# Soliton (optics)

An optical soliton is a pulse of light that propagates through a nonlinear medium, typically an optical fiber, without changing its temporal or spectral shape, because chromatic dispersion and the intensity-dependent refractive index (the [Kerr effect](https://www.edgepedia.org/kerr-effect)) exactly compensate each other. The concept was introduced to fiber optics by [Akira Hasegawa](https://www.edgepedia.org/akira-hasegawa) and Fred Tappert in 1973 and first observed experimentally by Linn F. Mollenauer, Robert H. Stolen, and James P. Gordon at Bell Laboratories in 1980.<sup>[1](https://www.nature.com/articles/s41566-023-01307-9)</sup><sup> • </sup><sup>[2](https://doi.org/10.1103/PhysRevLett.45.1095)</sup>

| Key fact | Detail |
|---|---|
| Theoretical prediction | Hasegawa and Tappert, paper submitted 12 April 1973, *Applied Physics Letters* **23**, 142 (1973)<sup>[1](https://www.nature.com/articles/s41566-023-01307-9)</sup> |
| First observation | Mollenauer, Stolen, and Gordon, *Physical Review Letters* **45**, 1095 (29 September 1980)<sup>[2](https://doi.org/10.1103/PhysRevLett.45.1095)</sup> |
| 1980 experiment | 7-ps pulses from a mode-locked color-center laser in a 700-m single-mode silica fiber at 1.55 μm; β₂ ≈ −20 ps²/km, γ ≈ 1.3 W⁻¹km⁻¹<sup>[2](https://doi.org/10.1103/PhysRevLett.45.1095)</sup><sup> • </sup><sup>[3](https://www.sciencedirect.com/topics/computer-science/optical-soliton)</sup> |
| Governing equation | The nonlinear Schrödinger equation (NLSE), the governing equation for nonlinear pulse propagation in fibers<sup>[4](https://epubs.siam.org/doi/10.1137/1.9780898717082)</sup> |
| Soliton condition | Peak power inversely proportional to the square of pulse width; the area theorem gives Eτ = 2\|β₂\|/γ<sup>[5](https://ecommons.cornell.edu/bitstreams/084da01c-045e-42e3-926b-7aa05a8b17e1/download)</sup> |
| Long-distance record | Soliton transmission over more than 4000 km using periodic Raman amplification (Mollenauer and Smith)<sup>[6](https://www.nokia.com/bell-labs/publications-and-media/publications/optical-solitons-for-high-bit-rate-all-optical-communication-systems/)</sup> |
| Modern form | Dissipative Kerr solitons in integrated microresonators, first reported in a CW-driven resonator by Herr et al. in 2014, generating low-noise frequency combs<sup>[7](https://www.sciencedirect.com/science/article/pii/S0079672722000623)</sup> |

## Physical principle

Two effects act on a short pulse in a fiber, and each alone would distort it. Chromatic dispersion makes different frequency components travel at different group velocities, so the pulse temporally broadens. [Self-phase modulation](https://www.edgepedia.org/self-phase-modulation), the Kerr effect, makes the refractive index depend on intensity, so the instantaneous frequency of the pulse is swept during propagation and the spectrum broadens.<sup>[8](https://doi.org/10.1364/JOSAB.28.0000A1)</sup>

In the anomalous dispersion regime, where the group-velocity dispersion parameter β₂ is negative, these two effects can cancel exactly. The pulse envelope is then a hyperbolic secant,

\[ |A(z,t)| = \left( \frac{|\beta_2|}{\gamma t_0^2} \right)^{1/2} \operatorname{sech}\left( \frac{t}{t_0} \right), \]

where γ is the nonlinear coefficient and t₀ a width parameter; the full width at half maximum is 1.7627 t₀.<sup>[9](https://phys.au.dk/fileadmin/site_files/publikationer/phd/Carsten_Krogh.pdf)</sup><sup> • </sup><sup>[5](https://ecommons.cornell.edu/bitstreams/084da01c-045e-42e3-926b-7aa05a8b17e1/download)</sup> The peak power required scales as the inverse square of the pulse width, so shorter solitons need proportionally more power. The soliton area theorem, Eτ = 2\|β₂\|/γ, fixes the product of pulse energy and duration for a given fiber: for each pair of β₂ and γ, a family of solitons with different energies and durations can coexist as long as they satisfy this relation.<sup>[5](https://ecommons.cornell.edu/bitstreams/084da01c-045e-42e3-926b-7aa05a8b17e1/download)</sup>

**Robustness is a defining property.** A launched pulse of sufficient energy whose shape is not exactly the soliton profile automatically sheds the mismatch as a small ordinary pulse and evolves during propagation into a perfectly shaped soliton; within a suitable range of peak power, an input pulse becomes a fundamental soliton after a few dispersion lengths.<sup>[4](https://epubs.siam.org/doi/10.1137/1.9780898717082)</sup><sup> • </sup><sup>[10](https://users.physics.ox.ac.uk/~rtaylor/teaching/Opto_lecture8.pdf)</sup> Two solitons that collide, or pass through each other, remain solitons afterward.<sup>[11](https://opg.optica.org/aop/fulltext.cfm?id=369003&uri=aop-9-3-429)</sup>

In the normal dispersion regime (β₂ > 0), bright solitons of this kind do not exist; instead, dark solitons, dips on a continuous-wave background, are the stable solutions.<sup>[12](https://www.sciencedirect.com/science/article/abs/pii/S0370157397000732)</sup>

## History

The idea of temporal solitons in fibers as a way to compensate dispersion-induced pulse broadening through the Kerr nonlinearity of silica was introduced by Hasegawa and Tappert in 1973, in a paper submitted on 12 April of that year.<sup>[1](https://www.nature.com/articles/s41566-023-01307-9)</sup> Hasegawa later recalled that he derived the light-wave envelope equation for glass fiber including Kerr nonlinearity and group dispersion in the form of the nonlinear [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation), while Tappert performed the computer simulations demonstrating soliton stability; the exact soliton solution of the NLSE had been obtained by Zakharov and Shabat in 1972, but the two were unaware of that Russian-language paper at the time.<sup>[13](https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.1044845/full)</sup>

The first experiments had to wait for two developments of the late 1970s: fibers with low loss at the wavelength where solitons are possible, and a suitable source of picosecond pulses, the mode-locked color-center laser.<sup>[14](https://www.cambridge.org/core/books/abs/optical-solitons/solitons-in-optical-fibers-an-experimental-account/95FF72CB321C5B6C02B2AB24C71CF73D)</sup> In the 1980 experiment, 7-ps pulses at 1.55 μm, a wavelength of minimum loss and large anomalous group-velocity dispersion, were sent through a 700-m single-mode silica fiber; at certain critical power levels the observed narrowing and splitting of the pulses was characteristic of solitons.<sup>[2](https://doi.org/10.1103/PhysRevLett.45.1095)</sup> The same group went on to invent the soliton laser, discover the soliton self-frequency shift, and study interaction forces between solitons.<sup>[14](https://www.cambridge.org/core/books/abs/optical-solitons/solitons-in-optical-fibers-an-experimental-account/95FF72CB321C5B6C02B2AB24C71CF73D)</sup>

## Higher-order solitons

A pulse with N² times the fundamental soliton peak power, where N² = γP₀T₀²/\|β₂\|, propagates as an Nth-order soliton.<sup>[15](https://gctjaipur.wordpress.com/wp-content/uploads/2015/08/fiber-optic-communication-systems.pdf)</sup><sup> • </sup><sup>[28](https://www.rp-photonics.com/tutorial_passive_fiber_optics12.html)</sup> Higher-order solitons do not hold a fixed shape: they undergo a periodic evolution, returning to their initial form after one soliton period. In a simulation of a fourth-order soliton, that period was 92 m, and the pulse was not fully reproduced at the fiber end because the fiber's third-order dispersion caused a deviation from ideal behavior, an effect more pronounced for shorter pulses.<sup>[16](https://www.rp-photonics.com/rp_fiber_power_demos_ho_solitons.html)</sup> The initial pulse for a third-order soliton carries 3² = 9 times the energy of the fundamental soliton.<sup>[17](https://www.rp-photonics.com/rp_propulse_demo_soliton_d.html)</sup>

**Real fibers perturb the ideal solution.** Solitons in practice evolve under linear loss, third-order dispersion, stimulated [Raman scattering](https://www.edgepedia.org/raman-scattering), and other effects absent from the integrable NLSE.<sup>[18](https://doi.org/10.1007/978-981-10-7087-7_17)</sup> The most prominent of these is the soliton self-frequency shift: intrapulse Raman scattering transfers energy from shorter to longer wavelengths within the pulse, causing a continuous downshift of its mean frequency. J. P. Gordon showed in 1986 that the effect varies roughly with the inverse fourth power of the pulse width; at 1.5 μm in a fiber with 15 ps/nm/km dispersion, a 250-fs soliton is predicted to shift by its own spectral width after about 100 m of propagation.<sup>[19](https://opg.optica.org/ol/abstract.cfm?uri=ol-11-10-662)</sup> The shift can be suppressed by bandwidth-limited amplification<sup>[20](https://opg.optica.org/ol/abstract.cfm?uri=ol-14-10-514)</sup> or strongly decelerated by third-order dispersion near a photonic bandgap edge, and it is exploited deliberately in fiber-integrated tunable subpicosecond pulse sources.<sup>[21](https://opg.optica.org/ol/abstract.cfm?uri=ol-28-24-2512)</sup>

## Solitons in optical communications

Because solitons resist the dispersion that limits conventional pulse transmission, they were proposed as carriers for long-distance fiber communications. Hermann A. Haus of MIT and colleagues reviewed this program, covering the causes of bit errors in long-distance soliton transmission and the perturbation theory used to analyze them.<sup>[22](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.68.423)</sup> A central limitation is the Gordon-Haus effect: amplified spontaneous emission noise is partly absorbed into the soliton and changes its central frequency, and dispersion converts this frequency jitter into arrival-time jitter, which can push a pulse into the wrong time slot.<sup>[11](https://opg.optica.org/aop/fulltext.cfm?id=369003&uri=aop-9-3-429)</sup> Conditions for practical soliton transmission include short amplifier spacing and low pulse duty cycle to minimize inter-soliton interaction.<sup>[11](https://opg.optica.org/aop/fulltext.cfm?id=369003&uri=aop-9-3-429)</sup>

The distance achieved grew dramatically: Mollenauer and Smith transmitted solitons without distortion over more than 4000 km using periodically injected Raman pump light, a distance two orders of magnitude larger than that of a comparable linear pulse.<sup>[6](https://www.nokia.com/bell-labs/publications-and-media/publications/optical-solitons-for-high-bit-rate-all-optical-communication-systems/)</sup> Nevertheless, soliton-based data transmission schemes investigated in the 1980s were later abandoned in favor of wavelength-division multiplexing, which is easier to implement and scales better to higher data rates.<sup>[23](https://www.nature.com/articles/nature22387)</sup> [Soliton](https://www.edgepedia.org/soliton) pulse delivery from mode-locked lasers has also not become common, essentially because of the severe limitation in pulse energy.<sup>[24](https://www.rp-photonics.com/case_study_soliton_experiments.html)</sup>

## Microresonator solitons and frequency combs

The most active modern form of the optical soliton is the dissipative Kerr soliton (DKS), a pulse circulating continuously in an optical microresonator. Unlike the conservative fiber soliton, a DKS relies on a double balance: dispersion against nonlinearity, and parametric gain against cavity loss. Its dynamics is described by the Lugiato-Lefever equation when higher-order effects are neglected.<sup>[7](https://www.sciencedirect.com/science/article/pii/S0079672722000623)</sup> Temporal dissipative solitons in a high-Q microresonator were reported by Herr and colleagues; the solitons arise spontaneously when the pump laser is tuned through the resonance, and single-soliton states correspond in the frequency domain to low-noise optical frequency combs with smooth spectral envelopes.<sup>[25](https://arxiv.org/abs/1211.0733)</sup>

These chip-integrated soliton combs turned solitons from a candidate communication channel into a component of one. In a silicon nitride microresonator, DKSs are generated via four-photon interactions mediated by the Kerr nonlinearity, and the continuous-wave tones of the associated frequency comb serve as carriers for massively parallel wavelength-division multiplexing, rather than the soliton pulse train itself.<sup>[23](https://www.nature.com/articles/nature22387)</sup> DKS combs have since been applied in coherent telecommunications, dual-comb spectroscopy, optical frequency metrology, and low-noise microwave generation, with repetition rates in the range of tens to hundreds of gigahertz.<sup>[26](https://www.nature.com/articles/s41467-017-00719-w)</sup> A remaining practical constraint is pump-to-comb conversion efficiency, which limits the comb power available in many applications.<sup>[27](https://www.nature.com/articles/s44310-024-00030-9)</sup>

## References

1. [The bright prospects of optical solitons after 50 years, Nature Photonics 17, 937 (2023)](https://www.nature.com/articles/s41566-023-01307-9)
2. [Mollenauer, Stolen, Gordon (1980). Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers, Phys. Rev. Lett. 45, 1095](https://doi.org/10.1103/PhysRevLett.45.1095)
3. [Optical Soliton, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/computer-science/optical-soliton)
4. [Mathematical Principles of Optical Fiber Communications, SIAM](https://epubs.siam.org/doi/10.1137/1.9780898717082)
5. [Femtosecond Pulse Generation in Fiber, Cornell eCommons thesis](https://ecommons.cornell.edu/bitstreams/084da01c-045e-42e3-926b-7aa05a8b17e1/download)
6. [Optical Solitons for High-Bit-Rate All-Optical Communication Systems, Nokia Bell Labs](https://www.nokia.com/bell-labs/publications-and-media/publications/optical-solitons-for-high-bit-rate-all-optical-communication-systems/)
7. [Photonic frequency microcombs based on dissipative Kerr and quadratic cavity solitons, Progress in Quantum Electronics](https://www.sciencedirect.com/science/article/pii/S0079672722000623)
8. [Nonlinear fiber optics: its history and recent progress, J. Opt. Soc. Am. B 28, A1 (2011)](https://doi.org/10.1364/JOSAB.28.0000A1)
9. [The Nonlinear Schrödinger Equation, PhD thesis, Aarhus University](https://phys.au.dk/fileadmin/site_files/publikationer/phd/Carsten_Krogh.pdf)
10. [Optical fibres lecture 8, University of Oxford](https://users.physics.ox.ac.uk/~rtaylor/teaching/Opto_lecture8.pdf)
11. [Performance limits in optical communications due to fiber nonlinearity, Advances in Optics and Photonics](https://opg.optica.org/aop/fulltext.cfm?id=369003&uri=aop-9-3-429)
12. [Dark optical solitons: physics and applications, Physics Reports](https://www.sciencedirect.com/science/article/abs/pii/S0370157397000732)
13. [Optical soliton: Review of its discovery and applications in ultra-high-speed communications, Frontiers in Physics (2022)](https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.1044845/full)
14. [Mollenauer, Solitons in optical fibers: an experimental account, in Optical Solitons, Cambridge University Press](https://www.cambridge.org/core/books/abs/optical-solitons/solitons-in-optical-fibers-an-experimental-account/95FF72CB321C5B6C02B2AB24C71CF73D)
15. [Fiber Optic Communication Systems (course text, fiber solitons chapter)](https://gctjaipur.wordpress.com/wp-content/uploads/2015/08/fiber-optic-communication-systems.pdf)
16. [Higher-order Soliton Pulses, RP Fiber Power demo](https://www.rp-photonics.com/rp_fiber_power_demos_ho_solitons.html)
17. [Solitonen höherer Ordnung, RP ProPulse demo](https://www.rp-photonics.com/rp_propulse_demo_soliton_d.html)
18. [Perturbations of Solitons in Optical Fibers, Handbook of Optical Fibers (Springer)](https://doi.org/10.1007/978-981-10-7087-7_17)
19. [Gordon (1986). Theory of the soliton self-frequency shift, Opt. Lett. 11, 662](https://opg.optica.org/ol/abstract.cfm?uri=ol-11-10-662)
20. [Gouveia-Neto, Gomes, Taylor (1989). Suppression and manipulation of the soliton self-frequency shift, Opt. Lett. 14, 514](https://opg.optica.org/ol/abstract.cfm?uri=ol-14-10-514)
21. [Chestnut, Taylor (2003). Soliton self-frequency shift in highly nonlinear fiber with extension by external Raman pumping, Opt. Lett. 28, 2512](https://opg.optica.org/ol/abstract.cfm?uri=ol-28-24-2512)
22. [Haus, Wong (1996). Solitons in optical communications, Rev. Mod. Phys. 68, 423](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.68.423)
23. [Microresonator-based solitons for massively parallel coherent optical communications, Nature 546, 274 (2017)](https://www.nature.com/articles/nature22387)
24. [Case Study: Numerical Experiments With Soliton Pulses in Fibers, RP Photonics](https://www.rp-photonics.com/case_study_soliton_experiments.html)
25. [Herr et al., Temporal solitons in optical microresonators, arXiv:1211.0733](https://arxiv.org/abs/1211.0733)
26. [Breathing dissipative solitons in optical microresonators, Nature Communications 8, 736 (2017)](https://www.nature.com/articles/s41467-017-00719-w)
27. [Advances in resonator-based Kerr frequency combs with high conversion efficiencies, npj Nanophotonics (2024)](https://www.nature.com/articles/s44310-024-00030-9)
28. [Tutorial Passive Fiber Optics, Part 12: Ultrashort pulses and signals in fibers](https://www.rp-photonics.com/tutorial_passive_fiber_optics12.html)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Waves and optics › Physical and wave optics › Classical light–matter interaction and nonlinear optics › Nonlinear propagation effects*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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