# Long-Term Variation of Helioseismic Far-Side Images and What Causes It

Junwei Zhao  Grace Y. Jiang  Ruizhu Chen  W. W. Hansen Experimental Physics
Laboratory, Stanford University, Stanford, CA 94305-4085, USA  Lynbrook High
School, San Jose, CA 95129, USA

###### keywords:

Sun: helioseismology; Sun: oscillations

##  1 Introduction

sec1

Seeing active regions (ARs) on the Sun’s far side is useful for monitoring
developments of solar magnetic activity and forecasting arrivals of large ARs
on the Sun’s earth-facing side, and local-helioseismic techniques have made
possible the detection of far-side ARs by using $p$-mode helioseismic waves.
Helioseismic holography first succeeded in detecting a large AR near the
center of the far side (Lindsey and Braun, 2000), and the method was later
further developed to include the entire far side (Braun and Lindsey, 2001) and
to enhance the signal-to-noise ratio (Lindsey and Braun, 2017). A
time–distance helioseismic technique, which utilized waves with three-, four-,
and five-skip waves, was also developed and validated using numerical
simulation (Zhao, 2007; Hartlep et al., 2008; Ilonidis, Zhao, and Hartlep,
2009). More recently, using Solar Dynamics Observatory / Helioseismic and
Magnetic Imager (SDO/HMI: Scherrer et al. 2012, Schou et al. 2012) full-disk
Doppler observations, Zhao et al. (2019) completely re-developed their
time–distance far-side imaging code and combined 14 measurement schemes that
utilized three-, four-, five-, six-, and eight-skip waves, substantially
enhancing the detection quality of the far-side ARs. The Sun’s helioseismic
far-side images were not just used to monitor the Sun’s magnetic activity, but
were also used to study the Sun’s long-term variations. Analyzing the
helioseismic-holography far-side images obtained using Solar and Heliospheric
Observatory/Michelson Doppler Imager (SOHO/MDI: Scherrer et al., 1995) data,
González Hernández, Scherrer, and Hill (2009) found that the mean helioseismic
phase-shifts, deficits of which indicate far-side ARs, varied substantially
with the phase of the solar cycle for Cycle 23. They attributed this long-term
phase-shift variation to the Sun’s seismic radius change with solar cycles.

In addition to the Sun’s photometric radius measured in visible light, the
Sun’s radius can also be measured through $f$-mode oscillations on its surface
(Schou et al., 1997). The radius measured this way is known as seismic radius
$\mathrm{R_{\odot}}$. Later studies found that $\mathrm{R_{\odot}}$ changed
with the phase of solar cycles, slightly larger during solar minima and
smaller during maxima with a variation of a few kilometers (Lefebvre and
Kosovichev, 2005; Kholikov and Hill, 2008). More recently, Kosovichev and
Rozelot (2018) analyzed 21 years of combined observations by SOHO/MDI and
SDO/HMI, and found that $\mathrm{R_{\odot}}$ reduces by $1-2$ km during the
activity maxima and the most significant radius change occurs at the depth of
$5\pm 2$ Mm. Despite a completely different approach, the long-term far-side
images that are primarily obtained from the low-degree deep-penetrating
$p$-mode waves also indicate a variation of $\mathrm{R_{\odot}}$ (González
Hernández, Scherrer, and Hill, 2009), which seems consistent with those
obtained by the global helioseismology analyses.

Having multiple sets of far-side images from different numbers of wave skips,
the newly-developed time–distance far-side imaging method (Zhao et al., 2019)
has an advantage over the helioseismic-holography images (González Hernández,
Scherrer, and Hill, 2009) in evaluating whether the measured acoustic phase-
shift (or travel-time shift for the time–distance helioseismic technique)
variation is due to the Sun’s $\mathrm{R_{\odot}}$ change or owing to other
factors. In this article, we examine the long-term variation of the travel-
time shifts in the time–distance far-side images measured from the SDO/HMI
observations during Solar Cycle 24, and investigate the physical causes of the
variation. We present our data-analysis procedure and results in Section 2,
discuss the results in Section 3, and give conclusions in Section 4.

##  2 Data Analysis and Results

sec2

###  2.1 Data

sec21

Figure 1.: Examples of the Sun’s helioseismic far-side images (Left) along
with near-side magnetograms (Right) when both sides of the Sun were
magnetically active (Upper) and quiet (Lower). Dark patches in the far-side
images indicate ARs, and the blue-green (orange-red) patches in the
magnetograms indicate positive (negative) magnetic regions. The near-side
magnetograms are plotted as the solar disk observed from near the Earth, and
the far-side images are plotted as if the solar disk were observed from a far-
side position on the extended Earth–Sun line.FS˙examples

The time–distance helioseismic far-side imaging technique that was recently
developed by Zhao et al. (2019) is applied to the SDO/HMI data from 1 May
2010, the first observing day after the instrument’s commissioning, through 30
April 2021 – a total of 11 years. Each far-side image is obtained from a
31-hour observing period, and two far-side images are obtained every day with
the middle time of the data period at 00:00 UT and 12:00 UT of the day. Figure
FS_examples shows two examples of far-side images when the Sun was very active
with large ARs on both the far and near sides, and when it was less active
with no visible ARs on either side. The helioseismic far-side images are
actually maps of the acoustic travel-time shifts measured relative to the Sun
of activity minimum years (1 January 2018 – 31 December 2019 for this study),
and dark patches in such images (negative travel-time shifts) indicate where
ARs are located. It is generally believed that the travel-time deficits
corresponding to far-side ARs are due to the Wilson depression associated with
magnetic regions (Lindsey, Cally, and Rempel, 2010).

For this time–distance far-side imaging method, each image is composed of and
overlapped by 14 different sets of partial-far-side images made from waves
experiencing different number of skips: three, four, five, six, and eight. In
the time–distance diagram, there are two branches corresponding to each
$n$-skip signal (refer to Figure 2 of Zhao et al. 2019), the lower one
corresponding to helioseismic waves traveling less than
$360\mbox{${}^{\circ}$}$ (hereafter, 1∘ represents 1∘ in great-circle distance
on the solar surface) and the upper one corresponding to waves traveling more
than $360\mbox{${}^{\circ}$}$. For a four-skip wave, e.g. if the wave starts
from the near side and is reflected twice before reaching the far-side target
point, then is reflected twice before traveling back to the near side, this
measurement scheme is called 2$\times$2-lower if the total travel distance of
the wave is less than $360\mbox{${}^{\circ}$}$, or 2$\times$2-upper if greater
than $360\mbox{${}^{\circ}$}$. Thus, the 14 sets of measurement schemes
include: 2$\times$2-lower, 2$\times$2-upper, 3$\times$3-lower,
3$\times$3-upper, 4$\times$4-lower, and 4$\times$4-upper, primarily used to
map the far-side central regions within $\approx 48\mbox{${}^{\circ}$}$ of the
far-side disk center; and 1$\times$2-lower, 1$\times$3-lower,
1$\times$3-upper, 2$\times$3-lower, 2$\times$3-upper, 2$\times$4-lower,
2$\times$6-lower, and 2$\times$6-upper, primarily used for mapping regions
within $\approx 45\mbox{${}^{\circ}$}$ of the far-side limbs. Because
different measurement schemes have different noise levels, not all the data
from these measurement schemes are used in this study.

###  2.2 Mean Travel-Time Shifts and Mean Magnetic Field

sec22

To study the temporal evolution of the travel-time shifts in the time–distance
far-side images during the 11-year period, we average the travel-time shift
measurements throughout the entire far side that covers
$180\mbox{${}^{\circ}$}$ in longitude and $120\mbox{${}^{\circ}$}$ in latitude
in longitude–sin(latitude) coordinates, and we call the averaged values
$\delta\tau_{\mathrm{mn}}$ hereafter. Mean magnetic-flux density
[$|B|_{\mathrm{mn}}$] is obtained daily by averaging the absolute values of
the near-side full-disk magnetic field. Figure long_time shows the 120-day
running averages of both $\delta\tau_{\mathrm{mn}}$ and $|B|_{\mathrm{mn}}$,
which show a very good anti-correlation, -0.995, between them.

Figure 2.: Long-term evolution of the Sun’s near-side mean magnetic flux
density [$|B|_{\mathrm{mn}}$] and far-side mean travel-time shifts
[$\delta\tau_{\mathrm{mn}}$]. Note that for the axis for
$\delta\tau_{\mathrm{mn}}$, values decrease upward.long˙time

It may not be surprising that $\delta\tau_{\mathrm{mn}}$ has a very good anti-
correlation with magnetic activity, because the travel-time deficits on far-
side maps are mostly caused by the far-side magnetic regions, and the mean
far-side magnetic-flux density is not expected to differ much from that of the
near side, particularly after a 120-day average. To examine whether the
contributions of far-side magnetic regions are the primary cause of the
general variation trend of the $\delta\tau_{\mathrm{mn}}$, we plot in Figure
NS_hemi the $\delta\tau_{\mathrm{mn}}$ averaged from the Sun’s entire far
side, as well as those averaged from the far-side northern and southern
hemispheres separately, together with the $|B|_{\mathrm{mn}}$ from the entire
near side as well as from the near-side northern and southern hemispheres. As
can be seen, the $\delta\tau_{\mathrm{mn}}$ from the far-side northern and
southern hemispheres, as well as the entire far side, are nearly identical,
while the magnetic-field strengths from the two near-side hemispheres show
dramatic differences. This indicates that the general trend in the
$\delta\tau_{\mathrm{mn}}$ variation that is seen in Figure long_time is
likely caused by a global-scale variation rather than by the magnetic-flux
variation at local scales.

Figure 3.: (a) Long-term evolution of $\delta\tau_{\mathrm{mn}}$ averaged from
the Sun’s far-side full disk as well as from both the far-side northern and
southern hemispheres. (b) Long-term evolution of the Sun’s $|B|_{\mathrm{mn}}$
obtained from the near-side full disk as well as from the near-side northern
and southern hemispheres.NS˙hemi

###  2.3 Mean Travel-Time Shifts and Number of Skips

sec23

The results shown in Figure long_time are essentially consistent with those
reported by González Hernández, Scherrer, and Hill (2009); however, the
time–distance far-side imaging technique has one advantage over theirs in that
we have multiple multi-skip measurements that provide us more data to track
down the causes of the general variation trend of the
$\delta\tau_{\mathrm{mn}}$. Comparing the $\delta\tau_{\mathrm{mn}}$ measured
from different measurement schemes, i.e. different combinations of wave-skips,
may shed a light on how we can understand the
$\delta\tau_{\mathrm{mn}}$variation with the solar cycle. Because the
$\delta\tau_{\mathrm{mn}}$ measured in the far-side central area has different
properties from that measured in the far-side limb area, we analyze these
measurements separately.

Figure 4.: (a) Temporal evolution of $\delta\tau_{\mathrm{mn}}$ obtained from
the far-side central areas using the measurement schemes marked in the panel.
(b) Same $\delta\tau_{\mathrm{mn}}$ as in Panel a but displayed after dividing
their respective number of wave skips, i.e. $\delta\tau_{\mathrm{mn}}/N$. Note
the measurements from the 2$\times$2-upper scheme are noisier and are not
shown, but their basic trends are consistent with the ones shown.time˙central

Figure time_centrala shows the temporal evolution of
$\delta\tau_{\mathrm{mn}}$ measured from the far-side central area using the
measurement schemes of 2$\times$2-lower, 3$\times$3-lower, 3$\times$3-upper,
4$\times$4-lower, and 4$\times$4-upper; Figure time_centralb shows
$\delta\tau_{\mathrm{mn}}/N$– $\delta\tau_{\mathrm{mn}}$ divided by their
respective number of skips used in the measurements. As can be seen, the
$\delta\tau_{\mathrm{mn}}$ from different measurement schemes show substantial
differences in values although all of them show a very similar general trend
of variations with the solar cycle; however, the $\delta\tau_{\mathrm{mn}}/N$
are essentially consistent with each other, indicating that a single skip from
each measurement scheme gets a similar amount of travel-time reduction.
Meanwhile, we want to emphasize that despite the close similarities among the
$\delta\tau_{\mathrm{mn}}/N$ for different measurement schemes, the small
differences between them are not negligible, and these important differences
will be discussed later in Section sec33.

Figure 5.: (a) Temporal evolution of $\delta\tau_{\mathrm{mn}}$ obtained from
the far-side limb areas using the measurement schemes marked in the panel. (b)
Same $\delta\tau_{\mathrm{mn}}$ as in Panel a but displayed after dividing
their respective number of wave skips, i.e., $\delta\tau_{\mathrm{mn}}/N$.
Note that the measurements from the 1$\times$2-lower and 1$\times$3-upper are
noisier and are not shown here, but their basic trends are consistent with the
ones shown.time˙limb

Similarly, Figure time_limb shows the $\delta\tau_{\mathrm{mn}}$ and
$\delta\tau_{\mathrm{mn}}/N$ measured from the limb areas of the far side,
including the measurement schemes of 1$\times$3-lower, 2$\times$3-lower,
2$\times$3-upper, 2$\times$4-lower, 2$\times$6-upper, and 2$\times$6-lower.
Similar to the result from the far-side central region, the
$\delta\tau_{\mathrm{mn}}$ from different measurement schemes differ
substantially but the $\delta\tau_{\mathrm{mn}}/N$ are quite consistent with
each other, again with non-negligible differences.

###  2.4 Mean Travel-Time Shifts and Latitude

sec24

Figure 6.: Temporal variations of far-side $\delta\tau_{\mathrm{mn}}$ for
selected latitudinal bands in the northern (a) and southern (b) hemispheres.
These data are from the final far-side images after combining all the
measurement schemes.lat˙var

Figure NS_hemi shows variations of $\delta\tau_{\mathrm{mn}}$ for both the
northern and southern hemispheres, and Figures time_centralb and time_limbb
show variations of the $\delta\tau_{\mathrm{mn}}/N$ for different measurement
schemes. All of these results exhibit very similar trends and amplitudes of
variations. However, as shown in Figure lat_var, the variation trends of the
$\delta\tau_{\mathrm{mn}}$, obtained from a few selected latitudinal bands in
the combined far-side images (rather than images of different measurement
schemes), show significantly different amounts of travel-time shifts, despite
that they still show similar variation trends. The $\delta\tau_{\mathrm{mn}}$
in the lower-latitude bands have greater magnitudes than those in the higher-
latitude bands. These results indicate that although the general trend of the
$\delta\tau_{\mathrm{mn}}$-variations seems to be caused by a global factor,
the variation amplitudes are actually disk-location dependent.

##  3 Discussion

sec3

###  3.1 What Causes the Long-Term $\delta\tau_{\mathrm{mn}}$-Variation?

sec31

The $\delta\tau_{\mathrm{mn}}$-variations shown in Section sec2 are
essentially consistent with those by González Hernández, Scherrer, and Hill
(2009), except that our results with different numbers of wave skips provide
more information that can help us better interpret the results. It may not be
surprising to see that the far-side $\delta\tau_{\mathrm{mn}}$ varies with the
phase of solar cycles, however, determining what is the physics behind these
measured variations may not be that straightforward.

Before understanding the cause of these variations, it is useful to review
some basic geometry used in the far-side imaging measurement schemes.
Helioseismic waves that are used to map the far-side ARs often experience a
few surface reflections during their propagation through the Sun. An example
of the 1$\times$3-lower measurement scheme is shown in Figure 1x3_scheme.
Waves starting from the near-side Sun experience a one-skip travel to reach
the far-side target, after which the waves continue with a one-skip and a two-
skip travel on the far side, and come back to the near side as three-skip
waves. The oscillatory signals observed in the near-side one-skip and three-
skip partial-annuli are used to measure travel-time shifts corresponding to
that far-side target. In this measurement scheme, the waves travel four skips
and touch the surface five times, and each touch of the surface has a
possibility of encountering an active region. Based on this knowledge of far-
side images’ measurement processes, we speculate that there may be four
possible causes that lead to the measured long-term
$\delta\tau_{\mathrm{mn}}$-variation.

  1. [i]

  2. 1.

Far-side ARs manifest themselves as travel-time deficits on the far-side maps.
With the waxing and waning of the number of ARs on the far side, one would
naturally expect that the far-side $\delta\tau_{\mathrm{mn}}$ varies with the
solar cycle.

  3. 2.

Far-side maps are calculated using the near-side Doppler velocities, and the
Doppler velocities observed inside the near-side ARs already carry travel-time
deficits with them. It is thus likely that the travel-time deficits measured
in the far-side images come from the near-side oscillatory signals in the
near-side ARs, whose travel-time deficits are well anti-correlated with the
near-side magnetic activity.

  4. 3.

As pointed out above, although all of the different measurement schemes focus
on the same far-side target, the waves actually get reflected a few times on
different surface locations before and after getting reflected at the far-side
target (see Figure 1x3_schemea). These surface-reflection locations,
distributed non-uniformly around the globe, have a probability of residing
inside magnetic regions, and travel-time deficits during such surface
reflections can be carried along the entire journey of the waves. The more
magnetic regions, the more likely that surface reflections occur inside a
magnetic region, and more travel-time deficits may be measured in
$\delta\tau_{\mathrm{mn}}$.

  5. 4.

As interpreted by González Hernández, Scherrer, and Hill (2009), the
$\delta\tau_{\mathrm{mn}}$-variation is due to the cyclic variation of the
Sun’s seismic radius $\mathrm{R_{\odot}}$, which is also in phase with the
Sun’s magnetic activity.

Not all these possible causes are completely independent from each other; for
instance, causes 1 and 2 are certainly parts of cause 3. But what we want to
find out here is the dominant factor that determines the
$\delta\tau_{\mathrm{mn}}$-variation trend. If cause 1 dominates, one would
expect that the $\delta\tau_{\mathrm{mn}}$ shows different trends in the
northern and southern hemispheres, similar to what we see in the magnetic-
field trends; but Figure NS_hemi shows that this is not the case. If cause 2
dominates, one would expect that the $\delta\tau_{\mathrm{mn}}$ with a
different number of wave skips show similar amount of variations because
essentially similar near-side oscillatory signals are used in all of these
calculations; however, Figures time_centrala and time_limba show that this is
unlikely to be the case. However, one also needs to be more cautious here,
because the waves with different numbers of skips are actually waves of
different oscillatory modes $(n,\ell)$, corresponding to waves penetrating to
different depths of the Sun. This difference likely leads to small
discrepancies seen in $\delta\tau_{\mathrm{mn}}/N$ in Figures time_centralb
and time_limbb rather than the larger differences seen in
$\delta\tau_{\mathrm{mn}}$ in Figures time_centrala and time_limba. Next, we
focus on examining which of causes 3 and 4 plays a dominant role in causing
the long-term $\delta\tau_{\mathrm{mn}}$-variation.

Figure 7.: (a) Great-circle slice showing the measurement scheme for
1$\times$3-lower case. Observations inside the one-skip and three-skip
partial-annuli on the near side, i.e. lower half of the circle, are used for
calculating travel-time shifts at the far-side target. (b) An example showing
the surface-reflection locations, in both near- and far-side disks,
corresponding to a far-side target at a longitude of $15\mbox{${}^{\circ}$}$
past the west limb and latitude of $20\mbox{${}^{\circ}$}$N. Single-skip
distance for this example is $75\mbox{${}^{\circ}$}$, one value from a range
of distances used in the calculations. The partial-annuli in the plot are
therefore a sample of much wider annuli.1x3˙scheme

###  3.2 Effect of Surface Magnetic Field or Change of $\mathrm{R_{\odot}}$?

sec32

To assess how the magnetic field, distributed around the global Sun, impacts
the $\delta\tau_{\mathrm{mn}}$-measurements, as listed as cause 3 above, we
need to assess the probability of acoustic waves, which carry far-side
information back to the near side, encountering a magnetic region on the solar
surface during their journey. We first assess for each measurement scheme the
probability density of a waves’ surface reflections as a function of latitude.
This probability density is also a function of longitude; however, because in
this study we only present the 120-day running-average results, the longitude
dependency can be omitted.

Figure 1x3_schemea shows one of the 14 measurement schemes that are used to
measure the travel-time shift corresponding to one far-side target, and Figure
1x3_schemeb shows one sample of a series of skip distances that are used in
this 1$\times$3-lower measurement scheme. As one can see, the reflection
points of the waves that are used to measure one far-side location are
distributed over most of the Sun’s latitudes. In fact, for one measurement
scheme and for one far-side target point, there are tens of thousands of
surface reflection locations associated with it even in our data with a low
spatial-resolution of $0.8\mbox{${}^{\circ}$}$ pixel-1, and the probability
density as a function of latitude $\lambda$ [$\mathrm{pdf}(\lambda)$] is a
measure of the distribution of these surface-reflection locations. These
surface-reflection locations can be found through spherical geometry
calculations, and the total number of reflections is then counted for each
$5\mbox{${}^{\circ}$}$-wide latitudinal bands from $-90\mbox{${}^{\circ}$}$ to
$+90\mbox{${}^{\circ}$}$. The same computation is repeated for each far-side
target for this measurement scheme, and then the same process is repeated for
all the 14 measurement schemes. The $\mathrm{pdf}(\lambda)$ can then be found
for each latitudinal band by normalizing relative to the total number of
reflections across all latitudes.

Likewise, for each day, we average the absolute values of the full-disk
magnetic field into $5\mbox{${}^{\circ}$}$-wide latitudinal bands, which is
$|B(\lambda,t)|$, where $t$ is time (or day in this case). We then calculate
an averaged magnetic-field strength at all of the waves’ reflection locations,
i.e. modulated by $\mathrm{pdf}(\lambda)$ following

$\langle|B(\lambda,t)|\rangle=\mathrm{pdf}(\lambda)|B(\lambda,t)|.$

$\langle|B|\rangle$ measures the averaged magnetic-field strength that the
waves encounter on the surface along their entire journey from the near side
to the far side and then back to the near side. These $\langle|B|\rangle$ are
then averaged again into 20∘-wide latitudinal bands for each day to compare
against the $\delta\tau_{\mathrm{mn}}$-measurements in Figure lat_var, and
Panels a and b of Figure meanT_estM show the comparison after a 120-day
running average. For the latitudinal bands of
$0\mbox{${}^{\circ}$}-20\mbox{${}^{\circ}$}$ and
$20\mbox{${}^{\circ}$}-40\mbox{${}^{\circ}$}$ in both hemispheres, the trend
and values of $\delta\tau_{\mathrm{mn}}$ and $\langle|B|\rangle$ match pretty
well, implying there is a single scaling factor between these two quantities.
However, the $\delta\tau_{\mathrm{mn}}$ in the
$40\mbox{${}^{\circ}$}-60\mbox{${}^{\circ}$}$ band are notably smaller than
$\langle|B|\rangle$ with that same scaling factor, and this will be further
discussed in Section sec33. Figure meanT_estMc shows that the
$\langle|B|\rangle$, integrated separately in both the northern and southern
hemispheres, are very similar to each other, just as one would expect if
comparing with Figure NS_hemia.

Figure 8.: Temporal variations of far-side $\delta\tau_{\mathrm{mn}}$ (solid
lines), plotted decreasing upward, and $\langle|B|\rangle$ (dashed lines) for
selected latitudinal bands in both northern (a) and southern (b) hemispheres.
(c) Comparison of the $\langle|B|\rangle$ integrated in the entire northern
and southern hemispheres separately.meanT˙estM

We thus believe that the long-term variations of $\delta\tau_{\mathrm{mn}}$
are caused collectively by the magnetic field distributed around the Sun,
whose interaction with the traveling acoustic waves at their surface
reflection locations around the Sun leaves a travel-time deficit on the far-
side acoustic maps. We tend to believe that the long-term
$\delta\tau_{\mathrm{mn}}$ variation is not caused by the seismic radius
$\mathrm{R_{\odot}}$ variation, because that can hardly explain the
latitudinal differences measured in $\delta\tau_{\mathrm{mn}}$. However, we
also want to stress that this does not necessarily indicate that the
$\mathrm{R_{\odot}}$ does not vary with the solar cycle. Our analysis only
demonstrates that the magnetic field around the Sun, through interacting with
the the waves’ surface reflections, is collectively the dominant factor that
causes the long-term variation measured in $\delta\tau_{\mathrm{mn}}$, and
what factor the $\mathrm{R_{\odot}}$ variation plays in the
$\delta\tau_{\mathrm{mn}}$ is not studied here, but it is believed not to play
a substantial role.

Meanwhile, we also recognize that the long-term variation of the
$\delta\tau_{\mathrm{mn}}$ is caused by magnetic field, but is mostly
materialized through the Wilson depression associated with magnetic field
(Lindsey, Cally, and Rempel, 2010) along with the interior thermal changes in
and around magnetic regions. Averaging local Wilson depressions across all
longitudes in indivdual latitudinal bands can result in the latitude-dependent
reductions of the solar radius, and this reduction can also in the same sense
explain the analysis results shown in Figure meanT_estM. However, this radius
reduction is mostly a local and near-surface phenomenon, not a global-scale
isotropic seismic-radius change in the relatively deeper interior.

###  3.3 A Few Discrepancies

sec33

So far, there are a few discrepancies in the
$\delta\tau_{\mathrm{mn}}$-measurements and in our explanation of measurements
that are not negligible. These include: i) Comparing Figures time_centralb and
time_limbb, one can find that the $\delta\tau_{\mathrm{mn}}/N$ have clear
differences in the near-center and near-limb measurements, around $-0.7$
seconds and $-0.9$ seconds, respectively, in the solar activity maximum years.
ii) As shown in Figure meanT_estM, the measured $\delta\tau_{\mathrm{mn}}$ in
the latitudinal bands of $40\mbox{${}^{\circ}$}-60\mbox{${}^{\circ}$}$ are
substantially smaller than $\langle|B|\rangle$ after scaling. iii) In either
Figure time_centralb or time_limbb, $\delta\tau_{\mathrm{mn}}/N$ is different
for different wave-skip numbers despite measuring the same area of the far
side.

The first discrepancy, the $\delta\tau_{\mathrm{mn}}/N$ measured in the near-
center area differs from that measured in the near-limb area by nearly 0.2
seconds, is likely due to the helioseismic center-to-limb effect (Zhao et al.,
2012; Chen and Zhao, 2018). The measurements for the far-side central area
often uses more observations from the near-side central area than the near-
side limb area, while the measurements for the far-side limb area often uses
more observations from the near-side limb area than from the near-side central
area. The helioseismic center-to-limb effect thus leaks into the far-side
measurements, causing the $\delta\tau_{\mathrm{mn}}$ measured in the far-side
limb area to be relatively smaller.

The second discrepancy, that the $\delta\tau_{\mathrm{mn}}$ and
$\langle|B|\rangle$ do not match well in the high-latitude bands, may be due
to the following reasons: First, similar to the first discrepancy, the
helioseismic center-to-limb effect causes the far-side
$\delta\tau_{\mathrm{mn}}$ measurements in higher latitude (limb area) to be
substantially smaller than in low-latitude areas (central area). Second, the
magnetic field’s effect on the travel-time shift is understandably not linear,
and weaker magnetic field in the high-latitude areas may have an even weaker
effect on travel-time shifts, resulting in smaller $\delta\tau_{\mathrm{mn}}$
in high latitudes.

The third discrepancy, the $\delta\tau_{\mathrm{mn}}/N$ measured from a
different number of wave skips, may reflect the real solar-interior
variations. Because these waves are of different oscillatory modes probing to
different depths of the solar interior, as pointed out in Section sec31, the
small differences in $\delta\tau_{\mathrm{mn}}/N$, an order of 0.1 seconds,
probably imply the interior physical changes associated with the solar cycle.
It is possible that the waves of slightly different modes may have slightly
different travel-time shifts when reflected from the same Wilson depression;
or it is possible that the solar-interior structure changes more near the
surface than in the deeper region so that waves with more skips, i.e.
higher-$\ell$ modes, experience slightly longer travel-time shifts. These
different scenarios are all worth further study.

##  4 Conclusion

sec4

The time–distance far-side images from 11 years of SDO/HMI observations
exhibit a systematic variation in the disk-averaged travel-time shifts, which
exhibit a very good anti-correlation with the near-side magnetic-flux density
that varies with the solar cycle. Our multiple multi-skip measurement schemes
that are used to image the far side of the Sun allow us to analyze the single-
skip travel-time deficits and the travel-time deficits measured in different
latitudinal bands. Our analysis demonstrates that the long-term variation in
the far-side travel-time shifts are primarily caused, collectively, by the
surface reflections, which unavoidably interact with the magnetic field that
is distributed non-uniformly around the whole Sun, rather than caused by the
seismic-radius variation. This analysis does not exclude the possibility that
the seismic radius may vary with the solar cycle, but the variation of the
seismic radius is not the dominant cause of the long-term variation of the
measured far-side travel-time shifts.

#### Acknowledgments

We thank an anonymous referee for his/her valuable comments and sugguestions
that help improve the quality of this article.

## Funding

This work was partly funded by the NOAA grant NA18NWS4680082.

## Data Availability

The far-side images used in this analysis are made by the SDO/HMI’s
time–distance far-side imaging pipeline, and can be downloaded from the
webpage: http://jsoc.stanford.edu/data/timed/. The near-side magnetograms are
from the SDO/HMI observations and can be downloaded from the webpage:
http://hmi.stanford.edu/magnetic/.

## Conflict of interest

The authors declare that they have no conflicts of interest.

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# A Modulation Format Correction Formula for the Gaussian Noise Model in the
Presence of Inter-Channel Stimulated Raman Scattering

Daniel Semrau, Eric Sillekens, Robert I. Killey, and Polina Bayvel This work
was supported by a UK EPSRC programme grant TRANSNET (EP/R035342/1) and a
Doctoral Training Partnership (DTP) studentship for Daniel Semrau.D. Semrau,
E. Sillekens, Robert I. Killey and P. Bayvel are with the Optical Networks
Group, University College London, London WC1E 7JE, U.K. (e-mail: {uceedfs;
e.sillekens; r.killey<EMAIL_ADDRESS>
###### Abstract

A closed-form formula is derived, which corrects for the modulation format
dependence of the Gaussian Noise (GN) model in the presence of inter-channel
stimulated Raman scattering (ISRS). The analytical result enables a rapid
estimate of the nonlinear interference (NLI) for arbitrary modulation formats
and avoids the need for complex integral evaluations and split-step
simulations. It is shown that the modulation format dependent NLI can be
approximated by two contributions, one originating from a single span and one
asymptotic contribution for a large number of spans. The asymptotic
contribution is solved in closed-form for an arbitrary link function, making
the result applicable for generic fiber systems using lumped, distributed or
hybrid amplification schemes. The methodology is applied to the ISRS GN model
and a modulation format correction formula in closed-form is derived which
accounts for an arbitrary number of spans, inter-channel stimulated Raman
scattering, arbitrary launch power distributions and wavelength dependent
dispersion and attenuation. The proposed formula is validated by numerical
simulations over the entire C+L band for multiple fiber types.

###### Index Terms:

Optical fiber communications, Gaussian noise model, Nonlinear interference,
Stimulated Raman Scattering, C+L band transmission, closed-form approximation,
modulation format correction

##  I Introduction

Analytical models experienced substantial popularity over the last decade, as
they provide a low complexity alternative to split-step simulations with good
agreement to experiments [1, 2, 3, 4, 5, 6, 7]. Perturbation models that
estimate nonlinear interference (NLI) are key for rapid and efficient system
design [8], achievable rate estimations of point-to-point links [9, 10, 11]
and physical layer aware network optimization [12, 13].

The Gaussian Noise (GN) model [14, 15, 16] is extensively applied throughout
research and industry, as it offers very low computational complexity while
being reasonably accurate for high cardinality modulation formats.
Additionally, the GN model offers approximations in closed-form for a variety
of optical transmission scenarios [14, 10, 17, 18, 16, 19, 20, 21, 22]. Such
closed-form approximations enable performance estimations within picoseconds
and are vital for real-time applications and on-the-fly optimizations.

Figure 1: Illustration of the NLI contribution of the Gaussian Noise (GN)
model and the modulation format correction on the channel of interest (COI)
originating from a single interferer (INT). The channels exhibit arbitrary
modulation formats, power levels, bandwidths and center frequencies. The shown
modulation format is taken from [23].

The major source of inaccuracy of the GN model stems from its signal
Gaussianity assumption, which assumes that the signal can be written as a
Gaussian process at the transmitter. As a consequence the model significantly
overestimates the NLI for low cardinality modulation formats and in optical
links with low accumulated dispersion. More complex models, in integral form,
have been introduced to account for non-Gaussian modulation formats [24, 25,
26, 27, 28, 29, 30]. However, for those models to be applicable in a real-time
environment and to avoid the high computational complexity of modulation
format aware approaches, approximations in closed-form are necessary. A
closed-form modulation format correction formula in the asymptotic limit of a
large span number and optical bandwidths limited to C-band has been derived in
[21].

All of the aforementioned modeling approaches do not account for inter-channel
stimulated Raman scattering (ISRS) and are therefore not applicable to ultra-
wideband transmission systems that operate at optical bandwidths beyond C-band
(5 THz). Extensions of the conventional GN model to account for ISRS, named
ISRS GN model, have been recently proposed in integral form [31, 32, 33, 34,
35, 36]. An approximation in closed-form has been first proposed in [31] which
recently has been significantly generalized to include arbitrary launch power
distributions, wavelength dependent dispersion and a more accurate description
of ISRS [37, 38]. A comparible formula, based on the same approach and with a
similar result, has been reported in [39].

However, the works [31, 32, 33, 34, 35, 36, 37, 38, 39] assume signal
Gaussianity and the impact of the modulation format has not been addressed
analytically in the context of ultra-wideband transmission systems, where ISRS
cannot be neglected. More importantly, a closed-form formula suitable for
real-time and yet accurate performance estimations of arbitrary modulation
formats has not been reported to date.

In this paper, a modulation format correction formula for the ISRS GN model is
derived. Is is shown that the modulation format correction can be, in general,
approximated by two contributions. One contribution that accounts for the
modulation format correction after one span and one accounting for the
correction in the asymptotic limit of a large span number. The asymptotic
contribution is solved for an arbitrary transmission system, described by a
link function, yielding a generic asymptotic modulation format correction
formula. Both contributions, yielding a formula for any number of spans, are
solved particularly for the ISRS GN model. The proposed formula accounts for
arbitrary launch power distributions, arbitrary number of spans, arbitrary
modulation formats, lumped amplification and ultra-wideband effects such as
inter-channel stimulated Raman scattering and wavelength dependent dispersion
and attenuation. The finding is validated by numerical simulations for a low
dispersive non-zero dispersion-shifted fiber (NZDSF) and a high dispersive
standard single mode fiber (SMF) over 10 THz optical bandwidth.

The paper is organized as follows: In Section II, the formalism to
analytically describe the impact of the modulation format on the NLI is
reviewed and the assumptions necessary to solve it in closed-form are
described. The methodology is then applied to the ISRS GN model in Section
II-C to obtain a modulation format correction formula in closed-form, the key
result of this work. The formula is validated by numerical simulations in
Sections III and IV.

##  II Modulation Format Correction

After coherent detection, electronic dispersion compensation and neglecting
the impact of transceiver noise, the signal-to-noise ratio (SNR) of the
channel of interest (COI) $i$ can be calculated as

$\begin{split}\text{SNR}_{i}\approx\frac{P_{i}}{P_{\text{ASE}}+\eta_{n}P_{i}^{3}},\end{split}$
(1)

where $P_{i}$ is the launch power of channel $i$ at the transmitter and
$P_{\mathrm{ASE}}$ is the accumulated amplified spontaneous emission (ASE)
noise originating from optical amplifiers. ASE noise inflation as a result of
gain equalization can be included by a channel dependent $P_{\text{ASE}}$. The
nonlinear interference coefficient $\eta_{n}\left(f_{i}\right)$ after $n$
fiber spans is dependent on the center frequency $f_{i}$ of the COI.

Although modulation format independent models, such as the GN model, are in
relative good agreement with experiments using high cardinality modulation
formats, the NLI has been shown to be dependent on the transmitted modulation
format. This dependence is particularly high for low cardinality modulation
formats and short to medium-haul transmission distances. Substantially more
complex models have been introduced in order to accurately model the impact of
arbitrary modulation formats [24, 26, 27, 25, 28, 29, 30]. Most results show
that the total NLI can be separated into two contributions, one ’GN-like’
modulation format independent contribution and a correction term that accounts
for the transmitted modulation format. The NLI coefficient can hence be
written as

$\begin{split}\eta_{n}\left(f_{i}\right)=\eta_{\text{GN,}n}\left(f_{i}\right)+\eta_{\text{corr.},n}\left(f_{i}\right),\end{split}$
(2)

where $\eta_{\text{GN,}n}$ is the GN model contribution and
$\eta_{\text{corr.},n}$ is the modulation format correction term. For Gaussian
modulated signals the correction term $\eta_{\text{corr.},n}$ vanishes and one
obtains the GN model description of NLI.

In [28, 29], a complete set of formulas in integral form has been derived to
correct for arbitrary modulation formats for self-phase modulation (SPM/SCI),
cross-phase modulation (XPM/XCI) and four-wave mixing (FWM/MCI) NLI
contributions. In the approach taken in this work, only the most dominant
contribution, the cross-phase modulation contribution, is taken into account.
It has been shown that this approach is sufficiently accurate with only a
minor loss in accuracy [24, 25, 26, 27, 21]. The approach is particularly
accurate in high dispersive fibers such as standard single mode fiber (SMF)
and systems with high symbol rates or channel spacings. Two conditions that
are prevailing in legacy systems and likely to be installed in next generation
optical transmission systems.

The XPM assumption, taking only XPM-like terms into account, evaluates the NLI
within the channel of interest originating from a single interferer (INT). The
XPM assumption is illustrated in Fig. 1. The total NLI is obtained by summing
over all COI-INT pairs present in the transmitted signal. Mathematically, the
total XPM modualtion format correction is

$\begin{split}&\eta_{\text{corr,}n}\left(f_{i}\right)=\sum_{k=1,k\neq
i}^{N_{\text{ch}}}\eta_{\text{corr,}n}^{\left(k\right)}(f_{i}),\end{split}$
(3)

where $\eta_{\text{corr,}n}^{\left(k\right)}(f_{i})$ is the XPM contribution
of a single interfering channel $k$ on channel $i$.

The dominant XPM modulation format correction term is given by [29, Eq. (17)]

$\begin{split}&\eta_{\text{corr,}n}^{\left(k\right)}(f_{i})=\frac{80}{81}\left(\frac{P_{k}}{P_{i}}\right)^{2}\frac{\gamma^{2}\Phi}{B_{k}^{3}}\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\\\
&\cdot\left|\int_{-\frac{B_{k}}{2}}^{\frac{B_{k}}{2}}\mu\left(f_{1}+f_{i},f_{2}+f_{i}+\Delta
f,f_{i}\right)\sum_{m=0}^{n-1}e^{jmf_{1}\left(f_{2}+\Delta
f\right)\phi}df_{2}\right|^{2},\end{split}$ (4)

where $B_{i}$ and $B_{k}$ are the bandwidth of COI $i$ and INT $k$,
respectively, $\Delta f=f_{k}-f_{i}$ is the channel frequency separation
between COI and INT, $\gamma$ is the nonlinearity coefficient and
$\phi=-4\pi^{2}\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k})\right]L$ is a phase
mismatch term accounting for coherent NLI accumulation, with the group
velocity dispersion (GVD) parameter $\beta_{2}$ and its linear slope
$\beta_{3}$ at the reference wavelength. In (4), it is assumed that the
variation of the wavelength dependent dispersion is negligible over a single
channel bandwidth. $\mu\left(f_{1},f_{2},f_{i}\right)$ is the link function of
a single span. The link function is addressed in more detail in the following
sections.

$\Phi=\frac{\mathrm{E}\left[\left|X\right|^{4}\right]}{\mathrm{E}^{2}\left[\left|X\right|^{2}\right]}-2$
is the excess kurtosis of the transmitted modulation format. The excess
kurtosis of a few selected modulation formats are listed in Table I. Most
modulation formats exhibit a negative excess kurtosis resulting in a negative
modulation format correction $\eta_{\text{corr,}n}^{\left(k\right)}(f_{i})$
and reduced nonlinear interference as a consequence. Loosely spoken, the
modulation format correction is smaller for more ’Gaussian-like’ modulation
formats. This property does not only extend to the coordinates of the
individual symbols but also to their respective probabilty of occurance [40].

TABLE I: Excess kurtosis of selected modulation formats. Modulation format | Excess kurtosis $\Phi$
---|---
uniform QPSK | -1
uniform 16-QAM | -0.6800
uniform 64-QAM | -0.6190
uniform 256-QAM | -0.6050
uniform $\infty$-QAM | -0.6000
geom. shaped 64-QAM (12 dB SNR, GMI)[23] | -0.3403
proba. shaped 64-QAM (12 dB SNR, MI) [41] | -0.1871
Gaussian modulation | 0

The aim of the remainder of this paper is to find an approximation in closed-
form of the modulation format correction in integral form (4). This gives the
advantage of increase prediction accuracy for non-Gaussian modulation formats
without sacrificing computation complexity and execution time.

###  II-A Modulation format correction for generic transmission systems in
closed-form

In this section, an approximation of the solution of Eq. (4) for a generic
transmission system is derived. An optical fiber transmission system that is
described by the nonlinear Schrödinger equation can be, to first-order,
described by a link function $\mu\left(f_{1},f_{2},f_{i}\right)$. The link
function describes the nonlinear perturbation (i.e. the NLI) of three
interacting frequencies on the frequency $f_{i}$ after propagation.

The approach in this work relies on two key assumptions. The first assumption
is that the channel separation of two interfering channels (COI and INT) is
much greater than the channel bandwidth, which allows to approximate
$f_{2}+\Delta f\approx\Delta f$ in (4). Mathematically, this conincides with
the zeroth-order solution of the inner integral running over $f_{2}$ in (4). A
detailed assessment of this assumption was carried out in [38, Appendix C] and
negligible impact on the total NLI was found. Applying the assumption results
in

$\begin{split}\eta_{\text{corr,}n}^{\left(k\right)}(f_{i})\approx&\
\tilde{\gamma}\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\left|\mu\left(f_{1}+f_{i},f_{k},f_{i}\right)\right|^{2}\\\
&\cdot\left|1+\sum_{m=1}^{n-1}\mathrm{sinc}\left(m\phi
f_{1}\frac{B_{k}}{2}\right)e^{jm\phi f_{1}\Delta f}\right|^{2},\end{split}$
(5)

with
$\tilde{\gamma}=\left(\frac{P_{k}}{P_{i}}\right)^{2}\frac{80}{81}\frac{\gamma^{2}\Phi}{B_{k}}$,
which is introduced for notational brevity.

Eq. (5) does not seem to have an analytical solution. However, in order to
find an approximate solution, we analyze its asymptotic behavior for a large
number of fiber spans $n$. For a large number of spans, the oscilating terms
in (5) can be approximated by a Dirac delta function $\delta\left(x\right)$ as

$\begin{split}&\lim_{n\to\infty}\left|1+\sum_{m=1}^{n-1}\mathrm{sinc}\left(m\phi
f_{1}\frac{B_{k}}{2}\right)e^{jm\phi f_{1}\Delta f}\right|^{2}\\\
\approx&1+\delta\left(f_{1}\right)\lim_{n\to\infty}C_{n},\end{split}$ (6)

with normalization coefficient

$\begin{split}&C_{n}=\int_{-\infty}^{\infty}df_{1}\left|\sum_{m=1}^{n-1}\mathrm{sinc}\left(m\phi
f_{1}\frac{B_{k}}{2}\right)e^{jm\phi f_{1}\Delta f}\right|^{2}.\end{split}$
(7)

The sinc function is defined as
$\mathrm{sinc}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{x}$. The
normalization coefficient $C_{n}$ is an immediate consequence of the property
$\int_{-\infty}^{\infty}\delta\left(x\right)dx=1$ of the Dirac delta function.

The approximation (6) is the second key assumption in this work and it can be
motivated as follows: For $f_{2}=0$, both sides of (6) yield infinity, making
the $1$ negligible and as a result it can be pulled out of the absolute square
term. For $f_{2}\neq 0$, the oscillating terms add mostly out-of-phase
(deconstructively) and they are further damped by the $\frac{1}{f_{1}}$ decay
in the $\mathrm{sinc}\left(x\right)$ function. As a consequence, the sum over
the oscillating terms can be approximated by a Dirac delta function, the
oscillating terms are negligible with respect to $1$ and the $1$ can be,
again, pulled out of the absolute square.

As a result of the asymptotic approximation (6), the NLI coefficient can be
approximated by only two contributions. One modulation format correction
originating in the first span and an asymptotic contribution originating in
the limit of a large span number. The NLI coefficient can be thus written as

$\begin{split}&\eta_{\text{corr,}n}^{\left(k\right)}(f_{i})\approx\underbrace{\eta_{\text{corr,}1}^{\left(k\right)}(f_{i})}_{\text{1st
span
corr.}}+\underbrace{\tilde{n}\cdot\eta_{\text{corr,}a}^{\left(k\right)}(f_{i})}_{\text{asympt.
corr.}}\end{split}$ (8)

with

$\begin{split}&\eta_{\text{corr,}1}^{\left(k\right)}(f_{i})=\tilde{\gamma}\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\left|\mu\left(f_{1}+f_{i},f_{k},f_{i}\right)\right|^{2}\end{split}$
(9)
$\begin{split}&\eta_{\text{corr,}a}^{\left(k\right)}(f_{i})=\tilde{\gamma}\left|\mu\left(f_{i},f_{k},f_{i}\right)\right|^{2}\cdot\lim_{n\to\infty}\frac{\partial}{\partial
n}C_{n}\end{split}$ (10)

and

$\begin{split}&\tilde{n}=\begin{cases}0,&\text{if }n=1\\\
n,&\text{otherwise}\end{cases}\end{split}$ (11)

The asymptotic contribution $\eta_{\text{corr,}a}^{\left(k\right)}(f_{i})$ is
valid in the limit for a large number of spans. In order to calculate the
modulation format correction for any number of spans, the asymptotic
contribution is approximation by a Taylor series with respect to the number of
spans and truncated to first-order. As the asymptotic contribution after one
span must be zero, only its slope (in the asymptotic limit) must be calculated
as in (10). This can be done exactly and the necessary identity is derived in
Appendix A.

Using the identity (17), derived in Appendix A, the asymptotic contribution
can be written exactly in closed-form as

$\begin{split}&\eta_{\text{corr,}a}^{\left(k\right)}(f_{i})=\tilde{\gamma}\left|\mu\left(f_{i},f_{k},f_{i}\right)\right|^{2}\\\
&\cdot\frac{2\pi}{\left|\phi\right|B_{k}^{2}}\left[\left(2\Delta
f-B_{k}\right)\mathrm{ln}\left(\frac{2\Delta f-B_{k}}{2\Delta
f+B_{k}}\right)+2B_{k}\right]\end{split}$ (12)

Eq. (12) represents a generic modulation format correction formula, valid in
the asymptotic limit of a large span number. Remarkably, the asymptotic
contribution does not require any further integration and it, therefore,
represents a closed-form correction formula for any optical fiber transmission
system described by a link function $\mu\left(f_{1},f_{2},f_{i}\right)$. Eq.
(12) can be universally applied and it is one of the main results in this
paper.

To further increase the accuracy for arbitrary span numbers, the integral,
describing the modulation format correction contribution after a single span
(9), must be solved. However, this integral resembles the mathematical
structure of the GN model contribution
$\eta_{\mathrm{GN,}n}\left(f_{i}\right)$ (cf. [38, Eq. (14)]) In other words,
if there exists a closed-form approximation for the GN model contribution
after one span for an arbitrary transmission system, a modulation format
correction formula in closed-form for any number of spans immediately follows
with the results in this paper using (8)(9)(12).

###  II-B Link function of the ISRS GN model

In this section, the link function of the GN model in the presence of inter-
channel stimulated Raman scattering is reviewed. Equations (8)(9)(12) are then
applied to the ISRS GN model link function in order to derive a modulation
format correction formula for arbitrary span numbers. The ISRS GN model
represents an extension of the conventional GN model to account for ultra-
wideband effects such as the power transfer between propagating frequencies
due to ISRS. The generic link function of the ISRS GN model is given by [34,
Eq. (9)]

$\begin{split}&\mu\left(f_{1},f_{2},f_{i}\right)=\int_{0}^{L}d\zeta\
\frac{P_{\text{tot}}e^{-\alpha\zeta-
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\cdot(f_{1}+f_{2}-f_{i})}}{\int
G_{\text{Tx}}(\nu)e^{-P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\nu}d\nu}\\\
&\cdot e^{j\tilde{\phi}\left(f_{1},f_{2},f_{i},\zeta\right)},\end{split}$ (13)

where $P_{\text{tot}}$ is the total optical launch power, $C_{r}$ is the slope
of the linear regression of the Raman gain function,
$L_{\text{eff}}=\frac{1-e^{-\bar{\alpha}\zeta}}{\bar{\alpha}}$ and
$\tilde{\phi}=-4\pi^{2}(f_{1}-f_{i})(f_{2}-f_{i})\left[\beta_{2}+\pi\beta_{3}(f_{1}+f_{2})\right]\zeta$.
Eq. (13) can be used to calculate the nonlinear perturbation on $f_{i}$ after
one span for any arbitrary frequency triplet $\left(f_{1},f_{2},f_{i}\right)$.
Eq. (13) accounts for all occuring nonlinear mixing products, namely self-
phase (SPM/SCI), cross-phase (XPM/XCI) and four-wave mixing (FWM/MCI)
products. However, the proposed formulas (8)(9)(12), only correct for the
dominant mixing products which are XPM products. This restricts the frequency
triplets to the XPM domain which is $\left(f_{1}+f_{i},f_{k},f_{i}\right)$
with $f_{1}\in\left[-\frac{B_{i}}{2},\frac{B_{i}}{2}\right]$.

In our previous work [38, Eq. (18)], an approximation of (13) has been derived
under the XPM assumption and a first-order description of ISRS. It was
demonstrated that the ISRS GN model link function is well approximated by

$\begin{split}&\mu\left(f_{1}+f_{i},f_{k},f_{i}\right)\approx-\frac{1+\tilde{T}_{k}}{-\alpha+j\phi_{i,k}f_{1}}+\frac{\tilde{T}_{k}}{-A+j\phi_{i,k}f_{1}}\end{split}$
(14)

where $\tilde{T}_{k}=-\frac{P_{\text{tot}}C_{\text{r}}}{\bar{\alpha}}f_{k}$,
$\phi_{i,k}=-4\pi^{2}\left(f_{k}-f_{i}\right)\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k}\right]$
and $A=\alpha+\bar{\alpha}$. If not specified otherwise, it holds that
$\bar{\alpha}=\alpha$. The parameter $\bar{\alpha}$ can be used to apply the
proposed closed-formula in more general cases. Such cases include improved
accuracy for non-uniform (tilted) launch power distributions, wavelength
dependent attenuation and even the extension of the formula beyond 15 THz i.e.
beyond the triangular region of the Raman gain spectrum. This is done by
reinterpreting $\alpha$, $\bar{\alpha}$ and $C_{r}$ as channel dependent
quantities. The parameters are then matched to reproduce the actual power
profile of each channel and the proposed modulation format correction formula
can be applied. The drawback of this strategy is larger complexity as the
Raman equations must be solved numerically and additional regression
operations are necessary in order to obtain the channel dependent $\alpha$,
$\bar{\alpha}$ and $C_{r}$.

In order to obtain the modulation format correction of the ISRS GN model in
closed-form, the approximated link function (14) must be inserted in (9) and
(12). As mentioned in Section II-A, the integral that needs to be executed in
(9) resembles the GN model contribution after one span. This integral has been
solved in our previous work [38]. Hence, the modulation format correction for
the ISRS GN model can be obtained using the results in this paper and the GN
contribution derived in closed-form in [38].

###  II-C Modulation format correction for the ISRS GN model in closed-form

Using the modulation correction formula for a generic system, derived in
Section II-A, combined with the approximated link function (14) of the ISRS GN
model, yields a modulation format correction formula for the ISRS GN model in
closed-form as

$\begin{split}&\eta_{\text{corr.,}n}\left(f_{i}\right)\approx\frac{80}{81}\Phi\sum_{k=1,k\neq
i}^{N_{\mathrm{ch}}}\left(\frac{P_{k}}{P_{i}}\right)^{2}\frac{\gamma^{2}}{B_{k}}\left\\{\frac{1}{\phi_{i,k}\bar{\alpha}\left(2\alpha+\bar{\alpha}\right)}\right.\\\
&\cdot\left[\frac{T_{k}-\alpha^{2}}{\alpha}\mathrm{atan}\left(\frac{\phi_{i,k}B_{i}}{\alpha}\right)+\frac{A^{2}-T_{k}}{A}\
\mathrm{atan}\left(\frac{\phi_{i,k}B_{i}}{A}\right)\right]\\\
&\left.+\frac{2\pi\tilde{n}T_{k}}{\left|\phi\right|B_{k}^{2}\alpha^{2}A^{2}}\left[\left(2\left|\Delta
f\right|-B_{k}\right)\log\left(\frac{2\left|\Delta
f\right|-B_{k}}{2\left|\Delta
f\right|+B_{k}}\right)+2B_{k}\right]\right\\},\end{split}$ (15)

with $\Delta f=f_{k}-f_{i}$,
$\phi_{i,k}=-2\pi^{2}\left(f_{k}-f_{i}\right)\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k}\right]$111Please
note that $\phi_{i,k}$ is different in Eq. (14) and in Eq. (15)(16). The
variable is redefined to be consistent with prior work [38, Appendix A]. and
$T_{k}=\left(\alpha+\bar{\alpha}-P_{\text{tot}}C_{\text{r}}f_{k}\right)^{2}$.
The formula is applicable for lumped-amplified links for optical bandwidths of
up to 15 THz, as the formula relies on the triangular Raman gain spectrum. For
larger optical bandwidths, the variables $\alpha$, $\bar{\alpha}$ and $C_{r}$
can be matched to the actual power profile in the fiber and the formula can be
applied (cf. Section II-A). The former summand in (15) corrects for the
modulation format within a single span, while the latter summand corrects the
modulation format contribution across multiple spans as described by (8). The
sum in (15) corrects for all interfering channels within the transmitted WDM
signal.

##  III Numerical Validation

In this section the proposed closed-form correction formula (15) is validated
by numerical simulations over the entire C+L band, covering 10 THz optical
bandwidth. The validation is carried out for two fiber types, one high
dispersive standard single mode fiber (SMF) and one low dispersive non-zero
dispersion-shifted fiber (NZDSF).

###  III-A Simulation Setup

Figure 2: The nonlinear interference coefficent as a function of fiber spans
for the channel centered at $f_{i}=-4.0$ THz. The results were obtained by
split-step simulations (markers) and the ISRS GN model [34, Eq. (9)] with the
modulation format correction in integral form (5) (solid lines). The ISRS GN
model with the modulation format correction in closed-form (15) is shown in
dashed lines. In the case of of QPSK, only the channel of interest exhibits
Gaussian modulation for validation purposes. Figure 3: The nonlinear
interference coefficent as a function of fiber spans for the channel centered
at $f_{i}=0$ THz. The results were obtained by split-step simulations
(markers) and the ISRS GN model [34, Eq. (9)] with the modulation format
correction in integral form (5) (solid lines). The ISRS GN model with the
modulation format correction in closed-form (15) is shown in dashed lines. In
the case of of QPSK, only the channel of interest exhibits Gaussian modulation
for validation purposes. For comparison the result of [21, Eq. (1)] is shown,
which proposed a modulation format correction formula in the absence of ISRS.

The validation was carried out by numerically solving the Manakov equation
using the well established split-step Fourier method (SSFM). Inter-channel
stimulated Raman scattering was included in the SSFM by applying a frequency
dependent loss at every linear step, so that the signal power profile altered
by ISRS, is obtained.

A logarithmic step size distribution was implemented, where $0.25\cdot 10^{6}$
simulation steps were found to be sufficient for the considered launch powers
and parameters of both fiber types. The launch powers were chosen to maximize
the performance of the central channel assuming a 5 dB noise figure erbium-
doped fiber amplifier (EDFA).

Gaussian symbols, drawn from a circular-symmetric Gaussian distribution and
uniform QPSK symbols were used for transmission. In the case of QPSK
modulation, the channels of interest, centered at $f_{i}=0$ THz and
$f_{i}=-4.0$ THz, exhibit Gaussian modulation while the rest of the channels
are modulated with QPSK symbols. This was done for validation purposes and
only in the results of this section III. This approach offers a more precise
validation, as the proposed formula only corrects for modulation formats of
interfering channels (XPM/XCI terms).

The receiver consisted of digital dispersion compensation, ideal root-raised-
cosine (RCC) matched filtering and constellation rotation. The SNR was ideally
estimated as the ratio between the variance of the transmitted symbols
$E[|X|^{2}]$ and the variance of the noise $\sigma^{2}$, where
$\sigma^{2}=E[|X-Y|^{2}]$ and $Y$ represents the received symbols after
digital signal processing. The nonlinear interference coefficient was then
estimated via Eq. (1). In order to improve the simulation accuracy, four
different data realizations were simulated and averaged for each transmission.

Ideal, noiseless amplifiers were considered to ease the NLI computation and
for a fair comparison between numerical simulation and modeling results.

TABLE II: System Parameters Parameters | SMF | NZDSF
---|---|---
Loss ($\alpha$) [dB/km] | 0.2 | 0.19
Dispersion ($D$) [ps/nm/km] | 17.0 | 4.5
Dispersion slope ($S$) [ps/$\text{nm}^{2}$/km] | 0.067 | 0.05
NL coefficient ($\gamma$) [1/W/km] | 1.2 | 1.3
Effective core area [$\mu\text{m}^{2}$] | 80 | 72
Raman gain slope ($C_{\text{r}}$) [1/W/km/THz] | 0.028 | 0.031
Raman gain ($C_{\text{r}}\cdot 14$ THz) [1/W/km] | 0.39 | 0.44
Channel Launch Power ($P_{i}$) [dBm] | 0 | -2
Total Launch Power ($P_{\text{tot}}$) [dBm] | 24 | 22
Symbol rate [GBd] | 40
Channel Bandwidth ($B_{i}$) [GHz] | 40.004
Channel spacing [GHz] | 40.005
Number of channels | 251
Optical bandwidth ($B_{\text{tot}}$) [THz] | 10.05
Reference Wavelength [nm] | 1550
Roll-off factor [%] | 0.01
Number of symbols [$2^{x}$] | 17
Simulation steps per span [$10^{6}$] | 2.5

###  III-B Results

The nonlinear interference coefficient as a function of span numbers is shown
in Fig. 2 and 3 for the channels with center frequencies $f_{i}=-4.0$ THz and
$f_{i}=0$ THz. The results are shown for both fiber types and for the case
with ISRS and without ISRS. The (unphysical) case of not considering ISRS is
shown for comparison.

Markers represent simulation results, while lines represent modelling results.
In the case of Gaussian modulation, the ISRS GN model in integral form [34,
Eq. (9)][36, Eq. (2)] was used and is shown in dotted lines. In the case of
QPSK modulation, the ISRS GN model in integral form was used with the
modulation format correction in integral form (4) (solid lines), as well as
the modulation format correction formula in closed form (15) (dashed lines).

The ISRS GN model has remarkable accuracy with numerical simulations
exhibiting a negligible modelling error. In the case of QPSK modulation, the
modulation format correction in integral form (5) models the impact of QPSK
with good accuracy, despite one of the key assumptions made in this work
($\Delta f\gg\frac{B_{k}}{2}$). The average deviation between the modulation
format correction in integral form and the numerical simulation is $0.26$ dB
throughout the shown results. The error mostly stems from the XPM assumption
and assumptions inherited by Eq. (4).The small impact of the assumption
$\Delta f\gg\frac{B_{k}}{2}$ has been mathematically shown in [34, Appendix C]
and is, therefore, not surprising.

Figure 4: The nonlinear interference coefficient as a function of channel
frequency for different modulation formats and fiber types after 6x100 km
spans. The results were obtained by split-step simulations and in closed-form
using Eq. (16). Figures a) and b) show the case without and with considering
inter-channel stimulated Raman scattering, respectively.

The modulation format correction formula in closed-form (15) shows good
accuracy throughout all number of spans. Throughout Fig. 2 and 3, the average
absolute error is 0.45 dB between closed-form and numerical simulation. The
majority of the mismatch can be traced back to the asymptotic assumption (6)
and its linear approximation of the asymptotic contribution (10) with respect
to the span number (see Section II-A). The mismatch is smaller for an
increasing number of spans. Particularly, the error vanishes in the case of
single span transmission and in the limit of infinte transmission spans. Not
shown in the figures, the mismatch between modulation format correction in
closed-form and integral form is $<0.1$ dB after 100 spans.

The result in this paper is the first modulation format correction formula in
the presence of ISRS. However, in the absence of ISRS (e.g. for optical
bandwidths of at most 5 Thz), modulation format correction formulas are
already available in the literature. To compare our results to previously
published works in the absence of ISRS, [21, Eq. (1)] is shown in Fig. 3a),
which proposes a modulation format correction formula in the absence of ISRS.
As the result in [21] has been derived in the asymptotic limit of a large
number of spans, it is rather inaccurate for the first few spans. The
comparison shows that (15) is not only capable of correcting the modulation
format in ISRS impaired systems but it also more accurate for than previously
published results in the absence of ISRS.

##  IV The ISRS GN model for arbitrary modulation formats in closed-form

In this section, we combine the modulation format correction formula (15) with
the ISRS GN model contribution in closed form derived in [38]. The result is
closed-form formula capable of predicting the total nonlinear performance for
arbitrary modulation formats. The proposed formula is once again validated by
numerical simulations for the two fiber types with parameters listed in table
1. The formula for the total NLI coefficent as in (2) is

$\begin{split}&\eta_{n}\left(f_{i}\right)\approx\frac{4}{9}\frac{\gamma^{2}}{B^{2}_{i}}\frac{\pi
n^{1+\epsilon}}{\phi_{i}\bar{\alpha}\left(2\alpha+\bar{\alpha}\right)}\\\
&\cdot\left[\frac{T_{i}-\alpha^{2}}{\alpha}\text{asinh}\left(\frac{\phi_{i}B_{i}^{2}}{\pi
a}\right)+\frac{A^{2}-T_{i}}{A}\text{asinh}\left(\frac{\phi_{i}B_{i}^{2}}{\pi
A}\right)\right]\\\ &+\frac{32}{27}\sum_{k=1,k\neq
i}^{N_{\mathrm{ch}}}\left(\frac{P_{k}}{P_{i}}\right)^{2}\frac{\gamma^{2}}{B_{k}}\left\\{\frac{n+\frac{5}{6}\Phi}{\phi_{i,k}\bar{\alpha}\left(2\alpha+\bar{\alpha}\right)}\right.\\\
&\cdot\left[\frac{T_{k}-\alpha^{2}}{\alpha}\mathrm{atan}\left(\frac{\phi_{i,k}B_{i}}{\alpha}\right)+\frac{A^{2}-T_{k}}{A}\
\mathrm{atan}\left(\frac{\phi_{i,k}B_{i}}{A}\right)\right]\\\
&\left.+\frac{5}{3}\frac{\Phi\pi\tilde{n}T_{k}}{\left|\phi\right|B_{k}^{2}\alpha^{2}A^{2}}\left[\left(2\left|\Delta
f\right|-B_{k}\right)\log\left(\frac{2\left|\Delta
f\right|-B_{k}}{2\left|\Delta
f\right|+B_{k}}\right)+2B_{k}\right]\right\\},\end{split}$ (16)

with $\phi_{i}=\frac{3}{2}\pi^{2}\left(\beta_{2}+2\pi\beta_{3}f_{i}\right)$,
$T_{k}=\left(\alpha+\bar{\alpha}-P_{\text{tot}}C_{\text{r}}f_{k}\right)^{2}$,
$\Delta f=f_{k}-f_{i}$,
$\phi_{i,k}=-2\pi^{2}\left(f_{k}-f_{i}\right)\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k}\right]$
and $A=\alpha+\bar{\alpha}$. Eq. (16) models the GN model contribution of the
SPM and XPM contributions, where the XPM terms are corrected for its
modulation format dependence using (15). The reader is referred to [38] for
more details about the GN model contribution.

The total NLI coefficient as a function of channel frequency with and without
ISRS is shown in Fig. 4. The results are shown for SMF and NZDSF. All
transmitted channels are entirely modulated with either Gaussian, 16-QAM or
64-QAM symbols. Overall, Eq. (16) shows very good agreement with the numerical
results. In both cases (with and without ISRS), the average mismatch between
the closed-form model (16) and numerical simulation is 0.3 dB and 0.2 dB for
SMF and NZDSF, respectively.

The analysis shows that the derived closed-form approximation (16) is capable
of predicting nonlinear performance of ultra-wideband optical transmission
systems considering arbitrary modulation formats with sufficient accuracy. A
key result in the real-time modelling of next generation ultra-wideband
transmission systems.

##  V Conclusion

A methodology was presented to derive the modulation format correction in
closed-form for arbitrary optical transmission systems that are described by a
link function. The generic approach is applied to the Gaussian Noise model in
the presence of inter-channel stimulated Raman scattering (ISRS GN model) to
derive a closed-form formula that predicts the impact of modulation formats on
the nonlinear interference in ultra-wideband transmission. The derived formula
accounts for arbitrary span numbers, arbitrary launch power distributions,
inter-channel stimulated Raman scattering and wavelength dependent dispersion
and attenuation.

The analytical result was validated by numerical simulations over the entire
C+L band (10 THz), with an average prediction error in nonlinear interference
of 0.3 dB for standard single-mode fiber (SMF) and 0.2 dB for non-zero
dispersion-shifted fiber (NZDSF).

The results in this paper are the first analytical investigation of the impact
of modulation formats in ultra-wideband optical transmission and the first
closed-form formula accounting for it. A significant result for the design of
optical communication systems and the real-time performance modeling of ultra-
wideband transmission systems.

##  Appendix A Proof

In this section, an identity for $\lim_{n\to\infty}\frac{\partial}{\partial
n}C_{n}$ is derived in order to obtain closed-form formula for the asymptotic
modulation format correction contribution (10). The asymptotic differential
normalization coefficient can be calculated exactly and no approximations are
needed.

For notational brevity, the identity is derived for arbitrary parameters $a$
and $b$ which reads

$\begin{split}C^{{}^{\prime}}_{\infty}&=\lim_{n\to\infty}\frac{\partial}{\partial
n}\int_{-\infty}^{\infty}dx\left|\sum_{m=1}^{n}\mathrm{sinc}\left(max\right)e^{jmbx}\right|^{2}\\\
&=\frac{\pi}{a^{2}}\left[\left(b-a\right)\mathrm{ln}\left(\frac{b-a}{a+b}\right)+2a\right].\end{split}$
(17)

Executing the derivatve with respect to the span number in (17) yields

$\begin{split}&C^{{}^{\prime}}_{\infty}=\lim_{n\to\infty}\frac{\partial}{\partial
n}\int_{-\infty}^{\infty}dx\left|\sum_{m=1}^{n}\mathrm{sinc}\left(max\right)e^{jmbx}\right|^{2}\\\
&=\lim_{n\to\infty}\int_{-\infty}^{\infty}dx\frac{1}{2a^{2}nx^{2}}\sum_{m=1}^{n}\frac{1}{m}\left\\{\right.\\\
&\left[\sin\left(mc_{2}x\right)+\sin\left(mc_{1}x\right)\right]\left[\sin\left(nc_{2}x\right)+\sin\left(nc_{1}x\right)\right]\\\
+&\left[\cos\left(mc_{2}x\right)-\cos\left(mc_{1}x\right)\right]\left.\left[\cos\left(nc_{2}x\right)-\cos\left(nc_{1}x\right)\right]\right\\},\end{split}$
(18)

with $c_{1}=a+b$ and $c_{2}=a-b$, which are defined for the sake of a concise
notation. Using trigonometric identities and the exact integral identity
$\int_{-\infty}^{\infty}dx\frac{\sin\left(ax\right)\sin\left(bx\right)}{x^{2}}=\frac{1}{2}\pi\left(\left|a+b\right|-\left|a-b\right|\right)$
yields

$\begin{split}&C^{{}^{\prime}}_{\infty}=\lim_{n\to\infty}\frac{\pi}{2a^{2}n}\sum_{m=1}^{n}\frac{1}{m}\left\\{\right.\\\
&-\left|mc_{2}-nc_{2}\right|+\left|mc_{1}+nc_{2}\right|+\left.\left|mc_{2}+nc_{1}\right|-\left|mc_{1}-nc_{1}\right|\right\\}.\end{split}$
(19)

To resolve the absolute value operations in (19), we identify that $b\geq 0$,
$a\geq 0$ and $b\geq a$, resulting in $c_{1}\geq 0$ and $c_{2}\leq 0$. The
conditions are met by default as the minimum feasible channel spacing is
$\Delta f\geq\frac{B_{k}}{2}$ and the result is invariant to the sign of $b$.
Applying the conditions results in

$\begin{split}C^{{}^{\prime}}_{\infty}&=\lim_{n\to\infty}\frac{\pi}{2a^{2}n}\sum_{m=1}^{n}\frac{1}{m}\\\
&\left\\{b\left(m-n\right)+\left(n+m\right)a+\left|mc_{1}-n\left|c_{2}\right|\right|\right\\}.\end{split}$
(20)

Eq. (20) can be written as two distinct series, which are solved separately in
the following. The first series is exactly written and solved as

$\begin{split}&\lim_{n\to\infty}\frac{\pi}{2a^{2}n}\sum_{m=1}^{n}\frac{1}{m}\left\\{b\left(m-n\right)+\left(n+m\right)a\right\\}\\\
&=\lim_{n\to\infty}\frac{\pi}{2a^{2}}c_{1}-\left|c_{2}\right|\mathrm{Har}\left(n\right),\\\
\end{split}$ (21)

with $\mathrm{Har}\left(n\right)$ being the n’th harmonic number. The second
series can be written and exactly solved as

$\begin{split}&\lim_{n\to\infty}\frac{\pi}{2a^{2}n}\sum_{m=1}^{n}\frac{1}{m}\left|mc_{1}-n\left|c_{2}\right|\right|\\\
&=\lim_{n\to\infty}\frac{\pi}{2a^{2}}\left(\sum_{m=1}^{\left\lfloor\frac{n\left|c_{2}\right|}{c_{1}}\right\rfloor}\frac{\left|c_{2}\right|}{m}-\frac{c_{1}}{n}+\sum_{m=\left\lceil\frac{n\left|c_{2}\right|}{c_{1}}\right\rceil}^{n}\frac{c_{1}}{n}-\frac{\left|c_{2}\right|}{m}\right)\\\
&=\lim_{n\to\infty}\frac{\pi}{2a^{2}}\left\\{\left|c_{2}\right|\mathrm{Har}\left(\left\lfloor\frac{n\left|c_{2}\right|}{c_{1}}\right\rfloor\right)-\left|c_{2}\right|\mathrm{Har}\left(n\right)\right.\\\
&+\left|c_{2}\right|\mathrm{Har}\left(\left\lceil\frac{n\left|c_{2}\right|}{c_{1}}\right\rceil\right)\left.+\frac{c_{1}}{n}\left(n-\left\lceil\frac{n\left|c_{2}\right|}{c_{1}}\right\rceil-\left\lfloor\frac{n\left|c_{2}\right|}{c_{1}}\right\rfloor\right)\right\\}\\\
&=\lim_{n\to\infty}\frac{\pi}{2a^{2}}\left[2\left|c_{2}\right|\mathrm{Har}\left(\frac{n\left|c_{2}\right|}{c_{1}}\right)-\left|c_{2}\right|\mathrm{Har}\left(n\right)+c_{1}-2\left|c_{2}\right|\right]\end{split}$
(22)

, where it was used that
$\lim_{n\to\infty}\left\lceil\frac{nc_{2}}{c_{1}}\right\rceil=\lim_{n\to\infty}\left\lfloor\frac{nc_{2}}{c_{1}}\right\rfloor=\lim_{n\to\infty}\frac{nc_{2}}{c_{1}}$.
Both series can be further simplified by recalling that
$\lim_{n\to\infty}\mathrm{Har}\left(n\right)=\lim_{n\to\infty}\log\left(n\right)+\gamma$,
with $\gamma$ being the Euler-Mascheroni constant.

Combining (LABEL:eq:series_1) and (22) yields the final result

$\begin{split}C^{{}^{\prime}}_{\infty}&=\lim_{n\to\infty}\frac{\pi}{a^{2}}\left\\{\left|c_{2}\right|\mathrm{Har}\left(\frac{n\left|c_{2}\right|}{c_{1}}\right)-\left|c_{2}\right|\mathrm{Har}\left(n\right)+2a\right\\}\\\
&=\frac{\pi}{a^{2}}\left[\left|c_{2}\right|\mathrm{ln}\left(\frac{\left|c_{2}\right|}{c_{1}}\right)+2a\right]\\\
&=\frac{\pi}{a^{2}}\left[\left(b-a\right)\mathrm{ln}\left(\frac{b-a}{a+b}\right)+2a\right]\end{split}$
(23)

which proves the identity in (17). Therefore, Eq. (17) can be used to write
(10) in closed-form as (12), without imposing any additional assumptions.

## Acknowledgment

Financial support from UK EPSRC through programme grant TRANSNET
(EP/R035342/1) and a Doctoral Training Partnership (DTP) studentship to Daniel
Semrau is gratefully acknowledged.

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Daniel Semrau (S’16) received the B.Sc. degree in electrical engineering from
the Technical University of Berlin, Berlin, Germany, in 2013, the M.Sc. degree
in photonic networks engineering from Scuola Superiore Sant’Anna, Pisa, Italy,
and Aston University, Birmingham, U.K., in 2015. In 2015, he joined the
Optical Networks Group, University College London, U.K., where he is currently
working toward the Ph.D. degree. In 2018, Daniel was presented with the
Graduate Student Fellowship award of the IEEE Photonics Society. His research
interests are mainly focused on channel modeling, nonlinear compensation
techniques, and ultra-wideband transmission coherent optical communications.
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Eric Sillekens (S’16) received his BSc and MSc in electrical engineering from
the Eindhoven University of Technology in 2012 and 2015 respectively, with his
research focussed on advanced coded modulation for optical fibre transmission
systems. He is currently a PhD research student in the optical networks group
at Univesity College London (UCL) and is supervised by Dr. R. Killey. He is
working to holistically optimise long-haul fibre transmission systems, where
his interest is in coded modulation and machine learning. He has designed
modulation formats with a trade off between fibre nonlinearity and shaping
gain.
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Robert I. Killey (SM’17) received the B.Eng. degree in electronic and
communications engineering from the University of Bristol, Bristol, U.K., in
1992, the M.Sc. degree from University College London (UCL), London, U.K., in
1994, and the D.Phil. degree from the University of Oxford, Oxford, U.K., in
1998. He is currently an associate professor with the Optical Networks Group
with UCL. His research interests include nonlinear fiber effects in WDM
transmission, advanced modulation formats, and digital signal processing for
optical communications. He has participated in many European projects,
including ePhoton/ONe, Nobel, BONE and ASTRON, and national projects. He is
currently a Principal Investigator in the EPSRC funded UNLOC project. He was
with the technical program committees of many international conferences
including European Conference on Optical Communication, Optical Fiber
Communication Conference ACP, and OECC. He was an Associate Editor of the
IEEE/OSA Journal of Optical Communications and Networking and is currently an
Associate Editor of the Journal of Lightwave Technology.
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Polina Bayvel (F’10) received the B.Sc. (Eng.) and Ph.D. degrees in electronic
and electrical engineering from UCL (University of London), in 1986 and 1990,
respectively. In 1990, she was with the Fiber Optics Laboratory, General
Physics Institute, Moscow, Russian Academy of Sciences, under the Royal
Society Postdoctoral Exchange Fellowship. She was a Principal Systems Engineer
with STC Sub- marine Systems, Ltd., London, U.K., and Nortel Networks (Harlow,
U.K., and Ottawa, ON, Canada), where she was involved in the design and
planning of optical fibre transmission networks. During 1994–2004, she held a
Royal Society University Research Fellowship at University College London
(UCL), London, U.K., where she became a Chair in Optical Communications and
Net- works. She is currently the Head of the Optical Networks Group, UCL,
which she set up in 1994. She has authored or coauthored more than 300
refereed jour- nal and conference papers. Her research interests include
wavelength-routed optical networks, high-speed optical transmission, and the
study and mitigation of fibre nonlinearities. She is a Fellow of the Royal
Academy of Engineering, IEEE, the Optical Society of America and the U.K.
Institute of Physics. She is Honorary Fellow of the Institution of Engineering
and Technology (FIET). She was a recipient the Royal Society Wolfson Research
Merit Award (2007–2012), the 2013 IEEE Photonics Society Engi- neering
Achievement Award, the 2014 Royal Society Clifford Patterson Prize Lecture and
Medal and 2015 Royal Academy of Engineering Colin Campbell Mitchell Award. She
leads the UK EPSRC Programme TRANSNET (2018-2024).
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