# Pioneering experiments on optical solitons in fibres

Bertrand Kibler, J.M. Dudley, Christophe Finot

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Bertrand Kibler, J.M. Dudley, Christophe Finot. Pioneering experiments on optical solitons in fibres. Photoniques, 2023, 122, pp.41-45. 10.1051/photon/202312241. hal-04314857

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# PIONEERING EXPERIMENTS
ON OPTICAL SOLITONS
IN FIBRES

Bertrand KIBLER<fnref n="1" />, John M. DUDLEY<fnref n="2" />, Christophe FINOT<fnref n="1" />,*

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Fifty years ago, the concept of solitons in optical fibres was proposed numerically by F. Tappert and A. Hasegawa from Bell Labs. Seven years later in 1980, experiments confirmed that dispersion and the Kerr nonlinearity could combine to yield a coherent ultrashort pulse able to propagate without distortion. The implications of this discovery go well beyond the sector of telecommunications as initially suggested, and solitons now play a major role in modern ultrafast nonlinear photonics.

https://doi.org/10.1051/photon/202312241

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## A FUNDAMENTAL CONCEPT
OF NONLINEAR PHYSICS

In August 1934, Scottish engineer John Scott Russell made the remarkable observation of what he called a “solitary wave” which propagated without deformation for miles along the Union Canal near Edinburgh. Russell was convinced that he had seen a fundamental-ly new form of wave, but his opinion was not universally accepted as it was incompatible with the linear wave models of the time. Although Russell’s observation was

reproduced by Bazin in Dijon in 1859, it wasn't until the theory of Boussinesq (1871) and Korteweg and De Vries (1895) that it was possible to understand how this localisation arose from the balance between nonlinear and dispersive propaga-tion effects to create what we now call a **soliton**. The soliton concept is general, and is not restricted only to hydrodynamics. Indeed, interest in solitons has spread to virtually all branches of physics, with studies be-nefiting from advances in numerical computing and the development of

advanced mathematical methods such as the inverse scattering trans-form [1].

The ability to study solitons of light propagating over long distances in optical fibre resulted from two major technical revolutions in the 1960s: the development of the laser provided a high power source of coherent light, and the development of low loss fibre waveguides resulted in the confinement of light in a mi-cron-scale core over long distances. Whilst much work in fibre optics was naturally focussed on the ●●●

<footnote n="1">¹ Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne, Dijon, France</footnote>

<footnote n="2">² Université de Franche-Comté, Institut FEMTO-ST, CNRS UMR 6174, Besançon, France</footnote>

<footnote>*christophe.finot@u-bourgogne.fr</footnote>
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**Figure 1.** (a) Longitudinal evolution of the temporal (panels 1) and spectral (panels 2) properties of a fundamental soliton in the regime of anomalous dispersion. (b) Same as (a) but for a higher-order soliton (second order soliton). (c) Longitudinal evolution of the temporal profile of a black soliton in a fiber with normal dispersion. The numerical simulations are inspired by the one involved in [3], losses being not included.

in the presence of nonlinearity, and the existence of the optical temporal soliton was predicted numerically in 1973 by Akira Hasegawa and Frank Tappert.

## BALANCING DISPERSION AND NON-LINEARITY

The propagation of an optical wave in a single-mode optical fibre is governed by the 1D nonlinear Schrödinger equation or NLSE (see box) that takes into account the simultaneous interaction between dispersion and third-order (Kerr) nonlinearity [2]. When considered independently, dispersion and non-linearity are both detrimental to pulse propagation: dispersion will typically broaden the pulse temporally while nonlinearity will generate new frequency components and broaden the corresponding

telecommunications potential, there was also interest in understanding the fundamental physics of how a short pulse of light would evolve if injected in a fibre at high power. This led to the emergence of a new theory of ultrashort pulse propagation

**Figure 2.** Pioneer experimental results (black lines) from Fig. 2 of Ref. [4]. Input pulse characteristics (spectrum and autocorrelation) coupled into the fibre are depicted in the upper left box. Other graphs show the resulting pulse characteristics (spectrum, autocorrelation, temporal power profile) at fibre output for distinct input powers. Four distinct propagation regimes (columns a-d) were observed. Additional numerical simulations based on the NLSE are here provided for comparison (red dotted lines) using experimental parameters from Ref. [4]. Theoretical soliton solution (blue solid line) is shown for regime (b).

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spectrum. In both cases, instantaneous frequencies are redistributed within the pulse (frequency chirp) and when dispersion and non-linearity act together, the temporal and spectral distortions can become highly complex. However, there is a special case: if the injected pulse has the temporal shape of a hyperbolic secant profile, the effect of non-linearity can exactly compensate the effects of dispersion, provided the pulse propagates in the anomalous dispersion regime so that the signs of the dispersive and nonlinear chirps are different. This yields a stationary wave able to propagate without distortion (Fig. 1a).

This is what Akira Hasegawa and Frederick Tappert (both with backgrounds in plasma physics) showed in 1973 [3]. Working at Bell Labs in the United States, they used theory and numerical simulation to consider the propagation of picosecond pulses with parameters typical of communication applications. They identified the bright ‘fundamental’ soliton as an ideal potential bit of information which would be robust against noise, collisions and other perturbations. However, they also recognized that the choice of power was crucial to achieve balance between dispersion and nonlinearity. Indeed, for higher powers, longitudinal periodic oscillations develop with alternating stages of temporal compression and splitting and broadening (Fig. 1b), consistent with theoretical predictions made by V. E. Zakharov and A. B. Shabat the year before in 1972. Significantly, Tappert’s numerical work was in itself pioneering, because solving the NLSE required that he developed a new and efficient method. This is now known as the split-step Fourier algorithm that remains the numerical tool of reference for the community 50 years later! In an accompanying publication, Hasegawa and Tappert showed that even propagation in the normal dispersion regime could yield stationary propagating structures, but this time they were “dark” solitons, a hole of light appearing in a continuous background (Fig. 1c).

## EXPERIMENTS

Although the theory and numerics were clear in 1973, it took until 1980 before fibre solitons were seen in experiment, with the work also performed at Bell Labs by Linn F. Mollenauer, Roger H. Stolen and James P. Gordon [4]. In order to propagate in the anomalous dispersion regime with low loss, they used a mode-locked colour centre laser that was specifically constructed by Mollenauer to operate around 1.55 µm. The delivered 7-ps pulses were launched into a 700 m-long single-mode silica fiber, and the output was measured in the time domain through nonlinear autocorrelation, and in the frequency domain by an optical spectrometer (Fig. 2). At low power, the pulse showed temporal broadening (Fig. 2a), but at a critical peak power around 1 W, the initial temporal profile was maintained as the input pulse was a fundamental soliton (Fig. 2b). As the power was further increased, the temporal profile showed up to a factor of 3 temporal compression and spectral broadening as expected for a higher-order soliton (Fig. 2c). Ultimately this resulted in pulse splitting (Fig. 2d). This
experiment was of major impor-
tance for several reasons. Firstly, it
confirmed that fundamental solitons
could sustain their shape with pro-
pagation, potentially of great inte-
rest for communications. Secondly,
it introduced a simple means of
temporally compressing a pulse to
shorter durations. And finally, the
fact that the experimental results
were well described by the predic-
tions of the NLSE validated this mo-
del for nonlinear fibre optics.

BEYOND
TELECOMMUNICATIONS

The pioneering experiments of
Mollenauer, Stolen and Gordon in-
volved distances of less than a kilo-
meter, so the influence of losses was
limited. However, if transcontinental
distances were to be covered, it was
essential to be able to compensate
for losses at regular intervals, notably
using optical amplifiers. Solitons pro-
ved to be a robust solution, and the 80s
and 90s saw impressive technological

demonstrations [5]. Nevertheless, af-
ter the burst of the internet bubble in
the 2000s, alternative technologies
such as coherent signal modulation
proved to be more economically sui-
table for increasing data rates.
But beyond telecommunications, the
soliton concept has proven central to
advance the field of ultrafast optics.
To cite just a few examples, soliton
dynamics are central to octave span-
ning supercontinuum generation,
and the generalised concept of the

The non-linear Schrödinger equation is a universal equation of physics that applies not only to fibre optics, but also to hydrodynamics, quantum or plasma physics. For single-mode fibres, this equation governs the longitudinal evolution of the complex envelope of the electrical field $\psi(z,t)$ for an optical pulse of a typical duration of several picoseconds:

$$ \frac{i\partial\psi}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2\psi}{\partial t^2} + \gamma|\psi|^2\psi = 0$$

It includes the effect of second-order dispersion ($\beta_2$ coefficient), leading in the spectrum to a linear delay of the frequency components. This translates into a temporal redistribution of instantaneous frequencies (chirp), resulting in a temporal broadening. In the case of a fibre with anomalous dispersion ($\beta_2 < 0$), blue frequencies components are shifted towards the front of the pulse, while

red ones are shifted towards the tail of the pulse. The Kerr non-linearity (associated with the nonlinear coefficient $\gamma$) manifests itself as a self-phase modulation term that generates new frequencies, thus broadening the pulse spectrum. The generated frequencies are proportional to the temporal intensity gradient. Therefore, for a bright pulse, red is generated on the leading edge and blue on the trailing edge. For a fundamental soliton (the right shape with the right input power), the changes induced by self-phase modulation can exactly cancel the effect of dispersion such that the pulse preserves its temporal and spectral characteristics. This condition results in the following simple relation for the critical peak power: $P = 3.11|\beta_2|/(\gamma T_p^2)$, where $T_p$ is the pulse duration (at half-maximum).

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**Figure.** Soliton basis: (a) Evolution of the temporal and spectral content of a Gaussian pulse when submitted to dispersive or nonlinear effects exclusively. (b) The perfect balance achieved in the case of the soliton where the instantaneous frequency changes induced by linear and nonlinear effects perfectly cancel each other.
dissipative soliton has enabled the
development of new classes of mode-
locked fibre lasers with dramatically
increased power and pulse quality.
In addition, transposed to the field of
integrated waveguides and micro-re-
sonators, soliton concepts underpin
our models of the generation of opti-
cal frequency combs and associated
applications in metrology.

CONCLUSION

The soliton concept proposed in
1973 and confirmed by experiments
in 1980 has opened up major funda-
mental and applied perspectives that
are still relevant today. The many stu-
dies of soliton physics over the last 50
years provide a remarkable illustra-
tion of how fruitful it can be to com-
bine knowledge from different fields
of physics. In particular, by exploiting
the extremely low level of loss when
compared to other physical media,
fibres have now become the test bed
of choice for the study of a much

The soliton concept proposed in 1973 and confirmed by experiments in 1980 has opened up major fundamental and applied perspectives that are still relevant today.

wider range of nonlinear phenome-
na, including analogies with event
horizon physics and the formation
of oceanic rogue waves. The soliton
also provides a stunning example of
how an initial problem developed
in an industrial context can inspire

a new paradigm in the fundamental
field of complex non-linear dyna-
mics. Research related to solitons and
their applications shows no sign of
slowing down, and soliton concepts
remain central to the most modern
applications of ultrafast photonics.

REFERENCES

[1] A.C. Scott, F.Y.F. Chu, D.W. McLaughlin, Proceedings of the IEEE 61, 1443 (1973).

[2] G.P. Agrawal, Nonlinear Fiber Optics, Fourth Edition, Academic Press, San Francisco, CA, 2006.

[3] A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).

[4] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).

[5] J.R. Taylor, *Optical solitons: theory and experiment*, Cambridge University Press (1992).