Optimal integrating techniques for supercontinuum simulations

ORIGINAL PAPER
Optimal integrating techniques for supercontinuum simulations
M S J AL-Taie*
Gifted Guardianship Committee, Directorate of Education Missan, Missan, Iraq
Received: 04 May 2023 / Accepted: 23 January 2024
Abstract: Study theoretical solutions of the generalized nonlinear Schrodinger equation, with a particular emphasis to
While modeling supercontinuum creation in optical photonic crystal fibers. The improvement in integration of the non-
linear section of the equation in the time domain (TD) compared to the frequency domain (FD) demonstrates why
integrating the nonlinear operation into the FD is more effective than integrating it into the TD. The effectiveness of several
adaptive step-size methods was examined in conjunction with the interaction graphic integration approach to demonstrate
how it substantially affects whether nonlinear operation integration is carried out in the FD or TD. One discovery is that the
newly invented preservation quantity error adaptive step-size algorithm and addressing the nonlinearity in the FD produce
the most effective method for supercontinuum modeling in optical.
Keywords: Photonic crystal fibers (PCFs); Frequency domain (FD); Time domain (TD); Step-size algorithm;
Supercontinuum modeling
1. Introduction
Various events that arise after interaction of strong light
with a substance are included in the field of nonlinear
optics. The field of nonlinear optics has steadily increased
in attention since the first experimental observation of
second harmonic production by Franken et al. in 1961 [1].
This is due in part to the field’s practical use in the creation
of revolutionary technologies. The field’s ongoing devel-
opment and thriving are mostly due to its extremely rich
physics and our growing comprehension of it. Primarily, it
advances the development of existing light sources, mate-
rial platforms, nanofabrication technological advances, and
the growing computational capacity for simulations. This
combination constantly pushes the boundaries of nonlinear
optical effects’ efficiency, controllability, comprehension,
and applications. In addition to producing a variety of
remarkable phenomena, the alteration of the material
properties in the presence of a light field—specifically, the
induced nonlinear polarization—allows for ultrafast and
versatile light manipulation and the creation of light in
unusual spectral windows via nonlinear conversion of
frequencies or extremely wide-spectrum broadening.
Supercontinuum generation (SCG) is the process by which
a relatively intense input laser pulse experiences significant
spectral broadening while retaining high brightness and
spatial coherence. It has been studied in many different
nonlinear media and has found uses in optical frequency
comb technologies, spectroscopy, imaging, and optical
coherence tomography, among other fields. Although
spectral broadening (SCG) may be easily observed when a
high-peak-power pulsed light source is available, the
medium’s linear and nonlinear characteristics have a major
role in controlling the SCG. In that regard, waveguiding
systems have transformed surface coherent graphene
(SCG) through the provision of improved light field con-
finement across comparatively extended propagation
lengths when contrasted with bulk systems, as well as the
ability to adjust the dispersion to optimize and regulate the
nonlinear interactions. While fluoride fiber-based middle
infrared (MIR) supercontinuum (SC) generators have also
been accessible, the majority of modern, commercially
available SC sources are based on silica glass photonic
crystal fibers (PCFs) [2–4].
Early on in the history of SCG, SCG was demonstrated
to function better than SC from bulk or traditional silica
fibers thanks to PCFs’ dispersion engineering capabilities.
The widest spectra are now produced in extremely non-
linear chalcogenide fibers, and much research is still con-
ducted in fiber-based SCG [5, 6]. Integrated waveguides
immediately emerged as a potent and alluring strategy for
*Corresponding author, E-mail: msjadr72@gmail.com
Indian J Phys
https://doi.org/10.1007/s12648-024-03108-4
Ó 2024 IACS

SCG, with the potential to enhance light field confinement
for more effective devices, to reduce power consumption of
power accordingly, and potentially enable dense integra-
tion with additional functionalities across compact chip-
based platforms. Now, integrated SC sources are finally
available, thanks to recent advancements in fabrication
technology. This assessment highlights the enormous
potential of this technology with several successful recent
demonstrations. The primary focus is on our present
approach to the creation of optical waveguide devices for
the whole optical communication industry. Since the pro-
cess of introducing light and causing it to propagate in a
particular manufacturing medium are both done using
optical waveguides, the large amounts of data that must be
processed increase the importance of calculation time [7].
Studying supercontinuum formation calls for quick simu-
lation techniques. Supercontinuum may be generated using
a variety of methods, such as CW, picoscale, femtoscale,
and nanoscale lasers. For supercontinuum generation,
nonlinear mediums include PCF, smooth glass fiber, gains
fiber, regular optical fiber, etc. The supercontinuum laser’s
wavelength range can readily expand to ultraviolet, mid-
infrared, and far-infrared regions, in addition to being
visible in nearly infrared region. Numerous useful uses for
supercontinuum light sources have also been achieved,
including nonlinear spectroscopy, optical coherence
tomography, accurate time and frequency measurement,
and optical fiber networking. This can be the scenario,
particularly in studies on viscous dissipation soliton reso-
nants [8] or optical renegade waves, which employ statis-
tical techniques [9], where a large number of mathematical
simulations must be run in order to determine the param-
eters that determine where certain solitons occur. Under
certain circumstances, traditional computing techniques
can become unworkable and sluggish [10].
A nonlinear propagation equation that incorporates self-
steepening, immediate and delayed Raman responses,
instantaneous and delayed Kerr effects, and dispersive
effects may be used to predict supercontinuum formation in
optical PCFs. Typically, this equation is written in the time
domain (TD) [11]
oA
oz
þ b1
oA
ot
þ ib2
o2
A
ot2
b3
o3
A
ot3
þ
¼ ic x
ð Þ 1 þ
i
xo
o
ot

ðA z; t
ð Þ r
þ1
1
Rð
tÞAðz; t
tÞd
tÞ ð1Þ
where
Rð
tÞ ¼ 1 fR
ð Þd t
ð Þ þ fRhRð
tÞ
and A(z, t) is the complicated envelope of the electric field.
The retracted time for an indicator framework moving at
the envelope’s group velocity is t. bk are often dispersed
related settings with the Taylor collection extension for the
propagating consistent b(x) in the vicinity of the frequency
of the carrier xo, c is the nonlinear component for the
original signal of the fiber, and the fractional contributions
of the delay Raman response hR are symbolized by
0 fR 1 is commonly approximated by
hR t
ð Þ / exp t=s2

sinðs1Þ. In this study, we choose
more precise function.
hRð
tÞ ¼ fa þ fc
ð Þha t
ð Þ þ fbhb t
ð Þ
hað
tÞ ¼ s1 s2
1 þ s2
2

exp s=s2

sin s=s2

hb t
ð Þ ¼
2sb t
s2
b

exp t=sb

ð2Þ
Here s1 = 12.2 fs, s2 = 32 fs, sb = 96 fs, fa = 0.75,
fb = 0.21, fc = 0.04, and fR = 0.24 [6]. The main
drawback of adopting (1) is that temporal variations
generate mathematical inaccuracy due to time frame
partitioning. When (1) is transformed into the frequency
domain (FD), these derivatives vanish, leaving just the
discrete longitudinal step size as a cause of numerical
inaccuracy. By specifying X ¼ x xo, A(z, X) determined
with the Fourier transformation on A(z, t), and removing
the evidence from both viewpoints A(z, X) and A(z, t), the
FD formulation of Eq. (1) turns into [12]
o
A
oz
þ i
A bðX
ð Þ bo b1 X
ð Þ ¼ ic 1 þ
X
xo

F 1 fR
ð ÞAjAj2
þ fRAF1
hRF jAj2
h i
h i
h i
ð3Þ
where F denotes the direct transformation function, F1 is
the inverse Fourier transform, and
hR ¼ F hR t
ð Þ
ð Þ. F and
F-1
are calculated numerically to use the quick Fourier
transformation (FFT) and inverse Fourier transform
(IFTT), respectively. As we will see in the following sec-
tion, the FD may benefit more from the adoption of
dynamic step-size techniques. Equation (3) has the added
benefit of including the frequency response of the nonlinear
component c(x) in a straightforward manner using the
simple substitution c ! c x
ð Þ. This technique only provides
an approximate answer, although it has been shown to
enable accurate modeling of a function of frequency losses,
dispersal, and non-linearity [13, 14].
2. The interactions of graphical technique algorithm
employ fourth-order Runge–Kutta
In recent years, the split-step Fourier technique (SSFM) has
been widely utilized to solve (1) [15, 16]. The FD solves
the dispersive component of the problem, whereas the
nonlinear part, the right-hand side (RHS) of (1), is there-
fore solved in the TD. A speedier integrated approach,
M S J AL-Taie

tightly related to the SSFM and initially designed in the
study of Bose–Einstein condensates, was recently found to
be more effective. In the interactive graphic technique, it is
known as the fourth-order Runge–Kutta (R–K) [17, 18].
A z þ h; t
ð Þ ¼ F1
exp
h
2

D

F A1 þ k1=6 þ k2=3 þ k3=3
h i
þ k4=6
ð4Þ
where
A1 ¼ F1
exp
h
2

D

A z; X
ð Þ

;
k1 ¼ F1
exp
h
2

D

F hN A z; t
ð Þ
ð Þ
½

;
k2 ¼ hN A1 þ
k1
2

; k3 ¼ hN A1 þ
k2
2

;
k4 ¼ F1
hN exp
h
2

D

F A1 þ k3
½

D ¼ i bðX
ð Þ bo b1 X
ð Þ
ð Þ and N A z; t
ð Þ
ð Þ is the
nonlinear operator, namely the RHS of (1) applying to
A(z, t). In addition, that (RK) could be utilized in the FD.

N
A z; x
ð Þ

is described as the use of the nonlinear operator
in the FD, i.e., the RHS of (3).

A z þ h; t
ð Þ ¼ exp
h
2

D

A1 þ
k1
6
þ
k2
3
þ
k3
3

þ
k4
6
ð5Þ
where

A1 ¼ exp
h
2

D

A z; X
ð Þ; k1 ¼ exp
h
2

D

½h
N
A z; t
ð Þ

;
k2 ¼ h
N
A1 þ
k1
2

k3 ¼ h
N
A1 þ
k2
2

; k4 ¼ h
N exp
h
2

D

ð1þk3Þ

In both situations, the total amount of FFTs produced using
(4) or (5) is 16. The improvement in IFFTs in (4) is mat-
ched by four extra FFTs in (5) that have to be calculated
when the nonlinear integration in the FD is implemented
which has a MATLAB version of these equations [19, 20].
Even though the nonlinear operator N is used in the TD, the
inversion in the RHS of (1) is normally done in the FD.
3. Techniques for adaptive step-size
The step size h may be modified dynamically to improve
simulation times. Its value may be determined by esti-
mating the error at each process level. For example, the
locally error methodology (LEM) [21] predicts the local
error by computing a rough solution with a full step h and
then separately computing a refined solution with two half-
steps. By comparing the two outcomes, the amount of the
inaccuracy may be calculated. Another strategy is used by
the conserved quantitative error methodology (CQEM)
[22, 23]. To ascertain the error, it computes the quantum
photon before and after the integration has been performed.
Due to the fact that (1) and (2) keep the photon count
constant, irrespective of change, it may be read as a
mathematical mistake and its size is determined. The
techniques used to estimate the step size, including both
CQEM and LEM, are described in [24, 25]. By simply
examining the imagined component of the nonlinear
operator, one approximated the maximal inaccuracy. This
assumption holds true for modest systems, but as we will
discover in this study, it suffers when large energies are
considered.
4. Results and discussion
By simulating supercontinuum production in optical fibers,
one evaluates the effectiveness of FD or TD integration or
the nonlinear operation N. We investigated a variety of
system settings and reported results in two example situa-
tions. The modeling settings are those for modeling fem-
tosecond pulse propagation in the abnormal dispersion
section of a highly nonlinear photonic crystal fiber (PCF).
The proper dispersion constants were determined using the
fiber zero-dispersion wavelength of 780 nm. The wave-
length for input pulse is 1550 nm, amplitude A 0; t
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Psech t=to

r
while to ¼ 28:4 fs,P ¼ 25; 75
ð Þ Kw. In the
beginning, it should be noted the effect of changing the
amount of input power on the output spectrum depending
on Eq. 1, which is represented by the ultra-wide spectral
expansion, as the results appear in Figs. 1 and 2.
The solution is found via TD or FD integration of the
nonlinear operator, followed by testing the LEM and
CQEM for adaptable modification of h. We compute the
global error for each simulation (e) to compare outcomes,
which is specified as [26].
e ¼
X
N
k¼1
jjAsim
k l2
jAtrue
k j2
j
Nmax jAtrue
k j2
ð6Þ
where n refers to number of field samples, Asim
the mod-
eled to the field for the photonic crystal fiber output, and
Atrue
is the a references solve computed with the best
attainable must-have. Nevertheless, this notion of global
error, which was also used in [24] and [26], is unaffected
Optimal integrating techniques

by phase faults. As a result, one confirmed simulation
convergence using an alternate global error definition that
accounts for phase errors [26] and achieved comparable
findings. One confirmed that the simulation results are
unaffected by the approach used to generate Atrue
as long as
the same accuracy is attained and the same FD or TD
mechanism is utilized. The computational cost of a par-
ticular scenario is determined by how many FFTs were
performed throughout the simulation’s duration, as previ-
ously published [23, 24]. This value must be proportionate
to the simulation time.
The collected findings allow us to identify the suit-
able approach in our present investigation, as indicated in
the following results for different power values.
Figures 3 and 4 illustrate that for P = 25 kW. For the
same quantity of FFTs, the CQEM delivers a degree
magnitude greater precision than the LEM. These findings
support [20]’s observation that the CQEM works better for
more complicated issues.
This restricts the minimum number of FFTs required by
a certain technique to achieve convergence. The field
diverges while attempting to conduct fewer FFTs by
reducing the desired local error. For P = 75 kW, the LEM
is far more effective than the CQEM-TD for such a given
global error when nonlinear integration is performed in the
TD, which is explained by the reality that, for greater
energy powers, the real part of the nonlinear technician
could be ignored and the estimate made in the execution of
the CQEM-TD no longer provides; Figs. 4 and 5 show that
The generalized nonlinear Schrodinger equation
(GNLSE) (1) TD expression comprises temporal instru-
ments such as derivatives, which are exclusively estimated
roughly when dealing with distinct amounts with a limited
supply amount of representative qualities. As a result, extra
speculative numerical errors occur that are independent of
the longitudinal step size and cannot be eliminated by an
adaptive step-size approach. By converting the GNLSE
into the FD (3), time implications may be prevented.
Because the only residual source of the constrained step
size results in numerical inaccuracies, which can be easily
managed by adaptive techniques, any of these approaches
are predicted to perform better in the FD. This is demon-
strated in our simulations. The effectiveness of the FD
counterparts of all examined adaptable step-size algorithms
Fig. 1 Potential applications for the supercontinuum generation process
500 1000 1500
Wavelength ( nm)
0
10
20
30
40
50
60
Spectral
intensity(dB)
-5 0 5
Delay ( ps)
-100
-50
0
50
Temporal
intensity(dB)
Fig. 2 Simulated supercontinuum generation when P = 25 kW
M S J AL-Taie

is higher than that of TD implementations, as shown in
Figs. 3 and 4.
For the methodologies studied, the performance gap
between TD and FD implementations varies greatly.
Another side effect of resolving the GNLSE in the FD is
that the FD becomes more stable. The solutions discovered
in TD and FD are not the same. The longitudinal step size
is determined at the maximum numerical accuracy for TD
and FD implementations. This difference was shown to be
relatively minimal for P = 25 kW. Applying TD or FD
nonlinear integration, the frequency range in Fig. 1 is
essentially identical; in other words, the spectrum dis-
played in P = 25 kW is essentially same whether TD or FD
nonlinear integration is used. For P = 75 kW, however, the
change is considerable. Because (1) is a 2-D issue, this is
the case. Since the sample size is fixed, the subset-related
errors related to t are constant even if the error rate in z is
lowered. The solution obtained in the TD remains less
accurate than the response found in the FD because
numerical derivative algorithms are approximated by
500 1000 1500
Wavelength ( nm)
10
20
30
40
50
60
Spectral
intensity(dB)
-5 0 5
Delay ( ps)
0
10
20
30
40
50
Temporal
intensity(dB)
Fig. 3 Simulated supercontinuum generation when P = 75 kW
Fig. 4 Simulated output waveforms, P = 25 kW, (a) using CQEM-FD integration, (b) using CQEM-TD integration
Fig. 5 Simulated output waveforms, P = 25 kW, (a) using LEM-FD integration, (b) using LEM-TD integration

character. By using additional points of measurement or
shorter periods of time, temporal precision may be
increased, fidelity of the analogues computations improves,
and the TD solution converges toward the FD solution
(lacking sampling points), as shown in P = 75 kW. When
(4) is used, i.e., the nonlinear operator is solved in TD, the
LEM is substantially more advantageous than the CQEM.
We relate this to the fact that the CQEM is susceptible to
mistakes generated by numerical derivative computed
errors in the TD solving N, as these mistakes modify the
photon number. As a result, in trying to compensate, the
algorithm maintains a small on size tiny, but because
mistakes are not connected due to the restricted size of
steps, this approach leads to diminished efficiency. The
LEM, on the other hand, is only sensitive to faults induced
by the discrete step size and hence operates more effective
when additional defects other than step size related are
present (Figs. 6 and 7).
5. Conclusions
As previously stated, resolutions to the GNLSE with non-
linearity distribution are easily calculated of the FD. In this
situation, both CQEM and LEM were efficiently used for
step-size adaptation. In this paper, it was demonstrated that
using the FD methodology integrated is preferable, even
though the nonlinear coefficient is expected to remain
constant across the whole spectral spectrum, because the
mathematical model in the FD is intrinsically more precise
as well as quicker than the TD strategy for a given
appropriate global error.
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Fig. 6 Simulated output waveforms, P = 75 kW, (a) using CQEM-FD integration, (b) using CQEM-TD integration
Fig. 7 Simulated output waveforms, P = 75 kW, (a) using LEM-FD integration, (b) using LEM-TD integration
M S J AL-Taie

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