# Timing and energy stability of resonant dispersive wave emission in gas-
filled hollow-core waveguides

Christian Brahms<EMAIL_ADDRESS>John C. Travers  School of Engineering and
Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

###### Abstract

We numerically investigate the energy and arrival-time noise of ultrashort
laser pulses produced via resonant dispersive wave emission in gas-filled
hollow-core waveguides under the influence of pump-laser instability. We find
that for low pump energy, fluctuations in the pump energy are strongly
amplified. However, when the generation process is saturated, the energy of
the resonant dispersive wave can be significantly less noisy than that of the
pump pulse. This holds for a variety of generation conditions and while still
producing few-femtosecond pulses. We further find that the arrival-time jitter
of the generated pulse remains well below one femtosecond even for a
conservative estimate of the pump pulse energy noise, and that photoionisation
and plasma dynamics can lead to exceptional stability for some generation
conditions. By applying our analysis to a scaled-down system, we demonstrate
that our results hold for frequency conversion schemes based on both small-
core microstructured fibre and large-core hollow capillary fibre.

##  I Introduction

Resonant dispersive wave (RDW) emission in gas-filled hollow-core waveguides
is a promising source of tuneable few-femtosecond pulses from the vacuum
ultraviolet (VUV) to the near infrared [1, 2, 3, 4, 5]. It is based on
resonant energy transfer to a phase-matched spectral band during soliton self-
compression of a laser pulse [6, 7]. This is fundamentally a dynamical and
strongly nonlinear process: anomalous group-velocity dispersion continually
compensates the positive chirp induced by the optical Kerr effect, compressing
the pump pulse as it propagates and increasing the intensity, which leads to
more rapid spectral broadening and, ultimately, the formation of sub-cycle
pulses (see Fig. 1) [4, 5]. This feedback loop lies at the heart of the unique
capabilites offered by RDW emission—most notably, wavelength tuneability and
the generation of pulses significantly shorter than the pump pulse. However,
it may also make the process more sensitive to changes in the pump pulse than,
for instance, simple spectral broadening in the normal-dispersion regime [8].

The noise properties of soliton effects have been studied in detail in the
context of supercontinuum generation in solid-core fibres, using both
numerical and experimental methods and considering practical as well as
fundamental limitations [9, 10, 11, 12, 13]. Experimental studies of soliton
self-compression in gas-filled hollow-core waveguides have found that both the
pulse energy and absolute phase of self-compressed few-cycle pulses can be
stabilised to the same degree as in normal-dispersion systems [14, 15].
Ultraviolet RDW emission has also been investigated purely in terms of the
spectral intensity noise, which was found to be very low for correctly chosen
generation parameters [4] but depend strongly on the pulse-energy noise of the
driving laser [16, 17]. However, these works investigate only one or at most a
few different combinations of generation parameters (gas pressure, nominal
pump energy and duration), so that the effect of changing, for instance, the
RDW wavelength remains unknown.

Since it enables the efficient generation of few-femtosecond pulses across the
vacuum and deep ultraviolet, where many chemical compounds have absorption
resonances, RDW emission holds particular promise in the field of ultrafast
spectroscopy. Experiments in this field require a high degree of timing
stability between two or more pulses used to excite, control and probe
ultrafast dynamics in a sample. In this respect, the dynamical nature of the
RDW emission process poses a challenge. In contrast to conventional frequency-
conversion schemes (for instance harmonic generation), the dispersive wave is
not automatically locked to the pump pulse. Instead, it is generated around
the point of maximal self-compression—the location of which depends
sensitively on the pump pulse—and then propagates under the influence of both
linear dispersion and further nonlinear interactions with the soliton. The
variable propagation length in combination with a difference in group velocity
means that the arrival time of the RDW pulse at the waveguide exit is coupled
to the pump pulse energy and duration [see Fig. 1(c)].

Figure 1: Example simulation of soliton self-compression and RDW emission. (a)
Evolution of the pulse spectrum along the waveguide when pumping a helium-
filled hollow-capillary fibre with $125\text{\,}\mathrm{\SIUnitSymbolMicro m}$
core radius and $1\text{\,}\mathrm{m}$ length with $7.5\text{\,}\mathrm{fs}$
pulses containing $225\text{\,}\mathrm{\SIUnitSymbolMicro J}$ of energy. The
labels point out the spectral bands of RDW emission and third-harmonic
generation (THG). (b) Corresponding evolution of the pulse envelope in time.
The white dashed line indicates the calculated point of maximal self-
compression (see section III.2). (c) Pulse profile of the RDW at the end of
the waveguide, filtered with $15\text{\,}\mathrm{\char 37\relax}$ relative
bandwidth around $180\text{\,}\mathrm{nm}$, for two different pump energies.

Here we use numerical simulations to study the noise properties of RDW
emission for a range of generation conditions, considering fluctuations in the
generated energy, the arrival time, and the central wavelength. We find that
for low pump energy, pump pulse noise is strongly amplified. However, at
higher energy, where the generation is saturated, the energy noise of the RDW
pulse can be significantly lower than that of the pump pulse. This effect is
consistent across different gas pressures (corresponding to different RDW
wavelengths) and for both a constant gas pressure and a longitudinal pressure
gradient, which can be used for direct delivery of compressed RDW pulses to a
vacuum system [18, 19]. We further find that, even for a conservative estimate
of the pump pulse noise, the arrival-time jitter of the RDW pulse remains well
below 1 femtosecond. Higher gas pressures (longer RDW wavelengths) generally
perform better in terms of timing jitter, but photoionisation dynamics lead to
exceptional stability for very low pressure. Comparison between energy-scaled
systems reveals that the energy and timing noise characteristics we identify
here apply to RDW emission in both small-core microstructured fibres and
large-core hollow-capillary fibres.

##  II Numerical model

###  II.1 Propagation equation

To accurately simulate the nonlinear propagation of laser pulses through a
gas-filled hollow-core waveguide, we use a carrier-resolved multi-mode model
which includes the effect of gas and waveguide dispersion, confinement loss,
the optical Kerr effect, and photoionisation and plasma dynamics. This model
achieves quantitative agreement with experimental results without any fitting
parameters [4]. We numerically solve the multi-mode unidirectional pulse
propagation equation [20, 4, 21],

$\partial_{z}E_{m}(\omega,z)=\left(i\beta_{m}(\omega,z)-i\frac{\omega}{v(z)}-\frac{1}{2}\alpha_{m}(\omega,z)\right)E_{m}(\omega,z)+i\frac{\omega}{4}P_{m}^{\mathrm{nl}}(\omega,z)\,,$
(1)

where $E_{m}(\omega,z)$ is the modal electric field in mode $m$ as a function
of angular frequency $\omega$ and propagation distance $z$,
$\alpha_{m}(\omega,z)$ and $\beta_{m}(\omega,z)$ are the frequency-dependent
attenuation and propagation constants of mode $m$, respectively, $v(z)$ is the
velocity of the reference frame (chosen as the group velocity at the central
wavelength of the pump pulse in the fundamental mode), and
$P_{m}^{\mathrm{nl}}(\omega,z)$ is the nonlinear polarisation. We consider
propagation in the radially symmetric and linearly polarised hybrid modes of
the capillary, HE1m, for which the attenuation and propagation constants are
obtained from the capillary model [22] as

$\beta_{m}(\omega,z)=\frac{\omega}{c}\sqrt{n_{\mathrm{gas}}^{2}(\omega,z)-\frac{c^{2}u_{1m}^{2}}{a^{2}\omega^{2}}}\,,\qquad\alpha_{m}(\omega,z)=\frac{c^{2}u_{1m}^{2}}{a^{3}\omega^{2}}\frac{\nu(\omega,z)^{2}+1}{\sqrt{\nu(\omega,z)^{2}-1}}\,,$
(2)

where $c$ is the speed of light in vacuum, $u_{1m}$ is the $m^{\mathrm{th}}$
zero of the Bessel function of the first kind $J_{0}$, $a$ is the core radius
of the capillary, $n_{\mathrm{gas}}(\omega,z)$ is the refractive index of the
filling gas, and $\nu(\omega,z)$ is the ratio of the refractive index of the
cladding and $n_{\mathrm{gas}}$,
$\nu(\omega,z)=n_{\mathrm{clad}}(\omega,z)/n_{\mathrm{gas}}(\omega,z)$.
$n_{\mathrm{gas}}(\omega,z)$ is calculated by scaling the refractive index,
obtained from Sellmeier expansions [3, 23], to the correct pressure using the
pressure-dependent number density $\rho(z)$ [24]. For the case of a decreasing
pressure gradient for direct coupling to vacuum, the pressure along the
waveguide $p(z)$ is described by

$p(z)=p_{0}\sqrt{1-\frac{z}{L}}\,,$ (3)

where $p_{0}$ is the fill pressure and $L$ is the length of the waveguide.

The modal nonlinear polarisation $P_{m}^{\mathrm{nl}}(\omega,z)$ is obtained
from the real-space polarisation $P^{\mathrm{nl}}\\!\left(t,r,\theta,z\right)$
by projecting onto the normalised modal fields $\hat{e}_{m}$ and taking the
Fourier transform,

$P_{m}^{\mathrm{nl}}(\omega,z)=\int_{-\infty}^{+\infty}\\!\\!\mathrm{d}\omega\int_{0}^{2\pi\\!\\!}\mathrm{d}\theta\int_{0}^{a}\\!\\!r\mathrm{d}r\,\hat{e}_{m}^{*}(r,\theta)\cdot
P^{\mathrm{nl}}(t,r,\theta,z)\mathrm{e}^{i\omega t}\,.$ (4)

To calculate the real-space polarisation, in turn, the real-space field
$E\\!\left(t,r,\theta,z\right)$ is constructed from the modal fields
$E_{m}(\omega,z)$,

$E\\!\left(t,r,\theta,z\right)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\\!\\!\mathrm{d}\omega\,\mathrm{e}^{-i\omega
t}\sum_{m}\hat{e}_{m}(r,\theta)E_{m}(\omega,z)\,.$ (5)

The real-space polarisation consists of two parts capturing the effects of
third-order (Kerr) nonlinearity and photoionisation and plasma, respectively:

$P^{\mathrm{nl}}(t,r,\theta,z)=P^{\mathrm{Kerr}}\\!\left(t,r,\theta,z\right)+P^{\mathrm{ion}}\\!\left(t,r,\theta,z\right)\,.$
(6)

The Kerr term is given by

$P^{\mathrm{Kerr}}\\!\left(t,r,\theta,z\right)=\rho(z)\epsilon_{0}\gamma^{(3)}E^{3}\\!\left(t,r,\theta,z\right)\,,$
(7)

where $\epsilon_{0}$ is the permittivity of free space and $\gamma^{(3)}$ is
the third-order hyperpolarisability of the gas, calculated from reference
values for the third-order susceptibility [25]. To suppress third-harmonic
generation, the Kerr term can be altered to

$P^{\mathrm{Kerr}}\\!\left(t,r,\theta,z\right)=\frac{3}{4}\rho(z)\epsilon_{0}\gamma^{(3)}\left|\hat{E}\\!\left(t,r,\theta,z\right)\right|^{2}E\\!\left(t,r,\theta,z\right)\,,$
(8)

where $\hat{E}$ is the analytic representation of the field, obtained from
$\hat{E}=E+i\mathcal{H}\left[E\right]$ with $\mathcal{H}$ denoting the Hilbert
transform. The photoionisation term is given by [26]

$P^{\mathrm{ion}}\\!\left(t,r,\theta,z\right)=I_{p}\int_{-\infty}^{t}\\!\\!\mathrm{d}t^{\prime}\frac{\partial_{t^{\prime}}\rho_{\mathrm{e}}(t^{\prime},r,\theta,z)}{E\\!\left(t^{\prime},r,\theta,z\right)}+\frac{e^{2}}{m_{\mathrm{e}}}\int_{-\infty}^{t}\\!\\!\mathrm{d}t^{\prime}\int_{-\infty}^{t^{\prime}}\\!\\!\mathrm{d}t^{\prime\prime}\rho_{\mathrm{e}}(t^{\prime\prime},r,\theta,z)E\left(t^{\prime\prime},r,\theta,z\right)\,,$
(9)

where $I_{p}$ is the ionisation potential of the gas,
$\rho_{\mathrm{e}}(t,r,\theta,z)$ is the density of free electrons, $e$ is the
elementary charge, and $m_{\mathrm{e}}$ is the mass of the electron.
$\rho_{\mathrm{e}}(t,r,\theta,z)$ is calculated from

$\rho_{\mathrm{e}}(t,r,\theta,z)=\rho(z)\left(1-\exp\left\\{-\int_{-\infty}^{t}\\!\\!\mathrm{d}t^{\prime}w\\!\left(\left|E\\!\left(t^{\prime},r,\theta,z\right)\right|\right)\right\\}\right)\,,$
(10)

where $w(E)$ is the ionisation rate after Perelomov _et al._ [27].

We solve eq. 1 with the Dormand-Prince method (an adaptive 4(5)th-order Runge-
Kutta method) [28] after transforming the propagation equation into the
interaction picture [29]. The modal projection in eq. 4 is carried out using a
p-adaptive cubature method to integrate over the core of the capillary. We
consider the first four hybrid modes of the capillary (HE11 to HE14). The grid
used in the simulations extends from $100\text{\,}\mathrm{nm}$ to
$3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in wavelength with a time window of
$450\text{\,}\mathrm{fs}$, resulting in 4096 samples. To accurately calculate
the nonlinear polarisation including third-harmonic generation and the plasma
response, the field is resampled onto a finer grid with a time resolution of
$55\text{\,}\mathrm{as}$. Convergence tests show that this is sufficient
resolution for all parameters we consider here.

###  II.2 Statistical sampling

The most direct way of modelling the noise properties of RDW emission is to
simply repeatedly simulate the process while randomly choosing input
parameters according to a given model of the pump pulse noise. This approach
is commonly used to analyse the coherence properties of supercontinuum
generation [10] and has previously been applied to RDW emission in gas-filled
hollow-core waveguides to investigate the effects of quantum noise [2] and
pump pulse energy fluctuations [16] for a fixed combination of waveguide and
nominal pump pulse parameters. However, for a more comprehensive study of the
influence of various experimental parameters on the RDW noise using an
accurate multi-mode model, the very large number of individual propagation
simulations required makes the computational cost of this method prohibitive.

To reduce the computational cost, here we take a different approach based on
re-sampling detailed parameter scans. Instead of running full simulations for
randomly chosen input parameters, we run a fixed set of simulations with small
increments in those same parameters [see Fig. 2(a)]. We then extract the
output quantities of interest—for instance, the energy of the RDW pulse—and
create an interpolant for each quantity. This allows us to analyse the output
noise by randomly choosing parameter values at which to sample the interpolant
[see Fig. 2(b)], which is several orders of magnitude faster than running a
full simulation.

Assuming that environmental conditions (temperature, laboratory air pressure
and humidity) and waveguide parameters (length, transverse dimensions, gas
fill pressure and species) only vary on timescales much longer than an
experiment, the two most important sources of noise are instabilities in the
energy and duration of the pump pulse. However, two additional instabilities
have to be taken into account. Firstly, in most pump laser systems the
absolute phase, or carrier-envelope phase (CEP), of the pump pulse varies
completely randomly from shot to shot. Secondly, quantum shot noise is
unavoidable; since soliton self-compression is intimately related to
modulational instability, which is seeded from such noise [30], it is
important to assess its influence.

Figure 2: (a) Simulated output spectra for a $1\text{\,}\mathrm{m}$ long HCF
with $125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core radius at a helium
pressure of $2.1\text{\,}\mathrm{bar}$ pumped with $7.5\text{\,}\mathrm{fs}$
pulses centred at $800\text{\,}\mathrm{nm}$. (b) Illustrative example of the
resampling approach to extracting noise characteristics: an interpolant of the
RDW energy as a function of pump energy is resampled at random points with a
standard deviation of $2\text{\,}\mathrm{\char 37\relax}$ around
$150\text{\,}\mathrm{\SIUnitSymbolMicro J}$ (bottom histogram), creating a
corresponding spread in the RDW energy (left histogram). (c, d) Comparison
between directly sampled and resampled methods to model RDW emission noise for
(c) total energy and (d) arrival time of the RDW pulse when and using a pump
energy standard deviation of $2\text{\,}\mathrm{\char 37\relax}$. Direct
sampling uses 500 independent propagation simulations for each mean pump
energy whereas resampling uses 10000 samples from interpolated data.

We validate our resampling approach by comparing it to the direct method for a
representative set of parameters. Figure 2(c) and (d) show the results of this
comparison for the total energy and arrival time of the dispersive wave
generated in a $1\text{\,}\mathrm{m}$ long HCF with
$125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core radius at
$2.1\text{\,}\mathrm{bar}$ of helium pressure when pumping with transform-
limited $7.5\text{\,}\mathrm{fs}$ Gaussian pulses centred at
$800\text{\,}\mathrm{nm}$. For each mean energy, the direct approach uses 500
independent simulations with an input energy that is randomly chosen from a
Gaussian distribution with $2\text{\,}\mathrm{\char 37\relax}$ standard
deviation around the respective mean. These simulations also include uniformly
random CEP and the effect of quantum shot noise using the one-photon-per-mode
model [10]. We choose a fixed spectral window to capture the whole spectral
band of the RDW; for the results in Fig. 2 the window spans
$185\text{\,}\mathrm{nm}265\text{\,}\mathrm{nm}$. The RDW energy is extracted
by integrating the filtered spectral energy density and summing over the
waveguide modes, and the arrival time is extracted by transforming the
filtered frequency-domain field to the time domain and calculating the first
moment (centre of mass) of the resulting pulse. In our resampling approach, we
run simulations with 800 uniformly spaced pump energies ranging from
$80\text{\,}\mathrm{\SIUnitSymbolMicro
J}220\text{\,}\mathrm{\SIUnitSymbolMicro J}$ (extending past the range of
interest shown in Fig. 2(c) and (d) to avoid edge artefacts) with fixed CEP
and no shot noise. We then apply the same analysis to extract the RDW energy
and arrival time and resample interpolants of the resulting data for a range
of mean pump energy values from $110\text{\,}\mathrm{\SIUnitSymbolMicro
J}210\text{\,}\mathrm{\SIUnitSymbolMicro J}$ using the same standard deviation
of $2\text{\,}\mathrm{\char 37\relax}$.

The excellent agreement between the two methods demonstrates that the
resampling approach can be used to analyse the noise properties of RDW
emission. The small discrepancy in the timing jitter at pump energies below
$100\text{\,}\mathrm{\SIUnitSymbolMicro J}$ is most likely due to CEP-
dependent interference between the RDW and the third harmonic (see section
III.1). The simulations for the resampling approach took 29 CPU hours as
compared to over 1600 hours for the direct method, making the study of wide
parameter ranges feasible.

Importantly, the agreement also shows that CEP fluctuations and quantum shot
noise do not significantly influence the results in the parameter range we
consider here. If this were not the case, the resampling approach would be
invalid, since quantum noise in particular cannot be modelled in this way.
That quantum noise has no effect is to be expected from previous numerical
studies of RDW emission, which found that the process is fully coherent
(insensitive to quantum noise) for typical parameters [2]. In this context, it
is important to note the difference between the coherence of any given real-
world source and the generation process itself. In particular, a frequency-
conversion or supercontinuum _source_ can be incoherent—that is, exhibit
random shot-to-shot phase fluctuations—due to noise in the pump source, even
if the _underlying process_ is fully coherent. In fact, because the majority
of laser systems exhibit random CEP, most practical sources are incoherent in
the strictest sense. The key distinction is that while pump-source noise can
be mitigated by technical means, incoherence due to quantum noise is a
fundamental limitation inherent to the process.

The use of a fixed frequency window makes the above comparison
straightforward, but to capture the whole band of the RDW at all pump
energies, it has to be more broadband than can realistically be achieved with
filtering optics. For the following analysis, we instead use a variable window
function with a fixed relative bandwidth of $15\text{\,}\mathrm{\char
37\relax}$. For each mean energy (each point on the horizontal axis in Fig. 2)
we find the spectral location of the RDW by calculating the first moment of
the square of the spectral energy density in the ultraviolet. We then create a
separate interpolant for each mean energy with windows centred on these
wavelengths. This approach emulates the limitations of existing filtering
optics while also taking into account the energy dependence of the RDW
wavelength [1], and thus avoids artefacts due to the RDW shifting into and out
of the filter pass-band [16, 17]. Since the generation parameters are usually
chosen such that the RDW spectrum fits within the pass-band of the optics, it
also mirrors experimental reality.

##  III Results and Discussion

###  III.1 Energy noise amplification

Figure 3: (a, b) Energy noise amplification of RDW emission as a function of
mean pump energy in a $1\text{\,}\mathrm{m}$ long HCF with
$125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core radius filled with different
pressures of helium at constant pressure (a) or with a decreasing gradient (b)
and driven with $7.5\text{\,}\mathrm{fs}$ pulses at $800\text{\,}\mathrm{nm}$.
Horizontal dashed lines indicate a noise amplification of $1$, that is, the
same noise in the pump and the dispersive wave. The vertical dashed line in
(b) indicates the energy for which direct simulations are shown in Fig. 4. (c,
d) RDW pulses at the mean energy at which the lowest energy noise is observed,
filtered with $15\text{\,}\mathrm{\char 37\relax}$ relative bandwidth.

Figure 3(a) and (b) show the energy noise amplification of RDW emission—the
standard deviation of the RDW energy, $\sigma^{\mathrm{(dw)}}_{E}$, normalised
by the that of the pump, $\sigma^{\mathrm{(pump)}}_{E}$. The pump energy
standard deviation is chosen as $2\text{\,}\mathrm{\char 37\relax}$, however
the noise amplification is insensitive to this choice even for values up to
$10\text{\,}\mathrm{\char 37\relax}$. The gas pressures are chosen so as to
generate a dispersive wave at a wide range of wavelengths from
$\sim\\!$140\text{\,}\mathrm{nm}$$ to $\sim\\!$320\text{\,}\mathrm{nm}$$. For
the decreasing gradient, the pressures are increased by a factor of $3/2$
[19]. The maximum energy is chosen so as to avoid excessive photoionisation
effects at the lowest pressure we consider here. To compare results with
different gas pressures, the pump energy is scaled by the pressure and
normalised to this maximum energy. In this way, a particular value of the
normalised energy corresponds to a fixed pressure-energy product. The energy
range shown in Fig. 3 spans from the approximate normalised energy at which
RDW emission begins to appear (0.45) to just below the maximum (1.0) to avoid
edge effects. For $0.8\text{\,}\mathrm{bar}$ static pressure this range is
$260\text{\,}\mathrm{\SIUnitSymbolMicro
J}555\text{\,}\mathrm{\SIUnitSymbolMicro J}$, whereas for
$4\text{\,}\mathrm{bar}$ it is $52\text{\,}\mathrm{\SIUnitSymbolMicro
J}111\text{\,}\mathrm{\SIUnitSymbolMicro J}$. Note that these ranges cannot be
arbitrarily scaled to any other system; they depend on several other
parameters, including the waveguide dimensions, the gas species, and the pump
pulse duration and wavelength.

The first important result is that the energy of the RDW generally becomes
more stable as the mean energy is increased. The second is that, surprisingly,
the energy noise of the RDW can be _lower_ than that of the pump pulse. In the
extreme case in our data, the standard deviation in the RDW energy is a factor
of 25 lower than that in the pump pulse energy. This goes counter to the
conventional assumption that nonlinear optical processes amplify noise. The
underlying physical reasons for this dramatic improvement in noise performance
are not completely clear from our simulations. However, it is likely that
saturation effects play an important role: at low pump energy, the pump pulse
maximally self-compresses at or just before the end of the waveguide, and the
rapid transfer of energy to the RDW band—which occurs around the self-
compression point—is interrupted by the pulse exiting the waveguide. A small
change in pump energy leads to earlier or later self-compression and thus has
a large effect on how much energy is converted before the waveguide end. This
amplifies the noise in the pump pulse. At high pump energy, the pump pulse
maximally self-compresses well before the end of the waveguide, and conversion
to the RDW band has ceased by the time the pulse reaches the end, making the
process less sensitive. In addition, self-compression with more energy leads
to more structure in the pulse at the point of RDW emission, which reduces the
conversion efficiency [31, 32]. This may act as a second saturation mechanism
leading to lower noise. Figure 3(c) and (d) show the temporal profiles of the
filtered RDW pulse at the point of lowest energy noise in Fig. 3(a) and (b),
respectively. Their short duration shows that, to obtain good noise
performance, RDW emission does not have to be driven so deep into saturation
that dispersive effects distort the generated pulse.

The effect is reversed when the energy is increased beyond a certain point and
the energy noise rises again. This is likely due to a combination of factors,
including the increased effect of (highly nonlinear) photoionisation and
plasma dynamics as well as coupling into higher-order modes. This is
particularly clear in the data at the lowest pressure in the constant-pressure
case, where the noise rises dramatically after the minimum.

The results with constant-pressure fill and a decreasing gradient are broadly
similar, but the use of a gradient presents several advantages: firstly, the
best energy noise achieved is lower than in the constant-pressure case for
most gas pressures. Secondly, the energy noise is low for a larger range of
pump energies, making such a system more flexible. Finally, the optimal energy
noise is generally obtained at higher energy, which means that a gradient-
filled system can generate more RDW energy before nonlinear instabilities
degrade the noise performance. This is reflected in the higher peak power of
the pulses in Fig. 3(d).

Because the wavelength of RDW emission depends on the pump pulse energy [1],
the noise characteristics depend on the choice of spectral filter function. In
the extreme case, placing the window on the edge of the RDW band results in
very large relative energy noise as the spectral wings of the pulse shift into
and out of the filter pass-band, even if the energy and spectrum of the RDW
are stable [16]. Experimentally, this is equivalent to detuning the RDW
wavelength with respect to the filtering optics used. In our simulations we
always find an energy at which the RDW energy noise is comparable to or better
than the pump noise, irrespective of the chosen filtering bandwidth, as long
as the window is centred on the spectral band of the RDW.

Figure 4: (a) Overlay of 50 individually simulated RDW spectra in a decreasing
gradient from $2.2\text{\,}\mathrm{bar}$ to vacuum with a nominal pump pulse
energy of $240\text{\,}\mathrm{\SIUnitSymbolMicro J}$,
$2\text{\,}\mathrm{\char 37\relax}$ energy noise, and including uniformly
random CEP as well as quantum noise. (b) Same as (a) but with fixed CEP. (c)
Same as (a) but with third-harmonic generation (THG) removed from the Kerr
term in the nonlinear polarisation. (d) Temporal profile of the RDW pulse for
the spectra shown in (a).

To verify that the unexpectedly low noise of the RDW energy is not an artefact
of our analysis, in Fig. 4 we show simulations using the direct approach
discussed above. Here we overlay 50 shots with low-opacity lines to build a
direct picture of the stability. The parameters are a gradient with
$2.2\text{\,}\mathrm{bar}$ fill pressure and a nominal pump energy of
$240\text{\,}\mathrm{\SIUnitSymbolMicro J}$, which corresponds to the centre
of the low-noise region around a normalised energy of 0.75 [see vertical
dashed line in Fig. 3(b)]. For the complete model [Fig. 4(a)], the spectrum
appears very unstable, which would seem to contradict our earlier findings.
However, the total energy noise is significantly lower than the fluctuations
in spectral energy density. Closer inspection shows that the spectrum contains
interference between the RDW and the third harmonic of the self-compressed
sub-cycle pulse. This acts as an $f$-$3f$ interferometer encoding the CEP in
the phase of the fringes [33]. Since the CEP is random in these simulations,
this explains the strong instability. The same set of simulations with fixed
CEP [Fig. 4(b)] yields much lower noise in the spectral energy density, though
it is still higher than would be expected from the extremely low overall
energy noise. This is still due to third-harmonic interference, but with the
phase shift from shot to shot coming purely from the different nonlinear
phase. Turning off third-harmonic generation but including random CEP [Fig.
4(c)] demonstrates this by drastically reducing the noise. Importantly, even
with all effects included, the RDW pulse in time is stable [see Fig. 4(d)].
This is because the peak power of the third harmonic is very low, so its
influence on the time-domain pulse is weak. Note that at low pump energy, when
the RDW is weak, the third harmonic can more strongly influence the shape of
the filtered pulse; this is likely the cause of the small discrepancy in the
timing jitter observed between the two methods compared in Fig. 2.

This representative example points to a subtlety in the analysis of source
noise in RDW emission: the most appropriate measure of the noise depends
critically on the intended application. For instance, despite the large
instability in the spectral energy density in Fig. 4(a), the RDW pulse in the
time domain is stable, and so a source with these noise characteristics is
perfectly usable for most ultrafast experiments. If, on the other hand, the
source is used directly for spectroscopy, the shot-to-shot fluctuations in the
fringe phase need to be removed by averaging. An assessment purely in terms of
the commonly used spectral relative intensity noise (RIN)—the standard
deviation in spectral energy density normalised to its mean value—obscures
this distinction [16]. Furthermore, the large difference between Fig. 4(a) and
(b) indicates that numerical simulations of the spectral RIN which ignore the
effect of random CEP are incomplete.

###  III.2 Timing jitter

Figure 5: (a, b) Timing jitter of RDW emission as a function of mean pump
energy in a $1\text{\,}\mathrm{m}$ long HCF with
$125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core radius filled with different
pressures of helium at constant pressure (a) or with a decreasing gradient (b)
and driven with $7.5\text{\,}\mathrm{fs}$ pulses at $800\text{\,}\mathrm{nm}$.
The lighter purple lines show the results for $0.8\text{\,}\mathrm{bar}$ and
$1.2\text{\,}\mathrm{bar}$, respectively, when photoionisation and plasma
dynamics are ignored in the simulations. (c) Group velocity difference $\Delta
v_{\mathrm{g}}$ with respect to $800\text{\,}\mathrm{nm}$ for the same
pressures as in (a). The circles mark the value of $\Delta v_{\mathrm{g}}$ for
the approximate wavelength of RDW emission as extracted from the simulations,
with the lightened dot and dashed grey line indicating the result without
plasma as for the light purple lines in (a) and (b). The dashed purple line
shows $\Delta v_{\mathrm{g}}$ for $0.8\text{\,}\mathrm{bar}$ pressure and a
free-electron density of ${10}^{16}\text{\,}{\mathrm{cm}}^{-3}$.

Figure 5(a) and (b) show the arrival-time jitter of the RDW pulse for the same
set of parameters considered in Fig. 3. The most important result here is that
the jitter is generally very small compared to the pulse duration, remaining
below $300\text{\,}\mathrm{as}$ for all parameters—well within the
requirements for ultrafast experiments with few-femtosecond resolution. In
contrast to the energy noise amplification, the modelled arrival-time jitter
depends on the chosen pump energy noise. We find that it is simply
proportional over the entire range of energy and pressure considered here, so
that halving the pump energy noise also halves the amount of timing jitter.
There are two overall trends: with the exception of the lowest pressure,
higher pressures (longer RDW wavelengths) generally perform better, and there
is a pronounced maximum in the timing jitter as a function of pump energy.

The first of these trends is somewhat counterintuitive. As the lines in Fig.
5(c) show, higher pressures lead to stronger higher-order dispersion and hence
a generally larger group-velocity difference with respect to the pump pulse.
However, the longer RDW wavelength resulting from these higher pressures more
than compensates for this effect, as demonstrated by the circles plotted in
Fig. 5(c). Despite being larger in general, the group-velocity difference at
the RDW wavelength is reduced. The VUV RDW generated with
$0.8\text{\,}\mathrm{bar}$ pressure breaks this trend, with the timing jitter
at all pump energies substantially lower than for other gas pressures. The
reason for this lies in the strong influence of plasma dynamics resulting from
the high pump pulse energy required at low pressure. Photoionisation by the
self-compressing pulse leads to soliton self-frequency blue-shift [34, 35,
36]. Because the pump and RDW wavelengths are inversely related [7], this
shifts the RDW to longer wavelengths, where the group-velocity difference is
reduced [see purple circle in Fig. 5(c)]. By switching off photoionisation in
the simulations, we can restore the trend observed for higher pressures, as
shown by the lighter purple lines in Fig. 5(a) and (b) and the lighter purple
circle in Fig. 5(c). This effect alone cannot account for the dramatically
lower timing jitter, however, since it only reduces the group-velocity
difference to the same level as for $3\text{\,}\mathrm{bar}$ pressure. It is
aided by the presence of free electrons. Since the plasma dispersion is
anomalous, it accelerates the VUV RDW and reduces the group-velocity
difference. For an electron density of ${10}^{16}\text{\,}{\mathrm{cm}}^{-3}$
(an ionisation fraction of $0.05\text{\,}\mathrm{\char 37\relax}$ at
$0.8\text{\,}\mathrm{bar}$), the group-velocity difference for the VUV RDW is
near zero [see open purple circle in Fig. 5(c)]. Free electrons are only
created when the intensity is very high, so that they are localised around
self-compression points. However, re-compression of the blue-shifted soliton
can maintain high intensity even after the RDW is generated [35]. In our
simulations at $0.8\text{\,}\mathrm{bar}$ pressure, the electron density
remains around ${10}^{16}\text{\,}{\mathrm{cm}}^{-3}$ even after the first
self-compression point for all energies at which an energetic RDW pulse is
created.

Figure 6: (a) RDW timing jitter for the same set of parameters as in Fig. 3(a)
and Fig. 5(a) as calculated using the simple model eq. 11. (b) Jitter in the
central wavelength of the RDW for the data in (a).

To better understand the change in timing jitter with pump energy, we consider
a simple model for the delay $\tau$ induced by the difference in group
velocity of the pump pulse and the RDW pulse,

$\tau=L_{\mathrm{prop}}\left[\beta_{1}(\omega)-\beta_{1}(\omega_{0})\right]=L_{\mathrm{prop}}\Delta\,,$
(11)

where $L_{\mathrm{prop}}$ is the propagation length of the RDW pulse after the
generation point, $\beta_{1}(\omega)=\partial_{\omega}\beta(\omega)$ is the
inverse of the group velocity, and $\omega$ and $\omega_{0}$ are the central
frequencies of the RDW and pump pulses, respectively. The propagation length
can be approximated by $L_{\mathrm{prop}}=L-L_{\mathrm{f}}$, where $L$ is the
waveguide length and $L_{\mathrm{f}}$ is the fission length—the propagation
length after which the pump pulse self-compresses and the RDW is generated.
The latter is given by

$L_{\mathrm{f}}=\sqrt{\frac{T_{0}^{2}}{\gamma\absolutevalue{\beta_{2}(\omega_{0})}P_{0}}}\,,$
(12)

where $T_{0}$ and $P_{0}$ are the pump pulse duration and peak power (assuming
a $\sech^{2}$ profile), $\gamma$ is the nonlinear coefficient of the gas-
filled waveguide, and $\beta_{2}(\omega)=\partial^{2}_{\omega}\beta(\omega)$
is the group-velocity dispersion [32]. The pump energy enters the two factors
in eq. 11 in different ways: $L_{\mathrm{prop}}$ is longer for higher energy
as self-compression happens more quickly ($L_{\mathrm{f}}$ decreases), whereas
the energy-dependent wavelength of the RDW affects the group-velocity
difference encoded in $\Delta$. We estimate the jitter in the arrival time by
calculating the delay $\tau$ using eq. 11 for 10000 randomly chosen pump
energies around each of the mean energy values for which we show the simulated
jitter in Fig. 5, while finding the central wavelength of the RDW in the same
manner as for our choice of window function.

Figure 6(a) shows the predictions for the timing jitter from this simple
model. The model closely reproduces the magnitude of the timing jitter, which
indicates that the group-velocity difference in combination with a changing
fission length is indeed the dominant factor. The general trend of longer-
wavelength RDWs at higher pressures being more stable is also captured.
Importantly, while the jitter is generally small for the lowest pressure, the
simple model does _not_ reproduce the extent to which the jitter is
suppressed, especially at higher energy. This is because only the refractive
index of the neutral gas is considered in eq. 11; the effect of the plasma is
not included. Figure 6(b) shows the jitter in the central wavelength of the
RDW. This is extracted from the simulations in an analogous manner to the
energy and timing, using the first moment of the filtered spectrum. The model
results reveal that the central-wavelength jitter is the origin of the
pronounced maximum observed in Fig. 5(a) and (b). For certain energy ranges,
the RDW very rapidly blue-shifts with increasing energy, and the group-
velocity difference increases correspondingly. Note that the maxima in
central-wavelength and timing jitter appear close to the minima in overall
energy noise [Fig. 3(a) and (b)]. Thus it may be preferable to tune the
generation parameters slightly away from this region for the best combination
of stable pulse energy and timing.

###  III.3 Effect of the pump pulse duration

Figure 7: Energy (a) and timing (b) noise for the same conditions as shown in
Figs. 3 and 5 but assuming $1\text{\,}\mathrm{\char 37\relax}$ fluctuations in
the pump pulse duration. The solid lines show results for a gradient from
$3.2\text{\,}\mathrm{bar}$ to vacuum and the dashed lines for a constant
helium pressure of $2.1\text{\,}\mathrm{bar}$.

As exemplified by the expression for the fission length in eq. 12, the
dynamics of soliton self-compression depend on the pump pulse duration as well
as the energy. The extent to which the duration of femtosecond laser pulses
fluctuates varies greatly with the generation method, as does the degree of
correlation with the pump pulse energy. For instance, in nonlinear pulse
compression, the output bandwidth is coupled to the input energy, but whether
the output duration increases or decreases with larger bandwidth depends on
the details of the subsequent dispersion compensation. Similarly, the duration
of laser pulses can change due to fluctuations in bandwidth, the spectral
phase, or both, again depending on the details of the generation mechanism. We
therefore treat the pulse duration fluctuations independently and only
consider a change in the duration of a perfectly compressed input pulse (that
is, changes in bandwidth but not spectral phase).

Figure 7 shows the energy and timing noise induced by a
$1\text{\,}\mathrm{\char 37\relax}$ fluctuation in the pump pulse duration for
one of the sets of parameters shown before. The behaviour with changing pump
energy is strikingly similar to that observed when the energy fluctuates
instead, particularly for the RDW energy noise. At low energies, the RDW
energy fluctuates severely, but as the pump energy is increased and RDW
generation saturates, the energy noise improves greatly; for the case of a
pressure gradient, it falls close to zero.

The induced timing jitter is of a similar magnitude to that shown in Fig. 5,
despite the smaller relative pump noise in our model ($1\text{\,}\mathrm{\char
37\relax}$ in duration as compared to $2\text{\,}\mathrm{\char 37\relax}$ in
energy). This is because the fission length depends more sensitively on the
duration than on the energy—taking into account that the peak power $P_{0}$ is
proportional to $E_{p}/T_{0}$, where $E_{p}$ is the pump energy, the fission
length scales as $T_{0}^{3/2}$ and $E_{p}^{-1/2}$, respectively, when the
waveguide parameters $\gamma$ and $\beta_{2}$ are fixed. Here, however, the
different possible relationships between pump duration and energy become
important: for a pump source in which $E_{p}$ and $T_{0}$ are anti-correlated
(that is, in which higher pump energy corresponds to shorter pump pulses),
these two coupled noise sources will result in larger energy and timing noise
in the dispersive wave. In the opposite case, the effect of an increase in
pump energy is partly counteracted by a corresponding increase in pump pulse
duration, resulting in lower overall noise.

###  III.4 Overall energy scaling

Nonlinear optical effects in gases, including soliton self-compression and
resonant dispersive wave emission, can be arbitrarily scaled in energy by
appropriately scaling the gas density and the longitudinal as well as
transverse dimensions (here, waveguide length and core radius, respectively)
[37, 4, 38]. In this way, RDW emission with the same temporal and spectral
properties can in principle be obtained in both small-core microstructured
fibres [1, 32, 2, 3, 39] and large-core hollow capillary fibres [4, 38, 5]. In
Fig. 8 we compare the noise characteristics of these two regimes. The data for
the HCF with $125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ core radius is the
same as shown in Figs. 3 and 5. We simulate propagation in a hollow-core PCF
with a core radius of $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ using the
same model but assuming negligible waveguide loss. The core radius scaling by
a factor of 5 results in 25 times lower overall energy, shorter length and
higher density. (Note that the _pressure_ is not scaled by exactly 25 when
taking into account the full equation of state for helium [24].)

Figure 8: Effect of overall energy scaling on the (a, b) energy and (c, d)
arrival-time noise of RDW emission for both a constant pressure (left column)
and a decreasing gradient (right column).

The overall trend of the energy and arrival-time noise is very similar for
both the small and large core, but the curves do not agree exactly. This is
due to the difference in propagation loss—the transmission of the
$1\text{\,}\mathrm{m}$ HCF is $\sim\\!$87\text{\,}\mathrm{\char 37\relax}$$ at
$800\text{\,}\mathrm{nm}$—which the general scaling law does not account for.
Switching off the waveguide loss in the large-core simulations confirms this
hypothesis. As the green dashed line in Fig. 8 shows, without loss, the noise
characteristics are exactly the same despite the large difference in overall
energy and the physical size of the system.

##  IV Conclusions

In summary, we have investigated the energy and arrival-time noise of RDW
emission in gas-filled hollow waveguides for a range of generation conditions
and with a focus on applications in ultrafast science. There are two key
results from this study. The first is that, when saturated, RDW emission can
be as stable as or even more stable than the pump source driving the process,
while still generating ultrashort pulses. This is the case for a wide range of
RDW wavelengths, so that the full flexibility of RDW-based sources can be
exploited. The second key result is that even for a conservative estimate of
the pump energy noise, the timing jitter induced by the wavelength-dependent
group velocity and resulting pulse walk-off remains well below one
femtosecond. As many ultrafast laser systems, even those including nonlinear
compression, perform far better than the $2\text{\,}\mathrm{\char 37\relax}$
energy noise we have considered here, a timing jitter below
$100\text{\,}\mathrm{as}$ should be achievable in experiments. There are clear
wavelength-dependent trends in the timing jitter and strong plasma generation
can dramatically improve the stability by reducing the group-velocity
mismatch. Instabilities in the pump pulse duration have a similar effect to
those in pump energy, though with larger timing jitter for the same relative
magnitude of noise. Finally, by scaling the model to parameters similar to
experiments in anti-resonant hollow-core fibres, we have demonstrated that our
results are widely applicable in those systems as well. We expect that our
results will aid in the design and implementation of versatile, stable
ultrafast light sources based on RDW emission.

## Acknowledgments

This work was funded by the European Research Council under the European
Union’s Horizon 2020 Research and Innovation program: Starting Grant agreement
HISOL, No. 679649; Proof of Concept Grant agreement ULIGHT, No. 899900.

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