as well (see LEFT]
(RIGHT) inset in Fig. 6 for a-CNT (c-CNT)). The effect of disorder in the
a-CNTs network on the electronic conduction active path was analyzed by
projecting the electronic conductivity into a spatial grid. This involved the
implementation of the space-projected conductivity (SPC) formalism which
exploits the Kubo-Greenwood formula to obtain information about the conduction
pathways in materials [56, 57]. Fig. 7 is a representation of the conduction
path observed in a-CNT. We found that a-CNT favored only axially directed
conduction paths along connected hexagonal carbon rings. The axially directed
conduction path is consistent with published results for crystalline nanotubes
[58, 59, 60]. While the presence of non-hexagonal rings in a-CNT restricts
electron flow in the network, this effect is not as pronounced as it is in
amorphous graphite [23]. As shown in Fig. 7, the 7-member ring does not
completely terminate the conduction path but rather acts as a bifurcation of
the electron transport on the yellow-colored C atoms into the red-colored C
atom path. We stress that the average electronic conductivity of a-CNT
decreased by a factor of order 102 when compared to crystalline CNT [59, 61].

Inter-layer cohesion in layered carbon structures derives from a combination
of Van der Waals forces and delocalized electrons originating from occupied
states near the Fermi level ($E<E_{f}$). These states are the linear
combination of $\pi$-orbitals that are projected in the normal direction in
the locality of each sp2 carbon atom. For the a-CNT models, the contribution
of the electrons in the $\pi$ subspace was decomposed from the total
electronic charge density and projected as isosurfaces on sliced partitions.
Fig. 8 shows the $\pi$-electron distribution for ten bands closest to the
Fermi level. Fig. 8 (a) shows the $\pi$-electron distribution on a single
slice of a 400 atom a-CNT model. The atoms contributing to that particular
slice are shown in (b). The red atoms (R1) are above the green plane which is
the slice being considered. there are 2 shades of black in (b), the lighter
(B1) and darker (B2) shades of black correspond to atoms that are below and
intersect the green plane respectively. The reader can find a 3D visualization
of contributing atoms from Fig S5 [31] which is a CW rotation of Fig. 8 (b).
In Fig. 8 (c), the $\pi$-electron density is normalized to the maximum value
in the slice and mapped to a mesh grid with zero values (colored as black) to
increase the contrast in the $\pi$-electron density distribution. We have
labeled the atoms positions in Fig. 8 (a) that corresponds to Fig. 8 (b). In
Fig. 8 (a), o, x and x∗ and corresponds to R1, B1 and B2 atoms respectively.
x- and o- are atoms within the region that do not contribute to the $\pi$
orbital. xo indicate a combined contribution of two atoms (x and o) to the
electron distribution. The figure shows a slice was taken at 1.9 Å from the
origin and Fig. S6 shows the slicing pattern implemented on the a-CNT400
model. The choice of 1.9 Å is arbitrary and we have provided an animation
(PiBandDensity_mov.mp4) in the supplementary material [31] that shows
variations in the $\pi$-electron distribution for multiple slices across the
a-CNT400 model discussed here. Rozp$\l$och and co-workers [62, 63] suggested
the cross-interactions related to the $\pi$-electrons as a contributor to the
cohesion between layers in the metallic model of Schmidt [64] based on the
analysis of the g-factor [63]. The low concentration of quasi-free
$\pi$-electrons in the gallery causes a weak metallic bond between the tubes.

For a-CNT, we found that the maximum value for the $\pi$-electron distribution
is $\approx$ 10 % of the maximum total charge distribution in all the bands
and $\approx$ 2.75 % of the maximum total charge distribution around the
center of the gallery. While the low charge densities in the gallery may lead
one to argue that the interaction between tubes in CNT originates purely from
Van der Waals interaction, the simulation temperature ($\approx$ 3000 K),
obtained average electrical conductivity, and (to be discussed) thermal
conductivity of our models, coupled with the absence of corrections for Van
der Waals interaction in the ML-GAP (or conventional DFT) potential proves
otherwise. In the simulation of amorphous graphite [23], the inclusion of Van
der Waals corrections in the VASP implementation [65] did not change the
average energy of the large systems obtained using ML-GAP. This support the
argument that the ”metallic” cross-interaction of the delocalized
$\pi$-electrons is involved in the inter-layer cohesion. Additionally, the
values observed at the center of the gallery in a-CNTs were lower than those
from amorphous multi-shell fullerenes ( $\approx$ $4.3\%$) [24] but larger
than amorphous graphite ($\approx$ $2.0\%$) [23]. The mid-gallery values for
these amorphous carbon allotropes are considered to be subjected to the
influence of the local atomic curvature on the degree of the linear
combination (cross-interaction) of the Kohn-Sham orbitals describing the
$\pi$-bands. It is noteworthy that in the amorphous phase of these layered
carbon structures, layering persists despite the absence of an exact graphite
stacking registry.

###  3.3 Vibrations and Thermal conductivity

Carbon nanotube samples that are produced in the laboratory are not always
perfect crystals and may include non-hexagonal carbon rings. Additionally,
there could be impurities (like fullerenes, graphitic and amorphous carbon,
carbon nanoparticles, etc.) in the samples [48] and this can lead to confusion
in the interpretation of experimental vibrational spectra. We present in Fig
9, a vibrational spectrum that is potentially an experimentally testable
fingerprint of a-CNT. The atomic vibrational density of states was calculated
within the harmonic approximation model as:

$g(\omega)=\frac{1}{3N}\sum_{i=1}^{3N}\delta(\omega-\omega_{i})$ (3)

where, $N$ and $\omega_{i}$ represent the number of atoms and the eigen-
frequencies of normal modes, respectively. The $\delta$ function (approximated
by a Gaussian with a standard deviation equal to 1.5% of the maximum
frequency) ensures that high-density values were assigned to vibration
frequencies that lie close to the normal modes. The (green) scatter plot in
Fig. 9 is the vibration inverse participation ratio (VIPR) that indicates the
extent of localization of each normal mode frequency [24]. Low values of VIPR
indicate vibrational mode evenly distributed among the atoms while higher
values imply that few atoms contribute at that particular eigen-frequency.
Additionally, the region where VIPR $\gtrsim$ 0.15 is considered the
transition frequency from diffusons to locons [66], and is found to be at
around 1582 cm-1 in a-CNT. The lack of periodicity in the a-CNT lattice
restricts vibrations to non-propagating modes (e.g. diffusons and locons) [67,
68] and does not allow for a rigorous classification of the vibrational modes
as purely acoustic (at low-frequency spectrum) or purely optical (at high-
frequency spectrum) as in crystals. However, the phase quotient ($Q_{p}$) of
Bell and Hibbins-Butler [69] provides a measure of how vibrations of
neighboring atoms are in-phase (acoustic mode) and out-of-phase (optical
mode). The normalized $Q_{p}$ is given as [70]:

$Q_{p}~{}=\frac{1}{N_{b}}\frac{\sum_{m}\mathbf{u}^{i}_{n}\cdot\mathbf{u}^{j}_{n}}{\sum_{m}|\mathbf{u}^{i}_{n}\cdot\mathbf{u}^{j}_{n}|}$
(4)

where $N_{b}$ is the number of valance bonds, $\mathbf{u}^{i}_{n}$ and
$\mathbf{u}^{j}_{n}$ are the normalized displacement vectors for the $n^{th}$
normal mode. The index, i, sums over all the C atoms and $j$ enumerates
neighboring atoms of the $i^{th}$ atom. The in-phase vibration of the bulk
material gives $Q_{p}$ = 1 (purely acoustic). Conversely, a value of -1 would
correspond to motion in the opposite direction between neighboring atoms
(purely optical). It then follows that positive (negative) $Q_{p}$ is more
“acoustic-like” (“optical-like”). Another important tool for understanding
vibrations is the bond-bending or bond-stretching character ($S(\omega)$)
given as:

$S(\omega_{n})~{}=\frac{\sum_{m}|\mathbf{u}^{i}_{n}-\mathbf{u}^{j}_{n}|\times\hat{\mathbf{r}}_{ij}}{\sum_{m}|\mathbf{u}^{i}_{n}-\mathbf{u}^{j}_{n}|}$
(5)

where, $\mathbf{u}^{i}_{n}$ is as the same as in Eq. 4and
$\hat{\mathbf{r}}_{ij}$ is the unit vector parallel to the mth bond.
$S(\omega_{n})$ is close to unity when the mode of vibration is predominantly
of bond-stretching type and will be close to 0 otherwise. Fig. 10 show $Q_{p}$
[TOP] and $S(\omega_{n})$ [BOTTOM] for a double-walled a-CNT (DWCNT) with 400
atoms. The relationship becomes that optic-like modes at the high-frequency
end have bond-stretching character and acoustic-like modes at the low-
frequency end have bond-bending character. We have provided animations to
visualize the vibrations in the supplementary material. The mid-frequency
region around 800 - 820 cm-1 show some localization (see inset in Fig. 9) and
a local minimum at $\approx$ 807 cm-1, which is also $\approx$ 0 in the for
$Q_{p}$ plot. It is worth mentioning that the behavior of $Q_{p}$ around 0
cannot be distinguished into acoustic- or optical-like modes. The animation,
freq_807.mp4, shows that the modes in this region are quasi-localized
“resonant modes” [71] since they are distributed over a number of atoms and
also are not diffuson to locon transition frequencies. Other notable
vibrations in the double-wall a-CNT analyzed are at 32, 44, 55 cm-1 which
correspond to twisting vibrations of the outer shell, translation, and
twisting vibration of the inner shell respectively (see supplementary material
[31] for animations of these modes). We emphasize that the “twisting” modes of
a particular tube at a given frequency cost very little energy. The electron
gas in the ”gallery” between layers is quite diffuse and it is weakly
correlated with particular atoms in the layer, so the energy dependence of the
system, to the first approximation, is on the distance between the shells and
not affected by the angle. The radial breathing modes (RBMs) which dominate
the low-frequency Raman spectrum in experiments are unique to specific MWCNTs
and much focus has been given to the research of RBM because of its importance
in the applications of structure and property characterizations [72, 73, 74].
We observed an in-phase, low-frequency RBM at around 81 cm-1 (see animation in
supplementary material). RBM in MWCNTs has been observed experimentally from
Raman-active peaks around 100 cm-1 [73], and in-phase RBM at around 94 (85)
cm-1 have been predicted for armchair DWCNT (SWCNT) [75]. We note here that
the similarities observed between a-CNT and the armchair CNT could be a path
worth exploring.

Finally, we shift focus to the contribution of the topological defect to the
thermal conductivity (TC) in a-CNTs. Heat conduction in carbon materials is
usually dominated by phonons (even in graphite [76]), this allows us to
consider the contribution of the heat flux (J) for each atom [77, 78] in a
2000 atom a-MWCNT and then relate an ensemble average of the auto-correlation
of J to the TC ($\kappa$) using the Green-Kubo formala given as [79, 80]:

$\kappa~{}=\frac{1}{3Vk_{B}T^{2}}\int_{0}^{\tau}\left<\textbf{J}(0)\cdot\textbf{J}(t)\right>dt$
(6)

where V, T, and $k_{B}$ are the system volume, temperature, and Boltzmann’s
constant, respectively. The upper limit of the integral was approximated by
$\tau$ (= 2 ns) which is the correlation time required for the heat current
auto-correlation to decay to zero. The TC was obtained by averaging the
integral in Eq. 6 from 10 independent models. The Nosé–Hoover thermostat [81,
82] was used for thermalization and equilibration at T = 300 K at a fixed
volume using a 1 fs time-step, and at the beginning of the simulation, initial
velocities were assigned to the atoms randomly from a Gaussian distribution.
Our result showed that the average TC calculated for a-CNT was 22.15 Wm-1K-1
at 300 K. The thermal conductivity of MWCNT is $\approx$ 3000 Wm-1K-1 [83,
84], so the thermal conductivity of a-MWCNT is an order of 102 less than
c-MWCNT, which is similar to what we have reported for the electrical
conductivity.

##  4 Conclusion

In this work, we explored the formation of a-CNT starting from an initially
random configuration of carbon atoms using the ML-GAP potential. The starting
configuration required terminating the periodic boundary conditions in the
$x$-$y$ plane only. The complexity involved in the choice of the important
variables to simulate realist a-CNTs was minimized by learning the important
features from a random forest implementation. The structure of the a-CNT
models was validated by conjugate gradient relaxation as implemented within
SIESTA and VASP. The diameter of the a-CNTs ranged from 0.5 nm in a-SWCNTs to
2 nm in a-MWCNTs, and the inter-wall spacing was $\approx$ 0.31 nm. The ring
defects in a-CNTs were discussed and a potentially new type of defect was
observed. Electronic structure analysis showed that there was no band-gap at
the Fermi level, which is a property observed in all armchair CNTs. The
delocalized $\pi$-electrons in the gallery were confirmed to be involved in
the inter-layer cohesion in a-CNTs. The density of state and corresponding
participation ratio for the phonon vibrations were analyzed, and the result
showed that a-CNTs have localized states only at the high-frequency end of the
vibration spectrum which is consistent with other amorphous structures. The
phase quotient and stretching character analysis further suggested that those
localized sites were from atoms participating in non-hexagonal rings. The
average thermal conductivity for a-CNT was 22.15 Wcm-1K-1 at room temperature
and is an order of 102 less than crystalline MWCNT, which was also similar to
the average thermal conductivity. In conclusion, the strong proclivity to form
layered structures, even with topological defects, from chaotic configuration
remains one of the wonders of carbon.

Supporting Information

Supporting Information is available in the supplementary material [31] for
this manuscript.

Acknowledgements

The authors thank Anna-Theresa Kirchtag for proofreading the manuscript. The
U.S. Department of Energy for support under Grant No. DE-FE0031981 and XSEDE
(supported by National Science Foundation Grant No. ACI-1548562) for
computational support under allocation no. DMR-190008P

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Figure 1: Figure showing some a-CNT models with 3-fold coordination for each atom. The colors indicate the number of tubes in the a-MWCNT. Figure 2: Examples of models that do not form [LEFT] and models that form [RIGHT] a-CNT based on their input variables Figure 3: The decision boundary from a decision tree model using the aspect ratio and system size as features. 1 means an acceptable a-CNT formed, while 0 indicates otherwise. The feature values were scaled by subtracting the mean of the distribution and dividing the result by the standard deviation. The mean and standard deviation used for AR and system size are shown in Table 1. Figure 4: Figure showing primary ring defects in a-CNT. These defects are (a) Stone-Wales (5665) (b) 5675 and (c) Stone-Thrower-Wales (5775) defects, which are the primary defects. The secondary ring defects in a-CNT are the (d) inverse Stone-Thrower-Wales (7557) + 57 (e) 7558 + 57 (f) 7557 + 7558 defects. The blue ring indicates the connecting vertex atoms (a-c) or shared basal atoms (d-f) in pentagon-forming rings, from which we classify the defects. Table 1: Table showing the important Feature index predicted from a random forest classifier with K=4 cross-validation set. It also shows the accuracy rate on the test set for each fold and the mean and standard deviation calculated for scaling the aspect ratio and system size for the decision boundary construction. See discussion in the supplementary material [31] | Kfold-1 | Kfold-2 | Kfold-3 | Kfold-4 | mean | std
---|---|---|---|---|---|---
Aspect Ratio (height / diameter) | 0.32 | 0.34 | 0.32 | 0.33 | 4.3 | 1.25
System Size (No. of atoms) | 0.18 | 0.20 | 0.19 | 0.23 | 1672 | 883
Cutoff Radius [Å] | 0.13 | 0.12 | 0.12 | 0.08 |  |
Vacuum in the $x$-$y$ plane [Å] | 0.11 | 0.12 | 0.12 | 0.13 |  |
Energy [eV/atom] | 0.25 | 0.22 | 0.25 | 0.23 |  |
Accuracy [%] | 87 | 92 | 87 | 89 |  |
Figure 5: Radial distribution function g(r) for different a-CNT models
compared with crystalline CNT (c-CNT) and amorphous Carbon (a-Carbon). The
inset shows the ring distribution obtained using King’s criterion for ring
statistics. Figure 6: Electronic density of states for a-CNT. The [LEFT] and
[RIGHT] insets compare the structure of a-CNT and armchair CNT. The periodic
images of the models were replicated in the $z$-direction. Figure 7: Spatially
projected electronic conductivity in a-CNT. The presence of the 7-member ring
defect causes a bifurcation of the electronic conduction path on the yellow-
colored C atom into the red-colored atom. Figure 8: The figure in (a) shows
the $\pi$-electron distribution on a single slice of a 400 atom a-CNT model.
The atoms contributing to that particular slice are shown in (b). In (b), the
red atoms (R1) are above the green plane (the slice being considered). There
are 2 shades of black in (b), the light (B1) and dark (B2) shades correspond
to atoms that are below and intersect the green plane respectively. In the
plot in (c), the $\pi$-electron density is normalized to the maximum value in
the slice and mapped to a mesh grid with zero values (colored as black) to
increase the contrast in the $\pi$-electron density distribution. We have
labeled the atoms positions in (a) that corresponds to (b). in (a) x and o
corresponds to B1 and R1 atoms respectively. x∗ are B2 atoms. x- and o- are
atoms within the region that do not contribute to the $\pi$ orbital, while xo
indicate a combined contribution of two atoms (x and o). The figure shows a
slice was taken at 1.9 Å from the origin. An animation, PiBandDensity_mov.mp4
in the supplementary material [31] shows variations in the $\pi$-electron
distribution for multiple slices across the a-CNT400 model discussed here.
Figure 9: Figure showing the total VDoS and VIPR for a-CNT calculated from the
harmonic approximation. The inset shows the VIPR around 797 cm-1 825 cm-1
Figure 10: Figure showing the [TOP] phase quotient and [BOTTOM] stretching
character of a-CNT


# Dual-comb mode-locked Yb:CALGO laser based on cavity-shared configuration
with separated end mirrors

Ruixin Tang 1,2  Ziyu Luo 1,2  Pengfei Li 1,2  Pengrun Ying 1,2  Haiyang Xie
1,2  Siyuan Xu 1,2  Hui Liu 1,2,3  Jintao Bai 1,2,4  1State Key Laboratory of
Energy Photon-Technology in Western China, Northwest University, Xi’an 710127,
China
2Institute of Photonics and Photon-Technology, Northwest University, Xi’an
710127, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Dual-comb spectroscopy typically requires the utilization of two independent
and phase-locked femtosecond lasers, resulting in a complex and expensive
system that hinders its industrial applications. Single-cavity dual-comb
lasers are considered as one of the primary solution to simplify the system.
However, controlling the crucial parameter of difference in repetition rates
remains challenging. In this study, we present a dual-comb mode-locked
Yb:CALGO laser based on a cavity-shared configuration with separated end
mirrors. We employ two pairs of end mirrors and two thin-film polarizers
angled at 45 degrees to the cavity axis, leading to separating the cross-
polarized laser modes. We achieve simultaneous operation of two combs at
approximately 1040 nm with pulse durations of around 400 fs and an average
power exceeding 1 W. The repetition rates are approximately 59 MHz and their
difference can be easily tuned from zero up to the MHz range. By effectively
canceling out common mode noises, we observe minimal fluctuation in the
repetition rate difference with a standard deviation of about 1.9 Hz over ten
minutes, while experiencing fluctuations in repetition rates as large as 90
Hz. We demonstrate the capabilities of this system by utilizing the free-
running dual-comb setup for asynchronous optical sampling on a saturable
absorber and measuring etalon transmission spectrum. This system allows for
simple and independent control of the repetition rates and their difference
during operation, facilitating the selection of optimal repetition rate
difference and implementation of phase-locking loops. This advancement paves
the way for the development of simple yet high-performance dual-comb laser
sources.

##  1 Introduction

Dual-comb laser (DCL), comprising two coherent optical frequency combs with
slightly different repetition rates, has found extensive applications. Among
these applications, the most notable one is the dual-comb spectroscopy
(DCS)[1], encompassing ultrabroadband IR spectroscopy [2, 3, 4, 5], mid-
infrared near-field microscopy [6], spectrally resolved Lidar [7], coherent
Raman spectroscopy [8, 9, 10], THz spectroscopy [11, 12], and optomechanical
spectroscopy[13]. When employed in pump-probe experiments, the DCL will fully
unleash its potential for precise material characterization and measurement,
contributing to advancements in THz time-domain spectroscopy[14, 15],
ultrafast photoacoustics [16, 17], and ultrafast magnetization dynamics [18,
19] . The field of laser ranging has greatly benefited from the utilization of
DCL [20, 21, 22], which simultaneously offers high precision and extended
ambiguity range.

Conventional dual-comb lasers typically operate using two independent and
phase-locked optical frequency combs (OFCs) [23, 24]. However, the high cost,
complex design, and large scale of such systems impede their widespread
adoption for practical applications. Therefore, there is an urgent need for a
cost-effective, simplified, and compact dual-comb laser (DCL). One promising
approach to achieve this is through a single-cavity or quasi-single-cavity DCL
that sustains two OFCs with different repetition frequencies within a
stringent or non-stringent shared resonator [25, 26, 27, 28]. By canceling out
common mode noises between the OFCs in the shared cavity, desirable mutual
coherence can be achieved without requiring a phase-locking device. As a
result, single-cavity DCLs have garnered significant attention. Although the
concept of single-cavity DCLs emerged as early as in the 1990s to synchronize
pulses with different wavelengths in modelocked lasers [29, 30], it is only in
recent years that these devices have experienced rapid development with
emphasis on asynchronous operation and high mutual coherence characteristics.

In 2015, the U. Keller research team achieved the realization of a single-
cavity dual-comb laser in a modelocked integrated external-cavity surface
emitting laser (MIXSEL) by incorporating a birefringent crystal into the
cavity, thereby introducing the concept of dual-comb modelocked laser [31]. In
2017, the same team demonstrated the feasibility of DCS using a free-running
dual-comb MIXSEL, significantly reducing the complexity of DCS laser systems
[32]. The application of this concept was later extended to a mode-locked
Yb:CaF2 laser [33]. In 2022, they realized a dual-comb mode-locked laser using
spatially multiplexed all-solid-state lasers [34]. T. Ideguchi and his
colleague reported in 2016 on a Ti:sapphire bidirectional ring dual-comb mode-
locked laser utilizing a ring cavity structure and showcased its potential for
broadband DCS applications [35]. This bidirectional ring approach was
subsequently adopted by other groups for fiber ring lasers as well [36, 37,
38, 39, 40]. Currently, various approaches have been explored to achieve
single-cavity dual-comb lasers by exploiting the degrees of freedom in
polarization, spatial mode, wavelength, and circular direction. These
strategies have demonstrated successful applications both in solid-state and
fiber lasers[41].

Despite these advancements, achieving flexible control over the repetition
rate difference (RRD) remains a challenge for single-cavity DCLs. In the case
of all-solid-state single-cavity dual-comb mode-locked lasers, adjusting the
RRD typically necessitates realigning the cavities or modification of other
parameters rather than simply translating a mirror [26, 33]. Consequently,
this approach offers only a very limited range of RRD control. Moreover, due
to the lack of independent control over repetition rates, enhancing coherence
or mutual coherence in single-cavity DCLs beyond that achieved under free-
running conditions poses a significant challenge. Therefore, single-cavity
DCLs in free-running generally exhibit moderate levels of mutual coherence. A
current trend in the field is the establishment of a dual-comb laser system
featuring a shared cavity and enhanced control over RRD. This has been
recently demonstrated using thin-disk lasers[27, 28] and fiber lasers[42].

In this study, we present an all-solid-state dual-comb mode-locked laser based
on a cavity-shared configuration with separated end mirrors, enabling flexible
control of the RRD. This dual-comb laser utilizes a bulk Yb:CALGO crystal and
generates each comb with an average power exceeding 1 W and a pulse duration
of approximately 400 fs. The two combs share all intracavity elements but
employ separate end mirrors. This design facilitates extensive tuning of the
RRD during mode-locking operation by longitudinally adjusting the position of
the end mirrors using a three-dimensional (3D) translation stage, allowing for
optimal and flexible selection of this crucial parameter. With fundamental
repetition rates approaching 59 MHz, the RRD can be easily tuned from zero up
to the MHz range. The RRD exhibits minimal fluctuations, with a standard
deviation of only 1.9 Hz over a period of 10 minutes, while the fluctuation
magnitude in repetition rate for each comb is approximately 90 Hz. We
demonstrate the spectroscopic measurement capability of this dual-comb system
in free-running mode by measuring the transmission spectrum of an etalon.
Furthermore, under free-running conditions, we showcase its usability in pump-
probe experiments through asynchronous optical sampling (ASOPS) on a
semiconductor saturable absorber mirror (SESAM) which enables rapid resolution
of absorption dynamics on picosecond timescales. If higher precision
measurements are required, phase-locking can be independently applied to each
comb within this co-cavity system to enhance coherence or mutual coherence.

##  2 Experimental setup

The dual-comb mode-locked laser, as depicted in Fig. 1, is constructed using a
bulk Yb:CALGO crystal (CASTEC) and optically pumped by a 50 W fiber-coupled
diode laser operating at 981 nm. The crystal has been doped with 3 atomic
percent of ytterbium ions and is c-cut with dimensions of 3 mm × 3 mm × 3 mm.
The pump beam passes through a telescope consisting of two lenses (75 mm × 150
mm) before being focused into the gain medium. To achieve dual-comb operation,
a birefringent crystal with a thickness of 2 mm (cut at an angle of 45° with
respect to the optical axis) is inserted after the fiber end (see setup in
Fig. 1 inset), resulting in the splitting of the pump single beam into two
beams that walk off transversely from each other. As a result, there are two
spatially separated pump spots within the gain medium, with a separation
distance of 200 $\upmu$m. It should be noted that this spatial separation of
pumping plays a crucial role in enabling dual-comb operation; otherwise, only
one comb output would occur due to gain competition.

Figure 1: Schematic drawing of the dual-comb laser system. SESAM,
semiconductor saturable absorber mirror; TFP, thin-film polarizer; DCM,
dichroic mirror; HR, high reflection; M, mirror; OC, output coupler; GTI,
Gires–Tournois interferometer .

To achieve soliton mode-locking in the negative dispersion regime, dispersive
mirrors (GTI1, GTI2, M1 and M2) are introduced, yielding a net group delay
dispersion of $-$5100 fs2 per round trip. The concave mirrors M1–M3 have
curvature radii of 500 mm, 300 mm, and 500 mm respectively. The DCMs represent
dichroic mirrors with high transmission in the wavelength range of 900–1000 nm
and high reflection (HR) in the range of 1030-1050 nm. The flat output
couplers (OCs) exhibit a transmittance of 2.4$\%$. For self-starting and
maintaining mode-locking, we employed a commercial SESAM (BATOP) with a
modulation depth of 1.2$\%$ and saturation fluence Fsat = 60 $\upmu$J/cm2. To
adjust the distance between M3 and the SESAMs, we mount the SESAMs on
micrometer-driven translation stages.

The dual-comb operation is achieved by the arrangement of two nearly identical
laser resonators, which share almost all cavity mirrors except for the SESAM
and OC. In the cavity, two 3-mm thin-film polarizers (TFP) (Edmund optics) are
positioned at 45° angle to enable polarization selection of the laser modes.
The TFP reflects s-polarization and transmits p-polarization. It should be
noted that simultaneous oscillation of cross-polarized modes with balanced
power is only possible when the crystal is c-cut, not if it is a-cut. By
separating the SESAMs and OCs, independent tuning of repetition rates for each
comb becomes possible. With this configuration, RRD can be easily tuned from
zero up to the MHz range by adjusting the end mirrors’ position. This provides
a significant advantage over solid-state single-cavity DCLs that rely on
spatially multiplexed or birefringent or bidirectional cavities, where
tunability of RRD is quite limited and typically requires realignment of
cavities or modification of other parameters rather than simple mirror
translation.

##  3 Dual-comb characteristics

Figure 2:  The performance evaluation of the dual-comb. (a) Laser output power
versus total pump power, (b) Optical spectra of comb 1 (blue) and comb 2
(red), (c)-(d) Autocorrelation trace with sech2 fit, (e)-(f) Laser output
dected by a fast photodiode and measured by an electrical spectrum analyzer
where the insets indicate signal-to-noise ratio with a resolution bandwidth
(RBW) of 300 Hz.

The output laser properties were characterized with self-started and stable
mode-locking of both combs. Here, we designate the two combs as comb 1
(p-polarized output) and comb 2 (s-polarized output). Figure 2(a) displays the
individual comb output power versus the total incident pump power, with
maximum average output powers of 1.07 W and 1.14 W for the two combs,
respectively. The optical spectra were measured using an optical spectrum
analyzer (OSA, Yokogawa, AQ6370B), as shown in Fig. 2(b), revealing a full-
width at half-maximum (FWHM) bandwidth of 3.2 nm (3.4 nm) at wavelength of
1043 nm (1040 nm). The pulse autocorrelation traces were obtained by an
intensity autocorrelator (APE, pulseCheck) at the highest output power level,
yielding a pulse duration of 407 fs (349 fs), as depicted in Fig. 2(c)-2(d).
After capturing the pulse sequences with a high-speed InGaAs detector
(Thorlabs, DET08CL/M), the repetition rates and harmonics were measured using
an electrical signal analyzer (ESA, Keysight, N9000A). A signal to noise ratio
of approximately 77 dB (76 dB), as shown in Fig. 2(e)-2(f), indicates stable
fundamental modelocking at a repetition rate of 59 MHz for both combs. The
difference between the combs in these characteristics is mainly attributed to
the weak polarization anisotropy within the gain crystal.

To demonstrate the high mutual coherence between the two combs, we
simultaneously measured their repetition rates. The repetition rates were
accurately determined using two counters (Stanford Research System, SR620)
with a precision of millihertz. Figure 3(a) shows the temporal evolution of
the two repetition frequencies and their difference. The repetition rates of
both lasers exhibit drifts of approximately 90 Hz; however, the difference
between them shows only weak fluctuations with a standard deviation of 1.9 Hz
over a duration of 10 minutes. This observation confirms the remarkable
suppression of common noises within this dual-comb laser system. The
longitudinal positions of the SESAMs can be continuously adjusted with 3D
translation stages. In this study, the translation of the SESAMs allows tuning
the RRD between the two combs from 0 to 1.2 MHz, as shown in Fig. 3(b), with a
negligible loss on the laser output performance. Additionally, control over
RRD can also be achieved by manipulating the OCs.

Figure 3: (a)Temporal evolution of the repetition rates and their difference.
(b) Repetition rate difference versus the SESAM longitudinal position.

##  4 Measurement of etalon transmission spectrum with free-running dual-comb
laser

Figure 4: Interferogram measurment setup. HWP, halfwave plate; PBS, polarizing
beamsplitter; LPF, low-pass filter; OSA, optical spectrum analyzer.

We measured the transmission spectrum of an etalon using a free-running dual-
comb to investigate its potential for spectroscopic measurements. The RRD was
set to 80 Hz. To capture the dual-comb spectrum, we constructed an
interferogram measurement setup as illustrated in Fig. 4. The two laser beams
from the combs were combined using a polarizing beam splitter (PBS) with
orthogonal polarization and then passed through a half-wave plate and another
PBS. As a result, the dual-comb laser was split into two collinear branches
where both lasers had identical polarization directions and interfered with
each other. One branch served as a reference and was directed towards an OSA
(Yokogawa, AQ6370B). The other branch was detected by a fast photodetector
(Thorlabs, DET08CL/M) followed by an 18 MHz low-pass filter and a digital
phosphor oscilloscope (Tektronix, DPO7254C). During the measurement of the
etalon’s transmission spectrum, the etalon was positioned upstream of the
second PBS. In this experiment, we utilized an uncoated YAG etalon that had a
thickness of 250 $\upmu$m.

Figure 5: (a) Single-shot interferogram (inset: zoom-in of the interferogram
in the blue box portion). (b) Comparison of the spectrum with and without
etalon measured with dual-comb spectroscopy. (c) Comparison of the spectrum
with and without etalon measured with the OSA. (d) Comparison of the etalon
transmission spectrum measured with the dual-comb system and OSA.

To measure the dual-comb spectrum, we acquired a single-shot 50 $\upmu$s
interferogram (Fig. 5(a)) and performed Fourier transformation into the
frequency domain using the built-in Fast-Fourier-Transform (FFT) function of
the oscilloscope (Fig. 5(b)). Simultaneously, the spectrum was recorded by an
OSA (Fig. 5(c)). By comparing spectra obtained with or without an etalon, we
derived the transmission spectrum (Fig. 5(d)). Although there was a slight
difference between the spectra measured by our dual-comb system and those
measured by OSA due to the partial overlap of comb spectra, excellent
agreement in transmission spectra validates our dual-comb system’s suitability
for spectroscopic measurements.

##  5 Measurement of the recovery time of SESAM based on ASOPS with free-
running dual-comb

Given the direct output of asynchronous high-power femtosecond pulses with
high mutual coherence, we employed this laser system in free-running to assess
the recovery time of a SESAM using the ASOPS method. To measure the
reflectivity of the commercially available BATOP SESAM, characterized by a
modulation depth of 30$\%$, saturation fluence of 23 $\upmu$J/cm2, and a
recovery time constant of 0.7 ps, we established a non-collinear pump-probe
setup as depicted in Fig. 6.

Figure 6: Setup of the ASOPS-based SESAM measurement. HWP, halfwave plates;
PBS, polarizing beamsplitter; LPF, low-pass filter. Figure 7: ASOPS-based
SESAM measurement at different pump fluences.

In the ASOPS-based pump-probe setup, comb 2 served as the pump laser, while
comb 1 was power-reduced to function as the probe laser. The SESAM surface was
targeted by focusing the pump and probe beams using a 100-mm focal length lens
with 1/e2 diameters of 100 × 90 $\upmu$m2 and 120 × 90 $\upmu$m2, respectively
(measured by Dataray Beam Profiler). The probe power was attenuated to 5 mW,
whereas the pump power was sufficiently high to saturate the SESAM. A InGaAs
photodetector (Thorlabs, DET20C2) detected the reflected probe laser signal
which underwent low-pass filtering before being digitized and recorded on a
digital phosphor oscilloscope (Tektronix, DPO7254C). Simultaneously,
triggering of the oscilloscope occurred through an interferogram signal. For
this experiment, we set RRD at 60 Hz resulting in an estimated step size for
delay scanning of $\Delta$frep / frep2 =17 fs; this value is smaller than the
duration of the probe pulse. To convert to delay time scale, lab time scale is
multiplied by a factor $\Delta$frep/ frep = (9.8 ×105)-1 . This ASOPS system
enables rapid measurements that are unattainable with mechanically delayed
scanning systems. Figure 7 illustrates measured results depicting reflectivity
variation of SESAM under different pumping fluences. We extracted a fast
recovery time constant of approximately 0.8 ps at a pump fluence level of 122
$\upmu$J/cm2. These findings demonstrate that our free-running dual-comb
system is well-suited for applications in pump-probe studies and its scanning
speeds surpassing those achievable with any mechanical delay.

##  6 Conclusion

In summary, we have demonstrated a cavity-shared configuration with separated
end mirrors for an all-solid-state mode-locked Yb:CALGO laser, resulting in
the realization of a dual-comb oscillator with excellent controllability of
the repetition rate difference and mutual coherence. The Yb:CALGO laser
crystal is c-cut to ensure good isotropy for polarization. Two pairs of end
mirrors and two TFPs angled at 45 degrees to the cavity axis are employed to
separate the cross-polarized laser modes. Compared to the single cavity
designs that share all elements, our design allows for more controllable
repetition rate differences and easier cavity alignment.

We achieved simultaneous self-starting modelocking at a center wavelength of
1043 nm (1040 nm) with average powers up to 1.07 W (1.14 W) and pulse duration
around 400 fs for each comb. The repetition rates are approximately 59 MHz,
and the RRD can be tuned from zero to 1.2 MHz without requiring realignment of
the cavity or modification of other parameters. Common mode noises are
effectively canceled out, resulting in minimal fluctuation of the RRD with a
standard deviation of about 1.9 Hz over ten minutes, while repetition rates
exhibit fluctuations as large as 90 Hz.

Considering its output parameters, it exhibits promising potential for
efficient nonlinear frequency conversion into spectral ranges that are
pertinent to spectroscopy applications, such as the mid-infrared, terahertz,
or ultraviolet regions. The power and pulse duration of the presented laser
will be enhanced through improved mode matching between the laser modes and
the pump laser, utilization of a pump laser with a more suitable wavelength,
implementation of Kerr-lens mode-locking action, and incorporation of an
optimized thin-film polarizer. Furthermore, although this dual-comb system is
currently not actively locked for high-coherence operation, we anticipate
higher performance in DCS or other precision-based measurements using dual-
combs in future studies by applying feedback to the end mirrors.

Funding This work is supported by National Natural Science Foundation of China
(62005215), Natural Science Foundation of Shaanxi Province(2024JC-YBMS-507),
Natural Science Basic Research Program of Shaanxi Province (2019JCW-03),
Science and Technology Program of Xi’an (202005YK01), Scientific Research
Program of Shaanxi Education Department (23JK0690).

Disclosures The authors declare no conflicts of interest.

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