A short pump pulse launched into a photonic-crystal fiber's tiny (≈2 μm) silica core experiences both group-velocity dispersion (GVD) and an intensity-dependent Kerr nonlinearity. In normalized retarded time τ = t/T₀ and distance ξ = z/LD (with LD = T₀²/|β₂|), the pulse envelope u(ξ,τ) obeys a generalized nonlinear Schrödinger equation:
∂u/∂ξ = i/2 ∂²u/∂τ² − δ₃ ∂³u/∂τ³ + i N² |u|² u
N² = γ P₀ T₀² / |β₂| (soliton order)
δ₃ = β₃ / (6 T₀ |β₂|) (third-order dispersion)
This 2D version solves the identical equation with the same split-step Fourier method, but instead of drawing an instantaneous bar chart in 3D, it plots the entire propagation history at once as two stacked waterfall maps: time-domain intensity |u(τ,ξ)|² above, spectral intensity |û(Ω,ξ)|² below — distance ξ runs down the vertical axis, so the supercontinuum's spectral broadening and the temporal soliton fission are both visible as a single 2D density field, the way these processes are actually plotted in nonlinear-fiber-optics papers (cf. Dudley, Genty & Coen, Rev. Mod. Phys. 2006).
- N (soliton order) — for integer N>1 in an ideal (δ₃=0) fiber, the pulse is a higher-order soliton that breathes periodically without changing shape. Real photonic-crystal fibers always carry some higher-order dispersion, which breaks that periodicity.
- δ₃ (third-order dispersion) — triggers soliton fission: the higher-order soliton splits into N individual fundamental solitons of different amplitude and width, each walking off in time, plus a dispersive wave. Their spectra spread out and overlap into a broadband supercontinuum — visible in the lower map as the spectral band fanning out with increasing ξ.
- ξ (propagation distance) — scrub or press Play to sweep a horizontal cursor line down both maps; the readouts update to the intensity/spectrum slice at that ξ.
- T₀ (pulse duration) — only rescales the physical distance readout (via a fixed representative PCF dispersion β₂ = −11 fs²/mm near an 800 nm Ti:sapphire pump); shorter pulses need a physically shorter fiber to reach the same normalized ξ.
Real-world relevance: this is the mechanism (Ranka et al. 2000) that makes photonic-crystal fiber supercontinuum sources — octave-spanning white light from a single near-IR laser — the workhorse light source for optical coherence tomography, frequency combs and hyperspectral microscopy.