††thanks: Equal contributions††thanks: Equal contributions††thanks: Equal
contributions

# Emergence of new van Hove singularities in the charge density wave state of
a topological kagome metal RbV3Sb5

Soohyun Cho  Center for Excellence in Superconducting Electronics, State Key
Laboratory of Functional Materials for Informatics, Shanghai Institute of
Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai
200050, China  Haiyang Ma  School of Physical Science and Technology,
ShanghaiTech University, Shanghai 201210, China  Key Laboratory of Artificial
Structures and Quantum Control (Ministry of Education), Shenyang National
Laboratory for Materials Science, School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China  Wei Xia  School of Physical
Science and Technology, ShanghaiTech University, Shanghai 201210, China
ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University,
Shanghai 201210, China  Yichen Yang  Center for Excellence in Superconducting
Electronics, State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology, Chinese Academy
of Sciences, Shanghai 200050, China  Zhengtai Liu  Center for Excellence in
Superconducting Electronics, State Key Laboratory of Functional Materials for
Informatics, Shanghai Institute of Microsystem and Information Technology,
Chinese Academy of Sciences, Shanghai 200050, China  Zhe Huang  Center for
Excellence in Superconducting Electronics, State Key Laboratory of Functional
Materials for Informatics, Shanghai Institute of Microsystem and Information
Technology, Chinese Academy of Sciences, Shanghai 200050, China  Zhicheng
Jiang  Center for Excellence in Superconducting Electronics, State Key
Laboratory of Functional Materials for Informatics, Shanghai Institute of
Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai
200050, China  Xiangle Lu  Center for Excellence in Superconducting
Electronics, State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology, Chinese Academy
of Sciences, Shanghai 200050, China  Center of Materials Science and
Optoelectronics Engineering, University of Chinese Academy of Sciences,
Beijing 100049, China  Jishan Liu  Center for Excellence in Superconducting
Electronics, State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology, Chinese Academy
of Sciences, Shanghai 200050, China  Center of Materials Science and
Optoelectronics Engineering, University of Chinese Academy of Sciences,
Beijing 100049, China  Zhonghao Liu  Center for Excellence in Superconducting
Electronics, State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology, Chinese Academy
of Sciences, Shanghai 200050, China  Center of Materials Science and
Optoelectronics Engineering, University of Chinese Academy of Sciences,
Beijing 100049, China  Jinfeng Jia  Key Laboratory of Artificial Structures
and Quantum Control (Ministry of Education), Shenyang National Laboratory for
Materials Science, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  Yanfeng Guo<EMAIL_ADDRESS>School of Physical Science and Technology, ShanghaiTech University, Shanghai
201210, China  Jianpeng Liu<EMAIL_ADDRESS>School of Physical
Science and Technology, ShanghaiTech University, Shanghai 201210, China
ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University,
Shanghai 201210, China  Dawei Shen<EMAIL_ADDRESS>Center for
Excellence in Superconducting Electronics, State Key Laboratory of Functional
Materials for Informatics, Shanghai Institute of Microsystem and Information
Technology, Chinese Academy of Sciences, Shanghai 200050, China  Center of
Materials Science and Optoelectronics Engineering, University of Chinese
Academy of Sciences, Beijing 100049, China

###### Abstract

Quantum materials with layered kagome structures have drawn considerable
attention due to their unique lattice geometry, which gives rise to flat bands
co-existing with Dirac-like dispersions. The interplay between strong Coulomb
correlations and nontrivial band topology in these systems results in various
exotic phenomena. Recently, vanadium-based materials with layered kagome
structures are discovered to be topological metals, which exhibit charge
density wave (CDW) properties, significant anomalous Hall effect, and unusual
superconductivity at low temperatures. Here we exploit high-resolution angle-
resolved photoemission spectroscopy to investigate the electronic structure
evolution induced by the CDW transition in a vanadium-based kagome material
RbV3Sb5. A remarkable band renormalization in the CDW state is observed, which
is consistent with first principles calculations based on an inverse star-of-
David superstructure. The CDW phase transition gives rise to a partial energy
gap opening at the Fermi level, a shift in the band dispersion, and most
importantly, the emergence of new van Hove singularities associated with large
density of states, which are absent in the normal phase and may be related to
superconductivity observed at lower temperatures. Our work would shed light on
the microscopic mechanisms for the formation of the CDW and superconducting
states in these topological kagome metals.

A kagome lattice, which is composed of a two-dimensional network of corner-
sharing triangles, provides a unique playground for electrons’ correlations
and nontrivial band topology. For example, spin systems with antiferromagnetic
intersite coupling on a kagome lattice are proposed as promising candidates
for realizing quantum spin-liquid states Freedman et al. (2010); Han et al.
(2012a); Fu et al. (2015); Yan et al. (2011); Wulferding et al. (2010); Han et
al. (2012b). Besides, kagome metals usually host electronic structures with
co-existing dispersionless flat bands, Dirac cones, and van Hove singularity
(VHS) due to the unique crystal structure, which results in topologically
nontrivial states Kang et al. (2020); Ye et al. (2018); Liu et al. (2018,
2019); Yang et al. (2020); Zhou et al. (2021); Wu et al. (2021); Hu et al.
(2021); Liang et al. (2021a).

Depending on electron’s filling number, a variety of novel quantum states have
been theoretically proposed to exist in the kagome lattice systems, including
charge fractionalization O’Brien et al. (2010); Rüegg and Fiete (2011),
density wave orders Guo and Franz (2009); Isakov et al. (2006); Wang et al.
(2013); Ortiz et al. (2021a), topological Chern insulators Xu et al. (2015),
and superconductivity Wang et al. (2013); Ko et al. (2009) etc. In particular,
when the band filling of a kagome metal is near the VHS, calculations show
that several saddle points in the band dispersions emerge near the Fermi level
($E_{F}$) Wang et al. (2013); Barros et al. (2014); Zhou et al. (2021); Wu et
al. (2021). Coulomb scatterings of electrons between these saddle points would
give rise to novel Fermi surface instabilities, which may lead to spin or
charge density wave (CDW) states Kiesel and Thomale (2012); Kiesel et al.
(2013); Nayak (2000).

Recently, a new family of materials AV3Sb5 (A = K, Rb, Cs) with layered kagome
lattice structures have been discovered Ortiz et al. (2019, 2020, 2021b); Yang
et al. (2020); Yin et al. (2021); Jiang et al. (2020); Liang et al. (2021b);
Chen et al. (2021); Li et al. (2021); Liu et al. (2021); Zhou et al. (2021);
Nakayama et al. (2021); Hu et al. (2021); Wang et al. (2021); Liang et al.
(2021a). Previous first principles calculations based on density functional
theory (DFT) suggest that the energy bands near the Fermi surfaces in this
family of materials are topologically nontrivial characterized by odd
$\mathbb{Z}_{2}$ indices Ortiz et al. (2020, 2021b). Moreover, these vanadium-
based kagome topological metals undergo CDW phase transitions with the
critical temperatures $T_{\rm CDW}$ $\approx$ 78 - 103 K, below which a new
$2\times 2$ inverse star-of-David (ISD) superlattice structure becomes
energetically favored Jiang et al. (2020); Li et al. (2021); Ratcli et al.
(2021); Tan et al. (2021), leading to a reduction of density of states near
$E_{F}$ Jiang et al. (2020); Liang et al. (2021b); Chen et al. (2021); Tan et
al. (2021). Superconductivity emerges from the CDW states if the temperature
is further decreased, and the superconducting transition temperature
$T_{c}\sim 2.5\,$K Ortiz et al. (2021b). Surprisingly, remarkable anomalous
Hall effect (AHE) is observed in the CDW states of these systems Yang et al.
(2020); Yu et al. (2021), although so far no clear evidence of long-range
magnetic order has been reported. Despite some early theoretical attempts Feng
et al. (2021); Park et al. (2021); Wu et al. (2021); Lin and Nandkishore
(2021); Denner et al. (2021); Ratcli et al. (2021), the microscopic mechanisms
behind these intriguing phenomena, such as the driving force for the CDW
states, the pairing mechanism and symmetry of the superconducting states, and
the origin of AHE, are still open questions.

Figure 1: (a) The crystal structure of RbV3Sb5 from top view. (b) The
resistivity versus temperature curve with the anomaly near the 101 K,
signifying the charge density wave phase transition. (c) Three-dimensional
hexagonal Brillouin zone of the layered kagome lattice. (d) Zoom-in plot of
the band dispersion in the red box on (g) near the $M$ point. The ARPES
spectra are divided by the Fermi-Dirac distribution function to visualize the
band dispersion above $E_{F}$. The green dotted line is guide for the eye. (e)
Experimentally obtained Fermi surface map at 19 eV and above $T_{\rm CDW}$.
The back line indicates the hexagonal BZ of the kagome lattice. (f) The energy
constant map taken from the band calculations corresponding to the Fermi level
of ARPES data.(g) The ARPES spectra above $T_{\rm CDW}$ along the
$\Gamma$-$M$-$K$-$\Gamma$ direction and (h) the band calculations with the
spin-orbital coupling along the same direction as (g).

In this work, combining angle-resolved photoemission spectroscopy (ARPES) and
first principles calculations, we study the electronic structure and the
microscopic mechanism for the CDW state in RbV3Sb5, which exhibits the highest
CDW transition temperature among the AV3Sb5 (A = K, Rb, Cs) materials Ortiz et
al. (2019, 2021b); Yin et al. (2021). In particular, we have performed
systematic measurements for the temperature evolution of the electronic
structure in RbV3Sb5. For the first time, we have observed strong band
renormalization effects at low temperatures $T<T_{\rm CDW}$. As a result of
the CDW transition, we observe a partial energy gap opening at $E_{F}$ and a
shift in the band dispersion to higher binding energy. More interestingly, new
VHSs around $M$ points emerge in the CDW phase, which were completely absent
in the normal phase. The new VHSs contribute to large density of states near
$E_{F}$, which may be related to superconductivity observed at lower
temperatures.

In addition, by measuring the zero-frequency joint density of states at the
Fermi surface, we find significant charge fluctuations at wavevectors
connecting two of the $M$ points. We further calculate the phonon dispersions
in the normal phase, and find unstable modes at both $M$ and $L$ points in the
Brillouin zone. These results imply that both the charge fluctuations at the
Fermi surface and the phonon instabilities in the pristine structure are
important in driving the system into the CDW state.

RbV3Sb5 crystallizes in a layered hexagonal lattice structure with space group
$P6/mmm$ (No. 191). As shown in Fig. 1(a), the system consists of vanadium
atoms forming layers of kagome lattices, which are intercalated by Sb atoms
forming honeycomb lattice. The vanadium kagome planes are separated by layers
of alkali Rb ions forming triangular networks. The in-plane hexagonal lattice
constant $a$ is 5.472 ${\AA}$, and the out-of-plane lattice constant $c$ is
9.073 ${\AA}$. In Fig. 1(b) we show the temperature dependence of the in-plane
resistivity for RbV3Sb5, which exhibits an anomaly around 101 K. This anomaly
in resistivity is consistent with previous reports Ortiz et al. (2021b); Yin
et al. (2021), and is believed to be a hallmark of the CDW phase transition.

Figure 2: (a) The image of vanadium-based kagome lattice under the 2 $\times$
2 charge density wave, which is consistent with the ISD superlattice. The
green line indicates the unit cell of 2 $\times$ 2 superstructure of ISD. (b)
The hexagonal Brillouin zone (black line) and the folded Brillouin zone of 2
$\times$ 2 superstructure (red dotted line). (c) ARPES spectra of CDW phase
along the $\Gamma$-$K$-$M$ direction taken with 55 eV photon energy at 10 K,
below $T_{\rm CDW}$, and (d) the calculated unfolded spectral function for the
ISD CDW phase. (e) Schematic picture of the saddle point near the VHS3. The
high symmetry cut along the $M$ to $K$ direction obtained from the normal
phase ($>$ $T_{\rm CDW}$) (f) and the CDW phase ($<$ $T_{\rm CDW}$) (g). The
second derivative spectra of the left panel to enhanced the visibility of the
band near the $M$ point. (h) The cut data for the vertical direction of
$M$-$K$ direction from cut 1 to 6 as marked in the (g). (i) The ARPES spectra
and the calculated unfolded spectra of 2 $\times$ 2 $\times$ 1 ISD along the
$M$-$VHS3$-$R$ direction. The green and red dotted line is guide for the eye.

We first focus on the normal phase of RbV3Sb5 ($T>T_{\rm CDW}$). According to
the detailed photon energy dependent ARPES measurements, we found that the
inner potential $V_{0}\\!=\\!15.6\,$eV, based on which the value of $k_{z}$,
as well as high-symmetry planes and points in the Brillouin zone (BZ) [Fig.
1(c)] can be precisely determined. Figure 1(e) illustrates the photoemission
intensity map in the $k_{x}$-$k_{y}$ plane measured at 200 K with 19 eV
photons, which corresponds to Fermi surface of RbV3Sb5 on the $\Gamma-M-K$
plane. Besides, in Fig. 1(g) we show the photoemission intensity plot along
the high-symmetry path connecting $\Gamma$-$K$-$M$-$\Gamma$ points. Combining
both Fig. 1(e) and 1(g), it can be clearly seen that the system possesses an
electron pocket around $\Gamma$, a Dirac-cone-like dispersion around $K$ point
and some relatively flat dispersions with large intensity near $M$ point.

Figure 3: (Color online). Temperature-dependent ARPES measurements taken from
the 19 eV near the $M$ point. (a) The hole-like band at $M$ point is shifted
to higher binding energy as decreasing the temperature. The green arrows
indicate the VHS at the schematic figure. (b) Corresponding EDCs at the $M$
point for data in (a) to reveal the evolution of the peak shift. The red mark
indicates the emergence of the peak below the CDW transition temperature.

In Figs. 1(f) and 1(h) we demonstrate the calculated Fermi surface map on the
$\Gamma-M-K$ plane and calculated band structure along the
$\Gamma$-$K$-$M$-$\Gamma$ path, respectively, both of which are in good
agreement with experiments. Neglecting spin-orbit coupling (SOC), calculations
show that the the Dirac point at $K$ point is fourfold degenerate (including
spin degeneracy), which is protected by $C_{3z}$, inversion, and time-reversal
symmetries of RbV3Sb5 in its normal state. Similar Dirac points in the absence
of SOC have been observed in other transition-metal based kagome lattices Ye
et al. (2018); Liu et al. (2019). Including SOC would open a gap $\sim$20 meV
at the Dirac point, which is indeed observed in our ARPES measurement. The
calculations further indicate that the Dirac cone near the $K$ point is mostly
consisted of vanadium $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals, while the
electron pocket around $\Gamma$ is mainly contributed by the antimony $p_{z}$
orbitals. Moreover, both ARPES measurements and DFT calculations indicate that
the $E_{F}$ intersects with two VHSs near $M$ point [marked as VHS1 and VHS2
in Fig. 1(h)], which exhibit upward dispersion along $M-\Gamma$ path while
downward dispersion along $M-K$ path, as shown in Fig. 1(d). These lead to
large density of states around $M$ points near $E_{F}$, which may introduce
Fermi surface instabilities at wavevectors connecting two $M$ points.

We then turn to discuss the electronic structure in the CDW phase of RbV3Sb5.
When $T<T_{\rm CDW}$, the system forms a $2\times 2$ ISD type superstructure
as shown in Fig. 2(a) Jiang et al. (2020); Tan et al. (2021); Ratcli et al.
(2021), and our first principles calculations indicate that such a structure
is energetically favored. It is worthy to note that recent studies suggest
there are additional cell doubling or even quadrupling along the $c$ axis
Liang et al. (2021b); Uykur et al. (2021); Park et al. (2021). However, since
the states around the VHSs near $M$ points and the Dirac point are of in-plane
orbital characters [Fig. 1(f)], the cell doubling/quadrupling along the $c$
axis does not significantly change the electronic structure around the $M$
point and the Dirac point, thus in this work we only consider the in-plane 2
$\times$ 2 ISD structure in the calculations. The BZ of the ISD supercell is
one quarter of the pristine BZ [Fig. 2(b)], therefore the energy bands from
the pristine lattice would be folded from the original BZ to the ISD supercell
BZ, which may open up gaps at the supercell BZ boundary. However,
experimentally we hardly discover any spectral weight of the folded bands as
shown in Fig. 2(c), suggesting a mild charge modulation in the CDW phase of
RbV3Sb5 with ISD structure.

Although the band-folding effect is weak, we indeed find that the CDW
transition would result in an orbital-selective variation of the band
structure compared to that of the normal phase. First, as shown in Fig. 2(c),
the Fermi wavevector $k_{F}$ of the electron pocket around $\Gamma$ increases
from 0.257 ${\AA}^{-1}$ in the normal state to 0.262 ${\AA}^{-1}$ with a
dramatic sinking of this band due to the CDW transition. As the electron
pocket around $\Gamma$ is mostly contributed by the out-of-plane $p_{z}$
orbitals, a strong energy shift at $\Gamma$ implies a possible cell doubling
or quadrupling along the $c$ axis. In sharp contrast, the Dirac band around
the $K$ point, mainly of in-plane orbital character, remains nearly intact
upon the phase transition. Such behavior is reminiscent of that in bulk
1$T$-TiSe2, which is well known as a 2 $\times$ 2 $\times$ 2 CDW material
Wegner et al. (2020). This finding implies the possible charge modulation
along the _c_ -axis in RbV3Sb5, as also suggested by some previous reports
Liang et al. (2021b); Uykur et al. (2021).

Besides, the CDW phase transition gives rise to significant change of spectra
around $M$ point, as shown by the direct comparison of the photoemission
intensity plots measured in the normal and CDW states [Figs. 2(f-g)].
Evidently, nearly the whole VHS1 in the normal state is pushed downwards in
binding energy, while only a small region of the higher VHS2 close to the $M$
point is gapped, resulting in the ‘M’-shaped band dispersion along $K$-$M$-$K$
path in Fig. 2(g). Here second derivative spectra are provided as well in
order to enhance the visibility of the CDW-induced spectral weight shift
around the $M$ point. Comparing the energy positions of the VHS1 saddle point
in the normal phase and in the CDW phase [indicated by the red arrows in Figs.
2(f-g)], we estimate an energy shift of about 200 meV induced by the CDW
transition. More interestingly, a new VHS (marked as VHS3 thereafter) emerges
in the CDW phase, and its saddle point is located somewhere close to $M$, as
illustrated in Fig. 2(g). Our experimental finding is fully supported by first
principles calculations based on a $2\times 2\times 1$ ISD structure, as shown
by the calculated unfolded spectral function in Fig. 2(d) and in the right
panel of Fig. 2(i). Taking into account of the large density of states
introduced by this emergent VHS3, it should be intimately relevant to the
superconductivity in RbV3Sb5 observed at lower temperatures.

Figs. 3(b-g) show the detailed temperature dependent evolution of VHSs around
$M$ across $T_{\rm CDW}$. When the temperature is above $T_{\rm CDW}$, as
schematically illustrated in Fig. 3(a), both VHS1 and VHS2 around the $M$
point are in the vicinity of $E_{F}$, and they almost remain intact with
decreasing temperature as shown in Figs. 3(b)-(d). However, once the
temperature is below $T_{\rm CDW}$, states around $M$ point are gradually
shifted to higher binding energy, eventually leading to a large downshift
$\sim\\!$ 200 meV for states at $M$, and a ‘M’-shaped band dispersion appears
at 10 K [Fig. 3(h)]. Such temperature dependence of the gapped spectral weight
around $M$ has unambiguously illustrated its CDW origin. Moreover, we also
present the photoemission energy distribution curves (EDCs) taken around $M$
in Fig. 3(i). We found that one more shoulder-like peak (marked by red
triangles) in the EDC suddenly emerges below $T_{\rm CDW}$, and is
continuously pushed to higher binding energy with the decrease of temperature,
which demonstrates a complete picture of the CDW gap development in RbV3Sb5.

In order to better understand the microscopic mechanism for the formation of
the CDW state, we calculate the zero-frequency joint density of states (DOS)
contributed by the Fermi surface:

$C(\mathbf{q})=\frac{\Omega{{}_{0}}}{(2\pi)^{3}}\int
d\mathbf{k}\,A(\mathbf{k},E_{F})\,A(\mathbf{k}+\mathbf{q},E_{F})$ (1)

, where $A(\mathbf{k},E_{F})$ is the spectral function at $E_{F}$ at
$\mathbf{k}$ point in the BZ, and $\Omega_{0}$ is the volume the primitive
cell. $C(\mathbf{q})$ describes the phase space for the scatterings of
electrons from the states at $\mathbf{k}$ to those at $\mathbf{k}+\mathbf{q}$
at the Fermi surface, which is a characterization of the Fermi-surface charge
fluctuations with wavevector $\mathbf{q}$. Therefore, it is expected that
$C(\mathbf{q})$ would show a peak at the corresponding ordering wavevector if
there exist any charge instability induced by charge fluctuations at the Fermi
surface. This autocorrelation of ARPES spectra in the normal state has been
demonstrated to give a reasonable count for charge-ordering instabilities of
various compounds Shen et al. (2007); Chatterjee et al. (2006); McElroy et al.
(2006); Shen et al. (2008). The $C(\mathbf{q})$ map extracted from APRES
spectra is shown in Fig. 4(a), which exhibit several peaks at wavevectors
$\mathbf{q}_{i}$ (i=1, 2 and 3). This is consistent with the calculated zero-
frequency joint DOS as shown in Fig. 4(b). We find that $\\{\mathbf{q}_{i}\\}$
are exactly the “nesting” wavevectors connecting the different saddle points
(around $M$) at the Fermi surface in the BZ, as schematically shown in Fig.
4(c). This indicates that the strong charge fluctuations due to the Fermi-
surface nesting is an important driving force for the formation of CDW state
in RbV3Sb5.

Moreover, we have also calculated the phonon band structure along the high-
symmetry path as shown in Fig. 4(d), which clearly shows two unstable modes
around $M$ and $L$ points respectively. The soft phonon mode at $M$ point and
the strong charge fluctuations at $\mathbf{q}_{1}$ would cooperate with each
other, both of which tend to drive the system into a CDW state that doubles
the primitive cell along the $\mathbf{q}_{1}$ direction in the kagome plane.
However, since there are three equivalent $\mathbf{q}_{1}$ wavevectors (three
equivalent $M$ points) in the BZ, these three $\mathbf{q}_{1}$ vectors may be
linked with the three sublattices in the kagome lattice, forming a ”triple-Q”
CDW state preserving $C_{3z}$ symmetry Feng et al. (2021), which is exactly
the $2\times 2$ ISD structure. The additional soft phonon mode at $L$ point
may be the origin of the cell doubling along the $c$ axis in the CDW phase.

Figure 4: (Color online). (a) Two dimensional joint DOS results on the saddle
band point regions. (b) Calculated joint DOS at the Fermi energy. (c)
Calculated Fermi surface projected to the $k_{x}$-$k_{y}$ plane. (d)
Calculated phonon spectrum along a high-symmetry path. The high symmetry
points are marked in Fig. 1(c).

To summarize, we have studied the electronic structures of a kagome
topological metal RbV3Sb5 both in the normal phase and in the CDW phase. Our
APRES measurements indicate strong band renormalization effects due to the CDW
phase transition. These include partial gap opening at $E_{F}$, orbital-
selective energy shift of the dispersion, and most importantly, the emergence
of new VHSs near $E_{F}$ induced by the CDW transition. The band
renormalizations around the $K$ and $M$ points, as well as the emergence of
new VHSs, can be fully captured by first principles calculations based on a
$2\times 2$ ISD supercell. The new VHSs in the CDW phase contribute to large
DOS near $E_{F}$, which may be relevant with the emergence of
superconductivity at lower temperatures. We further investigate the
microscopic mechanisms for the formation of the CDW state, and find that the
system exhibits strong charge fluctuations at wavevectors connecting two
adjacent saddle points at the Fermi surface. The Fermi-surface charge
fluctuations would cooperate with a soft mode at the same wavevector,
eventually driving the system into the CDW state observed in experiments. Our
work is a significant step forward in understanding the mysterious CDW phase
in the V-based topological kagome metal systems, and will provide useful
guidelines for future experimental and theoretical studies.

## Methods

###  .1 Single crystal synthesis and characterization.

The RbV3Sb5 crystals were grown by using a mixture of RbSb2 and RbSb precusors
as the flux. The pre-reacted RbSb2, RbSb and VSb2 were mixed in a molar ratio
of 2 : 2 : 1, placed into an alumina crucible, and then sealed into a quartz
tube. The assembly was heated up to 1000∘C in 10 h in a weel-type furnace,
maintained at the temperature for more than 30 h, and then slowly cooled down
to 450∘C at a temperature decreasing rate of 1.5∘C/h. Then the furnace was
shut down and naturally cooled to room temperature. Crystals with black shiny
metallic luster were obtained in the aluminium crucible. The crystallographic
phase and quality of the crystals were examined on a Bruker D8 VENTURE powder
diffractometer using Cu K$\alpha$ radioactive source ($\lambda$ = 1.5418
$\AA$) at room temperature. During measurement, the crystals were aligned
along the (001) plane. The resistivity of the crystals was measured by using a
standard four-wire method in a commercial DynaCool physical properties
measurement system from Quantum Design.

###  .2 Angle-resolved photoemission spectroscopy.

ARPES measurements were performed at both 03U beamline of Shanghai Synchrotron
Radiation Facility (SSRF) Yang et al. (2021) and 13U beamline of National
Synchrotron Radiation Laboratory (NSRL). These two endstations are both
equipped with Scienta-Omicron DA30 electron analyzers. The angular and the
energy resolutions were set to 0.2∘ and 8 $\sim$ 20 meV (dependent on the
selected probing photon energy). All samples were cleaved in an ultrahigh
vacuum better than 8.0 $\times$ 10-11 Torr.

###  .3 First principles calculations of the bulk electronic structure.

Density functional theory (DFT) calculations are performed with the Vienna ab
initio simulation package (VASP) which adopts the projector-augmented wave
method Kresse and Furthmüller (1996). The plane-wave energy cutoff is set at
400 eV. The exchange-correlation functional of the Perdew-Burke-Ernzerhof
(PBE) typePerdew et al. (1996); Blöchl (1994); Kresse and Joubert (1999) is
used for both structural relaxations and self-consistent electronic
calculations. The convergence criteria for the electronic iteration and
geometry optimization are set to 10-6 eV and 0.001 eV/Å, respectively. The BZ
is sampled by a 9$\times$9$\times$7 k-mesh for the normal phase and a
6$\times$6$\times$4 k-mesh for the CDW phase within the Gamma-centered scheme.
Phonon spectrum is calculated with PHONOPY code Togo and Tanaka (2015), using
density-functional perturbation theory (DFPT) method. Joint DOS is calculated
based on Wannier orbitals which are obtained through VASP2WANNIER90 interface
Mostofi et al. (2008), 60 orbitals are included with SOC being considered.

## Acknowledgements

This work was supported by the National Key R$\&$D Program of the MOST of
China (Grant No. 2016YFA0300204), the National Science Foundation of China
(Grant Nos. U2032208, 92065201, 11874264), and the Natural Science Foundation
of Shanghai (Grant No. 14ZR1447600). Y. F. Guo acknowledges the starting grant
of ShanghaiTech University and the Program for Professor of Special
Appointment (Shanghai Eastern Scholar). Part of this research used Beamline
03U of the Shanghai Synchrotron Radiation Facility, which is supported by ME2
project under contract No. 11227902 from National Natural Science Foundation
of China. J. P. Liu acknowledges the start-up grant of ShanghaiTech
University. The authors also thank the support from Analytical Instrumentation
Center (#SPST-AIC10112914), SPST, ShanghaiTech University.

## References

  * Freedman et al. (2010) D. E. Freedman, T. H. Han, A. Prodi, P. Muller, Q.-Z. Huang, Y.-S. Chen, S. M. Webb, Y. S. Lee, T. M. McQueen, and D. G. Nocera, J. Am. Chem. Soc. 132, 16185 (2010).
  * Han et al. (2012a) T. Han, S. Chu, and Y. S. Lee, Phys. Rev. Lett. 108, 157202 (2012a).
  * Fu et al. (2015) M.-X. Fu, T. Imai, T.-H. Han, and Y. S. Lee, Science 350, 655 (2015).
  * Yan et al. (2011) S. Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011).
  * Wulferding et al. (2010) D. Wulferding, P. Lemmens, P. Scheib, J. Röder, P. Mendels, S.-Y. Chu, T.-H. Han, and Y. S. Lee, Phys. Rev. B 82, 144412 (2010).
  * Han et al. (2012b) T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Nature 492, 406 (2012b).
  * Kang et al. (2020) M. Kang, L. Ye, S. Fang, J. S. You, A. Levitan, M. Han, J. I. Facio, C. Jozwiak, A. Bostwick, E. Rotenberg, et al., Nat. Mater. 19, 163 (2020).
  * Ye et al. (2018) L. Ye, M. Kang, J.-W. Liu, F. Von Cube, C. R. Wicker, T. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C. Bell, et al., Nature 555, 638 (2018).
  * Liu et al. (2018) E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, et al., Nature Physics 14, 1125 (2018).
  * Liu et al. (2019) D.-F. Liu, A.-J. Liang, E.-K. Liu, Q.-N. Xu, Y.-W. Li, C. Chen, D. Pei, W.-J. Shi, S. K. Mo, P. Dudin, et al., Science 365, 1282 (2019).
  * Yang et al. (2020) S.-Y. Yang, Y. Wang, B. R. Ortiz, D. Liu, J. Gayles, E. Derunova, R. Gonzalez-Hernandez, L. Šmejkal, Y. Chen, S. S. P. Parkin, et al., Sci. Adv. 6, eabb6003 (2020).
  * Zhou et al. (2021) X. Zhou, Y. Li, X. Fan, J. Hao, Y. Dai, Z. Wang, Y. Yao, and H.-H. Wen, arXiv preprint arXiv:2104.01015 (2021).
  * Wu et al. (2021) X. Wu, T. Schwemmer, T. Müller, A. Consiglio, G. Sangiovanni, D. D. Sante, Y. Iqbal, W. Hanke, A. P. Schnyder, M. M. Denner, et al., arXiv preprint arXiv:2104.05671 (2021).
  * Hu et al. (2021) Y. Hu, S. M. L. Teicher, B. R. Ortiz, Y. Luo, S. Peng, L. Huai, J. Z. Ma, N. C. Plumb, S. D. Wilson, J.-F. He, et al., arXiv preprint arXiv:2104.12725 (2021).
  * Liang et al. (2021a) Z. Liang, X. Hou, W. Ma, F. Zhang, P. Wu, Z. Zhang, F. Yu, J. J. Ying, K. Jiang, L. Shan, et al., arXiv preprint arXiv:2103.04760 (2021a).
  * O’Brien et al. (2010) A. O’Brien, F. Pollmann, and P. Fulde, Phys. Rev. B 81, 235115 (2010).
  * Rüegg and Fiete (2011) A. Rüegg and G. A. Fiete, Phys. Rev. B 83, 165118 (2011).
  * Guo and Franz (2009) H.-M. Guo and M. Franz, Phys. Rev. B 80, 113102 (2009).
  * Isakov et al. (2006) S. Isakov, S. Wessel, R. Melko, K. Sengupta, and Y. B. Kim, Phys. Rev. Lett. 97, 147202 (2006).
  * Wang et al. (2013) W.-S. Wang, Z.-Z. Li, Y.-Y. Xiang, and Q.-H. Wang, Phys. Rev. B 87, 115135 (2013).
  * Ortiz et al. (2021a) B. R. Ortiz, S. M. L. Teicher, L. Kautzsch, P. M. Sarte, J. P. C. Ruff, R. Seshadri, and S. D. Wilson, arXiv preprint arXiv:2104.07230 (2021a).
  * Xu et al. (2015) G. Xu, B. Lian, and S.-C. Zhang, Phys. Rev. Lett. 115, 186802 (2015).
  * Ko et al. (2009) W.-H. Ko, P. A. Lee, and X.-G. Wen, Phys. Rev. B 79, 214502 (2009).
  * Barros et al. (2014) K. Barros, J. W. Venderbos, G.-W. Chern, and C. Batista, Phys. Rev. B 90, 245119 (2014).
  * Kiesel and Thomale (2012) M. L. Kiesel and R. Thomale, Phys. Rev. B 86, 121105 (2012).
  * Kiesel et al. (2013) M. L. Kiesel, C. Platt, and R. Thomale, Phys. Rev. Lett. 110, 126405 (2013).
  * Nayak (2000) C. Nayak, Phys. Rev. B 62, 4880 (2000).
  * Ortiz et al. (2019) B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bordelon, J. S. Mangum, I. W. Oswald, J. A. Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, et al., Phys. Rev. Mater. 3, 094407 (2019).
  * Ortiz et al. (2020) B. R. Ortiz, S. M. Teicher, Y. Hu, J. L. Zuo, P. M. Sarte, E. C. Schueller, A. M. Abeykoon, M. J. Krogstad, S. Rosenkranz, R. Osborn, et al., Phys. Rev. Lett. 125, 247002 (2020).
  * Ortiz et al. (2021b) B. R. Ortiz, P. M. Sarte, E. M. Kenney, M. J. Graf, S. M. Teicher, R. Seshadri, and S. D. Wilson, Phys. Rev. Mater. 5, 034801 (2021b).
  * Yin et al. (2021) Q. Yin, Z. Tu, C. Gong, Y. Fu, S. Yan, and H. Lei, Chinese Phys. Lett. 38, 037403 (2021).
  * Jiang et al. (2020) Y.-X. Jiang, J.-X. Yin, M. M. Denner, N. Shumiya, B. R. Ortiz, J. He, X.-X. Liu, S.-T. S. Zhang, G.-Q. Chang, I. Belopolski, et al., arXiv preprint arXiv:2012.15709 (2020).
  * Liang et al. (2021b) Z.-W. Liang, X.-Y. Hou, W.-R. Ma, F. Zhang, P. Wu, Z.-Y. Zhang, F.-H. Yu, J.-J. Ying, K. Jiang, L. Shan, et al., arXiv preprint arXiv:2103.04760 (2021b).
  * Chen et al. (2021) H. Chen, H.-T. Yang, B. Hu, Z. Zhao, J. Yuan, Y.-Q. Xing, G.-J. Qian, Z.-H. Huang, G. Li, Y.-H. Ye, et al., arXiv preprint arXiv:2103.09188 (2021).
  * Li et al. (2021) H.-X. Li, T.-T. Zhang, Y.-Y. Pai, C. Marvinney, A. Said, T. Yilmaz, Q. Yin, C. Gong, Z. Tu, E. Vescovo, et al., arXiv preprint arXiv:2103.09769 (2021).
  * Liu et al. (2021) Z.-H. Liu, N.-N. Zhao, Q.-W. Yin, C.-S. Gong, Z.-J. Tu, M. Li, W.-H. Song, Z.-T. Liu, D.-W. Shen, Y.-B. Huang, et al., arXiv preprint arXiv:2104.01125 (2021).
  * Nakayama et al. (2021) K. Nakayama, Y. Li, M. Liu, Z. Wang, T. Takahashi, Y. Yao, and T. Sato, arXiv preprint arXiv:2104.08042 (2021).
  * Wang et al. (2021) Z. Wang, S. Ma, Y. Zhang, H. Yang, Z. Zhao, Y. Ou, Y. Zhu, S. Ni, Z. Lu, H. Chen, et al., arXiv preprint arXiv:2104.05556 (2021).
  * Ratcli et al. (2021) N. Ratcli, L. Hallett, B. R. Ortiz, S. D. Wilson, and J. W. Harter, arXiv preprint arXiv:2104.10138 (2021).
  * Tan et al. (2021) H.-X. Tan, Y.-Z. Liu, Z.-Q. Wang, and B.-H. Yan, arXiv preprint arXiv:2103.06325 (2021).
  * Yu et al. (2021) F.-H. Yu, T. Wu, Z.-Y. Wang, B. Lei, W.-Z. Zhuo, J.-J. Ying, and X.-H. Chen, arXiv preprint arXiv:2102.10987 (2021).
  * Feng et al. (2021) X.-L. Feng, K. Jiang, Z.-Q. Wang, and J.-P. Hu, arXiv preprint arXiv:2103.07097 (2021).
  * Park et al. (2021) T. Park, M.-X. Ye, and L. Balents, arXiv preprint arXiv:2104.08425 (2021).
  * Lin and Nandkishore (2021) Y.-P. Lin and R. M. Nandkishore, arXiv preprint arXiv:2104.02725 (2021).
  * Denner et al. (2021) M. M. Denner, R. Thomale, and T. Neupert, arXiv preprint arXiv:2103.14045 (2021).
  * Uykur et al. (2021) E. Uykur, B. Ortiz, S. Wilson, M. Dressel, and A. Tsirlin, arXiv preprint arXiv:2103.07912 (2021).
  * Wegner et al. (2020) A. Wegner, J. Zhao, J. Li, J. Yang, A. A. Anikin, G. Karapetrov, K. Esfarjani, D. Louca, and U. Chatterjee, Phys. Rev. B 101, 195145 (2020).
  * Shen et al. (2007) D.-W. Shen, B.-P. Xie, J.-F. Zhao, L.-X. Yang, L. Fang, J. Shi, R.-H. He, D.-H. Lu, H.-H. Wen, and D.-L. Feng, Phys. Rev. Lett. 99, 216404 (2007).
  * Chatterjee et al. (2006) U. Chatterjee, M. Shi, A. Kaminski, A. Kanigel, H. Fretwell, K. Terashima, T. Takahashi, S. Rosenkranz, Z. Li, H. Raffy, et al., Phys. Rev. Lett. 96, 107006 (2006).
  * McElroy et al. (2006) K. McElroy, G.-H. Gweon, S.-Y. Zhou, J. Graf, S. Uchida, H. Eisaki, H. Takagi, T. Sasagawa, D.-H. Lee, and A. Lanzara, Phys. Rev. Lett. 96, 067005 (2006).
  * Shen et al. (2008) D.-W. Shen, Y. Zhang, L.-X. Yang, J. Wei, H.-W. Ou, J.-K. Dong, B.-P. Xie, C. He, J.-F. Zhao, B. Zhou, et al., Phys. Rev. Lett. 101, 226406 (2008).
  * Yang et al. (2021) Y.-C. Yang, Z.-T. Liu, J.-S. Liu, Z.-H. Liu, W.-L. Liu, X.-L. Lu, H.-P. Mei, A. Li, M. Ye, S. Qiao, et al., Nucl. Sci. Tech. 32, 1 (2021).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996).
  * Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Togo and Tanaka (2015) A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).
  * Mostofi et al. (2008) A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008).


2022

[1]Wonkeun Chang

1]School of Electrical and Electronic Engineering, Nanyang Technological
University, 50 Nanyang Avenue, 639798, Singapore 2]Research School of Physics,
The Australian National University, Canberra, 2601, ACT, Australia

# Microjoule-level mid-infrared femtosecond pulse generation in hollow-core
fibres

Ang Deng  Trivikramarao Gavara  Muhammad Rosdi Abu Hassan  Md Imran Hasan
<EMAIL_ADDRESS>[  [

###### Abstract

We demonstrate a fibre-based approach that generates mid-infrared femtosecond
pulses in the $34\text{\,}\mathrm{\SIUnitSymbolMicro m}$ spectral region with
microjoule-level single pulse energy. This is realised in a piece of gas-
filled antiresonant hollow-core fibre that is pumped by a two-micron light
source. A rapid variation of the dispersion near a structural resonance of the
fibre creates a phase-matching point in the mid-infrared, which mediates the
frequency-down conversion. We generate femtosecond pulses centred at
$3.16\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength with the pulse energy
of more than $1\text{\,}\mathrm{\SIUnitSymbolMicro J}$, achieving the
conversion efficiency as high as $9.4\text{\,}\mathrm{\char 37\relax}$. The
wavelength of the radiation is determined solely by the dielectric wall
thickness of the cladding elements, while the yield is subject to other
experimental parameters. This, combined with high power-handling capability of
hollow-core fibres, makes it possible to power scale the mid-infrared output
by either increasing the pulse energy or repetition rate of the pump. The
technique presents a new pathway to build an all-fibre-based mid-infrared
supercontinuum source, which promises to be a powerful new tool for ultrahigh
sensitivity molecular spectroscopy.

##  1 Introduction

High-power ultrafast light sources have become an indispensable tool in
science and technology. Most powerful laser systems generating femtosecond
pulses operate around one-micron wavelength with available average output
power in the kilowatts level Mueller:2020:OL . The two-micron region is also
gaining a lot of attention lately, emerging as a new wavelength for high-
throughput ultrafast lasers Gaida:2018:OL . Towards the longer wavelength,
however, the available power drops dramatically. The lack of powerful
ultrafast sources in mid-infrared (mid-IR) stands in stark contrast to their
many applications. Highly concentrated mid-IR photons are attractive for
triggering extreme nonlinear effects such as high-harmonic generations
Popmintchev:2012:Science . Moreover, they can efficiently drive coherent mid-
IR supercontinuum generation Petersen:2014:NP , offering a unique opportunity
in molecular spectroscopy Ebrahim-Zadeh:2008:Book . With a powerful mid-IR
supercontinuum source, the “fingerprints” of various molecules can be
identified in a single measurement at an unprecedented sensitivity.

A common solution to generate femtosecond mid-IR pulses rely on optical
parametric processes in nonlinear crystals Seidel:2018:SciAdv ;
Elu:2017:Optica . While the technique is largely successful, delivering multi-
watt-level average output power, it has a major drawback of being prone to
vibrations and other environmental variations. Fibre-based systems have the
advantage in this regard, but silica—commonly used material for optical
fibres—is not transparent beyond $2.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
A recent progress made in modelocked lasers incorporating erbium-, holmium-,
or dysprosium-doped fluoride fibre yielding nanojoule pulses Huang:2020:Optica
; Bawden:2021:OL ; Antipov:2016:Optica ; Wang:2019:OL  marks a breakthrough in
the ultrafast mid-IR light source development.

Another platform that is attracting growing interest in mid-IR photonics is
hollow-core fibres Wheeler:2014:OL ; Yu:2019:APLPhton . In a hollow-core
fibre, the light is guided in its central hollow region, bypassing the
limitations imposed by intrinsic properties of the waveguide material. It
enables low-loss mid-IR transmission even in silica-based optical fibres. An
interesting prospect is then to use hollow-core fibre as a container for
matter particles to induce lasing at or frequency conversion to mid-IR
wavelengths Xu:2017:OL ; Aghbolagh:2019:OL ; Zhou:2022:LSA ; Adamu:2019:SR ;
Deng:2021:CLEO . Many demonstrations have been made in the last decade on the
use of gas-filled kagomé-lattice, antiresonant, or simple capillary hollow-
core fibres to achieve phase-matched conversion of near-infrared pump to,
mostly, the ultraviolet region Joly:2011:PRL ; Mak:2013:OE ;
Koettig:2017:Optica ; Travers:2019:NP ; Xiong:2021:PR . This is due to general
landscape of the dispersion in these fibres, which permits soliton-dispersive
wave phase matching only in the blue-side. It is reported that phase matching
to mid-IR is possible in gas-filled hollow-core fibre by exploiting the
photoionisation effect, albeit with rather poor efficiency Koettig:2017:NC .

One promising approach to generate mid-IR radiation is to utilise so-called
the band-edge effect in antiresonant hollow-core fibres (AR-HCFs) Tani:2018:PR
. It exploits the effect of the structural resonances in AR-HCF, which
dominates and overrules the dispersion profile around them. The concept has
been tested in a recent experiment, successfully frequency-downshifting the
pump at $800\text{\,}\mathrm{nm}$ to $1.45\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, i.e., to a wavelength determined by the fibre geometry Gavara:2020:OL .
Here we build on this idea to realise a fibre-based ultrafast mid-IR light
source. Namely, we produce microjoule-level femtosecond pulses centred at
$3.16\text{\,}\mathrm{\SIUnitSymbolMicro m}$ that emerge from an argon-filled
AR-HCF pumped at $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with the conversion
efficiency as high as $9.4\text{\,}\mathrm{\char 37\relax}$. We also
investigate the role of fibre geometry in the process by employing another
fibre of different dimensions, demonstrating shifting of the mid-IR radiation
to $3.61\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

##  2 Results

The frequency conversion is staged in a gas-filled AR-HCF. Figure 1a presents
a cross-section of the fibre used in the experiment. It consists of seven
silica tubular cladding elements, which have average outer diameter
$d=$24\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and wall thickness
$t=$1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The light is guided in the
hollow core of diameter $D=$73\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
Inhibition of the coupling between spatial modes in the hollow core and those
in the cladding ensures that light launched in the core is confined and
transmitted along the fibre with minimal leakage Debord:2017:Optica ;
Deng:2020:OE . It turns out dielectric modes in the cladding wall interact the
most strongly with the core modes, making the wall thickness $t$ one of the
key geometrical parameters in the AR-HCF guidance. Namely, $t$ governs
resonant bands where light in the core leaks out through coupling to the
dielectric modes. This occurs at wavelengths given by Archambault:1993:JLT :

$\lambda_{m}=\frac{2t}{m}\sqrt{n^{2}-1}\textrm{,}$ (1)

where $n$ is refractive index of the dielectric material, i.e., silica, and
$m=1$, $2$, $3$, … is the resonance order. The hollow guidance in the core
then happens in low-loss regions between the resonances.

Figure 1: AR-HCF characterisation. a. A cross-sectional image of the AR-HCF
taken with a scanning-electron microscope. The right-hand side is the
magnified image of a cladding element. The fibre has core diameter
$D=$73\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, average outer cladding element
diameter $d=$24\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and dielectric wall
thickness $t=$1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. b. Calculated
transmission loss of an idealised AR-HCF of the same geometry (blue-solid
line), and normalised transmission spectrum taken at the output of a
$2\text{\,}\mathrm{m}$-long fibre (grey-solid line). c. Calculated GVD of the
AR-HCF when evacuated (black-dotted line) and argon pressurised at
$16\text{\,}\mathrm{bar}$ (red-solid line). GVD of a simple dielectric
capillary of the same core size and argon pressurisation is also plotted for
comparison (grey-solid line). RB1, RB2, and RB3, denote the first three
$t$-induced resonant bands (magenta shades).

Figure 1b shows calculated transmission loss of the fundamental core mode of
an idealised AR-HCF that has the same cross-sectional dimensions as the one
shown in Fig. 1a. The presence of the resonant and antiresonant bands across
the spectrum is evident. The normalised transmission spectrum taken at the
output of a $2\text{\,}\mathrm{m}$-long fibre is also plotted in Fig. 1b. It
matches well with the calculated loss spectrum, as well as locations of the
resonances given by Eq. (1). The first three loss bands appear in $2.412.86$,
$1.381.44$, and $0.950.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
respectively. Hence, the $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$-pump lies
in the second transmission window of the fibre, and the first window covers
mid-IR with the calculated loss of less than
$1\text{\,}\mathrm{dB}\text{\,}{\mathrm{m}}^{-1}$ in the
$35.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ range. The transmission loss of
less than $0.35\text{\,}\mathrm{dB}\text{\,}{\mathrm{m}}^{-1}$ is measured at
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ using the cut-back method.

The resonances also largely influence group-velocity dispersion (GVD) in their
vicinity as shown in Fig. 1c. In the central part of each transmission band,
GVD closely follows that of a simple dielectric capillary of the same core
size and gas pressurisation Marcatili:1964:BSTJ . However, it deviates and
varies rapidly near the band edges. As we shall see, this dramatic GVD change
in the band edges is the key to the phase-matched frequency conversion to mid-
IR. Note that in the band edges, the $t$-induced resonances dominate the
dispersion, while the gas filling makes only a minor impact. The system
exhibits anomalous dispersion at $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$
when pressurised with argon at $16\text{\,}\mathrm{bar}$, which permits the
formation of optical solitons.

The experimental setup is illustrated in Fig. 2. A
$25\text{\,}\mathrm{cm}$-long AR-HCF is pressurised with argon. One of the two
fibre ends is pumped by $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ idler beam
from an optical parametric amplifier that is driven by a Ti:sapphire laser
operating at $1\text{\,}\mathrm{kHz}$ repetition rate. The pump pulse has
$65\text{\,}\mathrm{fs}$ full width at half maximum (FWHM) duration (See
Supplementary Note 1). The beam exiting from the output end of the fibre is
focused to diagnostics.

Figure 2: Experimental setup. A schematic of the experiment. OPA, optical
parametric amplifier; LP, long-pass filter; HWP, half-wave plate; TFP, thin-
film polariser; PM, power meter; SM, spectrometer; MC, monochromator; M,
mirror; L, lens.

Figure 3a presents an output spectrum when the fibre is pressurised with
$16\text{\,}\mathrm{bar}$ argon and the energy of the pulse exiting the fibre
is $11.7\text{\,}\mathrm{\SIUnitSymbolMicro J}$. The corresponding pump pulse
energy is $12.5\text{\,}\mathrm{\SIUnitSymbolMicro J}$ after accounting for
the loss in the fibre. We observe broadening of the pump spectrum, and more
importantly, appearance of a prominent mid-IR spectral band centred at
$3.16\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the blue-edge of the first
transmission band. Its peak is approximately $10\text{\,}\mathrm{dB}$ below
the main pump spectrum and the energy in the mid-IR calculated from the
intensity calibrated spectrum is $1.1\text{\,}\mathrm{\SIUnitSymbolMicro J}$.
This amounts to $8.6\text{\,}\mathrm{\char 37\relax}$ energy conversion from
the input. Both the total and mid-IR beams at the output have near-Gaussian
profiles as shown in the inset in Fig. 3a, indicating that they emerge from
the fundamental mode of the fibre.

Figure 3: Mid-IR generation. a. Measured (red-solid line) and simulated (grey
shade) spectra at the AR-HCF output. The fibre is pressurised with
$16\text{\,}\mathrm{bar}$ argon and the pump pulse energy is
$12.5\text{\,}\mathrm{\SIUnitSymbolMicro J}$. The pump spectrum is plotted
together for reference (blue-dashed line). The inset is intensity profiles of
the total (dotted lines) and mid-IR (solid lines) beams emerging from the
fibre output characterised using the knife-edge method. b. Simulated spectral
evolution along the fibre length. c. The corresponding soliton-dispersive wave
phase-matching diagram (blue-solid line). The dephasing of a simple dielectric
capillary of the same core size and argon pressurisation is included for
comparison (red-dashed line). Phase-matching points that originate from the
resonant bands (black crosses) and capillary model (red diamond) are marked in
the diagram. Only the first three resonant bands, denoted RB1, RB2, and RB3
(magenta shades), are considered in the dephasing diagram and band-edge
induced phase-matching points for clarity.

A numerical simulation of the pulse propagation is carried out for the same
set of parameters. Simulated output spectrum shown in Fig. 3a exhibits a high-
level of agreement with the measurement, reproducing most of the spectral
details including the strong mid-IR radiation and other prominent peaks around
$1.45$ and $0.35\text{\,}\mathrm{\SIUnitSymbolMicro m}$. These features appear
due to phase matching between soliton and dispersive waves which leads to
coherent transfer of energy from the pump Akhmediev:1995:PRA . A false colour
map of the spectral evolution along the fibre length is presented in Fig. 3b.
The $12.5\text{\,}\mathrm{\SIUnitSymbolMicro J}$-pump, which amounts to a
higher-order soliton of order $2.7$, initially undergoes temporal compression
and spectral broadening due to interplay between anomalous dispersion and
nonlinearity. Efficient frequency conversion takes place after
$15\text{\,}\mathrm{cm}$ of propagation in the fibre, when low-intensity
spectral tails of the broadened pump eventually extend to zero phase-mismatch
points. Here, these are given by the soliton-dispersive wave phase-matching
condition:

$\Delta\beta_{\textrm{DW}}=\beta-\beta_{0}-\beta_{1}\left(\omega-\omega_{0}\right)-\frac{3\omega^{2}}{8c^{2}\beta}\chi^{\left(3\right)}\lvert
E_{0}\rvert^{2}=0\textrm{,}$ (2)

where $c$ is the speed of light in free space and $\beta$ is a wavevector of
the propagating mode in the fibre. $\beta_{0}$ and $\beta_{1}$ are the zeroth
and first Taylor series expansion coefficients of $\beta$ at the pump,
$\omega_{0}$, which are related to phase and group velocities, respectively.
The last term in Eq. (2) is a nonlinear correction term, where
$\chi^{\left(3\right)}$ is the third-order susceptibility of the filling
material and $E_{0}$ is the peak electric field strength of the pump. Figure
3c is the soliton-dispersive wave phase-matching diagram that results from Eq.
(2). There are two distinctive origins of phase matchings in this diagram. The
first is from the general dispersion landscape of the hollow-core fibre in the
central parts of the transmission bands. It is responsible for the emission at
$0.35\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and the equivalent capillary
model also features the same phase matching. The other mechanism is caused by
presence of the resonances. The rapid GVD change around each resonance creates
a phase-matching point in the long-wavelength side of the resonance, i.e., in
the blue-edge of the transmission band of the same order. All phase-matching
points marked in Fig. 3c, except for $0.35\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, are due to the band-edge effect. This includes the one at
$3.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ that is promoting the mid-IR
radiation. The band-edge-induced phase matchings do not show up in the
capillary model.

We can characterise the mid-IR pulse from the numerical data. Its temporal
intensity and phase profiles are shown in Fig. 4. The pulse is centred at
$3.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength and has FWHM duration
of $123\text{\,}\mathrm{fs}$ and peak power of $10.6\text{\,}\mathrm{MW}$ at
the fibre output. After its generation at around the $15\text{\,}\mathrm{cm}$
mark, the mid-IR light travels linearly in the normal dispersion regime in the
remaining length of the fibre. This is accountable for the frequency chirp
across the pulse, which leaves a room for post pulse compression. The duration
and peak power of the mid-IR pulse at its Fourier limit are
$67\text{\,}\mathrm{fs}$ and $20.4\text{\,}\mathrm{MW}$, respectively.

Figure 4: Mid-IR pulse characterisation. Temporal intensity (blue-solid line)
and phase (red-dashed line) profiles of the simulated mid-IR pulse at the
fibre output. The pulse is centred at $3.1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ wavelength, and it has FWHM duration and peak power of
$123\text{\,}\mathrm{fs}$ and $10.6\text{\,}\mathrm{MW}$, respectively. At its
Fourier limit, they are $67\text{\,}\mathrm{fs}$ and
$20.4\text{\,}\mathrm{MW}$ (grey-dotted line).

One of the parameters that can be changed easily in the experiment is the pump
pulse energy. It alters peak intensity of the incoming pulse which in turn
affects nonlinear phase accumulation. Figure 5a is a collection of output
spectra measured at different pump pulse energies while fixing the other
settings the same as in Fig. 3. The band-edge-induced spectral features at
$3.16$ and $1.45\text{\,}\mathrm{\SIUnitSymbolMicro m}$ appear at the pump
energy as low as $8\text{\,}\mathrm{\SIUnitSymbolMicro J}$. The latter
exhibits a higher spectral intensity due to its location in the same
transmission window with the pump. As the energy further increases, the main
spectrum around $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ broadens, and we
start to observe rise of the $0.35\text{\,}\mathrm{\SIUnitSymbolMicro m}$ peak
brought by the capillary model-mediated phase matching. At the same time,
energy in the mid-IR gradually increases. Figure 5b shows energy in the mid-IR
spectral band, i.e., in the region above
$2.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$, obtained from the intensity
calibrated output spectrum and the corresponding conversion efficiency versus
the pump energy. The maximum mid-IR energy observed in the experiment is close
to $2\text{\,}\mathrm{\SIUnitSymbolMicro J}$ when the pump is
$30\text{\,}\mathrm{\SIUnitSymbolMicro J}$. A monotonic increase of the mid-IR
energy right up to the maximum pump used in the experiment suggests that mid-
IR pulses of higher energies are achievable by further increasing the pump.
The conversion efficiency, on the other hand, tops off at
$15\text{\,}\mathrm{\SIUnitSymbolMicro J}$-pump where
$9.4\text{\,}\mathrm{\char 37\relax}$ of the input energy is converted to mid-
IR. At higher pump energies, the mid-IR generation onsets at a shorter
propagation distance, and hence it travels a longer length in the fibre. Since
the transmission loss is higher in the mid-IR region than around the pump, the
conversion becomes less efficient. The pump energy has no meaningful influence
on spectral position of the mid-IR band as shown in Fig. 5c. This is because
the dispersion in the band-edge is largely dominated by the nearby resonance,
which depends solely on the geometrical parameter, $t$. Added to that, the
nonlinear correction term in Eq. (2) makes only a small contribution to the
overall phase matching. Therefore, the pump energy has a negligible effect on
locations of the band-edge-induced phase-matching points. On the contrary, it
noticeably affects bandwidth of the mid-IR pulse. The band generally becomes
broader with the rising pump energy. The enhanced broadening of the main
spectrum at higher pump pulse energies leads to the conversion over a wider
spectral range in mid-IR.

Figure 5: Pump energy and pressure tuning. a. Output spectra at different pump
pulse energies when the fibre is pressurised with argon at
$16\text{\,}\mathrm{bar}$. b. Energy of the mid-IR pulse (blue-solid line) and
the corresponding conversion efficiency (red-dashed line) versus pump energy.
c. The centre wavelength (blue-solid line) and the FWHM spectral bandwidth
(red-dashed line) of the mid-IR pulse versus pump energy. d. Output spectra at
different argon pressurisations when the pump pulse energy is
$18\text{\,}\mathrm{\SIUnitSymbolMicro J}$. The arrows mark emission due to
the capillary model-induced phase matching. e. Energy of the mid-IR pulse
(blue-solid line) and the corresponding conversion efficiency (red-dashed
line) versus argon pressure. f. The centre wavelength (blue-solid line) and
the FWHM spectral bandwidth (red-dashed line) of the mid-IR pulse versus argon
pressure.

The gas pressure is another experimental parameter that can be varied onsite.
Changing the pressure alters not only the nonlinearity but also the dispersion
of the system. Figure 5d presents output spectra at different argon filling
pressures when the pump pulse energy is $18\text{\,}\mathrm{\SIUnitSymbolMicro
J}$. We observe general broadening of the main pump spectrum as well as
generated spectral bands when the pressure is raised. The observation is
similar to increasing the pump energy in Fig. 5a, since they both enhance the
system nonlinearity. The energy in the mid-IR band is plotted against the
pressure in Fig. 5e together with the conversion efficiency. The energy
transfer to mid-IR increases rapidly when the pressure is varied from $7$ to
$12\text{\,}\mathrm{bar}$ and flattens afterwards arriving at
$1.6\text{\,}\mathrm{\SIUnitSymbolMicro J}$ when the pressure is
$25\text{\,}\mathrm{bar}$. One important remark is that spectral location of
the mid-IR emission shifts slightly—over a bandwidth of
$3.2\text{\,}\mathrm{THz}$ between $3.14$ and
$3.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$—with the pressure change as
evident in Fig. 5f. While the resonance is the main influence on the phase
matching in the band-edge, the pressure change can also cause small but
observable shift in the vicinity. In comparison, emission at the phase
matching supported by the capillary model, i.e., the spectral feature marked
with arrows in Fig. 5d, drifts across a much wider range—over a bandwidth of
$236\text{\,}\mathrm{THz}$ from $0.31$ to
$0.41\text{\,}\mathrm{\SIUnitSymbolMicro m}$—when the pressure is varied from
$10$ to $25\text{\,}\mathrm{bar}$.

To shift the mid-IR wavelength beyond the small stretch that is attainable
with the pressure tuning, we need to use an AR-HCF with a different cladding
wall thickness $t$. To this end, a new set of experiments is carried out with
another AR-HCF. Figure 6a is a cross-sectional image of the second fibre. The
core and seven silica cladding tubes are smaller than the former, having
diameters $D=$46\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$d=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, respectively. The key
parameter, $t$, is thicker at $1.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$. It
red-shifts the first resonant band to
$2.63.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, while still allowing the
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$-pump to propagate in the second
transmission window. This choice of fibre is interesting, not just for
studying the role of $t$, but also for understanding the impact, or lack
thereof, of $D$ in the mid-IR generation. Measured spectrum at the output of a
$25\text{\,}\mathrm{cm}$-long fibre filled with argon at
$15\text{\,}\mathrm{bar}$ is plotted in Fig. 6b. The output pulse energy is
$12.8\text{\,}\mathrm{\SIUnitSymbolMicro J}$ which amounts to
$15\text{\,}\mathrm{\SIUnitSymbolMicro J}$ at the input. Further measured data
at other pump energies and argon pressures are enclosed in Supplementary Note
2. A mid-IR band appears on the long-wavelength side of the first resonant
band centred at $3.61\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The energy in
the mid-IR band, i.e., $>$3.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, is
$124\text{\,}\mathrm{nJ}$. Simulated output is generally in good agreement
with the experiment. The mid-IR spectral feature especially is well
reproduced. The conversion to mid-IR onsets at a shorter propagation distance
in this fibre due to the smaller core size that leads to a higher launch pulse
peak intensity and earlier accumulation of the nonlinear phase shift. Here,
the mid-IR must travel a longer length in the fibre, exposing it to a
substantially higher loss than the pump due to the small core size
Deng:2021:Crystals . This results in a significantly poorer conversion
efficiency. The early formation of the long-wavelength band in the fibre is
also evident from the broader temporal profile of the mid-IR pulse at the
fibre output. Its FWHM duration and peak power characterised from the
numerical data are $254\text{\,}\mathrm{fs}$ and $1.1\text{\,}\mathrm{MW}$,
respectively. It can be compressed to $110\text{\,}\mathrm{fs}$ duration and
$2.8\text{\,}\mathrm{MW}$ peak power at its Fourier limit. The soliton-
dispersive wave phase matching diagram of the system is presented in Fig. 6c
with the band-edge-induced zero-mismatch appearing next to each resonant band.
Note that phase-matching due to the capillary characteristic of the dispersion
is located at $0.22\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Again, the
resonance plays a crucial role in creating the zero-mismatch point deep in the
mid-IR region.

Figure 6: Geometrical tuning of mid-IR wavelength. a. A cross-sectional image
of the antiresonant hollow-core fibre used in the second experiment. The
right-hand side is the magnified image of a cladding element. The fibre has
core diameter $D=$46\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, average outer
cladding element diameter $d=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and
dielectric wall thickness $t=$1.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. b.
Measured (red-solid line) and simulated (grey shade) spectra at the output of
a $25\text{\,}\mathrm{cm}$-long fibre. The system is pressurised with
$15\text{\,}\mathrm{bar}$ argon and the pump pulse energy is
$15\text{\,}\mathrm{\SIUnitSymbolMicro J}$. c. The corresponding soliton-
dispersive wave phase-matching diagram (blue-solid line). The dephasing of a
simple dielectric capillary of the same core size and argon pressurisation is
included for comparison (red-dashed line). The phase-matching points that
originate from the resonant bands are marked in the diagram (black crosses).
Only the first four resonant bands, denoted RB1, RB2, RB3, and RB4 (magenta
shades), are considered in the dephasing diagram and band-edge induced phase-
matching points for clarity.

##  3 Discussion

The generation of microjoule-level mid-IR femtosecond pulses is demonstrated
in gas-filled AR-HCFs. The technique exploits the band-edge effect, which
dominates the dispersion landscape around structural resonances of the fibre.
The wavelength of the mid-IR emission is dictated primarily by location of the
resonance, hence highlighting the main advantage of the approach; we can
induce radiation at a desired wavelength by controlling the wall thickness of
the cladding elements alone. Other system parameters, such as the pump pulse
energy and its duration, as well as the fibre length, core size and gas
filling, are still relevant for inducing nonlinear effect to mediate the
frequency mixing process. They dictate the extent of the conversion. This can
be further exploited, together with the high power-handling capability of
hollow-core fibres, to facilitate power-scalable mid-IR generation.

In the experiments, the pump is deliberately chosen at
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ considering the rapid advances in
high-performance ultrafast fibre lasers at this wavelength. The system
demonstrated herein can be integrated at the output of such a laser to realise
a high-power mid-IR fibre laser that has a small footprint, near-diffraction-
limited output beam quality, and exceptional stability. One interesting
prospect of the source is driving ultrabroadband mid-IR comb generation. The
phase-matched conversion maintains coherence between the spectral components
in the pump and mid-IR. Therefore, with the right choice of the pump laser,
the comb lines remain intact in the mid-IR pulses.

##  4 Methods

###  4.1 Fibre characterisation

The AR-HCFs used in the experiments are fabricated in Nanyang Technological
University. The normalised transmission spectrum is obtained with a
supercontinuum laser spanning $0.42.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
The broadband input is coupled into a $2\text{\,}\mathrm{m}$-long fibre, and
the output spectrum is acquired with two optical spectrum analysers to capture
the full spectral range. This is then divided by the source spectrum for the
normalisation. The transmission loss of less than
$0.35\text{\,}\mathrm{dB}\text{\,}{\mathrm{m}}^{-1}$ is measured at
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with a power meter by cutting-back
the fibre from $1.5$ to $0.2\text{\,}\mathrm{m}$.

Guiding properties of the fundamental core modes in the AR-HCFs are calculated
based on their idealised structures. They are constructed on the measured
geometrical dimensions in their cross-sectional images taken with a scanning-
electron microscope. From these, the complex mode indices are computed using
the finite-element method over a wide spectral range. The transmission loss
includes the confinement loss, absorption in the silica, and surface
scattering Fokoua:2012:OE . The linear dispersion takes into account the
waveguide portion obtained from the finite-element modelling, as well as the
pressure dependent argon gas contribution Boerzsoenyi:2008:AO .

###  4.2 Experiment

A piece of AR-HCF has two gas cells attached at the ends to provide the argon
pressurisation. Each gas cell has a $3\text{\,}\mathrm{mm}$-thick CaF2 window
for the optical access, of which the input side window has an antireflection
coating covering the $1.82.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ range. The
one in the output gas cell is uncoated.

The input at $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is an idler from an
optical parametric amplifier which is pumped by a Ti:sapphire laser at
$1\text{\,}\mathrm{kHz}$ repetition rate. A long-pass filter is placed at the
output of the optical parametric amplifier to filter out the signal at
$1.34\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The input energy is adjusted
with a half-wave plate and thin film polarizer. The light is passed through an
iris to enhance its spatial beam quality and coupled into the fibre using a
plano-convex lens. All input optics have $2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-coating to suppress reflections. The input pulse has FWHM duration of
$65\text{\,}\mathrm{fs}$ just before entering the gas cell as measured in the
frequency-resolved optical gating technique DeLong:1994:JOSAB  (See
Supplementary Note 1). Typical coupling efficiencies are
$>$80\text{\,}\mathrm{\char 37\relax}$$ for the fibre with
$73\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core diameter and around
$70\text{\,}\mathrm{\char 37\relax}$ for the one with
$46\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core diameter.

The beam leaving the output gas cell is focused to diagnostics. The total
power of the output beam is measured with a thermal power meter. The spatial
profiles of the total and mid-IR output beams are characterised using the
knife-edge method Araujo:2009:AO . For the spectral measurements, we employ a
fibre-coupled CCD-type spectrometer covering the
$0.21.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ range as well as two
monochromators. One of the monochromators has a silica photodiode and Peltier-
cooled lead selenide photodiode for the $0.21.1$ and
$15.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ regions, respectively. The other
monochromator has a Dewar-cooled mercury cadmium telluride photodiode that
covers the $212\text{\,}\mathrm{\SIUnitSymbolMicro m}$ range. The data
recorded on the CCD-type spectrometer is corrected between
$0.241\text{\,}\mathrm{\SIUnitSymbolMicro m}$ based on its grating efficiency
and photodiode responsivity. The monochromators are intensity calibrated in
the region $>$0.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ with high-power
calibration light sources. The spectral measurements from these devices are
cross-checked and merged. The signal-to-noise ratios of the spectrometer and
monochromators are $30$ and $50\text{\,}\mathrm{dB}$, respectively.

###  4.3 Pulse propagation simulation

The pulse evolution along the gas-filled hollow-core fibre is simulated by
solving a single-mode unidirectional field propagation equation. It is given
by Chang:2013:OL :

$\displaystyle\partial_{z}E\left(z,\omega\right)=$ $\displaystyle
i\left(\beta-\frac{\omega}{v}+i\frac{\alpha}{2}\right)E\left(z,\omega\right)+i\frac{\omega^{2}}{2c^{2}\beta}\mathcal{F}\\{\chi^{\left(3\right)}E\left(z,\tau\right)^{3}\\}$
$\displaystyle-\frac{\omega}{2\epsilon_{0}c^{2}\beta}\mathcal{F}\\{\partial_{\tau}N\left(z,\tau\right)\frac{I_{P}}{E\left(z,\tau\right)}+\frac{e^{2}}{m_{e}}\int_{-\infty}^{\tau}N\left(z,\tau^{\prime}\right)E\left(z,\tau^{\prime}\right)d\tau^{\prime}\\}\textrm{,}$
(3)

where $\epsilon_{0}$ is the vacuum permittivity, $\tau$ is the time frame
moving at a reference velocity, $v$, and $z$ is the propagation length.
$E\left(z,\omega\right)$ is the evolving optical field in the frequency domain
with $\omega$ denoting angular frequency. This is obtained by taking the
Fourier transformation of the fast oscillating electric field, i.e.,
$E\left(z,\omega\right)=\mathcal{F}\\{E\left(z,\tau\right)\\}$. The linear
dispersion of the fundamental core mode of the gas-filled hollow-core fibre is
accounted for in $\beta$. The term containing $\chi^{\left(3\right)}$ is
responsible for the third-order nonlinearity of the pressurised argon
Lehmeier:1985:OC . The last term describes photoionisation which has
negligible effect for the cases studied in this work, but is nevertheless
included for the sake of completeness. Here, $I_{P}$ is the first ionisation
energy, and $e$ and $m_{e}$ are the electron charge and mass, respectively.
For the calculation of local free-electron density, $N\left(z,\tau\right)$, a
model formulated by Ammosov et al. is adopted Ammosov:1986:JETP .

## Acknowledgments

This work is supported by the Ministry of Education, Singapore, under its
Academic Research Fund Tier 2 (2019-T2-2-026).

## Author contributions

AD, TG, MIH, and WC designed the experiments. AD and TG built the experimental
setup, performed the experiments, and processed the experimental data. MRAH
fabricated the fibres. AD and WC carried out the numerical analyses. AD and WC
prepared the first draft of the manuscript, and all authors contributed to its
development. WC supervised this work.

## References

  * (1) Müller, M., Aleshire, C., Klenke, A., Haddad, E., Légaré, F., Tünnermann, A. & Limpert, J. $10.4\text{\,}\mathrm{kW}$ coherently combined ultrafast fiber laser. Opt. Lett. 45, 3083–3086 (2020).
  * (2) Gaida, C., Gebhardt, M., Heuermann, T., Stutzki, F., Jauregui, C. & Limpert, J. Ultrafast thulium fiber laser system emitting more than $1\text{\,}\mathrm{kW}$ of average power. Opt. Lett. 43, 5853–5856 (2018).
  * (3) Popmintchev, T., Chen, M.-C., Popmintchev, D., Arpin, P., Brown, S., Ališauskas, S., Andriukatis, G., Balčiunas, T., Mücke, O. D., Pugzlys, A., Baltuška, A., Shim, B., Schrauth, S. E., Gaeta, A., Hernández-García, C., Plaja, L., Becker, A., Jaron-Becker, A., Murnane, M. M. & Kapteyn, H. C. Bright coherent ultrahigh harmonics in the $\mathrm{keV}$ x-ray regime from mid-infrared femtosecond lasers. Science 336, 1287–1291 (2012).
  * (4) Petersen, C. R., Møller, U., Kubat, I., Zhou, B., Dupont, S., Ramsay, J., Benson, T., Sujecki, S., Abdel-Moneim, N., Tang, Z., Furniss, D., Seddon, A. & Bang, O. Mid-infrared supercontinuum covering the $1.413.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ molecular fingerprint region using ultra-high NA chalcogenide step-index fibre. Nat. Photon. 8, 830–834 (2014).
  * (5) Ebrahim-Zadeh, M. & Sorokina, I. T. Mid-Infrared Coherent Sources and Applications. (Springer, 2008).
  * (6) Seidel, M., Xiao, X., Hussain, S. A., Arisholm, G., Hartung, A., Zawilski, K. T., Schunemann, P. G., Habel, F., Trubetskov, M., Pervak, V., Pronin, O. & Krausz, F. Multi-watt, multi-octave, mid-infrared femtosecond source. Sci. Adv. 4, eaaq1526 (2018).
  * (7) Elu, U., Baudisch, M., Pires, H., Tani, F., Frosz, M. H., Köttig, F., Ermolov, A., Russell, P. St.J. & Biegert, J. High average power and single-cycle pulses from a mid-IR optical parametric chirped pulse amplifier. Optica 4, 1024–1029 (2017).
  * (8) Huang, J., Pang, M., Jiang, X., Köttig, F., Schade, D., He, W., Butryn, M. & Russell, P. St.J. Sub-two-cycle octave-spanning mid-infrared fiber laser, Optica 7, 574–579 (2020).
  * (9) Bawden, N., Henderson-Sapir, O., Jackson, S. D. & Ottaway, D. J. Ultrafast $3.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ fiber laser. Opt. Lett. 46, 1636–1639 (2021).
  * (10) Antipov, S., Hudson, D. D., Fuerbach, A. & Jackson, S. D. High-power mid-infrared femtosecond fiber laser in the water vapor transmission window. Optica 3, 1373–1376 (2016).
  * (11) Wang, Y., Jobin, F., Duval, S., Fortin, V., Laporta, P., Bernier, M., Galzerano, G. & Vallée, R. Ultrafast Dy^3+:fluoride fiber laser beyond $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Opt. Lett. 44, 395–398 (2019).
  * (12) Wheeler, N. V., Heidt, A. M., Baddela, N. K., Fokoua, E. M., Hayes, J. R., Sandoghchi, S. R., Poletti, F., Petrovich, M. N. & Richardson, D. J. Low-loss and low-bend-sensitivity mid-infrared guidance in a hollow-core-photonic-bandgap fiber. Opt. Lett. 39, 295–298 (2014).
  * (13) Yu, F., Song, P., Wu, D., Birks, T., Bird, D. & Knight, J. Attenuation limit of silica-based hollow-core fiber at mid-IR wavelengths. APL Photonics 4, 080803 (2019).
  * (14) Xu, M., Yu, F. & Knight, J., Mid-infrared $1\text{\,}\mathrm{W}$ hollow-core fiber gas laser source. Opt. Lett. 42, 4055–4058 (2017).
  * (15) Aghbolagh, F. B. A., Nampoothiri, V., Debord, B., Gerome, F., Benabid, F. & Rudolph, W. Mid IR hollow core fiber gas laser emitting at $4.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Opt. Lett. 44, 383–386 (2019).
  * (16) Zhou, Z., Wang, Z., Huang, W., Cui, Y., Li, H., Wang, M., Xi, X., Gao, S. & Wang, Y. Towards high-power mid-IR light source tunable from $3.8$ to $4.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by HBr-filled hollow-core silica fibres. Light Sci. Appl. 11, 15 (2022).
  * (17) Adamu, A. I., Habib, M. S., Petersen, C. R., Lopez, J. E. A., Zhou, B., Schülzgen, A., Bache, M., Amezcua-Correa, R., Bang, O. & Markos, C. Deep-UV to mid-IR supercontinuum generation drive by mid-IR ultrashort pulses in a gas-filled hollow-core fiber. Sci. Rep. 9, 4446 (2019).
  * (18) Deng, A., Gavara, T., Hassan, M. R. A., Jin, Y., Wang Q. J. & Chang, W. Four-octave-spanning mid-infrared supercontinuum generation in a gas-filled hollow-core fiber. Conference on Lasers and Electro-Optics 2021, pp. 1–2.
  * (19) Joly, N. Y., Nold, J., Chang, W., Hölzer, P., Nazarkin, A., Wong, G. K. L., Biancalana, F. & Russell, P. St.J. Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber. Phys. Rev. Lett. 106, 203901 (2011).
  * (20) Mak, K. F., Travers, J. C., Hölzer, P., Joly, N. Y. & Russell, P. St.J. Tunable vacuum-UV to visible ultrafast pulse source based on gas-filled Kagome-PCF. Opt. Express 21, 10942–10953 (2013).
  * (21) Köttig, F., Tani, F., Biersach, C. M., Travers, J. C. & Russell, P. St.J. Generation of microjoule pulses in the deep ultraviolet at megahertz repetition rate. Optica 4, 1272–1276 (2017).
  * (22) Travers, J. C., Grigorova, T. F., Brahms, C. & Belli, F. High-energy pulse self-compression and ultraviolet generation through soliton dynamics in hollow capillary fibres. Nat. Photon. 13, 547–554 (2019).
  * (23) Xiong, D., Luo, J., Hassan, M. R. A., Wu, X. & Chang, W. Low-energy-threshold deep-ultraviolet generation in a small-mode-area hollow-core fiber. Photonics Res. 9, 590–595 (2021).
  * (24) Köttig, F., Novoa, D., Tani, F., Günendi, M. C., Cassataro, M., Travers, J. C. & Russell, P. St.J. Mid-infrared dispersive wave generation in gas-filled photonic crystal fibre by transient ionization-driven changes in dispersion. Nat. Commun. 8, 813 (2017).
  * (25) Tani, F., Köttig, F., Novoa, D., Keding, R. & Russell, P. St.J. Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonics crystal fibers. Photonics Res. 6, 84–88 (2018).
  * (26) Gavara, T., Hasan, M. I., Hassan, M. R. A., Deng, A. & Chang, W. Band-edge mediated frequency down-conversion in a gas-filled anti-resonant hollow-core fiber. Opt. Lett. 45, 6815–6818 (2020).
  * (27) Debord, B., Amsanpally, A., Chafer, M., Baz, A., Maurel, M., Blondy, J. M., Hugonnot, E., Scol, F., Vincetti, L., Gérôme, F. & Benabid, F. Ultralow transmission loss in inhibited-coupling guiding hollow fibers. Optica 4, 209–2017 (2017).
  * (28) Deng, A., Hasan, I., Wang, Y. & Chang, W. Analyzing mode index mismatch and field overlap for light guidance in negative-curvature fibers. Opt. Express 28, 27974–27988 (2020).
  * (29) Archambault, J.-L., Black, R. J., Lacroix, S. & Bures, J. Loss calculations for antiresonant waveguides. J. Light. Technol. 11, 416–423 (1993).
  * (30) Marcatili, E. A. J. & Schmeltzer, R. A. Hollow metallic and dielectric waveguides for long distance optical transmission and lasers. Bell Syst. Tech. J. 43, 1783–1809 (1964).
  * (31) Akhmediev, N. & Karlsson, M. Cherenkov radiation emitted by solitons in optical fibers. Phys. Rev. A 51, 2602 (1995).
  * (32) Deng, A. & Chang, W. Geometrical scaling of antiresonant hollow-core fibers for mid-infrared beam delivery. Crystals 11, 420 (2021).
  * (33) Fokoua, E. N., Poletti, F. & Richardson, D. J. Analysis of light scattering from surface roughness in hollow-core photonic bandgap fibers. Opt. Express 20, 20980–20997 (2012).
  * (34) Börzsönyi, A., Heiner, Z., Kalashnikov, M. P., Kovács, A. P. & Osvay, K. Dispersion measurement of inert gases and gas mixtures at $800\text{\,}\mathrm{nm}$. Appl. Opt. 47, 4856–4863 (2008).
  * (35) DeLong, K. W., Trebino, R., Hunter, J. & White, W. E. Frequency-resolved optical gating with the use of second-harmonic generation. J. Opt. Soc. Am. B 11, 2206–2215 (1994).
  * (36) de Araújo, M. A. C., Silva, R., de Lima, E., Pereira, D. P. & de Oliveira, P. C. Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis. Appl. Opt. 48, 393–396 (2009).
  * (37) Chang, W., Hölzer, P., Travers, J. C. & Russell, P. St.J. Combined soliton pulse compression and plasma-related frequency upconversion in gas-filled photonic crystal fiber. Opt. Lett. 38, 2984–2987 (2013).
  * (38) Lehmeier, H. J., Leupacher, W. & Penzkofer, A. Nonresonant third order hyperpolarizability of rare gases and N2 determined by third harmonic generation. Opt. Commun. 56, 67–72 (1985).
  * (39) Ammosov, M. V., Delone, N. B. & Krainov, V. P. Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field. Sov. Phys. JETP 64, 1191–1194 (1986).