# Generation of robust temporal soliton trains by the multiple-temporal-
compression (MTC) method

André C. A. Siqueira<EMAIL_ADDRESS>Guillermo Palacios  Albert
S. Reyna  Boris A. Malomed  Edilson L. Falcão-Filho  Cid B. de Araújo

###### Abstract

We report results of systematic numerical analysis for multiple soliton
generation by means of the recently reported multiple temporal compression
(MTC) method, and compare its efficiency with conventional methods based on
the use of photonic crystal fibers (PCFs) and fused silica waveguides (FSWs).
The results show that the MTC method is more efficient to control the soliton
fission, giving rise to a larger number of fundamental solitons with high
powers, that remain nearly constant over long propagation distances. The high
efficiency of the MTC method is demonstrated, in particular, in terms of
multiple soliton collisions and the Newton’s-cradle phenomenology.

###### keywords:

Multiple Solitons , Ultrashort pulses , Supercontinuum generation , Newton’s
cradle

††journal: Optics Communications

[label1] organization=Departamento de Física, Universidade Federal de
Pernambuco,city=Recife, postcode=50670-901, state=Pernambuco, country=Brazil

[label2]organization=Comissão Nacional de Energia Nuclear, Centro Regional de
Ciências Nucleares do Nordeste – CNEN/CRCN-NE,city=Recife, postcode=50740-545,
state=Pernambuco, country=Brazil

[label3]organization=Programa de Pós-Graduação em Engenharia Física, Unidade
Acadêmica do Cabo de Santo Agostinho, Universidade Federal Rural de
Pernambuco,city=Cabo de Santo Agostinho, postcode=54518-430, state=Pernambuco,
country=Brazil

[label4]organization=Department of Physical Electronics, School of Electrical
Engineering, Faculty of Engineering,postcode=69978, state=Tel Aviv,
country=Israel

[label5]organization=Instituto de Alta Investigación, Universidad de
Tarapacá,city=Casilla 7D, state=Arica, country=Chile

##  1 Introduction

In the course of the last decades, the investigation of temporal solitons has
been widely developed for the study of various ultrafast optical phenomena,
from the viewpoint of both fundamental research and applications [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Temporal solitons have been
studied in the context of optical communications, where they may be employed
as data bits [17, 18, 19, 20, 21, 22, 23]. The generation of self-confined
solitary waves is commonly achieved by balancing the group-velocity dispersion
(GVD) and self-phase modulation (SPM) in nonlinear (NL) media. For this
purpose, the power of the incident beam must exceed the self-confinement
critical level ($P_{crit}$), so that the nonlinearity allows the input pulse
to maintain its temporal envelope during the propagation [15]. Thus, if the
incident power is higher than $P_{crit}$, the input pulse evolves into a
fundamental soliton with the remaining energy shed off as dispersive waves. If
the incident power is much higher than $P_{crit}$, multiple fundamental
solitons may be generated, under specific conditions [24, 25, 26, 27, 28, 29,
30, 31, 32, 33].

One widely used method for generating temporal solitons is based on the
dispersion and nonlinearity engineering in photonic crystal fibers (PCFs) [9,
10, 11, 12]. This methodology has been useful for investigating the generation
of temporal solitons in the near-infrared and visible ranges, due to the
ability of PCFs to shift the anomalous dispersion region towards much lower
wavelengths than those attainable with standard fibers. It is relevant to
stress that, due to the high nonlinearity, low confinement loss and tunable
chromatic dispersion that PCFs exhibit, they have been commonly used to
investigate the evolution of fundamental and higher-order solitons. The first-
order (fundamental) solitons are the only ones that strictly maintain a
constant profile during the propagation, while higher-order solitons show a
periodically varying shape. Under the action of perturbations, higher-order
solitons split into several fundamental solitons, each centered on a different
wavelength, in the process called soliton fission [15]. Although soliton
fission works well in PCF, its efficiency in producing multiple-soliton trains
is poor. This occurs because during the fission strong intrapulse Raman
scattering (IRS) is induced as a result of significant temporal compression
around the central region of the input pulse, caused by the generation of the
first soliton exhibiting a very broad supercontinuum spectrum.The dissipative
effect of the IRS limits the soliton propagation over long distances, and
reduces the number of generated solitons, as a large portion of the energy
remains confined in the first soliton.

Recently, we reported a new methodology, called multiple temporal compression
(MTC), to generate multiple temporal ultrashort solitons in a more efficient
way [34, 35]. The MTC method is based on a specific version of the general
scheme of the nonlinearity management [36], viz., the propagation of a laser
pulse through a composite medium made up of alternating self-focusing and
self-defocusing segments, both with normal group-velocity dispersion. During
propagation in the first segment, the input pulse accumulates positive linear
($\phi_{disp}$) and NL ($\phi_{nl}$) phases due to the GVD and self-focusing
effects, respectively. Consequently, the pulse is positively chirped.
Subsequently, a partial compensation of $\phi_{nl}$ occurs, due to the self-
defocusing effect, when the pulse propagates in the second segment. This
compensation is not complete under to the accumulation of $\phi_{disp}$ that
occurs in both segments due to the action of the normal dispersion. Thus,
instead of the compensation of the positive chirp in the second segment in the
course of the propagation, it generates pairs of temporal solitons that are
temporally symmetric with respect to the pulse’s center. Furthermore, in the
course of the longer propagation, more temporal solitons are generated from
the leading and trailing edges, and eventually from the central region of the
pulse.

In this work, we numerically solve the generalized NL Schrödinger equation to
compare the efficiency in generating multiple temporal solitons by using the
MTC method [34, 35] and previous methods based on the use of photonic crystal
fibers (PCFs) and fused silica waveguides (FSWs). The efficiency of the
methods is compared according to the number of temporal solitons formed from
the same input conditions, as well as considering the energy distribution in
the multiple-soliton train during the fission and the ability of each soliton
to maintain its properties unchanged in the course of the subsequent
propagation.

##  2 Numerical analysis of the multiple temporal solitons process

The generation and propagation of temporal solitons is studied in the
framework of the generalized NL Schrödinger equation (NLSE), written as [13,
14, 15, 16]:

$\displaystyle\frac{\partial A}{\partial z}-\left(\sum\limits_{n\geqslant
2}\beta_{n}\frac{i^{n+1}}{n!}\frac{\partial^{n}A}{\partial
T^{n}}\right)=i\gamma_{0}\left(1+\frac{i}{\omega_{0}}\frac{\partial}{\partial
T}\right)(\left(1-f_{R}\right)A|A|^{2}+\hskip 28.73712pt$
$\displaystyle+f_{R}A\int_{0}^{\infty}h_{R}(\tau)|A(z,T-\tau)|^{2}d\tau),$ (1)

where the term in the parentheses on the left-side of the Eq. (1) describes
the high-order dispersion, while the right-side of the Eq. (1) describes the
third-order NL optical effects with
$\gamma_{0}=\omega_{0}n_{2}(\omega_{0})/cA_{eff}$ being the third-order NL
coefficient. The temporal derivative in the first parenthesized factor on the
right-hand side represents effects such as the self-steepening and optical
shock formation. In the second factor, the fractional contribution of the
delayed Raman response to the NL polarization is represented by $f_{R}$, with
the first term representing the temporal SPM and instantaneous Raman
contribution. The second term represents the convolution integral describing
the delayed Raman response, which leads to effects such as the IRS and the
soliton self-frequency shift.

To perform the numerical solutions of Eq. (1), we use the fourth-order Runge-
Kutta in the interaction picture (RK4IP) method, which was reported to be more
accurate compared to other methods such as the conventional split-step Fourier
method [14]. The simulations were started for input pulses with the hyperbolic
secant shape, pulse duration time $T_{FWHM}=90fs$, and the input peak power
varying from $P_{0}=2.5kW$ to $30kW$. For the purpose of the comparison the
central pulse’s wavelength ($\lambda_{0}$) is chosen in such a way that the
materials available for the realization of the MTC, PCF and FSW methods
exhibit close absolute values of the GVDs with opposite signs. The NL
parameters of the medium used in each procedure are described below.

The generation and propagation of temporal solitons using the PCF, which
implies the pulse propagation in the single medium (with length $L=150mm$),
was considered in the framework of the well-established model [13, 14]. In
this case, for the central pulse wavelength $850nm$, we take GVD coefficients
in Eq. (1) as $\beta_{2}=-1.27\cdot 10^{-2}ps^{2}m^{-1}$ (the anomalous-
dispersion regime), $\beta_{3}=8.11\cdot
10^{-5}ps^{3}m^{-1},\beta_{4}=-1.32\cdot
10^{-7}ps^{4}m^{-1},\beta_{5}=3.03\cdot
10^{-10}ps^{5}m^{-1},\beta_{6}=-4.19\cdot 10^{-13}ps^{6}m^{-1},$ and
$\beta_{7}=2.57\cdot 10^{-16}ps^{7}m^{-1}$.

The MTC method was also analyzed considering the same central pulse
wavelength, $\lambda_{0}=850nm$, but in the normal-dispersion regime, by using
fused silica as the matrix for the two segments of the NL medium. By using
pure fused silica in the first segment, a positive NL refractive index is
obtained, with the GVD coefficients $\beta_{2}=+3.22\cdot
10^{-2}ps^{2}m^{-1},\beta_{3}=2.94\cdot
10^{-5}ps^{3}m^{-1},\beta_{4}=-1.67\cdot
10^{-8}ps^{4}m^{-1},\beta_{5}=4.61\cdot
10^{-11}ps^{5}m^{-1},\beta_{6}=-1.28\cdot 10^{-13}ps^{6}m^{-1},$ and
$\beta_{7}=4.31\cdot 10^{-16}ps^{7}m^{-1}$, which were calculated from the
derivation of the Sellmeier expression [37]. On the other hand, the second
segment of the NL medium includes metal nanoparticles in fused silica to
achieve the self-defocusing behavior [27, 28, 29, 30, 31, 32, 38, 39, 40, 41],
while maintaining the medium in the normal-dispersion regime. Another strategy
may use birefringent NL crystals, such as LiNbO3, BBO, and KTP, in the second
segment of the NL medium, as negative second-order cascade effects are strong
enough to compensate for excessive positive Kerr nonlinearity. In both cases,
the second segment with an effective defocusing nonlinearity, $n_{2,eff}<0$,
and normal dispersion is used [42, 43, 44, 45, 46, 47].

For completeness, we also consider the temporal soliton generation procedure
that uses a $150mm$ length FSW. In this case, it was not possible to obtain a
normal-dispersion regime. However, for comparison with the MTC method, the
central pulse wavelength of $1585nm$ is chosen for use in the framework of the
FSW method so that the absolute value of GVD is the same considered in both
methods. Thus, from the derivation of the Sellmeier expression, for fused
silica at $1585nm$, we obtain: $\beta_{2}=-3.22\cdot
10^{-2}ps^{2}m^{-1},\beta_{3}=16.56\cdot
10^{-5}ps^{3}m^{-1},\beta_{4}=-5.63\cdot
10^{-7}ps^{4}m^{-1},\beta_{5}=2.73\cdot
10^{-9}ps^{5}m^{-1},\beta_{6}=-1.61\cdot 10^{-11}ps^{6}m^{-1},$ and
$\beta_{7}=1.11\cdot 10^{-13}ps^{7}m^{-1}$ [37].

For correct comparison between the three methods,
$|\gamma_{0}|=0.045W^{-1}m^{-1}$ is adopted as the magnitude of the NL
coefficient of all involved media, which corresponds to the reference value
reported for a PCF [13, 14]. In the PCF- and FSW-based methods, one has
$\gamma_{0}>0$ and the NL medium length $L=150mm$. However, in the framework
of the MTC method, we use the first segment of length $10mm$ with the NL
coefficient $\gamma_{0}^{(1)}=|\gamma_{0}|$, and the second segment of length
$140mm$ with $\gamma_{0}^{(2)}=-|\gamma_{0}|$ .

The Raman parameters used to model the influence of the IRS were obtained from
Refs. [13, 14, 15], as all the NL media considered in this work are based on
fused silica. Thus, the fractional contribution of the delayed Raman response
to the nonlinear polarization is taken as $f_{R}=0.18$, and the Raman response
function is
$h_{R}(\tau)=\tau_{1}(\tau_{1}^{-2}+\tau_{2}^{-2})exp(-\tau/\tau_{2})sin(\tau/\tau_{1})$,
where $\tau_{1}=12.2fs$ and $\tau_{2}=32.0fs$.

##  3 Results and discussions

Figure 1 shows the generation and propagation of solitons produced by the
temporal evolution of the pulse intensity, as obtained from simulations of Eq.
(1) adjusted to modeling the PCF and FSW methods. Figures 1(a) and 1(c),
obtained for input peak powers $5kW$, show that both methods generate the
first soliton after the incident pulse has traveled the distance $\approx
20mm$ ($\approx 25mm$ for PCF, once we have
$|\beta_{2}^{(PCF)}|<|\beta_{2}^{(FSW)}|$ ). It is clearly seen that the first
soliton carries a large part of the input beam power, with the slight
excessive power carried by dispersive waves ejected around the soliton. For
higher input powers, the energy remaining after the creation of the first
soliton drives the generation of secondary solitons, i.e., the higher-order
soliton breaks up into fundamental solitons through fission. For instance,
Figs. 1(b) and 1(d) show that, respectively, five or three temporal solitons
are generated by the PCF or FSW method, for input power $25kW$. Although the
PCF method has been shown to be more efficient than the FSW, both demonstrate
low peak powers of the secondary solitons, as well as power loss in the course
of propagation under the action of the strong IRS.

(a)

(b)

(c)

(d)

Figure 1: The temporal evolution of the high-order soliton fission produced by
the (a,b) PCF, and (c,d) FSW methods. In the left column, the input peak power
is $5kW$, and in the right one it is $25kW$.

Figure 2 shows a pronounced trend for all solitons to lose their peak powers
in the course of propagation over long distances. By analyzing the soliton
propagation between $50mm$ and $150mm$, it is observed that the peak power of
the first soliton is reduced by $\sim 39\%(\sim 41\%)$, while the second
soliton experiences the decrease in the peak power of $\sim 51\%(\sim 37\%)$,
when generated from the input power of $25kW$, by using the PCF (FSW) method.
Due to the mechanism by which multiple-soliton trains are generated, the first
soliton tends to appear in the temporally isolated form, being subject to
power loss via IRS due to the generation of optical phonons, while the central
wavelength of the soliton features a red shift [15]. This loss mechanism tends
to limit the propagation of robust solitons over long distances, as the NL
effects is weakening. Secondary solitons are also influenced by the IRS.
However, as the secondary solitons are generated in the region where the
energy remaining after the creation of the first soliton is concentrated,
these solitons are more likely to undergo collisions with each other, giving
rise to energy transfer between them.

(a)

(b)

Figure 2: The evolution of the peak power of the first, second, and third
solitons in the course of the propagation, in the case of using the PCF
(black) or FSW (blue) method for input powers equal to (a) $5kW$, and (b)
$25kW$.

Contrary to the results produced by the PCF and FSW methods, the solution of
the generalized NLSE corresponding to the MTC method provides interesting
results for the generation and propagation of temporal solitons over long
distances. As shown in Fig. 3, the MTC method begins by generating a pair of
solitons at edges of the pulse, labeled as the leading-edge soliton ($S_{LE}$)
and one at the trailing edge ($S_{TE}$). Subsequently, new pairs of secondary
solitons are generated closer to the central region of the pulse. As expected,
the number of generated temporal solitons increases with the increase of the
input peak power. However, remarkably, the secondary solitons show powers
similar to that of the first soliton, indicating more efficient energy
distribution during the soliton fission. While the MTC method is efficient in
generating a large number of high-power temporal solitons, we focus here on
the second ($2_{nd}$) and third ($3_{rd}$) $S_{LE}$ and $S_{TE}$.

A notable difference between the three methods studied here is the dispersion
regime in which solitons are generated. In the MTC method, the normal-
dispersion regime provides acceleration of each soliton, as IRS causes the red
shift of its central wavelength. On the contrary, the anomalous-dispersion
regime in the PCF and FSW causes deceleration of the soliton in the course of
propagation. This difference is the reason why the solitons in Figs. 1 and 3
propagate in opposite temporal directions. Furthermore, the soliton
acceleration induced by the MTC method creates a propitious configuration to
observe phenomena ranging from soliton collisions to the Newton’s cradle (NC)
[3]. The latter phenomenon can be clearly seen in Fig. 3(b), where the second
$S_{TE}$ collides with the first $S_{TE}$, and then collides again with the
newly formed solitons in the central region of the pulse. Finally, as a result
of the multiple inelastic collisions, the NC soliton ($S_{nc}$) is ejected
with a higher peak power. A similar dynamical behavior, but with a larger
number of collisions, is observed in the case of formation of the NC soliton
at higher input powers, see Fig. 3(c).

Interestingly, in Fig.3(c) we observe the NC effect in the direction opposite
to that in Fig.3(b). The central region of the pulse generates a soliton which
slows down and collides, consecutively, with the third, second, and first TE
solitons, and then escapes.

(a)

(b)

(c)

(d)

(e)

Figure 3: (a-c) The dynamics of the soliton fission produced by the MTC
method, for different input powers. (d-e) The evolution of the peak powers of
the first, second and third $S_{TE}$ and $S_{LE}$ generated in (a-c) in the
course of the propagation.

The dependence of the soliton peak power on the propagation distance, shown in
Figs. 3(d,e), also shows advantages offered by the MTC method, in comparison
to the PCF and FSW. In the case of the MTC method, the initial energy
distribution is not only more efficient to generate multiple-soliton trains
with high powers during the soliton fission, but it is also observed that each
soliton has the ability to maintain its peak power during the propagation.
This happens due to the cooperative process of energy exchange with dispersive
waves generated during the soliton fission, or by collisions with other
neighboring solitons, that allows the temporal solitons to travel longer
distances compared to the case of the PCF and FSW. For example, Fig. 3(d)
shows that the first $S_{TE}$ starts its propagation with a lower peak power,
compared to the pair of the first $S_{LE}$. However, as the first $S_{TE}$
accelerates, the central region of the pulse transfers energy to it (see Fig.
3(a)), making its peak power comparable to that of the first $S_{LE}$. On the
other hand, for higher input peak powers, the energy exchange through multiple
collisions of the first $S_{TE}$ with the secondary $S_{TE}$ (Fig. 3(b)), or
with the soliton NC (Fig. 3(b,c)) allows its peak power to increase or
maintain its value over long propagation distances. These energy exchange and
feedback processes are responsible for the fact that $S_{TE}$ demonstrates
higher power than $S_{LE}$ for longer propagation distances, despite the fact
that the initial energy distribution was opposite.

In addition, the efficiency of the PCF, FSW and MTC methods can be compared by
examining the number of solitons generated with the increase of the input peak
power, as shown in Fig. 4. For the same input pulse, the MTC method shows the
ability to generate a larger number of temporal solitons for all peak powers,
while the PCF and FSW methods exhibit poorer results. Note that, for the input
peak power higher than $15kW$, the number of solitons generated by the MTC
method is almost double compared to the PCF and FSW.

Figure 4: Numbers of solitons ejected from soliton fission as a function of
the input peak power, as provided by the PCF, FSW, MTC methods.

Thus, the MTC method emerges as a powerful tool to generate multiple temporal
solitons, with high peak power, which is highly desired for long-propagation
distances. However, the efficiency of the MTC declines considerably when the
goal is the generation of multiple solitons in the visible regime, as the
method seems to be ineffective for generating supercontinuum spectra that
reach the visible range. As shown in Fig. 5, the PCF method, widely used for
soliton generation in the visible regime [9, 10, 11, 12, 13, 14, 15, 16],
provides a supercontinuum spectrum ranging from $480nm$ to $1360nm$, while the
MTC and FSW are limited to the spectral region from $720nm$ to $2600nm$. Thus,
the spectral region where the MTC method starts to work efficiently
corresponds to the near-infrared, taking into regard that the IRS contribution
shifts the central wavelength of the solitons up to $1200nm$. Another
interesting feature of the spectral analysis shown in Fig. 5 (c) is that the
observed frequency shift in the region between $1200nm$ and $2600nm$ is not
due to the IRS. Such giant near-infrared emission ($>1200nm$) seems to be
correlated with the accumulation of soliton collisions, similarly to Refs.
[24, 25, 48, 49]. Further studies are underway to better understand the
present setup in this spectral regime.

(a)

(b)

(c)

Figure 5: The pulse’s spectral evolution, for the input peak power $25kW$, as
produced by the (a) PCF, (b) FSW and (c) MTC methods.

##  4 Conclusions

In summary, we demonstrate, through systematic numerical calculations of the
generalized nonlinear Schrödinger equation, the high efficiency that the
multiple temporal compression (MTC) method provides for the generation of
multiple temporal solitons from a single optical pulse, in comparison to the
well-known PCF and FSW methods. The numerical results show that the MTC method
allows to generate twice as many solitons as the conventional ones, for the
input peak powers studied here, with each soliton exhibiting a high power that
remains approximately constant during the propagation. While the efficiency
offered by the MTC technique decreases in the visible region, it appears as a
useful tool for the studies of multiple soliton collisions in the near-
infrared region, as well as for demonstrating the realization of the multi-
soliton Newton’s cradle.

##  5 Acknowledgments

This work was supported by the Brazilian agencies Conselho Nacional de
Desenvolvimento Científico e Tecnológico - CNPq (Grant: 431162/2018-2) and the
National Institute of Photonics (INCT program) - Grant: 465.763/2014, Fundação
de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE), and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). The
doctoral scholarship of A.C.A. Siqueira was provided by the CNPq. G. Palacios
thanks a fellowship from CNPq Process No. 381191/2022-2. The work of B.A.
Malomed was supported, in part, by the Israel Science Foundation through grant
No. 1695/22.

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