Example 5: Higher-Order Soliton Compression

Reproducing the periodic breathing and temporal compression of N=3 solitons (Mollenauer, Stolen & Gordon, 1980)

DOI: 10.1103/PhysRevLett.45.1095

Physical Background

When a sech² pulse with N > 1 is launched in the anomalous-dispersion regime, it forms a higher-order soliton that undergoes periodic compression. At each integer multiple of the soliton half-period $z_{1/2} = \frac{\pi}{2} L_D$, the pulse returns to its original shape. At intermediate distances it compresses dramatically, reaching a minimum duration near:

\[z_{\min} \approx 0.32 \frac{L_D}{N}\]

with a compression ratio approximately:

\[F_c \approx 4.1 N\]

The N=3 soliton was the first experimentally observed higher-order soliton in an optical fiber.

Simulation

using GNLSE

# ─── Corning SMF-28 telecom fiber ───────────────────────────────────────────
medium = commercial_fiber("Corning_SMF28", length=100.0, lambda0=1550e-9)

# ─── N=3 higher-order soliton pulse ─────────────────────────────────────────
grid  = create_grid(2^12, 40e-12, medium.lambda0)
pulse = sech_pulse(grid, 500.0, 1e-12) # 500 W peak, 1 ps FWHM

# ─── Solve GNLSE ─────────────────────────────────────────────────────────────
params = SimParams(; medium=medium, z_saves=300, raman_model=nothing)
sol = solve(pulse, params; progress=false)

z_fiss, peak_power_z, _ = track_solitons(sol)
println("N=3 Soliton Peak Power Compression: ", round(maximum(peak_power_z)/peak_power(pulse); digits=1), "x")

N=3 Soliton Peak Power Compression: 6.3x

Expected Results

Higher-Order Soliton Gallery

Run all soliton orders 1–4 and compare:

FWHM  = 1e-12                          # same pulse duration as above
T0    = FWHM / 1.7627                  # sech FWHM -> T0
beta2 = medium.dispersion.betas[1]     # [s²/m]
gamma = medium.gamma                   # [1/(W·m)]

for N in 1:4
    P0_N = N^2 * abs(beta2) / (gamma * T0^2)
    pulse_N = sech_pulse(grid, P0_N, FWHM)
    sol_N   = solve(pulse_N, SimParams(; medium=medium, z_saves=200,
                                        raman_model=nothing, self_steepening=false))
    peaks_N = [maximum(abs2, sol_N.At[:, i]) for i in axes(sol_N.At, 2)]
    Fc = maximum(peaks_N) / P0_N
    println("N=$N: max compression = $(round(Fc; sigdigits=3))x  (theory ≈ $(4.1*N)x)")
end

Soliton Fission vs. Ideal Compression

In practice, the clean higher-order soliton evolution is disrupted by:

• Third-order dispersion β₃: breaks the periodicity
• Raman scattering: shifts individual solitons in frequency (see Example 2)
• Self-steepening: distorts the pulse shape

Enable these to observe realistic (imperfect) soliton dynamics:

L_D    = T0^2 / abs(beta2)             # dispersion length [m]
Zhalf  = (π / 2) * L_D                 # soliton half-period [m]

medium_realistic = Medium(;
    length  = Zhalf,
    gamma   = gamma,
    loss    = 0.0,
    betas   = [beta2, 5e-41],    # add β₃
    lambda0 = medium.lambda0,
)
sol_real = solve(pulse, SimParams(;
    medium          = medium_realistic,
    z_saves         = 500,
    raman_model     = BlowWood(),
    self_steepening = true,
))

References

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095–1098 (1980). DOI: 10.1103/PhysRevLett.45.1095

G. P. Agrawal, Nonlinear Fiber Optics, 6th ed., Academic Press (2019). Chapter 5: "Optical Solitons"