Optical Solitons in Fibers

A picosecond pulse launched into an optical fiber inherits two facts from earlier in this chapter, and a soliton is what happens when they are forced into an exact standoff. From the article on chromatic and modal dispersion the pulse carries the group-velocity dispersion parameter $\beta_2$ — the reason a short pulse, built from a band of frequencies, ordinarily smears out in time. From the article on waveguide modes it carries a single confined transverse mode of effective area $A_{\mathrm{eff}}$ , packing the light into a few square microns — tightly enough that a second, normally negligible effect becomes decisive: the Kerr nonlinearity, in which the refractive index of silica rises slightly with intensity, $n=n_0+n_2 I$ , so an intense pulse writes a time-varying phase onto itself and generates new frequencies (self-phase modulation). These two effects impose chirps of opposite sign in the anomalous-dispersion regime. For one special pulse shape — a hyperbolic secant — and one special peak power, they cancel not just on average but at every instant across the pulse. The result is a soliton: a pulse that propagates indefinitely, in a lossless fiber, without changing its temporal or spectral profile. The governing equation is the nonlinear Schrodinger equation (NLSE), which is completely integrable by the inverse-scattering transform — the deep reason solitons are robust, particle-like objects rather than delicate coincidences. They are attractors: launch something close and the fiber sheds the mismatch as radiation and settles onto the exact soliton. Every real fiber then perturbs this picture — loss, higher-order dispersion, the Raman shift — so understanding solitons means understanding both the integrable model and its perturbations.

Key takeaways

An optical soliton keeps its shape as it propagates because the temporal broadening of anomalous group-velocity dispersion (GVD) is cancelled by the self-phase modulation (SPM) of the Kerr nonlinearity.
The governing equation is the nonlinear Schrodinger equation; its fundamental bright soliton is a hyperbolic-secant pulse whose peak power is locked to its width by the condition $N=1$ .
Bright solitons require anomalous dispersion ( $\beta_2<0$ , silica beyond $\approx1.3\,\mu\mathrm{m}$ ); in normal dispersion the surviving structure is a dark soliton — an intensity notch on a bright background.
A single number, the soliton order $N=\sqrt{L_D/L_{NL}}$ , decides the outcome: $N=1$ is stationary, integer $N\ge2$ is a breather recovering every period $z_0=(\pi/2)L_D$ .
Solitons are attractors and behave like particles, surviving collisions — which is why they underlie soliton fiber lasers, supercontinuum sources, and dissipative Kerr microcombs.
Key idea: a pulse becomes a soliton when the down-chirp from anomalous GVD is exactly cancelled by the up-chirp from Kerr self-phase modulation; for a sech pulse this happens at a single peak power fixed by $N=\gamma P_0 T_0^2/|\beta_2|=1$ .

The core insight

A pulse becomes a soliton when the down-chirp imposed by anomalous GVD is exactly compensated, at every instant, by the up-chirp from Kerr self-phase modulation; for a sech pulse this balance occurs at one peak power set by $N=\gamma P_0 T_0^2/|\beta_2|=1$ .

Where this lives today

For a decade the soliton was optics' great hope for transoceanic communication — a pulse that fights its own broadening, needing only periodic amplification to cross an ocean of fiber. That programme ran into the Gordon–Haus timing jitter — amplifier noise nudges each soliton's centre frequency, and dispersion turns those kicks into arrival-time jitter that caps the bit-rate–distance product — and coherent detection with digital signal processing eventually won the long-haul market. But the soliton's descendants thrive. The most active is the dissipative Kerr soliton in a microresonator: in a driven, damped silicon-nitride microring, the conservative dispersion–nonlinearity balance is supplemented by a balance of parametric gain against cavity loss, and a soliton circulates at the round-trip rate. Its output is a broadband, perfectly equidistant frequency comb — a "microcomb" — whose line spacing is the free spectral range. These chip-scale combs, heirs of Hasegawa's fiber soliton, now clock optical atomic clocks, calibrate exoplanet-hunting spectrographs, and drive terabit transmitters — the same self-organizing pulse, on a chip a millimetre across.

The physical picture

Think of two chirps racing to cancel. Anomalous GVD lets the blue (higher) frequencies run ahead of the red, imposing a chirp of one sign. Self-phase modulation does the opposite: because the Kerr index rises with instantaneous power, the leading edge is red-shifted and the trailing edge blue-shifted, an up-chirp. At the right power these two patterns are equal and opposite at every instant, so the instantaneous frequency stays flat — and a pulse with flat instantaneous frequency has no reason to spread: a soliton. The sech shape is not decorative: it is the unique profile for which the nonlinear phase $\propto|A|^2$ has exactly the same curvature in time as the dispersive phase, so the cancellation holds everywhere, not just near the peak. Launch a Gaussian instead and the mismatched wings radiate away as the pulse reshapes into the nearest sech — the first face of the soliton's robustness.

Figure

Chirp cancellation in a fundamental soliton: self-phase modulation writes an up-chirp with red light on the leading edge and blue on the trailing edge, while anomalous GVD with β₂ ≪ 0 writes the equal and opposite chirp, so the net instantaneous frequency stays flat and the sech pulse propagates frozen at N = 1

Solitons also behave like particles: two well-separated fundamental solitons pass straight through each other and emerge unchanged apart from a phase and a small position shift — a consequence of the NLSE being integrable, and the signature that solitons are exact eigen-objects of the dynamics. Because the balance ties peak power to width, a soliton carries a specific price in power — the number a fiber engineer reaches for, plotted in the second figure.

Figure

Fundamental-soliton peak power versus pulse width in standard single-mode fiber at 1550 nm with β₂ ≈ −21.7 ps²/km and γ ≈ 1.3 W⁻¹km⁻¹: the balance law makes peak power scale as 1 over T², so a 1 ps soliton needs ≈ 52 W but a 20 ps soliton needs only ≈ 130 mW

Energy management is everything in a real fiber. Losing half the peak power (as loss inevitably does) would require the pulse to double its width to stay a soliton — which it cannot do smoothly over short distances, so it sheds energy as dispersive waves instead. This is why practical systems demand periodic amplification, the seam along which every real-world complication enters.

The core equations

The starting point is the equation of motion for the pulse envelope. Expanding the propagation constant $\beta(\omega)$ to second order gives the dispersive term, the Kerr index the nonlinear term; together they are the nonlinear Schrodinger equation for the envelope $A(z,T)$ , in the lossless limit.

i\,\frac{\partial A}{\partial z} - \frac{\beta_2}{2}\frac{\partial^2 A}{\partial T^2} + \gamma |A|^2 A = 0 \qquad (1)

where: $A(z,T)$ — slowly varying complex envelope, normalized so $|A|^2$ is power in watts; $z$ — distance along the fiber; $T=t-z/v_g$ — retarded time in the group-velocity frame; $\beta_2$ — group-velocity dispersion, negative for anomalous, units $\mathrm{s^2/m}$ ; $\gamma$ — nonlinear coefficient, units $\mathrm{W^{-1}m^{-1}}$ . The second term spreads the pulse, the third focuses it; a soliton is their standoff. The fundamental bright soliton is the exact solution of (1) in which those terms cancel — a sech pulse acquiring only a uniform phase along the fiber.

A(z,T)=\sqrt{P_0}\,\mathrm{sech}\!\left(\frac{T}{T_0}\right)\exp\!\left(\frac{i z}{2 L_D}\right) \qquad (2)

where: $P_0$ — peak power; $T_0$ — characteristic width (the measurable FWHM is $1.763\,T_0$ ); $L_D=T_0^2/|\beta_2|$ — dispersion length. Notice what is absent: no $z$ -dependence in the amplitude or width — the pulse is frozen, and only its overall phase evolves. Whether a launched pulse becomes a soliton is decided by one dimensionless ratio — the soliton order — comparing the length scales dispersion and nonlinearity act over.

N^2=\frac{L_D}{L_{NL}}=\frac{\gamma P_0 T_0^2}{|\beta_2|},\qquad L_D=\frac{T_0^2}{|\beta_2|},\quad L_{NL}=\frac{1}{\gamma P_0} \qquad (3)

where: $N$ — soliton order ( $N=1$ fundamental, integer $N\ge2$ higher-order); $L_D$ — distance over which GVD noticeably broadens the pulse; $L_{NL}$ — distance for one radian of SPM phase. When the two lengths are equal, $N=1$ and the effects balance exactly. Higher-order solitons do not sit still. An integer- $N$ soliton is a bound state that breathes — narrowing, then exactly recovering its shape — with a period set only by the dispersion length.

z_0=\frac{\pi}{2}L_D=\frac{\pi}{2}\frac{T_0^2}{|\beta_2|} \qquad (4)

where: $z_0$ — soliton period, the distance over which a higher-order soliton returns to its launch profile. This single length governs soliton-effect compression and many fiber-laser cavities. Turning the balance around gives the working formula every practitioner uses — the peak power a fundamental soliton of a given width must carry, picking up $1.763^2$ when written in the measured FWHM.

P_0=\frac{|\beta_2|}{\gamma T_0^2}=\frac{3.11\,|\beta_2|}{\gamma\,T_{\mathrm{FWHM}}^2} \qquad (5)

where: $T_{\mathrm{FWHM}}=1.763\,T_0$ , so $1.763^2\approx3.11$ . The steep $1/T^2$ scaling is the whole content of the second figure: short solitons are expensive, long ones cheap. The coefficient $\gamma$ is set by the material Kerr index and the tightness of the guided mode — where this article draws on the waveguide-modes result for $A_{\mathrm{eff}}$ .

\gamma=\frac{n_2\,\omega_0}{c\,A_{\mathrm{eff}}}=\frac{2\pi n_2}{\lambda\,A_{\mathrm{eff}}} \qquad (6)

where: $n_2\approx2.6\times10^{-20}\,\mathrm{m^2/W}$ for silica — the Kerr coefficient; $A_{\mathrm{eff}}$ — effective mode area; $\lambda$ — vacuum wavelength; $\omega_0=2\pi c/\lambda$ . A tighter mode raises $\gamma$ and lowers the soliton power — why highly nonlinear fibers exist. Measuring distance in dispersion lengths, time in pulse widths, and field in peak amplitude strips the constants out, leaving the canonical integrable form.

i\,\frac{\partial u}{\partial \xi} + \frac{1}{2}\frac{\partial^2 u}{\partial \tau^2} + |u|^2 u = 0 \qquad (7)

where: $u=A/\sqrt{P_0}$ ; $\xi=z/L_D$ ; $\tau=T/T_0$ . Here the fundamental soliton is simply $u=\mathrm{sech}(\tau)\,e^{i\xi/2}$ , and the inverse-scattering transform applies with no fiber parameter in sight. In normal dispersion the sign flips and no bright soliton forms — but the same nonlinearity supports a robust intensity dip on a CW background, the dark soliton.

u_d(\xi,\tau)=\tanh(\tau)\,e^{i\xi} \qquad (8)

where: $\tanh(\tau)$ — amplitude with a zero at $\tau=0$ , a notch on a bright background; $e^{i\xi}$ — the background's phase evolution. The notch carries a $\pi$ phase step across its centre and propagates without spreading.

Derivation AThe sech ansatz solves the nonlinear Schrodinger equation

The goal is to prove that the sech pulse of equation (2) is an exact solution of the lossless NLSE, and to see where the balance condition comes from — not as an assumption but as a forced algebraic consequence. We substitute a sech carrying only a $z$ -dependent phase, use the standard second-derivative identity for the hyperbolic secant, and demand the equation hold for all $T$ ; since $\mathrm{sech}$ and $\mathrm{sech}^3$ are independent, their coefficients must vanish separately.

Step 1. Write $A=\sqrt{P_0}\,\mathrm{sech}(T/T_0)\,e^{i\phi(z)}$ with $\phi$ depending only on $z$ . Then $i\,\partial_z A=-\phi'(z)\,A$ , a term proportional to $A$ . Why: a purely longitudinal phase cannot change the pulse shape, so it contributes a term with the same time profile as $A$ — what makes a stationary solution possible.

Step 2. Use $\partial_T^2\,\mathrm{sech}(u)=[\mathrm{sech}(u)-2\,\mathrm{sech}^3(u)]/T_0^2$ with $u=T/T_0$ , so the dispersion term generates both a $\mathrm{sech}$ and a $\mathrm{sech}^3$ piece. Why: the curvature of a sech is what dispersion feeds on, and the cubic piece it produces is exactly the shape the nonlinearity must match.

Step 3. The nonlinear term $\gamma|A|^2A=\gamma P_0\,\mathrm{sech}^3(u)\,e^{i\phi}$ is pure $\mathrm{sech}^3$ . Collecting gives $\big(-\phi'-\tfrac{\beta_2}{2T_0^2}\big)\mathrm{sech}(u)+\big(\tfrac{\beta_2}{T_0^2}+\gamma P_0\big)\mathrm{sech}^3(u)=0$ . Why: since $\mathrm{sech}$ and $\mathrm{sech}^3$ are independent in $T$ , each bracket must vanish on its own — two equations from one.

Step 4. The $\mathrm{sech}^3$ bracket gives $\gamma P_0 T_0^2=|\beta_2|$ (using $\beta_2=-|\beta_2|$ ), the soliton condition $N=1$ ; the $\mathrm{sech}$ bracket gives $\phi'=1/(2L_D)$ , the phase-winding rate. Why: the cubic balance fixes the power the pulse needs, the linear balance fixes how fast its phase advances — together the entire fundamental soliton. *Verified: substituting $\gamma P_0T_0^2=|\beta_2|$ and $\phi'=1/(2L_D)$ into equation (1) makes it hold identically; matches Agrawal Ch. 5.*

Go deeper · Derivation B · 1 minSelf-phase modulation writes a chirp opposite to anomalous GVD

Derivation BSelf-phase modulation writes a chirp opposite to anomalous GVD

The aim is to make the physical picture quantitative: to show that the Kerr effect and anomalous dispersion impose frequency chirps of opposite sign, and that for a sech pulse they cancel point by point rather than only on average. We follow the phase each effect adds to a pulse $A=\sqrt{P(T)}\,e^{i\theta}$ , differentiate to get the local frequency shift, and compare — the intuition behind the whole subject, and why the sign of $\beta_2$ cannot be changed.

Step 1. The Kerr index adds $\phi_{NL}(z,T)=\gamma\,P(T)\,z$ , a phase proportional to instantaneous power over distance. Why: the index $n_2 I$ changes the optical path in step with the pulse's own power profile, so the pulse phase-modulates itself — hence self-phase modulation.

Step 2. The instantaneous frequency shift is $\delta\omega_{\mathrm{SPM}}=-\partial\phi_{NL}/\partial T=-\gamma z\,\partial P/\partial T$ . On the leading edge $\partial P/\partial T>0$ gives red; on the trailing edge, blue — an up-chirp. Why: frequency is the rate of phase advance, so rising power writes lower frequencies first and higher last, broadening the spectrum.

Step 3. Anomalous GVD ( $\beta_2<0$ ) advances higher frequencies over lower, $\Delta T_{\mathrm{GVD}}=\beta_2 z\,\delta\omega$ — the opposite chirp, blue at the front and red at the back. Why: dispersion reshuffles the colours SPM created, and only $\beta_2<0$ opposes SPM instead of reinforcing it — why normal dispersion supports no bright soliton.

Step 4. For a sech the two patterns are not merely opposite in sign but identical in shape, so at the balance power they cancel at every $T$ ; the net frequency is flat and the pulse neither chirps nor spreads. Why: matching only at the peak (as a Gaussian would) leaves residual chirp in the wings that radiates away — the sech is special because the match is exact everywhere. *Verified: consistent with Agrawal Ch. 5 — the SPM up-chirp cancels the anomalous-GVD down-chirp exactly only for the sech at $N=1$ .*

Go deeper · Derivation C · 1 minThe soliton period from the inverse-scattering eigenvalues

Derivation CThe soliton period from the inverse-scattering eigenvalues

This derivation explains why integer-order solitons are periodic and pins down the period $z_0=(\pi/2)L_D$ that governs pulse compression and laser cavity design. The canonical NLSE is integrable, and the inverse-scattering transform associates to an $N$ -soliton a set of $N$ discrete bound-state eigenvalues. The pulse is a superposition of these states; because each evolves at its own phase rate, the intensity beats at the difference of the two slowest — a recurrence returning the pulse to its start.

Step 1. The transform gives eigenvalues $\eta_n=(2n-1)/2$ for $n=1,\dots,N$ , each contributing a phase $\eta_n^2\,\xi$ in normalized distance $\xi$ . Why: solitons are bound states of the scattering problem, so their internal clock rates are set by the squares of these eigenvalues, as energy levels set quantum phase rates.

Step 2. The slowest beat is between the two lowest states, $\Delta(\eta^2)=(3/2)^2-(1/2)^2=2$ . Why: the profile recurs only when every pair of components realigns, and the closest-spaced pair sets the fundamental recurrence length.

Step 3. Intensity depends on twice the phase, so the profile returns when the relative phase advances by $\pi$ : $\Delta(\eta^2)\,\xi_0=\pi$ , giving $\xi_0=\pi/2$ . Why: intensity is amplitude squared, folding the $2\pi$ phase recurrence down to a $\pi$ advance — the origin of the $\pi/2$ .

Step 4. Restoring units with $\xi=z/L_D$ gives $z_0=\tfrac{\pi}{2}L_D=\tfrac{\pi}{2}\,T_0^2/|\beta_2|$ . Why: the normalized period becomes physical distance once the dispersion length returns — why an $N=2$ soliton compresses and perfectly recovers every $z_0$ . *Verified: reproduces the standard higher-order soliton period; matches Agrawal Ch. 5.*

Limiting cases

The single number $N^2=\gamma P_0 T_0^2/|\beta_2|=L_D/L_{NL}$ — dispersion length over nonlinear length — classifies every regime of pulse evolution.

Case	Result	Meaning
Dispersion-dominated ( $N\ll1$ )	$L_D\ll L_{NL}$	Nonlinearity is negligible; the pulse broadens by GVD as in ordinary linear propagation. No soliton forms.
Fundamental soliton ( $N=1$ )	$\gamma P_0 T_0^2=\vert\beta_2\vert$	Exact balance; the sech pulse propagates unchanged apart from a uniform phase. Peak power is locked to width and fiber parameters.
Higher-order soliton ( $N=2,3,\dots$ )	$z_0=\tfrac{\pi}{2}\,T_0^2/\vert\beta_2\vert$	The pulse breathes: it narrows, develops structure, and recovers its input shape every soliton period. Basis of soliton-effect compression.
Near-integer launch ( $0.5<N<1.5$ )	relaxes to $N=1$	The soliton is an attractor; the fiber sheds the mismatch as dispersive radiation and settles onto the fundamental soliton.
Normal dispersion ( $\beta_2>0$ )	no bright soliton	SPM and normal GVD both broaden a bright pulse; the stable structure is a dark soliton — an intensity notch on a CW background.

Common trapA soliton needs a perfectly shaped sech pulse at exactly the right power, or it falls apart

It does not, and believing so misses the deepest property of solitons. They are attractors: launch a pulse whose order lies anywhere in roughly $0.5<N<1.5$ and the fiber sheds the excess (or rings up the deficit) as dispersive radiation and relaxes onto the exact fundamental soliton, sech shape and all. This robustness, a consequence of integrability, is why solitons are useful rather than fragile. The one genuinely non-negotiable requirement is the sign of the dispersion: bright solitons exist only for anomalous GVD ( $\beta_2<0$ ). In normal dispersion no peak power produces one, since SPM and dispersion then broaden the pulse in the same sense; what survives is the dark soliton. Get the sign wrong, not the shape, and you have no soliton at all.

Worked examples

1 · The peak power of a 1 ps soliton (confidence builder)

A $1\,\mathrm{ps}$ FWHM fundamental soliton is to propagate in standard single-mode fiber (SMF-28) at $1550\,\mathrm{nm}$ . Using textbook constants, find its required peak power.

Step 1. Convert dispersion to $\beta_2$ : with $D=17\,\mathrm{ps\,nm^{-1}km^{-1}}$ and $\lambda=1550\,\mathrm{nm}$ , $|\beta_2|=D\lambda^2/(2\pi c)\approx2.17\times10^{-26}\,\mathrm{s^2/m}=\mathbf{21.7\ ps^2/km}$ (negative, anomalous).

Step 2. Nonlinear coefficient from (6): $\gamma=2\pi n_2/(\lambda A_{\mathrm{eff}})$ with $n_2=2.6\times10^{-20}\,\mathrm{m^2/W}$ and $A_{\mathrm{eff}}=80\,\mu\mathrm{m}^2$ gives $\mathbf{1.3\ W^{-1}km^{-1}}$ .

Step 3. FWHM to sech parameter: $T_0=1/1.763=\mathbf{0.567\ ps}$ , so $T_0^2=0.321\,\mathrm{ps^2}$ .

Step 4. Apply (5): $P_0=|\beta_2|/(\gamma T_0^2)=21.7/(1.3\times0.321)\approx\mathbf{52\ W}$ (equally from $3.11\times21.7/1.3$ ). A 1 ps soliton in standard SMF needs about 52 W of peak power — not "a few milliwatts." The steep $1/T^2$ law is why ultrashort solitons are made in highly nonlinear fibers, not ordinary SMF.

2 · A telecom soliton: energy and length scales

Now the practical communications case — a $20\,\mathrm{ps}$ FWHM soliton in the same SMF. Find its peak power, pulse energy, dispersion length, and soliton period.

Step 1. $T_0=20/1.763=\mathbf{11.34\ ps}$ , $T_0^2=128.6\,\mathrm{ps^2}$ .

Step 2. Peak power from (5): $P_0=21.7/(1.3\times128.6)\approx\mathbf{130\ mW}$ — twenty times wider than example 1, so $400$ times less power (the amber marker on the second figure).

Step 3. Sech pulse energy $E=2P_0T_0=2(0.130)(11.34\times10^{-12})\approx\mathbf{2.9\ pJ}$ — picojoule-scale at modest power, why solitons are practical carriers.

Step 4. Length scales: $L_D=128.6/21.7\approx\mathbf{5.9\ km}$ , $z_0=(\pi/2)L_D\approx\mathbf{9.3\ km}$ . So a 20 ps telecom soliton carries ≈130 mW peak, ≈2.9 pJ, with a ≈6 km dispersion length — comfortably longer than the amplifier spacing in a real link, which is what makes soliton transmission workable. (Launch the same pulse at four times this power, $520\,\mathrm{mW}$ , and $N^2=(1.3\times0.520\times128.6)/21.7\approx4$ : an $N=2$ breather that compresses and recovers every $z_0\approx9.3\,\mathrm{km}$ — the basis of soliton-effect pulse compression.)

Physics of the portrait

Optical solitons sit at the confluence of two stories: a nineteenth-century solitary water wave, and a twentieth-century discovery that certain nonlinear equations hide exact, particle-like solutions. The person who welded that mathematics onto a glass fiber is the natural portrait.

Origins · 1 minAkira Hasegawa and how this was discovered

Akira Hasegawa

b. 1934, theoretical physicist at Bell Laboratories · nonlinear fiber optics and plasma physics

That a light pulse could travel a fiber without spreading was, in 1973, pure theory. Akira Hasegawa, a Japanese theoretical physicist at Bell Laboratories whose main work was plasma turbulence (his name survives in the Hasegawa–Mima equation of drift-wave physics), recognized that the envelope of an intense pulse in a single-mode fiber obeys the very nonlinear Schrodinger equation whose soliton solutions Zakharov and Shabat had integrated two years earlier. With Frederick Tappert he published two Applied Physics Letters papers predicting both the bright soliton, in the anomalous-dispersion window beyond $1.3\,\mu\mathrm{m}$ , and the dark soliton in normal dispersion — a complete forecast made before low-loss fiber could test it. The prediction waited seven years, until Mollenauer, Stolen and Gordon observed the fiber soliton at the same laboratory in 1980. Hasegawa's leap — that an abstract integrable model of a plasma wave was literally the equation obeyed by light in a telecom fiber — is one of optics' cleaner cases of deep mathematics turning out to have been applied physics all along.

History

1834

— John Scott Russell observes a solitary "wave of translation" on the Union Canal

1895

— Korteweg and de Vries derive the shallow-water equation whose solitary waves bear their name

1965

— Zabusky and Kruskal coin the term "soliton" from numerical study of the KdV equation

1971

— Zakharov and Shabat integrate the nonlinear Schrodinger equation by the inverse-scattering transform

1973

— Hasegawa and Tappert predict bright and dark optical solitons in dielectric fibers

1980

— Mollenauer, Stolen and Gordon observe the fiber soliton experimentally at Bell Labs

1988

— Mollenauer and Smith transmit solitons over 4000 km using distributed Raman gain

Going further

Texts. Agrawal, Nonlinear Fiber Optics, 5th ed., Ch. 5 — the definitive treatment · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 24 and Ch. 10 · Hasegawa and Kodama, Solitons in Optical Communications · Hecht, Optics, 5th ed. · Demtroeder, Laser Spectroscopy.

Provenance. The NLSE, sech soliton, soliton order and lengths, period, peak-power scaling, nonlinear coefficient, and dark soliton follow Agrawal Ch. 5 and Saleh and Teich Ch. 24. Numerical examples use standard SMF-28 constants at 1550 nm: $D=17\,\mathrm{ps\,nm^{-1}km^{-1}}$ giving $|\beta_2|\approx21.7\,\mathrm{ps^2/km}$ , $\gamma\approx1.3\,\mathrm{W^{-1}km^{-1}}$ , $n_2=2.6\times10^{-20}\,\mathrm{m^2/W}$ , $A_{\mathrm{eff}}\approx80\,\mu\mathrm{m}^2$ ; $T_{\mathrm{FWHM}}/T_0=1.763$ and $1.763^2=3.11$ are textbook. A verifier should recompute $\beta_2$ from $\beta_2=D\lambda^2/(2\pi c)$ , $\gamma$ from $2\pi n_2/(\lambda A_{\mathrm{eff}})$ , and $P_0$ from $|\beta_2|/(\gamma T_0^2)$ — giving $\approx52\ \mathrm{W}$ for a 1 ps FWHM soliton and $\approx130\ \mathrm{mW}$ for a 20 ps one. (Note: the balance law gives tens of watts for a 1 ps soliton in SMF; the milliwatt regime belongs to tens-of-picosecond pulses.)

Builds on. Chromatic and Modal Dispersion in Fibers · Waveguide Modes and the Wave Equation · Single-Mode and Multimode Operation · Fiber Attenuation and Loss Mechanisms. Leads to. Dispersion Management and Compensation · Microring Resonators and Optical Cavities on Chip · Integrated Photonic Waveguides and Circuits.

Ask the tutor: Give me a fiber's dispersion and nonlinear coefficient and a pulse width, and I will walk you through computing the soliton order, the peak power, and whether the pulse settles, breathes, or simply disperses.

Is a soliton really lossless, or does it just spread very slowly?

In the idealized lossless NLSE a fundamental soliton is genuinely stationary — it propagates forever with no change in shape or width, not merely a slow spreading. That is an exact mathematical property, not an approximation. Real fibers have loss, so the balance holds only while the power stays at the soliton value. As attenuation drains the power the nonlinearity weakens, the balance tips toward dispersion, and the pulse broadens — which is why real systems place amplifiers periodically to top the power back up before the pulse drifts off the soliton condition.

Why must the pulse be a sech and not a Gaussian?

The sech is the one profile whose nonlinear phase has the same time-curvature as the dispersive phase at every point across the pulse, so the two cancel everywhere at once. A Gaussian matches only near its peak; its wings leave residual chirp. In practice you can still launch a Gaussian and get a soliton — the fiber radiates the mismatched part away and the core reshapes into a sech. So the sech is not a strict entry requirement but the fixed point the system relaxes to, which is why the mathematics singles it out.

What is the soliton order N, physically?

$N=\sqrt{L_D/L_{NL}}=\sqrt{\gamma P_0 T_0^2/|\beta_2|}$ compares how strongly the pulse disperses to how strongly it self-focuses. $N=1$ is the exact balance — the stationary fundamental soliton. $N\ll1$ means dispersion wins and the pulse broadens as in linear optics. Integer $N\ge2$ launches a bound "breather" that compresses and recovers with period $z_0=(\pi/2)L_D$ . Non-integer launches near 1 relax to the fundamental soliton by radiating the surplus. So one number tells you both whether a soliton forms and what kind.

What is a dark soliton, and when does it appear?

A dark soliton is the structure that survives in the normal-dispersion regime ( $\beta_2>0$ ), where no bright soliton can exist. It is a localized dip in intensity on a continuous-wave background — a notch that propagates without filling in or spreading, carrying a $\pi$ phase step across its centre, with profile $u_d=\tanh(\tau)\,e^{i\xi}$ . The same Kerr nonlinearity that broadens a bright pulse in normal dispersion stabilizes the notch. Dark solitons appear in fiber and in wider nonlinear-wave systems, including matter waves in Bose–Einstein condensates, where the same NLSE (there Gross–Pitaevskii) applies.

How does fiber loss actually destroy a soliton?

Loss lowers the peak power $P_0$ , hence the soliton order since $N^2\propto P_0$ . Once $\gamma P_0 T_0^2$ falls below $|\beta_2|$ the nonlinearity can no longer supply the up-chirp that cancels dispersion, and the pulse broadens and radiates like a linear one — it cannot adiabatically widen to stay balanced over the short distances loss acts on, so it sheds energy instead. Periodic amplifiers restore the average power; the cost is amplified-spontaneous-emission noise, which shifts the soliton's centre frequency and, through dispersion, its arrival time — the Gordon–Haus timing jitter.

Are optical solitons connected to solitons elsewhere in physics?

Deeply. The word "soliton" was coined in 1965 for solutions of the Korteweg–de Vries equation of shallow-water waves, whose ancestor is Russell's 1834 canal wave. What ties these to optics is a class of integrable nonlinear equations solvable by the inverse-scattering transform, of which the NLSE is a member. The same NLSE governs light in a fiber, matter waves in a Bose–Einstein condensate (as the Gross–Pitaevskii equation), and certain plasma waves. So the optical soliton is one instance of a universal phenomenon: whenever nonlinearity balances a spreading tendency in an integrable system, robust particle-like waves appear — which is why intuitions transfer freely between fiber optics, hydrodynamics, plasma physics, and cold atoms.